Helping Children Learn Mathematics, 10th Edition Solution Manual

Helping Children Learn Mathematics, 10th Edition

By Reys, Lindquist, Lambdin, Smith

Instructor’s Resource Guide, Chapter 1 Sandi Cooper |1

Chapter 1 —School Mathematics in a Changing World Chapter Overview In this chapter, the student is asked to examine what mathematics is and what determines the mathematics being taught. A brief historical perspective of the mathematics curriculum is discussed with a clear presentation of the underlying Principles developed by the National Council of Teachers of Mathematics [NCTM]. Three general factors (needs) are presented as influences on the mathematics curriculum. The chapter concludes with a discussion of resources teachers can use to learn more about teaching mathematics.

Student Objectives After reading the chapter, the students will be able to: 1.

Describe five views of mathematics.

2.

Summarize three needs that influence the mathematics curriculum.

3.

Summarize recommendations from NCTM’s six underlying Principles of school mathematics.

4.

Identify resources for learning more about mathematics teaching.

Key Vocabulary For this chapter, many of the key terms involve connections to names of persons and organizations. Students may have encountered these names and terms in earlier courses with others being specifically tied to the mathematics subject area.

Journal for Research in Mathematics Education [JRME] learning Teaching Children Mathematics [TCM] assessment Mathematics Teaching in the Middle School [MTMS] technology Principles and Standards for School Mathematics [PSSM] new math field theory (Gestalt) Committee of Seven meaning and understanding (Brownell) teaching mathematical proficiency (NRC) equity social utility (Wilson) curriculum connectionism (Thorndike) construct knowledge National Council of Teachers of Mathematics [NCTM] mental discipline National Assessment of Educational Progress [NAEP] incidental learning Trends in Mathematics and Science Study [TIMSS] No Child Left Behind [NCLB] Common Core State Standards Initiative [CCSSI] .S o n s

Instructor’s Resource Guide, Chapter 1 Sandi Cooper |2 Supplemental Lecture Topics 1.

Provide more background information about NCTM's Principles and Standards for School Mathematics, Professional Standards for Teaching Mathematics, Assessment Standards for School Mathematics and/or Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. The booklet, Principles and Standards for School Mathematics: An Overview, the Principles and Standards Outreach CD 2nd Ed., the Quick Reference Guide: Table of Standards and Expectations, and Answers to Frequently Asked Questions about PSSM, and A Research Companion to Principles and Standards for School Mathematics are available from NCTM and provide useful information.

2.

Provide information concerning NCTM's student memberships and starter kits for students. NCTM has packets of information that can be distributed to students. The Member Services and Public Relations Department may be reached at (703)620-9840 ext.

113. Have students examine sample copies of the NCTM Educational Material Catalog, Teaching Children Mathematics (formerly Arithmetic Teacher), Mathematics Teaching in the Middle School, Mathematics Teacher, and Journal for Research in Mathematics Education (JRME).

3.

Discuss mandates and recommendations for mathematics education that are currently being enacted in your state or region. For example, implementation of the Common Core State Standards, state goals, grade level objectives, state assessment instruments, NCLB,etc. Discuss the influence these mandates have on classroom instruction. Invite a practicing teacher to discuss his/her view of initiatives.

4.

Share results from the Trends in Mathematics and Science Study (TIMSS), from the books, Standards & Curriculum: A View from the Nation, Results and Interpretations of the 1990 through 2000 Mathematics Assessments of the National Assessment of Educational Progress, or Mathematics Education in the United States 2004.

5.

The NCTM 2000 Yearbook, Learning Mathematics for a New Century and the 2004 Yearbook, Perspectives on the Teaching of Mathematics, includes several timely articles about issues in mathematics education. Use these as discussion starters or assign various students to read and report to the class.

6.

Implementing Standards-Based Math Instruction: A Casebook for Professional Development and Reflecting on NCTM’s Principles and standards in Elementary and Middle School Mathematics: Readings from NCTM’s School-Based Journals are available from NCTM. Thinking like Mathematicians: Putting the NCTM Standards into Practice is available from Heinemann. They provide research-based cases articles that illustrate reform efforts and ideas for group discussions. Use these cases or articles to help students envision how reform occurs in classrooms.

7.

Share more information on school mathematics history from A History of School Mathematics, Volumes I and II.

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Instructor’s Resource Guide, Chapter 1 Sandi Cooper |3 Student Textbook Activities 1.

As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and on PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2.

Involve students in a discussion about what mathematics is, what it has been and what it should be. These questions are central to the discussion of the mathematics curriculum and the forces affecting its form. Most prospective teachers and elementary children, based on their past experiences, may tend to view mathematics as arithmetic.

Mathematics is often viewed as a quest for the "correct answer". Have students examine Figure 1-1, Vision of the mathematics content distribution through the grade levels and have them relate this to their own school experiences.

3.

PPT Slide #3, What is Mathematics? describes mathematics in five different ways - as discussed in the text. You may want to start by having students compile a list of words they associate with mathematics. Using these, lead the discussion to the five views of mathematics and have the class brainstorm examples that fit within each definition.

4.

PPT Slides #6-12, Principles for School Mathematics, provides the five underlying principles of school mathematics as outlined in NCTM’s Principles and Standards for School Mathematics. These may also be found in the text on pages 6-7. Encourage students to discuss each principle and its implications for them as future elementary teachers.

5.

In small groups, have students develop a vertical timeline (1900, 1910...1990, 2000).

Have each group locate key items from chapter 1 and place them on the timeline.

Discuss the results as a class. A suggested timeline follows which may be cut apart for groups to locate in the text. Follow up with a discussion of PPT Slides #13-14, Major Trends in School Mathematics.

Late 1800's 1900 1920's 1930's 1930's 1931 1940s 1957 1950's, 60's 1960s 1970's 1973 1980’s .S o n s

Mental Discipline Thorndike-connectionism progressive movement, incidental learning Gestalt-field theory Brownell-meaning and understanding Committee of Seven Wilson-social utility Sputnik New Math Movement concern for disadvantaged students Minimal-Competency Movement NAEP begins Problem Solving Movement

Instructor’s Resource Guide, Chapter 1 Sandi Cooper |4 1989 1990s 1991 1995 1996 2000 2000 2001 2001 2003 2006 2010

NCTM's Curriculum and Evaluation Standards for School Mathematics Standards movement NCTM's Professional Standards for Teaching Mathematics NCTM's Assessment Standards for School Mathematics TIMSS results NAEP results from the Seventh Mathematics Assessment released NCTM's Principles and Standards for School Mathematics released Adding it Up released by the National Research Council No Child Left Behind Trends in Mathematics and Science Study (TIMSS) NCTM’s Curriculum Focal Points for Grades K-8 Common Core State Standards Initiative

7.

Use PPT Slides #15-16, Resources for Teachers, to discuss the suggestions provided in the text. Encourage students to share specific examples for each that they have seen or used.

8.

The Math Links in this chapter provide students with additional resources for this chapter. In class, or as an outside assignment, have students explore these web sites.

9.

Provide students with the opportunity to discuss unresolved concerns about the elementary school mathematics curriculum and preview the upcoming chapters where the identified concerns will be addressed in greater detail. Use the following discussion starters to stimulate dialogue and encourage brainstorming in small groups.

Starter #1: Brainstorm concerns which will need to be addressed by the mathematics education community as curriculum change is implemented.

Starter #2: Brainstorm what classroom teachers can do as they prepare to change the elementary mathematics curriculum.

Student Supplemental Activities 1.

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Have students complete an introductory survey on the first day of class. These surveys may be returned to students on the last day of class and changes that occurred may be discussed. The survey may include items such as: • Math is...

• When it comes to math, I feel...

• I think my math ability is...

• Elementary math should be...

• It is most important that elementary math students learn...

• Name one thing you remember from your math classes in elementary school.

• Describe what you hope to learn from this course.

• How I feel about teaching mathematics is…

Instructor’s Resource Guide, Chapter 1 Sandi Cooper |5 2.

As students introduce themselves, have them share information from one of their survey items such as a memory (good or bad) from their math classes in elementary school.

Discuss implications for their future teaching of mathematics or preview how the items discussed will be addressed in the course.

3.

Present to the students a list of myths about mathematics as presented in the article by Martha L. Frank, What Myths about Mathematics are Held and Conveyed by Teachers?

(Arithmetic Teacher, 37(5): 10-12 - Jan 1990). Allow them to discuss in small groups and then share as a whole group. Discuss how certain beliefs might be conveyed unintentionally through instruction.

4.

Start a collection of magazine and newspaper articles related to mathematics education, student performance, the accountability or reform movement. For example, titles such as "Study: U.S. Students poor in math and science", or "U.S. Students Math Scores Up"...

Divide students into groups to discuss the articles and then summarize results for the class. Start a display and encourage students to bring in articles throughout the course.

5.

Have the class build "get acquainted" line plots or graphs using materials such as grid paper, Post-It Notes®, links or paper clips, or connecting cubes (See Figures 17-3 and 17-4 in Chapter 17). Prior to collecting the data, have students discuss and predict the outcomes. You may have students estimate the number of candies in a jar or respond to questions such as: • How do you feel about teaching mathematics? Stack your cube under the happy face, so-so face, or sad face.

• What grade do you most want to teach? Hang your link on your favorite grade.

• What month were you born? Place your Post-it Note® under the correct month.

• What is the first letter of your first name? Mark an x on the line plot above the letter.

• How many letters in your first name? Estimate the average number of letters in first names in this class.

Examine the variety of ways elementary students can easily display data. Have students brainstorm other questions which could be used as get acquainted mathematics activities with elementary students such as name your favorite color, measure your height in centimeters...

6.

Have students work in groups to discuss items from Things to do: From What You've Read, Things to do: Going Beyond this Book (page 11).

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences, 4th edition. The following activities have been designed to be used with Chapter 1: .S o n s

Instructor’s Resource Guide, Chapter 1 Sandi Cooper |6 1.

Learning about the School and Its Resources: Curriculum Guide, Mathematics in the School

2.

Interviewing the Teacher and Students: Interviewing the Teacher about Equity

Additional Resources Adding It Up: Helping Children Learn Mathematics. National Research Council, National Academy Press, by Jeremy Kilpatrick, Jane Swafford, and Bradford Findell. Shares recommendations drawn from research and defines mathematical proficiency.

Answers to Frequently Asked Questions about Principles and Standards for School Mathematics.

Provides answers to commonly asked questions about PSSM. Available from NCTM. A History of School Mathematics, edited by George M.A. Stanic and Jeremy Kilpatrick, provides a record of the history of mathematics education in the United States and Canada. Available from NCTM. Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development, by Mary Kay Stein, Margaret Schwan Smith, Marjorie A. Henningsen, and Edward A. Silver. Shares cases and materials for use in classes and workshops. Available from Teachers College Press and NCTM. Learning Mathematics for a New Century (2000 Yearbook), provides several articles about issues in mathematics education. Available from NCTM. Lessons Learned from Research. Edited by Judith Sowder and Bonnie Schapelle. This book shares research in a teacher friendly format. Available from NCTM. Making the Grade in Mathematics: Elementary School Mathematics in the United States, Taiwan, and Japan--By Harold W. Stevenson, Max Lummis, Shin-Ying Lee, and James W. Stigler. Compares mathematics performance of elementary school students. Focuses on the differences in mathematics teaching methods and attitudes. Available from NCTM. Mathematics Education in the United States 2004, edited by Zalman Usiskin and John A. Dossey, is a report from the Tenth International Congress on Mathematical Education. Available from NCTM. An Overview of Principles and Standards for School Mathematics, A booklet which presents the main ideas that are presented in the latest Standards document. Available from NCTM. Perspectives on the Teaching of Mathematics (2004 Yearbook), focuses on teaching mathematics. Available from NCTM. Principles and Standards for School Mathematics Outreach CD, 2nd ed. Contains Powerpoint presentations, handouts, and video clips. Available from NCTM. .S o n s

Instructor’s Resource Guide, Chapter 1 Sandi Cooper |7 Putting Research into Practice in the Elementary Grades: Readings from Journals of the NCTM, edited by Donald L. Chambers, includes 70 articles from the research departments of the Arithmetic Teacher and Teaching Children Mathematics.

The Quick Reference Guide: Table of Standards and Expectations, three 11”x17” sheets present the goals and expectations for the six principles and ten standards. Available from NCTM. Reflecting on NCTM’s Principles and Standards in Elementary and Middle School Mathematics: Readings from NCTM’s School-Based Journals and Other Publications, edited by Ann R. Teppo, is two volumes of articles that illustrate the vision of the Principles and Standards.

Available from NCTM. A Research Companion to Principles and Standards for School Mathematics, edited by Jeremy Kilpatrick, W. Gary Martin, and Deborah Schifter. This book includes research about school mathematics as related to Principles and Standards from School Mathematics. Available from NCTM. Results and Interpretations of the 1990 through 2000 Mathematics Assessments of the National Assessment of Educational Progress. Reports and interprets trends in NAEP data collected between 1990-2000. Available from NCTM. Standards & Curriculum: A View from the Nation, edited by Johnny W. Lott and Kathleen Nishimura. Provides insights into where we appear to be headed as a nation in school mathematics. Available from NCTM. Thinking Like Mathematicians: Putting the NCTM Standards into Practice, by Thomas Rowan and Barbara Bourne. This book helps teachers see how their teaching can reflect NCTM’s recommendations and support children’s active involvement in mathematics. Available from Heinemann.

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Instructor’s Resource Guide, Chapter 2 Sandi Cooper |1

Chapter 2 —Helping Children Learn Mathematics with Understanding What This Chapter Is About A survey of teachers asked, “How do children learn?” Although many teachers provided insightful responses to this difficult yet important question, almost half of the teachers said, “I have not thought about it.” This chapter is designed to focus attention on a few critical things known about how children learn mathematics. Ways to support diverse learners are presented.

Key research findings on cognitive development are consolidated and discussed in an effort to translate relevant ideas and their implications for instruction into a form that is meaningful and understandable. Behaviorism and constructivism are introduced. Four recommendations for helping children make sense of mathematics are developed and presented.

Student Objectives After reading the chapter, the students will be able to: 1.

Summarize five ways to support the diverse learners in our mathematics classrooms.

2.

Differentiate between procedural and conceptual knowledge.

3.

Differentiate between behaviorist and constructivist learning theories.

4.

List four common observations about how children learn as characterized by Piaget, Bruner, and Dienes and differentiate among them. Summarize recommendations for helping children make sense of mathematics as based on these observations about how children learn.

Key Vocabulary A number of key terms related to learning, mathematics learning in particular, are used in this chapter. Some of them build upon concepts learned earlier in educational psychology but many of them are unique to mathematics.

Behaviorism Constructivism zone of proximal development procedural knowledge conceptual knowledge multiembodiment metacognition .S o n s

mathematics anxiety learned helplessness retention cognitive development physical development social development

Instructor’s Resource Guide, Chapter 2 Sandi Cooper |2 Supplemental Lecture Topics 1.

Discuss some current trends in education that have behaviorism as a base such as drill and practice curricula or instructional materials, and instructional design work by Gagne.

2.

Provide additional information about constructivist work in mathematics education being conducted by researchers such as Kamii, Fosnot, and Cobb. In Beyond Classical Pedagogy: Teaching Elementary School Mathematics, share with students research from three disciplinary perspectives. In What’s Happening in Math Class? Vol. I, share essays written by constructivist teachers. Sample constructivist activities may be found in About Teaching Mathematics, 2nd. Ed by Marilyn Burns and available from Math Solutions.

3.

Provide additional information about Piaget, Bruner, and Dienes and their theories. See the end of book reference list for books they have authored.

4.

Provide additional information about children’s cognitive, social, and physical development.

5.

Discuss procedural and conceptual understanding in greater detail. Excellent sources by Hiebert are included in the end of book resources. In addition, see Making Sense listed in additional resources.

6.

Currently, most preservice and inservice students did not learn elementary mathematics with manipulatives. Provide additional information about what research says about the use of manipulatives and recommendations for teaching with manipulatives. A classic source of information is "Considerations for teachers using manipulative materials" by Robert Reys in Arithmetic Teacher, December, 1971. In this article he defines manipulatives, provides a rational for the use of manipulatives, discusses selection criteria, and gives do's and don'ts for use.

7.

Students also need to become familiar with commercial manipulatives that are available for teaching elementary mathematics. While it is assumed that these will be included throughout the semester, the study of chapter two would be an appropriate time to introduce to students several of the more common commercial manipulatives. Some states, such as Texas, have lists of recommended manipulatives for each elementary grade and those could be duplicated for students.

Manipulatives which might be introduced: attribute blocks, base ten blocks, fraction and decimal models, geoboards, interlocking centimeter cubes, interlocking counting cubes such as unifix or multilink cubes, pattern blocks, tangrams, and pentominoes. A good resource is The Super Source from Cuisenaire Company. (See additional resources.)

Bring an assortment of catalogs from which manipulatives may be ordered (and websites) and as you display each item, let students handle it and look up the price. Follow up with a discussion of ways to cut costs such as the use of overhead versions for demonstration,

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Instructor’s Resource Guide, Chapter 2 Sandi Cooper |3 purchasing for small groups rather than individuals, and alternatives which may be constructed by the teacher or students (see Appendix B in text).

8.

Provide additional information about diverse learners and the teaching of mathematics.

The article by Fennema et al. in the chapter two resources would be a useful source as well as the 1997 NCTM Yearbook-Multicultural and Gender Equity in the Mathematics Classroom, the Changing the Faces of Mathematics series from NCTM, Fiesty Females: Inspiring Girls to Think Mathematically available from Heinemann, and How to Encourage Girls in Math and Science: Strategies for Parents and Educators available from Dale Seymour Publications.

9.

Discuss metacognition in greater detail. A classic article by Garafalo in the May 1987 issue of the Arithmetic Teacher is a good place to start. In addition, Garafalo and Lester's article, Metacognition, cognitive monitoring, and mathematical performance, may be found in the 1985 (issue 3) Journal for Research in Mathematics Education and provides information concerning metacognitive knowledge of person, task, and strategy and metacognitive self-regulation.

10.

Discuss the six standards for teaching mathematics found in NCTM's Professional Standards for Teaching Mathematics, 1991 (See NCTM.org). You could organize the discussion by using a “jigsaw” method, where students are in “home” groups of six and each student is assigned specific standard to prepare. After they have read their standard individually, they meet with their “expert” group to discuss the assigned standard in more detail (expert group refers to all students with the same standard). Then, students join their “home” group to allow each to share the standard they “studied.” You may also use lesson Snapshot video clips found throughout this text as examples to follow this discussion.

11.

Provide more information about math anxiety. A couple of good sources include Math: Facing an American Phobia by Marilyn Burns, available from Math Solutions and Overcoming Math Anxiety (available from W.W. Norton and Company) and Succeed with Math: Every Student’s Guide to Conquering Math Anxiety (available from The College Board) both by Shelia Tobias.

12.

Provide more information about teaching mathematics to young children. Good resources include The Young Child and Mathematics by Juanita V. Copley (Available from NCTM) and Growing Mathematical Ideas in Kindergarten by Linda Schulman, Dacey and Rebeka Eston (Available from Math Solutions).

Student Textbook Activities 1.

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As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and on PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

Instructor’s Resource Guide, Chapter 2 Sandi Cooper |4 2.

Have students review the Chapter 2 Snapshot of a Lesson (pp. 12-13). Discuss how this lesson illustrates some of the shifts in classroom environment (Table 2-1, p. 15).

3.

Using PPT Slide #3, How Can we Support the Diverse Learners in our Classrooms?, discuss the five strategies a teacher can use in his/her math classroom. Talk about potential benefits and challenges that come when teaching classrooms of diverse learners.

4.

Use PPT Slides 4-7, Procedural and Conceptual Knowledge to review the definition of the two terms. Then one at a time, display the provided examples of student knowledge and have the class classify them as procedural or conceptual.

5.

Use PPT Slides #8-12, Behaviorism and Constructivism, to allow students to compare and contrast the behaviorist and constructivist theories of learning. Or have them fill in a 3-column chart with headings Behaviorism, Constructivism, and Both.

Possible responses may include: Behaviorism-behavior may be shaped with reinforcement, direct instruction and drill and practice are necessary, specific goals or objectives-often simple and short term are identified for teachers and students, teachers develop task analysis as skills are learned in a specific order Constructivism-knowledge is actively invented by the learner, new knowledge is integrated into existing mental structures, learning is a social process, exploration and discovery are necessary Both-are evident in today's textbooks

6.

Use Tables 2-2 and 2-3, Characteristics of Learners, and have small groups think of other suggestions for teachers.

7.

Have students work through In the Classroom 2-1, Different Kinds of Four-Sided Figures. Discuss how this activity can be used to determine students’ conceptual understanding.

8.

Have students use interlocking cubes to complete In the Classroom 2-2, Patterns with Blocks. Discuss with students how the completion of this activity could not only encourage problem solving and exploration but also lead to the discovery of the formula for volume of a box (right prism). You may also want to show a clip from the video, Mathematics With Manipulatives: Six Models that shows students filling boxes with cubes.

Student Supplemental Activities 1.

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Bring in sample goals and assessment items being used for mathematics in your state.

Have students examine them and determine if they seem to be based on behaviorism, constructivism, or some combination.

Instructor’s Resource Guide, Chapter 2 Sandi Cooper |5 2.

Have students examine and compare elementary texts from the late 1970s and the early 2000s. Encourage students to identify behaviorist and constructivist influences. Discuss how 1970s edition was published during the "back to the basics" era. Have students answer the question, "Are textbooks less behavioristic in the 2000s than the 1970s?"

3.

Using those same textbooks mentioned in supplemental activity 2, have students select one lesson and identify the conceptual and procedural knowledge to be learned. Have students discuss whether texts seem to emphasize one type of knowledge over another.

4.

Assign groups of students to research and report on persons who have been prominent influences on mathematics curriculum such as Piaget, Bruner, and Dienes.

5.

Choose a mathematical topic and demonstrate how it could be taught from a behaviorist or a constructivist point of view. For example, in the book, About Teaching Mathematics 2nd edition, by Marilyn Burns, an excellent example of a constructivist lesson is provided on the relationship between a circle's circumference and diameter. After students have experienced the two lessons, have them discuss the pros and cons of each approach.

6.

To further illustrate the concept of multi-embodiment, bring in a variety of fraction models for parts of a whole such as fraction pies and fraction strips and parts of a set such as color tiles. Discuss the importance of multi-embodiment and have students brainstorm other materials that could be used to illustrate examples and non-examples for fraction ideas.

7.

PPT Slides #19-21 shows another example of a powerful formula that can be developed naturally with pictures. These pictures encourage students to build on what they know about finding the area of rectangles and apply it to find the sum of the first n counting numbers.

Materials: PPT Slides #19-21, grid paper (1/2 inch or larger, see Appendix B), scissors • Share as a class in considering the first two specific examples pictured on the transparency.

• Assign students to work either individually, in pairs, or small groups to use grid paper and scissors to form staircases for the first 8 counting numbers, first 10 counting numbers, etc.

• Have each group report on their findings, or have them mount their evidence on a large sheet of butcher paper or bulletin board to allow the results to be viewed easily.

• Consider the general case for the sum of the first n counting numbers as illustrated on the transparency. The staircases cut from grid paper should serve to convince any skeptics that 1 + 2 + 3 + 4 + . . . . . + n = n (n+1)/2 .

This problem involves students in problem solving, provides the opportunities for many connections, and illustrates several principles for learning mathematics discussed in this chapter. Encourage students to write about and discuss their

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Instructor’s Resource Guide, Chapter 2 Sandi Cooper |6 learning experience. It would also be meaningful to have students examine their metacognitions during the experience.

8.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book, and Annotated Resources. (pp. 30-31)

9.

The Book Nook for Children, found on pages 31 of the text, provides some children's literature that may be used. Read some of these books and have students develop lessons to go with particular titles. Discuss how children's literature provides a context for developing conceptual understanding or examining historic figures.

10.

Using the References at the end of the chapter, have students locate additional information on topics that interest them such as constructivism, human development or teaching recommendations.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 2: 1.

Learning about the School and Its Resources: Classroom Manipulatives-Finding Them, Classroom Manipulatives-Using Them

2.

Observing the Teacher and Students: Children’s Development, Equity for Gender in Whole Class Lessons, Equity in Single-Gender Group Work, Equity in Mixed-Gender Group Work, Equity for Students with Special Needs, Equity for English Language Learners

3.

Interviewing the Teacher and Students: Teaching & Learning, Attitudes about Mathematics

4.

In the Classroom Lessons: Patterns with Blocks

Additional Resources About Teaching Mathematics, 2nd edition by Marilyn Burns provides many activities presented from the constructivist perspective. Available from Math Solutions.

Beyond Classical Pedagogy: Teaching Elementary School Mathematics by Terry Wood, Barbara Scott Nelson, and Janet Warfield share research from three disciplinary perspectives on alternative forms of instruction. Available from Lawrence Erlbaum. .S o n s

Instructor’s Resource Guide, Chapter 2 Sandi Cooper |7 Changing the Faces of Mathematics is a series of books from NCTM that focuses on equity issues and recommended practices for teaching African American, Asian American and Pacific Islander, Indigenous People of North America, Latinos, and male and female students.

Considerations for teachers using manipulative materials by Robert Reys may be found in the Arithmetic Teacher, December 1971. This classic article provides an excellent overview concerning teaching with manipulatives. Available from NCTM. Fiesty Females: Inspiring Girls to Think Mathematically by Karen Karp, E. Todd Brown, Linda Allen, and Candy Allen, provides literature-based mathematics instruction that includes strong female role models. Available from Heinemann.

Growing Mathematical Ideas in Kindergarten by Linda Schulman Dacey and Rebeka Eston, provides a good example of mathematics instruction with young children. Available from Math Solutions.

How to Encourage Girls in Math and Science: Strategies for Parents and Educators by Joan Skolnick, Carol Langbort, and Lucille Day provides activities, project ideas, and techniques for teaching math and science to girls. Available from Dale Seymour Publications.

Making Sense: Teaching and Learning Mathematics with Understanding by James Hiebert, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piet Human. Included in this book is a summary of research on how classrooms can support students’ mathematical conceptual understanding. Available from Heinemann.

Math: Facing an American Phobia by Marilyn Burns discusses why two-thirds of Americans hate math. It also shows how we can help prevent students from developing negative attitudes.

Available from Cuisenaire Company.

Multicultural and Gender Equity in the Mathematics Classroom: The Gift of Diversity (1997 Yearbook)-edited by Janet Trentacosta contains numerous ideas for providing powerful K-12 mathematics programs for students of any race, ethnicity, language, gender, or socioeconomic situation. Available from NCTM. Overcoming Math Anxiety (available from W.W. Norton and Company) and Succeed with Math: Every Student’s Guide to Conquering Math Anxiety (available from The College Board) both by Shelia Tobias, provide strategies for overcoming anxieties related to math experiences.

The Super Source is a collection of teacher resource books organized by three levels (grades K-2, 3-4, and 5-6). Each book focuses on a particular manipulative including Color Tiles, Cuisenaire Rods, Geoboards, Pattern Blocks, Snap Cubes, and Tangrams. Available from Cuisenaire Co.

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Instructor’s Resource Guide, Chapter 2 Sandi Cooper |8 What’s Happening in Math Class? Vol. I Edited by Deborah Schifter. Shares essays written by teachers who are teaching with constructivist methods. Available from Teachers College Press.

Videos Several videos illustrate different ways children learn mathematics. Each of the following videos would be appropriate to help bring some of the ideas discussed in this chapter into sharper focus.

These videos complement the content of this chapter and would stimulate additional discussions related to different ways mathematics learning occurs.

Mathematics with Manipulatives -- series of six 20 minute videotapes which demonstrate the use of various manipulatives in teaching a variety of mathematical topics in grades K - 6. Five of the tapes focus on one type of manipulative (Base Ten Blocks, Color Tiles, Cuisenaire Rods, Geoboards, and Pattern Blocks). A sixth tape, Six Models, includes several other manipulatives.

All of the tapes highlight different ways in which children learn, including the role of concrete materials and communication. Each of these tapes also includes some print materials to facilitate their use. Available from Cuisenaire Company.

Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 videos and 3, 5-8 videos.

Videos include lessons illustrating Standards 1-4. Each video contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept.

TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples

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Instructor’s Resource Guide, Chapter 3 Sandi Cooper |1

Chapter 3— Planning for and Teaching Diverse Learners Chapter Overview The importance of planning and instruction is the focus of the chapter, opening with a discussion of questions teachers should ask before planning begins. These include: Do I understand the mathematics I am teaching? Where are my students developmentally? What do my students know? What kinds of tasks will I give my students? How will I encourage my students to communicate? What kinds of questions will I ask? What materials will we use? Reasons for planning and the different levels of planning - daily, unit, year-long - are outlined. Basic components and approaches to a lesson plan are reviewed for the investigative plan, direct instruction plan, and exploration plan. Suggestions for planning to meet the needs of students from other cultures, students who are learning English, and students with special needs are included. Other planning and assessment considerations are summarized in the chapter material.

Student Objectives After reading the chapter, the students will be able to: 1.

Summarize six questions teachers ask before planning lessons. (These include: Do I understand the mathematics I am teaching? Where are my students developmentally?

What do my students know? What kinds of tasks will I give my students? How will I encourage my students to communicate? What kinds of questions will I ask? What materials will we use?)

2.

Differentiate among levels of planning and three types of lessons (investigative, direct instruction, and exploration) as well as discuss why it is also important to assess and reflect after teaching.

3.

Summarize recommendations for meeting the needs of all students during mathematics lessons.

Key Vocabulary Preparation for teaching is stressed within this chapter and involves the use of the vocabulary listed below. Students will encounter these terms as they progress through their learning and teaching experiences and should be comfortable with these terms. developmentally appropriate decisions manipulative materials Standards-Based curricula Adaptations investigative lesson direct instruction lesson exploration lesson .S o n s

technology drill and practice software tutorial software simulation software educational game software problem-solving software tool software

spiral approach scope-and-sequence equity strategic moment context or theme assessment analysis

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Supplemental Lecture Topics 1.

To provide more information on recommended instructional practices, discuss the NCTM Professional Standards for Teaching Mathematics. Topics may include the six professional standards, evaluation of the teaching of mathematics, professional development of teachers of mathematics and support of mathematics teachers and teaching.

2.

Share examples from Empowering Students by Promoting Active Learning in Mathematics: Teachers Speak to Teachers and Now I Get It: Strategies for Building Confident and Competent Mathematicians, K-6. Discuss how veteran teachers learn about current teaching recommendations and adjust their instructional practices.

3.

To provide more background on equitable instruction, assign groups of students to read articles from the February 2001 Focus Issue of Teaching Children Mathematics, Mathematics and Culture, October 2004 Focus Issue of Teaching Children Mathematics, Teaching Mathematics to Special Needs Students, Adapting Curriculum and Instruction in Inclusive Classrooms kit, or Windows of Opportunity: Mathematics for Students with Special Needs.

4.

Have the class try some activities for children from NCTM’s Multicultural Mathematics Materials, 2nd edition. Discuss how learning activities from another culture might help develop an appreciation for diversity.

5.

Discuss the role of small groups in the teaching of mathematics. The book, Cooperative Learning in Mathematics: A Handbook for Teachers by Neil Davidson provides useful information.

6.

Use video clips in to show students clips of teaching and learning in classrooms. (See videos under additional resources and Snapshots of a Lesson at the beginning of many chapters in the text.)

7.

