____ THE ______ MATHEMATICS____ _________ EDUCATOR _____ Volume 17 Number 2

Fall 2007
MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editors Kelly W. Edenfield Ryan Fox
Dear TME Reader, Along with my coeditor, Ryan Fox, and the rest of the editorial team, I am happy to present a new issue of The Mathematics Educator, the second and final issue of Volume 17. In it, I hope you will find pieces that are educational, thought provoking, and promote continued dialogue on issues of interest to the mathematics education community.
Associate Editors Tonya Brooks Allyson Hallman SooJin Lee Diana May Kyle Schultz Susan Sexton Catherine Ulrich
As is a tradition of TME, this issue presents a variety of viewpoints on topics in mathematics education. However, the first half of this issue does have a common focus: problem solving. This focus on problem solving is apropos, especially in the state of Georgia as our K12 mathematics curriculum is in the process of changing to a performance standards based curriculum that challenges both students’ and teachers’ thinking about mathematics and what it means to do mathematics. Steve Benson, in a guest editorial, starts off by presenting his view that mathematics is learned through problem solving and continues by sharing teaching methods that have helped his students become better problem solvers. In his theoretical analysis, Jamin Carson argues for the importance of a knowledge base and transferability of knowledge in the teaching of problem solving. José Contreras completes the problem solving section by presenting his problemposing framework, a framework for guiding the modification of given problems to create other interesting, worthwhile mathematics problems.
Advisor Dorothy Y. White
MESA Officers 20072008 President Rachael Brown VicePresident Nick Cluster
The final three pieces provide glimpses into two research studies and into the Turkish teacher education curriculum. In the first, Brian Evans examines the attitudes, conceptions, and achievement of college students enrolled in undergraduate statistics courses, revealing some unexpected findings. Janet Frost and Lynda Wiest continue by studying how a girls’ mathematics and technology camp affected the confidence of the participants, paying particular attention to girls of color and those of low socioeconomic status. The final article by Mine Isiksal, Yusuf Koc, Sarure Bulut, and Tulay AtayTurhan outlines the new elementary mathematics teacher education in Turkey, including the motivations for the changes and the present status of the revisions.
Secretary Kelly Edenfield Treasurer Susan Sexton Colloquium Chair Dana TeCroney NCTM Representative Kyle Schultz
The production of TME requires the help and support of numerous groups of people. As a studentrun and studentproduced journal, TME is responsible for recruiting and training reviewers, associate editors, and editors, primarily from the ranks of present graduate students. I greatly appreciate the time and energy put forth by these members of our staff. I would like to thank all the reviewers, authors, and faculty members whose work and advice have enabled us to present the readers with this issue. In particular, I extend my gratitude to our training seminar participants and to the editorial staff for all of their hard work.
Undergraduate Representative Dan Davis Kelli Parker
Kelly W. Edenfield 105 Aderhold Hall The University of Georgia Athens, GA 306027124
tme@uga.edu www.coe.uga.edu/tme
About the Cover This issue’s cover features two GSP sketches from the article by José Contreras. Dr. Contreras invites us into his problemposing framework through an example; he takes a straightforward problem about isosceles triangles and poses additional interesting problems about triangles along with other geometric shapes.
This publication is supported by the College of Education at The University of Georgia
____________ THE ________________ ___________ MATHEMATICS ________ ______________ EDUCATOR ____________ An Official Publication of The Mathematics Education Student Association The University of Georgia
Fall 2007
Volume 17 Number 2
Table of Contents 2 Guest Editorial… Problem Solving by Analogy / Problem Solving as Analogy STEVE BENSON 7 A Problem With Problem Solving: Teaching Thinking Without Teaching Knowledge JAMIN CARSON 15 Unraveling the Mystery of the Origin of Mathematical Problems: Using a ProblemPosing Framework With Prospective Mathematics Teachers JOSE CONTRERAS 24 Student Attitudes, Conceptions, and Achievement in Introductory Undergraduate College Statistics BRIAN EVANS 31 Listening to the Girls: Participant Perceptions of the ConfidenceBoosting Aspects of a Girls’ Summer Mathematics and Technology Camp JANET HART FROST & LYNDA R. WIEST 41 An Analysis of the New Elementary Mathematics Teacher Education Curriculum in Turkey MINE ISIKSAL, YUSUF KOC, SAFURE BULUT, & TULAY ATAYTURHAN 52 Submissions information 53 Subscription form
© 2007 Mathematics Education Student Association All Rights Reserved
The Mathematics Educator 2007, Vol. 17, No. 2, 2–6
Guest Editorial… Problem Solving by Analogy / Problem Solving as Analogy Steve Benson Everyone talks about how important it is for a young quarterback to sit on the bench and watch the game. But instead of learning how to play, all they learn is how to sit and watch. (A paraphrase of Mike Ditka on ESPN’s Sunday NFL Countdown, November 18, 2007)
If current frameworks, standards, and assessments are any indication, there is international consensus that students should be able to solve new (to them) mathematical problems (“realworld” and otherwise) in addition to knowing specific facts and performing basic calculations. Problem solving as a part of the mathematics curriculum has gone in and out of favor for several decades, perhaps due to the range of ways it has been approached in textbooks and classrooms. Too often, problem solving is taught very algorithmically and, as mentioned in another article in this issue, is seen as independent of mathematical content. In fact, I believe that mathematics is learned through problem solving, so when taught well, mathematical content and problem solving can’t really be separated. A number of problem solving “habits of mind” are taught explicitly in mathematics courses at all levels. Many of these ways of thinking can be traced to suggestions from How to Solve It and other publications by the father of modern problem solving, George Pòlya (1945, 1954). (In fact, in Volume I of his Mathematics and Plausible Reasoning, he wrote extensively about analogy in mathematics.) I won’t restate these suggestions here since most have become part of the present day mathematical lexicon, but I would like to present some methods and ideas that I have found promising in helping my students become more successful problem solvers.
After spending 7 years at Education Development Center in Newton, MA, creating and facilitating contentbased professional development materials, Steve Benson is now an associate professor of mathematics at Lesley University in Cambridge, MA. He earned his Ph.D. at the University of Illinois and has taught at St. Olaf College, Santa Clara University, University of New Hampshire, and University of Wisconsin—Oshkosh. He has been a coDirector of the Master of Science for Teachers program in the UNH mathematics department since 1997 and was lead author on "Ways to Think About Mathematics: Activities and Investigations for Grade 612 Teachers," published by Corwin Press. 2
Analogy in Problem Solving A common trait of expert problem solvers is their ability to recognize connections between two or more problem situations and their solution methods. That is, by reposing a problem in another context, the problem is often made more tractable. The Handshake Problem: The twenty members of the math club met last Tuesday to plan next month’s annual banquet. A tradition of the club is to start each meeting by having the members shake hands with each other. How many handshakes will occur? There are two common strategies students (and others) use to approach this problem. The first is motivated by the observation that we could arrange the club members in some order (from first to twentieth). The counting of the handshakes usually starts something like this: The first member shakes hands with each of the other 19 members, while the second member shakes hands with 18 others, the third members shakes hands with 17 people, and so on, until the 19th person shakes one person’s hands and the 20th person doesn’t shake any. Therefore, the total number of handshakes is 19 + 18 + 17 + … + 2 + 1 = 190. (Of course, the 2nd member is involved in a total of 19 handshakes, like all the others, but the handshake with the first member has already been counted, so it would be more correct to say that the 2nd member was involved in 18 more handshakes, the 3rd member (who had already shaken hands with members 1 and 2) was involved in 17 more handshakes, and so on.) The second strategy usually goes something like the following: Each of the 20 math club members would have shaken the hands of the other 19 members, for a total of 20•19 = 380 hands being shaken. But each handshake requires two hands (or each handshake gets counted twice), so there are 380/2 = 190 handshakes in all. Problem Solving and Analogy
Later, when asked to determine a closed form solution for 1 + 2 + 3 + … + n (determining the nth triangular number, for example), many students recognize that this sum corresponds to using the first method of solving the handshake problem for a group of n + 1 people. Since the second solution method (n +1) " n , the students arrive involves the calculation 2 at a closed form solution to the sum by taking advantage of this connection. Of course, there are several alternative strategies that would give! rise to the solution, but the example serves as an illustration of the point that the ability to make connections between apparently dissimilar problems is something we, as educators, hope to promote in our classes. In each class I teach, I try to convince my students that solving mathematical problems is an exercise in reasoning by analogy. In fact, we could argue that since each new problem is solved by somehow connecting it to our existing knowledge, this reasoning by analogy is always happening in one way or another. Another (possibly less familiar) example came up recently in my Number Theory course for inservice elementary and middle school teachers. How many ways can you spell NUMBERS? Starting at the top and always moving to adjacent letters in the row directly below, how many paths can you take to spell NUMBERS? N U U M M M B B B B E E E E E R R R R R R S S S S S S S
As anticipated, many students observed that, starting with the N in the first row, there are 2 choices (right or left) when you move down to the second row. Similarly, there are 2 choices when moving from each subsequent letter, so there are 26 = 64 different paths that spell NUMBERS. However, several students used a strategy I didn’t expect; however, in retrospect, I should have foreseen it. These students kept track of the number of ways to get to each individual letter using a method we had used earlier to count taxicab paths between lattice points in the coordinate plane, labeling each letter as shown below. Steve Benson
N1 U U1 1 M M2 M1 1 B B3 B3 B1 1 E E4 E6 E4 E1 1 R R5 R10 R10 R5 R1 S1 S6 S15 S20 S15 S6 S1 The superscript for each letter represents the number of paths that start at the top N and lead to that letter. 1
Adding up the number of ways to get to each letter on the bottom row, they also arrived at the total of 64 paths that spell NUMBERS. One student spoke up, “Hey, that gives us a way to prove that conjecture we made earlier—that the sum of the entries in a row of Pascal’s Triangle is a power of 2!” Of course, I was very happy that she made this connection, but I was even more pleased since I knew it had occurred naturally; I hadn’t “telegraphed” the strategy for them since it hadn’t occurred to me that anyone would solve the problem that way. The surprising connections continued when I asked my students to consider the same question with the following configuration. My goal was to give the other students an opportunity to use the student’s method we had just discussed. N U U M M M B B B B E E E R R S
I had assumed that they would recognize that there were 20 ways to get to the S in the middle of the 7th row, as determined in the previous problem. Most did as I expected, but one student looked at it another way. She noticed that she could label the first 4 rows as before. N1 U U1 1 M M2 M1 1 B B3 B3 B1 E E E R R S 1
But then she started from the S at the bottom and worked up, noticing there were the same number of ways to go down to each B as there were ways to get up to them. 3
N1 U U1 1 M M2 M1 1,1 B B3,3 B3,3 B1,1 1 E E2 E1 1 R R1 S1 1
She then computed the sum 1•1 + 3•3 + 3•3 + 1•1 = 20, using the following explanation. There is 1 way (1•1=1) to spell NUMBERS by going through the first (leftmost) B. That is, there is 1 way to get to that B from the top and only 1 path from that B to the S at the bottom. There are also 3•3=9 ways to go through the second B, i.e., 3 paths down to the B and 3 paths from there to S. Identically, there are 3•3=9 ways to go through the third B. Finally, only 1•1=1 way exists by going through the last B. This strategy yields a total of 1 + 9 + 9 + 1 = 20 paths that spell NUMBERS. Of course, the students aren’t the only ones who are learning in a class that is based on problem solving. As the student was explaining her solution, it occurred to me that she had just outlined a very natural, elementary proof of the following theorem: The sum of the squares of the entries in row n of Pascal’s Triangle is the middle term of row 2n. n
That is,
n 2 k
"( ) k= 0
=
( ). 2n n
Following the student’s lead, I was able to reason by analogy to see the connection with another problem and make a connection I had never seen before. When I ! gushed about how cool I thought her solution was, she was taken aback by my surprise, saying, “I thought this is what you wanted us to do! I was just trying to use what we did before.” This was gratifying because a common mantra I repeat in classes I teach is, “How is this like something I’ve done before?” By this, I do not mean to give the impression that problem solving is remembering how to solve a problem. Rather, being successful at problem solving requires the ability to sift through all the stuff you know and pick out those facts and techniques and strategies that might apply to the problem at hand. MetaProblem Solving Discussion of the actual process of problem solving is missing from the vast majority of mathematics textbooks, which tend to focus on the final process of proof. How the author figured out the solution is often missing, thus leaving the impression that the author just knew how to solve the problem. (One reason for this could be the traditional textbook style, while another is the need of the publisher to 4
reduce costs by limiting the amount of text and, therefore, paper.) I know that if I’m not careful, I can also give this impression in my classes, especially if I’m in a hurry. (When are we not in a hurry, though?) When leading a discussion of a problem, I attempt to “fill in the blanks” with explanations of my thinking: What I was thinking here was… or I noticed that this problem felt similar to … or I wasn’t sure what to do, so I decided to try … I also elicit suggestions from the class so they won’t think the whole discussion was “scripted.” We often pursue strategies that I’m pretty sure (or I know) won’t be successful, because much can be learned from “dead ends,” as well. This seems to convince students that I really am figuring out the problem with them rather than remembering (or just knowing) how to do it. They see that it’s okay if you don’t know what to do when you encounter a problem and that they, too, can figure out the problems. These discussions of metaproblem solving—the processes of trying and failing, trying again and making partial progress, and so on, that problem solving entails—inevitably involve statements like Doing mathematics is like hiking through the woods; Problem solving is like driving a car; Problems are just really big Sudoku puzzles. Analogies of Problem Solving The use of analogy in solving problems allows the solver to connect the familiar (a previously used method, strategy, or context) to the unfamiliar (a new problem). Therein lies the power of analogy. Even when I try to share my thinking during problem solving discussions, students have difficulty applying the methods and strategies they’ve previously used to solve a new problem. I had believed this was due to unfamiliarity with having to solve problems, rather than exercises designed to practice a particular skill or use a specific, easily identified, concept. As previously mentioned, I have always tried to model the problem solving process in classes, but realize that at the end of a problem discussion, even when insisting on student input for direction and strategy, I am often the one who solved the problem. I have repeatedly shared with students what I consider one of the most important facts about problem solving: you’re not expected to know how to solve a problem; you’re supposed to figure out how to solve it. But even after providing them with many problem solving experiences, often in more than one course, many still have trouble getting started, saying that they’re not sure what the “right way” to solve the problem is. I believe that this is due to the fact that they don't have a sense for what actual Problem Solving and Analogy
problem solving feels like. By building explicit connections to other activities with which they might be familiar, I hope my students are able to feel more comfortable with the processes (and inherent uncertainty) of problem solving. None of these analogies are perfect, and it is easy to stretch them to their breaking points, but I have found that each of them serves as a touchstone for the problem solving process. I’ll explain one in detail. A Mathematical Hike in the Woods There are at least three different ways to go hiking in the woods. One is to be led down a previously created path, often by an expert who’s taken the path before. Another is to follow a path with which you are already familiar, perhaps after being led along the path several times. The third is to be willing to leave a familiar path to try a completely new trail when the need arises (or just for the heck of it). Being led on the hike is efficient, if your goal is to get to the end of the trail. You can see some sights but only those you are led to. It is probably the most comfortable method for the novice hikers, because it isn’t necessary for them to keep track of where they are. However, this also doesn’t help them learn to get to places that are not on the trail. Traveling alone on a path with which you are familiar is less efficient, since now you don’t have an expert to keep you on the trail. However, it’s definitely more interesting, since you can choose when and where to stop and how quickly to walk. Of course, in order get to a location you haven’t already been to, you must be willing to stray off the path and possibly blaze a new trail every now and then. Some of these new locations might be just off the path, while others may be far away from your comfort zone, but these lesstraveled sights are often the most interesting (and educational). And each new trail you forge provides you with new locations you know how to get to (and return to later). But many hikers aren’t natural explorers. It isn’t likely that someone who has always been led through the woods will stray far from the known path. It takes a rare person to feel confident enough to take over leading the group to the end of a trail (or even back to where they started) or to choose to lead the group entirely off the path just to explore, unless he had been given the chance to explore in the past. Unless it is your responsibility to get everyone back to the trailhead, your mind is usually focused on following the leader.
