NCSSSMST JOURNAL
The National Consortium for Specialized Secondary Schools of Mathematics, Science and Technology
Fall 2013
Volume 18
Issue 1
www.ncsssmst.org 1
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NCSSSMST Journal | Fall 2013
Table of Contents
Consortium Board
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President’s Note: What Am I Doing About This? - Dr. Tim Gott
Carol Martin Gatton Academy of Mathematics and Science in Kentucky
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Director’s Note: Greetings from Your New Executive Director - Todd Mann
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The Social Role of Science through Henrietta Lacks - Arthur S. Williams, PH.D.
10 What’s Really Different About These Schools? - Jerald Thomas, Ed.D.
Dr. Tim Gott, President
Crystal Bonds, President Elect High School for Math, Science and Engineering at City College (NY)
Dr. Susan Caffery, LPC, Secretary Academy of Science and Technology (TX)
Hungsin Chin, Treasurer Alabama School School of Fine Arts - Russell Math and Science Center (AL)
Dr. Jerald (Jay) Thomas, Past President Aurora University (IL)
Todd S. Mann, Executive Director Washington, DC
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Don’t Blame the Fish - Dr. Craig Gaska
Amanda Baskett
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Motivating and Challenging Highachieving Students in Mathematics
Charmain Brammer SUCCESS Academy (UT)
- Dr. Jeff Wiesman
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Calculus Reshuffled - Dr. Vince Matsko
Alison Earnhart Liberal Arts and Science Academy of Austin
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Using the STAT Manual - Jane Hemelt
Dr. William F. Elliott Carnegie Mellon University (PA)
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Virtual Field Trips: Using Technology to Enhance Learning - Dr. Steve Canipe
Rockdale Magnet School for Science and Technology (GA)
Russell Davis Bergen County Academies (NJ)
Mark Godwin South Carolina Governor’s School for Science and Mathematics (SC) Bob Gregory Arkansas School for Mathematics, Science and the Arts (AR)
Cheryl Hach Kalamazoo Area Mathematics and Science Center (MI)
Rosemarie Jahoda The Bronx High School of Science (NY)
On the Cover
Christopher Kolar
Photo Credit: Fufei Chen
Illinois Mathematics and Science Academy (IL)
Fufei is a 2013 graduate of The Maine School of Science and Mathematics (MSSM). He is currently a pre-med student at University of Maine in Orono. He serves as a photographer and designer at the University of Maine’s Advanced Structures and Composites Center.
Letita Mason
Know a talented student photographer? Suggest they enter the 2013-2014 NCSSSMST Journal Student Science Photo Competition. Students should email their photo and a brief
Dr. Heather Brandon Sondel
North Carolina School of Science and Mathematics (NC)
Luke Shorty Maine School of Science and Mathematics (ME) Thomas Jefferson HS for Science and Technology (VA)
biography to ncsssmsteditor@gmail.com.
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NCSSSMST Journal is the official publication of the National Consortium for Specialized Secondary Schools of Mathematics, Science and Technology. STAFF Todd Mann Executive Director NCSSSMST Todd.mann@ncsssmst.org Amanda Baskett Interim NCSSSMST Journal Editor ncsssmsteditor@gmail.com
The NCSSSMST Journal (ISSN 1084-6522) is published twice a year. Copyright 2013 by the National Consortium of Specialized Secondary Schools of Mathematics, Science and Technology (NCSSSMST). All rights reserved. Editorial materials published herein is the property of the NCSSSMST unless otherwise noted. Opinions expressed in the NCSSSMST Journal do not necessarily reflect the official position of the NCSSSMST. Permissions: Copyrighted material from the NCSSSMST Journal may be reproduced for noncommercial purposes provided full credit acknowledgements and a copyright notice appear on the reproduction. Other requests for reprinting should be directed to the Journal Editor. Submissions: Manuscripts for feature articles and teacher practice summaries are invited. Author guidelines are found at www.ncsssmst.org>publications>journal. The NCSSSMST Journal assumes no responsibility for unsolicited manuscripts. Students’ research papers are encouraged.
PAST STAFF Dr. Thomas Morgan, Founding Editor
Website: www.ncsssmst.org
Dr. Jerald Thomas, Past Editor Dr. Arthur S. Williams, Past Editor Dr. Martin Shapiro, Past Editor Dr. Richard W. Shelly, Past Associate Editor Gary L. White, Past Co-Editor Dr. Ron Laugen, Past Editor
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Subscriptions: Individual subscription price is $50 per year US dollars and $75.00 per year for international subscriptions with postage at an additional cost. Institutional pricing is available by contacting NCSSSMST.
From the Desk of the President
What Am I Doing About This? Recently, I saw a billboard in a major city that said something about laser eye surgery, “completed in a femtosecond”. Being a math-oriented person, this caught my attention. More to the point, what was a femtosecond? My high school son, with his technology-minded approach, took out his smart phone, googled the term, and found, almost in a femtosecond, that the word means one quadrillionth of a second. Now, the message may have been hyperbole, but in that moment, my universe changed some. I realized that the world we live in is moving and operating on scales that have previously been unknown. We are accepting and interacting with technologies daily that are ground-breaking, yet many of us are completely oblivious of the mathematical and scientific foundation from which these are built. Reflecting on this, I began to assess to what degree are educational leaders aware of this dilemma and doing something about it. As a director of a specialized STEM school, I had to ask myself, “What am I doing about this?”
world.
Our Mission The mission of NCSSSMST, the nation’s alliance of secondary schools and programs preparing students for success and leadership in science, technology, engineering, and mathematics is to serve our members’ students and professionals, foster collaborations, influence STEM policy and advocate transformation in education.
NCSSSMST Purposes • Encourage, promote, and share emergent ideas and innovations in curriculum, teaching, learning, and technology. • Collaborate with our members to develop and offer special programs, activities, and resources to members’ students and professionals. • Invite affiliate membership from colleges, universities, and other organizations that support our vision and mission. • Function as a clearinghouse of resources and services to new and developing schools and programs that share our mission and vision. • Collaborate with all other organizations in order to speak with a unified voice on policy issues.
As a director of a specialized STEM school, I had to ask myself, “What am I doing about this?”
So, now as president of NCSSSMST, I feel compelled to ask this question on a broader scale. What should we be doing as a national alliance of schools that are blazing the trail in STEM education? In the past couple of years, the NCSSSMST board of directors has sought to clearly define the organizations purpose and vision. The following statements were developed and adopted as guiding documents:
Our Vision The Consortium will serve as a catalyst for transforming education by empowering students, teachers, and communities to meet the demands of a technologically advanced
Member Purposes Our members have unique responsibilities to: • develop and model transformative courses and relationships in science, technology, engineering, and mathematics; • prepare students for entrance to and success in postsecondary education; • provide student research opportunities; • promote opportunities for career exploration, internships, and competitions;
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• develop students as leaders; and • identify, recruit, and support students from underrepresented populations. One word that stands out to me in these statements is transformation. As leaders in STEM education, we bear a substantial responsibility for being the voice for the need to change. We need to be intentional in our efforts to impact the present circumstances. We have strong evidence that our schools are setting the standards for excellence in math and science curriculum and instruction. Now, we should help guide the development of new schools and programs and the reform of many existing ones. In doing so, we will play a meaningful role in the ever-evolving world around us. NCSSSMST stands ready to help create a broader and more influential community of educational leaders who can meet this challenge. As we look to move forward in this exponentially changing environment, we need to be preparing our students for femtoseconds, petaflops, nanotechnologies, and zettabytes. Together, we can help prepare the coming generations for this rapidly shifting arena. —Tim Gott
Dr. Tim Gott, Director, Gatton Academy of Mathematics and Science in Kentucky tim.gott@wku.edu
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From the Desk of the Director
Greetings From Your New Executive Director It’s a pleasure to introduce myself to you as the new Executive Director of NCSSSMST. It’s an honor to be a part of its mission to advance and transform mathematics, science, and technology education for secondary schools and their students. I’d like to tell you a little bit about my background and outline my goals for the organization over
the next year. I grew up in central Massachusetts and attended high school in the suburbs of Chicago. My undergraduate years brought me back to the East Coast to attend Vassar College, where I was in the second class of men to matriculate. I majored in Russian Studies, having spent 8 years studying the language. I deferred attending graduate school to accept a job offer on Capitol Hill, and then never looked back. I did return to academe, though, as a Trustee of Vassar. Sitting on committees for budget and finance as well as one for student admissions, provided me with a college’s administrative perspective that I trust will come in handy in my new NCSSSMST role. Rounding out my academic career were financial executive courses at Emory University’s Goizueta Business School.
work successfully in disparate industries: financial services, hospitality, healthcare, and construction. I knew little about these fields before assuming leadership positions in them, but was aided by being a fast learner and synthesizing challenges quickly. I enjoy applying best practices to nonprofit management and governance, thinking differently about how I can bring an organization unique partnerships that will benefit the membership, and applying my business acumen to an organization’s finances. Most recently, I was COO of a $24 million nonprofit trade association representing the construction industry. While there, I led the development of a strategic plan for the organization. I also directed a “refreshing” of its governance structure, which hadn’t been addressed in 15 years. The construction industry faced a big challenge: an anticipated gap of 1.5 million skilled craftspeople entering the workforce by 2015. When I learned that the highest demographic for comic book readers was 20 to 30 year-olds, I created a partnership with Marvel to customize a comic book in which newly trained craftspeople were featured as super heroes. Distribution was targeted at high schools and community colleges with strong vocational orientations. While certainly not a panacea, this vehicle represented a unique and vibrant effort to interest young people in the industry.