Use the latest Technology Counts issue of Education Week to discuss the status of technology integration in the United States Have small groups share information from Guidelines for Evaluating Computerized Instructional Materials, the December 2005 Focus Issue of Educational Leadership, the February 2002 Focus Issue of Teaching Children Mathematics, Learning and Teaching Mathematics with Technology, NCTM’s 2005 Yearbook, Technology-Supported Mathematics Learning Environments, or Learning Math with Calculators.

8.

Demonstrate some of the activities from Math is Language Too: Talking and Writing in the Mathematics Classroom. Brainstorm ways to use talking as a way to support learning.

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9.

Introduce and demonstrate the following types of software found on pages 43-44: drill and practice, simulation, problem solving, tutorial, tool, and game. Use PPT Slides #910, Educational Software, to define each type. Discuss criteria for evaluating software.

10.

Discuss the specific lesson plan model or format students will be expected to use when student teaching. If there is a specific format used throughout the state, display and discuss it. Use some textbook lesson plans and show examples of investigative lessons and direct instruction lessons.

11.

Research has shown that detailed lesson plans are helpful to novice teachers. Discussion may be found in: Leinhardt, G. (1989). Math lessons: A contrast of novice and expert competence. Journal for Research in Mathematics Education, 20(1), 52-75.

12.

Bring in additional resources for integrating children’s literature into mathematics. A good list of books are included under additional resources.

13.

Discuss the role of communication in learning mathematics. Good resources (in additional resources list) include the 1996 Yearbook-Communication in Mathematics K12 and Beyond the February 1995 issue of Teaching Children Mathematics, the April 2000 issue of Mathematics Teaching in the Middle School, Writing in Math Class by Marilyn Burns, and Math is Language Too: Talking and Writing in the Mathematics Classroom.

14.

Provide additional information about teaching mathematics to young children using resources such as Developmentally Appropriate Practice in Early Childhood Programs, (See additional resources.)

15.

Discuss strategies for supporting students who are learning English using resources such as Fifty Strategies for Teaching English Language Learners (book found in references) and the December 2004 Focus Issue of Educational Leadership, Educating Language Learners.

Student Textbook Activities 1.

As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2.

Have students review the Chapter 3 Snapshot of a Lesson (pp. 32-33). Have them use a lesson plan template to create a lesson plan for the snapshot.

3.

Discuss questions teachers should ask before planning lessons. Use PPT Slides #3-5, Preparing to Teach, to summarize key ideas.

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4.

Discuss PPT Slide #6, Effective Mathematical Tasks. Have students look through the entire text on the various In the Classroom Lesson Cards. Challenge them to locate activities that represent the bullets on PPT Slide #6.

5.

Use PPT Slide #7: Questions, to highlight qualities of good questions. Then have students write questions for each question type.

6.

Continue a discussion concerning the role of manipulatives in mathematics lessons. Use PPT Slide #8, Manipulatives, to summarize points made in the text. Have students brainstorm ideas for accomplishing items 2-4.

7.

Discuss each of the three levels of planning: year, unit, day. Use PPT Slides #11-14, Levels of Planning, to provide illustrations for year and unit planning. Specific observations concerning the planning examples follow.

Year: Examine text, district curriculum guide, state requirements, current research and recommendations (ex. NCTM Principles and Standards) and the children's needs.

Identify the units or topics, the sequence of units and the amount of time spent on each.

The article, The Mathematics Textbook: How Can it Serve the Standards? by Rosemary Schmalz, may be found in the February 1994 Arithmetic Teacher. She provides a description of how the teacher can adjust the text sequence to include more new material when planning.

Discussion for PPT Slide #11 example: • The teacher assessed the class and determined that most of the students had mastered text Ch. 1-Place Value and text Ch. 2-addition and subtraction so he chooses to begin with Chapter 3-multiplication, including reviews of Ch 1,2 content while in Ch. 3. Three to four weeks are saved and can be used as needed later in the year. Problem solving strategies, estimation, and mental computation will also be included in daily warm-up activities throughout the year.

• The teacher also identified required topics from the district and state curriculum guides.

The district had an additional computer unit to be completed by all sixth graders and the grade level team planned a mini project around the fall election. State curriculum topics had to be included prior to the state assessment in February so Ch. 9,10 and part of Ch. 7 were moved up. Since the state assessment does not require multiplying fractions and statistics but the district curriculum does, a section of Ch. 7 and Ch. 11 were taught after the state assessments were completed.

• The teacher also knew that division and fractions were the most difficult for students and heavily emphasized on the state assessment so he scheduled the most days for those two units. Following the NCTM Standards recommendations, he plans to emphasize understanding with lots of hands on explorations. Measurement and geometry were lighter topics and provided a break from the computational instruction. They were also good topics to study during December when holiday activities can be integrated. He

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decided to teach metric and customary measurement together although the text separated them into different chapters.

Unit: Identify specific objectives, sequence of objectives, the amount of time spent on each objective and how student knowledge will be assessed. It would also be appropriate to discuss the potential of interdisciplinary planning in elementary classrooms.

Discussion for PPT Slides #13-14 example: The teacher decided to use the text as a resource for the first two days of the unit while providing information about collecting, displaying, and interpreting data. He will use a concrete model and a real life application when introducing mean, median, mode and range. For the rest of the unit, students will work in small groups and apply the skills to a real, meaningful project. Computers will be used to collect and display the data. Groups will present and display their findings. On the last day, students will complete journal entries, update their portfolios and complete a performance assessment.

Day: Identify specific objectives, identify how students will be grouped, the types of learning activities to be included, the materials to be used, amount of time spent and how student progress will be evaluated. PPT Slides #15-16 may be reviewed. Using the lesson plan format students will be required to follow, a lesson plan from the sample chapter 11 may be developed and discussed.

8.

Review Masters PPT Slides #4-6. Then starting with a textbook lesson, demonstrate how it can be used as a base for developing a lesson plan. As examples, use PPT Slide #17 to create an investigative lesson plan, PPT Slide #18 to create a direct instruction lesson plan, and PPT Slide #19 to create an exploration lesson plan.

9.

Use PPT Slide #24, Adaptations, to discuss how a lesson plan may be geared up or down to meet the needs of all students. Provide students with a lesson plan (You might use the direct instruction lesson on page 51 of the text.) and challenge them to create adaptations for low achievers, high achievers, and English language learners. Examples of adaptations are provided in Figure 3-9 on page 57.

10.

Review PPT Slides #20-21: Teaching English Language Learners, and PPT Slides #2223: Potential Barriers for Students with Special Needs. For each key idea, brainstorm examples.

Student Supplemental Activities 1.

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Have students examine several current textbook series (traditional and standards-based).

Have them identify evidence of the NCTM Principles and Standards recommendations, the use of manipulatives and technology, the use questioning and communication, the use of adaptations, and assessment techniques.

Instructor’s Resource Guide, Chapter 3 Sandi Cooper |6

2.

Elementary students are expected to communicate and reason about mathematics. For this to occur, preservice elementary teachers need to have experiences that involve them in mathematics in a similar fashion. The following problem provides an opportunity for students to verbalize their understandings and reasoning about one of many patterns to be found in the hundred board.

Materials: PPT Slides #26-27: Find a Pattern, copies of hundred boards (Appendix B), color tiles, calculators, copy of hundred board to display.

• Start by making sure that students have a hundred board to use and access to a calculator for computing the averages necessary.

• Share the problem and steps for establishing a pattern using the overhead transparency of the problem.

• Allow time for students to work and share together. Take advantage of the chance to observe your students as they work.

• Bring the group together to discuss the pattern that was observed and share their understandings of why this pattern is occurring. Encourage their participation by communicating to others their understanding of the situation using words, symbols, or pictures/models. Use of an hundred board transparency may be helpful as students provide explanations.

3.

Use PPT Slides #9-10: Educational Software to review types of software. Have students review computer software that could be used in teaching mathematics. Include at least one example each for the 5 software types: drill and practice, simulation, tutorial, problem solving, and game. Discuss how higher-level thinking can be incorporated.

Working in this setting provides prospective teachers with experience as learners being taught in the way they are to teach. Have them turn in a written review in which they discuss the pros and cons of one of the pieces reviewed.

4.

Have students participate in a small group learning activity that utilizes the "jigsaw" strategy. Sample activities may be found in resource books such as Get It Together: Math Problems for Groups by Tim Erickson, published by the Lawrence Hall of Science.

After completing the activity, debrief and allow students to discuss their impressions.

5.

Provide each small group with the teacher's guide from a textbook series with a particular topic or lesson marked. Have the group plan a lesson based on what they have learned from this and earlier chapters. How could problem solving, technology, and manipulative materials be incorporated into the topic they are considering? How can the lesson reflect the recommendations of the NCTM Teaching Standards? Have them write the lesson using PPT Slides #17-19 as templates. Allow time for the groups to share their ideas for approaching the variety of topics selected.

6.

Have students examine samples of state and district curriculum guides and textbook scope and sequence charts. Have students outline a plan for a year or a unit plan similar to the one provided on PPT Slides #12-14.

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7.

Allow students to examine lesson plans written in various formats with varying levels of detail. For example, provide some traditional textbook lessons, standards-based textbook lessons, teacher plan book lessons, commercial lessons such as the ones found in books from Math Solutions. Have students discuss in groups the pros and cons of each lesson plan type.

8.

Invite a practicing teacher or administrator to class to talk about lesson planning. Or have students interview a teacher or administrator. Discuss the types of plans that are required, the level of detail required, the time spent on planning, etc.

9.

Locate the textbook pages from which the sample lesson plans, Figures 3-3, 3-4, 3-5, and 3-6 were created. Allow students to compare the text pages with the lesson plans and discuss critical features.

10.

The Math Links in this chapter provide students with additional resources for this chapter about communicating, manipulatives, and English Language Learners. In class, or as an outside assignment, have students explore these web sites.

11.

The Book Nook for Children, found in the text on pages 60 lists several teacher resource books for using children’s literature with mathematics. Bring several of these books to class and have students brainstorm ideas that could be used in a math lesson.

12.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book. (pp. 59-60)

13.

Using the References list at the end of the chapter, have students locate additional information on topics which interest them such as cooperative learning and understanding mathematics.

14.

Current elementary textbooks may also be used to show examples of a spiral curriculum.

Have various small groups choose specific content, such as place value or multiplication of whole numbers, and examine how the level of difficulty increases as the grade level increases. Discuss the pros and cons of a spiral curriculum mentioned in the text.

15.

One way for students to begin to understand how children learn mathematics is to observe children in action. Have your students pose a problem to an individual child or small group. One interesting approach would be to vary the use or nonuse of manipulative materials as children attempt to find a solution to the situation presented.

Students can access Teachscape 3.1 and 3.2 to view videos.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 3: .S o n s

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1.

Learning about the School and Its Resources: Classroom Sketch, Children’s Literature in the Mathematics Classroom, Mathematics Textbook Lesson, Mathematics Textbook Unit

2.

Observing the Teacher and Students: The Learning Environment, Teaching a Mathematics Lesson, Focusing on Individuals, Checking on Technology, Analyzing Classroom Discourse, Analyzing Textbook Use

3.

Interviewing the Teacher and Students: Teaching a Topic, Grouping in the Classroom

4.

Helping Children Learn with Technology: Reviewing Software, WWW Lesson Plan

Additional Resources Adapting Curriculum and Instruction in Inclusive Classrooms: Staff Development Kit, by Ebeling, Deschenes, and Sprague, helps teachers understand the rationale for making adaptations and provides nine suggested types of adaptations. Available from the Institute for the Study of Developmental Disabilities in Bloomington, Indiana.

Choosing a Standards-Based Mathematics Curriculum by Goldsmith, Mark, and Kantrov, is a resource for districts considering adopting and implementing a Standards-based mathematics curriculum. It provides an overview of the 13 programs endorsed by the Educational Development Center.

Communication in Mathematics, K-12 and Beyond (1996 Yearbook), edited by Portia C. Elliott, emphasizes the importance of communication in implementing math education reform. Shows how classrooms and schools can become "discourse communities". Available from NCTM. Cooperative Learning in Mathematics, edited by Neil Davidson, provides a collection of essays concerning small group learning in mathematics.

Developmentally Appropriate Practice in Early Childhood Programs, edited by S. Bredekamp and C. Copple, provides recommended practices for the teaching of young children. Available from the National Association for the Education of Young Children (NAEYC).

Educating Language Learners, is a Focus Issue of Educational Leadership, Dec. 2004/Jan. 2005.

It discusses issues related to supporting students in our classrooms who are learning English.

Available from ASCD. Empowering students by Promoting Active Learning in Mathematics: Teachers Speak to Teachers is edited by Dorothy Buerk and describes how five teachers worked to implement NCTM’s standards and empower their students. Available from NCTM. Exploring Mathematics through Literature: Articles and Lessons for Prekindergarten through Grade 8, edited by Diane Thiessen, provides articles and lesson examples of how to use children’s literature to teach mathematics. Available from NCTM.

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Fifty Strategies for Teaching English Language Learners, by A. Herrell, provides teachers with a wide variety of instructional strategies that classroom teachers can use to support the students in their classrooms who are learning English. Available from Merrill.

Get it Together: Math Problems for Groups by Tim Erickson is available from the Lawrence Hall of Science. Each problem has 6 clues for the group. Group members must work together and share their clues to reach a solution.

Guidelines for Evaluating Computerized Instructional Materials by Heck, Johnson, and Kansky provides suggestions for wise software selection. Contains reproducible evaluation instruments and outlines for using them. Available from NCTM. How to Use Children's Literature to Teach Mathematics, 2nd ed. by Rosamond WelchmanTischler. Provides a collection of children's books with accompanying math activities.

Available from NCTM. Integrating Children’s Literature and Mathematics in the Classroom: Children as Meaning Makers, Problem Solvers, and Literary Critics by Michael Shiro illustrates how children’s literature can be used to communicate mathematical concepts as well as provides set of standards for assessing the mathematical and literary quality of children’s books. Available from Teachers College Press.

Just for Favorite Manipulatives-from Creative Publications includes 10 books (4 primary and 6 intermediate) on using specific manipulatives. Manipulatives highlighted include: pattern blocks, linker cubes, teddy bears, coins, base ten, fraction circles, rainbow cubes, tangrams, and geoboards.

Learning and Teaching Mathematics with Technology, is a Focus Issue of Teaching Children Mathematics, February 2002. It gives a good picture of how technology is being used in elementary classrooms. Available from NCTM. Learning in the Digital Age, is a Focus Issue of Educational Leadership, Dec. 2005/Jan. 2006. It discusses issues related to the use of technology in today’s classrooms. Available from ASCD. Learning Math with Calculators: Activities for Grades 3-8, by Sparrow and Swan, addresses questions and concerns teachers have about using calculators. It also contains a variety of calculator activities that are designed to develop children’s number sense and problem solving ability. Available from Math Solutions.

The Math, Literature, and Nonfiction Series is a set of six books that shows teachers how to use children’s books in math lessons. Available from Math Solutions.

Mathematical Communication-A special April 2000 focus issue of Mathematics Teaching in the Middle School. Available from NCTM.

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Mathematics and Culture, is a Focus Issue of Teaching Children Mathematics, February 2001. It includes a nice variety of articles on equitable mathematics instruction. Available from NCTM. Mathematics and Literature, is a Focus Issue of Mathematics Teaching in the Middle School, April 2005. It shows how children’s literature may be used to teach mathematics in middle school classrooms. Available from NCTM. Math is Language Too: Talking and Writing in the Mathematics Classroom by David J and Phyllis Whitin, describes how fourth graders were encouraged to inquire through writing and talking. Also includes activities to use with children. Available through NCTM. Math Lessons: A contrast of novice and expert competence by G. Leinhardt may be found in the 1989 (volume 20) issue 1 of the Journal for Research in Mathematics Education. In this article, the value of detailed planning for novices is discussed.

Multicultural Mathematics Materials: 2nd edition by Marina C. Krause includes games and activities from around the world that can be used with children. Promotes an appreciation of cultural diversity. Available from NCTM. New Visions for Linking Literature and Mathematics by David Whitin and Phyllis Whitin, offers criteria for evaluating mathematics-related books and provides a wide range of books and strategies for using them. Available from NCTM. Now I Get It: Strategies for Building Confident and Competent Mathematicians, K-6 by Susan O’Connell, provides many examples of best practice in mathematics instruction including the role of the teacher, how to promote thinking, how to engage students, and how to incorporate math talk. Available from Heinemann.

Professional Standards for Teaching Mathematics, NCTM, provides additional information concerning the role of the teacher, student, task, and environment. Available from NCTM. Teaching Mathematics to Special Needs Students, is a Focus Issue of Teaching Children Mathematics, October 2004. It gives a collection of articles on how students with special needs may be taught in elementary classrooms. Available from NCTM. Technology-Supported Mathematics Learning Environments, 2005, 67th NCTM Yearbook with CD, edited by William J. Masalski, provides overviews of research on the impact of technology on mathematics instruction. Available from NCTM. The Handbook of Research on Mathematics Teaching and Learning, edited by D. Grouws, provides up to date research findings on topics such as planning, gender issues, etc.

The Wonderful World of Mathematics: A Critically Annotated List of Children's Books in Mathematics edited by Diane Thiessen, Margaret Matthias, and Jacquelin Smith provides an annotated bibliography of books for young children which may be used to teach mathematics.

Available from NCTM. .S o n s

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Windows of Opportunity: Mathematics for Students with Special Needs edited by Carol Thornton and Nancy Bley, is a resource for teachers who work with students with disabilities or are gifted or talented in mathematics. Available from NCTM. Writing in Math Class-by Marilyn Burns presents the case for making writing a part of math instruction. Five types of writing, helpful suggestions for including writing, and student work samples are included. Available from Cuisenaire Co. Videos Attaining Excellence: A TIMSS Resource Kit Video provides examples of classroom teaching.

Available from NCTM. Children’s Mathematics: Cognitively Guided Instruction includes a book and two accompanying CD’s with video clips illustrating teaching and learning in real classrooms.

Mathematics for Middle Grades: The Role of the Teacher -- one from a series of three 21minute videotapes targeted toward middle grades. Portions of mathematics lessons in grades 6-8 are shown to illustrate how problem situations can be created, and how children can work cooperatively in learning mathematics. Available from Cuisenaire Company.

Fostering Children’s Mathematical Development: The Landscape of Learning by Maarten Dolk and Catherine Twomey Fosnot, is a kit that contains a facilitator’s guide and CD with video clips showing children’s ideas and strategies. This material is part of a larger set of professional development materials called, Young Mathematicans at Work, Grades Prek-3. Available from Heinemann.

Talking Mathematics Resource Package by Corwin, Storeygard, Price, and Smith, contains a staff development guide and videotape with seven video clips from elementary classrooms and for teacher staff development. Available from Heinemann.

Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 tapes and 3, 5-8 tapes. Tapes include the four process standards of problem solving, communication, reasoning, and connections. Each tape contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Available from: The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800-864-9846.

Teaching Mathematics Effectively -- a 21-minute videotape which reports characteristics of effective mathematics lessons. These characteristics are discussed and illustrated by scenes from actual mathematics lessons. Available from ASCD. Mathematics: Teaching for Understanding, is a set of 3, 20-minute videos. Part 1 models teacher-directed, whole-class lessons. Part 2 features small groups of students working independently on a menu of activities. Part 3 shows how communication in math lessons, supports children's learning. The set is available from Cuisenaire Company. .S o n s

Instructor’s Resource Guide, Chapter 4 Sandi Cooper |1

Chapter 4— Assessment: Enhanced Learning and Teaching What This Chapter Is About This chapter looks at student assessment as an integral part of teaching. Recommendations from NCTM's Assessment Principle and Assessment Standards for School Mathematics are highlighted. Assessment of learning (summative) and assessment for learning (formative) are discussed and the four phases of assessment are illustrated. Various ways of gathering evidence, recording, analyzing, and communicating information from assessments are shared. Stressed is the importance of collecting sufficient data to be used in positive ways to support students learning, modify instruction, and inform parents.

Student Objectives After reading the chapter, the students will be able to: 1.

Distinguish between and provide examples of assessment of learning (summative) and assessment for learning (formative).

2.

Summarize the NCTM Assessment Standards recommendations and shifts in student assessment.

3.

Summarize methods of gathering evidence and making judgments about student learning.

4.

Summarize methods of keeping records and communicating about assessments.

5.

Identify three groups with which the teacher communicates.

Key Vocabulary The chapter stresses the importance of using appropriate forms of assessment and discusses each of the terms below. It is important that students understand the different possibilities and be able to use the correct terms when discussing assessment. observation rubric questioning performance indicator interviewing peer assessment performance tasks Assessment Principle self-assessments checklists work samples class records portfolios student files writings teacher-designed written tests Assessment Standards for School Mathematics .S o n s

summative assessment high-stakes assessments No Child Left Behind(NCLB) curriculum mapping disaggregation Formative assessment standardized achievement tests

Instructor’s Resource Guide, Chapter 4 Sandi Cooper |2 Supplemental Lecture Topics 1.

The Assessment Standards for School Mathematics (in additional resources and PPT Slides #9-11) provides information about NCTM's recommendations concerning assessment.

2.

Display and discuss assessment instruments and techniques used in your state's assessment program. Invite a practicing teacher to discuss implications for instruction.

3.

Discuss specific assessment techniques and display student work samples. The NCTM 1993 Yearbook, Assessment in the Mathematics Classroom, Exploring Classroom Assessment in Mathematics, Learning from Assessment: Tools for Examining Assessment through Standards, Mathematics Assessment: Cases and Discussion Questions for Grades K-5, Mathematics Assessment Sampler, and Measure for Measure: Using Portfolios in K-8 Mathematics are good sources.

4.

Display and discuss specific examples of alternative assessment instruments.

Mathematics Assessment: Myths, Models, Good Questions, and Practical Suggestions, (in additional resources) provides great examples.

5.

Display and discuss PPT Slides #22-26. They are work samples taken from children in grades three and four. All of the children were asked to solve the following problem: Nineteen children are going on a field trip. Five children can ride in each car. How many cars will be needed to get all 19 children? Problem source: Models of problem solving processes by Carpenter et al. (See additional resources.) The problems were read to the students and reread as many times as needed. The samples are intended to show the variety of approaches children will use when they are not restricted to a particular method. They also show that children are often intuitively able to solve problems with manipulatives or drawings for which they have not had formal instruction. Specific observations concerning each sample follow: PPT #22 Alex used color tiles to successfully solve the problem. He does not attempt to write a division equation.

PPT #23 Ben successfully draws a picture, then uses the division symbol in an equation. He knows the answer to the problem is four but does not consider the remainder.

PPT #24 Carl uses repeated subtraction to successfully solve the problem. He does not consider the remainder.

PPT #25 Andrea successfully solves the problem. She uses tally marks. First, she places 4 in each car, then corrects and uses five. She places the writes the symbols in the format of a subtraction problem.

PPT #26 Amy used color tiles to successfully solve the problem, then sketched this solution. She notes there will be a remainder of one seat left.

Encourage students to share observations about the work samples and what the children know. Students could also use the holistic scoring scale on p. 76 or the analytic scoring scale on page 75 to evaluate the work samples.

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Instructor’s Resource Guide, Chapter 4 Sandi Cooper |3 6.

Review video clips included in Integrating Mathematics and Pedagogy to Illustrate Children’s Reasoning (IMAP) CD and have students discuss children’s thinking, brainstorm possible reasons for misconceptions, and suggest instructional strategies that might be effective for these children.

7.

Select a chapter or two from The having of wonderful ideas and other essays on teaching and learning by Eleanor Duckworth to share highlights with your students to share great stories about children’s thinking.

8.

Provide additional information about the No Child Left Behind act and what impact this has had on various educational expectations in your state.

9.

Provide additional information about the National Assessment of Educational Progress (NAEP). A good resource is NCTM’s Results and Interpretations of the 1990-2000 Mathematics Assessments of the National Assessment of Educational Progress or the NAEP website listed in the chapter Math Links.

Student Textbook Activities 1.

As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #3. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2.

Have students review the Chapter 4 Snapshot of a Lesson (pp. 61-62). The clip may be viewed at NCTM.org (see Math Links 4.4). Discuss how different types of assessment can help us learn what students know and can do.

3.

Discuss the difference between assessment of learning and assessment for learning. Use PPT Slides #4-6: Two Types of Assessment, to highlight key ideas.

4.

Discuss the NCTM Assessment Principle and Assessment Standards for School Mathematics. Use PPT Slides #9-11, to highlight the key ideas.

5.

Review the methods of assessment discussed in the text. Use PPT Slides #12-13, Ways to Assess, to guide the discussion. Have the class brainstorm pros and cons of each method.

6.

Have students use In the Classroom 4-1 to conduct an assessment interview or Figure 4-5 to complete an observation. Discuss their results.

7.

Have students examine the student work samples found in Figure 4-9. Then have them use Figure 4-2, Figure 4-3, or Figure 4-10 to evaluate the work samples. Additional samples may be found in A Collection of Performance Tasks and Rubrics and Mathematics Assessment Sampler, Grades 3-5.

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Instructor’s Resource Guide, Chapter 4 Sandi Cooper |4 8.

Discuss various methods for recording assessment information. Use PPT Slides #19-20 to guide the discussion. Have students critique the forms found on pages 79-80.

9.

Discuss the issues addressed in the Cultural Connections at the end of the chapter.

Student Supplemental Activities 1.

Collect a variety of actual assessment instruments from a variety of sources. Give each small group one or two instruments to discuss in light of their experiences and topics brought up for consideration in this and earlier chapters. Classify them as whether they could be used for assessment for learning or assessment of learning. Are there items they think are valuable for assessing the understanding of children? Why? Are there items that are not useful? Why? How could items be improved? (See additional resources for good sources of instruments.)

2.

Have students read the articles, Learning to question: categories of questioning used by preservice teachers during diagnostic mathematics interviews by Moyer & Milewicz (2002) and Interviews: a window to students’ conceptual knowledge of the operations by Huinker (1993). Lead a group discussion of important information from these articles, especially as a preparation for conducting interviews with children.

3.

Have students prepare a basic plan for conducting an interview. Place students in pairs to role play the interview process based on the written outline prepared. If students are involved in a field experience, have them audiotape or videotape their interview with a child. After either experience, have students reflect on their interview. Were there questions they should have included? Were there responses that were unexpected? How would they refine the interview outline and questions in preparation for the next interview experience? In the Classroom 4-1 may be used as an example.

4.

You may choose to have students solve problems from chapter 5, then trade and practice using various assessment methods highlighted on PPT Slides #12-13, Ways to Assess, to evaluate the solutions.

5.

Have students examine and evaluate the assessment instruments found in current elementary textbook series. Have students report on how closely the instruments follow the NCTM recommendations.

6.

Have students interview practicing teachers concerning assessment in their classroom.

Develop a class graph to determine the types of assessment most frequently used.

7.

Show the video, Mathematics: Assessing Understanding or A Look at Children's Thinking (listed in additional resources). Discuss what we can learn from interviewing children.

8.

Show the video, Mathematics Assessment: Alternative Approaches, or Mathematics Assessment: A Video Library, K-12 (in additional resource list) and discuss the alternative

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Instructor’s Resource Guide, Chapter 4 Sandi Cooper |5 methods illustrated in the tape.

9.

The Math Links in this chapter provide students with additional resources and video examples for assessment. In class, or as an outside assignment, have students explore these web sites.

10.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

11.

The Book Nook for Children, lists several children’s stories that focus on assessment.

Share one or more and discuss the implications.

12.

Using the References list at the end of the chapter, have students locate additional information on topics that interest them such as NAEP.

13.

Divide students into groups of 4. Have each person in the group read and report on one article from the November 2005 issue of the Kappan.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 4: 1.

Learning about the School and its Resources: Testing

2.

Observing the Teacher and Students: Proctoring a Test or a Classroom Assignment, Focusing on Individuals, Analytic/Holistic Problem Solving Scoring, Performance Task

3.

Interviewing the Teacher and Students: Assessment, Primary Interview on Place Value

Additional Resources A Collection of Performance Tasks and Rubrics-by Charlotte Danielson is available in four volumes (lower elementary, upper elementary, middle school, and high school). These books provide performance tasks and scoring rubrics for a variety of topics in mathematics. Also includes several samples of student work. Available from Eye on Education.

Assessment to Promote Learning, a focus issue of The Kappan, November 2005. Available from Phi Delta Kappa. A variety of articles on assessment are provided.

Assessment in the Mathematics Classroom, NCTM 1993 Yearbook, edited by Webb.

The yearbook is divided into 5 sections: general classroom assessment, grades K-4, 5-8, and 9-12 assessment, and classroom assessment issues. .S o n s

Instructor’s Resource Guide, Chapter 4 Sandi Cooper |6 Assessment Standards for School Mathematics is the third standards document from NCTM. It contains recommendations for assessment practices and assessment systems.

Exploring Classroom Assessment in Mathematics: A Guide for Professional Development by Deborah Bryant and Mark Driscoll. Guides teachers through steps such as deciding the purpose of the assessment, gathering evidence of student learning, and using data about student’s learning. Available from NCTM. California Mathematics Council & EQUALS. (1989). Assessment alternatives in mathematics: An overview of assessment techniques that promote learning. Available from EQUALS. Huinker, D. (1993). Interviews: a window to students’ conceptual knowledge of the operations.

In N.L. Webb (Ed.), Assessment in the mathematics classroom (pp. 80-86). Reston, VA: National Council of Teachers of Mathematics.

Learning from Assessment: Tools for Examining Assessment through Standards by Tania Madfes. Contains middle school mathematics professional development materials which may be used with cooperative or independent study opportunities. Available from NCTM. Mathematics Assessment: A Practical Handbook for Grades k-2, 3-5 and 6-8 are three in a series of six books that complement the Assessment Standards for School Mathematics. Contain examples, tips, and teacher-developed assessment tasks. Available from NCTM. Mathematics Assessment: Cases and Discussion Questions for Grades K-5 and 6-12 are two more in a series of six books. They contain cases and helpful hints for discussion. Available from NCTM. Mathematics Assessment: Myths, Models, Good Questions, and Practical Suggestions, edited by Jean Kerr Stenmark, provides assessment models as well as instructions on how to use portfolios and other techniques. Available from NCTM. Mathematics Assessment Sampler Grades 3-5 provides formative assessment problems for the NCTM content strands, student work with comments, teacher notes, objective questions, samples rubrics, and resources. This is one of four in a series from NCTM. Measure for Measure: Using Portfolios in K-8 Mathematics, by Therese M. Kuhs provides practical suggestion for using portfolios and explaining them to students and parents. Available from Heinemann.

Moyer, P.S., & Milewicz, E. (2002). Learning to question: categories of questioning used by preservice teachers during diagnostic mathematics interviews. Journal of Mathematics Teacher Education, 5 (4): 293-315.

Results and Interpretations of the 1990-2000 Mathematics Assessments of the National Assessment of Educational Progress edited by Kloosterman and Lester is available from NCTM. .S o n s

Instructor’s Resource Guide, Chapter 4 Sandi Cooper |7 The having of wonderful ideas and other essays on teaching and learning by Eleanor Duckworth provides some great real life situations that allow students to have a glimpse into children’s thinking. Available from Teachers College Press.

Videos Integrating Mathematics and Pedagogy to Illustrate Children’s Reasoning (IMAP), organized by Randy Phillip, Candace P. Cabral and the San Diego State University Foundation. This CD contains 25 video clips of children working through mathematical tasks. Available from Pearson Education, Inc. A Look at Children's Thinking by Kathy Richardson is a set of 2, 25-minute videos. Children are interviewed with a focus on how children think and what they understand.

Mathematics Assessment: A Video Library, K-12-contains 5 tapes which show informal and formal assessment options and demonstrate the link between instruction and assessment.

Available from Annenberg, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800864-9846.

Mathematics Assessment: Alternative Approaches, directed by Terese Kuhs, uses a panel discussion and classroom dramatizations to explore alternative approaches to assessment.