Steve Benson
The second method (taking a path with which you are familiar but without a leader) is more work, since you need to pay closer attention to where you are and which branches of the path you need to choose. You might feel limited to taking the particular path you are on, but the real fun comes when you veer from the known path and explore, knowing that if you choose the left fork and arrive at a dead end, you can always find your way back and choose the other fork. When you’re the leader of the hike (or at least an active participant in the decision making), whenever you break off a familiar path and look for new sights to see, you must be aware of where the familiar path is (in relation to your current location) so you don’t get lost. Learning to be comfortable with straying off the path (or becoming a trailblazer) comes from experience, but that experience need not be a solo effort. A good hike leader will point out trail markers and share his or her decisions with fellow hikers, letting them in on the thought processes being used as options are chosen and decisions are made. And novices could be encouraged to take charge under the watchful eye of the hike leader, who allows them to make the decisions. Even if the novices get lost, the leader can keep track of their location and bring them back to familiar territory (or help them solve the problem of finding their way back themselves). As novices become more comfortable (and experienced) with making decisions and realizing that every decision is reversible, they are more willing (and able) to explore on their own. If novice hikers ever find themselves in unfamiliar territory, they will have a very difficult time finding their way out if their only hiking experiences involved having been led or traveling on familiar paths. Therein lies the key connection with problem solving. If we are to help students learn to solve new problems that they haven’t seen before, they need experiences—guided and otherwise—that allow them to try and fail, try something else, and eventually arrive at a solution to the problem. They aren’t alone, though, because the teacher/hike leader is there as a safety net—not to solve the problem for them, but to serve as a mirror, reflecting their strategies and progress, asking probing questions that encourage the novices to think through their options. Experienced hikers, like expert problem solvers, are able to keep track of where they are, where they need to be, and the options available to them at any given moment. By explicitly discussing these options and decisions with novices (hikers and problem solvers), the novices gain an understanding of the process and are more likely to be able to navigate the 5
paths themselves. Pòlya (1954) spoke of the importance of intellectual courage—being ready to revise ones beliefs. I like to expand this notion a little further to include a willingness to persevere even when you don’t know whether your strategy will lead to a solution or to a dead end. Isn’t that the goal we have for all our students? Other Analogies of Problem Solving I leave it to the reader to elaborate on further analogies and to propose new ones. Of course, the list of analogies is endless, for what is life, but a series of “problems” to solve and situations to explore? Learning to solve/learning to drive. Many of the analogies with hiking can be transferred to finding your way around city streets. I have found many problem solving opportunities while driving in the Boston area. You never know when a road, bridge, exit ramp, or tunnel will be blocked off, often without detour signs to help you find your way back to your usual path. And until you are behind the wheel on your own, you don’t realize how little you learn by sitting in the passenger seat and watching someone else drive! Mathematics as a big Sudoku puzzle. There are many connections with Sudoku and problem solving.
6
As you solve more puzzles, you begin to notice patterns (and analogous configurations), develop useful strategies, and become more comfortable with each succeeding puzzle, recognizing that every puzzle is different and you never know which of the many techniques will be the most useful on the next one. And sometimes you just have to make a guess and keep track of the consequences of your assumption. If it turns out you run into a contradiction, you just backtrack (there’s that hiking analogy again!), change your guess, and continue. Mathematics as a video game. I conclude by stating the first “analogy of problem solving” I ever used in my teaching—a statement I’ve long included on course syllabi: Mathematics is like a video game; if you just sit and watch, you’re wasting your quarter (and semester).
References Pòlya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press. Pòlya, G. (1954). Mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.
Problem Solving and Analogy
The Mathematics Educator 2007, Vol. 17, No. 2, 7–14
A Problem With Problem Solving: Teaching Thinking Without Teaching Knowledge Jamin Carson
Problem solving theory and practice suggest that thinking is more important to solving problems than knowledge and that it is possible to teach thinking in situations where little or no knowledge of the problem is needed. Such an assumption has led problem solving advocates to champion contentless heuristics as the primary element of problem solving while relegating the knowledge base and the application of concepts or transfer to secondary status. In the following theoretical analysis, it will be argued that the knowledge base and transfer of knowledge—not the contentless heuristic—are the most essential elements of problem solving.
Problem solving theory and practice suggest that thinking is more important in solving problems than knowledge and that it is possible to teach thinking in situations where little or no knowledge of the problem is needed. Such an assumption has led problem solving advocates to champion contentless heuristics as the primary element of problem solving while relegating the knowledge base and the transfer or application of conceptual knowledge to secondary status. Yet if one analyzes the meaning of problem solving, the knowledge base and the transfer of that knowledge are the most essential elements in solving problems. Theoretical Framework Problem solving is only one type of a larger category of thinking skills that teachers use to teach students how to think. Other means of developing thinking skills are problembased learning, critical thinking skills, creative thinking skills, decision making, conceptualizing, and information processing (Ellis, 2005). Although scholars and practitioners often imply different meanings by each of these terms, most thinking skills programs share the same basic elements: (1) the definition of a problem, (2) the definition of problem solving, (3) algorithms, (4) heuristics, (5) the relationship between theory and practice, (6) teaching creativity, (7) a knowledge base, and (8) the transfer or the application of conceptual knowledge. The Definition of a Problem The first element of the theory of problem solving Dr. Jamin Carson is an assistant professor of curriculum and instruction at East Carolina University. He teaches the theory and practice of instruction as well as classroom management and discipline. His primary research interest is the epistemology of curriculum and instruction. Jamin Carson
is to know the meaning of the term problem. This theoretical framework uses the definition of problem presented by Stephen Krulik and Jesse Rudnick (1980) in Problem Solving: A Handbook for Teachers. A problem is “a situation, quantitative or otherwise, that confronts an individual or group of individuals, that requires resolution, and for which the individual sees no apparent or obvious means or path to obtaining a solution” (p. 3). The Definition of Problem Solving Krulik and Rudnick (1980) also define problem solving as the means by which an individual uses previously acquired knowledge, skills, and understanding to satisfy the demands of an unfamiliar situation. The student must synthesize what he or she has learned, and apply it to a new and different situation. (p. 4)
This definition is similar to the definition of the eighth element of problem solving, transfer: “[w]hen learning in one situation facilitates learning or performance in another situation” (Ormrod, 1999, p. 348). Problem Solving is Not an Algorithm One of the primary elements of this framework is that problem solving is not an algorithm. For example, Krulik and Rudnick (1980) say, The existence of a problem implies that the individual is confronted by something he or she does not recognize, and to which he or she cannot merely apply a model. A problem will no longer be considered a problem once it can easily be solved by algorithms that have been previously learned. (p. 3)
7
Table 1 Types of Problem Solving
Steps in Problem Solving
John Dewey (1933)
George Polya (1988)
Stephen Krulik and Jesse Rudnick (1980)
Confront Problem
Understanding the Problem
Read
Diagnose or Define Problem
Devising a Plan
Explore
Inventory Several Solutions
Carrying Out the Plan
Select a Strategy
Looking Back
Solve
Conjecture Solutions
Consequences
of
Test Consequences
Additionally, advocates of problem solving imply that algorithms are inferior models of thinking because they do not require thought on a high level, nor do they require deep understanding of the concept or problem. Algorithms only require memory and routine application. Further, they are not useful for solving new problems (Krulik & Rudnick, 1980). Problem Solving is a Heuristic Advocates of problem solving argue that educators need to teach a method of thought that does not pertain to specific or presolved problems or to any specific content or knowledge. A heuristic is this kind of method. It is a process or a set of guidelines that a person applies to various situations. Heuristics do not guarantee success as an algorithm does (Krulik & Rudnick, 1980; Ormrod, 1999), but what is lost in effectiveness is gained in utility. Three examples of a problem solving heuristic are presented in Table 1. The first belongs to John Dewey, who explicated a method of problem solving in How We Think (1933). The second is George Polya’s, whose method is mostly associated with problem solving in mathematics. The last is a more contemporary version developed by Krulik and Rudnick, in which they explicate what should occur in each stage of problem solving. I will explain the last one because it is the more recently developed. However, the three are fundamentally the same. The following is an example of how the heuristic is applied to a problem. Problem: Twelve couples have been invited to a party. The couples will be seated at a series of small square tables, placed end to end so as to form 8
Review and Extend
one large long table. How many of these small tables are needed to seat all 24 people? (Krulik & Rudnick, 1987, pp. 29–31)
The first step, Read, is when one identifies the problem. The problem solver does this by noting key words, asking oneself what is being asked in the problem, or restating the problem in language that he or she can understand more easily. The key words of the problem are small square tables, twelve couples, one large table, and 24 people. The second step, Explore, is when one looks for patterns or attempts to determine the concept or principle at play within the problem. This is essentially a higher form of step one in which the student identifies what the problem is and represents it in a way that is easier to understand. In this step, however, the student is really asking, “What is this problem like?” He or she is connecting the new problem to prior knowledge. The student might draw a picture of what the situation would look like for one table, two tables, three tables, and so on. After drawing the tables, the student would note patterns in a chart. (See below.) The third step, Select a Strategy, is where one draws a conclusion or makes a hypothesis about how to solve the problem based on the what he or she found in steps one and two. One experiments, looks for a simpler problem, and then conjectures, guesses, forms a tentative hypothesis, and assumes a solution. The fourth step is Solve the Problem. Once the method has been selected the student applies it to the problem. In this instance, one could simply continue the chart in step three until one reached 24 guests.
Problem Solving
Step 2: Explore. Draw a diagram to represent the problem.
!
1
Number of guests
4
2
3
6
4
8
.
10
.
.
Problem Solving Teaches Creativity
.
.
.
Pattern: As we add a table, the number of guests that can be seated increases by 2.
Real world situations require creativity. However, it has often been claimed that traditional classrooms or teaching approaches do not focus on developing the creative faculty of students. Advocates of problem solving, by contrast, claim that problem solving develops the studentsâ€™ creative capacities (Frederiksen, 1984; Slavin, 1997). Successful Problem Solvers Have a Complete and Organized Knowledge Base
Step 3: Select a Strategy. Number of tables
1
2
3
4
5
6
7
Number of guests
4
6
8
10
12
14
16
Form a tentative hypothesis. Since the pattern seems to be holding true for 16 guests, we can continue to add 1 table for every additional guest until we reach 24. Therefore, we add 4 additional tables for the additional guests (16 + 8 = 24). Hypothesis: It will take 11 tables to accommodate 24 guests. Step 4: Solve the Problem Number of tables
1
2
3
4
5
6
7
8
9
10
11
Number of guests
4
6
8
10
12
14
16
18
20
22
24
Jamin Carson
Problem Solving Connects Theory and Practice A perennial charge brought against education is that it fails to prepare students for the real world. It teaches theory but not practice. Problem solving connects theory and practice. In a sense this element is the same as the definitions of problem solving and transfer, only it specifically relates to applying abstract school knowledge to concrete real world experiences (Krulik & Rudnick, 1980).
Make a chart, record the data, and look for patterns. Number of tables
The final step, Review and Extend, is where the student verifies his or her answer and looks for variations in the method of solving the problem; e.g., n "2 , where represents the number of tables. Or we t= 2 could ask for a formula to determine how many guests we can seat given the number of tables. For example, n = 2t + 2 or n = 2(t + 1).