I hope to offload some of the tedious chores from NCSSSMST officers so that they can focus on the organization’s mission.
More personally, I have been married for almost 30 years to Susan Stockdale, an author and illustrator of children’s picture books about nature. We have two grown children, a daughter working in the advertising industry in Chicago and a son working in finance in Boston. Susan and I live just across the DC border in suburban Maryland. My professional life combines 18 years of nonprofit leadership and 17 years of business experience in the U.S. and abroad. I have developed a skill set that has enabled me to
While heading Business Development at the National Restaurant Association, the nonprofit association representing the restaurant industry, I met Ted Turner. He had just launched a chain of restaurants specializing in bison (Ted’s Montana Grill). One of the foundations he established focuses on environmental issues. My grant application to this foundation resulted in $350,000 for the first sustainability initiative in the restaurant industry. We used the funds in innovative ways, educating members on everything from how 7
to recycle grease from stove traps to using vegetable oil to fuel food delivery vans. These years of nonprofit leadership have prepared me for the ED role of NCSSSMST. I rounded out my preparation by qualifying as a certified association executive (CAE), a professional level that required seven years of nonprofit management; a four-hour exam on nonprofit law, finance, membership, and operations; and continuing education requirements. Before my years of association work in the DC area, I spent 17 years in Atlanta. My time there represented a long stretch of heading start-up companies. The first one, very low tech but high touch, created the concept of placing full-service bank branches inside supermarkets. We told banks they could gain a lot more customers by being positioned next to the produce department than by occupying a corner of real estate that people drove by. We also started the trend of banks being open evening and weekend hours. The next group of start-ups was all high tech. I am very excited to be the new Executive Director of NCSSSMST. My first priority is to streamline the administrative processes of the organization. I hope to offload some of the tedious chores from NCSSSMST officers so that they can focus on the organization’s mission. Reaching out and forming alliances with organizations that can enhance and aid the mission of NCSSSMST is also important. My location in the DC area should be a plus for NCSSSMST, as so many organizations relevant to its mission are based here. Being located here will give me an opportunity to develop and grow relationships and partnerships, both casually and more formally. This outreach will be part of a longer-term goal of helping to elevate the organization to the next level. Naturally, taking NCSSSMST further will likely involve increased funding. The organization will need to have greater financing in order to add programs and other offerings that assist in the transformation of STEM. I intend to use my experience and time to bring in these resources.
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As a member of the American Society of Association Executives (yes, there is an association for everything!), I attend seminars and trade shows. I expect to bring to NCSSSMST some of the latest ideas and technological approaches to aid our operations and growth. The networking with other nonprofit leaders will also benefit us as we add partnerships and relationships that further our cause. As of this writing, I have been on the job for just over two weeks, and I already owe a debt of gratitude to several people. Tim Gott has been an incredible resource to me, available on a 24/7 basis. Randell Barclay has also been extremely generous with his time and has been a fount of information and support. I am sure that when Steve Canipe returns from abroad, he will be just as valuable in transferring his institutional knowledge to me. In the near future, I will be reaching out to each of the NCSSSMST Board members. I want to hear more about where they see the organization going, and learn about their perspectives on its challenges and strong suits. This will help accelerate my own learning curve. I will follow this effort with conversations with non-Board members. I am a very strong believer in the mission of NCSSSMST. I tip my hat to those of you who have dedicated yourselves to advancing the efforts of schools to prepare students for leadership in science, technology and math. I look forward to doing everything I can to support you. I thank the Board and all of you for this wonderful opportunity. I am very excited to be a part of NCSSSMST.
Todd Mann, NCSSSMT Executive Director todd.mann@ncsssmst.org
Art’s Corner
The Social Role of Science through Henrietta Lacks Rebecca Skloot’s acclaimed book, The Immortal Life of Henrietta Lacks, has garnered attention in schools and universities while Skloot herself has become a hot ticket on the lecture circuit. The book has made the rounds of Consortium schools and is now under discussion at my own. It’s not hard to understand what all the attention is about because The Immortal Life of Henrietta Lacks is a readable and provocative mix of science, history of science, bioethics, sociology, and creative non-fiction. Such a generic gumbo may be better suited for stirring things up than for settling complex issues. Although Skloot avoids making overt judgments, her sympathies are clearly more with Henrietta Lacks and her family than with the medical establishment that treated —and arguably exploited — her.
is that of Henrietta Lacks and her family, descendants of slaves and slave owners in southern Virginia. This part of The Immortal Life of Henrietta Lacks is virtually an anatomy of the pernicious effects of poverty and racism. Although clearly sympathetic to the Lacks family, Skloot does not gloss over the social maladies visited upon them, including incest, venereal disease, madness, and murder. For me the book’s most compelling issues center upon the relationship between individuals and science (including medicine) as an institution. Henrietta Lacks and her family suffered because those who treated and profited from her effaced her human identity — an identity that Rebecca Skloot labors to recover. But they also suffered emotionally frominvasion of privacy when Henrietta’s medical records were published in an article. Apparently science has a difficult balancing act if it is to both acknowledge the full humanity of an individual and at the same time maintain his or her privacy.
The book’s most compelling issues center upon the relationship between individuals and science...
For the benefit of anyone still unfamiliar with it, Skloot’s book puts two quite different stories into juxtaposition. One story concerns cells taken in 1951 from the cervix of Henrietta Lacks, a young black woman dying of cervical cancer at the Johns Hopkins Hospital. The scientific interest of the book stems from the fact that cells continued to grow outside the body and became a staple of cellular research throughout the world. Skloot’s narrative follows developments in cellular biology up to recent times. Most of the ethical questions posed by the book arise from the fact that the cells were taken without permission, that the identity of Henrietta Lacks remained obscure for many years, and that many people other than the Lacks family eventually profited from the sale of the cells and from technologies that they helped to develop. The other story, a remarkable one in its own right,
Although something of a baggy monster in its overall design, The Immortal Life of Henrietta Lacks is a valuable contribution to discussions about the social role of science and the relationship between science and the humanities. — Arthur S. Williams, Ph.D.
Dr. Arthur S. Williams has taught English at the Louisiana School for Math, Science, and the Arts since 1984. He may be reached at awilliams@lsmsa.edu.
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Exam Schools Review
What’s Really Different About These Schools? What makes a Consortium school relevant? Necessary? When educational policy often seems more informed by test scores than by an understanding of motivation and learning, are specialized STEM schools facing a Sisyphean task of constantly justifying innovative educational practices? If so, what evidence do we have that specialized schools are graduating students who are qualitatively different? These are two of the insistent questions addressed by Chester Finn, Jr., and Jessical Hockett in Exam Schools: Inside America’s Most Selective Public High Schools (2012, Princeton University Press). In their rich profiles of eleven selective high schools (four of which are NCSSSMST member schools) Finn and Hockett present the first exploration of the histories, governance, curriculum, and, to the extent possible, outcomes of schools with selective admission policies.
prise an array of learning models, from statewide residential high schools to urban college preparatory schools to a “conversion charter” school, and the schools face similar institutional quandaries – diversity in recruitment, objectivity and equity in admissions policies, human and financial resources, and charges of elitism. Hockett and Finn address these issues both in the individual school profiles and in the introductory and concluding chapters, and in their analysis, point to several significant observations. Diversity While applicant pools and admissions criteria differ from school to school, diversity was an insistent concern across most schools. The overall demographic profile of exam schools yields several noteworthy trends, namely that African Americans are significantly over-represented relative to the general population (30% to 17%), Hispanic students tend to be under-represented (13% to 20%), Asian students are over-represented by a factor of 4 (21% to 5%), and white students are significantly under-represented (35% to 56%).