Mathematics: Assessing Understanding. is a set of 3, 20-minute videos which shows students, ages 7-12, being interviewed. Available from Cuisenaire Company.

Mathematics: Teaching for Understanding, is a set of 3, 20-minute videos. Part 1 models teacher-directed, whole-class lessons. Part 2 features small groups of students working independently on a menu of activities. Part 3 shows how communication in math lessons, supports children's learning. The set is available from Cuisenaire Company.

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Instructor’s Resource Guide, Chapter 5 Sandi Cooper |1

Chapter 5 — Processes of Doing Mathematics What This Chapter Is About This chapter provides an overview of NCTM’s five Process Standards: Problem Solving, Reasoning and Proof, Communication, Connections, and Representations. Problem solving is a process central to the mathematics curriculum. Problems should challenge and engage students in learning. Reasoning involves justifying mathematical conjectures. Communication allows children to share ideas and deepen their knowledge. Connections are made among mathematical ideas, symbols and procedures, and contexts outside mathematics.

Student Objectives After reading the chapter, the students will be able to: 1. Summarize NCTM’s five Process Standards.

2. Describe the eight mathematical practices highlighted in the Common Core State Standards Initiative.

3. Describe how teaching mathematics through problem solving is different from simply teaching students to solve problems.

Key Vocabulary The Process Standards and problem solving are important parts of the mathematics curriculum.

Students should understand what a problem is and be aware of the different strategies and approaches used in problem solving.

NCTM Process Standards problem solving reasoning and proof conjecture communication connections representation

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Instructor’s Resource Guide, Chapter 5 Sandi Cooper |2 Supplemental Lecture Topics 1.

Some NCTM yearbooks provide excellent information on the process standards.

Connecting Mathematics across the Curriculum (1995), Communication in Mathematics, K-12 and Beyond (1996), Developing Mathematical Reasoning in Grades K-12, (1999) and The Roles of Representation in School Mathematics (2001), are good sources of information. Two other resources include Mathematical Reasoning: Analogies, Metaphors, and Images and Language and Communication in the Mathematics Classroom (see additional resources).

2.

The classic chapter, Developing Understanding in Mathematics via Problem Solving, by Schroeder and Lester (listed under additional resources) provides an interesting discussion of three approaches to teaching problem solving. Discuss the three approaches (teaching about, for, and via problem solving) and demonstrate examples of the three approaches. For example: Teaching a specific problem-solving strategy would be an example of teaching about problem solving. Teaching how to multiply, then providing problems involving multiplication would be an example of teaching for problem solving. The banquet table problem, found in Mouse & Elephant, The Middle Grades Mathematics Project, (additional resources) would be an example of teaching via problem solving.

3.

Provide additional examples of supporting children’s talk by using resources such as Talking Mathematics: Supporting Children’s Voices (book and video) and Let’s Talk Math.

4.

Provide additional information about Cognitively Guided Instruction. The book, Children’s Mathematics: Cognitively Guided Instruction (additional resources) and accompanying CD provide insight into teachers and children implementing the strategy.

5.

Preservice teachers may have the misconception that young children aren’t able to do problem solving and reasoning. Share examples from Little Kids-Powerful Problem Solvers, Math is Language Too, Show and Tell (see additional resources).

6.

Show additional video clips from the text’s Snapshot of a Lessons or from video resources found under Additional Resources. Discuss the process standards being applied.

Student Textbook Activities 1.

As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2.

Have students review the Chapter 5 Snapshot of a Lesson. Have them identify all of the different process standards that were used.

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Instructor’s Resource Guide, Chapter 5 Sandi Cooper |3 3.

Discuss the NCTM Process Standards. Use PPT Slides #3-14, Process Standards, to highlight the NCTM recommendations.

4.

Have students brainstorm what the process standards might specifically look like in their classrooms. Use PPT Slides #15-16: Supporting Mathematics Learning with the Process Standards, to record their ideas. A few general ideas are included to get them started however encourage them to be more specific. For example, under communication, they might list “write in math journals twice a week”.

5.

Discuss different types of representations. Use PPT Slide #18: Five Ways to Represent Mathematical Ideas. Select a variety of concepts such as one-half, division, greater than, triangle, mode. Divide the class into five groups and assign each a different representation. Have them share their representation for each concept.

6.

Once you have completed chapters 1-6, the students will have received a good overview of recommended mathematics instruction in elementary classrooms. Chapter 7 will begin the second section of the text when individual mathematics topics will be addressed.

Give the students an opportunity to reflect on all they have learned so far and identify ideas they find to be most important. Use PPT Slide #19: Big Ideas for Teaching Mathematics and have them list 3-5 “big ideas” they plan to use in their own classrooms.

For each, have them list specific methods or materials they would use and also explain why this big idea is beneficial to children’s learning. Many of the reasons that support these methods may be found in chapters 1-6. An example is provided here that may help them get started. Once you have completed this exercise here, you might want to go back to it after you complete other chapters. For example, after completing Chapters 8-11, have students make a chart listing 3-5 Big Ideas for Teaching Number and Operations for Whole Numbers or after Chapter 17, make a chart with 3-5 Big Ideas for Teaching Data, Statistics and Probability. Having students complete these charts is also helpful when preparing for the Case History Exam Essay Questions.

EXAMPLE: Recommendation-Use a variety of manipulative materials Specific Methods/Materials-Keep a variety of counters and other manipulatives in tubs so small groups of students or individuals can use them when needed. Always begin development of new math concepts with concrete models students can handle.

Reasons why beneficial-We know children’s learning moves from concrete to abstract.

They need to be actively involved in learning.

7. In small groups, have students play the game, Rolling the Dice, on In the Classroom 5-1.

Give each group time to analyze the game using the questions on the card. Then have each student complete “Writing about the Game”. Finally, have them examine what the fourth-grade students wrote in Figure 5-2. Discuss the benefits of oral and written communication in mathematics learning. Identify the process standards that were used while completing this activity.

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Instructor’s Resource Guide, Chapter 5 Sandi Cooper |4 8. Have students work in groups to complete the logical reasoning problems in Figure 5-5.

Share the children’s responses described in the chapter. Then discuss the connections to the process standards.

Student Supplemental Activities 1.

The use of the problem shown on PPT Slides #20-22: Pentominoes, provides students with a problem-solving situation. The setting is not a typical word problem from textbook mathematics but considers a more interesting, nonroutine problem. Prospective teachers need to experience problem solving firsthand and need to be encouraged to reflect on their approaches and solutions as Polya's problem-solving model suggests.

After the activity has been complete, discuss the process standards that were used while completing the activity.

Materials: PPT Slides #20-22, color tiles, grid paper, crayons or colored pencils • Show the examples and nonexamples of pentominoes using the PPT files and ask each student to write a definition of pentomino on the back of the provided grid paper. (Do not display the last question so that students remain focused on the definition step.)

• Have students share their definition with another student to allow comparison and clarification of definitions. Share several as whole class and decide on class definition of pentomino.

• Share the last question on the PPT. You will need to discuss what it means to be different. One way to gain agreement on this is to use color tiles on the overhead to present two pentominoes and ask if they are the same. Have the students supply reasons why or why not. Agree that if a figure has been slid, flipped, or turned, it is the same figure.

• Have students work in pairs to experiment using their color tiles to form the different pentominoes, recording their set of pentominoes on the grid paper. The use of color, rather than pencil or ink, makes it easier to distinguish the pentominoes on the grid paper.

• As the students work, circulate and observe how they are proceeding. Are they just randomly trying things or do some pairs seem to have developed a strategy?

• Come back together as a whole class and ask how many different pentominoes were found. Have students provide the 12 possible pentominoes making use of the overhead so all students have all 12 possibilities.

• Engage the group in a discussion of their experiences as learners in a problem-solving situation. Ask about the teacher's role during this time.

For further information about pentominoes and other activities involving their use, consider the article "Pentominoes Revisited" by Barry Onslow in the May 1990 Arithmetic Teacher. In addition, the book, Pentomino Activities, Lessons, and Puzzles

by Henri Picciotto provides over 200 Pentomino activities (listed in additional resources).

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Instructor’s Resource Guide, Chapter 5 Sandi Cooper |5 2.

Have students complete activities from the following NCTM books listed under Additional Resources. After they are finished, debrief and discuss the process standards they applied. You may select from Navigating through Problem Solving and Reasoning, Mission Mathematics K-6, or Games and Puzzles for Elementary and Middle School Mathematics.

3.

The Math Links in this chapter provide students with additional resources and video examples for the process standards. In class, or as an outside assignment, have students explore these web sites.

4.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

5.

Using the References at the end of the chapter, have students locate additional information on topics that interest them such as the use of communication and connections.

6.

The Book Nook for Children, provides a list of children's literature that may be used.

Read some of these books and have students develop lessons to go with particular titles.

Discuss how children's literature provides a context for the process standards.

7.

Divide students into groups of 4. Have each person in the group read and report on one article from the following journals. The April 2000 focus issue of Mathematics Teaching in the Middle School, is on Mathematical Communication. The February 1998 focus issue of Teaching Children Mathematics is about Linking Mathematics Learning with Parents, Communities, and Business and Industry.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 5: 1.

Observing the Teacher and Students: Mathematical Processes

2.

Interviewing the Teacher and Students: Doing Math, Problem-Solving Strategies

Additional Resources Beyond the Classroom: Linking Mathematics Learning with Parents, Communities, and Business and Industry is a focus issue of Teaching Children Mathematics (February 1998).

Shows real-life connections between mathematics and the world. Available from NCTM.

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Instructor’s Resource Guide, Chapter 5 Sandi Cooper |6 Children’s Mathematics: Cognitively Guided Instruction by Carpenter, et al. shares research on teaching and assessing children’s learning using CGI. Two accompanying CD’s provide clips of teachers and students implementing the strategy. Available from NCTM. Communication in Mathematics, K-12 and Beyond, the 1996 NCTM yearbook provides a nice variety of chapters about integrating communication into mathematics class. Available from NCTM. Connecting Mathematics across the Curriculum, the 1995 NCTM yearbook provides suggestions for connecting mathematics topics with other mathematics topics, other disciplines, and other settings. Available from NCTM. Developing Mathematical Reasoning in Grades K-12, the 1999 NCTM yearbook provides good background information about reasoning and problem solving. Available from NCTM. Developing Understanding in Mathematics via Problem Solving by Schroeder and Lester discusses three types of problem solving teaching. (Teaching about, for and via problem solving) This chapter may be found in the 1989 NCTM Yearbook, New Directions for Elementary School Mathematics, edited by Paul Trafton.

Games and Puzzles for Elementary and Middle School Mathematics: Readings from the Arithmetic Teacher edited by Smith and Backman, includes more than 100 articles on the use of games and puzzles. Available from NCTM. Language and Communication in the Mathematics Classroom (1998) edited by Steinbring, Bargolini Bussi, and Sierpinska builds on a series of papers first presented at the Sixth International Congress of Mathematics Education in Quebec in 1992. Available from NCTM. Let’s Talk Math: Encouraging Children to Explore Ideas by Pat Lilburn and Pam Rawson, shows teachers how to encourage children to explore, discuss, and reflect. Available from Heinemann.

Little Kids-Powerful Problem Solvers by Andrews and Trafton, provides a collection of problem solving stories from a kindergarten classroom. Available from Heinemann.

Mathematical Communication is a focus issue of Mathematics Teaching in the Middle School, that provides many classroom examples. April 2000. Available from NCTM. Mathematical Reasoning: Analogies, Metaphors, and Images by Lyn D. English is available from Lawrence Erlbaum Associates.

Math is Language Too: Talking and Writing in the Mathematics Classroom by David Whitin and Phyllis Whitin describes what happened in a fourth-grade classroom over a period of four years. Available from NCTM.

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Instructor’s Resource Guide, Chapter 5 Sandi Cooper |7 Mission Mathematics K-6, edited by Mary Ellen Hynes and Mission Mathematics 5-8, edited by Vincent O’Conner and Michael C. Hynes provide hands-on activities that integrate mathematics and space science. Available from NCTM. Mouse & Elephant by Glenda Lappan and others is part of The Middle Grades Mathematics Project. Topics include measuring growth, area, perimeter, surface, area, and volume.

The book is available from Dale Seymour Publications.

Navigating through Problem Solving and Reasoning is a series of books from NCTM. Investigations for students are provided that encourage use of the process standards. Each grade level book comes with a CD containing interactive electronic activities to use with students, printable activity pages and NCTM articles.

Pentomino Activities, Lessons, and Puzzles by Henri Picciotto provides over 200 Pentomino activities. The book is available from Dale Seymour Publications.

Pentominoes Revisited by Barry Onslow in Arithmetic Teacher (vol. 37, no. 9, p. 5-9) provides additional pentomino activities.

Show and Tell: Representing and Communicating Mathematical Ideas in K-2 Classrooms by Dacey and Eston, demonstrates how young children deepen their understanding when representing and communicating their thinking. Classroom vignettes are included. Available from Math Solutions.

Talking Mathematics: Supporting Children’s Voices by Rebecca B. Corwin with Judith Storeygard and Sabra L. Price, discusses the role of talk in children’s mathematics and shows ways to support children as they work. A video resource package is also provided (below).

Available from Heinemann.

The Roles of Representation in School Mathematics the 2001 NCTM yearbook, edited by Albert A. Cuoco. Available from NCTM. Videos Mathematics for Middle Grades: The Role of the Teacher -- one from a series of three 21minute videotapes targeted toward middle grades. Portions of mathematics lessons in grades 6-8 are shown to illustrate how problem situations can be created, and how children can work cooperatively in learning mathematics. Available from Cuisenaire Company.

Mathematics: Teaching for Understanding is a set of 3, 20 minute videos. Part 1 models teacher-directed, whole-class lessons. Part 2 features small groups of students working independently on a menu of activities. Part 3 shows how communication in math lessons, supports children's learning. The set is available from Cuisenaire Company.

Talking Mathematics Resource Package is a video tape that focuses on important aspects of children’s talk and provides unedited classroom episodes and a summary of a Talking .S o n s

Instructor’s Resource Guide, Chapter 5 Sandi Cooper |8 Mathematics seminar. A professional Development guide is also provided. Available from Heinemann.

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Instructor’s Resource Guide, Chapter 6 Sandi Cooper |1

Chapter 6 — Helping Children with Problem Solving What This Chapter Is About In this chapter, problem solving and types of problems are defined. Key points for teaching mathematics through problem solving are summarized. Factors from research for success in problem solving are presented. The importance of providing children with a variety of problemsolving experiences is stressed. Specific problem-solving strategies are presented with supporting sample problems.

Student Objectives After reading the chapter, the students will be able to: 1.

Differentiate between a problem and an exercise, routine and nonroutine problems.

2.

Summarize signposts or recommendations for teaching mathematics through problemsolving.

3.

Summarize what research tells us about factors for success in problem solving including characteristics of children and problems.

4.

Summarize recommendations for choosing appropriate problems, finding problems, having students pose problems, and using technology.

5.

Demonstrate problem-solving strategies.

6.

Summarize how to help all students with problem solving including managing time, classroom routines, and student needs.

Key Vocabulary The Process Standards and problem solving are important parts of the mathematics curriculum.

Students should understand what a problem is and be aware of the different strategies and approaches used in problem solving.

Problem Exercise routine problem nonroutine problem open ended problems problem solving strategies 4-stage model [Polya]

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act it out strategy make a drawing or diagram strategy look for a pattern strategy construct a table strategy guess and check strategy work backword strategy solve a simpler or similar problem strategy

Instructor’s Resource Guide, Chapter 6 Sandi Cooper |2 Supplemental Lecture Topics 1.

Present a summary of the National Assessment of Educational Progress (NAEP) results for problem solving. (See Kloosterman and Lester in end of book resources)

2.

The article, Models of Problem Solving: A Study of Kindergarten Children's ProblemSolving Processes, by Carpenter et al. (listed under additional resources) demonstrates that children have many valuable, intuitive notions about mathematics and can often perform at a higher level than expected. Display the problems presented to the kindergarten children and discuss the results of the study. Another good source of children’s problems and work samples may be found in Children are Mathematical Problem Solvers from NCTM. If possible, have the class present some of these problems to elementary children and discuss their performance.

3.

Display and discuss PPT Slides #17-23, Student Interviews. They are work samples taken from children in grades K-4. Problem source: Models of problem solving processes by Carpenter et al. (See additional resources.) The problems were read to the students and reread as many times as needed. The samples are intended to show a wide range of understandings among children. They also show that children are often intuitively able to solve problems with manipulatives or drawings for which they have not had formal instruction. Specific observations concerning each sample follow:

PPT #17 PPT #18 PPT #19 PPT #20 PPT #21 PPT #22 PPT #23

Meredith successfully solves the problem although she has not yet been introduced to the symbolic representations.

Allan successfully solves the problem. He knows the symbols for seven and four but does not place them in a number sentence.

Clint successfully solves the problem although he has not yet been introduced to the symbolic representations for division. He uses addition.

Bill successfully solves the problem. He uses a known fact of addition.

Darla knows how to add and adds the two numbers in the problem, then draws 18 circles to match her answer. When she is finished she says, "But I don't think he had 18 jars."

Ellen successfully solves the problem by placing an equal number of guppies in each jar. She has not yet been introduced to the symbolic representation for division and places the numbers in the format of a subtraction problem.

Allison successfully solves the problem using color tiles. Then she sketches her solution. She makes no attempt to represent it symbolically.

Encourage students to share observations about the work samples and what the children know.

4.

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Additional lecture material on problem solving may be found in Teaching and Learning: A Problem-Solving Focus, Teaching Mathematics through Problem Solving, Word Problems, Problem-solving Techniques Helpful in Mathematics and Science. The 1980 NCTM Yearbook provides an overview of classic problem solving research.

Instructor’s Resource Guide, Chapter 6 Sandi Cooper |3 5.

Have students work together to solve the problems found in the text. Encourage them to use the recommended strategy listed in the book. Selected solutions are provided below.

Solutions to Selected Problems The following solutions for selected Chapter 6 problems are provided with the assistance of Mary Kabiri, Lincoln University, Jefferson City, Missouri. In some cases more than one solution is possible and not all are necessarily provided.

1.

6 + 5 = 11

2.

Use small paper hearts or cubes to act out the problem.

Start with 2 students, A and B. A <→ B, 2 Valentines 3 students: A → B →C

B→ C → A

C → A Show each person’s Valentines. → B We see 3 groups with 2 Valentines, total of 6.

4 students: → B A→C →D

→ A B →C →D

→ A C → B → D

→ A 4 groups with 3 Valentines, total 12 D → B → C

Continue with 5 and get 5 groups of 4 exchanges for 20 Valentines. The pattern would indicate that 24 students would each have 23 exchanges for a total of 552 Valentines. # of people 1 2 3 4 5 6 7

# of Exchanges 0 1 3 6 10 15 21

# of Valentines 0 2 6 12 20 30 42

3.

Use play money to act out the problem.

Suppose each person starts with $100.

Buyer: 100 - 60 +70 - 80 + 90 = $120, gain of $20 Note: The gain would be the same no matter what initial amount was used.

4.

See Lizzie’s drawing in Figure 6-6.

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Instructor’s Resource Guide, Chapter 6 Sandi Cooper |4 5.

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 cm

end Day 1 2 cm

Day 2 4 cm

Day 3 6 cm

Day 4 8 cm

Day 5 10 cm

Day 6 12 cm

Day 7 out

Day 8

6.

Answers vary.

7.

See Figure 6-7.

8.

time passed # of people Total in minutes who heard 0 1 1 15 3 4 2 30 3 groups of 3 = 3 =9 13 45 32 groups of 3 = 33 = 27 40 1hr.

60 34 = 81 121 75 35 = 243 364 6 90 3 = 729 1093 7 105 3 = 2187 3280 2hr 120 38= 6561 9841 135 39 = 19,683 29, 524 2.5hr 150 310 = 59,049 88, 573 165 So by the time 165 min has passed, all 90,000 3hrs 180 townspeople will have heard the rumor.

Note: You could also use formula for geometric sum S = a1 (rn - 1) (r - 10) In this problem r = 3, n = 11, a1 = 1 S = 1(311 - 1) (3 - 1) = 88,573

9.

Ways to make change for a quarter : 12 ways.

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Instructor’s Resource Guide, Chapter 6 Sandi Cooper |5 # of

Dimes 2 2 1 1 1 1 0 0 0 0 0 0

Nickels 1 0 3 2 1 0 5 4 3 2 1 0

# of Minutes 20 = 1 21 = 2 22 = 4 23 = 8 24 = 16

Pennies 0 5 0 5 10 15 0 5 10 15 20 25

#of coins 3 7 4 8 12 16 5 9 13 17 21 25

10.

Day 1 2 3 4 5

Assume 10 school days in two weeks 10 days 29 = 512 min Day 10 8hrs. 32 min.

11.

Plan 1 Day # Amt in cents 1 20 = 1 2 21 = 2 3 22 = 4 4 23 = 8

Plan 2 Day # Amt in $ 1 1 2 2 3 3 4 4

. . .

. . .

15 214 = 16,384 15 15 2 14 1 + 2 + 3 + ... + 15 = (15 x 16)/2=$120 The sum = 1 + 2 + 2 + ... 2 = 1 (215 - 1) (2-1) = 32767 cents = $327.67 Plan 1 earns $207.67 more than Plan 2. .

12.

Postcards 1 2 3 4 5 6 7

letter 11 10 9 8 7 6 5

cost 4.30 4.16 4.02 3.88 3.74 3.60 3.46

To accompany Helping Children Learn Math9e, Reys et al. ©2009 John Wiley & Sons

Instructor’s Resource Guide, Chapter 6 Sandi Cooper |6 8 13.

4

Flavors A Apple B Banana C Cherry D Dark Chocolate E Elegant Peach F French vanilla G Green mint H Hearty mango

3.32 AB

AC BC

AD BD CD

AE BE CE DE

AF BF CF DF EF

AG BG CG DG EG FG

7 + 6 + 5 + 4 + 3 + 2 + 1 = 28 combinations 14. 18 + 4 + 2 + 1 15 + 7 + 2 + 1 15 + 6 + 3 +1 15 + 5 + 4 + 1 15 + 5 + 3 +2 12 + 10 + 2+ 1 12+9+3+1 12+8+4+1 12+8+3+2 10+9+5+1 10+9+4+2 10+8+6+1 10+8+5+2 10+8+4+3 10+7+6+2 10+7+5+3 9+8+7+1 9+8+6+2 9+8+5+3 8+7+6+4 15.

1 5 3

16.

6 4

2

+ 4 0 3 7 1 9 8 12 8 11 15 9 17 6 10 6 9 13 7 15 2 6 2 5 9 3 11 4 8 4 7 11 5 13 5 9 5 8 12 6 14

To accompany Helping Children Learn Math9e, Reys et al. ©2009 John Wiley & Sons

AH BH CH DH EH FH GH

7 combos 6 combos 5 combos 4 combos 3 combos 2 combos 1 combo

Instructor’s Resource Guide, Chapter 6 Sandi Cooper |7 17.

We know each of the 3 sisters got 4 cookies, so they got 12 cookies. This was exactly half of the total. Sue baked 24 cookies.

18.

A simpler problem could involve 4, 5, 6, 8, or 16 players If there are 4 players, round 1: 2 games, 2 lost, round 2: 1 game, 1 lost, 3 games total If there are 5 players, round 1: 4 play 2 games, 1 watches, 2 lost, round 2: 2 play 1 game, 1 watches, round 3: 2 play 1 game, 1 lost, 4 games total If there are 6 players, round 1: 3 games, 3 lost, round 2: 2 play 1 game, 1 watches, 1 lost, round 3: 2 play 1 game, 1 lost, 5 games total If there are 8 players, round 1: 4 games, 4 lost, round 2: 2 games, 2 lost, round 3: 1 game, 1 lost, 7 games total If there are 16 players, round 1: 8 games, 8 lost, round 2: 4 games, 4 lost, round 3: 2 games, 2 lost, round 4: 1 game, 1 lost, 15 games total The pattern indicates the number of games is always 1 less than the number of players, so if 64 players, 63 games

19.

32.7 mpg x 14 gallons = 457.8 miles.

A simpler problem would be 30 mpg x 15 gallons

20.

Fish flavor is 12-3 = 9 cents on sale. 6 cans x 9 cents is 54 cents.

21.

List the numbers 1-19: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Notice if 19 goes in the center, the others can be paired (1,18) (2,17) etc. to give sums of 19. Add this to the center of 19 to give 38.

There are two other solutions. (a). Place 1 in center, then pair the others to sum to 22.

The total will be 23. (b). Place 10 in the center. Pair (1,19) (2,18), etc. Sum is 30.

22.

Find three different integers a,b,c such that 1/a + 1/b + 1/c is an integer.

1/a + 1/b + 1/c = 1 a=6, b=2, c=3 1/6 = 1/2 + 1/3 =1/6 + 3/6 = 2/6 = 1 Choose 3 different whole numbers. Take their reciprocals and add them together. The sum should be a whole number. For example choose 2,5,8. Their reciprocals are 1/2,1/5,1/8. Their sum is 1/2 + 1/5 + 1/8. The sum is 33/40. These choices are incorrect because I wanted their sum to be a whole number.

23.

If I buy 7 items for $.07 each then the price is $.49. Similarly 10 items for $.10 each costs $1.00 or 100 cents. If I buy p items for p cents then p x p = 225. So p = 15 items for $.15.

24.

20 divided by 2 is 10 with no leftovers 20 divided by 3 is 6 with 2 leftovers 20 divided by 4 is 5 with no leftovers 20 divided by 5 is 4 with no leftovers

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Instructor’s Resource Guide, Chapter 6 Sandi Cooper |8 25.

Some number divided by 2, 3, 4, 5, 6 has no leftovers. The number must be a multiple of each of these numbers. Look for the least common multiple. It is 60.

6 12 18 24 30 36 42 48 54 60 5

10

15

20

25

30

35

40

45

50

55

60

4

8 52

12 56

16 60

20

24

28

32

36

40

44

48

3

6 39

9 42

12 45

15 48

18 51

21 54

24 57

27 60

30

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36

2

4 26 48

6 28 50

8 30 52

10 32 54

12 34 56

14 36 58

16 38 60

18 40

20 42

22 44

24 46

Student Textbook Activities 1.

As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2.

Hand out paper hearts or cubes and have students work in groups on the Valentine problem. Discuss the strategies used. Have students review the Chapter 6 Snapshot of a Lesson. The clip may be viewed at www.learner.org/resources/series32.html (video 42).

Discuss how acting out the problem helped children make sense of it and solve it.

3.

Use PPT Slide #3, Problem Types, to review the vocabulary introduced in the text.

Through discussion, students should determine that problem and nonroutine problem are synonyms and exercise and routine problem are synonyms. Use PPT Slides #4-6, Are these Problems or Exercises?, to give students practice in recognizing different problems and exercises.

4.

Use PPT Slide #7, Signposts for Teaching Mathematics through Problem Solving, to discuss considerations the teacher should keep in mind. Tie these recommendations back to the Snapshot of a Lesson, Valentine Exchange.

5.

Research findings concerning problem solving are discussed in the text. Use PPT Slides #8-9, Factors for Success in Problem Solving, to highlight and discuss the research findings.

6.

Problem solving is often thought of as solving word problems. By carefully planning for problem solving, teachers can provide students with the variety of experiences that are necessary to build a solid base for approaching any problem situation. PPT Slide #10, Choosing Appropriate Problems, presents a list of suggested problem types. Bring in

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Instructor’s Resource Guide, Chapter 6 Sandi Cooper |9 texts and resource books and have students locate and share problems that match the criteria. You could assign various content strands and grade levels. This could also be an out of class assignment with time to share findings in class. Students could also bring in copies for the class so a problem solving file is assembled.

7.

Have students work on the traditional and open ended problems in Figure 6-3 and 6-5, or In the Classroom 6-1, Which Rectangle is Biggest? Next, bring in some traditional texts.

Have students locate a traditional problem for each NCTM content strand. Then have them work together to turn them into open ended problems. These could be duplicated for the class file.

8.

Discuss the principles for helping students pose problems, found in the text. Then bring in some children’s books and have the class work in small groups to create some problems.

9.

Recommendations concerning the use of technology with problem solving are presented in the text. It would be useful to present calculator and computer activities that demonstrate the recommendations.

10.

The text identifies Polya’s 4 stage problem solving model and 7 specific problem-solving strategies and provides examples of problems that can be solved using those strategies.

PPT Slides #11-12, Problem Solving Strategies, provides a list of the strategies. Discuss each strategy and have the class provide further examples to fit each category.

11.

The text presents a variety of problems to illustrate 7 specific strategies. Begin with a demonstration of the strategy, "act it out". Use PPT Slide #13 to present the horse problem that was in the text. After students have worked the problem independently, record suggested solutions on the board. There will probably be 3-4 different solutions provided. Next, have two volunteers come to the front of the class and act out the problem according to the class' direction. If each actor starts with 10 "ten dollar bills", they can complete the necessary transactions and count the remaining bills. The buyer ends up with $120 and the seller ends up with $80. Discuss the results and the value of acting out the problem.

Assign students to work on several problems from the text, using the recommended strategy. Encourage them to reflect on how they approached the problem and to share alternate approaches to finding an appropriate solution.

Student Supplemental Activities 1.

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Provide each group with a current newspaper or magazine and have the group construct several problem situations. This activity will help to reinforce the need to give students problem-solving experiences which use real world information and take advantage of students' interests.

Instructor’s Resource Guide, Chapter 6 S a n d i C o o p e r | 10 2.

Have students examine one or two elementary textbook series. Have students report on the texts' approach to problem solving, the teaching of problem solving strategies, and the use of routine and non-routine problems. You may want to have them compare a “Standards-based” text vs. a “traditional” text.

3.

Have students collect a file of problems that may be used with elementary students.

Encourage students to locate problems of varying types, difficulty levels, and content area applications.

4.

Bring a variety of commercial problem solving materials to class for students to examine.

For example: Dale Seymour's problem solving poster and book sets such as Problem Play and Problem Parade, or The Lane County Mathematics Project, Tops Resource Books, Tops Card Decks , The I Hate Mathematics! Book or Math for Smarty pants books, Math at a Glance, NCTM’s Menu Collection, Navigating through Problem Solving, or Creative Publication's The Problem Solver series. (These and others are listed in additional resources.) Have students brainstorm the pros and cons of each.

5.

Have students view a problem solving video such as The Adventures of Jasper Woodbury by the Cognition and Technology Group at Vanderbilt University, HRM Video's Detective Stories for Math Problem Solving, or Math Monsters by NCTM (see additional resources). Discuss the implications of using multimedia in mathematics.

6.

Have students complete some calculator or computer problem solving activities. For example, use activities from How to Develop Problem Solving with a Calculator by Janet Morris or Spreadsheet Activities in Middle School Mathematics, 2nd ed. By John C. Russell. (available from NCTM)

7.

If possible, have students observe elementary students engaged in problem solving activities. Discuss how the observations illustrate other points made during the study of chapter 6.

8.

Bring sample inventories and checklists which may be used for problem solving assessment. For example, How to Evaluate Progress in Problem Solving by Charles, et al. (available from NCTM) includes some samples. Have students evaluate the various formats.

9.

Share examples from Little Kids-Powerful Problem Solvers: Math Stories from a Kindergarten Classroom, to show how young children can effectively problem solve.

10.

The Math Links in this chapter provide students with additional problem resources and video examples for problem solving. In class, or as an outside assignment, have students explore these web sites. Other video sources are available in the Additional Resources list.

11.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

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Instructor’s Resource Guide, Chapter 6 S a n d i C o o p e r | 11 12.