A knowledge base consists of all of the specific knowledge a student has that he or she can use to solve a given problem. For example, in order to solve algebraic problems, one not only needs to know information about numbers and how to add, subtract, multiply, and divide, but one must also possess the knowledge that goes beyond basic arithmetic. A knowledge base is what must accompany the teaching of a heuristic for successful problem solving to occur. Problem Solving Teaches Transfer or How to Apply Conceptual Knowledge Transfer, or the application of conceptual knowledge, is the connecting of two or more reallife problems or situations together because they share the same concept or principle. Transfer or the application of conceptual knowledge helps students see similarities and patterns among seemingly different problems that are in fact the same, or similar, on the conceptual level. Some research about problem solving claim that it is more effective than traditional instruction (LunykChild, et al., 2001; Stepien, Gallagher, & Workman, 1993), that it results in better longterm retention than 9
traditional instruction (Norman & Schmidt, 1992), and that it promotes the development of effective thinking skills (Gallagher, Stepien, & Rosenthal, 1994; Hmelo & Ferrari, 1997). On the other hand, in Research on Educational Innovations, Arthur Ellis (2005) notes that the research base on problem solving lacks definition, possesses measurement validity problems and questionable causality, and it fails to answer the claim that successful problem solvers must have a wealth of contentspecific knowledge. Ellis further notes that there is “no generally agreedon set of definitions of terms” (p. 109), that thinking skills are notoriously difficult to measure, and that given these first two problems, it is impossible to trace cause back to any specific set of curricular instances. Ellis states, [t]he idea that thinking skills are content specific and cannot be taught generically must be seriously entertained until it is discredited. We don’t think that will happen. And if this is so, how does one construct contentfree tests to measure thinking skills? (pp. 109–110)
The conclusions of Ellis and other research studies I will cite later state that it would be impossible to reinvent solutions to every problem that develops without recourse to past knowledge. This recourse to past knowledge is evidence, in itself, that one must not completely construct reality. One must apply knowledge that has already been formed by others and understand that knowledge, or else not solve the problem. It is this critique that I will invoke in the following treatment of problem solving. What I hope to show is that the heuristic for problem solving cannot be successful if one holds strongly to the theoretical framework in which it is often situated. Rather, one must accept that already formed knowledge is essential to problem solving. In fact, the meanings of problem solving found in articles and textbooks often convey this contradiction. On the one hand, it is argued that problem solving is the antithesis of a contentcentered curriculum. On the other hand, a successful problem solver must possess a strong knowledge base of specific information, not merely a generalizeable heuristic that can be applied across several different situations. The Problem With Problem Solving The main problem with problem solving lies in the fourth element listed above: problem solving is a heuristic. Recall that a heuristic is a guideline that may or may not yield success but, unlike an algorithm, it does not depend on knowledge of the problem to be 10
successful. Heuristic is a method of thought that does not pertain to any specific problems or content. The element is problematic because it contradicts three other elements within the theory: the definition of problem solving, successful problem solving requires a knowledge base, and problem solving enables learners to transfer knowledge. Each of these three elements implies that previously learned knowledge of the problem is necessary to solving the problem, whereas use of a heuristic assumes no knowledge is necessary. I argue, like Peikoff (1985), that there is no way to separate thinking or problem solving from knowledge. Just like instruction and curriculum, these concepts imply one another and cannot be discussed separately for long. Likewise, to acquire knowledge, one must think. This is not to say that students cannot construct knowledge as they solve a given problem, only to say that often the problems they are presented only require them to apply existing knowledge. From this perspective, it must be assumed that students do not construct all of the knowledge in a given curriculum. Yet problem solving as a heuristic is the most cherished aspect of problem solving because it is contentless. For example, in the preface to Mathematical Discovery, George Polya (1962), one of the foremost thinkers on problem solving says, I wish to call heuristic the study that the present work attempts, the study of means and methods of problem solving. The term heuristic, which was used by some philosophers in the past, is halfforgotten and halfdiscredited nowadays, but I am not afraid to use it. In fact, most of the time the present work offers a downtoearth practical aspect of heuristic. (p. vi)
Instructional textbooks sometimes play off this process versus content dichotomy: a teacher can either teach students to be critical thinkers and problem solvers or she can teach students more content knowledge. The authors of one textbook say, Too often children are taught in school as though the answers to all the important questions were in textbooks. In reality, most of the problems faced by individuals have no easy answers. There are no reference books in which one can find the solution to life’s perplexing problems. (Gunter, Estes, & Schwab, 2003, pp. 128–129)
The dichotomy implies that thinking and knowledge are mutually exclusive, when in fact critical thinking and problem solving require a great deal of specific content knowledge. Problem Solving
Problem solving and heuristics cannot be contentless and still be effective. Critical thinking, problem solving, and heuristics must include a knowledge base (Fredricksen, 1984; Ormrod, 1999). Including the knowledge base enables the principle cognitive function of problem solvingâ€”the application of conceptual knowledge, or transferâ€”to occur (Peikoff, 1985). However, the degree to which Dewey and Polya actually believed that a heuristic could be completely contentless and still be effective is not clear. Further, many instructional textbooks actually stress the importance of content knowledge in solving problems (Henson, 2004; Kauchak & Eggen, 2007; Lang & Evans, 2006). The Elements of Problem Solving Revised Each of the above elements of problem solving will be reviewed again in light of the relationship between thinking and knowledge and the research base on problem solving. Element one, the definition of a problem, implies that one must have some knowledge of the problem to solve it. How can one solve a problem without first knowing what the problem is? In fact, identification of the problem is what is called for in the first two steps, Read and Explore, of the heuristic. In this step, the student first becomes aware of the problem and then seeks to define what it is or what the problem requires for its solution. Awareness and definition comprise the knowledge that is essential to solving the problem. Consider the effectiveness of students relative to their respective experiences with a given problem. The student more familiar with the problem will probably be better able to solve it. In contrast, the student new to the problem, who has only studied the heuristic, would have to reinvent the solution to the problem. So the first two steps of the heuristic imply that one needs a great deal of knowledge about the problem to be an effective problem solver. In fact, if one wants to solve the problem for the long term, one would want to thoroughly study the problem until some kind of principles were developed with regard to it. The final outcome of such an inquiry, ironically, would yield the construction of an algorithm. The second element, the definition of problem solving, also implies a connection between thinking and knowledge. It says that problem solving is essentially applying old knowledge to a new situation (Krulik & Rudnick, 1987). However, if knowledge or a problem is genuinely new, then the old knowledge would not apply to it in any way. Ormrod (1999) suggests that the socalled new situation is really the Jamin Carson
same as the old in principle. For example, the principle of addition a student would use to solve the problem 1 + 2 = 3 is essentially the same principle one would apply to 1 + x = 3. The form may be different but ultimately the same principle is used to solve both problems. If this is the case, then a more proper element of problem solving would be number eight, the transfer of knowledge or application of conceptual knowledge. The third and fourth elements algorithms and heuristics are problematic. Krulik and Rudnick (1980) distinguish between algorithms and heuristics. Unlike employing an algorithm, using a heuristic requires the problem solver to think on the highest level and fully understand the problem. Krulik and Rudnick also prefer heuristics to algorithms because the latter only applies to specific situations, whereas a heuristic applies to many as yet undiscovered problems. However, an algorithm requires more than mere memorization; it requires deep thinking too. First, in order to apply an algorithm, the student must have sufficient information about the problem to know which algorithm to apply. This would only be possible if the student possessed a conceptual understanding of the subject matter. Further, even if a student could somehow memorize when to apply certain algorithms, it does not follow that he or she would also be able to memorize how to apply it (Hu, 2006; Hundhausen & Brown, 2008; Johanning, 2006; Rusch, 2005). Second, algorithms and problem solving are related to one another. Algorithms are the product of successful problem solving and to be a successful problem solver one often must have knowledge of algorithms (Hu, 2006; Hundhausen & Brown, 2008; Johanning, 2006; Rusch, 2005). Algorithms exist to eliminate needless thought, and in this sense, they actually are the end product of heuristics. The necessity to teach heuristics exists, but heuristics and algorithms should not be divided and set against one another. Rather, teachers should explain their relationship and how both are used in solving problems. A secondary problem that results from this flawed dichotomy between algorithms and heuristics is that advocates of problem solving prefer heuristics because algorithms only apply to specific situations, whereas heuristics do not pertain to any specific knowledge. If one reflects upon the steps of problem solving listed above one will see that they require one to know the problem to be successful at solving it.. Consider the sample problem above to which the heuristic was applied. If one knows the heuristic process and possesses no background knowledge of 11
similar problems, one would not be able to solve the problem. For example, in the first step of the heuristic one is supposed to Read the problem, identify the problem, and list key facts of the problem. Without a great deal of specific content knowledge how will the student know what the teacher means by “problem,” “key facts,” and so on? The teacher will probably have to engage the student in several problems. Without extensive knowledge of facts, how does the student know what mathematical facts are, and how they apply to word problems, for example? In the second step, Explore, the problem solver looks for a pattern or identifies the principle or concept. Again, how can one identify the pattern, principle, or concept without already possessing several stored patterns, principles, and concepts? Indeed, to a student with very little mathematical knowledge, this problem would be extremely difficult to solve. The heuristic would be of little help. The heuristic says to draw a diagram, presumably to make the problem more concrete and therefore more accessible to the student, but without already knowing what the concept the problem exhibits this would very difficult, if not impossible. Using the chart with the data as an example, it would require previous knowledge in mathematics to be able to construct it. It seems that the heuristic in this problem is in reality just another algorithm that the teacher will have to teach as directly and as repetitively until the students learn how and when to apply it, which is the very opposite of what advocates of problem solving want. The same is also true of step five, Review and Extend. Presumably if a student could represent this problem in algebraic form, he or she should also be able to solve the same problem without recourse to drawing diagrams, recording data, etc. One could simply solve the problem right after step one. The sample problem illustrates what scientists have discovered about novices and experts. In studies that examined expert and novice chess players, researchers found that their respective memories were no different in relation to random arrangements of chess pieces. When the pieces were arranged in ways that would make sense in a chess game, the experts’ memories were much better. The theory is that an expert chess player is not a better problem solver, he or she just has a more extensive knowledge base than a novice player. He or she is past the rudimentary hypothesis testing stage of learning, past the problem solving heuristic stage and is now simply applying algorithms to alreadysolved problems (Ross, 2006). The same could be said for students applying a heuristic to the above 12
problem. The only ones who could solve it would be those who use an algorithm. Even if a teacher taught the heuristic to students, he or she would essentially be teaching an algorithm. Advocates of problem solving are not solely to blame for the misconception between thinking and knowledge and between heuristics and algorithms. The misconception is likely due to teachers that have overused algorithms and never shown students how they are formed, that they come from heuristics, and that one should have a conceptual understanding of when they should be used, not merely a memorized understanding of them. The fundamentally flawed dichotomy within problem solving probably stems from thinking in terms of “eitherors.” One side defines appropriate education as teaching algorithms by having students memorize when to use them but not why. The other side, by contrast, emphasizes that thinking for understanding is preferable to simply memorized knowledge. Perhaps what has happened in the shift from the former to the latter practices is the instructional emphasis has shifted from content to thinking so much that the knowledge base has been wiped out in the process. Ironically, eliminating knowledge from the equation also eliminates the effectiveness of problem solving. The dichotomy between knowledge and thinking has also affected elements five and six. Number five states that problem solving connects theory and practice. At the core of this element is yet another flawed dichotomy. Many educators hold that education should prepare students for the real world by focusing less on theory and more on practice. However, dividing the two into separate cognitive domains that are mutually exclusive is not possible. Thinking is actually the integration of theory and practice, the abstract and the concrete, the conceptual and the particular. Theories are actually only general principles based on several practical instances. Likewise, abstract concepts are only general ideas based on several concrete particulars. Dividing the two is not possible because each implies the other (Lang & Evans, 2006). Effective instruction combines both theory and practice in specific ways. When effective teachers introduce a new concept, they first present a perceptual, concrete example of it to the student. By presenting several concrete examples to the student, the concept is better understood because this is in fact the sequence of how humans form concepts (Bruner, Goodnow, & Austin, 1956; Cone 1969; Ormrod, 1999; Peikoff, 1993). They begin with two or more concrete particulars and abstract from them the essential Problem Solving
defining characteristics into a concept. For example, after experiencing several actual tables a human eventually abstracts the concept a piece of furniture with legs and a top (Lang & Evans, 2006). On the other hand, learning is not complete if one can only match the concept with the particular example of it that the teacher has supplied. A successful student is one who can match the concept to the as yet unseen examples or present an example that the teacher has not presented. Using the table as an example, the student would be able to generate an example of a new table that the teacher has not exhibited or discussed. This is an example of principle eight, the transfer of knowledge or applying conceptual knowledge. The dichotomy between theory and practice also seems to stem from the dichotomous relationship between the teaching for contentknowledge and teaching for thinking. The former is typically characterized as teaching concepts out of context, without a particular concrete example to experience through the five senses. The latter, however, is often characterized as being too concrete. Effective instruction integrates both the concrete and abstract but in a specific sequence. First, new learning requires specific real problems. Second, from these concrete problems, the learner forms an abstract principle or concept. Finally, the student then attempts to apply that conceptual knowledge to a new, never before experienced problem (Bruner, Goodnow, & Austin, 1956; Cone, 1969; Ormrod, 1999; Peikoff, 1993). The theory vs. practice debate is related to problem solving because problem solving is often marketed as the integration of theory and practice. I argue, however, it leaves out too much theory in its effort to be practical. That is, it leaves out the application of conceptual knowledge and its requisite knowledge base. Element six, problem solving teaches creativity, is also problematic. To create is to generate the new, so one must ask how someone can teach another to generate something new. Are there specific processes within a human mind that lead to creative output that can also be taught? The answer would depend at least in part on the definition of create. When an artist creates, he or she is actually recreating reality according to his or her philosophical viewpoint, but much, if not all, of what is included in the creation is not a creation at all but an integration or an arranging of already existing things or ideas. So in one sense, no one creates; one only integrates or applies previously learned knowledge. No idea is entirely new; it relates to other ideas or things. The theory of relativity, for Jamin Carson
example, changed the foundational assumptions of physics, but it was developed in concert with ideas that already existed. There may be no such thing as pure creativity, making something from nothing. What seems like creativity is more properly transfer or the application of concepts, recognizing that what appears like two different things are really the same thing in principle. On the other hand, it is possible to provide an environment that is conducive to creativity. Many problemsolving theorists have argued correctly for the inclusion of such an atmosphere in classrooms (Christy & Lima, 2007; Krulik & Rudnick, 1980; Slavin, 1997; Sriraman, 2001). I only object to the claim that problem solving teaches creativity defined as creating the new. It can, however, teach creativity defined as the application of previously learned principles to new situations. Element seven, problem solving requires a knowledge base, although not problematic is only neglected within the theory of problem solving. This is ironic given how important it is. Jeanne Ormrod (1999) says, “Successful (expert) problem solvers have a more complete and better organized knowledge base for the problems they solve” (p. 370). She also relates how one research inquiry that studied the practice of problem solving in a high school physics class observed that the high achievers had “better organized information about concepts related to electricity” (p. 370). Not only was it better organized, the students were also aware of “the particular relationships that different concepts had with one another” (Cochran, 1988, p. 101). Norman (1980) also says, I do not believe we yet know enough to make strong statements about what ought to be or ought not to be included in a course on general problem solving methods. Although there are some general methods that could be of use…I suspect that in most real situations it is…specific knowledge that is most important. (p. 101)
Finally, element eight, problem solving is the application of concepts or transfer, is also not problematic; it too is only neglected within the theory of problem solving. Norman Frederiksen (1984) says, for example, “the ability to formulate abstract concepts is an ability that underlies the acquisition of knowledge. [Teaching how to conceptualize] accounts for generality or transfer to new situations” (p. 379). According to this passage, it is the application of conceptual knowledge and not the heuristic alone that as Frederiksen says, “accounts for generality or
13
transfer,” (p. 379) which the advocates of problem solving so desire.
Hundhausen, C. D., & Brown, J. L. (2008). Designing, visualizing, and discussing algorithms within a CS 1 studio experience: An empirical study. Computers & Education, 50, 301–326.
Conclusion
Johanning, D. (2006). Benchmarks and estimation: A critical element in supporting students as they develop fraction algorithms. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 384–386). Mérida, Mexico: Universidad Pedagógica Nacional.
Problem solving would be more effective if the knowledge base and the application of that knowledge were the primary principles of the theory and practice. Currently, it seems that a contentless heuristic is the primary principle, which, as I have argued, is problematic because it dichotomizes thinking and knowledge into two mutually exclusive domains. In fact, in the course of solving any problem one will find themselves learning of all things not a heuristic, but an algorithm. In other words, teachers must not only teach students the heuristic and set their students free upon the problems of everyday life. Rather, teachers must, in addition to teaching students sound thinking skills, teach them what knowledge in the past has been successful at solving the problems and why. References Alexander, P. A., & Judy, J. E. (1988). The interaction of domainspecific and strategic knowledge in academic performance. Review of Educational Research, 58, 375–404. Bruner, J., Goodnow, J., & Austin, G.(1956). A study of thinking. New York: Wiley. Christy, A. D., & Lima, M. (2007). Developing creativity and multidisciplinary approaches in teaching Engineering problemsolving. International Journal of Engineering
Education, 23, 636–644. Cochran, K. (1988). Cognitive structure representation in physics. Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA. Cone, E. D. (1969). Audiovisual methods in teaching (3 rd ed.). New York: Holt, Rinehart and Winston. Dewey, J. (1933). How we think. Boston: D. C. Heath. Ellis, A. K. (2005). Research on educational innovations (4th ed.). Larchmont, NY: Eye on Education. Fredriksen, N. (1984). Implications of cognitive theory for instruction in problem solving. Review of Educational Research, 54, 363–407.