Are the overabundant applicant pools suggestive of higher quality (or even qualitatively different) experiences for the students?
“Exam schools,” according to Finn and Hockett’s definition, account for approximately 1% of American public high schools. Indeed, following six specific criteria, Finn and Hockett identify only 165 such schools. Exam schools, first and foremost, are schools that maintain some form of admissions criteria – admissions tests, portfolios, interviews, etc. – and out of the exclusivity of the schools come the book’s first critical questions: Why are we willing to accept the exclusivity in the college application process and less so for high schools? Not surprisingly, the exam schools profiled in the book are oversubscribed. But are the overabundant applicant pools suggestive of higher quality (or even qualitatively different) experiences for the students? The evidence is not so apparent. Exam schools com-
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Teachers The most compelling finding among teacher profiles is that, among exam schools, teachers with an earned doctorate comprise 11% of the teacher population, compared to 2% among public school teachers. Further, exam school teachers with a master’s degree outnumber the general population by 11%. Does expertise matter? While Finn and Hockett indirectly address this question, this should be a prime question for research among exam schools and specialized schools. Why, for example, is the doctorate requisite in higher education and not a higher priority in secondary education? Advanced curriculum Although very few of the schools profiled have cause to worry about their stu
Exam Schools Review dents’ performance on statewide standardized exams (and several are not, in fact, required to administer them) the specter of AP exams looms in the background of nearly all. Finn and Hockett suggest that the pressure to perform on AP exams … “press[es] on students, parents, teachers, and entire schools in ways that are plausibly said to discourage experimentation, risk taking, unconventional thinking, unique courses, individualized research, and independent study….” (p.184-185) Indeed, the “AP tiger” (p.186) too often defines the academic expectations and frustrates innovative teachers and administrators. Is the proper (and perhaps most ethical) response to the AP question to establish a system of assessment that mitigates the ceiling effect of other standardized tests and allows for demonstration of value-added? The authors argue not. Then how to demonstrate efficacy and valueadded of an exclusive education?
gitudinal study has been conducted across a sample of Consortium schools, and more recently, a large scale outcomes study conducted by Rena Subotnik, Robert Tai, Rochelle Rickoff, and John Almarode, which responds to a number of the questions posed by Hockett and Finn. “Exam Schools” speaks to our efforts to transform STEM education, but it is also a clear call to demonstrate not only what we do best but how we go about it. As an example, several Consortium schools have recently administered the Collegiate Learning Assessment (CLA), a measure of critical thinking, and will be sharing findings and soliciting wider participation. Such innovative measures would allow us to respond directly to the paucity of research so often cited in Hockett and Finn’s book. I have remarked many times that if you take bright, curious students and put them in a classroom with an engaging and innovative teacher, good things happen. It is our task to figure out how that works. Perhaps NCSSSMST should write the definitive text on what works and how we know.
I have remarked many times that if you take bright, curious students and put them in a classroom with an engaging and innovative teacher, good things happen. It is our task to figure out how that works.
As an educational researcher and as a member of the NCSSSMST community for over twenty years, I find myself frequently ruminating on the persistent questions posed by Finn and Hockett. Indeed, they emphatically suggest in their conclusion that the effectiveness of exam schools is the great unknown. How well would our students have done without the intervention of a specialized school? Does an enriched curriculum predict educational attainment and professional success? Finn and Hockett suggest that exam schools often trade on their reputation without much solid evidence of student growth. “They can proudly demonstrate intricate research projects, display cases full of academic prizes, science fair and robotics competition ribbons, National Merit lists, and messages from grateful alums. But they have access to little ‘value-added’ data.” (p.178-179). Finn and Hockett suggest at the outset that very little (if any) systematic research has been conducted on exam schools, which is true. But I take this assertion as a challenge to NCSSSMST. To be sure, one smaller scale lon-
— Jerald Thomas, Ed.D.
Jerald (Jay) Thomas, Ed.D., is an associate professor of education at Aurora University and is Past President of NCSSSMST.
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Don’t Blame the Fish by Dr. Craig Gaska, Gaska Consulting, Sugar Grove, IL
Abstract This article discusses the main ideas presented by Richard Stiggins in his Assessment Manifesto: A Call for the Development of Balanced Assessment Systems and the support for prescriptive teaching practices the author believes compliment and are a natural outgrowth from the use of formative assessments as suggested by Stiggins, especially for gifted students. The author defines prescriptive instruction as tailoring instruction to the individual needs of the students
using formative assessment data. This idea opposes assumptive instruction whereby all students are exposed to what is assumed should be learned, needed or not. Stiggins suggests that the past 20 years of research on formative assessment positions us to use assessment in productive ways. The thesis of this article suggests that perhaps this research also positions us to use more effective instructional practices in gifted education as well.
The comparative testing that we are so familiar with has its roots in the Army Alpha test developed by Arthur Otis. Its purpose was to sort recruits and identify potential officer candidates for WWI. This successful army-testing program became the model for a number of educational tests still embraced by educators today (Popham, 2006). The almost exclusive use of testing information for sorting and grading is a disservice to our students and a misuse of valuable information.
process, not just an end-of-activity test, and a majority of students preferred assessments throughout the learning process rather than only at its end. Lastly, and most importantly, as a result of their experience with formative assessment, students experienced a higher level of self-confidence and motivation.
more useful role in helping students learn it should be moved into the middle of the teaching and learning process instead of being postponed as only the end-point of instruction” (Shepard, 2000, p. 101). In a study of embedded assessment in a junior high school science course for gifted students, Miedijensky and Tall (2009) found that formative assessment students realized that they are partners in the learning process and, therefore, have a degree of autonomy when involved in a formative assessment process. Further, students reported a realization that assessment is a cyclic
Educators of gifted students, and all educators, need to pay attention to Stiggins when he shares his frustration regarding how students in American schools are currently assessed. He is correct when he observes that education has “reached a tipping point in the evolution of our schools when we must reevaluate, redefine, and redesign assessments’ role in the development of effective schools” (Stiggins, 2008, p.2). In addition, current assessment practices are a gross misuse of data in making educational decisions and an injustice to the students!