Using some of the references listed here under additional resources, have students locate additional information on topics which interest them with problem solving. For example, assign articles from the February 2000 Focus Issue or October 2005 Focus Issue of Teaching Children Mathematics.

13.

The Book Nook for Children, found in the text, provides a list of children's literature that may be used. Read some of these books and have students develop lessons to go with particular titles. Discuss how children's literature provides a context for problem solving.

14.

Divide students into groups of 4. Have each person in the group read and report on one article or chapter from the References listed related to one of the problem solving issues discussed in the chapter. For example, one student may report from How to Choose and Create Good Problems for Primary Children, one from Introduction to Problem Solving: Strategies for the Elementary Math Classroom, one from Problem Solving: Tips for Teachers, and one from Teaching Mathematics through Problem Solving: Prekindergarten-Grade 6.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences, 4e. The following activities have been designed to be used with Chapter 6: 1.

Learning about the School and its Resources: Problem Solving in the Textbook and Classroom (found in Chapter 1)

2.

Observing Students: Analytic/Holistic Problem Solving Scoring (found in Chapter 2)

3.

Interviewing the Teacher and Students: Problem-Solving Strategies (found in Chapter 3)

4.

Helping Children Learn with Technology: Problem-Solving on the Web and ProblemSolving Web Sites (found in Chapter 5)

Additional Resources Children Are Mathematical Problem Solvers edited by Lynae E. Sakshaug, Melfried Olson, and Judith Olson provides 29 problems from the Problem Solvers column of Teaching Children Mathematics. Solutions and children’s work samples are included.

Children as Mathematicians is a focus issue of Teaching Children Mathematics (February 2000).

Articles provide several examples of the process standards at work in classrooms. Available from NCTM. .S o n s

Instructor’s Resource Guide, Chapter 6 S a n d i C o o p e r | 12 How to Choose and Create Good Problems for Primary Children by Nelson and Worth, guides the teacher in designing and selecting mathematical problems that draw very young children into the problem-solving way of thinking. Also includes ideas for organizing the classroom around problem-solving activities. Available from NCTM. Introduction to Problem Solving: Strategies for the Elementary Math Classroom by Susan O’Connell provides background information about effective problem-solving instruction and introduces specific problem solving strategies. Available from Heinemann.

Little Kids-Powerful Problem Solvers: Math Stories from a Kindergarten Classroom by Angela G. Andrews and Paul Trafton shares a problem for each month of the school year and shows what young children can accomplish. Available from Heinemann.

Math at a Glance: A Month-by-Month Celebration of the Numbers Around Us by Susan Ohanian provides anecdotes and celebrations on a calendar with investigative situations for students to solve. Available from Heinemann.

Math for Smarty Pants and The I Hate Mathematics! Book by Marilyn Burns provide interesting math riddles, games, puzzles, and activities. Available from Cuisenaire Company.

Menu Collection, edited by C. Patrick Collier, includes over 200 problems from NCTM’s journal, Mathematics Teaching in the Middle School. Problems are organized by topic or theme.

Models of Problem Solving: A Study of Kindergarten Children's Problem-Solving Processes by Carpenter, Ansell, Franke, Fennema, and Weisbeck demonstrates that children can often do more than we think they can. This article may be found in the Journal for Research in Mathematics Education, volume 24, number 5, November 1993.

Navigating through Problem Solving and Reasoning books are part of the Navigations series available from NCTM. A variety of investigations in each of the content strands are provided for various elementary grade levels.

Posing and Solving Problems is a focus issue of Teaching Children Mathematics (October 2005).

Articles provide several examples of problem solving in classrooms. Available from NCTM. Problem Play (grades 1-3) by Stephen Currie and Problem Parade (grades 4-6) by Dale Seymour provide reproducible problems that require both logical and creative thinking. Include teaching suggestions, directions, answers, and extensions. Available from Dale Seymour Publications.

Problem Solving in Mathematics: The Lane County Mathematics Project, directed by Brannan and Schaaf, is a problem solving program which provides grade level resource books for grades 4-9. Each book has at least 100 pages of blackline masters plus teacher commentary. Available from Dale Seymour Publications.

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Instructor’s Resource Guide, Chapter 6 S a n d i C o o p e r | 13 Problem Solving in School Mathematics, the 1980 NCTM Yearbook, edited by Krulik and Reys provides a good summary of problem solving research during that era.

The Problem Solver-A series of graded volumes for gr. 1-6 by Goodnow, Hoogeboom, Moretti, Stephens, and Scanlin. Children learn to use ten different problem-solving strategies and a fourstep approach to solving problems. Each volume contains a strategies, practice problems, and solutions section. Available from Creative Publications.

Problem-solving Techniques Helpful in Mathematics and Science by Reeves, illustrates with sample problems how to teach the different problem-solving strategies in elementary school.

Available from NCTM. Problem Solving: Tips for Teachers by O'Daffer spotlights teacher-tested strategies to help children become better problem solvers. Available from NCTM. Teaching and Learning: A Problem-solving Focus edited by Curcio focuses on George Polya, his ideas, and influence on problem solving. Available from NCTM. Teaching Mathematics through Problem Solving: Prekindergarten-Grade 6 is edited by Frank Lester and Randall Charles. The book is divided into four main sections: Issues and Perspectives, In the Classroom, The Role of Technology, and Research. It provides teachers with directions for how to teach mathematics through problem solving. Available from NCTM. Word Problems by Stephen K. Reed is a volume in the Studies in Mathematical Thinking and Learning Series. Available from Lawrence Erlbaum.

Videos Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 tapes and 3, 5-8 tapes. Tapes include lessons illustrating Standards 1-4. Each tape contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept.

TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org.

Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples Challenge of the Unknown is a set of 7 video tapes appropriate for middle school and high school students. The films provide real-life examples of people solving problems and using math to make their jobs easier. Made possible by a grant from the Phillips Petroleum Company to the American Association for the Advancement of Science, schools can schedule free loans of the videotape masters and get permission to make free copies. Teacher's guides are also provided.

Available from W.W. Norton & Company, Inc., New York, NY. .S o n s

Instructor’s Resource Guide, Chapter 6 S a n d i C o o p e r | 14 HRM Video Company, 175 Tompkins Avenue, Pleasantville, NY (1-800-431-2050) publishes a catalogue of problem solving videos for mathematics.

Math Monsters is a set of 12 animated videos appropriate for young children and include realworld problems. A teachers guide and black line masters are included. Available from Slim Goodbody Educational Materials.

Reaching Higher – 28-minute videotape illustrates with actual classroom scenes a problemsolving approach to elementary school mathematics. Support materials include sample lessons for K-5. Available from NCTM.

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Instructor’s Resource Guide, Chapter 7 Sandi Cooper |1

Chapter 7 — Developing Counting and Number Sense in Early Grades Chapter Overview This chapter addresses the important role that number sense plays in the development of computational concepts and skills. This understanding of numbers zero to ten is basic to a students' future with larger numbers and the concept of place value. Three pre-number concepts, consisting of classification, patterns, comparisons, and two early number development concepts consisting of conservation, and group recognition are introduced. Counting and its various stages and strategies are outlined together with 4 counting principles. The relationship between a set of objects, the number name, and the written symbol is noted. Cardinal, ordinal, and nominal numbers are defined. Recommendations for teaching the writing of numerals are presented.

Student Objectives After reading the chapter, the students will be able to: 1.

Summarize characteristics of children with good number sense and provide examples of observable number sense behaviors.

2.

Summarize the pre-number concepts of classification, patterns, comparisons, and early number development concepts of conservation, and group recognition as well as provide examples of activities that promote their development.

3.

Identify counting principles, stages, and strategies as well as provide examples of activities that promote their development.

4.

Identify number benchmarks and visual patterns for connecting groups of objects with the oral name and symbolic representation.

5.

Differentiate among cardinal, ordinal, and nominal numbers.

6.

Summarize recommendations and provide examples of activities for teaching the writing of numerals.

Key Vocabulary The terms listed here are important to the understanding of the important role of numbers and counting in developing future mathematical concepts. Students need to gain familiarity with these terms and use them appropriately. number sense classification patterns comparisons conservation group recognition .S o n s

rote counting rational counting counting on counting back skip counting number benchmarks

subitizing ordinal number prenumber concept nominal number early number concept cardinal number attribute blocks continuous quantities discrete objects counting principles ten-frame

Instructor’s Resource Guide, Chapter 7 Sandi Cooper |2 Supplemental Lecture Topics 1.

Display and discuss PPT Slides # 32-38, Student Work Samples. They are writing samples taken from kindergarten children during the third month of school. The children were asked to begin with one and write their numbers to ten. The samples are intended to show the wide range of writing skills encountered within one class of students. Specific observations, focusing on what the child can do, follow: PPT # 32 Chad makes figures to symbolize the numerals.

PPT # 33 Cody indicated that the first row was 1-5 and the second row was 6-10.

He writes horizontally from left to right and makes a discrete symbol for each numeral.

PPT # 34 Steven repeated the use of several symbols in a variety of combinations.

The directionality of the numeral two varies.

PPT # 35 Elizabeth writes the numerals 1-5 horizontally, from left to right. All numerals are pointing to the right.

PPT # 36 Andrea writes the numerals 1-12 horizontally, from left to right. All numerals are pointing to the right and the numerals 6 and 9 are omitted.

PPT # 37 Jason correctly writes the numerals 1-10 and also draws the number of blocks each numeral represents.

Display PPT #38 to show that it is often common for older children to reverse numerals and not a cause for concern. Joshua is a second grader asked to write the numbers 1-25.

2.

Provide more detailed information about number sense using resources such as Teaching Number Sense, Number SENSE: Simple Effective Number Sense Experiences, Young Mathematicians at Work, Research Ideas for the Classroom: Early Childhood.

3.

See the Additional Resources list for other books for teaching mathematics in early childhood such as The Young Child and Mathematics, Mathematics in the Early Years from NCTM.

Student Textbook Activities 1.

As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and on PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2.

Use PPT Slides #4-5, Number Sense, to discuss this concept that is difficult to define.

Have the class brainstorm examples for each bullet item.

3.

Use PPT Slide #3, Understand numbers, ways of representing numbers, relationships among numbers, and number systems, to highlight the key recommendations related to number in grades PreK-2 from NCTM's Principles and Standards. These may also be found in the textbook in Chapter 7 and Appendix A.

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Instructor’s Resource Guide, Chapter 7 Sandi Cooper |3 4.

Have students review the Chapter 7 Snapshot of a Lesson. It may be viewed at www.learner.org, video #3. Discuss the counting and early number concepts displayed.

5.

Display PPT Slide #7, Pre-Number Concepts. Allow the class to brainstorm activities and materials that may be used to develop the five concepts (classification, patterns, comparisons, conservation, group recognition) discussed in the chapter.

6.

Classification is an important concept to be developed for all elementary students.

Duplicate and cut apart the attribute pieces (Appendix B). Or use commercial attribute blocks or other sortable objects such as colored pasta, cereal, beans, buttons, ceramic tiles, etc. Have students complete In the Classroom 7-1 and 7-2. Finally, refer the students to Figure 7-5. Have them make two intersecting circles with yarn. A leader selects a mystery attribute for each circle. Others in the group take turns placing pieces.

The leader confirms whether a piece is placed correctly. Play continues until someone correctly guesses the mystery attributes. You may wish to begin by modeling the activities and have students work together in small groups. It is appropriate to discuss the different levels of involvement and understanding experienced in a whole class demonstration versus a small group, hands on setting.

7.

Discuss the four types of pattern activities discussed in the chapter (copying a pattern, finding the next one, extending a pattern, making a pattern), then provide an opportunity for students to experience the activities. Materials which could be used include pattern blocks, color tiles, connecting cubes, and other sortable objects such as colored pasta, cereal, beans, buttons, ceramic tiles...

8.

PPT Slides #12-14, Conservation of Number, provides hypothetical conservation interview excerpts with two children. Have students use the information provided in the text to determine the children's conservation of number. Student A is not able to conserve number and Student B is able to conserve number.

9.

PPT Slides #18-20, Counting, summarizes the counting principles, stages, and strategies.

Use counters on the projector to provide examples for each.

10.

PPT Slides #26-29, Counting Principle Examples, provides examples of children's counting. Have students use the information provided in the text to analyze the children's use of the counting principles. Encourage students to focus on what the children know rather than what they do not know.

• Student A is displaying the stable order rule and the cardinality rule.

• Student B is displaying the one to one rule, the stable order rule, and the cardinality rule.

• Student C is displaying the order irrelevance rule and the cardinality rule.

• Student D is displaying the cardinality rule.

11.

Have students work together on In the Classroom 7-3, Numbers on a Calendar. Then have them make up additional problems and share them with the class.

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Instructor’s Resource Guide, Chapter 7 Sandi Cooper |4 12.

Have students use calculators to complete In the Classroom 7-4, Counting On…and On…and On, and 7-5, Counting On…and Back, in the text. Discuss other calculator counting activities that could be done with children.

13.

Once they have learned to count on the calculator, have students complete In the Classroom 7-6, Hunting for Numbers, and 7-7, Skip Counting. Then have them use the calculator and crayons to shade some of the 100's charts in Appendix B and look for patterns that emerge.

14.

Have students examine Figure 7-18, then use the five- and ten-frames in Appendix B. Students can place connecting cubes, color tiles, or beans in the frames. Challenge the students to find more variations for the numbers shown and to continue the examples for the numbers 11 to 20.

Student Supplemental Activities 1.

PPT Slides #30-31, Construct a Pattern, involves students in the creation of patterns following specific conditions. The slide outlines the directions for the activity.

Materials: PPT Slides #30-31, color tiles (at least 20 per student), color spinner (may be constructed from the spinner patterns in Appendix B), and timer.

• Present the directions for the pattern activity. Provide an example to clarify any questions about how the activity will proceed.

• Spin the spinner twice to determine the two colors to be used, remind the students that the pattern they construct must repeat at least twice, set timer and let students create!

• At the end of the allotted time, have several students present their patterns. This could be done using an overhead set of color tiles or using construction paper squares on floor or chalkboard. Determine if anyone had a unique pattern.

• Vary the number of colors to be used, the number of times the pattern must repeat, or use different types of manipulatives. The use of attribute blocks would allow more diversity in developing patterns by using color, shape, size in setting criteria for building patterns.

• Discuss the role that patterning plays in the mathematical development of learners. What are other activity ideas that could be used?

2.

Bring the class a collection of available counting books. Distribute them randomly to each small group for discussion of similarities and differences. For example, not all counting books choose to address the concept of zero. As a follow-up, have each group construct their own counting book, picture or slide collection using the best of what they have seen. For example, read the book, Ten Black Dots, then have groups create their own book pages using adhesive dots. Another option would be to assign each group a different number and have them represent that number in as many different ways as possible.

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Instructor’s Resource Guide, Chapter 7 Sandi Cooper |5 3.

Give each small group a set of objects and ask them to find at least 3 different ways to classify the items. Suggestions for objects: the right shoe of all students in the class, a box of assorted books, items from your middle desk drawer, or a handful of buttons or seashells. Classification is a necessary skill not only for mathematics but for other subject areas, such as science.

4.

Share some other activities that encourage the development of pre-number concepts from the classic book, Mathematics Their Way, by Mary Baratta-Lorton.

5.

Have students view the tapes, Number Sense Now in the additional resource list or review the activity books, Number SENSE, in the additional resources list. Have the students generate a list of activities which would help develop number sense.

6.

Show a clip from the video, Six Models. First graders are making attribute block trains.

7.

Have students complete activities from Navigating through Numbers and Operations in Prekindergarten-Grade 2.

8.

The Math Links in this chapter provide students with additional electronic manipulatives and video examples for early number sense. In class, or as an outside assignment, have students explore these web sites.

9.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

10.

The Book Nook for Children, found in the text, provides a list of children's literature which may be used. Read some of these books and have students develop lessons to go with particular titles. Discuss how children's literature provides a context for early number topics.

11.

Divide students into groups of 4. Using the additional resources list below, have selected students report on and react to chapters addressing topics discussed in the chapter.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 7: 1. Learning about the School and Its Resources: Counting in the Textbook and Classroom (found in Chapter 1) 2. Observing the Teacher and Students Content: Focusing on Counting and Number Recognition (found in Chapter 2) 3. Interviewing the Teacher and Students: Conservation and Counting, Counting, Early Number Sense (found in Chapter 3) .S o n s

Instructor’s Resource Guide, Chapter 7 Sandi Cooper |6 4. Helping Children Learn with Games: Buzz-A Counting Game, Dots and Numbers (found in Chapter 4) 5. Helping Children Learn with Technology-Counting On and On and On, Skipping Around, Counting Backward (found in Chapter 5) 6. Helping Children Learn with In the Classroom Lessons: Who am I?, Alike-andDifference Trains, Hunting for Numbers, Skip Counting, Decide If It’s Up or Down, Counting on a Hundred Chart, Finding 10s (found in Chapter 6) Additional Resources Mathematics for the Young Child is edited by Joseph N. Payne. It is filled with suggestions for preschool through grade 4. Available from NCTM. Mathematics in the Early Years is edited by Juanita V. Copley. Presents historical, theoretical, and social pictures of early childhood mathematics and discusses mathematics content for young children. Available from NCTM. Navigating through Number and Operations in Prekindergarten-Grade 2 provides counting and ordering activities for children. Available from NCTM. Number Sense: Simple Effective Number Sense Experiences by McIntosh et. al. is a series of books for grades 1-8 that provides a variety of number sense activities. Available from Dale Seymour.

Research Ideas for the Classroom: Early Childhood is part of a 3-volume set from NCTM. Teaching Number Sense is a 3-volume set for grades K, 1, and 2. Detailed lesson plans and descriptions of how the lesson went in a real classroom are included. Available from Math Solutions.

The Young Child and Mathematics by Juanita V. Copley is co-published by NCTM and NAEYC. It focuses on activities and classroom vignettes for children from ages 3-8.

Ten Black Dots--A children's counting book by Donald Crews. Available from Scholastic Books.

Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction by Catherine Twomey Fosnot and Maarten Dolk, focuses on children, between the ages of four and eight, as they develop understanding of number. Available from Heinemann.

Videos Number Sense Now-- Directed by NCTM President, Francis "Skip" Fennell. Three 25-minute videotapes are designed to promote a greater understanding of number sense in grades K-6. The first tape What is Number Sense? shows children doing number sense activities as they study mathematical topics across different grade levels. The second tape Number Sense: A Way of .S o n s

Instructor’s Resource Guide, Chapter 7 Sandi Cooper |7 Teaching illustrates ways in which number sense can be creatively and naturally incorporated into mathematics lessons. The third tape Number Sense Connections illustrates ways in which number sense is an integral part of real world mathematical applications. Available from NCTM. Six Models--One of a set of six, twenty-minute video tapes designed to show how to use manipulative materials to teach mathematics in K-6 classrooms. Available from Cuisenaire.

Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 tapes and 3, 5-8 tapes. Tapes include lessons illustrating Standards 1-4. Each tape contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tape 2 contains lessons on number sense and numeration. Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800-8649846 or may be viewed at www.learner.org. Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples

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Instructor’s Resource Guide, Chapter 8 Sandi Cooper |1

Chapter 8 — Extending Number Sense: Place Value What This Chapter Is About This chapter stresses the importance of understanding our number system in order to develop mathematical literacy. Characteristics of our number system are reviewed. The use of concrete models, ungrouped and pre-grouped, both proportional and non-proportional, is presented as a necessary bridge to the more abstract concepts of place value. The connections between place value and rounding are discussed. Recommendations for teaching the reading and writing of numbers are included. Examples of activities that help to develop place value concepts are provided throughout the chapter.

Student Objectives After reading the chapter, the students will be able to: 1.

Summarize characteristics of our numeration system and place value.

2.

Summarize place value grouping, trading, patterns, and modeling including the use of both ungrouped and pre-grouped materials, and proportional and non-proportional models as well as provide examples of activities which promote their development.

3.

Summarize place value instruction including moving from concrete to symbolic place value representations.

4.

Describe reading and writing numbers as well as provide examples of activities and models that promote their development.

5.

Describe rounding as well as provide examples of activities and models that promote its development.

Key Vocabulary Number sense is firmly based on the concepts of place value. Students should be aware of the characteristics of our number system and discuss place value concepts using the correct terminology. place value grouping or trading proportional models non-proportional models symbolic rounding

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ungrouped materials pre-grouped materials benchmarks rounding

Instructor’s Resource Guide, Chapter 8 Sandi Cooper |2 Supplemental Lecture Topics 1.

Use PPT Slides #3-4, Understand numbers, ways of representing numbers, relationships among numbers, and number systems, to highlight information from NCTM’s Principles and Standards for School Mathematics, the Number and Operation Standard.

2.

Provide background information concerning one of the earliest place value models, the abacus. The book, Activities for the Abacus: A Hands-On Approach to Arithmetic (see additional resources) provides activities which can be used with the abacus.

3.

Provide information concerning the place value results from the 7th National Assessment for Educational Progress (NAEP). See "Whole Number Properties and Operations" by Kouba and Wearne (additional resources).

4.

Provide additional information concerning estimation of quantity. A useful resource is Developing Skills in Estimation by Dale Seymour. Available from Dale Seymour Publications.

5.

Show video clips from Taking Inventory (additional resources) and discuss developing place value ideas.

6.

Display and discuss PPT Slides #22-27, Student Interviews. They are work samples taken from children in grades one through four. First, in PPT Slides # 22, 24, and 267, the children were asked to represent a two-digit number (14 or 16) and then circle the number of cubes represented by the numeral one in the tens place. Second, in PPT Slides # 23, 25, and 27, the same children wrote several numbers that were read to them. The samples are intended to show the wide range of understandings among children. Specific observations concerning each sample follow: PPT #22 PPT #23 PPT #24 PPT #25 PPT #26 PPT #27

Abbie shows that the numeral one in 16 represents one cube.

Abbie can accurately write two-digit numbers.

Clay shows that the numeral one in 14 represents one cube.

Clay can accurately write numbers through ten thousands.

Elsa shows that the numeral one in 14 represents ten cubes.

Elsa can accurately write numbers through ten thousands.

Have students brainstorm other questions or activities that would further clarify the child's understanding for the teacher.

Student Textbook Activities 1.

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As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

Instructor’s Resource Guide, Chapter 8 Sandi Cooper |3 2.

Have students review the Chapter 8 Snapshot of a Lesson. It may be viewed at www.learner.org, video #4. Discuss the types of place value models used.

3.

The numeration system we use has four very important characteristics that are listed in the text. Review each characteristic and have students provide examples to illustrate each.

4.

Children need a thorough understanding of place value in order for computation instruction to be meaningful for them. In Chapter 8, the two key ideas on which place value rests are presented. They include: 1.) Explicit grouping or trading rules are defined and consistently followed. 2.) The position of a digit determines the number being represented. Discuss each key idea and demonstrate them with the use of a place value model such as base-ten blocks.

5.

Discuss and demonstrate the difference between ungrouped and pre-grouped place value materials. For example use connecting cubes to represent numbers both ungrouped and pre-grouped.

6.

Proportional and non-proportional place value models are illustrated in Figure 8-4. PPT Slides #6-8, Proportional and Non-proportional Place Value Models, provides definitions and examples for each. Before showing the Master to students, use a variety of materials to demonstrate how each type of model may be used to model whole numbers. Provide students with some color tiles or other colored counters and ask them to see if they can illustrate place value in both a proportional and non-proportional manner. Confirm their models with PPT Slides #6-8.

7.

Have the students use a place value mat and base-ten blocks to play Race-to-Flat, In the Classroom 11-4 from Chapter 11. Discuss with students how this is an example of a proportional place value model. Next, using a relabeled place value mat, play the game again using color tiles. The new game may be called Race-to-a-Green and the headings on the mat may be changed to Green, Red, Yellow. Once students have collected 10 yellows, they may trade for one red tile. Once they have 10 reds they may trade for a green tile. Discuss how this is an example of a non-proportional model.

8.

It is important that children are assisted in making the connection between concrete place value models and abstract symbols. Figures 8-7 and 8-8 illustrate how the connection may be made. PPT Slide #9, Moving from Concrete to Symbolic Models, provides an illustration for Figure 8-7 and may be used to highlight the connection.

9.

Give students experiences using a variety of manipulative materials to model numbers.

Consider using a variety of both ungrouped and pre-grouped, proportional and nonproportional materials for students to explore using trading mats. Use Figure 8-11 to provide an additional example of regrouping. This opportunity for hands-on experience with these manipulative materials allows students to make decisions about appropriateness for use and provides a background for further teaching experiences. It is

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Instructor’s Resource Guide, Chapter 8 Sandi Cooper |4 also appropriate to introduce homemade alternatives to commercial materials such as bean sticks or paper models (Appendix B).

10.

Many place value ideas may be developed through the use of hundreds charts. Use the charts (found in Appendix B) to complete In the Classroom 8-1 and In the Classroom 8-4, found in the text.

11.

An important place value idea, which prepares students to regroup during computation, involves representing numbers in more than one way. In the Classroom 8-2 illustrates how students can be encouraged to represent numbers with various combinations of ones, tens, and hundreds. Ask students to find several solutions for the last two targets. Share the results and discuss other variations that might be used with students.

12.

The calculator is another valuable tool that may be used to assist students in identifying place value patterns. Figure 8-9 demonstrates how the calculator may be used to count be ones, tens, etc. and Figure 8-10 shows how to count back and illustrate integers. On most four-function calculators, you can enter a starting number, plus sign, an addend, and then repeatedly press the equal sign to continue adding the addend. For example, 0 + 1 = = = = will progressively display the numbers from 1 to 5. Have students also examine In the Classroom 8-3 to see another idea for using calculators to investigate place value concepts. The students may also use the calculator to play the game "Wipe-Out", described in the text.

13.

Some concrete models for rounding numbers are provided in the text. Allow students to use those materials to model the rounding of whole numbers. Then discuss the pros and cons of each model. Also encourage students to develop a model for rounding to the nearest ten on a hundreds chart (Appendix B). Could the model be extended to allow students to represent rounding to the nearest hundred?

14.

The Math Links in this chapter provide students with additional electronic manipulatives and video examples for place value. In class, or as an outside assignment, have students explore these web sites.

15.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

16.

The Book Nook for Children, found in the text, lists several children's books which may be used when studying place value. Bring several of these books to class and have students brainstorm ways in which they could be used in a math lesson.

Student Supplemental Activities 1.

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Three children's books which help establish meaning for concepts of a million, billion, and trillion are If You Made a Million, How Much Is a Million? and On Beyond a Million

Instructor’s Resource Guide, Chapter 8 Sandi Cooper |5 all by David M. Schwartz. Share the books with the class and discuss ways they could be implemented into a place value lesson.

2.

Have students investigate the development of our numeration system. A look at other early number systems can serve to clarify the uniqueness of our system. This investigation connects history to mathematics and allows students to appreciate the role that mathematics plays in past and present civilizations.

3.

Have students examine elementary textbooks to see how the concepts of place value and estimation of quantity are developed.

4.

Show the tape, Mathematics with Manipulatives: Base Ten Blocks, by Marilyn Burns (Available from Cuisenaire) or videl #3 from the Annenberg Teaching Math: A Video Library, K-4. Discuss the place value concepts children are developing.

5.

Provide a variety of calculator activities for students to try. The text provides several examples of activities and references of other sources with interesting calculator ideas.

Do several as a whole class activity. Have students work on certain activities individually, and have them work as pairs to play any of the many calculator games.

After they have spent some time working with the calculator, engage the students in a discussion about their experiences. Did they have to think while using the calculator?

What other skills were they using during their work time?

6.

Bring some teacher resource books to class that provide place value activities. Allow students to review the books and preview some of the activities. For example, The Place Value Connection, Navigating through Number and Operations, Lessons for Introducing or Extending Place Value (see additional resources) may be used.

7.

The Math Links in this chapter provide students with additional electronic manipulatives, video, and lesson plan examples for place value. In class, or as an outside assignment, have students explore these web sites.

8.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

9.

Using the References listed at the end of the chapter, have students locate additional information on topics that interest them such as modeling numbers.

10.

Divide students into groups of 4. Have each person in the group read and report on one article from the chapter reference list related to one of the place value issues discussed in the chapter.

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Instructor’s Resource Guide, Chapter 8 Sandi Cooper |6 Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 7: 1.

Learning about the School and Its Resources: Place Value in the Textbook and Classroom (found in Chapter 1)

2.

Interviewing the Teacher and Students: Place Value, Developing Place Value Concepts (found in Chapter 3)

3.

Helping Children Learn with Games: Wipe Out, Hit the Target (found in Chapter 4)

4.

Helping Children Learn with Technology: Base Blocks on the Internet, Chip Trading on the Internet, Hitting Hundreds (found in Chapter 5)

5.

Helping Children Learn with In the Classroom Lessons: Finding Tens, The Power of 10 on the Thousand Chart (found in Chapter 6)

Additional Resources Activities for the Abacus: A Hands-on Approach to Arithmetic (2nd ed.) by Joan A Cotter, (1988) is available from Activities for Learning Company, Hutchinson, MN. Developing Skills in Estimation by Dale Seymour. Provides activities to practice rounding numbers, using estimates in computation, judging approximate measurements and costs, estimation techniques. Available from Dale Seymour Publications.

Lessons for Introducing Place Value, Grade 2 and Lessons for Extending Place Value, Grade 3 provide detailed lessons that emphasize number sense and problem solving. Also included are classroom vignettes and samples of student work. Available from Math Solutions.

Navigating through Number and Operations in Prekindergarten-Grade 2 includes lessons on representing one, two, and three digit numbers. Available from NCTM. The Place Value Connection by Diana A D'Aboy (1985) is available from Dale Seymour Publications.

Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress. Available from NCTM. Taking Inventory: The Role of Context is part of a video-based professional development series.

Included is a facilitator’s book and CD with video clips and multimedia files. From the Young Mathematicians at Work series, available from Heinemann. .S o n s

Instructor’s Resource Guide, Chapter 8 Sandi Cooper |7 Videos Mathematics with Manipulatives: Base Ten Blocks -- a 20-minute videotape which demonstrates the use of Base Ten Blocks. Children are shown using these blocks as they discuss and explore important concepts related to place value. Available from Cuisenaire Company.

Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 tapes and 3, 5-8 tapes. Tapes include lessons illustrating Standards 1-4. Each tape contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tape 3 is on numeration and Tapes 18 and 19 are estimation lessons. Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org. Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples

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Instructor’s Resource Guide, Chapter 9 Sandi Cooper |1

Chapter 9 — Operations: Meanings and Basic Facts What This Chapter Is About Whole-number computation remains a focus of elementary school mathematics programs. The four prerequisites for whole number operations of counting, concrete experiences, problemsolving context, and language are discussed. Models and mathematical properties for each operation are provided. The basic facts are presented with a focus on beginning from where children are and developing skill using experiences ranging from concrete to pictorial to symbolic. Specific suggestions for drill, activities for practice and mastery, and thinking strategies for recall of basic facts are provided.

Student Objectives After reading the chapter, the students will be able to identify and discuss: 1.

Summarize prerequisites for whole number operations including facility with counting, concrete experiences, problem contexts, and talking and writing.

2.

Demonstrate developing the meanings for whole-number operations of addition, subtraction, multiplication, division, and mathematical properties as well as provide examples of activities and models that promote their development.

3.

Identify the basic facts and what children know.

4.

Summarize research recommendations for developing basic fact fluency as well as provide examples of activities and models which encourage fluency.

5.

Demonstrate using thinking strategies to present the basic facts as well as provide examples of activities and models that promote their development.

Key Vocabulary Computation is a focus of most elementary mathematics programs. Students should be familiar with terms below as they relate to beginning whole-number operations.