Kauchak, D. P., & Eggen, P. D. (2007). Learning and teaching: Researchbased methods. Boston: Allyn and Bacon. Krulik, S., & Rudnick, J. A. (1987). Problem solving: A handbook for teachers (2nd ed.). Boston: Allyn and Bacon. Lang, H. R., & Evans, D. N. (2006). Models, strategies, and methods for effective teaching. Boston: Pearson Education. LunykChild, O. L., Crooks, D., Ellis, P. J., Ofosu, C., O'Mara, L., & Rideout, E. (2001). Selfdirected learning: Faculty and student perceptions. Journal of Nursing Education, 40, 116– 123. Norman, D. A. (1980). Cognitive engineering and education. In D. T. Tuma & F. Reif (Eds.), Problem solving and education: Issues in teaching and research (pp. 97–107). Hillsdale, NJ: Erlbaum. Norman, G. R., & Schmidt, H. G. (1992). The psychological basis of problembased learning: A review of the evidence. Academic Medicine, 67, 557–565. Ormrod, J. (1999). Human learning (3 rd ed.). Upper Saddle River, NJ: Prentice Hall. Peikoff, L. (1985). The philosophy of education [CD Lecture Series]. Irvine, CA: Second Renaissance. Peikoff, L. (1993). Objectivism: The philosophy of Ayn Rand. New York: Meridian. Polya, G. (1954). Mathematics and plausible reasoning: Vol. 1. Induction and analogy in mathematics. Princeton, NJ: Princeton University Press. Polya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving, Vol. 1. New York: John Wiley & Sons. Polya, G. (1988). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton, NJ: Princeton University Press. Ross, P. E. (2006, August). The expert mind. Scientific American, 64–71.
Gallagher, S. A., Stepien, W. J., & Rosenthal, H. (1994). The effects of problembased learning on problem solving. Gifted Child Quarterly, 36, 195–200.
Rusch, T. L. (2005). Step one for developing a great mathematics lesson plan: Understand the mathematics. Ohio Journal of
Gunter, M. A., Estes, T. H., & Schwab, J. (2003). Instruction: A models approach (4th ed.). Boston: Pearson Education,
Slavin, R. E. (1997). Educational psychology: Theory and practice (5th ed.). Boston: Allyn and Bacon.
Henson, K. T. (2004). Constructivist teaching strategies for diverse middlelevel classrooms. Boston: Pearson Education.
Sriraman, B. (2004). Understanding mathematical creativity: A framework for assessment in the high school classroom. In D. E. McDougall & J. A. Ross (Eds.), Proceedings of the 26th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 350–352). Toronto, Canada: OISE/UT.
Hmelo, C. E., & Ferrari, M. (1997). The problembased learning tutorial: Cultivating higher order thinking skills. Journal for the Education of the Gifted, 20, 401–422.
Hu, C. (2006). It’s mathematical after all—the nature of learning computer programming. Education & Information Technologies, 11, 83–92.
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School Mathematics, 51, 25–34.
Stepien, W. J., Gallagher, S. A., & Workman, D. (1993). Problembased learning for traditional and interdisciplinary classrooms. Journal for the Education of the Gifted, 16, 338–357.
Problem Solving
The Mathematics Educator 2007, Vol. 17, No. 2, 15–23
Unraveling the Mystery of the Origin of Mathematical Problems: Using a ProblemPosing Framework With Prospective Mathematics Teachers José Contreras In this article, I model how a problemposing framework can be used to enhance our abilities to systematically generate mathematical problems by modifying the attributes of a given problem. The problemposing model calls for the application of the following fundamental mathematical processes: proving, reversing, specializing, generalizing, and extending. The given problem turned out to be a rich source of interesting, worthwhile mathematical problems appropriate for secondary mathematics teachers and high school students.
As a student and teacher of mathematics, I was intrigued about the origin of mathematical problems, especially nontrivial problems whose solutions are not obtained by a formula or algorithm, problems that somehow extend the frontiers of our personal mathematical knowledge. When, as a student, I solved nonroutine textbook problems, I thought, not only of devising a plan or a solution, but also of how the textbook authors and mathematicians generated mathematical problems. Their origin remained an enigma to me until later when, as a teacher, my curiosity led me to read The Art of Problem Posing (Brown & Walter, 1990). This book provided insights into the origin of mathematical problems and motivated me to examine more closely the relationships among related problems. In their book, Brown and Walter propose the “Whatif” strategy as a generic means to modify a given problem to create additional related problems. Because I am a geometry lover, I first applied Brown and Walter’s “Whatif” problemposing strategy to geometric problems. As a result, I developed a problemposing framework that has guided my students and me to pose mathematical problems systematically. This problemposing framework calls for the application of the following prototypical problemposing strategies: proof problems, converse problems, special problems, general problems, and extended problems. I often use the Geometer’s Sketchpad (GSP) (Jackiw, 2001) to verify the reasonability of the resulting conjectures. The main purpose of this article is twofold. First, I José N. Contreras teaches mathematics at The University of Southern Mississippi. He is interested in enhancing prospective teachers’ abilities to generate problems, conjectures and theorems.
José Contreras
model how the framework can be used to generate nonroutine mathematical problems from a given problem and, as a consequence, to discover mathematical patterns and relationships. Second, I discuss some of the difficulties that my students, prospective secondary mathematics teachers, experience when generating mathematical problems. The approach described below reflects the approach that I have followed in class. Because the focus of this article is on problem posing, proofs for most of the resulting theorems are not provided. The ProblemPosing Framework in Action: Generating Problems From a Problem Involving Isosceles Triangles and Medians Many mathematical problems are rich sources of additional related problems. I illustrate the use of the problemposing framework with the following problem: What special property do the medians corresponding to the congruent sides of an isosceles triangle have? Because problem posing and problem solving go hand in hand, it is not only important to pose problems but also to solve them. Before engaging ourselves in generating problems from this given problem, let us solve it. As Figure 1 suggests, the medians corresponding to the congruent sides of an isosceles triangle seem to be congruent. A straightforward proof shows that, indeed, the conjecture is true, and, therefore, it is a mathematical theorem. Of course, we can derive other conclusions: The medians of the congruent sides of an isosceles triangle divide each other in the ratio 2:1, they create congruent triangles, etc.
15
Figure 1. Medians of an isosceles triangle How can we generate additional mathematical problems from a given problem? Some problems can be modified to generate new problems by applying the following fundamental mathematical processes: proving, reversing, specializing, generalizing, and extending. Such processes are fundamental to mathematics because they are common means of generating or establishing mathematical knowledge. By applying these processes, we generate the following types of problems: proof problems, converse problems, special problems, general problems, and extended problems (Figure 2). Every problem that can be modified to generate related problems is called a base problem. Mathematical Situation
Base Problem
Proof Problem
Converse Problem
Special Problem
General Problem
Extended Problem
Figure 2. A problemposing framework As indicated in Figure 2, in some cases we may face a mathematical situation that does not contain a mathematical problem, e.g., we could have been asked to generate problems based only on the geometric diagram displayed in Figure 1. In these cases, our first task is to formulate a mathematical problem using information contained in the situation. Posing Proof and Converse Problems As indicated by the framework, an important type of problem to generate is a proof problem (i.e., a problem asking for a proof). In some cases, a problem that is not written in proof form can be reformulated as a proof problem. Proving is a critical activity of doing mathematics. First, proof is the fundamental mathematical process by which mathematicians establish the validity of their claims, and, as a consequence, the means by which mathematical
knowledge is enlarged. Second, a proof allows us to gain insights into why a mathematical theorem holds. Finally, a proof guaranties that our proof problem is well posed in the sense that is solvable. That is, reformulating a problem as a proof problem involves more than changing the syntactic structure of the problem: it implies that we either know that a proof exists or we can develop a proof. Of course, if this is not the case, we can always formulate a problem beginning with the phrase “If possible, prove that….” Our original problem above can be reformulated as a proof problem as follows: Prove that the medians corresponding to the congruent sides of an isosceles triangle are congruent. We can also reverse a known and an unknown of a problem to generate a converse problem. In other words, we formulate a converse problem when we substitute a known attribute of a base problem by an unknown attribute and vice versa. Why should we consider formulating a converse problem? Generating and investigating the converse of a problem is also a valuable mathematical activity. The converse of a mathematical problem is often a potential avenue for discovering new mathematical relationships, thus expanding mathematical knowledge. While a direct problem allows us to investigate the necessary conditions or properties of a mathematical object, a converse problem allows us to investigate its sufficient conditions or properties. In this way, we gain a more complete characterization of the properties of a mathematical object. Every problem has the potential for generating one or more converse problems. Of course, in many situations the resulting converse conjecture does not hold. In some of these situations we may need to impose additional conditions or restrictions for the converse theorem to hold. In any event, we often broaden our mathematical knowledge by investigating the converse of a problem. The converse of our medians problem follows: Prove that a triangle with two congruent medians is isosceles or, more specifically, prove that, in a triangle with two congruent medians, the sides corresponding to the medians are congruent. As it is often the case, the proof of the corresponding converse theorem is more challenging than the proof of the original theorem. This case is no exception. A proof follows: Let AD and BE be two congruent medians of triangle ABC (Figure 3). Since AD and BE are medians we have that AE = EC and BD = DC. Euclid’s allows constructing through B the !fifth postulate ! parallel line to AD . Let F !be the point ! of intersection
16
Problem Posing
!
!
of this parallel line to AD and line ED . Since D and E are the midpoints of two sides of a triangle, we know that EF is parallel to AB . Therefore, quadrilateral ABFD is a parallelogram. !As a consequence, BF = AD. ! Since AD = BE, we know that BF = BE. Since the opposite angles of a parallelogram are congruent we ! ! have that "BAD # "BFD . On the other hand, "BFD # "BEF by the isosceles triangle theorem. By the alternate interior angle theorem, "BEF # "ABE . Thus, "BAD # ABE . Therefore, "ABD # "BAE by ! the SAS congruence criterion. As a consequence, AE = BD , and, hence, AC = BC . That is, "ABC is ! isosceles. ! !
!
!
!
Figure 3. A triangle with two congruent medians Posing Special Problems Another potential way to generate mathematical problems is through specialization. We accomplish this by substituting for a mathematical object or attribute of the base problem with a particular example or case of the original mathematical object or attribute. In a special problem we impose additional restrictions on one or more of the attributes. A compelling reason for generating special problems is that, in some situations, specializing a problem allows us to detect implicit relationships among mathematical concepts that are not apparent at first sight. In other cases, specializing a problem permits us to find stronger relationships among the involved problem attributes. By detecting implicit relationships or finding stronger relationships between or among problem attributes, our mathematical knowledge becomes deeper. Prove that the medians corresponding to the congruent sides of an isosceles triangle are congruent. Angle bisectors Equilateral Parallelogram Altitudes Right Rhombus Perpendicular bisectors Scalene
Figure 4. Potential changes to the original base problem How can we generate a special problem? A useful idea is to underline the attributes of the base problem that can be changed, list some possible changes, and examine which attributes have special cases (Figure 4). For this problem, the attribute that has a special case is isosceles. Therefore, we can pose a problem involving JosĂŠ Contreras
equilateral triangles because an equilateral triangle is a special case of an isosceles triangle. Our special problem can be formulated as follows: Prove that the medians of an equilateral triangle are congruent (Figure 5).
Figure 5. The medians of an equilateral triangle We can also formulate and prove the converse of the previous problem but I will leave this task to the reader. Of course, other special problems can be generated (e.g., prove that the medians corresponding to the congruent sides of a right isosceles triangle are congruent). The above special problem is exemplary because an isosceles triangle is defined as a triangle with at least two congruent sides while an equilateral triangle is defined as a triangle in which all sides are congruent. In addition, the corresponding theorem for equilateral triangles reveals additional properties about the medians (i.e., the three medians are congruent) while the corresponding theorem for right isosceles triangles does not reveal any additional or implicit properties about the medians of the congruent sides. Posing General Problems Another potential source of mathematical problems is generalization. We create a general problem by substituting a mathematical object or attribute of the base problem with another for which the original is an example. A compelling reason for formulating a general problem is that in some general cases the same relationship holds while in others a weaker or more subtle relationship exists. Of course, in other general cases, there is not a relationship at all. In any case, mathematicians, as searchers of mathematical patterns, want to discover all possible relationships and the conditions under which a relationship exists or does not exist. As we relax our original conditions, we gain a more complete understanding of the properties of mathematical objects.
Figure 6. The medians of a triangle 17
!
Because I proved that a triangle is isosceles if and only if two medians are congruent, I will pose the general problem in an openended format: Is there a relationship among the medians of a triangle? As Figure 6 shows, there is not an apparent relationship among the medians of a scalene triangle. However, by measuring the lengths of the sides of the triangle a more elusive, subtle relationship may be noticed: In any triangle, the longest median corresponds to the shortest side and vice versa. Sure, this relationship involves more than medians, but it is still a valid generalization from the original statement. By specializing this new general relationship for isosceles triangles, we obtain our former relationship involving the congruence of two of the medians of an isosceles triangle as the following argument shows. Let ABC be an isosceles triangle with AC = BC and medians AD and BE (Figure 1). If AD > BE then AC > BC , which contradicts our hypothesis. A !similar contradiction exists if AD !< BE . Therefore, AD = BE . ! ! ! I should confess that I did not discover this general ! relationship by myself. I discovered it when I was ! looking for a proof of a related challenging problem (presented below) that I generated by modifying other attributes of the original problem. Posing Extended Problems Another potential source of mathematical problems is extension. We pose an extended problem when we substitute a mathematical object or attribute of a base problem by another similar or analogous mathematical object or attribute. In this case, none of the mathematical objects is a special case of the other. Why should we consider generating extended problems? After all, at this point we might have already formulated a general relationship as well a special one. A compelling reason is that, in some situations, the same relationship exists for an extended case while in others a similar or analogous relationship exists. If the extended situation is a special case of the general case, we may discover a stronger relationship than the general relationship or we may find an implicit relationship. If the extended situation is not a special case of the general case, we may find that the same or a similar relationship exists. In any event, mathematicians, as searchers of mathematical patterns and relationships, want to discover, examine, or characterize all possible relationships that exist between specific mathematical objects. Extension is a common means of enlarging mathematical knowledge.