This is especially true of the gifted student population. Testing and assessment, as it is currently used in schools today, is of little value in the learning process. Such assessments simply confirm what is already known. Educators must advocate for the use of more formative assessments with gifted children. Formative assessments provide much more valuable information to both teachers and students. In addition, using formative assessments with the gifted capitalizes on their metacognitive abilities and increases their motivation to learn. “In order for assessment to play a 12
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Change in education comes slowly. Dr. Richard Stiggins continues to advocate for a paradigm shift from a focus on summative uses for assessment to formative. His assessment manifesto reflects his belief that a tipping point has been reached that requires a reassessment of assessment. Schools in the past were places where some succeeded and others did not. We now expect all students to meet a predetermined set of standards (Stiggins, 2008). The following discussion explores prescriptive teaching practices supported by the paradigm shift for which Stiggins is calling. As assessment needs to shift to a focus on formative uses, instruction needs to shift to prescriptive rather than assumptive practices. Prescriptive instruction uses formative assessment to tailor instruction to the individual needs of the students rather than assumptive curricular practices that assume all students need to be exposed to the same concepts, at the same time, regardless of students’ abilities or achievement. A doctor does not assume what treatment is needed before carefully examining a patient; rather medication is prescribed only after evaluating the patient’s condition. So should be the case in the classroom. Formative assessment sets the stage for using data, before the fact, to prescribe the most appropriate instructional intervention. The sole use of summative assessment is judgmental and creates a cycle of failure for many students. Our almost total reliance on normreferencing does nothing more than judge, track and grade; doing very little to actually help students learn. There is a significant body of literature that suggests significant gains can be realized through the use of
formative assessments, which have all but been ignored for the past 20 years. Dr. Stiggins refers to two compelling studies to support his thesis: The Power of Feedback (Hattie & Timperley, 2007) and Inside the Black Box: Raising Standards Through Classroom Assessment (Black & William, 1998). In their comprehensive metaanalysis of “feedback” and discussion on assessment, Hattie and Timperley conclude that feedback occurs too rarely, yet it is one of the most powerful influences on learning. Black and William cite research that supports the power of formative feedback and further discusses the problems with, and negative impact of, assessment as commonly practiced. Stiggins (2008) also calls for a metamorphosis of assessment priorities from norm-referenced to criterionbased assessment and suggests a focus on assessment for learning rather than assessment of learning. In other words, the focus should be placed on formative rather than summative assessments. Doug Reeves (2000) offers the medical analogy that formative assessment is to summative assessment as a physical exam is to an autopsy. Rather than keeping the patients (students) healthy by making educational decisions based on timely information (the physical exam) used to improve learning, we merely assess what went wrong after it’s too late (autopsy). Why are formative assessments so important? “With assessment, purpose is everything” (Stiggins, 2008, p. 3). Again, Stiggins is calling for a major paradigm shift in our thinking and a metamorphosis in our practices. He suggests that the true primary purposes of assessment are to inform instructional decisions and motivate
students. Too many students find assessments degrading, judgmental, ego deflating, and stressful, which causes them to feel disenfranchised from school. This results in the struggling learner to give up rather than try harder. The message to the student is, “You can’t do it and this assessment proves the point.” The message should be, “You can do it, and this assessment shows us how and what we need to do to help you!” A positive outcome of “No Child Left Behind” is the mandate that all children can and will learn. Educators are forced to take responsibility for learning and can no longer wash their hands of student failure. As a result, many new initiatives such as response to intervention (RtI), use of the professional learning community (PLC) model, guided teaching practices, scaffolding, etc. are changing the way we teach. We also need to change the way we assess. Educators need to embrace a paradigm shift from reliance on normreferenced testing to criterion-based testing (creating a system in which all students can succeed), and an emphasis on formative rather than summative assessment. In discussing the first purpose for assessment, which is the gathering of evidence to inform instructional decisions, Stiggins suggests three levels of use: classroom, program, and institution. The following three questions need to be asked at each level: 1. What are the instructional decisions to be made? 2. Who will be making those decisions? 3. What information will help make good decisions? (Stiggins, 2008). With multiple examples of forma13
tive evidence at the classroom level, teachers, students, and sometimes parents will be looking at continuous evidence of each student’s progress in reaching the standard being learned. At the program level, teacher teams, teacher leaders, principals, and curriculum personnel will be looking at periodic, but frequent, evidence aggregated across classrooms to determine which standards are not being mastered. At the institutional level, superintendents, school boards, and legislators will use annual summaries of standards mastered on accountability tests to determine if enough students are meeting required standards (Stiggins, 2008). Notice that criterion and formative assessment are emphasized at the first two levels and only at the institutional level is there a focus on summative results. So obvious, yet so elusive, is the second purpose for assessing - motivating students to learn. If assessment is going to be motivating for students, the focus needs to shift to assessment for learning, which includes goal setting, progress monitoring, and intervention planning (Stiggins, 2008). As the saying goes, “If the fish are sick, don’t blame the fish - change the water.” For too long the blame has been placed on the fish and no one changed the water. Acknowledgment of the critical importance of instructional decisions made by students, reliance on repeated self-assessments by both the students and teacher, and designing carefully drawn learning progressions based on sound pedagogy are the first three important steps. The paradigm shift from summative to formative assessment is of utmost importance and urgency. “We must replace the grossly out-of-balance assessment systems of the past with 14
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those that honor the information needs of all assessment users - systems that both support and verify learning from the classroom to the boardroom” (Stiggins, 2008, p. 10). Stiggins further notes that the past two decades of research and development positions us to use assessment in productive ways. Perhaps the practice of formative assessment will position us to use all the other accumulated research in education to foster prescriptive practices in the classroom as well. This will not be easy, but it is doable. In addition to the great body of research supporting formative assessments, there is a great body of research supporting educational models that complement it. Building a common knowledge base and implementing prescriptive models and strategies promoted by educational experts and supported by research will enable schools to create a comprehensive package for success. Prescriptive models such as professional learning communities, curriculum mapping, and understanding by design; supported by strategies such as differentiation, balanced literacy practices, and response to intervention, to name a few, will lead to success for all. Many of these models need to be packaged to create a comprehensive educational program that is supportive rather than judgmental of students and prescriptive rather than assumptive in meeting their needs. There is not a single road to success; every school needs to plot its own course. However, schools will need a compass for this journey. The appropriate use of formative assessments, as called for by Dr. Stiggins in concert with prescriptive instruction, is indeed the compass needed to plot a course for success for every student.
References
Black, P., & Dylan, W. (1998). Inside the Black Box: Raising Standards through Classroom Assessment. Phi Delta Kappan. 80(2) Retrieved December 30, 2012 from http://www.pdkintl.org/ kappan/kbla9810.htm Eaker, R., DuFour, R., & DuFour, B. (2002) Getting Started: Reculturing Schools to Become Professional Learning Communities, National Educational Service, Bloomington, IN. Hattie, J. & Timperley, H. (2007) The Power of Feedback, Review of Educational Research, 77(81) Retrieved December 30, 2012 from http://www.aera.net Miedijensky, S., & Tal, T. (2009). Embedded Assessment in ProjectBased Science Courses for the Gifted: Insights to Inform Teaching All Students. International Journal Of Science Education, 31(18), 24112435. Popham, W. J. (2006), Assessment for Educational Leaders, Pearson Education Inc., Boston, MA. Reeves, (2000), Accountability in Learning, Advanced Learning Centers, Denver, CO Shepard, L. A. (2000). The Role of Assessment in a Learning Culture. Educational Researcher, 29(7), 4-14. Stiggins, R. (2008) Assessment Manifesto: A Call for the Development of Balanced Assessment Systems, Educational Testing Service, Portland, OR Retrieved December 30, 2012 from http://www.nmsa. org/portals/0/pdf/advocacy/other_resources/AssessmentManifesto08.pdf
Motivating and Challenging Highachieving Students in Mathematics by Dr. Jeff Wiesman, Wheaton Warrenville South High School, Wheaton, IL
Abstract Reports document underachievement among high-achieving and gifted students in mathematics. In addition, teachers often do not differentiate and adapt instruction to challenge bright kids. Therefore, educators must identify the needs of advanced learners and incorporate more appropriate motivational and pedagogical techniques. The purpose of this study was to determine how to most effectively motivate and engage high-achieving freshmen math students. Students from a public high school in twentyone different math classes were surveyed re-
Introduction There are numerous research reports documenting underachievement among high-achieving and gifted students in America’s classrooms. Many also suggest that our schools have to do a better job of engaging and challenging our advanced learners. Educators must identify their needs and incorporate more appropriate curricula and pedagogical techniques. The National Council of Teachers of Mathematics indicated that educators who differentiate mathematical tasks will enhance participation and motivation among gifted students (Chval and Davis, 2008). Unfortunately, teachers and administrators often pay less attention to highachieving children and, instead, the majority of time and effort is spent determining how to get lower achieving students to meet state standards to avoid being labeled as a failing school based on national mandates. According to Jolly and Makel (2010), educators repeatedly ignore bright students in order to focus on those who are not proficient in mathematics. Since high achieving students are likely to get good grades or score well on achievement tests, they do not always warrant the appropriate concern amongst educators (Morisano and Shore, 2010). Moon, Brighton, and Callahan (2003) conducted a study on the effects of state tests and noted that teachers often do
garding their perceived effectiveness of various motivational constructs (N = 448). The data from freshmen enrolled in an honors course was disaggregated and used for this study (n = 103). Results indicated that advanced students were inherently motivated to understand new concepts and they were most motivated when they developed mastery-oriented goals. When educators strive to understand how to best motivate high-achieving math students, they will improve instruction and challenge our top students to greater heights. not differentiate and adapt instruction to challenge high-achieving learners. In their study, one participant stated, “The gifted student does not have to do all of the drilling, but they often have to follow the same timeline and curriculum as the other kids. So they are losing out because I don’t think they are being challenged” (p. 54). Ironically, this outcome has emerged despite the fact that the No Child Left Behind act suggests that we should address the needs of every single child. In fact, Morisano and Shore (2010) declared that the needs of high achieving children should warrant as much time and effort as spent with low performing students or those with learning disabilities. Educators frequently fail to spend the requisite time and, consequently, many students with gifted mathematical minds underachieve. According to Morisano and Shore, up to half of the gifted and high-achieving youth are underperforming and do not reach their full potential. Furthermore, in Frederick, Alfeld, and Eccles’ (2010) study of gifted high school students, they found inadequate levels of passion among the participants, and to make matters worse, schools often did not do anything to foster passion. In addition, male students often will not put forth maximum effort because they believe they can do well without having to do all the work (Hong and Aqui, 2004). It certainly is unfortunate to both the individual 15
and our society if our brightest are not pushed to their highest potential. Studies have also shown that both the academic interest and motivation of adolescents decline as they progress through junior and senior high school (Lepper, Corpus, and Iyenger, 2005); Wigfield, Eccles, and Rodriquez (1998) found this to be especially true in mathematics and the natural sciences. The purpose of this study, therefore, was to determine how to most effectively motivate gifted and high-achieving mathematics students. This investigation aimed to address the aforementioned problems by identifying ways to fully engage them during their first year of high school. This study specifically strives to (a) identify the types of methodologies educators should incorporate into their lessons to motivate high-achieving freshmen; (b) discover types of teacher characteristics that will motivate these students; and (c) identify which motivational constructs are the most effective. The results of this study have practical implications for high school educators, college professors who work with pre-service teachers, and individuals who are involved with teacher in-service or mentoring programs. If educators can understand how to motivate high-achieving math students, then they can improve instruction and challenge our top students to greater heights.