Computational fluency basic facts thinking strategies commutativity doubles counting on counting back .S o n s

adding to 10 and beyond repeated addition splitting the product skip counting separation comparison part-whole

combinations to 10 array and area measurement partition equal groups patterns

Instructor’s Resource Guide, Chapter 9 Sandi Cooper |2 Supplemental Lecture Topics 1.

Use PPT Slides #3-4, Understand meanings of operations and how they relate to one another, and PPT Slide #5, Compute fluently and make reasonable estimates, to highlight the key recommendations related to basic facts in grades Pre-K-5 from NCTM's Principles and Standards. These may also be found in the textbook in Figure 9-1, and in Appendix A

2.

The February 2003 focus issue of Teaching Children Mathematics is on computational fluency. Use these articles to provide additional information for students.

3.

Summarize Arthur Baroody’s review of two contrasting theories of how children learn basic facts, the absorption theory (traditionally believed) and the cognitive theory (currently believed). This may be found in Children’s Mathematical Thinking.

4.

Several titles listed under Additional Resources provide workshop materials that could be used to supplement Chapter 9. These include Number Sense and Operations in the Primary Grades, Teaching Arithmetic Series, Understanding Addition and Subtraction in the Primary Grades, Young Mathematicians at Work.

Student Textbook Activities 1.

As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2.

Have students review the Chapter 9 Snapshot of a Lesson. Discuss the children’s strategies for solving the basic facts.

3.

The text highlights four prerequisites for whole number operations. They include counting, concrete experiences, problem contexts, language. For each, have students brainstorm examples of how the prerequisite provides the necessary foundation for computation. The text provides examples.

4.

It is important that children have the opportunity to move from concrete to abstract experiences when studying the basic facts. On PPT Slide #11, Developing Meanings for the Operations, three suggestions are provided for teaching with models. Discuss these with the class and demonstrate the progression with some basic fact "problems".

5.

Students need to recognize that the whole number operations often have several meanings. PPT Slide #12, Meanings for Operations, displays the meanings discussed in the text. Have students use concrete materials such as unifix cubes, color tiles, or beans and develop real-life problems for each of the computational meanings. Compile a class file of problems. There are also appropriate children’s books that can be used to illustrate the various meanings. For example, (See additional resources for complete reference.)

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Instructor’s Resource Guide, Chapter 9 Sandi Cooper |3 Addition: Animals on Board Subtraction: Elevator Magic, Mouse Count Multiplication: Amanda Bean’s Amazing Dream, Each Orange Had 8 Slices, What Comes in 2’s, 3’s, & 4’s?

Division: Divide and Ride, The Doorbell Rang 6.

Helping children develop fluency with their basic facts is an important task for all elementary teachers. PPT Slide #23, When to Develop Basic Fact Fluency, summarizes recommendations found in the text. Demonstrate and discuss how a teacher could assess whether a child was ready to memorize the facts. The complete classic chapter by Ashlock and Washbon may be found in chapter 3 of the 1978 NCTM Yearbook, Developing Computational Skills.

7.

PPT Slides #25-28, Thinking Strategies for Basic Facts, defines the basic facts for addition, subtraction, multiplication, and division and lists the related thinking strategies which are discussed in the text. For each of the strategies, have students practice solving several basic facts.

8.

Using basic fact charts (Appendix B) have students discuss and fill in the facts which may be solved using a particular thinking strategy. Use a different color for each strategy. (See Figures 9-5 to 9-10, 9-12 to 9-18.)

9.

The text summarizes the recommendations concerning drill of basic facts. PPT Slides #29-30, Principles for Basic Fact Drill, provides a list of the recommendations. Discuss these recommendations and brainstorm ways these principles of drill may be carried out by classroom teachers. The complete classic chapter by Davis, may be found in chapter 4 of the 1978 NCTM Yearbook, Developing Computational Skills.

10.

In the Classroom 9-1 and Figure 9-15 in the text provides activities that develop the concept of multiplication using the array model. Have students complete the activities.

Also, discuss which meanings for multiplication are being developed (PPT Slide #12, Meanings for Operations). Have students brainstorm activities that might be introduced after these activities have been completed.

11.

Have students discuss the two meanings of division, introduced in the text. Challenge them to create a problem context for each meaning.

12.

In the Classroom 9-2, Games for Practicing Basic Facts, provides activities for practicing the basic facts. Assign groups of students to try an activity and to report on its strengths and weaknesses as a drill activity. Also have them brainstorm ways the activities may be adapted.

13.

Have students try Figure 9-19, Finger Multiplication. Discuss the usefulness of this strategy for children.

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Instructor’s Resource Guide, Chapter 9 Sandi Cooper |4 14.

The Book Nook for Children, found in the text, provides a list of children's literature that may be used with the study of basic facts. Read some of these books and have students develop basic fact lessons to go with particular titles. Encourage students to bring in additional children's literature books that help develop the basic facts. For example, 12 Ways to Get to 11 by Merriam develops addition combinations to 11, Bunches and Bunches of Bunnies by Mathews illustrates multiplication facts, Each Orange Had 8 Slices develops multiplication as repeated addition. These books and others are available from Scholastic Books. Discuss how children's literature provides a context for solving mathematical problems.

Student Supplemental Activities 1.

PPT Slide #31, 21 or Bust! Provides a calculator game that allows student to use logic and practice basic addition facts. After students have played a couple of times, have them determine a strategy that will help them win.

2.

Have students construct drill activities for one of the sets of basic facts. To get students started, provide each small group with a particular material. For example, how could a deck of cards or a set of dominoes be used to drill basic addition facts? Remind students to keep the drill principles discussed in mind as they develop their activity. Develop a class file.

3.

Have groups develop a set of visual representations for "doubles" facts in addition. For example, a photo or drawing of 2 hands could represent the fact 5 + 5 = 10. How could the other facts of this form be shown? This set of representations serves to bridge the movement from the concrete to the symbolic. Times Tables the Fun Way also provides oral and visual clues for the facts.

4.

Encourage students to locate and review other sources for fact drill activities. For example, Facts of Math: Addition, The Balance Book, and Box Cars & One-Eyed Jacks (see additional resources), provide teaching suggestions and games which can be used at school or home. The journals, Teaching Children Mathematics and the Arithmetic Teacher also provide valuable ideas and are available from NCTM.

5.

Calculators may also be used to drill the basic facts. Books such as Keystrokes and Math Mate Activity Books (see additional resources) provide games and activities. Allow students to participate in some of the activities, then discuss the pros and cons of using the calculator to drill.

6.

Assign students to preview and evaluate computer software which helps children drill the basic facts. Discuss the motivational characteristics of the disks. Compare disks that develop operation concepts versus those which drill for speed and accuracy.

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Instructor’s Resource Guide, Chapter 9 Sandi Cooper |5 7.

Have students brainstorm ways to encourage parent involvement in the memorization of basic facts. Also discuss ways to assist children who do not have help at home when memorizing the facts.

8.

Have students read and discuss "Timed Tests" by Marilyn Burns (see additional resources). Generate a list of pros and cons concerning the use of timed basic fact tests.

Students will likely have their own memories of basic fact tests and can share those as well. In addition, discuss alternative assessment techniques that may be used in place of timed tests. Encourage students to keep in mind the recommendations summarized in the text. As an alternative, have students read and discuss “Multiplication Games: How we made and used them” by Kamii and Anderson.

9.

Show and discuss some video clips of primary children involved in lessons which develop the concept of whole number operations. (See additional resources).

10.

Have students review resources that provide basic fact activities. For example, Teaching Number Sense, Navigating through Number and Operations in Prekindergarten-Grade 2.

(See additional resources)

11.

The Math Links in this chapter provide students with additional electronic manipulatives, video, and lesson plan examples for basic facts. In class, or as an outside assignment, have students explore these web sites.

12.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

13.

Divide students into groups of 4. Have each person in the group read and report on one article from Teaching Children Mathematics, related to basic facts or conceptual understanding of operations.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 8: 1.

Learning about the School and Its Resources: Operation Meaning in the Textbook and Classroom, Basic Facts in the Textbook and Classroom (found in Chapter 1)

2.

Observing the Teacher and Students: Timed Tests (found in Chapter 2)

3.

Interviewing the Teacher and Students: Basic Fact Difficulty, Basic Fact Fluency (found in Chapter (found in Chapter 3)

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Instructor’s Resource Guide, Chapter 9 Sandi Cooper |6 4.

Helping Children Learn with Games: Clear the Board!, Fill the Frame, The Fact Game, Roll a Sum, Roll a Product (found in Chapter 4)

5.

Helping Children Learn with Technology-Calculator Fact Practice, Guess My Fact, 21 or Bust!(found in Chapter 5)

6.

Helping Children Learn with In the Classroom Lessons: Literature Context for Facts, Dot Sticks, Rectangles and More Rectangles, How Could it Happen? (found in Chapter 6)

Additional Resources Animals on Board by Stuart Murphy, Harper Collins, 1998.

Amanda Bean’s Amazing Dream by Cindy Neuschwander, Scholastic, 1999.

The Balance Book by Lee Jenkins (1974) is available from Activity Resources Co. Inc. It provides activities for using a number balance. Basic fact learning is just one of the many topics included.

Box Cars and One-Eyed Jacks, Volumes I, II, III-By Joanne Currah and Jane Felling. Contains math games which use playing cards and dice. Available from Cuisenaire.

Children’s Mathematical Thinking by Arthur Baroody, Teachers College, Columbia University.

The Doorbell Rang by Pat Hutchins, Scholastic, 1986.

Developing Computational Skills, the 1978 NCTM Yearbook edited by Marilyn Suydam, provides a collection of chapters concerning the teaching of basic facts and computational algorithms. Available from NCTM. Divide and Ride by Stuart J. Murphy, Scholasstic, 1997.

Each Orange Had 8 Slices by Paul Giganti, Jr. The Trumpet Club, Inc. 1992.

Elevator Magic by Stuart J. Murphy, Harper Collins, 1997.

Facts of Math: Addition, by Tamara Busch and Bonnie Stoops, is a resource for teachers in grades 1-3. It provides a system for teaching the fact strategies and includes reproducible sheets.

Available from Dale Seymour Publications.

Keystrokes Calculator Activities for Young Children (4 books) by R. Reys et al. are available from Creative Publications. Specific books include Counting and Place Value, Addition and Subtraction, Multiplication and Division, and Exploring New Topics. .S o n s

Instructor’s Resource Guide, Chapter 9 Sandi Cooper |7 The Math Mate Activity Book (Levels 1-3) by D. Williams is published by Stokes Publishing Company and is available from Dale Seymour Publications.

Mouse Count by Ellen Stoll Walsh, The Trumpet Club, Inc. 1991.

“Multiplication Games: How We Made and Used them”, by Kamii and Anderson is in the November 2003 issue of NCTM’s Teaching Children Mathematics. They describe using games to help children develop multiplication fact fluency.

Navigating through Number and Operations in Prekindergarten-Grade 2 provides basic fact strategies and activities for children. Available from NCTM. Number Sense and Operations in the Primary Grades: Hard to Teach and Hard to Learn? Is part of a series of Mathematics Teaching Cases. A facilitator’s guide with sample discussion questions, materials, and a starter problem are provided. Available from Heinemann.

Teaching Arithmetic Series presents detailed lessons, classroom vignettes, and student work samples. Appropriate for Chapter 9 would be Lessons for Addition and Subtraction Grades 2-3, Lessons for Introducing Multiplication Grade 3, and Lessons for Introducing Division Grades 34. Available from Math Solutions.

Teaching Number Sense is a 3-volume set for grades K, 1, and 2. Detailed lesson plans and descriptions of how the lesson went in a real classroom are included. Available from Math Solutions.

"Timed Tests", by Marilyn Burns, is in the March 1995 issue of NCTM's Teaching Children Mathematics. In this article, Burns discusses some of the negative effects of having children take timed basic fact tests.

Times Tables the Fun Way-Presents pictures and stories designed to aid in retention of the basic multiplication facts. Also available for addition. Available from Key Publishers, Inc., 6 Sunwood Lane, Sandy, UT 84092, 1-800-585-6059.

Too Many Dinosaurs by Bob Barner, 1995, Rooster Books of Bantam Doubleday Dell Publishing Group, Inc. New York.

Understanding Addition and Subtraction in the Primary Grades is part of the Supporting School Mathematics series, a parent workshop series. Provides key ideas and handouts on CD. Available from Heinemann.

What Comes in 2’s, 3’s, & 4’s? by Suzanne Aker, Aladdin Paperbooks, an Imprint of Simon and Schuster, 1990.

Young Mathematicians at Work, Grades K-3 is a series of books that show how to approach computation instruction through inquiry, problem solving, and construction. Working with the Number Line and Addition and Subtraction Minilessons would be appropriate to use with chapter .S o n s

Instructor’s Resource Guide, Chapter 9 Sandi Cooper |8 9. A CD is provided with video clips of teachers and children in action. Available from Heinemann.

Videos Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 tapes and 3, 5-8 tapes. Tapes include lessons illustrating Standards 1-4. Each tape contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tapes 4 and 5 contain lessons on developing concepts of whole number operations. Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org. Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples

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Instructor’s Resource Guide, Chapter 10 Sandi Cooper |1

Chapter 10 – Computational Methods: Calculators, Mental Computation, and Estimation What This Chapter Is About In this chapter, the computational alternatives of estimation, mental computation, and calculator computation are discussed. The importance of balanced instruction and encouraging appropriate method choices are stressed. Suggestions for when to use the calculator as a computational tool and an instructional tool are provided. A rationale for helping children to develop mental computation is provided as well as guidelines for providing mental computation instruction.

Guidelines for providing estimation instruction as well as the computational estimation strategies of front-end, compensation, flexible rounding, compatible numbers, and clustering are discussed.

Student Objectives After reading the chapter, the students will be able to: 1.

Summarize recommendations concerning alternative computational methods and computational choice.

2.

Summarize recommendations concerning the use of a calculator as a computational tool and an instructional tool as well as provide examples of activities which promote its use.

3.

Summarize recommendations for teaching mental computation as well as provide examples of activities and models which promote its development.

4.

Summarize recommendations for teaching computational estimation strategies as well as provide examples of activities which promote their development.

Key Vocabulary As in the previous chapter, computation is the focus of most elementary mathematics programs.

The terms below are key to extending computation beyond basic understanding and facts. computational tool instructional tool mental computation computational estimation front-end estimation

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adjusting compatible numbers flexible rounding clustering

Instructor’s Resource Guide, Chapter 10 Sandi Cooper |2 Supplemental Lecture Topics 1.

Use PPT Slides #3-4, Number and Operations Standard: Compute fluently and make reasonable estimates, and PPT Slides #5-6, Number and Operations Standard: Understand numbers, ways of representing numbers, relationships among numbers, and number systems, to highlight the key recommendations related to computational alternatives in grades K-5 from NCTM's Principles and Standards. Discuss these recommendations.

2.

Discuss the role of calculators in the teaching of mathematics. Calculators in Mathematics Education, the 1992 NCTM Yearbook, has many useful articles. Research findings may also be introduced. An excellent overview, "Research on Calculators in Mathematics Education" by Hembree and Dessart is provided in the same yearbook.

3.

In the NCTM book, Mathematics for the Young Child, (see additional resources) Chapter 8, Whole Number Computation, provides additional material which may be used.

Written computation as well as computational estimation and mental computation are discussed. Examples of mental computation strategies such as bridge tens and doubles are provided as well as the estimation strategies of reference points, nice numbers, frontend estimation, and rounding.

4.

The 1986 NCTM Yearbook, Estimation and Mental Computation, provides additional information about teaching computational estimation and mental computation. Use this book to bring up additional issues concerning the teaching of these topics.

5.

A broad view of computation is provided in the book, Computational Alternatives for the Twenty-first Century: Cross-Cultural Perspectives from Japan and the United States(1994), edited by R. Reys and N. Nohda. (Available from NCTM.) The topics of mental computation, computational estimation, and the use of calculators are all discussed and may be used to stimulate discussion on those topics.

6.

Most of the undergraduate students have not had formal instruction in using a variety of strategies for mental computation and computational estimation (other than rounding).

Bring in books such as Mental Math in the Primary Grades, Computational Estimation, and Number SENSE which provide lessons teachers may use to teach specific mental computation and estimation strategies. Introduce several of the strategies and discuss introducing them to elementary children.

Student Textbook Activities 1.

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As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

Instructor’s Resource Guide, Chapter 10 Sandi Cooper |3 2.

Have students review the Chapter 10 Snapshot of a Lesson. It may be viewed at www.learner.org. Discussion should connect the Snapshot to NCTM Standards recommendations for computation.

3.

Discuss with students, Figure 10-2. The NCTM Principles and Standards recommend a balanced approach to computation. Compare the recommendation with the current practice found in most elementary classrooms. Have students examine Figure 10-3.

Today, many teachers continue to emphasize paper-and-pencil computation at the expense of the other types of computation. If the recommendation for computational decision making were supported by those teachers, how would their instruction change?

4.

Use PPT Slides #8-10 Myths and Facts of Calculators, to summarize the use of the calculator as a computational tool and as an instructional tool. Provide examples of activities that illustrate each recommendation. Visit the Texas Instrument web site for ideas. The Keystrokes books (in chapter selected references) are also good resources for instructional tool activities.

5.

Mental computation has often been ignored in the elementary curriculum. PPT Slides #21-23, Guidelines for Teaching Mental Computation, highlights the recommendations made in the text. Discuss these points with students and allow them to brainstorm ways mental computation could be regularly included in an elementary math class.

6.

Most students today learned only one computational estimation strategy in elementary school-rounding. PPT Slides #31-32, Guidelines for Teaching Estimation, provides suggestions for estimation instruction. PPT Slide #33, Computational Estimation Strategies, highlights the five strategies discussed in the text. Discuss each strategy and have students provide problems which might be solved using that strategy.

7.

In the Classroom 10-1 may be used to demonstrate appropriate use of a calculator.

Divide the class into two groups and conduct the activity. Discuss potential uses with children.

8.

In the Classroom 10-2 and 10-3 provide examples of a "choice" activity. Have students brainstorm which types of problems might be appropriate for each of the three computational methods. They can also discuss how the appropriateness of each tool depends on each individual's skill and confidence.

9.

In the Classroom 10-4 encourages students to use front-end estimation skills. Encourage students to brainstorm how the card may be adapted for children of different ages.

10.

The Book Nook for Children, found in the text, lists several children's books which may be used when studying computational methods. Bring several of these books to class and have students brainstorm ways in which they could be used in a math lesson. Encourage students to locate additional children's literature titles that may be used in computation instruction. For example, Counting on Frank by Rod Clement encourages students to estimate quantities. This title is available from Cuisenaire Publications.

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Instructor’s Resource Guide, Chapter 10 Sandi Cooper |4 Student Supplemental Activities 1.

PPT Slides # 11-14, Calculator Test Items, presents 6 test items for class discussion. For each item, have students decide if students should use a calculator, should not use a calculator, or it does not matter. Using their responses and ideas, discuss how assessment changes in the face of available technology. The article, "Using Calculators on Achievement Tests" (additional resources) provides additional information on the subject.

2.

PPT Slides # 15-16 presents the Three-Step Challenge Problem which uses calculators to explore a challenge and arrive at multiple solutions. Estimation skills are also being used in trying to develop a problem-solving strategy for determining a solution. The problem involves students with discovering how to move from a starting number to a target number using only certain operations and a limited number of steps. The activity comes from "IDEAS from the Arithmetic Teacher: Grades 4-6 Intermediate School", a set of activities compiled by Fennell and Williams (NCTM, 1986, p. 11-12). Materials needed for this include PPT Slides #15-16 and calculators.

• Distribute calculators or make sure that students have access to their own calculators.

• Present the challenge outlined on the overhead transparency. To make sure there are no questions, you may want to present one solution.

• Allow students to work and as you observe students finishing the first challenge stated, encourage them to develop their own challenge. Have the students who are prepared then exchange their constructed challenges.

• Discuss the experience and have several students share their challenge questions with the class as other variations of the original challenge.

This activity allows students to use technology in connection with developing a sense of operations and how they are interrelated.

3.

Encourage students to use resource books and textbooks to locate activities where the calculator may be used as a computational tool and an instructional tool. For example, find a problem solving activity, a pattern activity, a concept development activity, etc.

View the PRIME video (see additional resources) and discuss how the calculator is being used in elementary classrooms.

4.

Students need the opportunity to see for themselves how a problem may be solved in several different ways and how class discussions allow students to verbalize their strategies and learn about other strategies from their classmates. PPT Slides # 24-26, Mental Computation, provides three problems for which an exact answer may be calculated mentally and two which may be estimated.

Review the difference between mental computation and computational estimation (mental computation is exact and estimation is approximate). Then, one at a time, briefly

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Instructor’s Resource Guide, Chapter 10 Sandi Cooper |5 display the problems. Have volunteers explain how they solved the problems. See how many different solution processes were used.

Various mental computation strategies may be discussed. Figure 10-6 provides several examples for addition. The estimation strategies discussed in the text (PPT Slide #33, Computational Estimation Strategies) may also be reviewed at this time. Review how the NCTM Principles and Standards are also encouraging multiple solution processes for written computation as well. Discuss the implications for classroom teachers.

5.

PPT Slides #27-29, A Student’s View of Mental Computation, presents how a "typical" middle grade student feels about mental computation. Share the student's view and have students discuss the kind of math classroom this student might be in. Also have them brainstorm additional activities and experiences this student might benefit from.

6.

Have students examine several elementary textbooks to see how mental computation, calculator computation and estimation are introduced and presented. How do the texts reflect the NCTM recommendations? Are concrete materials used? Are students encouraged to invent solution processes? How large are the numbers in the problems that students are asked to solve?

7.

Have students preview computer software which has students working with calculator computation, mental computation, or estimation. Discuss the strengths and weaknesses of each.

8.

The Math Links in this chapter provide students with additional electronic manipulatives for computational algorithms. In class, or as an outside assignment, have students explore these web sites.

9.

Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

10.

Using the References listed at the end of the chapter, have students locate additional information on topics that interest them concerning computational alternatives. Divide students into groups of 4. Have each person in the group read and report on one article.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 9: 1. Learning about the School and its Resources: Computation in the Textbook and Classroom (found in Chapter 1) 2. Observing the Teacher and Students: Focusing on Computation (found in Chapter 2) 3. Interviewing the Teacher and Students: About How Much?, Do It Mentally (found in Chapter 3) .S o n s

Instructor’s Resource Guide, Chapter 10 Sandi Cooper |6 4. Helping Children Learn with Games: Calculators-Versus Mental Computation (found in Chapter 4) 5. Helping Children Learn with Technology: Integrating Calculators into the Math Classroom, Calculator Web Sites, Calculator Fact Practice, How Would You Do It? for Whole Numbers, How Would You Do It? for Fractions (found in Chapter 5) 6. Helping Children Learn with In the Classroom Lessons: Finding 100, How Could It Happen?, Sorting Products (found in Chapter 6) Additional Resources Calculators in Mathematics Education, NCTM's 1992 Yearbook, is a good resource for current findings concerning the use of calculators in mathematics.

Computational Estimation by R. Reys, Trafton, B. Reys, and Zawojewski provides lessons which teach computational estimation strategies such as front-end, adjusting, and rounding.

There is a book for grade 6, 7, and 8. Available from Dale Seymour Publications.

Mathematics for the Young Child, (1990) edited by Joseph Payne, is available from NCTM. Chapter 8 discusses whole number computation including paper-and-pencil computation as well as mental computation and computational estimation.

Mental Math in the Primary Grades by Hope, Leutzinger, B. Reys, and R. Reys includes 36 lessons which teach mental computation strategies. Also available are Mental Math in the Middle Grades and Mental Math in Junior High. Available from Dale Seymour Publications.

Number SENSE by McIntosh, Reys, and Reys (in Chapter 6 selected references) A 4-volume set (Gr. 1&2, 3&4, 4-6, and 6-8) contains many excellent mental computation and estimation activities. Available from Dale Seymour Publications.

Using Calculators on Achievement Tests by Long, Reys, and Osterlind, may be found in the Mathematics Teacher, volume 82, number 5. Discusses the use of calculators on the first administration of the Missouri Mastery and Achievement Test in 1987.

Videos Double-Column Addition: A Teacher Uses Piaget’s Theory –20 minute videotape by Constance Kamii that illustrates thought processes of second-grade children as they compute two-digit column addition. Several student-invented (constructed) algorithms are demonstrated. Available from Teachers College Press.

Multidigit Division—20 minute videotape by Constance Kamii which shows second graders “invent” division by sharing a package of crackers and third graders finding ways to divide 275 by 20. Available from Teachers College Press.

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Instructor’s Resource Guide, Chapter 10 Sandi Cooper |7 PRIME Calculators, Children and Mathematics --27-minute video tape which shows children ages 4-9 in various mathematical problem solving activities where calculators are used.

Teachers are shown discussing different aspects of teaching and some roles calculators might play. The tape also shows parents discussing some of their concerns about children using calculators as well as the effect of this project on them and their children. Guidebook and tape are available from Simon and Schuster Ltd. Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 tapes and 3, 5-8 tapes. Tapes include lessons illustrating Standards 1-4. Each tape contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tape 17 is the Snapshot of a Lesson. Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org. Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples

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Instructor’s Resource Guide, Chapter 11 Sandi Cooper |1

Chapter 11 —Standard and Alternative Computational Algorithms What This Chapter Is About In this chapter, the NCTM Principles and Standards recommendations concerning changes in computation instruction are discussed. The importance of the use of manipulative materials and place value ideas in developing computational algorithms are stressed. Various algorithms for operations are presented although encouraging children to develop solutions which make sense to them is emphasized.

The chapter closes with discussion of the importance of choosing appropriate means of calculating and developing computational proficiency.

Student Objectives After reading the chapter, the students will be able to: 1. Summarize recommendations concerning the teaching of computation.

2. Demonstrate written algorithms and supporting place value concepts for addition, subtraction, multiplication, and division as well as provide activities and models which promote their development.

3. Describe a variety of ways children might invent strategies for computation.

Key Vocabulary As in the previous chapter, computation is the focus of most elementary mathematics programs. The terms listed below are key to extending computation beyond basic understanding and facts. algorithms compatible numbers higher-decade combinations partial-sum addition algorithm decomposition algorithm partial-difference subtraction algorithm partial products multiplication algorithm lattice multiplication algorithm distributive algorithm subtractive algorithm

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Instructor’s Resource Guide, Chapter 11 Sandi Cooper |2 Supplemental Lecture Topics 1. Provide information from the February, 2003 Focus Issue of Teaching Children Mathematics on computational fluency.

2. To bring a historical perspective concerning computation, summarize points made by William Brownell (1947) found in the end of book in References.

4. The 1998 NCTM Yearbook, The Teaching and Learning of Algorithms in School Mathematics, provides important information about computational algorithms. For a historical perspective, the 1978 NCTM Yearbook, Developing Computational Skills, contains several chapters which discuss computational algorithms. Use these chapters to provide additional information for students. (see additional resources) 5. The book, Analysis of Arithmetic for Mathematics Teaching (1992), edited by Leinhardt, Putnam, and Hattrup, provides chapters concerning whole number addition, subtraction, multiplication, and division. (see additional resources) 6. The constructivist approach to teaching and learning computation is being advocated by many mathematics educators. Discuss key points made by Kamii, Lewis, and Livingston in the article "Primary Arithmetic: Children Inventing Their Own Procedures" (see Chapter 10 Annotated Resources) and Making Sense (1997) in additional resources.

7. Show video clips from the Children’s Mathematics: Cognitively Guided Instruction CD’s or from Kamii’s videos that illustrate children solving computational problems with algorithms that make sense to them. (see additional resources) 8. Provide information concerning the whole number operation results Results from the Seventh Mathematics Assessment of the National Assessment of Educational ProgressEdited by Edward A. Silver and Patricia Ann Kenney. Available from NCTM. 9. Share examples of teachers’ efforts to help children solve problems in ways that make sense to them. Good resources include Children’s Mathematics: Cognitively Guided Instruction, and Learning Through Problems. (see additional resources) Student Textbook Activities 1. As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and on PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2. Have students review the Chapter 11 Snapshot of a Lesson. Use the discussion on the previous page to connect the Snapshot to recommendations in the NCTM Standards.

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Instructor’s Resource Guide, Chapter 11 Sandi Cooper |3 3. NCTM recommends a change in the way computation instruction is provided. You may want to use PPT Slide #4, Balancing Conceptual Understanding and Computational Proficiency and PPT Slide #28, Building Computational Proficiency to focus on recommendations for written computation. Encourage students to discuss how these recommended changes will affect the teaching of elementary arithmetic and to identify both potential pros and cons.

4. Students may be unsure how to model computational algorithms with concrete materials.

Bring in a variety of manipulative materials to be used by small groups. Provide each group with a set of manipulative materials and PPT Slides #5-7, Modeling Written Algorithms with Concrete Materials. You may want to provide a book such as Building Understanding with Base Ten Blocks by Mary Laycock et al. (see additional resources) as a resource book. Once the group has decided on a way to model the algorithm, they can consult the text pages for additional ideas. Have students discuss: Do some types of materials work better with a particular operation? Any problems encountered in their use that would need to be considered in working with children? You may also use base-ten blocks with In the Classroom 11-4, Racing with Base-Ten Blocks, to illustrate the idea of trading.

5. Solving computational problems using a variety of strategies may be new to many students. One at a time, display the problems on PPT Slides #29-30, How Many Strategies? Have students generate a variety of ways to solve each using mental and/or written computation. Once they have completed their suggestions, they can consult the text pages for additional ideas. Discuss how concrete materials can also be used to represent invented algorithms as well. Students may also be encouraged to investigate more unfamiliar processes such as the complement technique for subtraction or the subtractive algorithm for division. A small group of students may also be assigned to "learn" one of these "unfamiliar" processes such as lattice multiplication and teach it to the rest of the class.

6. Use PPT Slide #20, Lattice Multiplication, to demonstrate how to find products using a lattice. Begin by walking them through the example on page 249, then have students make up another problem to try.

7. Have the class work in small groups to play In the Classroom 11-1, Capture. Then discuss the activity questions. Talk about how this game would work with children.

8. Use In the Classroom 11-3, Look for Patterns and 11-5, What’s Missing to encourage students to look for patterns in addition and multiplication. Can these patterns be modeled with concrete materials? Have students discuss the benefits of exposing students to such patterns.

9. Use In the Classroom 11-2, Starters, 11-6, The Missing Digits Game, 11-7, Making Examples, and 11-8, Easy Does It! to encourage students to examine the parts of an arithmetic problem in order to find a missing number, find a solution, or discover a rule.

Have students discuss how these types of activities help develop students' understanding. .S o n s

Instructor’s Resource Guide, Chapter 11 Sandi Cooper |4 Discuss the role of a calculator and manipulatives in these activities. For example, the calculator and number tiles may support the use of the trial and error strategy.

10. Use In the Classroom 11-9, The Remainder Game and 11-10, What Number am I! to encourage students to use mental computation and estimation strategies to enhance written computation. Have students discuss the benefits of exposing students to such patterns.

11. The Book Nook for Children, found in the text, lists some children's books which may be used in the study of computation. Encourage students to locate additional children's literature titles which may be used in computation instruction.

Student Supplemental Activities 1. Encourage students to locate activities where the calculator may be used to develop written computation. Discuss how the calculator may be used to enhance understanding of written computation. View the PRIME video (see additional resources) and discuss how the calculator is being used in elementary classrooms.

2. Have students preview computer software which has students working with computational algorithms. Discuss the strengths and weaknesses of each.