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Since a right triangle is not a special case or a general case of an isosceles triangle, we may say that a right triangle is an extended case of an isosceles triangle. As I reflected on how to pose a problem involving a right triangle, I remembered that the median corresponding to the hypotenuse is half as long as the hypotenuse and that the hypotenuse is the longest side of a right triangle. As a result of this thinking, I generated the following extended problem: Prove that the medians of a right triangle are greater than or equal to half the length of the hypotenuse. As we can notice, the problem takes advantage of properties of both scalene and right triangles, resulting in a stronger relationship for right triangles than for generic scalene triangles. Posing Further Extended Problems So far we have posed problems involving special, general, and extended cases of an isosceles triangle. We can continue generating problems by modifying other problem attributes as indicated in Figure 4. To distinguish these new problems from the extended problem generated previously, I call the new generated problems further extended problems. Notice, however, that we can distinguish between extended and further extended problems only when (a) one of the problem attributes has special, general, and extended cases, and (b) there is another attribute that can be changed. In the present situation, (a) an isosceles triangle has special cases (e.g., an equilateral triangle), general cases (i.e., a scalene triangle), and extended cases (e.g., a right scalene triangle) and (b) other attributes can be changed (e.g., medians.) Instead of medians, we can consider related attributes such as altitudes, angle bisectors, and perpendicular bisectors. However, since these geometric figures are lines or rays, they do not have a finite length. To circumvent this obstacle, and at the same time salvage our potential problem, we can consider the length of an altitude as the distance from the corresponding vertex to the opposite (extended) side (e.g., BE in Figure 7) of a triangle. The length of an angle bisector can be defined in a similar way (e.g., AD in Figure 7). The length of a perpendicular bisector, however, is more troublesome because a perpendicular bisector intersects both of the other two (extended) sides of a nonright triangle. To elude this problem, I defined such a length as the distance between the midpoint of the segment and the point of intersection of the perpendicular bisector with the adjacent side or its extension following a clockwise direction (e.g., FG in Figure 7). Notice that for right Problem Posing
triangles, the length of the perpendicular bisector of one leg is still infinite.
Prove that the altitudes corresponding to the congruent sides of an isosceles triangle are congruent. If a triangle is isosceles, prove that the lengths of the angles bisectors of the congruent angles are equal. Prove that the perpendicular bisectors of the congruent sides of an isosceles triangle are congruent (Figure 8). Figure 9. Some further extended problems of the original base problem
Figure 7. Examples of lengths of altitudes (BE), angle bisectors (AD), and perpendicular bisectors (FG) associated to a triangle.
As suggested by the framework, each of the further extended problems displayed in Figure 9 can be taken as a base problem to generate additional converse, special, general, and extended problems. Figure 10 displays some resulting theorems related to triangles and altitudes. Similar problems can also be posed for angle bisectors and perpendicular bisectors. Figure 11 displays a further extended problem and its solution that, with guidance, my students have solved.
Using this definition, I was still not able to find any significant relationship between the lengths of the perpendicular bisectors of the congruent sides of an isosceles triangle. However, as suggested by Figure 8, there is, indeed, a relationship involving the perpendicular bisectors of the sides of an isosceles triangle: EG = DF and EI = DH. This discovery led me to redefine the length of the perpendicular bisectors of the congruent sides of an isosceles triangle in two ways as suggested by Figure 8. One way involves defining the length of the perpendicular bisector of a congruent side as the distance between the corresponding midpoint and the point of intersection with the other congruent side or its extension (DF and EG in Figure 8). Another way involves defining such as length as the distance between the corresponding midpoint and the point of intersection with the (extended) base of the triangle (DH and EI in Figure 8). Notice that in each case the length of the perpendicular bisector of one of the congruent sides is measured to the adjacent side clockwise while the length of the perpendicular bisector of the other congruent side in measured to the adjacent side counterclockwise.
A triangle is isosceles if and only if the altitudes corresponding to the congruent sides are congruent. A triangle is equilateral if and only the three altitudes are congruent. In any triangle, the longest altitude corresponds to the shortest side and vice versa. Figure 10. Theorems related to triangles and altitudes Problem: Let DF and EG be the perpendicular bisectors of two sides of a triangle as indicated on the figure. If DF = EG , prove that "ABC is isosceles.
!
!
!
!
Since ED is a midsegment of "ABC we know that ED is parallel to AB . Construct EI and DH perpendicular to AB as indicated on the figure. ! "EGI # "DFH by the!HL congruence criterion. This implies that !"AEG # "BDF ! by the ASA ! congruence criterion. Therefore, "EAG # "DBF , which implies ! that "ABC is isosceles. Figure 11. A problem involving a triangle with ! congruent perpendicular bisectors and its solution !
Figure 8. Lengths of the perpendicular bisectors of the congruent sides of an isosceles triangle ! As a result of defining the lengths of altitudes, angle bisectors, and perpendicular bisectors as ! described above, we can pose some further extended problems of our original base problem (Figure 9).
! JosĂŠ Contreras
19
A challenging problem that I myself was not able to solve is as follows: If a triangle has two congruent angle bisectors, prove that it is isosceles. Because all of my attempts were futile, I proposed this problem to some of my colleagues, one of which is a mathematician, but none could solve it. An internet search revealed that this problem is a classic and challenging problem in Euclidean geometry with an interesting history. This problem is known as the SteinerLehmus problem. A short historical sketch can be found in Lewin (1974), and a simple and elegant proof is provided by Coxeter (1969). Coxeterâ€™s proof uses the theorem stating that if a triangle has two noncongruent angles, then the greater angle has the shorter angle bisector. This theorem inspired me to formulate the subtle generalization among the medians and sides of a generic triangle mentioned above. We can continue generating problems by considering the exterior angles of a triangle and defining the length of an angle bisector as the distance from the vertex of the angle to the point of intersection of the exterior angle bisector with the extension of the side opposite that vertex (e.g., AD in Figure 12). One such problem can be formulated as follows: Prove that the angle bisectors of the exterior angles of an isosceles triangle are congruent (Figure 12). Again, we can take this problem as a base problem to generate additional problems. For example, we can formulate its converse as follows: Two exterior angle bisectors of triangle are congruent. Is the triangle necessarily isosceles? Justify your response. I formulated this problem in an openended form because, even though I was â€œsureâ€? that a triangle with two congruent exterior angles is isosceles, I was not able to develop a proof in spite of strenuous efforts. Again, I proposed this problem to some of my colleagues but the solution remained elusive. (I challenge the reader to solve this problem before continuing reading. Hint: Use interactive geometry software!) I was so sure that the triangle is isosceles that I did not attempt immediately to find a counterexample with GSP. After working frantically on this problem for a couple of weeks, I did use GSP and I was amazed for what I discovered: The mystery of the equal exterior angle bisectors problem was revealed before my eyes. As Figure 13 suggests, there are some nonisosceles triangles with congruent exterior angle bisectors. This discovery inspired me to pose the following question that I am still trying to investigate: Is there a necessary and sufficient condition for a triangle to have two congruent exterior angle bisectors? If so, what is it?
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Figure 12. Diagram for the equal exterior angle bisector problem
Figure 13. A triangle with two congruent exterior angle bisectors To continue generating additional related problems we can extend some of the previous ideas to geometric figures other than triangles (e.g., parallelograms, trapezoids, etc.). We can also challenge our definitions for the lengths of medians, altitudes, angle bisectors, etc. For medians, we can consider the distance from the vertex of a triangle to the centroid or to the circumscribed circle along the median (Figure 14). As I did this, I was able to generate additional problems and, as a consequence, I was able to discover and prove more mathematical relationships. Needless to say, I am still trying to solve some of these problems. Without a doubt, generating problems may become interminable when each new problem becomes the source of additional problems.
Figure 14. Congruent segments related to medians of an isosceles triangle
Problem Posing
Some Prospective Secondary Mathematics Teachers’ Thinking and Difficulties When Working on ProblemPosing Tasks The problemposing tasks described in this article have been implemented with prospective secondary mathematics teachers enrolled in a college geometry course designed for them. I have implemented the problemposing tasks using two formats: a studentcentered approach and an instructorcentered approach. I use the studentcentered approach after the students have had some experience with at least one problemposing task. In this approach students pose most of the problems using the problemposing framework as a guide. Performing the problemposing tasks described in this article usually lasts more than two class periods (about 3 hours). I use the instructorcentered approach when students have not had any experience posing problems. In this case I model how to pose problems using the problemposing framework as a guide. Prospective secondary mathematics teachers’ abilities to generate problems are underdeveloped (Contreras & MartínezCruz, 1999). Their approaches to generate problems tend to be unsystematic, adhoc, and nongeneralizable. In addition, their generated problems tend to be trivial and unproductive to pursue (Knuth, 2002). For example, I asked my students to modify the following problem to pose different but related mathematical problems or questions: Prove that the medians corresponding to the congruent sides of an isosceles triangle are congruent. Among the problems posed were: What is a median? How many congruent sides does an isosceles triangle have? How difficult is it to prove? The problemposing strategies described in this article are prototypical strategies that can be used systematically in a variety of problem situations to generate worthwhile mathematical problems. Yet, as I have noticed with my students, without adequate experiences, students rarely use these prototypical strategies to generate problems. Therefore, there seems to be a need to provide students with experiences in generating proof problems, converse problems, special problems, general problems, and extended problems. Prospective secondary mathematics teachers’ thinking and difficulties with each of these problem types are elaborated below. Proof Problems As stated above, proving is a fundamental mathematical process that permeates mathematical thinking and research. In fact, a proposition for which a proof has not been developed is called a conjecture and José Contreras
not a theorem. Even though proof is a vital part of mathematics, my students are often reluctant to pose proof problems. For example, Contreras and MartínezCruz (1999) asked 17 prospective secondary mathematics teachers to pose problems related to each of four given geometric situations. The researchers found that only one student out of 17 generated a proof problem for one geometric situation. Even after instruction, students avoid developing a proof corresponding to a proof problem. In addition, many of them do not use the full power of a proof when adapted to a special case. In other words, they do not establish the truth of special theorems as corollaries of more general theorems. To illustrate, after my students proved that the medians corresponding to the congruent sides of an isosceles triangle are congruent, most of them did not use this theorem to prove that the three medians of an equilateral triangle are congruent. Instead, they provided a proof from scratch. Only a couple of students used an argument along the following lines: Since AC = BC (Figure 5) we know that AD = BE because the medians corresponding to the congruent sides of an isosceles triangle are congruent. Applying the same theorem again, we conclude that BE = CF because AB = AC. In conclusion, AD = BE = CF. This research and personal experience suggest that students should have extensive experiences posing and solving proof problems. Converse Problems From a problemposing perspective, the critical mathematical process of reversing involves generating a converse problem. Whereas converse problems permeate mathematical thinking and research, it is not natural for students to generate them. For example, Contreras and MartínezCruz (1999) found that students generated only one converse problem out of more than 68 potential converse problems. Ideally, each of the 17 students could have generated a converse problem for each of the four geometric situations. In addition, formulating the converse of a problem is challenging for some students. Some of my students have formulated the converse of the problem “the medians corresponding to the congruent sides of an isosceles triangle are congruent” as follows “If the medians corresponding to the congruent sides of a triangle are congruent, prove that the triangle is isosceles.” Notice that these students are assuming that the triangle is already isosceles. Examples like this emphasize how critical it is that students have a wide 21
range of experiences posing converse problems of varying difficulty. Special Problems Specializing a mathematical problem allows us to investigate implicit or stronger relationships between or among the corresponding problem attributes. Yet, my experience has shown that students rarely consider specializing as a problemposing strategy. In addition, the way a problem is formulated affects the quality of the generated special problem. The following problem was posed to a group of secondary mathematics majors enrolled in a geometry class: “If AD and BE are the medians corresponding to the congruent sides of an isosceles triangle ABC , prove that AD " BE .” When asked to generate a special problem, all students formulated a problem like ! ! this: “If AD and BE are the medians corresponding to ! the!congruent sides of an equilateral triangle ABC , prove that AD " BE .” While this may be a wellposed !problem,!it does not prompt us to investigate whether a stronger relationship holds for the special ! case, which is one of the assets of generating a special problem. ! Such evidence suggests that students should be given a broad variety of experiences in generating special problems. General Problems Despite the importance of generalizing, Contreras and MartínezCruz (1999) found that the 17 students only generated 38 general problems out of more than 100 possible general problems. That is, each student had the opportunity to generate at least 6 general problems for the four geometric situations. In addition, generating wellposed general problems is challenging for some students. While students generating illposed problems provides pedagogical opportunities, these problems often reveal that students do not fully understand the connections among the discovered relationships or the structural aspects of the problem. For example, some of my students formulated the following general problem related to the previous problem: Prove that the medians of a triangle are congruent. This illposed problem was generated after my students and I established that a triangle with two congruent medians is isosceles. Of course, we can reformulate the illposed problem as a wellposed problem as follows: “Does a (generic) triangle have congruent medians? Prove your answer” or “Is there a relationship among the medians of a (generic) triangle? Justify your response.” If our students have adequate experiences in posing general 22
problems, they may gain the expertise necessary to overcome their difficulties and, as a result, more frequently pose general problems. Extended Problems My experience suggests that students do not often extend mathematical problems. When they do, they sometimes generate illposed extended problems. For example, some students have generated the following extended problem: prove that the medians of a right triangle are congruent. Thus, providing students with opportunities to pose extended problems is essential. Reflection and Conclusion Mathematicians (e.g., Halmos, 1980; Polya, 1954), mathematics educators (e.g., Brown & Walter, 1990; Freudenthal, 1973), the National Council of Teachers of Mathematics (NCTM, 1989, 2000) and the National Research Council (Kilpatrick, Swafford, & Findell, 2001) consider problem posing as a core element of mathematical proficiency. The Principles and Standards for School Mathematics (NCTM, 2000), for example, calls for teachers to regularly ask students to pose interesting problems based on a wide variety of situations. From this, a pedagogical problem arises: How can we teach our prospective secondary mathematics teachers to pose mathematical problems so that they, in turn, can teach their students to pose problems? As a student of mathematics, I was never given the opportunity to pose problems, let alone interesting problems. Most of the problems that I solved came from the textbook and, on rare occasions, from the teacher. I was certainly content with this situation because I considered posing problems as a creative endeavor beyond my reach. Calls for teachers to ask students to pose problems (e.g., NCTM, 1989) challenged me to find ways of teaching my students how to pose mathematical problems. Brown and Walter’s (1990) The Art of Problem Posing motivated me to engage in creating problems and, as a result, I developed the problemposing framework described here. This problemposing framework calls for the systematic generation of problems using the following mathematical processes: proving, reversing, specializing, generalizing, and extending. These processes are essential means for discovering new mathematical patterns or relationships. Posing and solving mathematical problems are worthwhile but challenging activities for prospective teachers. As with any other worthwhile mathematical Problem Posing
activity, prospective teachers need to be engaged actively and reflectively in the problemposing process so they can generate nontrivial, productive, and wellposed mathematical problems. I truly believe that all of us—mathematics educators, teachers, and students— should experience the joy of generating problems and discovering mathematical relationships, even if they are new only to us. In this process, we develop a better appreciation and understanding of the origin of mathematical problems. References Brown, S., & Walter, M. I. (1990). The art of problem posing (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates. Contreras, J., & MartínezCruz, A. (1999). Examining what prospective secondary teachers bring to teacher education: A preliminary analysis of their initial problemposing abilities within geometric tasks. In F. Hitt & M. Santos (Eds.), Proceedings of PMENA XXI (Vol. 2, pp. 413–420). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, Netherlands: Reidel. Halmos, P. R. (1980, August–September). The heart of mathematics. American Mathematical Monthly, 87, 519–524. Jackiw, N. (2001). The Geometer’s Sketchpad (Version 4.0) [Computer software]. Emeryville, CA: KCP Technologies. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, D.C: National Research Council. Knuth, E. J. (2002). Fostering mathematical curiosity. Mathematics Teacher, 95, 126–130. Lewin, M. (1974). On the SteinerLehmus theorem. Mathematics Magazine, 47(2), 87–89. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. Polya, G. (1954). Mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.