Intrinsic & Extrinsic motivation Intrinsically motivated students take action because they are inherently interested in or enjoy learning a topic (Eccles and Wigfield, 2002). Wigfield et al. (1998) more specifically claimed that a person’s intrinsic motivation can be affected by his or her individual or situational interest. Individual interest, which is relatively stable, consists of someone’s inherent desire to increase knowledge, value, and positive feelings. Whereas situational interest, which may not last very long, develops when environmental factors generate interest, such as with surprise or novelty (Linnenbrink and Pintrich, 2002). While intrinsic motivation comes from within, extrinsically motivated individuals are driven to act because of external forces, such as pay, rewards, or coercion (Eccles and Wigfield, 2002). Some argue that extrinsic motivation, a motivational strategy used quite regularly in schools, does not work (Erwin, 2003). Furthermore, Erwin noted that one could unintentionally create stress and damage intrinsic motivation when motivating others by coercion or by offering rewards. Vlachopoulos et al. (2000) suggested, however, that individuals would have higher levels of motivation when students are both intrinsically and extrinsically inspired. Pierce, Cameron, Banko, and So (2003) agreed and suggested that individuals may develop an inherent interest even though they were originally motivated because of an external force. Goal Achievement Theory & Self Efficacy
Jeff Wiesman, Ed.D., is currently a mathematics teacher at Wheaton Warrenville South High School, which is located in a western suburb of Chicago. He has taught for over 18 years, including numerous honors-level courses. He received a doctorate of education in curriculum and instruction from Aurora University in 2007. He is now serving as an adjunct professor at Aurora University, teaching a math methods course.
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Goal theory is another motivational construct that explains why individuals choose to take action. When individuals create goals, they create a blueprint that gives them meaning, direction, and purpose (Covington, 2000). Researchers categorize achievement goal theory into two classes. The first type, mastery or learning goals, refers to the desire to increase one’s competency, develop new skills, or improve one’s level of understanding and appreciation (Elliot and Dweck, 1988). The second type of goal orientation, performance or ego, refers to individuals’ desire to outperform others, improve their status, prove their ability, or receive public acknowledgment for their superior performance (Eccles and Wigfield, 2002). Goal theory is a powerful motivational construct, but it does not address why someone would choose one goal over another (Covington, 2000). No matter which type of goal an individual establishes, how-
ever, many believe that when goals are specific, proximal, and challenging, individuals are more likely to accomplish them (Eccles and Wigfield, 2002). When considering goal orientation, researchers have also determined that students will regularly set social goals in hopes of gaining the respect of others and to achieve a sense of belonging (Wigfield et al., 1998). McInerney and McInerney (1998) suggested that the social component of school, which includes interactions with parents, teachers, and peers, could affect students’ attitudes toward school and their motivation to learn. Teenagers with better peer relationships have attitudes that are more positive towards school; if they associate with others who are high achieving, their motivation will likely improve (Strauch, 2003). Conversely, motivation could decline if adolescents join low achieving peer groups (Strauch, 2003). Indeed, school contexts are primarily designed to provide an academic education, but since teenagers value interpersonal relationships and acceptance by their peers, it is important for educators to consider adolescents’ social needs as well. In effect, when teachers meet their students’ psychological needs for love and the esteem of others, they will enhance academic motivation (Eccles et al., 1993). The self-efficacy theory examines human motivation as it relates to an individual’s expectancy beliefs. Eccles and Wigfield (2002) defined expectancies as a person’s belief as to how effective he or she will be on a given task or activity. Bandura (1989) more specifically described self-efficacy as a person’s confidence in his or her ability to accomplish a particular task. That is, if individuals believe a task is achievable, then motivation will increase and they will be more likely to execute their plan. Conversely, if individuals do not have confidence in their ability to accomplish a task, then their degree of motivation will diminish (Eccles and Wigfield, 2002). In addition, Bandura (1989) and Margolis and McCabe (2004) discovered that self-efficacy levels positively correlate with an individual’s willingness to set goals and sustain goal-oriented activity. Motivation among High-Achieving Students An advanced student enrolled in a mathematics course is likely to engage in academic activities for different reasons than students in other subgroups. High achieving adolescents are more likely to be intrinsically motivated than their peers (Hoekman, McCormick, and Barnett, 2005), and they more likely to be more self-efficacious in mathematics (Hong and Aqui, 2004). In addition, advanced math stu-
dents are more likely to be motivated if teachers integrate the disciplines, ask open-ended questions, demand the use of multiple reasoning methods, and require the formation of inquiry and generalizations (Greene, 1997). Greene further posited that motivation among gifted math students will increase when providing challenges where collaboration is required to investigate, brainstorm, and answer questions. Alternatively, tedium and the lack of a rewarding challenge may lead to a decrease in engagement among high-achieving students (Hoekman, McCormick, and Barnett, 2005). For instance, worksheets that focus on skill development often are not the recommended practice for advanced learners. However, this is a practice that is commonly employed in heterogeneous classes to aid lower performing students. Siegle and McCoach (2005) suggest that a lack of motivation among gifted students could also be the result of a lack of study skills, a poor attitude, or a mismatch between the student and the classroom environment. Educators must continue to research which motivational constructs are the most the effective with high-achieving mathematics students.
method A survey was used to gather data regarding students’ background information and their perceptions on the effectiveness of various motivational constructs. This study included a cluster sample of students from a public high school located in DuPage County, Illinois. Students from twentyone different mathematics classes, in grades nine through twelve, were surveyed (N = 448). The data from freshmen who were enrolled in an honors or advanced-level course was disaggregated and used for this study (n = 103). Of those participants, 51 (49.5%) were female students and 52 were male (50.5%); a third of the participants reported that they earned predominately A’s on their grade reports while 57% said they received mostly A’s and B’s. Approximately 85% of the surveyed were white, 2% were African-American, 4% were Asian-American, and 9% were Hispanic. The instrument used for this research study took sections from the following three surveys: the Inventory of School Motivation (ISM) by McInerney and Sinclair (1992); the Colorado High School Senior Survey by Susan McAlonan, Steve Kennedy, and Nanci Avitable (1999); and the Patterns of Adaptive Learning Scales (PALS) by Carol Midgley et 17
al. (2000). Questions from the ISM measured students’ perceptions of intrinsic motivation, extrinsic motivation, performance-oriented goals, mastery-oriented goals, and social goals. Questions from the PALS measured students’ perceptions of student self-efficacy; and finally, questions from the Colorado High School Survey measured the participants’ perceptions on the motivational effectiveness of teacher characteristics and instructional methodologies. All three surveys, which the researchers used in previous studies to determine perceived motivational factors, tested to be valid and reliable. Alpha scores for the ISM ranged from .66 to .82, and the alpha score for the self-efficacy construct in the PALS is .78. This researcher obtained permission to use all three surveys. The survey used for this study contained three sections. The first part of the survey gathered background information on the participants. In the second section, the participants rated—using a Likert scale where 6 represented strongly agree and 1 represented strongly disagree—the perceived effectiveness of the eight motivational constructs. Each motivational construct had five different questions on the survey, totaling forty questions in this section. In the final section of the survey, the participants stated which motivational construct was most likely to motivate them. Students completed the survey, which took approximately 15 minutes to finish, during their math class. All measures were taken to protect the rights of the students; participation was voluntary, those under the age of 18 received parental permission, and all students signed an informed consent form.