3. Introduce the students to the NCTM Addenda books for the elementary and middle grades. Particularly applicable to this chapter is Number Sense and Operations for the elementary grades. (See additional resources.) Have students complete some of the activities in this resource book and identify how they reflect the NCTM recommendations.

4. Many publishers are providing self-contained units for particular mathematics topics. For example, available from Cuisenaire is the Math by All Means series. Particularly applicable to this chapter is Multiplication, Grade 3, Division, Grades 3-4. Or have students examine Constructing Number Sense, Teaching Arithmetic series. Have students examine these units and compare them to "traditional" textbook units.

5. Show and discuss some video clips from Teaching Math: A Video Library, K-4. The videos, Amazing Equations, Domino Math, and Products and Sums show students engaged in whole number computation. These may be viewed at www.learner.org. (see additional resources) 6. Have students use base-ten blocks to illustrate invented algorithms described by children in the Kamii videos listed in additional resources.

7. The Math Links in this chapter provide students with additional electronic manipulatives for computation. In class, or as an outside assignment, have students explore these web sites. .S o n s

Instructor’s Resource Guide, Chapter 11 Sandi Cooper |5 8. Using the Cultural Connection as a starting point, have students explore algorithms used by other countries.

9. Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

10. Using the References listed at the end of the chapter, have students locate additional information on topics that interest them such as modeling numbers. Divide students into groups of 4. Have each person in the group read and report on one article from the chapter reference list related to one of the computation issues discussed in the chapter.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 11: 1. Learning about the School and Its Resources: Computation in the Textbook and Classroom (found in Chapter 1) 2. Observing the Teacher and the Student: Focusing on Computation (found in Chapter 2) 3. Interviewing the Teacher and Students: Basic Fact Difficulty, Basic Fact Fluency, Addition with Regrouping, Transitional Algorithms (found in Chapter 3) 3. Helping Children Learn with Games: Capture 5, Race-to-a-Flat, Sum or Difference Game, The Remainder Game (found in Chapter 4) 4. Helping Children Learn with Technology: Zeros Count (found in Chapter 5) 5. Helping Children Learn with In the Classroom Lessons: Finding 100, Sorting Products, Number Net, Look for Patterns (found in Chapter 6) Additional Resources Analysis of Arithmetic for Mathematics Teaching (1992), edited by Gaea Leinhardt, Ralph Putnam, and Rosemary A. Hattrup, is published by Lawrence Erlbaum Associates. Chapters devoted to whole number computation are included.

Building Understanding with Base Ten Blocks by Mary Laycock et al. is available for both primary and middle grades. Both books show how to use base ten blocks to model whole number computation. Available from Dale Seymour Publications. .S o n s

Instructor’s Resource Guide, Chapter 11 Sandi Cooper |6 Children’s Mathematics: Cognitively Guided Instruction by Carpenter, Fennema, Franke, Levi, and Empson is based on more than 20 years of research and provides a framework for assessing children’s thinking in whole number arithmetic. Two accompanying CD’s provide clips of students and teachers. Available from NCTM and Heinneman.

Constructing Number Sense, Addition and Subtraction and Construction Multiplication and Division by Catherine Twomey Fosnot and Maarten Dolk, focus on how to help children make sense of computation. Available from Heinemann.

Developing Computational Skills, the 1978 NCTM Yearbook edited by Marilyn Suydam, provides a collection of chapters concerning the teaching of basic facts and computational algorithms. Available from NCTM. Learning Through Problems by Paul Trafton and Diane Thiessen is a practical resource for teachers wanting to support children’s efforts to make sense of computation. Available from Heinemann.

Making Sense: Teaching and Learning Mathematics with Understanding (1997). by James Hiebert, Thomas Carpenter, Elizabeth Fennema, Karen Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piet Human. Current research-based ideas on how to design classrooms that help students learn mathematics with understanding by constructing their own procedures. Four research projects related to arithmetic in elementary school are summarized.

Math By All Means is a series of replacement units published by Cuisenaire. The units include manipulatives and children's literature. Multiplication (grade 3), provides a view of multiplication from geometric, number, and real-life perspectives. Division (grades 3,4), has students examine patterns, analyze data, solve problems with money, and use division in realworld contexts.

Number Sense and Operations is one of the Addenda Series of books published by NCTM. It includes activities for students in grades K-6.

Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress-Edited by Edward A. Silver and Patricia Ann Kenney. Reports results from the 1996 NAEP test of students in grades 4, 8, and 12. Available from NCTM. Teaching Arithmetic is a series of books providing detailed lessons for a particular grade level.

Also included are classroom vignettes and samples of student work. Titles appropriate for chapter 11 include Lessons for Extending Multiplication: Grades 4-5, and Lessons for Extending Division: Grades 4-5. Available from Math Solutions.

The Teaching and Learning of Algorithms in School Mathematics, NCTM’s 1998 yearbook includes a variety of chapters addressing algorithms and the issues surrounding them.

Young Mathematicians at Work is a professional development series for teachers. Included are workshop materials and a CD with video clips. Titles that apply to chapter 11 include Turkey .S o n s

Instructor’s Resource Guide, Chapter 11 Sandi Cooper |7 Investigations: A Context for Multiplication, Exploring Soda Machines: A Context for Division, Multiplication and Division Minilessons. Available from Heinemann.

Videos Double-Column Addition: A Teacher Uses Piaget’s Theory –20 minute videotape by Constance Kamii that illustrates thought processes of second-grade children as they compute two-digit column addition. Several student-invented (constructed) algorithms are demonstrated. Available from Teachers College Press.

Multidigit Division—20 minute videotape by Constance Kamii which shows second graders “invent” division by sharing a package of crackers and third graders finding ways to divide 275 by 20. Available from Teachers College Press.

PRIME Calculators, Children and Mathematics - a 27-minute video tape which shows children ages 4-9 in various mathematical problem solving activities where calculators are used.

Teachers are shown discussing different aspects of teaching and some roles calculators might play. The video also shows parents discussing some of their concerns about children using calculators as well as the effect of this project on them and their children. Guidebook and tape are available from Simon and Schuster Ltd. Teaching Math: A Video Library, K-4 and 5-8-. These videos include lessons illustrating Standards 1-4. Each video contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800864-9846 or may be viewed at www.learner.org. Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples Web Resources National Library of Virtual Manipulatives for Interactive Mathematics (http://nlvm.usu.edu) provides online Base-10 Blocks, color chips for addition, number line bars for addition, subtraction, multiplication, and addition, number line bounce (an addition and subtraction game), and using an area representation for multiplication and division.

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Instructor’s Resource Guide, Chapter 12 Sandi Cooper |1

Chapter 12 — Fractions and Decimals: Concepts and Operations What This Chapter Is About The focus of this chapter is on how to meaningfully approach fractions and decimals.

Meaningful teaching makes use of models and language to develop understanding of fractions and decimals before moving to the symbols used for representation. The point is made that just as with whole-number operations, operations with these numbers must be given meaning and not simply presented as rules to be followed. Partitioning, equivalence, benchmarks, and three meanings of fractions are introduced. Four models for the part-whole meaning of fractions and models for fraction algorithms are provided. The relationship of decimals to fractions and place value are discussed as well as decimal algorithms.

Student Objectives After reading the chapter, the students will be able to: 1. Explain and illustrate partitioning, equivalence, and three meanings of fractions.

2. Provide examples for the four models of the part-whole meaning of fractions.

3. Summarize steps for helping children make sense of fractions.

4. Demonstrate, concretely and symbolically, ordering and equivalence of fractions, benchmarks, mixed numbers, and improper fractions as well as provide examples of activities and models which promote their development.

5. Demonstrate, concretely and symbolically, the fraction algorithms for addition, subtraction, multiplication, and division as well as provide examples of activities and models which promote their development.

6. Summarize the relationship of decimals to fractions and place value.

7. Demonstrate, concretely and symbolically, ordering and rounding decimals and decimal algorithms for addition, subtraction, multiplication, and division as well as activities and models which promote their development.

Key Vocabulary The role of fractions and decimals in our number system is an important one. These terms are central to the teaching and learning of concepts and applications involving fractions and decimals. partitioning region common fractions area equivalence congruent decimal fractions equivalent fractions quotient length part-whole least common denominator ratio set benchmarks

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Instructor’s Resource Guide, Chapter 12 Sandi Cooper |2 Supplemental Lecture Topics 1.

Use PPT Slides #4-8, Number Sense Strand by Grade Level, to highlight key recommendations related to fractions and decimals from NCTM's Principles and Standards. These may also be found in the textbook in Figure 12-1 and in Appendix A.

2.

Share recommendations and case studies about teaching fractions and decimals using resources such as Mathematics Teaching Cases, Teaching Fractions and Ratios for Understanding, Making Sense of Fractions, Ratios, and Proportions, Supporting School Mathematics series, and Young Mathematicians at Work series, all listed in Additional Resources at the end of this section.

3.

Chapter 6 of Results from the Seventh Mathematics Assessment, highlights fraction and decimal items from the 7th National Assessment of Educational Progress (NAEP).

Display some of these items and their results, then discuss the implications for fraction and decimal instruction. Also, Table 12-1 provides additional examples. The book is listed in additional resources in the Chapter 11 Instructor’s Notes.

4.

In the book, Analysis of Arithmetic for Mathematics Teaching, (see additional resources) edited by Leinhardt, Putnam, and Hattrup, Chapter 5 by James Hiebert concerns decimal fractions and Chapter 6 by Thomas E. Kieren concerns rational and fractional numbers.

Both summarize research findings and provide implications for instruction.

5.

Provide additional ideas and activities for teaching fractions and decimals. Some good sources include Constructing Fractions, Decimals, and Percents and Understanding Rational Numbers and Proportions listed in additional resources.

6.

Display and discuss PPT Slides #32-34, Student Interviews. These include fraction work samples taken from children in grade four. The children were asked to compare fractions and add fractions. The samples are intended to show the wide range of understandings among children. Specific observations concerning each sample follow: PPT #32

Charles is able to represent the fractional symbols with pictures and successfully solves the comparison problems.

PPT #33 Amanda has not yet studied fractions in the fourth grade. She uses her knowledge of whole numbers to solve the problems. No fraction pictures are used. In the comparison problems, she identifies the fraction with the largest numeral in the denominator as largest. In the addition problems, she adds the numerators and adds the denominators.

PPT #34 Amy successfully adds two fractions with like denominators. When solving addition with unlike denominators, she is unsure what to do. She attempts to draw a picture but is unsuccessful so she adds the numerators like the previous problem and uses four as a denominator.

Ask students to brainstorm other questions or activities which would further clarify the child's understanding for the teacher. What activities could be provided for Amanda and Amy? .S o n s

Instructor’s Resource Guide, Chapter 12 Sandi Cooper |3 Student Textbook Activities 1. As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2. Have students review the Chapter 12 Snapshot of a Lesson. It may be viewed at www.learner.org. Discuss the area model for fractions.

3. Use PPT Slide #10, Three Meanings of Fractions, to discuss the part-whole, quotient, and ratio meanings for fractions. Have students brainstorm additional real-life examples for each and record them on chart paper or the board.

4. Using PPT Slide #11, Models of Part-Whole Meaning, discuss the four models used in developing the part-whole meaning for fractions. Have students brainstorm additional real-life examples for each and record them on a transparency or the board.

5. Discuss key ideas used when helping children make sense of fractions (partitioning, words, counting, symbols, and models and beyond) using PPT Slide #12, Making Sense of Fractions. These key ideas are listed in the text. Discuss the examples provided in the text and have students generate others.

6. In the Classroom 12-1, Three-Fourths, 12-2, Fraction Bars, and 12-3, Sharing Rods, each include activities with a concrete fraction models. Have students compare the three models and discuss their pros and cons.

7. In the Classroom 12-4, Comparing Models, and 12-5, Whole Hog, include activities that emphasize equivalence of fractions. Have students discuss the insights students can develop by participating in the various activities.

8. In the Classroom 12-6, A New Twist on Old Rhymes, contains rhymes for operations of fractions. Have students write more rhymes to share with the class.

9. In the Classroom 12-7, Can you Beat the Toss?, allows students to see the relationship between fractions and decimals. Have students brainstorm how concrete models could be used with this activity.

10. In the Classroom 12-8, What’s Your Answer? presents decimal computation activities.

Discuss the role of the calculator in these activities.

11. The Book Nook for Children, found in the text provides a list of children's literature that may be used with the study of fractions. Read some of these books and have students develop lessons to go with particular titles. Encourage students to locate other pieces of children's literature which contribute to the study of fractions and decimals. For example Alexander, Who Used to be Rich Last Sunday by Judith Viorst .S o n s

Instructor’s Resource Guide, Chapter 12 Sandi Cooper |4 includes the use of coins. Eating Fractions by McMillan shows children sharing food.

Discuss how children's literature provides a context for solving mathematical problems.

Student Supplemental Activities 1. Have students use grid paper, color tiles, markers, or other materials to form designs in an 8 by 8, 6 by 12, 10 by 10, or other dimensioned area. Consider questions such as: What fraction of your design is red? other color? Which is a greater fraction of your design, red or yellow? Have students compare fractions by using their sets of fraction bars to represent the fractions found in the design formed. If students work within a 10 by 10 framework, relationships among fractions, decimals, and percents can be illustrated.

2. Have students divide into small groups and give each group a different fraction. Ask them to represent that fraction in as many different ways, in as many different representations as possible. Have manipulatives, grid paper, paper, markers, and other supplies available for their use. To share their fractional presentations, have each group construct a bulletin board or other visual display of the many models found. Allow groups to rotate through each display to discover the creative discoveries of others groups in the class and extend the ideas available for use in teaching children.

3. Consider involving the class in an activity which uses calculators which have a fraction key. These calculators allow students to move from fractional representations to decimal representations and back again. Exploration using this type of technology will help students further clarify the role that technology can play in the teaching and learning of mathematics at the elementary school level. A resource book which provides ideas for using this type of calculator is Using the Math Explorer Calculator: A Sourcebook for Teachers by Bitter and Mikesell (see additional resources).

4. Have groups of students experiment with a variety of fraction and decimal manipulatives to model the computational algorithms. Small groups may take turns teaching each other how to use the manipulatives. For example, they may use fraction circles, strips, or bars.

Or they may use decimal squares, base ten blocks, or coins. Also be sure to refer students to the fraction and decimal models found in Appendix B. Online manipulative catalogs are another source of ideas for manipulatives for fractions and decimals.

5. The Marilyn Burns books, A Collection of Math Lessons, include several problemsolving lessons related to fractions. For example in the book for grades 3-6, a lesson on sharing cookies is presented. The Annenberg video series and nctm.org (see additional resources) also contains fraction and decimal lessons. Have students review several of these lessons and share a portion with the class.

6. Have the class design some interview questions they could use to assess children's understanding of fractions and decimals. If possible, have students interview some children and collect work samples similar to those provided illustrated in PPT Slides #3234. .S o n s

Instructor’s Resource Guide, Chapter 12 Sandi Cooper |5 7. There are many good calculator activities which develop decimal concepts. Share some of these with the class. Have students bring sample activities to display in the classroom.

8. Have students try some of the activities in Number SENSE, Fraction Circle Activities, Fraction Bars Starter set and Decimal Squares Starter set, Teaching Arithmetic series (additional resources) which develop number sense for fractions, decimals, and percents.

9. Show some video clips of fraction and decimal lessons. In the Teaching Math: A Video Library (found at www.learner.org), possible video clips include Fraction Strips, Arrays and Fractions, Everyday Decimals, Fractions with Geoboards (in K-4) and Fraction Tracks (in 5-8).

10. The Math Links in this chapter provide students with additional electronic manipulatives and video clips of lessons for fractions and decimals. In class, or as an outside assignment, have students explore these web sites.

11. Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

12. Using the References listed at the end of the chapter, have students locate additional information on topics which interest them such as using games to teach fractions or fractions in early childhood. Divide students into groups of 4. Have each person in the group read and report on one article or chapter.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 12: 1. Learning about the School and Its Resources: Fractions in the Textbook and Curriculum Guide (found in Chapter 1) 2. Interviewing the Teacher and Students: Three-Fourths (found in Chapter 3) 3. Helping Children Learn with Games: Fraction Shape Puzzle, Whole Hog, Can you Beat the Toss?, Make a Match (found in Chapter 4) 4. Helping Children Learn with Technology: How Would you do It?, Plug-In Puzzles, Decimal Maze (found in Chapter 5) 5. Helping Children Learn with In the Classroom Lessons: Composing Fractions, Fraction Strips, Comparing Models (found in Chapter 6). .S o n s

Instructor’s Resource Guide, Chapter 12 Sandi Cooper |6 Additional Resources Analysis of Arithmetic for Mathematics Teaching (1992), edited by Gaea Leinhardt, Ralph Putnam, and Rosemary A Hattrup is published by Lawrence Erlbaum Associates. Chapters 5 and 6 are devoted to decimals and fractions.

Constructing Fractions, Decimals, and Percents by Catherine Twomey Fosnot and Maarten Dolk focuses on how students in grades 5-8 construct fraction and decimal knowledge.

Decimal Squares Starter Set by Albert Bennet is a kit which contains vinyl decimal squares in tenths, hundredths, and thousandths as well as a teacher's guide with lessons and games for developing decimal concepts. Available from Cuisenaire.

Fraction Bars Starter Set by Albert Bennett and Patricia Davidson is a kit which contains vinyl fraction bars and a teacher's guide with lessons and games for developing fraction concepts.

Available from Dale Seymour Publications.

Fraction Circle Activities by Barbara Berman and Fredda Friederwitzer is a resource book which includes a wide range of activities ranging from names for unit fractions to addition and subtraction of fractions. Available from Dale Seymour Publications.

Making Sense of Fractions, Ratios, and Proportions, edited by Bonnie Litwiller, is NCTM’s 2002 Yearbook. Available from NCTM. Mathematics Teaching Cases: Fractions, Decimals, Ratios, & Percents edited by Carne BarnettClarke, Donna Goldenstein, and Babette Jackson provides cases describing experiences of elementary and middle school teachers. Available from Heinemann.

Number SENSE by McIntosh, Reys, and Reys is a set of 4 books (Gr. 1&2, 3&4, 4-6, and 6-8) which contain number sense activities. Several of the activities pertain to fractions and decimals.

Supporting School Mathematics: How to Work with Parents and the Public series provides workshop material for presenting on various math topics. The title, Understanding Fractions Across the Grades, would be appropriate for this chapter. Available from Heinemann.

Teaching Fractions and Ratios for Understanding by Susan Lamon, provides content knowledge and instructional strategies for teachers. Available from Lawrence Erlbaum.

Teaching Arithmetic is a series of books providing detailed lessons for a particular grade level.

Also included are classroom vignettes and samples of student work. Titles appropriate for chapter 12 include Lessons for Introducing Fractions: Grades 4-5, Lessons for Extending Fractions, Grade 5, Lessons for Multiplying and Dividing Fractions: Grades 5-6, and Lessons for Decimals and Percents. Available from Math Solutions.

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Instructor’s Resource Guide, Chapter 12 Sandi Cooper |7 Understanding Rational Numbers and Proportions, Grades 5-8 by Fran Curcio and Nadine Bezuk is one of NCTM’s Addenda books. Available from NCTM. Using the Math Explorer Calculator: A Sourcebook for Teachers by Gary Bitter and Jerald Mikesell is for grades 4-8. It contains instruction on using the Math Explorer plus problem solving activities, lesson plans, and transparency masters. Published by Addison Wesley and available from Dale Seymour Publications.

Young Mathematicians at Work is a professional development series for teachers. Included are workshop materials and a CD with video clips. Titles that apply to chapter 12 include Sharing Submarine Sandwiches: Grades 5-8, Working with the Ratio Table: Grade 6. Available from Heinemann.

Video Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 videos and 3, 5-8 videos. These video clips include lessons illustrating Standards 1-4. Each video contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept.

TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org.

Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples.

Web Resources National Library of Virtual Manipulatives for Interactive Mathematics (http://nlvm.usu.edu) provides online Fraction Bars, Fractions – Naming, Fractions – Parts of a Whole, Fractions – Visualizing, and Number Line Bars - Fractions.

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Instructor’s Resource Guide, Chapter 13 Sandi Cooper |1

Chapter 13 — Ratio, Proportion, and Percent: Meanings and Applications What This Chapter Is About This chapter begins with a discussion of ratios, the comparison of two or more numbers. The different forms and applications of ratios are presented. The opportunity to practice computational skill and problem solving together is seen in work with ratios and proportions.

The study of percent is a natural extension of ratio ideas with a comparison base of 100. Models for representing percent are introduced as well as the ratio and equation method for solving percent problems. Throughout the chapter, opportunities for developing number sense are highlighted.

Student Objectives After reading the chapter, the students will be able to: 1. Define ratios as well as provide examples of activities and models which promote their development.

2. Define proportions as well as provide examples of activities and models which promote their development.

3. Define percents as well as provide examples of activities and models which promote their development.

4. Demonstrate solving percent problems using the ratio, equation, and mental computation methods.

Key Vocabulary Understanding of each of the terms below will allow students to use correct terminology when discussing or teaching concepts of comparison. ratio proportion percent benchmark ratio method equation method ratio table two-sided number lines

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Instructor’s Resource Guide, Chapter 13 Sandi Cooper |2 Supplemental Lecture Topics 1. Use PPT Slides #3-5, Expectations from the Number and Operations Standard to highlight key recommendations related to ratio, proportion, and percent from NCTM’s Principles and Standards. These may also be found in the textbook in Figure 13-1 and in Appendix A. 2. Discuss research involving ratio, proportion, and percent. For example, the chapter, "Rational Number, Ratio, and Proportion", by Behr and others, may be found in the Handbook of Research on mathematics Teaching and Learning, edited by Douglas Grouws and available from NCTM. 3. The April 2003 focus issue of Mathematics Teaching in the Middle school is on proportional reasoning. Several articles in the Chapter 13 Annotated Resources provide several excellent ideas for developing the concepts of ratio, proportion, and percent. Use them to provide additional information for your students.

4. Share additional activities and case studies from sources such as Making Sense of Fractions, Ratios, and Proportions, Constructing Fractions, Decimals, and Percents, or Mathematics Teaching Cases: Fractions, Decimals, Ratios, & Percents (listed in additional resources).

Student Textbook Activities 1. As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2. Have students review the Chapter 13 Snapshot of a Lesson. This may be viewed at www.mmmproject.org. Discuss the various topics in mathematics that are included.

3. In this chapter, the concepts of ratio, proportion, and percent are developed. PPT Slides #6-7,10-11, and 14-17, provides a short definition and example for each. Use the slides to introduce the discussion, then have students generate additional examples which would be meaningful for children. In addition, have students brainstorm a list of prerequisite skills which students would need before studying these topics.

4. Pose the proportion problem shown on PPT Slides #24-27. Then show and discuss the three seventh grader’s solutions. These students had not done any formal work in solving proportions. Specific observations include: PPT #24 PPT #25

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Trevor recognizes the two ratios are equivalent to 2/3.

Deanna recognizes that 20:30 is equivalent to 40: 60 and can be changed to 40:60 by multiplying.

Instructor’s Resource Guide, Chapter 13 Sandi Cooper |3 PPT #26

Yang incorrectly believed that the relationship between doors and walls could be found by addition, rather than multiplication.

5. Have a class discussion using the percent questions on pages 291-292. PPT Slides #2835 are work samples taken from students in grade seven who had not yet formally studied percents. The students were asked to answer each question and tell why they answered the way they did. The samples are intended to show a range of understandings about percents. Specific observations concerning each sample follow: PPT #27-28 PPT #29-30

Jacob is able to connect percents to fractions.

Bradley is able to connect percents to both fractions and decimals (money). He has some difficulty with an increase of 150%.

PPT #31-32 Allan is able to connect percents to fraction words and pictures. His rationale for increasing a price by 50% is correct but does not match his example.

PPT #33-34 Jessica is able to connect percents to both fractions and decimals (money).

She is able to provide supporting examples.

Ask students to brainstorm other questions or activities which would further clarify the student's understanding for the teacher. What activities could be provided for Allan and Bradley?

6. The chapter concludes with a discussion concerning solving percent problems. Have the class generate real-life percent problems such as sale prices, tips, tax, etc. Allow students to try solving them using the methods mentioned in the text. Mental methods may be new for students and may need to be the focus of the discussion. Ask students to share the method they remember learning in school.

7. In the Classroom 13-1, Know Your Coins, provides a ratio and proportion activity. Have students examine this activity, then brainstorm a list of other real-life examples of ratio that could be used with children.

8. In the Classroom 13-2, Using Percents and 13-3, Estimating Percents provide models for representing and estimating percents. Have students examine these models and the additional models for percent found in the chapter. Discuss the pros and cons of each.

Have students work together to develop additional concrete models for percent.

9.

The Book Nook for Children, found in the text provides a list of children's literature which may be used with the study of ratio, proportion, and percent. Read some of these books and have students develop lessons to go with particular titles. Discuss how children's literature provides a context for solving mathematical problems. Another book,

by David Schwartz, If You Hopped like a Frog, develops the concept of ratio, with fun, real-life examples.

Student Supplemental Activities

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Instructor’s Resource Guide, Chapter 13 Sandi Cooper |4 1. Students should also be encouraged to use what they know about the properties of multiplication when solving percent problems. David Glatzer's article, "Teaching Percentage: Ideas and Suggestions", (in Chapter 12 selected references) provides several excellent percent activities. PPT Slides #21-22, Percent Shortcuts, offers some of Glatzer's suggestions for easily solving percent problems mentally which are interesting for students. For example: A% of B = B% of A (commutative property) While 28% of 50 would be difficult to solve mentally, 50% of 28 is quite manageable and yields the same result as the more difficult problem.

(K x A)% of B = K(A% of B) (associative property) Students may not be able to find 20% of 35 mentally. However, if they know how to find the benchmark 10%, the problem becomes easier. 20% of 35 is the same as (2 x 10)% of 35 which is also equivalent to 2(10% of 35). I can find 10% of 35 which is 3.5, then I double it to get 7.

(A+C)% of B=(A% of B) + (C% of B) (distributive property) The classic dilemma of calculating a 15% tip after a $60 meal in a restaurant can be reduced to a simple mental calculation. 15% of $60 is the same as (10+5)% of $60 which can also be expressed as (10% of $60) + (5% of $60). I know 10% of $60 is $6.

I also know that 5% is half of 10% and that would be 3. I add $6+$3 to get $9.

2. Tangrams are useful manipulative materials for geometry topics but this problem connects them to ideas of percent. Materials: PPT Slide #23, Tangrams and Percent, sets of tangram pieces, rulers, grid paper.

• Distribute materials or make sure each student has brought the necessary materials for the activity.

• Present the directions using the overhead transparency and answer any questions about the directions. To calculate the areas of the various tangram pieces, student may want to measure lengths of sides and altitudes, or they could use grid paper of an appropriate size by tracing the pieces onto the grid and calculating the needed area.

• Allow students time to complete the activity before sharing the results.

• As results are shared, ask if everyone got the same answers. Some students will use different units of measure in calculating the areas of the tangram figures, does this affect their calculations of percentage of area for each tangram piece?

While you have the tangram pieces available, discuss other ways in which tangrams can be used. What kinds of questions can be asked? What kind of figures can be constructed using the pieces? This and other activities by Edna F. Bazik are included in "Projects to Enrich School Mathematics: Level 2" (NCTM, 1988, p.42-50).

3. Divide the class into small groups and send them outside of the classroom to find ratios.

For example: ratio of open doors to closed doors in the hallway, ratio of sneakers to dress shoes, ratio of cars to bikes, or ratio of eye color. Have supplies available for the students to construct visual representations of the data gathered. Share the creative ratios found by the class and the chosen representations. .S o n s

Instructor’s Resource Guide, Chapter 13 Sandi Cooper |5 4. Have students use 10 by 10 grid paper to work with concepts related to percent. Have them construct a design that uses 73% or some other chosen percentage of the square.

Have the group brainstorm other activities that could use the grid representation for percent.

5. Collect maps of different states, countries, and cities and rulers for use in this activity.

Calculating the distance traveled on the map involves the use of ratios. Have each pair of students share a map and pose different questions. For example: From your chosen starting point, travel 150 miles North. Where are you? Another activity would allow you to set expense limits for a trip to be planned by the student pairs. In solving the problem students would have to consider distance, cost of gasoline, food, lodging, and other travel expenses. Additional questions might include, What percent of your budget was spent on food? on travel?

6. Teaching Arithmetic: Lessons for Decimals and Percents, NCTM's Addenda books for the Middle Grades, Understanding Rational Numbers and Proportions, and Developing Number Sense in the Middle Grades, provide activities for developing the concepts of ratio, proportion, and percent. Have students work together to review these books and present an activity to the class.

7. Marilyn Burns offers several activities which develop the concept of ratio and percent using Cuisenaire Rods and other materials. For examples, see her book, A Collection of Math Lessons From Grades 6-8, and the videos, Mathematics: for Middle School, Part 1 and Part 3. All are available from Cuisenaire Company. Allow students to view the video segments, then actively experience the activities themselves.

8. Bring in an example of materials which can be used to develop a replacement unit for ratio/proportion/percent. Have students discuss how they would organize the unit. For example, Similarity and Equivalent Fractions, from the Middle Grades Math Project, or Ratio, Proportion, and Scaling from the Mathematics Resource Project.

9. The Math Links in this chapter provide students with additional electronic manipulatives and video clips. In class, or as an outside assignment, have students explore these web sites. The National Library of Virtual Manipulatives has an applet for percentages (in 3-5 and 6-8).

10. Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

11. Using the References listed at the end of the chapter, have students locate additional information on topics which interest them such as proportional reasoning. Divide students into groups of 4. Have each person in the group read and report on one article.

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Instructor’s Resource Guide, Chapter 13 Sandi Cooper |6 Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 13: 1. Interviewing the Teacher and Students: Proportions, Percents (found in Chapter 3) 2. Helping Children Learn with Games: Make a Match (found in Chapter 4) 3. Helping Children Learn with In the Classroom Lessons: Know your Coins (found in Chapter 6) Additional Resources A Collection of Math Lessons from Grades 6-8 by Marilyn Burns and Cathy Humphreys provides several lessons concerning ratio, proportion, and percent. Available from Cuisenaire Company.

Constructing Fractions, Decimals, and Percents by Catherine Twomey Fosnot and Maarten Dolk focuses on how students in grades 5-8 construct fraction, decimal, and percent knowledge.

Developing Number Sense in the Middle Grades is one of the NCTM Addenda Books for the Middle Grades. Includes activities for grades 5-8. Available from NCTM. Making Sense of Fractions, Ratios, and Proportions, edited by Bonnie Litwiller, is NCTM’s 2002 Yearbook. Available from NCTM. Mathematics Teaching Cases: Fractions, Decimals, Ratios, & Percents edited by Carne Barnett, Donna Goldenstein, and Babette Jackson provides cases describing experiences of elementary and middle school teachers. Available from Heinemann.

Proportional Reasoning is the focus issue for Mathematics Teaching in the Middle School, April 2003. Available from NCTM. Teaching Arithmetic is a series of books providing detailed lessons for a particular grade level.

Also included are classroom vignettes and samples of student work. The title appropriate for chapter 13 is Lessons for Decimals and Percents. Available from Math Solutions.

Teaching Fractions and Ratios for Understanding by Susan Lamon, provides content knowledge and instructional strategies for teachers. Available from Lawrence Erlbaum.

Understanding Rational Numbers and Proportions by Curcio and Bezuk is one of the NCTM Addenda Books for the Middle Grades. Includes activities for grades 5-8. Available from NCTM. .S o n s

Instructor’s Resource Guide, Chapter 13 Sandi Cooper |7 Videos Mathematics: for Middle School, by Marilyn Burns are available from Math Solutions at www.mathsolutions.com. These 20-minute videos show problem solving lessons with students in grades 6-8. Of particular interest for this chapter is Part 1: Ratio and Proportion with Cuisenaire Rods, and Part 3: Percents: Sense or Nonsense.

Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 videos and 3, 5-8 videos.

Videos include lessons illustrating Standards 1-4. Each video contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tape Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org. Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples Web Resources National Library of Virtual Manipulatives for Interactive Mathematics (http://nlvm.usu.edu) provide an online applets for percentages.

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Instructor’s Resource Guide, Chapter 14 Sandi Cooper|1

Chapter 14—Algebraic Thinking What This Chapter Is About Using the mathematics curriculum to develop algebraic thinking in the elementary school is introduced. Problems, patterns, and relations can all be used to build algebraic thinking.

Repeating and growing patterns are defined and explored. Relations including properties of numbers, functions, and equality are discussed. Using the language and symbols of algebra for representation and analysis are demonstrated. Relationships, including equality and inequality, variables, and expressions and equations are each reviewed. Finally, modeling, generalizing, and justifying mathematical relationships are reviewed.

Student Objectives After reading the chapter, the students will be able to: 1. Define the role of algebra in the elementary school and summarize how algebraic thinking can be developed through problems, patterns, and relations.

2. Identify and construct repeating and growing patterns as well as provide examples of activities and models that promote their development.

3. Summarize how the relations of properties of numbers, functions, and equality can be used to develop algebraic thinking.

4. Summarize how the language and symbols of algebra can be learned.

5. Summarize how children can use the processes of modeling, generalizing, and justifying.

Key Vocabulary The role of pattern and relationship in mathematics is an important one. The key terms provided are central to discussion in these areas. algebraic thinking repeating pattern growing pattern routine problems nonroutine problems core repeating element recursive expression generalizations .S o n s

relationships representations variables odds and evens prime factorization equivalence equality inequality number theory

number properties change formulas functions modeling justifying explicit equation equation expression

reversibility

Instructor’s Resource Guide, Chapter 14 Sandi Cooper|2 Supplemental Lecture Topics 1. Use PPT Slides #4-7 to highlight the key recommendations related to patterns and algebra in grades Pre-K-8 from NCTM's Principles and Standards. These may also be found in the textbook in Figure 14-1 and in Appendix A. 2. Highlight algebra and functions items from the 7th National Assessment of Educational Progress (NAEP). Display some of these items and their results, then discuss the implications for instruction.

3. A variety of resources are available which discuss the development of algebraic thinking in children. For example, NCTM’s Navigating Through Algebra and the 1997 focus issue of Teaching Children Mathematics. (see additional resources) Use these to provide additional information for your students.

Student Textbook Activities 1. As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2. Have students review the Chapter 14 Snapshot of a Lesson which can be viewed at www.mmmproject.org. Discuss their ideas about algebra as a result of viewing this video.

3. Activities related to patterns are discussed in Chapter 14. These include repeating and growing patterns. Have students generate examples for each that could be used with children.

4. In the Classroom 14-1, Thinking about Patterns, allows students identify, describe and extend patterns. Have students brainstorm other patterns they could use with children.

5. Discuss relations, including properties of numbers and functions. Try the problems and examples in the chapter. In the Classroom 14-3, In-Out Machine provides a function machine activity. In the Classroom 14-5, From a Rule to a Graph provides a more advanced function activity.

6. In the Classroom 14-2, Balance Me addresses the concept of equality. Discuss misconceptions children may have about this concept.

7. In the Classroom 14-4, Painting Cubes provides an example of a problem that has students model and generalize. Discuss how the use of a concrete model helps develop understanding.

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Instructor’s Resource Guide, Chapter 14 Sandi Cooper|3 8. The Book Nook for Children, found in the text, lists some children's books which may be used in the study of patterns and algebra. Encourage students to read and review these and others that may be appropriate.

Student Supplemental Activities 1. This problem allows student to use manipulatives to model a pattern. Then they are encouraged to make a generalization and then use symbols to represent the pattern.

Materials: PPT Slides #15-16, Growing Patterns, color tiles, calculators • Display the PPT Slide and ask students to investigate as directed in the problem. Have color tiles available for student use.

• Have students get in small groups and share ideas. Then have a whole class discussion on modeling, generalizing and justifying.

Use T Charts to show some of the relationships that can be found.

Square Width (W) 1 2 3 4 5 . . .

10 20 E = 2W + 1

tiles used to enlarge (E) 3 5 7 9 11

? ?

Square width (W) 1 2 3 4 5 . . .

10 20

Total tiles in square (T) 1 4 9 16 25

? ?

T = W x W or W 2 (square numbers)

2. This problem involves investigating number patterns with ninths and elevenths.

Materials: PPT Slides #13-14, Patterns in Fractions, calculators, computer with spreadsheet software (optional).

• Display PPT Slides #13-14 and ask students to investigate as directed in the problem. Have calculators available for student use.

• Do not specify how they are to work the problem or how they should proceed.

Observe the class as they begin. Do they work alone, do they choose to use the available calculators?

• Bring the class back together and discuss what they have discovered. Ask what the focus of the problem was - to divide two numbers or to look for patterns?

Does technology make sense for use in this situation? Why or why not? Ask them to think about how they approached the problem, did they assume they were to work alone? Why? Did they reach for the calculator or begin to work with pencil-and-paper? What would they expect their students to do? .S o n s

Instructor’s Resource Guide, Chapter 14 Sandi Cooper|4 • You may want to set up the problem using a computer spreadsheet and demonstrate how it becomes a simple matter to look at any denominator, see many cases on the screen, and be able to hypothesize about the existence or nonexistence of a pattern. An example of the setup, using EXCEL, is provided below. With a little experimentation, the spreadsheet programs available are easily accessed and can be used in many situations. Your students may have had experience with spreadsheets and this may be an opportunity for them to share that knowledge with the rest of the class and with you.

A

Numerator 1 =A2+1 =A3+1 =A4+1 =A5+1 =A6+1 =A7+1 =A8+1 =A9+1

1 2 3 4 5 6 7 8 9 10

B

Denominator 9 =B2 =B3 =B4 =B5 =B6 =B7 =B8 =B9

C

Decimal Equivalent =A2/B2 =A3/B3 =A4/B4 =A5/B5 =A6/B6 =A7/B7 =A8/B8 =A9/B9 =A10/B10

3. The Palindrome Problem involves the use of place value ideas within an interesting problem situation. Materials: PPT Slides #23-25, Palidromes, copies of hundred boards, colored markers or crayons, calculators (optional).

• Begin by considering word palindrome examples given on PPT Slide #23. See if the students can suggest other words which are palindromes. Challenge them to form a sentence which is a palindrome - reading the same forward and backward.

One example, Able was I ere I saw Elba.

• Transfer from word palindromes to number palindromes with the next question.

Have students provide a definition of palindrome.

• Turn to the hundred board and have students circle in one color all the palindromes found on the board. This step can be done individually or as a class group. Is there a pattern?

• Consider a number that is not a palindrome, for example 57. Can we transform 57 in some way to form a number that is a palindrome? Add 75 to 57 and the answer is 132 which is still not a palindrome. Add 231 to 132 and the result is 363 - a palindrome! We then call 57 a two-step palindrome. Mark 57 on the hundred board with a color to represent two-step palindromes.

• Try this process on various other numbers to see if there are patterns which seem to develop.

• This chapter addresses the topic of place value. Ask the students how this problem relates to this topic, how the idea of palindromes could be used in classrooms.

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Instructor’s Resource Guide, Chapter 14 Sandi Cooper|5 4. Start by having students give definitions of even and odd numbers. Have them think about the process of understanding this concept. What concrete and pictorial experiences could be used to develop understanding of even and odd numbers? Using inch grid paper and color tiles or connecting cubes, investigate the shape of even versus odd as discussed in the text. Have students discuss how this setting can be used or extended to connect to other number ideas, such as factors of numbers.

5. Have groups or pairs of students construct patterns using color tiles, centimeter dot paper, other size grids or other materials (See Appendix B). Have them prepare an example of a repeating pattern and an example of a growing pattern. This provides you with an opportunity for assessment of their understanding of the two types of patterns discussed in the text and an opportunity to discuss assessment.

6. NCTM's Addenda books, Patterns, and Patterns and Functions both provide activities for developing pattern and number theory concepts. Historical Topics for the Mathematics Classroom provides number theory activities. Lessons for Algebraic Thinking from Math Solutions provides additional lessons. Have students work together to review these books and present an activity to the class. Students could also view video lessons on patterns and relationships from Teaching Math: A Video Library such as People Patterns, All Sorts of Buttons, Products and Sums, and Valentine Exchange (grades K-4) and Building Rafts with Rods (grades 5-8).

7. The resource book, Mathematics Their Way, (see additional resources) provides many examples for how young children may be introduced to patterns. Have students demonstrate some of the activities it contains.

8. Challenge groups of students to participate in a “patterns in the real world” scavenger hunt. Develop a bulletin board to display their findings.

9. Encourage students to locate other pieces of children's literature which contribute to the study of patterns and algebraic thinking. For example, One Hundred Hungry Ants by Pinczes, introduces factors of 100, One Grain of Rice by Demi introduces the classic problem of what happens if an item is doubled each day, and Two Ways to Count to Ten

by Ruby Dee investigates factors and multiples. All are available from Cuisenaire Company. Following is an example of how literature can be incorporated into a lesson.

10.

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Literature presents an interesting way in which to introduce new mathematical topics or explore mathematical ideas in a new representation or situation. An example of the use of literature in connection with technology can be seen in the use of the book, The King's Chessboard by David Birch . In this book, a proud king grants a wise man one wish. His wish is that he receive a grain of rice on the first day, and for each following day the amount of rice is double the previous day. This is to continue for each day of the chessboard, sixty-four days. Materials: The King's Chessboard by David Birch (Dial, 1988), 8 by 8 grids for student use, masking tape to outline an 8 by 2 grid on floor, manipulatives to represent rice grains, and calculators.

Instructor’s Resource Guide, Chapter 14 Sandi Cooper|6 • Begin by sharing the book to the class.

• Consider the first 2 rows of the chessboard. Have students act out the story on an 8-by -2 grid on the floor using beans, blocks, or other manipulative materials to represent the grains of rice. What kinds of problems surface? How quickly does the wise man have over 100 grains of rice?

• Have the students fill in the amounts for the first two rows on a provided 8-by8 chessboard chart using calculators. Ask them to estimate when the wise man would have more than 500,000 grains of rice, over a million, etc.

• Let students check their estimates and experiment with the idea of doubling using calculators. Have them discover the constant feature of the calculator, if it has one.

• Ask the students if they would rather have the amount of rice given on the last day or all the rice received during the first 63 days. The memory of many fourfunction calculators will not allow the computation of 2 raised to the 64th power.

If this is the case, how do we make the decision? Once again, a pattern may need to be investigated. Compare Day 4 to the total of the first 3 days, Day 5 to total of previous 4 days, etc. Construct a table and based on that information, what decision should the wise man make?

• To close the activity, ask the students if they can suggest other ways the story could be used. Are there other mathematical topics that could be connected beyond estimation and understanding the idea of doubling?

10. The Math Links in this chapter provide students with online manipulatives. In class, or as an outside assignment, have students explore these web sites. The National Library of Virtual Manipulatives includes an entire section on Algebra for each grade level that includes online applets for Algebra Tiles and a Function Machine.

11. Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

12. Using the References listed at the end of the chapter, have students locate additional information on topics that interest them such as sources of patterns. An additional NCTM resource, Algebraic Thinking, Grades K-12, provides additional articles. Divide students into groups of 4. Have each person in the group read and report on one article.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 14: 1. Interviewing the Teacher and Students: Thinking about Patterns, Equality, Algebraic Thinking (found in Chapter 3) .S o n s

Instructor’s Resource Guide, Chapter 14 Sandi Cooper|7 2. Helping Children Learn with Games: Factor Me Out (found in Chapter 4) 3. Helping Children Learn with Technology: Divisibility Discovery, Completing Patterns (found in Chapter 5) 4. Helping Children Learn with In the Classroom Lessons: Who Am I?, Alike and Difference Trains, Hunting for Numbers, Look for Patterns, Balance Me, In Out Machine, Do You Believe That?, What Do You See in Me? (found in Chapter 6) Additional Resources The Algebraic Thinking focus issue of Teaching Children Mathematics, February 1997, provides numerous ideas on how to build a good foundation and encourage young children to think algebraically.

Algebraic Thinking, Grades K-12: Readings from NCTM’s School-Based Journals and Other Publications edited by Barbara Moses. A comprehensive collection of 59 selected articles.

Available from NCTM. Historical Topics for the Mathematics Classroom is for teachers in grades K-12 and is available from NCTM. It provides the "why" and "how" of using historical mathematics topics in today's classrooms. It also contains a reference list of additional sources.

Lessons for Algebraic Thinking, is a series of 3 books available from Math Solutions. The activities build on arithmetic instruction. Grade levels include Grades K-2, 3-5, and 6-8.

Mathematics Their Way by Mary Baratto-Lorton is a classic resource for teachers of young children and contains over 200 classroom activities using familiar objects and materials.

Available from Cuisenaire Company.

Navigating Through Algebra for grades Pk-2, 3-5, and 6-8 are available from NCTM. Patterns is one of the NCTM Addenda books for the elementary grades. It includes activities for grades K-6. Available from NCTM. Patterns and Functions is one of the NCTM Addenda books for the middle grades. It includes activities for grades 5-8. Available from NCTM. Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress, edited by Silver and Kinney, is available from NCTM. Chapter 10 is on Algebra and Functions.

Video Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 videos and 3, 5-8 videos.

Videos include lessons illustrating Standards 1-4. Each video includes 2-3, 10-15 minute clips .S o n s

Instructor’s Resource Guide, Chapter 14 Sandi Cooper|8 of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tape Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept.

TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org.

Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples Web Resources National Library of Virtual Manipulatives for Interactive Mathematics (http://nlvm.usu.edu) provide online applets for Algebra Tiles and a Function Machine.

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Instructor’s Resource Guide, Chapter 15 Sandi Cooper |1

Chapter 15— Geometry What This Chapter Is About Geometry as an important yet often neglected topic in the curriculum is discussed in this chapter.

Properties of two- and three-dimensional shapes, locations, movements, and spatial relationships, transformations, visualization and spatial reasoning are illustrated through activity ideas and suggestions which reflect learning theories studied in earlier chapters. The importance of processes, properties, relationships, and classification schemes to the study of geometry are presented. The text offers specific activities together with other resources to allow beneficial, informal experiences with geometry.

Student Objectives After reading the chapter, the students will be able to: 1. Summarize properties of two- and three-dimensional shapes as well as provide examples of activities and models which promote the development of these concepts.

2. Describe locations, movements, spatial relationships, and transformations as well as provide examples of activities and models which promote the development of these concepts.

3. Describe visualization and spatial reasoning as well as provide examples of activities and models which promote their development.

Key Vocabulary Geometry is an important topic in the elementary mathematics curriculum. Students should be able to use the terminology that is accurate for the topic.

Van Hiele model for geometry understanding Three-dimensional shape Solid Face model Edge model Two-dimensional shape Symmetry Convex Concave transformations congruence similarity .S o n s

Instructor’s Resource Guide, Chapter 15 Sandi Cooper |2 Supplemental Lecture Topics 1. Spatial reasoning is an important skill not only in geometry but also in other areas of mathematics. Chapter 10 of Mathematics for the Young Child (see additional resources) and “Developing Spatial Abilities in the Early Grades” in Teaching Children Mathematics, September 1995, includes more information about this important skill.

2. Provide additional information about the computer language, LOGO. An article by Batista is included in the Chapter 15 Selected References.

3. The 1987 NCTM Yearbook, Learning and Teaching Geometry, K-12, February 1999 issue of Teaching Children Mathematics, and March-April 1998 issue of Mathematics Teaching in the Middle School contain several chapters and articles which would provide supplemental lecture and discussion.

4. A chapter by Martin and Strutchens in Results from the Seventh Mathematics Assessment, highlights measurement items from the 7th National Assessment of Educational Progress (NAEP). Display some of these items and their results and discuss the implications for geometry instruction.

Student Textbook Activities 1. As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2. Have students try the paper folding activity in the Chapter 15 Snapshot of a Lesson.

Then have them review the lesson at www.learner.org and discuss the role of the process standards in this lesson.

3. Use PPT Slides #3-5 to highlight the key recommendations related to geometry from NCTM's Principles and Standards. These may also be found in the textbook in Figure 15-1 and in Appendix A. 4. Two Dutch educators, the van Hieles, developed a model for describing levels of geometric understanding. Provide additional information about this model. Sources of information include Chapter 10 of Mathematics for the Young Child, and in the Crowley chapter, in the NCTM 1987 Yearbook, Learning and Teaching Geometry, K-12 (see selected references). Figure 15-2 may be used to illustrate the stages children go through.

5. Several properties of shape are outlined in the chapter section on 2-D geometry. They are listed PPT Slide #15, Properties of Shape. Have the class approach the properties from the viewpoint of 3-D geometry. Are these same ideas important to the recognition of solids? Do any change? .S o n s

Instructor’s Resource Guide, Chapter 15 Sandi Cooper |3 6. There are many activities in this chapter related to the geometric processes of describing and sorting, constructing, exploring, and discovering. Have the students try some of these and analyze which level of the van Hiele model would apply.

7. Constructing 3-dimensional figures is a valuable learning activity. In the Classroom 151, Solid Mystery, 15-2,What can you Discover?, 15-3, Gumdrops and Toothpicks, and 154, Build your own Pyramid provide a variety of construction methods. Have students experiment with these construction techniques and identify other materials that can be used to construct figures.

8. In the Classroom 15-6, Less is Best and 15-8, Show my Sides include a game and a handson activity which encourage students to examine the number of sides of various figures.

Have students use pattern blocks and geoboards to complete these cards.

9. In the Classroom 15-7, How Many Lines? examines lines of symmetry. This activity can be extended to letters of the alphabet. Mirrors or Mira may also be introduced.

10. In the Classroom 15-9, How Many Degrees in a Quadrilateral allows students to examine angles. The activity may be adapted for triangles.

11. In the Classroom 15-10, Piezles reinforces parallel and perpendicular lines. Have students use pattern blocks to complete the activity.

12. In the Classroom 15-11, What’s my Altitude? provides a hands-on activity involving altitude. Have students complete the activity and discuss any insights they gained.

13. In the Classroom 15-5, Find Me, 15-12, Classify Me, 15-13, Can you Find It? and 15-15, Shape Maker provide activities for classifying and labeling shapes. Discuss the various insights that may be gained.

14. In the Classroom 15-14, Shapes on a Grid allows students to explore shapes on a coordinate grid.

15. The Book Nook for Children, found in the text lists some children's books that may be used in the study of geometry. Encourage students to read and review these and others that may be appropriate.

Student Supplemental Activities 1. PPT Slides #21-22, Geoboard Quadrilaterals, presents a problem that involves the use of geoboards to explore the possible quadrilaterals that can be formed. This type of handson exploration illustrates the importance of this informal approach as opposed to the abstract, definitional approach often encountered. This problem incorporates many of the ideas about mathematics teaching and learning discussed at earlier points in the text.

Students are placed in a nonroutine problem-solving situation, using manipulative .S o n s

Instructor’s Resource Guide, Chapter 15 Sandi Cooper |4 materials in a hands-on setting to discover a geometry concept.

Materials: PPT Slides #21-22, Geoboards, rubber bands, Geoboard recording paper (see Appendix B).

• Pose problem using the provided transparency master. Answer any questions about the problem.

• Have students begin working in pairs to construct all 16 possible figures -working from geoboard to recording paper.

• If students finish before others, ask them if the 16 figures can be classified in any manner. How? • To close the activity, have students share the figures. You may want to have an overhead transparency of geoboard recording paper to be used in presenting the figures discovered.

• Ask at what grade level this problem could be used. Do we need to use the terminology noncongruent quadrilaterals?

Note that if geoboards are not available for use, the geoboard template provided in the text (Appendix B) can easily be used for exploring the problem.

2. Use pentominoes to investigate connections between area and perimeter. Each pentomino has an area of 5 square units, do they all have the same perimeter? What about the opposite situation? Suppose you have figures which all have a perimeter of 10 units, do they cover equal areas? Can you place the pentominoes together in such a way as to form a rectangle? What is its area? Students may use one-inch grid paper (Appendix B) as a work mat and color tiles to continue their investigation.

3. Allow students to work with different types of available computer software involving geometrical topics. Small groups working together in tackling technology can increase student willingness to experiment and experience what the software available can do.

Students need to experience cooperative learning and small group learning if these are methods we would like to see used in their classrooms. For example, Tessellmania! Is available from Dale Seymour Publications.

4. The topic of tessellations brings together geometry and art. The article "Thinking Geometrically" by Frederick and Williams (chapter annotated resources) provides all the necessary information, materials, and resources to allow an in-depth tessellation project on the web or an introductory experience for your students.

5. Have students examine the various commercial manipulative materials available for geometry. Many manipulative catalogs that students may also examine are available on the web. Brainstorm geometric concepts that may be reinforced by the various materials.

6. Encourage students to locate other pieces of children's literature which contribute to the study of geometry. For example, Grandfather Tang's Story by Ann Tompert, and The Greedy Triangle by Marilyn Burns. Both are available from Cuisenaire.

7. Demonstrate how geometry can be studied through real-life contexts and be multicultural.

For example, in the Chapter 15 annotated resources, Barkley and Cruz discuss beadwork .S o n s

Instructor’s Resource Guide, Chapter 15 Sandi Cooper |5 designs, Bradley discusses making Navajo blanket designs, Lipka, et al. discuss an Eskimo example, and Zaslavsky discusses round houses and American folk art.

8. Show students the patterns found in Appendix B for hands on geometry. For example, they include geoboards, dot paper, grid paper, attribute pieces, and a tangram.

9. Encourage students to learn more about the LOGO Computer language. See articles found in the Chapter 15 annotated resources by Battista.

10. NCTM's Navigating Through Geometry and the Addenda Books, Geometry and Spatial Sense, and Geometry in the Middle Grades provide activities for developing geometry concepts. Math by All Means from Math Solutions has two books on geometry. Have students work together to review these books and present an activity to the class.

11. Have students investigate paper folding for developing geometric concepts. Boxes, Squares, and Other Things and Paper Folding for the Mathematics Class (see additional resources) provide additional ideas.

12. The Math Links in this chapter provide students with additional electronic manipulatives, lesson plans, and video clips for geometry. In class, or as an outside assignment, have students explore these web sites. The Teaching Math: A Video Library includes videos such as Thanksgiving Quilt, Pattern Blocks, Shapes from Squares, A Rocket Shape (grades K-4) and Hexominos and Building Viewpoints (grades 5-8). The National Virtual Manipulatives Library includes numerous online applets for geometry including pattern blocks, tangrams, pentominos, and geoboards.

13. Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

14. Divide students into groups of 4. Have each person in the group read and report on one article from the chapter References list related to one of the geometry issues discussed in the chapter.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 15: 1. Interviewing the Teacher and Students: Compare Shapes, Find Me, Classify Me (found in Chapter 3) 2. Helping Children Learn with Games: Less is Best, Solid Shake (found in Chapter 4) 3. Helping Children Learn with Technology: Make me Different (found in Chapter 5) .S o n s

Instructor’s Resource Guide, Chapter 15 Sandi Cooper |6 4. Helping Children Learn with In the Classroom Lessons: Different Kinds of Four-Sided Figures, Rectangles and More Rectangles, Solid Activities, Gumdrops and Toothpicks, Show My Sides (found in Chapter 6) Additional Resources Geometry is a focus issue of Mathematics Teaching in the Middle School March-April 1998.

Geometry and Geometric Thinking is a focus issue of Teaching Children Mathematics, February 1999.

Geometry and Spatial Sense is one of the NCTM Addenda Books for the elementary grades. It includes activities for grades K-6.

Geometry in the Middle Grades is one of the NCTM Addenda Books for the middle grades. It includes activities for grades 5-8.

Math by All Means, A series of teaching units by Math Solutions include classroom tested lessons, problem solving activities incorporating manipulatives and children’s literature. The units that apply to chapter 15 are Geometry: Grades 1-2 and Grades 3-4.

Mathematics for the Young Child, (1990) edited by Joseph Payne, is available from NCTM. Chapter 10 discusses geometry including spatial sense and the Van Hiele levels of geometric understanding.

Navigating through Geometry is available from NCTM for grades PK-2, 3-5, and 6-8.

Paper Folding for the Mathematics Class (1957) by Donovan Johnson shows how to fold basic constructions and explores geometric concepts. Reissued by NCTM. Videos Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 videos and 3, 5-8 videos.

Videos include lessons illustrating Standards 1-4. Each video contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tapes available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org. Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples Web Resources National Library of Virtual Manipulatives for Interactive Mathematics (http://nlvm.usu.edu) provide online applets for pattern blocks, tangrams, geoboards, attribute blocks and much more. .S o n s

Instructor’s Resource Guide, Chapter 16 Sandi Cooper |1

Chapter 16 — Measurement What This Chapter Is About Measurement is an integral part of the elementary mathematics curriculum. Measurement is present in everyday life, helpful in learning other mathematical topics, useful in other areas of the curriculum, involves students, and can be approached through problem solving. Emphasized are the importance of identifying the attribute, choosing a unit of measurement, comparing the object to the unit, determining the number of units, and reporting the result. Issues related to each step, the role of formulas and measuring instruments are discussed. Connections between attributes, equivalencies, conversions, and measurement estimates are also addressed. Many suggestions and activities for including measurement in mathematics class are provided.

Student Objectives After reading the chapter, the students will be able to: 1. Summarize five reasons for teaching measurement.

2. Name the steps of measuring.

3. Define the attributes of measurement including length, capacity, weight, area, volume, angle, temperature, and time as well as provide examples of activities which promote their development.

4. Summarize the concepts related to measurement units.

5. Demonstrate measuring in units including standard units typically taught in particular elementary grades.

6. Demonstrate the use of measuring instruments as well as provide examples of activities which promote their development.

7. Demonstrate the use of measurement formulas as well as activities which promote their development.

8. Identify measurement equivalencies, conversions, estimation, and connections between attributes as well as provide examples of activities which promote their development.

Key Vocabulary Measuring is an important topic in the mathematics curriculum and is connected with many other topics of study within the subject. The terms below should be familiar to students and be used appropriately when discussing measurement ideas. measurement area conversion time conservation of area arbitrary unit equivalence capacity angle iteration length volume weight unitizing perimeter temperature chunking compare to a referent .S o n s

Instructor’s Resource Guide, Chapter 16 Sandi Cooper |2 Supplemental Lecture Topics 1. Use PPT Slide #3, The Measurement Standard, to highlight the key recommendations related to measurement from NCTM's Principles and Standards. These may also be found in the textbook in Figure 16-2 and in Appendix A. 2. A chapter by Martin and Strutchens in Results from the Seventh Mathematics Assessment, highlights measurement items from the 7th National Assessment of Educational Progress (NAEP). Display some of these items and their results and discuss the implications for measurement instruction.

3. In the April 2004 issue of Mathematics Teaching in the Middle School are a variety of articles about measurement. Young Mathematicians at Work provides professional development information on working with arrays. Use this information for supplemental lecture.

4. Chapter 11 of the NCTM book, Research Ideas for the Classroom: Early Childhood Mathematics, edited by Robert J. Jensen, highlights research results and recommendations concerning the teaching of measurement. Included is a discussion of Piaget's work with children's concepts of measurement and difficulties children have with measurement. Use this chapter to point out the importance of considering children's developmental levels when planning instruction.

Student Textbook Activities 1. As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2. Have students review the Chapter 16 Snapshot of a Lesson which may be viewed at www.learner.org. Have students identify the measurement attributes, tools, and units used in the segment. PPT Slide #21, Elementary Measurement, highlights examples of each.

3. Four reasons for including measurement in the elementary mathematics curriculum are discussed in the text. These reasons are outlined PPT Slide #4, Why Teach Measurement?. Divide students into four groups. Have each group focus on one of the reasons and brainstorm activities which would illustrate the reason. Conclude with a whole class discussion and sharing of the brainstormed activities.

4. PPT Slide #5, How to Teach Measurement, includes recommendations from a measurement study by Wilson and Osborne (1988). Share these recommendations and contrast them with textbooks' measurement lessons. Discuss how a textbook lesson can be supplemented or adapted. .S o n s

Instructor’s Resource Guide, Chapter 16 Sandi Cooper |3 5. PPT Slides #6-7, The Measurement Process, outlines the steps of the measurement process. Choose an attribute, such as length, and demonstrate how these steps may be carried out. For example, have groups of students use a ruler to measure their table or desk. Then have them select a nonstandard unit and measure the same table. Discuss their results and compare with the measurement steps from the text.

6. PPT Slides #8-15 include descriptions of the eight measurement attributes commonly taught in elementary school and these are discussed in Chapter 16. Highlight each attribute and refer students to Table 16-1 so they may see for each attribute which units are introduced in particular elementary grade levels.

7. A list of the concepts related to units that must be developed in learners over time is provided in the chapter. As the list is discussed, have students suggest activities that would help children in each area. Point out the activities provided in the text for further support of these concepts.

8. One of the concluding topics in Chapter 16 concerns estimating measurements. Begin with a review of Figure 16-1 which compares measuring and rounding. Then introduce three strategies for estimating measurements using PPT Slide #24, Estimating Measurements. For each strategy, engage the class in an estimation activity which illustrates the strategy. For example, to compare to a referent, have students use their known height to estimate the height of other students in the class. For chunking, have students use a known number of miles between two area cities to estimate the distance between the first city and a third city beyond the second city. For unitizing, have students measure a string that they estimate is the length of a desk or table in the classroom. Then have students cut a string that is four "tables" long.

9. In the text, three types of comparisons are discussed. In the Classroom 16-1, Perceptual Comparison of Lengths, 16-2, Direct Comparison to Length, and 16-3, Indirect Comparison of Length each illustrate one of the comparisons. After students have examined these three activities, have them develop one or two other activities for each type of measurement comparison.

10. In the Classroom 16-5, Are we the Same Size?, 16-8, Breaking up is Not Hard to Do, 169, Solving Conversions Problems with Square Units, and 16-11, With my Area, I can change my Shape! all develop the attribute of area. Figure 16-9 also shows a strategy for developing the area formula. In the Classroom 16-11, With my Area, I Can Change my Shape, is similar to 16-12, Same Volume, Different Shape except the second one is volume. Have students compare these activities and discuss the benefits of each.

11. In the Classroom 16-6, Measuring Length with Arbitrary Units develops the concept of arbitrary units. Have students develop additional activities which would support this development.

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Instructor’s Resource Guide, Chapter 16 Sandi Cooper |4 12. In the Classroom 16-7, Make Your Own shows how students can make their own instruments for measuring capacity. Have students brainstorm how homemade instruments for other attributes such as length or weight can be constructed.

13. Figure 16-8 provides examples of elapsed time problems. Discuss potential difficulties students might have in solving these problems. Then have students develop additional problems related to telling time.

14. In the Classroom 16-9, Solving Conversions Problems with Square Units presents the concept related to equivalence and conversions. Have students examine this activity and discuss other activities which could develop additional equivalence and conversion concepts.

15. In the Classroom 16-10, Ideas for Estimation presents seven ideas for developing measurement estimates. Have students try several of the activities, then discuss the benefits of encouraging estimation while measuring. Summarize the recommendations concerning estimating measurements.

16. In the Classroom 16-4, Comparing Height and Circumference, 16-13, Do You Know How to Connect Perimeter and Area? and 16-14, What is the Connection between Volume and Area? each show how measurement attributes can be connected. Have students try one of the activities and report what they discovered. Encourage students to find other connections.