Coxeter, H. S. M. (1961). Introduction to geometry (2 nd ed.). New York: John Wiley & Sons.
José Contreras
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The Mathematics Educator 2007, Vol. 17, No. 2, 24–30
Student Attitudes, Conceptions, and Achievement in Introductory Undergraduate College Statistics Brian Evans The purpose of this study was to measure student attitudes toward and conceptions about statistics, both before and after an introductory undergraduate college statistics class. Relationships between those attitudes and conceptions, as well as their relationships to achievement in statistics, were also studied. Significant correlations were found between student attitudes and achievement, both at the beginning and end of the course. A low, but significant, correlation was found between positive attitudes toward statistics and accurate conceptions about statistics in the posttest but not in the pretest. Although it was found that there was no change in student attitudes and conceptions over the course of the semester, students did have more positive attitudes and correct conceptions than expected. Additionally, instructor interviews revealed that the most common technique employed to improve student attitudes and conceptions was the use of realworld applications.
Students typically enter college mathematics classes with certain attitudes and preconceptions toward the subject. Upon completion of the course, some students may exit with new or changed attitudes and conceptions due to instructor intervention and exposure to content. This study, originally a doctoral dissertation (Evans, 2005), focuses on student attitudes toward statistics, conceptions about statistics, and achievement in introductory undergraduate college statistics classes. Although student attitudes and achievement in statistics have previously been investigated separately, the correlation between student attitudes and conceptions in this subject has not been as widely documented (Barkley, 1995). Also, no research was found regarding instructors’ methods for improving student attitudes and changing student misconceptions in the statistics classroom. Student attitudes toward statistics incorporates elements from Thurstone’s (1970) definition and is a measure of students’ positive and negative feelings toward the subject of statistics in terms of relevance and value, difficulty and selfefficacy, and general impression toward the subject. The phrase student conceptions about statistics—combining ideas from Davis and Palladino (2002) and Barkley (1995)— addresses student ideas and beliefs about statistics. This includes assessing the degree of student understanding of what questions can or cannot be answered using probability and statistics, as well as Dr. Brian R. Evans is Assistant Professor in The School of Education at Pace University in New York. His specific focus is mathematics education. He is also involved with the alternative certification programs Teach for America and New York City Teaching Fellows at Pace University. 24
applying probabilistic and statistical concepts in appropriate situations. It is important, however, to differentiate conceptions from achievement. Achievement is defined by student ability in computations and solving problems, which can normally be measured by written tests. Conceptions deal more with deeper understanding. For example, it is one thing to know how to calculate a Pearson correlation coefficient, but it is quite another to know that a Pearson correlation coefficient is a measure of the strength of the linear relationship between two variables. The connection between attitudes and achievement in mathematics has a rich history. Aiken (1970, 1974, 1976) found significant relationships between attitudes toward mathematics and achievement. Specifically, he showed that attitudes and achievement in mathematics are reciprocal: students who have better attitudes towards mathematics demonstrate higher achievement, and students who have higher achievement exhibit better attitudes. These results confirmed earlier work by Neale (1969), who found a low, but significant, relationship between attitudes toward mathematics and achievement. Ma and Kishor’s (1997) metaanalysis on the relationship between attitude toward and achievement in mathematics found a statistically significant positive relationship between the two variables. Although this relationship was found to be reliable, it was not found to be strong. Similarly in statistics, Schultz and Koshino (1998) showed that there exists a consistent positive relationship between attitudes toward statistics and achievement in statistics, confirming results from Arkaki and Schultz (1995); Elmore and Lewis (1991); Student Attitudes, Conceptions, and Achievement
Elmore, Lewis, and Bay (1993); Roberts and Saxe (1982); Sutarso (1992); and Wise (1985). Furthermore, Feinberg and Halperin (1978) reported that measures of anxiety and attitudes, among other variables, were significantly related to course outcome. Additionally, Gal and Ginsburg (1994) found that negative attitudes and beliefs about statistics can impede the learning of statistics. Sutarso (1992) said that it would be useful for instructors to know student attitudes toward the subject so that instructors could use better teaching strategies to overcome such problems. Finally, Rogness (1993) agreed that understanding student attitudes toward statistics can assist in creating an individualized intervention strategy to help remove some of the anxiety for a specific student. According to Gredler (2001), Bandura’s Social Cognitive Learning Theory says, “Selfefficacy is the learner’s belief in his or her capabilities to successfully manage situations that may include novel or unpredictable elements” (p. 330). Gredler, summarizing Bandura’s conclusion, says, “Selfefficacy beliefs affect human functioning indirectly by influencing individuals’ cognitive, motivational, affective, and selection processes. Individuals with high selfefficacy construct success scenarios, set challenging goals, persist in the face of difficulties, and control disturbing thoughts” (p. 330). This gives educational research on student attitudes a theoretical perspective while also providing a clear justification for improvement of attitudes regarding student selfefficacy. Shaughnessy (1992) claimed, “We need to develop some standard, reliable tools to assess our students’ conceptions of probability and statistics” (p. 489). These instruments should build on previous research and investigate students’ conceptions of probability and statistics for many different grade levels. Responding to this need, Hirsch and O’Donnell (2001) developed and used an instrument to measure student misconceptions. They determined that, despite a formal course in statistics, students continued to demonstrate misconceptions. Other researchers (delMas & Bart, 1987; Garfield, 1981; Garfield & Ahlgren, 1988; Huck, 2007; Kahneman & Tversky, 1972, 1973; Konold, 1991; Landwehr, 1989) have also identified student misconceptions. The National Council of Teachers of Mathematics (NCTM), in the Data Analysis and Probability section of Principles and Standards for School Mathematics (2000), states, “Misconceptions that arise because of students’ representations of data offer situations for new learning and instructions” (p. 113). If instructors were aware of the common Brian Evans
misconceptions students have before, and retain after, participation in an introductory statistics course, their practices could be adapted to assist students in improving their understanding of statistics (Barkley, 1995; Landwehr, 1989; Shaughnessy, 1977, 1992). Research Questions The following research questions guided this study. 1. To what extent do student attitudes toward statistics and conceptions about statistics change after taking an undergraduate introductory statistics course? 2. What differences exist with regard to student prior, as well as post, attitudes and conceptions toward statistics among the different departments offering an introductory undergraduate college statistics class (mathematics, psychology, and sociology)? 3. Is there a correlation between positive attitudes toward statistics and accurate conceptions about statistics prior to and after completing an undergraduate introductory statistics course? 4. Do student attitudes and conceptions significantly predict achievement in an undergraduate introductory statistics course? 5. What types of interventions have instructors utilized to improve student attitudes toward statistics and improve student conceptions about statistics throughout the course of the semester? Methodology The methodology of this study involves a nonexperimental student survey and instructor interview. The sample in this study consisted of students in six randomly selected statistics classes from the mathematics (n = 30 from two classes), psychology (n = 43 from three classes), and sociology (n = 42 from one doublesized class1) departments and their instructors (n = 5) at a large urban university in the northeastern United States. There were 80 female and 35 male students. All three departments offered introductory statistics courses addressing the same topics. Students completed a survey instrument entitled “Student Attitudes and Conceptions in Statistics” (STACS) (Evans, 2005) that focused on both attitudes (30 items) and statistical concepts (14 items). Both the attitudinal and conceptual sections used a 5point Likerttype scale coded with a range of 0 to 4, with a 25
score of 4 indicating strong agreement, a score of 2 indicating neutral agreement/disagreement, and a score of 0 indicating strong disagreement. For negative attitudes and incorrect conceptions the scoring was reversed. This survey instrument was based upon preexisting survey instruments (Barkley, 1995; Gilovich, Vallone, & Tversky, 1985; Huff, 1954; Rogness, 1993; Shaughnessy, 1992; Sutarso, 1992; Wise, 1985). Items for attitudes on STACS consisted of statements such as “I enjoy statistics” and “I would rather not be taking statistics.” Items for conceptions on STACS consisted of statements such as “If a commercial claim is that a study found 9 out of 10 dentists prefer All Clean Brand toothpaste, then the claim is very questionable” and “If I flip a coin nine times and get nine tails, the next flip will more likely be heads” (Evans, 2005). Students completed the questionnaire twice, once at the beginning and once at the end of the semester. The results of the STACS were used to answer research questions 1 through 4. To answer research question 5, the instructors were interviewed by the researcher at the end of the semester using a list of direct openended questions pertaining to what instructors did to improve attitudes and eliminate misconceptions for their students. Student achievement was measured by final course grade, which used a standard 100point grading scale. Final course grades in all departments were primarily determined by standard written tests involving statistical computations. Results To determine internal consistency, Cronbach’s alpha for the STACS questionnaire was .92 for the attitudes section and .59 for the conceptions section. These values indicate that the attitudes section of the instrument has high internal consistency, whereas the conceptions section has questionable internal consistency. Therefore, the results found using the conceptions section should be interpreted more cautiously than those found using the attitudes section of the instrument. Research questions 1 and 2, regarding changes in attitudes and conceptions and differences between the academic departments, were answered using a 3 x 2 repeated measures MANOVA. One independent variable is academic department offering a statistics course; this variable has three levels (mathematics, psychology, and sociology). Another independent variable is score on the attitudes and conceptions survey, with two levels (pretest and posttest). The 26
MANOVA determines whether or not significant differences exist among the students’ mean scores for attitudes and conceptions, by department, as well as if significant differences exist between the mean scores on the pretest and posttest for attitudes and conceptions. There were no overall statistically significant differences between the students’ pretests and posttests for attitudes and conceptions, F(1, 228) = 0.166, p = .684. Therefore, these students exhibited no significant change in attitudes and conceptions toward statistics over the course of the semester. However, statistically significant differences were found for attitudinal and conceptual scores among the different departments (mathematics, psychology, and sociology) with F(2, 227) = 9.193, p = .000. Separate ANOVAs were used to find differences for attitudes and conceptions with F(2, 227) = 30.412, p = .000, and F(2, 227) = 6.119, p = .003, respectively. A post hoc test (Tukey HSD) determined exactly where the means differed among departments for both attitudes and conceptions. Overall, students in the sociology department demonstrated more positive attitudes toward statistics than did students in the mathematics and psychology departments with p = .000. Students in the sociology department also demonstrated more correct conceptions than did students in the mathematics department with p = .002. There were no other statistically significant differences found. The descriptive statistics are summarized in Table 1. Because there were no significant changes in attitudes and conceptions, an additional analysis determined if the participants had significantly more positive than neutral attitudes towards statistics and more correct conceptions about statistics, as measured by their responses on STACS2. For both the pretest and posttest, an independent samples twotailed t test revealed that students did, in fact, have statistically significant positive attitudes and correct conceptions. The results are summarized in Tables 2 and 3. Despite having positive attitudes towards statistics, student responses indicated that they would rather not be taking statistics and would not enjoy taking other statistics courses in the future. Furthermore, students claimed they would not have taken statistics if it were not required; they did not find statistics very valuable in their programs of study. Although students held more correct conceptions than expected by the researcher, many students held the misconception that results can be extrapolated into the distant future. Students also failed to realize that the size of a population does not influence the sample size needed Student Attitudes, Conceptions, and Achievement
Table 1
Table 2
Means and Standard Deviations for Pretests and Posttests for Attitude and Conceptions Scores
Overall Results for the Pretest
Test and Department Attitudes Pretest Mathematics Psychology Sociology Total
Mean
Standard Deviation
2.06 2.23 2.52 2.29
0.439 0.423 0.369 0.446
Attitudes Posttest Mathematics Psychology Sociology Total
2.02 2.14 2.61 2.28
0.522 0.476 0.355 0.514
Conceptions Pretest Mathematics Psychology Sociology Total
2.26 2.37 2.46 2.37
0.259 0.408 0.278 0.335
Conceptions Posttest Mathematics Psychology Sociology Total
2.22 2.32 2.42 2.33
0.263 0.425 0.300 0.350
to use inference techniques as long as the population is much larger than the sample size. A Pearson correlation addressed research question 3, regarding the relationship between attitudes and conceptions. No significant correlation existed between positive attitudes toward statistics and accurate conceptions about statistics for the pretest, r = .143, n = 115, p = .127. However, the posttest revealed a low, but significant, correlation between positive attitudes toward statistics and accurate conceptions about statistics, r = .197, n = 115, p = .035. Because this correlation is low, it is questionable if attitudes toward statistics can be used to predict conceptions in statistics after an introductory statistics class. Additional analyses determined if significant correlations existed between positive attitudes toward statistics and accurate conceptions about statistics for the pretests and posttests for the individual departments. The mathematics department exhibited the only significant correlations for the pretests and posttests, with r = .451, n = 30, p = .012, and r = .431, n = 30, p = .018, respectively. The predictability of achievement using the measures of attitudes and conceptions, the focus of the Brian Evans