Results Goal Orientation In this descriptive study, high-achieving math students were surveyed to determine the perceived effectiveness of various motivational constructs in school. With regard to masteryoriented goals, both male and female students believed they were especially motivated so they could have a good future (M = 5.76, SD = 0.61 and M = 5.78, SD = 0.60, respectively). They also thought they would be motivated when they believed they were proficient at a task (M = 5.27, SD = 0.81 and M = 5.25, SD = 1.04, respectively). Female advanced learners agreed that motivation levels were high when they saw their work improving (M = 5.12, SD = 0.91) and when they were becoming better at their work (M = 5.20, SD = 0.82). Males were not as motivated as females 18
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because of improved work (M = 4.35, SD = 0.91) and when they were becoming better at their work (M = 4.69, SD = 0.82). Finally, as shown in Table 1, neither male nor female students thought they would be as motivated when solving problems (M = 3.90, SD = 1.06 and M = 4.08, SD = 0.95, respectively). As indicated in Table 2, students were not as driven to engage when they set performance-oriented goals as compared to mastery-oriented goals. However, all the participants were motivated to receive good grades; male students
had a mean of 5.31 (SD = 0.84) and females a mean of 5.49 (SD = 0.78). High-achieving female students thought they would be more likely to be motivated when they were praised (M = 4.31, SD = 1.18) than did the male students (M = 3.79, SD = 1.41). Similarly, males were less likely to be motivated when noticed by others (M = 3.69, SD = 1.38) than females (M = 4.14, SD = 1.24). Male and female advanced learners agreed that they were only somewhat motivated when competing with others (M = 4.46, SD = 1.38 and M = 4.39, SD = 1.36, respectively) and when doing better than other students (M = 4.54, SD = 1.17 and M = 4.43, SD = 1.35, respectively). Advanced math learners were also less likely to be motivated with social goals than compared with mastery-oriented goals. If asked to work cooperatively, however, male and female students thought they would be most motivated when
working with friends (M = 4.75, SD = 1.20 and M = 4.69, SD = 1.34, respectively). As shown in Table 3, students were not as likely to be motivated when working in groups that did not include their friends. Female students believed they would be more motivated when helping others (M = 4.67, SD = 1.08) and when showing concern for others (M = 4.37, SD = 1.07) than did male students (M = 3.96, SD = 1.20 and M = 3.75, SD = 1.22, respectively).
parents, but females would likely be driven to act to get praise from their parents (M = 4.96, SD = 1.10) and their teachers (M = 4.49, SD = 1.33). Male freshmen were less likely to be motivated to receive praise from their parents and teachers (M = 4.06, SD = 1.60 and M = 3.81, SD = 1.48, respectively). Finally, boys were more likely to engage to get some kind of reward (M = 4.62, SD = 1.08) than girls (M = 4.10, SD = 1.21).
Intrinsic and Extrinsic Motivation & Self-Efficacy
As evident in Table 6, high-achieving math students were undoubtedly motivated when they were confident in their ability to complete the work. The male and fe-
High achieving male and female math students were intrinsically driven to improve and become stronger math
students as evident by their response to “I like to see that I am improving in my schoolwork� (M = 5.38, SD = 0.79 and M = 5.73, SD = 0.45, respectively). Similarly, male and female learners would likely engage when they have a chance to redo a task and improve their work (M = 4.87, SD = 1.18 and M = 5.25, SD = 0.93, respectively). Moreover, female children thought they would be motivated to understand new concepts (M = 5.14, SD = 0.89); males generally agreed (M = 4.54, SD = 1.08), but not to the extent as the female participants (see Table 4).
male participants also believed they could do all the work if they do not give up (M = 5.00, SD = 1.00 and M = 5.29, SD = 0.87, respectively) and they will attempt even the most difficult problems (M = 5.02, SD = 1.12 and M = 4.92, SD = 1.04, respectively). Furthermore, female students generally believed that they would be motivated when they were certain they can figure out the difficult problems (M = 4.96. SD = 1.20); the male participants agreed (M = 4.81, SD = 1.13).
The data reveal the importance of establishing a
Table 5 illustrates that bright kids were not as extrinsically motivated as compared to their inherent desire to learn. Neither engaged in the classroom for presents from their
positive rapport with high-achieving math students. With a mean of 5.33 (SD = 0.88), girls thought that they were more
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likely to be motivated when they liked the teacher than boys (M = 4.94, SD = 0.89). Furthermore, both male and female students believed they were likely to be motivated when teachers incorporated humor into their lessons (M = 4.94, SD = 1.13, and M = 4.96, SD = 0.98, respectively). The female participants also thought they were likely to engage when they have an enthusiastic teacher (M = 4.90, SD = 1.12); the male students did not agree to the same extent (M = 4.43, SD = 1.24). When considering various instructional strategies, highachieving math students were most likely to be motivated when class instruction includes student choice; males had a mean score of 4.63 (SD = 1.40) and females a mean of 4.73 (SD = 0.91). Males and females also thought they would be just as motivated when the teacher incorporates handson activities (M = 4.58, SD = 1.26 and M = 4.90, SD = 1.12, respectively). In addition, girls were more likely to be motivated when the teacher uses a variety of pedagogical techniques (M = 4.61, SD = 1.01) than boys (M = 4.08, SD = 1.17). As shown in Table 8, the participants thought that solving real life problems and the use of technology is not as likely to motivate as with other instructional approaches.
evident from Figure 1 that the goal achievement theory is an effective motivational construct for all high-achieving learners. They strongly desire to have a good future and they want to be properly equipped for future endeavors. Furthermore, they will work to become excellent math students not only to receive good grades but also to better themselves. They are intrinsically motivated to be highachieving students, and given the proper curriculum and instruction, they will strive to improve their mathematical abilities. Finally, if bright children are challenged, they will likely be inherently motivated to accept the challenge.
discussion and implications Figure 1. What Motivates High-achieving Math Students the Most?
The final research question asked the participants which construct affects motivation the most in class. It is clearly
The primary aim of this article was to identify strategies that will motivate high-achieving students so they will reach their full potential. Results clearly indicated that advanced learners are most motivated when they develop masteryoriented goals. Teachers, therefore, must actively teach them how to become proficient goal-setters and then engage in dialogue to facilitate student aspirations. Educators should monitor student goals and offer suggestions regarding appropriate times to seek out more advanced objectives so they will continually increase aptitude. Gallagher (2005) stated that one of the four main reasons why high-achieving learners underachieve is because of a lack of goals. However, since these children strongly desire a good future, educators must tap into that motivation by discussing goals that will lead to a more accomplished mathematical mind, which, in turn, could lead to a more successful career in college or the workplace. High-achieving students certainly want to excel in mathematics and they are inherently motivated to learn. Data from this study suggests that educators can bolster intrinsic
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motivation by implementing four simple strategies. First, ensure that each lesson incorporates a topic, theorem, or application that is completely new to your advanced learners. Curriculum often spirals in mathematics, and while that may be a necessity for some, advanced students do not need to spend time reviewing concepts they likely remember. If a new topic is not introduced, that is a recipe for boredom and, as a result, underachievement. Similarly, math educators must be careful when assigning homework. For instance, bright students likely do not need to practice five problems of the same type. Instead, teachers should give them a variety of problems that are unique, where each question requires students to apply different mathematical concepts or processes. Third, intelligent math students are motivated to improve their mathematical abilities, and if given a chance, are driven to improve their work. Therefore, teachers will bolster motivation by assigning challenging, difficult problems, and then allowing opportunities to correct errors or redo tasks. This can be done with homework assignments, projects, or summative assessments. Finally, high-achieving freshmen are often bored with long lectures, but instead, they engage when they have an opportunity to struggle through math questions during class—individually or with a friend. Therefore, it is beneficial to give students time to work in class, as opposed to lectures throughout the entire period. Gifted students are more likely to be self-efficacious than their non-gifted peers (Hong and Aqui, 2004). Consequently, it is not surprising that the participants in this study were motivated to achieve because they were confident in their abilities. If educators continue to scaffold and equip bright children with appropriate strategies and techniques, and if the assignments are difficult enough to promote improvement, then they will be more likely to engage with the material. High-achieving students believed they can complete difficult assignments if they try and, as a result, math teachers should not be overly concerned about losing students by assigning difficult problems. Additionally, high-achieving students will work on challenging math problems for long periods of time because they are driven to persevere with their course work. There are numerous implications centering on the results that demonstrate the motivational worth of various teacher characteristics and the instructional activities they incorporate. Students believed they were motivated when they liked the teacher and, therefore, teachers must be intention-
al about developing a positive rapport by taking a personal interest in their students, using humor, and truly caring for them. Another finding suggested that teachers can likely motivate bright students if they allow for more student choice with the instructional activities they incorporate. For example, teachers can provide students with a list of problems to choose from when assigning homework. Some of the more gifted learners often select the most challenging problems because they yearn for a challenge. Integrating choice within an activity, lesson, or assessment is an excellent motivational technique for high-achieving students. The participants in this study were not as likely to be motivated when teachers incorporated extrinsic motivators. Consequently, educators should carefully consider the reasons why they give candy, extra credit, or other rewards. Not only were extrinsic motivators deemed relatively ineffective in this study, but also, since they might negatively influence intrinsic motivation (Deci, Koestner, & Ryan, 2001), teachers should use caution when attempting to extrinsically motivate students. As well, high-achieving students were not as likely to be motivated with cooperative lessons unless they can complete work with friends. This research aligns with French, Walker, and Shore’s (2011) study which indicated that gifted students do not necessarily prefer to work alone. They established that bright children would work with others if the educational conditions were appropriate for the given task. Educators do need to interpret the results of this study cautiously in light of its potential limitations. While this research may be transferable to suburban schools with similar demographics, sampling from other schools could yield different results. This is especially true if researchers completed a similar study in an urban, rural, or other school with greater diversity. Educators should also note the delimitations of this research study. Since the sole instrument used for this study was a survey, the gathered information only provides a snapshot of student perceptions. Researchers would gain a more accurate picture if they incorporated interviews in conjunction with the questionnaire. Nevertheless, the results are indicative of adolescents in a predominately white, suburban high school and the data reveals interesting data regarding perceptions of motivation among high-achieving math students. In sum, it is important for educators to consider the needs of our high-achieving children. Since they strongly desire
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to improve their mathematical abilities, our curriculum and instruction must be at the appropriate difficulty levels. While each student might respond differently to the various motivational practices, teachers can maximize their ability to engage high-achieving freshmen when utilizing certain constructs. Advanced learners are motivated because of their personal aspirations and their inherent interest in mathematics and, therefore, educators must develop pro-
grams, lesson plans, and activities that utilize the goal and intrinsic motivational theories.