17. The Book Nook for Children, found in the text provides a list of children's literature which may be used with the study of measurement. Read some of these books and have students develop measurement lessons to go with particular titles. Discuss how children's literature provides a context for solving mathematical problems.

Student Supplemental Activities 1. PPT Slides #26-27, Paint Problem presents a problem involving the measurement of volume. Students construct the figure using manipulative materials and are asked to consider how they arrived at their solution. This provides another chance for students to experience learning of the type to be used in elementary classrooms. Materials: PPT Slide #26 - Paint Problem, sets of Cuisenaire rods.

• Distribute the sets of rods so that each student has at least one of each size.

This may be an excellent opportunity to review the important points about distribution and storage of manipulative materials addressed in Chapter 3.

• Share the directions for the problem using the PPT Slide.

• Answer any questions about the construction of the rod tower. Encourage students to think about how they are solving the problem as they work toward a solution.

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Instructor’s Resource Guide, Chapter 16 Sandi Cooper |5 • After students have worked and seem to have reached solutions, have students volunteer their solution methods. If no one presents the method provided on PPT Slide #27, share that as another possible approach.

• Encourage students to suggest other problems of this sort that could be posed using the manipulative materials provided.

2. PPT Slides #28-29, Calendars, provides an overhead to go with the following Connections to History information. Mathematics is created by humans. Our calendar evolved over many years and reflects changes initiated by different people. This example highlights how Roman superstition about numbers affects each of us today. Making "connections" between cultures and mathematics is interesting and often encourages students to engage in further study to learn more about these phenomena. PPT Slides #28-29 can be used to guide the discussion about the history of time measurement.

Connections to History: The first calendar was established in 46 BC by Julius Caesar.

The odd numbered months had 31 days and the even numbered months had 30 days.

February had 29 days in normal years and 30 days every fourth year. Augustus, an adopted son of Julius Caesar and the first Roman emperor named the eighth month after himself, but with 30 days it was considered unlucky. He wanted "his month" to be lucky, so he solved the problem by taking a day from February and giving it to August. This action created a calendar with three consecutive 31 day months. Now he was concerned about having three consecutive lucky months, so he decreed that the months September through December be changed to 30, 31, 30 and 31 days, respectively.

As you discuss the first calendar, several "almost" patterns are suggested. The odd numbered months (1st, 3rd, 5th, etc.) had 31 days and the even numbered months (4th, 6th, etc.) had 30 days. The second month was devoted to the devil and therefore had only 29 days. Our calendar has undergone many changes over the years. Here is one interesting series of changes that uses patterns as well as odd and even numbers. Are you superstitious? Do you think 13 is unlucky? Do you have a lucky number? In the Roman times, months with even numbers of days were believed to be unlucky. Months with an odd number of days were thought to bring good luck.

3. Measurement is a topic that is closely related to the science subject area. It might be interesting to work together with a science methods instructor to devise an activity that would involve both subject areas and actively involve students in the integrated experience.

4. Have students examine elementary textbooks and identify grade levels when particular units are introduced. Compare these findings with Table 16-1. In addition have students examine the presentation of metric versus customary units.

5. Many measurement instruments can be homemade. Not only are they inexpensive, but the students often gain understanding of the units by making the instruments. Share ideas for homemade instruments with students. Ideas may be found in the sources listed below, encourage students to think of others. For example: 1-inch ceramic tiles are inexpensive .S o n s

Instructor’s Resource Guide, Chapter 16 Sandi Cooper |6 from carpet and tile stores. Measuring tapes may be cut from a vinyl table cloth. Lumber yards sell four foot strips of wood lath very inexpensively. Soft drink cups from fast food cups often come in cup, pint, etc. sizes. In the text, Appendix B also contains masters that may be used with measurement activities. Here are some classics: • Shaw, J.(1983). Let's Do It Student-made Measuring Tools. Arithmetic Teacher, November.

• Thompson, C. & Van de Walle, J. (1985). Let's Do It Learning About Rulers and Measuring. Arithmetic Teacher. April.

• Stenmark, J., Thompson, V. & Cossey, R. (1986). Family Math. Berkley, CA: Lawrence Hall of Science. (available from Dale Seymour Publications) 6. Have students identify how manipulatives such as base ten blocks, Cuisenaire rods, color tiles, connecting cubes, and links may be used for measurement activities.

7. Have students bring other children's books which may be used when studying measurement. For example, The Grouchy Ladybug, by Carle (time), Thunder Cake by Polacco (time and recipes), Cloudy with a Chance of Meatballs by Barrett (temperature, weather), When This Box is Full by Lillie (months of the year).

8. Bring in sample measurement replacement units for students to examine and use. Mouse and Elephant: Measuring Growth, from the Middle Grades Mathematics Project and Measuring: From Paces to Feet, from the Used Numbers series, are both available from Cuisenaire. Area and Perimeter is available from Math Solutions. See additional resources.

9. Show and discuss video clips (additional resources) which depict children doing measurement activities. Teaching Math: A Video Library includes videos such as Windows, Dinos, and Ants, How Long is a Minute?, Balloon Travel, Meter Cords, and Pencil Box Staining (grades K-4) and The Largest Container (grades 5-8).

10. NCTM's Navigating through Measurement, and the Addenda book for the middle grades, Measurement in the Middle Grades, provides activities which reflect the Standards recommendations. Have students examine the book and try some of the activities it contains.

11. The Math Links in this chapter provide students with additional electronic manipulatives, lesson plans, and video clips for measurement. In class, or as an outside assignment, have students explore these web sites. The National Library of Virtual Manipulatives includes online applets for clocks, converting units, and geoboards.

12. Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

13. Using the References listed at the end of the chapter, have students locate additional information on measurement topics which interest them. .S o n s

Instructor’s Resource Guide, Chapter 16 Sandi Cooper |7 14. Divide students into groups of 4. Have each person in the group read and report on one article from the April 2004 focus issue of Mathematics Teaching in the Middle School.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 16: 1. Learning about the School and Its Resources: Measurement Tools (found in Chapter 1) 2. Observing the Teacher and Students: Focusing on Measurement (found in Chapter 2) 3. Interviewing the Teacher and Students: Snake and Strip Unit Patterns, Measuring (found in Chapter 3) 4. Helping Children Learn with Games: Race to a Yard (or Meter) (found in Chapter 4) 5. Helping Children Learn with In the Classroom Lessons: Measuring Length with Arbitrary Units, Comparing Height and Circumference, How are we Alike? Different?, Make Two of Me!, What is the Connection Between Volume and Surface Area?, What is the Connection Between Perimeter and Height? (found in Chapter 6) Additional Resources April 2004 Focus Issue of Mathematics Teaching in the Middle School, is “Measurement”.

Math by All Means, A series of teaching units by Math Solutions include classroom tested lessons, problem solving activities incorporating manipulatives and children’s literature. The unit that applies to Chapter 16 is Area and Perimeter for grades 5-6.

Measurement in the Middle Grades is one of the NCTM Addenda Books for the Middle Grades.

It contains activities for grades 5-8.

Measuring: From Paces to Feet, a replacement unit from the Used Numbers series, has students measure and investigate characteristics of themselves and the classroom environment. Students are asked to collect, display, and interpret data using different scales of measurement. Available from Cuisenaire Company.

Mouse and Elephant: Measuring Growth, a replacement unit from the Middle Grades Mathematics Project, develops the concepts of area, perimeter, volume, and surface area through a problem-solving project. Available from Cuisenaire Company.

The series, Navigating through Measurement, from NCTM provides student measurement activities. There is a book for Prekindergarten-Grade 2, Grades 3-5, and grades 6-8. .S o n s

Instructor’s Resource Guide, Chapter 16 Sandi Cooper |8 Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress, edited by Silver and Kinney, is available from NCTM. Chapter 8 is on Geometry and Measurement.

Young Mathematicians at Work is a professional development series for teachers. Included are workshop materials and a CD with video clips. The title that applies to chapter 16 includes Working with the Array for grade 4. Available from Heinemann.

Videos Mathematics: With Manipulatives, Six Models by Marilyn Burns is available from Cuisenaire Company and Math Solutions (www.mathsolutions.com). The 20-minute video shows a variety of manipulative materials being used by children. The last two segments show lessons involving volume and surface area.

Mathematics: for Middle School, Parts 1-3 by Marilyn Burns are available from Cuisenaire Company and Math Solutions (www.mathsolutions.com). The 20-minute videos show problem solving lessons with students in grades 6-8. Of particular interest for this chapter: Part 1: segments on angles and the relationship between area and perimeter, Part 2: area and perimeter relationships.

Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 videos and 3, 5-8 videos.

Videos include lessons illustrating Standards 1-4. Each video contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tape Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept. TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org. Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples Web Resources National Library of Virtual Manipulatives for Interactive Mathematics (http://nlvm.usu.edu) provide online applets for clocks, converting units, and geoboards.

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Instructor’s Resource Guide, Chapter 17 Sandi Cooper |1

Chapter 17 — Data Analysis, Statistics, and Probability What This Chapter Is About This chapter addresses the topic of data analysis and the growing importance of the topic within the elementary school mathematics curriculum. Reasons for teaching statistics and probability are presented. Data analysis provides opportunities for connection with other mathematical topics such as fractions, decimals, ratios, and critical thinking. The point is made that this topic presents an opening to stimulate children's learning and a way to link school mathematics to real world mathematics. The three steps of data analysis, graphs, descriptive statistics and plots, are introduced as well as probability concepts.

Student Objectives After reading the chapter, the students will be able to: 1. Summarize why it is important to teach statistics.

2. Summarize the three steps of data analysis.

3. Identify, describe, and construct graphs and plots introduced in elementary school.

4. Summarize key ideas about data sense and data analysis.

5. Define descriptive statistics as well as provide examples of activities and models which promote their development.

6. Define probability concepts as well as provide examples of activities and models that promote their development.

Key Vocabulary Data analysis is an important topic in current elementary school mathematics programs.

Students should be comfortable with the terminology of the areas of probability and statistics. line graph stem-and-leaf plot mode picture graph box plot real graph bar graph probability of an event histogram pie graph event or outcome line plot descriptive statistics sample space data sense average randomness survey mean independence of events population median conditional probability sample range

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Instructor’s Resource Guide, Chapter 17 Sandi Cooper |2 Supplemental Lecture Topics 1. Use PPT Slide #2 to highlight the key recommendations related to statistics and probability from NCTM's Principles and Standards. These may also be found in the textbook in Appendix A. 2. Highlight statistics and probability items from the 7th National Assessment of Educational Progress (NAEP). Display some of these items and their results and discuss the implications for statistics and probability instruction.

3. Discuss research involving probability and statistics. For example, Reflections on Statistics, the chapter, "Research in Probability and Statistics: Reflections and Directions", by Shaughnessy, may be found in the Handbook of Research on Mathematics Teaching and Learning, edited by Douglas Grouws and available from NCTM. "Measurement, Probability, Statistics and Graphing" by Bright and Hoeffner may be found in Research Ideas for the Classroom: Middle Grades Mathematics edited

by Owens. (Available from NCTM.) Reflections on Statistics: Learning, Teaching, and Assessment, edited by Susanne Lajoie is available from Lawrence Erlbaum.

4. The February 1996 issue of Teaching Children Mathematics and the March 1999 issue of Mathematics Teaching in the Middle School are on data exploration. The book, Investigating Real Data in the Classroom, also provides important insight into classrooms. Use these resources to provide additional information for students.

Student Textbook Activities 1. As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #3. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2. Have students review the Chapter 17 Snapshot of a Lesson. It may be viewed at www.mmmproject.org. Discuss which reasons for teaching statistics and probability on are being illustrated.

3. There are several reasons why statistics and probability topics should be part of the elementary mathematics curriculum. PPT Slide #4, Why Teach Statistics and Probability? highlights the four reasons discussed in the text. Space has been provided so activities brainstormed by students for each reason may be listed. In addition, encourage students to develop other reasons for studying statistics and probability.

4. Chapter 17 lists the nine types of graphs and plots typically used by elementary school students. Divide students into nine groups. Each group should review the text concerning that type of graph and report to the rest of the class. Encourage students to find examples of each of the nine types in elementary textbooks and display them on a class bulletin board. .S o n s

Instructor’s Resource Guide, Chapter 17 Sandi Cooper |3 5. PPT Slide #5, Three Steps of Data Analysis, provides the three steps of data analysis discussed in the text. Choose a question or problem to be answered (In the Classroom 17-1 is a good source for ideas) to model the process which is suggested. Encourage students to bring in their own examples.

6. Review the concept of descriptive statistics with students. PPT Slide #18, Analyzing Data: Descriptive Statistics presents a brief definition for the statistics typically introduced in the elementary school. Have students use a concrete material to model each definition. Space has been provided so that students may add real-life problems that elementary students might solve involving each of the descriptive statistics.

7. There are several important terms related to the concept of probability. The terms discussed in Chapter 17 are listed on PPT Slide #22, Probability. Use some simple probability experiments with dice or spinners to demonstrate each term. Have students design additional experiments and use the terms to describe their experiment.

8. In the Classroom 17-1, Let’s Find Out and 17-2, Sharing Data show how children can collect and display data to answer real-life problems or use real-life data to answer questions. Have students locate or make up additional problems which would interest elementary children.

9. In the Classroom 17-3, Peanuts and 17-4, What’s the Average? give experiences with descriptive statistics. Have students work through these activities and discuss their usefulness with children.

10. In the Classroom 17-5, What are the Chances?, 17-6, What’s more Likely?, 17-7, Rolling and Recording, 17-8, Are you a Winner?, and 17-9, Can you make Predictions? introduce a variety of probability activities for children. Have students examine and compare the activities. Discuss how they might be included in a probability unit. Have students select a particular activity, carry it out, and present findings to the class.

11. Have students collect a set of data. Then have them examine the graphs presented in Figure 17-17. Have various groups display the data using the variety of graphs. Discuss how it is sometimes useful to display a set of data in more than one way. Students can experiment with taking one type of graph, such as a bar graph, and changing the scale to make the display look different.

12. Have students review the averages-mean, median, and mode in the text. Then have students use some of the concrete materials described (blocks, paper strips, cards, etc.) to model these averages. Discuss other materials that could be used to model the averages concretely.

13. The Book Nook for Children, found in the text lists some children's books which may be used in the study of probability and statistics. Encourage students to read and review these and others that may be appropriate. .S o n s

Instructor’s Resource Guide, Chapter 17 Sandi Cooper |4 Student Supplemental Activities 1. PPT Slides #29-31, Mystery Bags, presents a problem which involves the use of manipulatives to introduce the basic ideas of probability. Students try to predict the contents of the bag based on samples taken from the container. The problem provides an opportunity for students to participate in a probability experiment. As a part of an activity of this type, there is opportunity to introduce the correct terminology and connect it to a concrete example. Students are involved in trying to predict the contents based on random samples taken of the contents. Materials: PPT Slides #29-31, color tiles, paper lunch bags.

• Have each student prepare a mystery tile bag as outlined on the overhead.

• Have students deposit their bag into a large box or other container. Distribute the bags randomly so that each student receives a bag.

• Review the remaining directions on the overhead for the sampling procedures and questions of interest. Answer any questions about the process.

• Allow time for students to conduct their sampling experiment and record the results. To make the recording process easier, you may want to construct a worksheet with a table to organize their results as well as additional questions which may be of interest.

• Have students open their bag of tiles and compare their results with the actual color totals contained in the bag. Discuss the results. How many were correct in their hypothesis? Why did some results not match the actual situation? How do we improve our guesses of color distribution? The discussion can then turn to the ideas of probability based in this situation. Given the actual contents of the bag, what is the probability the tile drawn would be red? What was the probability based on your experimentation?

This situation can then be used to discuss terms, such as sample space, randomness, and equally likely events. Have students think about the range of activities that can be provided to help children at all levels learn probability and statistics concepts.

2. Most students will not be familiar with the box-and-whisker plots and the stem-and-leaf plots discussed in Chapter 17. Provide them with experience in what these representations are and how to construct each type. This is an opportunity for the students to gather data in an area of interest and participate in forming an appropriate representation. The use of technology can be integrated into the discussion through access to a piece of elementary graphing software, for example The Graph Club by Tom Snyder Productions, that allow students to easily experiment with a variety of data representations.

3. Have students consider the question of the average number of chocolate chips in a given brand of cookies, or a comparison of chips contained in several brands. This same question could be asked using snack-size bags or boxes of raisins, M&M's, conversation hearts, candy corn or other material to fit the season or interests of the student group.

Other interesting questions which involve the students in actively gathering data and determining the method of analysis and interpreting their results can be developed. .S o n s

Instructor’s Resource Guide, Chapter 17 Sandi Cooper |5 4. Have students bring in examples of data use from current publications: newspapers, magazines, or books. Divide the class into small groups to share their samples of data presentations. After each student in the group has share his/her contribution, have the group develop an activity for classroom use based on one of the examples shared. It might be challenging to have the students develop two activities based on the same sample, one appropriate for lower grades and one appropriate for upper grades. Have each small group share their ideas through presentations to the larger group.

5. NCTM's Navigating through Data Analysis and Probability for grades Pk-2, 3-5, and 6-8 and Addenda books, Making Sense of Data and Dealing with Data and Chance provide activities for developing the concepts of statistics and probability. Have students work together to review these books and present an activity to the class.

6. Several textbook replacement units are available for teaching statistics and probability.

For example Used Numbers by Susan Jo Russell, Rebecca Corwin, and Susan Friel, Math

By All Means: Probability, Grades 1-2 and Grades 3-4 by Marilyn Burns, Investigations in number, Data, and Space developed by TERC, and The Middle Grades Mathematics Project: Probability by Phillips, Lappan, Winter, Fitzgerald, and Shroyer. All are available from Cuisenaire Company. Have small groups of students examine these units and report on them to the class. Compare the presentation of probability and statistics in these units with those found in elementary textbooks.

7. Teacher resource materials for data and graphing are available from several other sources. The United States Census Bureau publishes free materials for teachers. USA Today newspaper has teacher resource materials available with the purchase of classroom subscriptions. See additional resources for addresses. Make these resources available for students to review.

8. The concept of fairness can be developed through games. The chapter by Bright, Harvey and Wheeler in the Chapter 17 Selected References provides ideas for using games to develop the concept.

9. The computer is a valuable tool when studying probability and statistics. It can be used to develop various types of graphs and plots quickly and easily. It can also be used to simulate probability experiments and collect and organize data for example, Probability Tool Kit by Ventura Educational Systems. Have students review various software pieces which can be used with elementary students.

10. Marilyn Burns offers several activities that develop the concepts of probability. For examples, see her book, A Collection of Math Lessons From Grades 6-8, and the video, Mathematics: for Middle School, Part 1. Both are available from Cuisenaire Company.

Math Teaching: A Video Library, K-4 also offers segments on statistics and probability such as Ladybugs, Woodpecker Habitat, Bubblegum Contest, Dice Toss, and Questioning Data (grades K-4) and The Location (grades 5-8). Allow students to view the video segments, then actively experience the activities themselves. .S o n s

Instructor’s Resource Guide, Chapter 17 Sandi Cooper |6 11. The Math Links in this chapter provide students with additional electronic manipulatives and video clips for data analysis, probability, and statistics. In class, or as an outside assignment, have students explore these web sites. The National Library of Virtual Manipulatives includes online applets for a variety of graphs, spinners, and coin tossing.

12. Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

13. Using the References listed at the end of the chapter, have students locate additional information on topics that interest them such as probability simulations or graphs and plots. Divide students into groups of 4. Have each person in the group read and report on one article or chapter.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities have been designed to be used with Chapter 17: 1. Learning about the School and Its Resources: Probability Models in the Classroom (found in Chapter 1) 2. Observing the Teacher and Students: Focusing on Probability and Data (found in Chapter 2) 3. Interviewing the Teacher and Students: Graphing, Statistics, Probability (found in Chapter 3) 4. Helping Children Learn with Games: Can you Beat the Toss?, Race to 20, Skunk (found in Chapter 4) 5. Helping Children Learn with Technology: Build a Graph, Build a Spinner (found in Chapter 5) 6. Helping Children Learn with In the Classroom Lessons: How Could it Happen?, Let’s Find Out, Peanuts, What are the Chances? Rolling and Recording, Can you make Predictions? (found in Chapter 6) Additional Resources A Collection of Math Lessons from Grades 6-8 by Marilyn Burns and Cathy Humphreys provides several lessons concerning probability. Available from Cuisenaire Company.

Census Education Project Box 901390 Kansas City, MO 64191-1390 .S o n s

Instructor’s Resource Guide, Chapter 17 Sandi Cooper |7 Data Exploration Focus Issue of Teaching Children Mathematics, February 1996.

Data and Chance Focus Issue of Mathematics Teaching in the Middle School, March 1999.

Dealing with Data and Chance is one of the NCTM Addenda books for the middle grades. It includes activities for grades 5-8.

Investigating Real Data in the Classroom is edited by Richard Lehrer and Leona Schauble, is a collaboration between researchers and classroom teachers. Available from Teachers College Press.

Making Sense of Data is one of the NCTM Addenda books for the elementary grades. It includes activities for grades K-6.

Math by All Means, A series of teaching units by Math Solutions include classroom tested lessons, problem solving activities incorporating manipulatives and children’s literature. The units that apply to chapter 17 are Probability: Grades 1-2 and Probability: Grades 3-4.

Navigating through Data Analysis and Probability, grades pk-2 and 3-5, 6-8, is a series of books with meaningful activities for children. Available from NCTM. Reflections on Statistics: Learning, Teaching, and Assessment in Grades k-12 is edited by Susanne P. Lajoie and is available from Lawrence Erlbaum.

Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress, edited by Silver and Kinney, is available from NCTM. Chapter 9 is on Data and Chance.

USA Today Classline 1000 Wilson Blvd.

Arlington, VA 22229 Teaching Statistics and Probability, the 1981 NCTM Yearbook edited by Albert Shulte, provides a collection of chapters concerning the role of statistics and probability in the curriculum and provides ideas for classroom activities.

Videos Mathematics: for Middle School, Part 1, by Marilyn Burns is available from Cuisenaire Company. The 20-minute video shows problem solving lessons with students in grades 6-8. Of particular interest for this chapter are two lessons on probability.

Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 videos and 3, 5-8 videos.

Videos include lessons illustrating Standards 1-4. Each video contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tape Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 .S o n s

Instructor’s Resource Guide, Chapter 17 Sandi Cooper |8 Dept. TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org. Several of this text’s Snapshot of a Lessons originated from this collection.

Additional video clips are available on the NCTM web site: http://standards.nctm.org/document/eexamples Web Resources National Library of Virtual Manipulatives for Interactive Mathematics (http://nlvm.usu.edu) provide online applets for online applets for a variety of graphs, spinners, and coin tossing.

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Instructor’s Resource Guide, Chapter 18 Sandi Cooper |1

Chapter 18 — Number Theory What This Chapter Is About This chapter addresses the topic of number theory in the elementary school. Number theory is a branch of mathematics, mainly concerned with integers, that has been a topic of study for centuries. The number theory topics presented in this chapter are organized as only a sampling, but include those that are in most elementary mathematics textbooks.

Student Objectives After reading the chapter, the students will be able to: 1. Summarize why it is important to include number theory in elementary mathematics.

2. Identify which number theory topics are appropriate in the elementary classroom.

3. Describe how number theory complements the teaching and learning of mathematics in elementary school.

4. Describe and identify number theory topics such as odds and evens, factors and multiples, primes and composites, divisibility, polygonal and triangular numbers.

Key Vocabulary Number theory topics should be considered in elementary school mathematics programs.

Students should be comfortable with the terminology of a variety of number theory topics. odd and even numbers factors multiples prime numbers composite numbers divisibility polygonal numbers triangular numbers prime factorization Sieve of Eratosthenes modular arithmetic Pascal’s Triangle Pythagorean Triples Fibonacci Sequence Supplemental Lecture Topics 1. Using information on PPT Slides #3-4, have students discuss the definition of number theory and the importance of including number theory activities in elementary mathematics instruction.

2. Select a couple of activities from the I Hate Mathematics! or the Math for Smarty Pants books (found in additional resources) and allow students to work on possible solutions.

Discuss the importance of finding numeracy connections in real-life situations.

To accompany Helping Children Learn Math10e, Reys et al. ©2011 John Wiley & Sons

Instructor’s Resource Guide, Chapter 18 Sandi Cooper |2 3. Highlight the four reasons that number theory should be included in the elementary mathematics curriculum, as outlined in the Introduction of the chapter. Allow students to discuss each reason and propose possible instructional ideas that would promote important connections for elementary students.

Student Textbook Activities 1. As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.

2. Have students review the Snapshot of a Lesson. It may be viewed at http://pbskids.org/cyberchase/forreal/301_for_real_hi.html.

3. Using PPT Slides #5 and #13, and In the Classroom 18-1, What Do You See in Me?, allow students to explore patterns in Pascal’s Triangle using the illustration. You might organize the class into small groups to discuss the questions included in the text and then discuss the proposed ideas as a whole group.

4. Allow students to explore magic squares by organizing the activity described In the Classroom 18-4, Magic Squares. In addition, you may share historical information about Benjamin Franklin and his amusement with finding magic squares (books suggested in additional resources).

5. Have students organize in teams to play the game described In the Classroom 18-5, A Game of Odds and Evens. After playing the game, use PPT Slide #6 to discuss ideas learned from this experience.

6. Using In the Classroom 18-9, Sieve: The Primes Remain, have students generate the prime numbers from 2 to 102 using the chart provided. Allow them to discuss in small groups the questions included in the activity, then share ideas with the entire group.

7. Using PPT Slide #9, Number Theory in Elementary School Mathematics: Divisibility, discuss divisibility rules. Then, provide students with a calculator and allow them to complete the chart found with In the Classroom 18-10, Divisibility Discovery. Facilitate a class discussion about divisibility rules and how this might help elementary students develop concepts in number theory.

8. Using PPT Slide #15, Other Number Theory Topics: Fibonnaci Sequence and information from the text, provide students with an overview of the Fibonacci Sequence.

You might locate some activities in Fibonacci Fun: Fascinating Activities With Intriguing Numbers (found in additional resources) to facilitate with small groups.

To accompany Helping Children Learn Math10e, Reys et al. ©2011 John Wiley & Sons

Instructor’s Resource Guide, Chapter 18 Sandi Cooper |3 9. The Book Nook for Children, found in the text lists some children's books which may be used in the study of probability and statistics. Encourage students to read and review these and others that may be appropriate. Some books to consider include Rabbits Rabbits Everywhere: A Fibonacci Tale by Ann McCallum, The Great Number Rumble: A Story of Math in Surprising Places by Cora Lee and Gillian O'Reilly, What's Your Angle, Pythagoras? A Math Adventure by Julie Ellis and the award-winning book, Go Figure!: A Totally Cool Book About Numbers by Johnny Ball. Another interesting book to consider is Ben Franklin and the Magic Squares by Frank Murphy.

Student Supplemental Activities 1. Math Teaching: A Video Library, K-4 also offers segments on number theory. You could consider organizing the activities from the lesson for the students to experience first, then allow them to view and analyze the video lesson. Clips to consider include Beans, Beans, Beans, Animals in Yellowstone, and Ladybugs (K-4).

2. The Math Links in this chapter provide students with additional electronic manipulatives for number theory. In class, or as an outside assignment, have students explore these web sites. The National Library of Virtual Manipulatives (http://nlvm.usu.edu/) includes applets for the Sieve of Eratosthenes, Fibonacci Sequence, the Golden Ratio, Exploring Number Patterns, and Pascal’s Triangle.

3. Have students work in groups to discuss items from Things to do: From What You've Read, and Things to do: Going Beyond this Book.

4. Using the References listed at the end of the chapter, have students locate additional information on topics that interest them. Divide students into groups of 4. Have each person in the group read and report on one article or chapter.

Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences. The following activities might be used with Chapter 18: 1.

Observing the Teacher and Students: Children’s Development (found in Chapter 2)

2.

Interviewing the Teacher and Students: Early Number Sense (found in Chapter 3)

3.

Helping Children Learn with Games: Capture 5, Factor Me Out (found in Chapter 4)

4.

Helping Children Learn with Technology: Divisibility Discovery (found in Chapter 5)

5.

Helping Children Learn with In the Classroom Lessons: What Do You See in Me?, Do You Believe That? (found in Chapter 6)

To accompany Helping Children Learn Math10e, Reys et al. ©2011 John Wiley & Sons

Instructor’s Resource Guide, Chapter 18 Sandi Cooper |4 Additional Resources Fibonacci Fun: Fascinating Activities With Intriguing Numbers by Trudi Hammel Garland (Author), Rachel Gage (Illustrator) and available from Dale Seymour Publications. This resource book includes reproducible activities and projects that provide a nice introduction to Fibonacci numbers and the golden ratio.

The Elementary Math Teacher's Book of Lists: With Ready-to-Use Patterns and Worksheets by Sonia M. Helton and Stephen J. Micklo and available from Jossey-Bass. This resource book includes some helpful activities for rules of divisibility, prime factorization, making factors trees, finding the square roots.

Benjamin Franklin's Numbers: An Unsung Mathematical Odyssey by Paul C. Pasles and available from Princeton University Press. This book provides some interesting historical information about how Franklin didn't do well in math at his school, but became proficient at using numbers for his printing business. From the book you will learn how he had to be able to calculate for some of the tables in his famous almanacs and he used numbers for predictions of population statistics, but the magic square was his mathematical amusement.

Elementary Number Theory by Gareth A. Jones and Josephine M. Jones and is available from Springer. This is a great resource for number theory and includes background information on divisibility, primes and congruences and more advanced subjects like Euler's functions, quadratic residues, Riemann zeta function, with a final chapter on Fermat's Last Theorem.

On the Shoulders of Giants: New Approaches to Numeracy by Mathematical Sciences Education Board, National Research Council and Lynn Arthur Steen (Editor) and available from National Academies Press. This book includes essays that expand on the idea of mathematics as the language and science of patterns, allowing the reader to realize the importance of facilitating hands-on experiences that will enable students to apply their knowledge to diverse numerical problems.

I Hate Mathematics! by Linda Allison, Marilyn Burns, and David Weitzman and the Math for Smarty Pants by Marilyn Burns both available from amazon.com. These books includes some great mathematical problem solving puzzles that allow for making numerical connections in the real-world.

Videos Teaching Math: A Video Library, K-4 and 5-8-includes 24, K-4 videos and 3, 5-8 videos. Videos include lessons illustrating Standards 1-4. Each video contains 2-3, 10-15 minute clips of actual teachers and their students engaged in teaching and learning activities that reflect the NCTM Standards. A guidebook and questions for discussion are included. Tape Available from: www.learner.org or The Annenberg/CPB Math and Science Collection, PO Box 2345 Dept.

TMB.S, Burlington, VT 05407-2345, 1-800-864-9846 or may be viewed at www.learner.org.

Web Resources To accompany Helping Children Learn Math10e, Reys et al. ©2011 John Wiley & Sons

Instructor’s Resource Guide, Chapter 18 Sandi Cooper |5 National Library of Virtual Manipulatives for Interactive Mathematics (http://nlvm.usu.edu) provide online applets for Sieve of Eratosthenes, Fibonacci Sequence, the Golden Ratio, Exploring Number Patterns, and Pascal’s Triangle.

Additional applets are available on the NCTM web site at http://standards.nctm.org/document/eexamples. The Understanding the Pythagorean Relationship Using Interactive Figures includes an online applet that allows students to explore a dynamic demonstration of the Pythagorean relationship.

To accompany Helping Children Learn Math10e, Reys et al. ©2011 John Wiley & Sons