Mean Attitudes
2.292
Standard Deviation 0.446
Conceptions
2.373
0.335
t value
Significance
7.019
.000*
11.926
.000*
Note. Student attitudes and conceptions scores were compared to a neutral value coded as 2 on the instrument (from a range of 0 to 4). * p < .05, two tails
Table 3 Overall Results for the Posttest Mean Attitudes
2.280
Standard Deviation 0.514
Conceptions
2.330
0.350
t value
Significance
5.852
.000*
10.110
.000*
Note. Student attitudes and conceptions scores were compared to a neutral value coded as 2 on the instrument (from a range of 0 to 4). * p < .05, two tails
fourth research question, was addressed using a Pearson correlation and linear regression. Neither initial student conceptions and course grades nor final student conceptions and course grades revealed significant correlations. However, significant correlations were found between initial and final student attitudes and course grades, r = .203, R2 = .04, n = 115, p = .030, and r = .247, R2 = .06, n = 115, p = .008, respectively. A simple regression equation was calculated for initial student attitude level and final course grades: y = 76.045 + 4.324x, where x represents initial attitudinal level and y represents final course grade. A simple regression equation was also calculated for final student attitude level and final course grades: y = 75.526 + 4.574x, where x represents final attitudinal level and y represents final course grade. However, because the Rsquared values were fairly low in both cases, caution should be taken when interpreting the regression equations. Although a significant linear relationship was found, practicality and utility are of questionable consequence; course grades cannot necessarily be predicted from the initial or final student attitudes towards statistics. Instructor interviews informed research question 5, regarding instructor interventions to improve attitudes 27
and eliminate misconceptions. Instructors generally said that they try to link the material from the statistics courses to realworld problems. This was the most commonly used technique for generating interest, improving student attitudes, and eliminating misconceptions in statistics. One instructor demonstrated the link between statistics and future courses the students would study. Another instructor used humor, optimism, and enthusiasm for statistics to generate more interest. To improve attitudes and eliminate misconceptions, one instructor had students gather survey data regarding opinions on an upcoming political election. This served as a realworld example to help students better understand statistical conceptions and their reallife applications. The added dimension of data collection allowed students to use their own data for statistical analysis. Discussion
Based on research and prior experience, it was expected that there would be a change in student attitudes and conceptions towards statistics over the course of the semester. It was also expected that students would not have positive attitudes toward statistics and that students would have a number of misconceptions in statistics. The results of the study presented evidence refuting these expectations. With these findings taken into consideration, the fact that no significant change occurred in attitudes and conceptions is not as disappointing. Students already held generally positive attitudes and generally correct conceptions in statistics and thus it would be quite difficult to improve already positive attitudes and correct conceptions. This indicates that the commonly held belief in education that students have negative attitudes toward statistics and many misconceptions about the nature of statistics (Paulos, 1988) may in fact no longer be true. It may be that since the mean age of the subjects surveyed in this study was 21 years old, many of these students learned mathematics from teachers who may have followed recommendations from the NCTM (1989, 2000) Curriculum and Evaluation Standards for School Mathematics and Principles and Standards for School Mathematics. Since the Standards emphasize statistics and probability, todayâ€™s students may have a better appreciation for and understanding of statistics than students in the past. Much of the prior research studied students who were in school before the Curriculum and Evaluation Standards for School Mathematics was published and implemented (Heaton & Mickelson, 2002). This could explain why student attitudes and conceptions in the present study were generally better 28
than previously believed. The attitudes and conceptions of students who experienced data analysis and probability in their earlier schooling, compared with those who lacked those experiences, provides another area for future research. Another unexpected result was that students taking statistics in the sociology department had more positive attitudes toward statistics than did students taking statistics in the mathematics and psychology departments. Also, students taking statistics in the sociology department had more correct conceptions about the nature of statistics than did students taking statistics in the mathematics department. This difference may be attributed to the fact that statistics classes in the mathematics department include students from a variety of majors, whereas the sociology statistics classes are designed for sociology majors. Students taking statistics in the mathematics department are fulfilling a required university course as part of a core curriculum. Classes offered in the sociology department are geared toward the work of sociologists. Since sociologists make extensive use of statistics in their literature and research, students may have had more interest in and exposure to statistics. However, there are two possible problems with this explanation. First, this does not explain why sociology students held more positive attitudes than psychology students since psychologists also make extensive use of statistics in their literature and research. However, this researcher believes that many students, when unsure of what major to choose, major in psychology. This may not be the case in sociology. Second, because over half of the students in this study were college sophomores, they had only begun to gain exposure to literature in their fields. Differences in attitudes and conceptions in statistics among students from different departments should be further studied. Despite having overall positive attitudes, students still reported some negative attitudes toward statistics. Some students might generally find statistics to be a worthy area of study and to be a subject in which they believe they can perform well. However, other students might feel statistics is something they would rather not be taking at the present time and not need in their own careers. The students in the latter group might feel that statistics is a useful subject for others but not for them specifically. Future research could investigate why students hold these beliefs. The NCTM (2000) recommended that K12 students be encouraged to make statistical and probabilitybased predictions and to compare those predictions with actual outcomes. Students will be Student Attitudes, Conceptions, and Achievement
challenged to confront misconceptions through discrepancies in their predictions. Similarly, a more “handson” approach to statistics such as that recommended for K12 students might be beneficial for college students, more so than just using realworld problems from the textbook. One instructor in this study used a survey approach to statistics in which the students surveyed other students on voting and political issues for an upcoming election. After collecting data, the students analyzed the data and developed conclusions. Perhaps this type of investigative approach to statistics could further improve attitudes and eliminate misconceptions. Students, both at the beginning and end of the course, reported relatively positive attitudes about their own selfefficacy in statistics, but ways of improving the view of the utility of statistics are needed. Student views could be further improved by making connections to various careers that use statistics, providing students with opportunities to see the versatility of statistics. This appeared to be the area where students had the poorest attitudes. The type, quality, and quantity of realworld applications that instructors use to generate interest, improve attitudes, and eliminate misconceptions should be further studied. Questions on the nature of these realworld problems should be investigated. How “realworld” are these problems? Do they come straight from the textbook or are they constructed by the instructors? How many of the realworld applications use actual data collection and analysis? Are the realworld problems authentic or developed specifically for the course? How much class time is dedicated to realworld applications? Answering these questions could contribute to improving college students’ statistical education. References Aiken, L. R. (1970). Attitudes toward mathematics. Review of Educational Research, 40, 551–596. Aiken, L. R. (1974). Two scales of attitude toward mathematics. Journal for Research in Mathematics Education, 5, 67–71. Aiken, L. R. (1976). Update on attitudes and other affective variables in learning mathematics. Review of Educational Research, 46, 293–311. Araki, L. R., & Schultz, K. S. (1995, April). Students’ attitudes towards statistics and their retention of statistical concepts. Paper presented at the annual meeting of the Western Psychological Association, Los Angeles, CA. Barkley, C. A. (1995). A study of misconceptions in statistics and statistics anxiety: Affect and performance in an undergraduate math course (Doctoral dissertation, University of Denver, 1995). Dissertation Abstracts International, AAI9543983. Brian Evans
Davis, S. F., & Palladino, J. J. (2002). Psychology (3rd ed.). Upper Saddle River, NJ: Prentice Hall. delMas, R. C., & Bart, W. M. (1987, April). The role of an evaluation exercise in the resolution of misconceptions in probability. Paper presented at the annual meeting of the American Educational Research Association, Washington, DC. Elmore, P. B., & Lewis, E. L. (1991, April). Statistics and computer attitudes and achievement of students enrolled in applied statistics: Effects of a computer laboratory. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL. Elmore, P. B., Lewis, E. L., & Bay, M. L. (1993, April). Statistics achievement: A function of attitudes and related experience. Paper presented at the annual meeting of the American Educational Research Association, Atlanta, GA. Evans, B. R. (2005). Student attitudes, conceptions, and achievement in introductory undergraduate college statistics (Doctoral Dissertation, Temple University, 2005). Dissertation Abstracts International, AAI3178774. Feinberg, L. B., & Halperin, S. (1978). Affective and cognitive correlates of course performance in introductory statistics. Journal of Experimental Education, 46(4), 11–18. Garfield, J. (1981). An investigation of factors influencing student attainment of statistical competence (Doctoral Dissertation, University of Minnesota, 1981). Dissertation Abstracts International, 42, AA18211476. Garfield, J., & Ahlgren, A. (1988). Difficulties in learning basic concepts in probability and statistics: Implications for research. Journal for Research in Mathematics Education, 19, 44–63. Gal, I., & Ginsburg, L. (1994). The role of beliefs and attitudes in learning statistics: Towards an assessment framework. Journal of Statistics Education, 2(2), 1–16. Gilovich, T., Vallone, R., & Tversky, A. (1985). The hot hand in basketball: On the misperception of random sequences. Cognitive Psychology, 17, 295–314. Gredler, M. E. (2001). Learning and instruction: Theory into practice (4th ed.). Upper Saddle River, NJ: Merrill Prentice Hall. Heaton, R. M., & Mickelson, W. T. (2002). The learning and teaching of statistical investigation in teaching and teacher education. Journal of Mathematics Teacher Education, 5, 35– 59. Hirsch, L. S., & O’Donnell, A. M. (2001). Representativeness in statistical reasoning: Identifying and assessing misconceptions [Electronic version]. Journal of Statistics Education, 9(2). Huck, S. W. (2007) Reading statistics and research (5th ed.). Boston: Allyn & Bacon. Huff, D. (1954). How to lie with statistics. New York: Norton. Kahneman, D., & Tversky, A. (1972). Subjective probability: A judgment of representativeness. Cognitive Psychology, 3, 430– 454. Kahneman, D., & Tversky, A. (1973). On the psychology of prediction. Psychological Review, 80, 237–251. Konold, C. E. (1991). Understanding students’ beliefs about probability. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 139–156). The Netherlands: Kluwer. 29
Landwehr, J. (1989, September). A reaction to alternative conceptions of probability. In J. Garfield (Chair), Alternative conceptions of probability: Implications for research, teaching, and curriculum. Symposium conducted at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, New Brunswick, NJ. Ma, X., & Kishor, N. (1997). Assessing the relationship between attitude toward mathematics and achievement in mathematics: A metaanalysis. Journal for Research in Mathematics Education, 28, 26–47.
Shaughnessy, J. M. (1977). Misconceptions of probability: An experiment with a smallgroup, activitybased, model building approach to introductory probability at the college level. Educational Studies in Mathematics, 8, 295–316. Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and directions. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 465– 494). New York: Macmillan. Schultz, K. S., & Koshino, H. (1998). Evidence of reliability and validity for Wise’s attitude toward statistics scale. Educational and Psychological Measurement, 82, 27–31.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
Sutarso, T. (1992, November). Students’ attitudes toward statistics (STATS). Paper presented at the annual meeting of the MidSouth Educational Research Association, Knoxville, TN.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Thurstone, L. L. (1970). Attitudes can be measured. In G. F. Summers (Ed.), Attitude measurement (pp. 127–141). Chicago: Rand McNally.
Neale, D. C. (1969). The role of attitudes in learning mathematics. Arithmetic Teacher, 16, 631–640. Paulos, J. A. (1988). Innumeracy: Mathematical illiteracy and its consequences. New York: Vintage Books. Roberts, D. M., & Saxe J. E. (1982). Validity of a statistics attitude survey: A followup study. Educational and Psychological Measurement, 42, 907–912. Rogness, N. T. (1993). The development and validation of a multifactorial instrument designed to measure student attitudes toward a course in statistics (Doctoral Dissertation, University of Northern Colorado, 1993). Dissertation Abstracts International, AAI9413644.
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Wise, S. L. (1985). The development and validity of a scale measuring attitudes toward statistics. Educational and Psychological Measurement, 45, 401–405. 1
The doublesized sociology class was one class consisting of two sections intended to be twice the size of a regular oneclass section. 2
On STACS, a 2 indicates a neutral response, with higher numbers representing positive attitudes or correct conceptions.
Student Attitudes, Conceptions, and Achievement
The Mathematics Educator 2007, Vol. 17, No. 2, 31–40
Listening to the Girls: Participant Perceptions of the ConfidenceBoosting Aspects of a Girls’ Summer Mathematics and Technology Camp Janet Hart Frost Lynda R. Wiest Females, students of color, and students of low socioeconomic status (SES) are often underserved or marginalized in mathematics education. However, some instructional approaches and intervention programs have been shown to educate these students more equitably. This study examines how girls of diverse racial/ethnic and socioeconomic backgrounds perceived the characteristics of one such intervention program as inspiring the development of greater confidence in their mathematics skills. This article explains the similarities and differences of the perceptions of each group, as well as the implications for classroom environments and further research.