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Erwin, J. (2003). Giving students what they need. Educational Leadership, 61(1) 19–23. Fredericks, J. A., Alfeld, C. & Eccles, J. (2010). Developing and fostering passion in academic and nonacademic domains. The Gifted Child Quarterly, 54(10), 18-30. French, L. R., Walker, C. L., & Shore, B. M. (2011). Do gifted students really prefer to work alone? Roeper Review, 33(3), 145-159. Gallagher, J. J. (2005). The role of race in gifted education: According to Jim Gallagher. Roeper Review, 27(3), 135-138. Glasser, W. (1986). Control theory in the classroom. New York: Harper & Row Publishers. Greenes, C. (1997). Honing the abilities of the mathematically promising. The Mathematics Teacher, 90(7), 582-586. Hoekman, K., McCormick, J. & Barnett, K. (2005). The important role of optimism in a motivational investigation of the education of gifted adolescents. The Gifted Child Quarterly, 49(2), 99-110. Hong, E. & Aqui, Y. (2004). Cognitive and motivational characteristics of adolescents gifted mathematics: Comparisons among students with different types of giftedness. The Gifted Child Quarterly, 48(3), 191201. Jolly, J. L. & Makel, M. C. (2010). No Child Left Behind: The inadvertent costs for high-achieving and gifted students. Childhood Education, 87(1), 35-40. Lepper, M., Corpus, J., & Iynengar, S. (2005). Intrinsic and extrinsic motivation orientations in the classroom: Age differences and academic correlates. Journal of Educational Psychology, 97(1), 184–196.
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Calculus Reshuffled
by Dr. Vince Matsko, Illinois Mathematics and Science Academy, Aurora, IL
What dictates the order of topics in calculus text-
are let naturally to the need to compute limits, and upon
books? Order there must be, if only for the sake of pagina-
considering the efficacy of these computations, to a discus-
tion. But textbooks are written and courses are taught by
sion of continuity. The order of these topics in our text-
those who know the subject well – those who appreciate the
book, however, is continuity, limits, then differentiation.
structure of The Calculus, and who consequently impart
their own teleology to its presentation. Unfortunately, high
engaged, I was engaged, and – in my mind, increasingly
school students understand little about mathematical struc-
more important – not once did I need to utter that stultify-
ture, and even less about teleology.
ing phrase, “We’re studying this now because you’ll need it
later.”
To understand the structure of a discipline requires
a sophistication often not acquired until graduate school.
Was this reshuffling successful? My students were
On to derivatives as linear approximations:
So how useful is it to tell students they must study limits in order to understand derivatives – when they don’t know what a derivative is?
We typically begin calculus with a qualitative dis-
cussion of derivatives, thinly disguised as velocity and ac-
(This is discussed several sections later in our textbook.) This is simply an intuitive rewrite of the limit calculation
celeration. We discuss such correspondences as local extrema on the displacement graph and zeros of the velocity
for small h.
graph. A typical exercise is to show students three graphs
Why now? Consider f(x) = x sin(x) We write
Thus, f ’(x) = sin(x) + xcos(x) Moreover, the use of
and have them determine which are displacement, velocity, and acceleration.
Can we be more quantitative? Graphing utilities
make the move from average velocity to instantaneous velocity easy to visualize – and allow an intuitive introduction to the idea of a limit. Slopes of lines tangent to the graph of f(x)=x2 can easily be calculated with only the briefest discussion of the process of taking a limit.
At this point, students can begin exploring deriva-
tives of other common functions – and encounter a few troublesome limits along the way. Now we need to know how to compute limits.
the symbol “≈” may be made rigorous using limits, and so its use is completely justified. At this point, rigor impedes intuition.
In class, I showed one example of using this tech-
nique, then handed out practice problems for in-class work.
We are far from satisfied, however, because the
After a few specific examples, students encountered h(x) =
procedure simply does not always work. When does it fail?
f(x)g(x) – and as a result were able to derive the product rule
A discussion of continuity is not out of place at this point.
as an exercise during class.
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By beginning to calculate derivatives right away, we NCSSSMST Journal | Fall 2013
to effectively calculate derivatives which would have been
Now consider
far too unwieldy using a limit approximation. Finally, the
.
product and quotient rules may be derived by the students We have
themselves.
.
Perhaps these few examples illustrate the point that
much of beginning calculus can be motivated by considerWhen h is small, we may consider
ing questions that arise naturally rather then resorting to vague teleology. Although I have only recently tried this
as the sum of an infinite geometric sequence, and so approximate
approach (in the Fall 2012 semester), I felt students (and instructor!) were more engaged than usual.
Do I recommend this particular approach for use
by all calculus instructors? Quite the contrary. Instructor engagement is key to student engagement, and this ap
The idea of a first-order approximation is empha-
proach reflects my own understanding of calculus and my
sized here – with a comment that higher-order approxima-
peculiar predilections. But I would suggest this – the next
tions lie in the realm of Maclaurin series. Thus,
time you hear yourself telling your students “you’ll need this later,” wonder if there isn’t some other way. (The problem with precalculus is especially pernicious, but save that for a later discussion.)
immediately giving f1(x) = -1/x2.
After working out a few
specific examples on their own, students encountered h(x) = f(x)/g(x) – and derived the quotient rule themselves. In fact, some students used the form
on their exams since this was the form their derivation took. Moreover, students were expected to be able to differentiate f(x) = tan(x) by approximating tan(x+h)
using the usual
trigonometric formula.
The chain rule was somewhat problematic, so we
worked it out during class. If h(x) = f(g(x)), we write g(x+h) ≈ g(x)+hg’(x), so that f(g(x+h)) ≈ f(g(x)+hg’(x)). Then, since hg’(x) is small, we write f(g(x+h)) ≈ f(g(x)) + hg’(x)f’(g(x)), giving the chain rule.
The emphasis on linear approximation was delib-
erate, as this concept is central to a study of calculus. But as importantly, using linear approximations allows students
Dr. Vince Matsko teaches mathematics at the Illinois Mathematics and Science Academy. vince.matsko@gmail.com
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Using the STAT Manual by Jane Hemelt, Prince George’s County Public Schools, Greenbelt, MD
The STAT Manual was originally created for both students and teachers, in the Science and Technology Programs in Prince Georges County, whose courses required a statistical analysis component. In our Research Practicum course, the students are required to do a year-long research project guided by teachers who are well grounded in their scientific fields. However, the teachers were not well grounded in statistical analysis. The STAT Manual took away the anxiety the teachers had for this aspect of the course. In fact, both students and teachers found The STAT Manual to be an enjoyable way to learn. While it was never intended to be the principal text for a statistics course, it has been used as such in middle school settings and even for a college level introductory statistics course. The book is laid out in a dual format: textually, as most statistics books are, and graphically with far more detailed and annotated graphs that can be found in an ordinary statistics book. In fact, it is possible to learn the statistics just by perusing the graphs. Each topic is presented in a plot-type format with villains (those who misinterpret the statistics) and heroes (those who come in with the proper statistical analysis to save the day). Our heroes include Principal Raybo Smith of High Q High School in New Jersey, Toby Mathias, an autistic custodian who saves an entire research project at a prominent pharmaceutical company, a spider named Cleopatra, Running Fox, who was responsible for the founding of an independent top-level science facility, Sandy Kandinsky who counted ants, Temp, a teen-age girl who became the real Santa Claus and many others, including astronauts and guardians who control
time and space. The villains include a menagerie of interesting characters. In fact, you might enjoy the stories whether or not you need to learn the statistics. With help from Howard Hughes Medicine Institute, the mis-
plained, interpreted, and defended. Over the years, we have noticed that students, who simply pushed the buttons and used canned statistical analysis software, had no idea what the output meant. They often published output that had absolutely no relevance to their project and was in fact misleading. Essentially they had no ability to explain what they had done, because they had omitted an important component of their research. It is crucial that we, as teachers, insist that students understand what they are doing. Students who use The STAT Manual have hands-on experience and actually do understand what they are doing. They have the ability to explain their results to teachers, other students, and adjudicators.
It is crucial that we, as teachers, insist that students understand what they are doing.
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NCSSSMST Journal | Fall 2013
sion began. Seven years later, the book was finally published. Along the way, there were teachers’ workshops, presentations at conferences, even testing done by a student focus group. In an educational setting, the mathematical underpinnings of scientific research are themselves an important part of the research. Mathematics is the common accepted element by which all research can be ex-
An unsolicited endorsement came from one of our graduates who was tutoring students: “I totally used The STAT Manual to learn statistics in ~2 hours! I picked up a new student a few months ago, and had to give myself a whirlwind review...I think I learned more reading that book than I did in several YEARS of data analysis at Caltech! Just thought you should know.”
Jane Hemelt, jhemelt@pgcps.org
Virtual Field Trips: Using Technology to Enhance Learning by Dr. Steve Canipe, Program Director, Minneapolis, MN
Remember those fun and educational trips outside the school building that students looked forward to taking? With retail gasoline prices hovering around $4 per gallon; the cost for a bus driver increasing; missing school time for unproductive travel; losing time which could be spent on Common Core and other test-related activities; concerns over safety and insurance costs; many schools have either totally curtailed or severely limited these educational trips.
By following the fairly simple procedures included in this article, teachers can once again expose their students to the world outside their school classroom without the issues mentioned earlier being major concerns. Is this virtual experience the same as an actual trip? The answer is obviously “not the same” but virtual experiences can help fill in the experiential gap and create a more level field for all students.
been found to work well. Teaching is part science and part art, as has been noted by many educators over the years. Creating VFTs is no different;
National Science Teaching Standards around the Structure and Function in Living Systems. Living systems at all levels of organization demonstrate the complementary nature of structure and function. This map was created with the free X-Mind software (http://www.xmind.net/). There are numerous other idea-mapping software programs available. Regardless of the idea-mapping tool selected, one of the most important things to remember is that ideas may change as you create the VFT experience.
Once a teacher creates one or two of these virtual experiences, students often emulate the techniques and technology in subsequent student projects; thereby gaining valuable technology skills, and also creating meaningful, authentic products, which can be used over and over in the classroom or posted online.
The idea of copyright is a very important concept to present to students and teachers in this day of easy “copy and paste”.
As with any lesson, the first task is to decide on what outcomes are going to be taught and then decide if a virtual field trip (VFT) can be used effectively in helping students achieve the objectives. Once this preliminary question about using a VFT has been answered, it will be possible to begin the multistep process of creating an effective and rewarding learning experience for students, as well as teachers. The procedures mentioned are certainly not the only way to create these learning experiences but over many years of trial and error, they have
if you find a better way to accomplish learning objectives and have fun in learning, please share your efforts with others. Once the specific objective is determined, you can begin to decide what resources are needed and where they can be found. A simple idea map is a good place to start because it helps in a practical way to ensure that activities are congruent and support the teaching/learning objectives. A sample of a simple idea map for a desert VFT is shown in Figure 1. It is based on the
Whenever a VFT is used, whether created by the teacher or by students, it is very important to make sure that all legal, ethical, and fair use issues are considered. The idea of copyright is a very important concept to present to students and teachers in this day of easy “copy and paste”. If students appear in the project, get parental clearance, even if your school has a blanket permission form that goes home on the first day of school. Applicable school board poli27
cies must always be followed. Using images and videos taken from the internet requires consideration of copyrights. Some individuals and websites give educational uses special permission regarding use; however, care must be taken to ensure compliance with fair use. Some online sites provide copyright free images, while others are provided in a copyright friendly mode. Copyright friendly generally means that as long as the material is being used for non-commercial, educational purposes the requirement is only to properly cite the owner. The best policy is to use sites that are either copyright free or copyright friendly. One favorite copyright friendly site, which has thousands of static images, is Pics4Learning (www. pics4learning.com). Others can be found via an internet search. VFTs, just like actual field trips, should not be undertaken just to do
them--consult your curriculum and syllabus. A danger that some teachers fail to see is that since VFTs are relatively easy to do they are done without a clear learning objective. This type of thinking is detrimental to learning and an inappropriate use of student and teacher time. Starting with the outcomes from the curriculum allows focus on the specifics that need to be included. A wellthought out VFT can be as simple as using a website, a PowerPoint presentation, a video stored on a DVD, or virtual reality images. To help inform these decisions, it is important to think how students might use the experience and enhance learning. With increasing prevalence of personal devices like a Smartphone, tablet, or similar device, it is possible to have a greater impact on learning. Whether the ultimate use is to stimulate and enhance learning, the presentation
should focus on the format: linear, branching, static, interactive, etc. Static images like a photograph or moving images like a video can now be easily created with narration using widely available software like iMovie™ and MovieMaker™. Some minimal animation is also easily possible. Using an interactive process called 3D-VR (three dimensional virtual reality) is now possible. Many digital cameras and Smartphones have the ability to create 3D-VR giving a 360° view. This is the same technology that allows a consumer to examine a product, a car, or even a house from many different angles. Delivery choices for the VFT creator/designer are nearly limitless and increasing almost weekly with new software and apps being created. The purpose of this paper is not to provide a tutorial on the various directions that a teacher might take in creating a VFT. The primary purpose is to encourage the VFT designer to simply get out there and try the process paying close attention the caveats mentioned. Thus far it has been noted that the creator of a VFT should: 1. Decide on the purpose matching the purpose to a curricular learning objective/outcome 2. Assemble various images (still or moving) which are needed to create the project 3. Decide on the specific delivery method (PowerPoint; video, internet; etc) 4 Produce the VFT and try it 5. Evaluate the process and make modifications as needed
Figure 1. Desert Field trip idea map created using X-Mind software. 28
NCSSSMST Journal | Fall 2013
The most efficacious way to begin
is to follow the KISS principle (keep it simply simple) and not to try too many different things at first. The available processes now allow the VFT creation to include static images, websites, videos, 3D-VR, panoramic images, and others. So what should be in a VFT creator’s tool box? The types of programs/ tools are listed in Figure 2 as suggestions only. Specific hardware and software is left to the VFT creator. Figure 2: Suggested Tool Box Items
Dr. Steve Canipe is the Program Director for Science, Mathematics, and Instructional Design and Technology in the Richard W. Riley College of Education and Leadership at Walden University, Minneapolis, MN. steve.canipe@waldenu.edu
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Notes
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NCSSSMST Journal | Fall 2013
Notes
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NCSSSMST Journal | Fall 2013