Mathematics serves as a powerful gatekeeper in the American culture (Moses & Cobb, 2001; Schoenfeld, 2002). In our increasingly quantitatively and technologically oriented society, an individual’s level of mathematics understanding can affect his or her ability to function effectively as a consumer and as a citizen (Carnevale & Desrochers, 2003; Meier, 2003; Orrill, 2001). A student’s understanding of school mathematics and the ability to demonstrate that understanding influences opportunities for K12 school progress, undergraduate and graduate college attendance, and access to many careers (Moses & Cobb, 2001; Parrott, Spatig, Kusimo, Carter, & Keyes, 2000). On standardized tests, mathematics achievement varies among selected student groupings, such as those formed by race/ethnicity, socioeconomic status (SES), and gender. Scores on these tests, including the National Assessment of Educational Progress (NAEP), Scholastic Assessment Test (SAT), and Graduate Record Exam (GRE), reveal that, as early as grade 4, both students of low SES and those who are African Janet Hart Frost is an Assistant Professor of Education at Washington State University Spokane. After teaching middle and high school mathematics for 13 years, she now researches topics related to diversity and equity in mathematics education and the influences that help to shape mathematics teachers' instructional approaches. Lynda R. Wiest is an Associate Professor of Education at the University of Nevada, Reno. Her professional interests include mathematics education, educational equity, and teacher education. Dr. Wiest is founder and Director of the Girls Math & Technology Program, conducted for Northern Nevada girls since 1998. Janet Hart Frost & Lynda R. Wiest
American, Hispanic, American Indian and Alaska Native are likely to have lower scores in mathematics than White or Asian American students or students of higher SES. The differences in achievement scores for these groups increase with age. On the 2006 NAEP, by grade 8, Hispanics, Native Americans, Alaska Natives, and African Americans scored 26%, 25%, and 30%, respectively, lower than their White counterparts in achieving or exceeding proficiency levels. Students of low SES scored 26% lower than those of higher SES. Additionally, in both grades 4 and 8, females scored statistically lower than males in most areas of mathematics (Lee, Grigg, & Dion, 2007). Although these score differences are small (1 to 5 points), the differences grow on the college and graduate school examinations used for school admissions. On the quantitative portions of the 2006 SAT, females’ scores were 34 points lower than males; on the quantitative portion of the 1999–2000 GRE, females’ average scores were 97 points below males (National Center for Fair and Open Testing, 2007). In addition to lower scores, females are also more likely to have negative attitudes toward mathematics than males (Goodell & Parker, 2001). Even when girls and boys have equivalent test scores, girls indicate lower levels of confidence than boys, and they are more likely than boys to attribute failure to lack of ability (Vermeer, Boekarts, & Seegers, 2000). While the issues of inequities in mathematics, science, technology, and engineering exist for girls of all racial and SES backgrounds, they may be more severe for girls of color and girls of low SES (Daisey & JoseKampfner, 2002; Lim, 2004; Parrott et al., 2000; Thompson, Smith, & Windschitl, 2004). For these 31
reasons, girls of color and/or low socioeconomic status face double or even triple jeopardy in being successful in mathematics. Many authors describe changes in educational methods that can help to address these inequities (Gavin & Reis, 2003; Gilbert, 2001; Gilbert & Gilbert, 2002; Goodell & Parker, 2001; Perez, 2000). In particular, numerous authors have described informal interventions that helped female students develop greater confidence, motivation, and achievement (Karp & Niemi, 2000; Parrott et al., 2000; Peterson, 2004; Wiest, 2004). One such intervention program is the Girls Math & Technology Camp, which is available to Northern Nevada middle school girls. The main component is a fiveday residential summer camp held on an urban university campus. The girls who attend this camp come from urban and rural areas, and they have a broad range of socioeconomic backgrounds and mathematics skills. In Wiest’s 2004 study, this camp was shown to improve participants’ confidence, knowledge and skills, motivation, and effort in mathematics. Of these program outcomes, one area that is worthy of deeper research is the positive effect on participants’ confidence in mathematics. Confidence, which some authors equate with selfesteem (Erkut, Marx, Fields, & Sing, 1998), can be considered in general terms or in terms specifically related to a subject area such as mathematics. Mathematical confidence is believed to help girls with perseverance and independence in their mathematical efforts, as well as their anticipation of success as an outgrowth of ability–anticipation that in turn improves achievement (Fennema & Peterson, 1985). For these reasons, the impact of a mathematics intervention program on participant confidence warrants more indepth study. Although many studies that discuss impacts on student confidence and selfesteem do not differentiate the research results according to race/ethnicity or SES (Karp & Neimi, 2000; Peterson, 2004; Wiest, 2004), a few do identify factors of race/ethnicity and/or SES as salient (Birenbaum & Nasser, 2006; Erkut, Marx, & Fields, 2001; Erkut et al., 1998; Greene & Way, 2005; Lim, 2004; Parrott et al., 2000). In particular, these studies suggest that levels of selfesteem and factors that contribute to confidence vary according to race/ethnicity and SES. Birenbaum and Nasser (2006) demonstrated that ethnicity and gender contribute to students’ attitudes toward mathematics and discussed the possibility that an ethnic group’s cultural standing within a society plays a part in students’ attitudes. Similarly, Lim (2004) suggested that the girls of color 32
and low SES backgrounds in the classroom she observed had particularly fragile levels of confidence and motivation in mathematics. These levels of confidence and mathematics were easily eroded by the White middle class teacher’s instructional approach and the caring she showed to White middle class students that she did not extend to students of color or low SES. Therefore, the following questions are of interest: How does an intervention program impact the mathematics confidence of girls with different racial/ethnic and SES backgrounds? What similarities and differences do girls of different racial/ethnic and SES backgrounds express about the characteristics of the program? What do these similarities and differences suggest about the kinds of instructional approaches within which girls of color and low SES feel most comfortable and validated? Background of Study This section begins with a discussion of situated learning theory as the theoretical framework for this study. The discussion includes commentary on the kinds of issues females confront in mathematics, particularly if they are students of color or of low SES. Finally, the Girls Math & Technology Camp (hereafter referred to as “Math Camp”), from which participants for this study were drawn, is described. Situated Learning Theory In his discussion of situated learning, Wenger (1998) discussed the process of creating a sense of identity within a community. In this community context, learning takes the form of moving from a position of limited understanding of, responsibility for, and participation in the community activities to increasing levels of understanding, participation, and responsibility. Learning is thus a process of transformation both in the individual’s role or identity and in the way that the individual interacts with others in the community. Although a person’s identity is always evolving, Wenger’s description suggests that the interaction between an individual and the communities in which she or he is involved plays an essential role in that evolution. In the case of the girls who participate in a particular mathematics intervention program, they bring both the history of their prior mathematics learning experiences and the values, beliefs, and experiences that are part of their personal, family, and cultural backgrounds. Past experiences and resulting attitudes and beliefs interact with elements of the new Math/Technology Camp Impact
environment to create unique directions for each girl’s identity development as a mathematics student. In particular, student perceptions of themselves and others in terms of mathematics competence, comfort level with mathematical tasks presented, and sense of connection with others in the mathematics learning environment all play a part in their sense of identity as a mathematics learner. Wenger (1998) notes, “Engagement in practice gives us certain experiences of participation, and what our communities pay attention to reifies us as participants” (p. 150). For this reason, different learning environments are likely to leave students with different perceptions about their identity as mathematics students. “We know who we are by what is familiar, understandable, usable, negotiable; we know who we are not by what is foreign, opaque, unwieldy, unproductive” (p. 153). Marginalizing Characteristics in Mathematics Classrooms As described above, situated theory suggests that learning does not occur in an isolated, individual form, but that the context of the learning plays an essential part in determining the knowledge that is acquired (Boaler, 2002a; Lave & Wenger, 1991; Thompson et al., 2004; Wenger, 1998). For example, Boaler (2002b) explained that the difficulties some females experience in mathematics, such as anxiety and lack of confidence, are not inherent female qualities but are instead often engendered by the nature of the classrooms in which they learn mathematics. Similarly, she suggested that those who believe that students of color or low SES struggle because of qualities inherent in their race or culture ignore the ways in which some methods of teaching mathematics disenfranchise or exclude these students from equitable learning opportunities. Wenger (1998) described situations such as these as keeping some participants in “marginal positions” (p. 166) that essentially close the possibility of success. For these reasons, it is important to consider the classroom qualities that may offer greater support and learning opportunities for females, particularly females of color and/or low SES. Both teaching methods and teacher attitudes in mathematics classrooms can pose problems for many students, particularly those in marginalized groups. Boaler (2002b) discussed girls’ desire for conceptual and connected understanding that is thwarted when they are taught mathematics in traditional ways, e.g., using direct instructional approaches, individual competition, procedural emphases, decontextualized and meaningless problems, and lecture and Janet Hart Frost & Lynda R. Wiest
demonstration rather than handson approaches. The result of this mismatch between girls’ interests and the instructional approach led to disinterest in continuing studies in mathematics. Similarly, Schoenfeld (2002) cited several studies that demonstrated that when students of color and low SES were taught mathematics in a way that did not connect with their lives or the real world, they often failed and/or discontinued taking mathematics courses. In contrast, when they learned mathematical concepts with an instructional approach that emphasized connections with their experiences and world and communication about these concepts, students of color and low SES were more likely to find academic success. Similarly, teachers often convey lower expectations for and stereotypes about students of color and low SES (Daisey & JoseKampfner, 2002; Parrott et al., 2000; Rousseau & Tate, 2003). This may be particularly true for females in mathematics classes (Gavin & Reis, 2003; Lim, 2004). When students are aware of these lower expectations and stereotypes, their ability to learn or find success in mathematics is compromised (Johns, Schmader, & Martens, 2005; Quinn & Spencer, 2001). Perhaps even more noteworthy, teachers’ beliefs about students’ ability to achieve success in mathematics may result in students’ assignment to lowertrack classes that are often inferior to higher tracks in methodology and content (Achinstein, Ogawa, & Speiglman, 2004; Gamoran, 2001; Parrott et al., 2000). Moreover, some White and middle class teachers do not establish the same rapport or positive, nurturing relationships with their students of color or low SES that they do with students with whom they share a similar racial/ethnic and class background (Lim, 2004). Intervention Programs Several researchers have described intervention programs such as afterschool or weekend programs or summer camps that were particularly successful with girls (Gavin & Reis, 2003; Karp & Niemi, 2000; Peterson, 2004; Volpe, 1999; Wiest, 2004). By occurring outside of school, these programs generally have the benefits that students participate by choice and the activities involve little or no pressure related to grades, tests, or time constraints. For these reasons, female students often find these programs less threatening than the traditional classroom, allowing them the opportunity to move beyond potential mathematics anxieties and toward greater risktaking (Gavin & Reis, 2003). Participation in these intervention programs often results in increased 33
comfort with mathematics, including confidence, motivation, engagement, and achievement (Thompson et al., 2004). The Girls Math & Technology Camp The camp attended by the girls interviewed in this study was held in Northern Nevada. As described above, the program centers on a fiveday residential summer camp for girls who will enter grade 7 or 8 the following fall. The goals of the camp are to improve girls’ knowledge, skills, dispositions, and participation in mathematics and technology. Although past research has shown that the camp has impacted all of these areas, the largest and most consistent impact has been in the area of the girls’ dispositions (Wiest, 2004). Advertising for the camp is sent to all public, private, and Native American schools in Northern Nevada. The participants have a wide range of mathematics ability as well as SES and race/ethnicity (Wiest, 2004). Free and reduced lunch status was used as an indicator of low SES. Instead of paying the full $350 cost of the program, girls paid $25 if they qualified for free lunch and $50 if they qualified for reduced lunch. In 2004, there were 29 girls entering grade 7 and 5 girls entering grade 8. Table 1 shows a breakdown of 2004 participants by race/ethnicity and ! participation in free/reduced lunch. Table 1 2004 Participant Demographics SES Indicator Received free/reduced lunch Did not receive free/reduced lunch
Girls of color 6
White girls 6
5
17
Figure 1. Example of problem used for group conjecture and debate in a geometry lesson. Angles a, b, and c are interior angles; angles d, e, and f are exterior angles. The girls were asked to make and defend conjectures about patterns in the sums a + b + c and d + e + f . In addition to the lessons conducted at the camp, ! the girls were provided with female role models. Role models at the camp included program staff members and a guest speaker who uses mathematics and/or technology in her job. The girls also learned about historically famous women in mathematics and computer science. Context and Methods Research Design and Researcher Role
Topics studied during the week include problem solving (2.5 hours), spatial tasks (1.5 hours), geometry (7.5 hours), data analysis and probability (younger girls only, 7.5 hours), and algebra (older girls only, 7.5 hours). The girls also participated in 4.5 hours of computer classes designed to support the mathematics objectives, and they used fourfunction and/or graphing calculators as mathematics tools. Most work was completed in cooperative groups, changing members daily. Key pedagogical strategies included handson activities, mixedability cooperative group work, realworld applications, and problem solving and investigation in a supportive learning environment. For example, geometry lessons began with group conjecture, discussion, and debate about the sums of interior and exterior angles of triangles, based on triangles the girls drew. (See Figure 1.) Subsequent 34
geometry lessons included identifying and discussing other patterns in angles of geometric figures as well as using models to identify patterns in the numbers of vertices, edges, and faces in prisms and pyramids. Other handson activities and group work included data analysis lessons in which the girls found the mean, median, and mode of their heart rates.
For this study, the first author chose girls from the attendees at the Math Camp according to their racial/ethnic and socioeconomic background. In contrast to Wiest’s 2004 study, more indepth interviews were conducted to focus on the impact the camp had on girls’ confidence in mathematics. Two semistructured interviews were conducted with each of these girls. The first interview occurred within two weeks after the camp, and the second occurred approximately six months later. These interviews generally occurred at the girls’ homes or at another location that was convenient for the participants. In a few cases in which the participants lived in a remote rural area, the interviews were conducted by phone. The first author had been the Math Camp Program Assistant and one of the instructors, so both the girls and their parents were familiar with her. This role allowed her to establish rapport with each of the girls before the interviews. It is possible that this familiarity Math/Technology Camp Impact
could have influenced this author’s interpretation of the girls’ responses, although the interviews were the only extended conversations she had with them. Participants Using stratified sampling, 16 girls were chosen from the Math Camp participants with four from each of the following groups: girls of color who received free or reduced lunch, girls of color who did not receive free or reduced lunch, White girls who received free or reduced lunch, and White girls who did not receive free or reduced lunch. The presence or absence of participation in free or reduced lunch was used as a convenient indicator of socioeconomic status because these data were readily available. However, a variety of circumstances shifted the number of participants. For example, because one girl chosen for the study carpooled with another girl who was not chosen, the request was made and granted to include both girls. One girl of color, who was identified as a person who did not receive free or reduced lunch, was later moved to the group of girls of color who received free or reduced lunch. This change was made because it was found that the free or reduced program did not exist at her reservation school. Because her family could not afford to pay the tuition, her fees had been paid by the Native American community of which she was a member. Finally, one higher SES girl of color declined to participate in the second interview. The demographic breakdown of the participants who were included in the study is shown in Table 2. Table 2 Research Study Participant Demographics SES Indicator Received free/reduced lunch Did not receive free/reduced lunch
Girls of color 6
White girls 5
3
5
Data Gathering Procedures Two interviews were conducted with each girl. Each interview lasted approximately one hour. The interviews were recorded and later transcribed. The openended questions included questions about the effect Math Camp attendance had on the girls’ dispositions toward mathematics and their school mathematics classes, as well as on strategies they use for dealing with problems in mathematics. These questions were designed to provide data about how and why the Math Camp experience might have had an effect on students’ confidence as mathematics learners. Janet Hart Frost & Lynda R. Wiest
The intent was to determine if there was a change in the girls’ confidence about their mathematics work, including evidence of increased optimism about their work in mathematics classes and their perseverance and ideas about available resources when they encountered difficult problems. The following questions were posed. 1. (One month and six months after camp) Did attendance at Math Camp have an effect on your confidence in mathematics? If so, why? What qualities of Math Camp helped you feel more confident? 2. (One month after camp) How do you feel about going to your math class in the fall? Do you think going to Math Camp had an effect on this? Why or why not? 3. (Six months after camp) How confident do you feel about math class this year compared to last year? Do you think going to Math Camp had an effect on this? Why or why not? 4. (Six months after camp) If you are having difficulty in math or with a specific problem, what do you do? How do you address the problem? Has that changed since you went to Math Camp? If so, do you think Math Camp had an effect on that? If so, why? Data Analysis Procedures Interview responses were coded using a grounded theory approach (Ryan & Bernard, 2000). Lists were made of specific ideas mentioned by individuals, such as comments about cooperative group work, and were compared to similar comments made by other girls. These responses were categorized according to the racial/ethnic and SES groups to which the girls had been assigned. The data were then examined to discover trends in the frequency of these responses, either across all groups or by racial/ethnic group and SES status. Responses that seemed especially salient to the impact of the camp or the girls’ experiences in learning mathematics, even those made by only one or two girls, were also noted. Results In the first set of interviews, 16 of the 19 participants said that Math Camp attendance had increased their confidence in mathematics in general and in regard to their fall mathematics class participation. In the second set of interviews, all 19 said that Math Camp had improved their confidence. In elaborating on this improvement, one girl said, “Last 35
year I wouldn’t attempt [math]. This year I’m getting a little better at it. Algebra is easy.” Another said,
that there were other people out there who were just like me.
I think I feel more successful because, like before, I didn’t think I could like get anything right in math, but after going to math camp, and after explaining it and everything to me, I feel like I can mostly do anything now.
Another girl said, “I can understand what I’m doing more and see what mistakes I’m making. ... I knew I wasn’t the only one having trouble, so I wasn’t afraid to work on it.” These and similar comments indicated that a sense of commonality with others was reassuring and helped the girls feel more confident, even when they did not immediately know how to solve a problem; this confidence directly impacted their perseverance. The Math Camp environment gave the students an opportunity to take the risk of acknowledging what they did not understand or found difficult without any negative repercussions. One girl accompanied her comments on this topic by expressing appreciation for the way other students helped her when she had difficulty, explaining that her friends at school just told her the answers, but these girls helped her to understand how to do the problems. Among those who mentioned the curriculum, 42% spoke specifically about the handson geometry lessons, which, as described above, included drawing and using manipulative models as the basis for making conjectures and debating angle measures and relationships. The girls said that they felt these lessons contributed to their improved confidence and understanding because the lessons were “fun” and allowed them to develop an understanding of terms that were new to them or for which they had not grasped the meaning in their previous studies. As stated above, these comments were made across all groups and seemed to reflect the girls’ appreciation for an atmosphere in which they felt a connection with the others in the group rather than feeling isolated or in competition with them. Additionally, their comments about geometry indicated that they enjoyed the chance to learn, in a handson way, about a topic that they either had not learned or had not mastered in the past.
The qualities of the Math Camp experience that each group believed had positively impacted their confidence are discussed below. Because of the openended nature of the questions, mention of a topic by two or more girls was used as an indicator that it warranted attention. Common topics across all groups included teaching methods, curriculum, and peer interactions. Differences among groups were noted in participant comments about the value of particular curriculum topics or experiences and of group presentations (sharing small group work in a wholeclass setting), as well as strategies used when facing difficulties in mathematics work. Common Topics Across All Groups Some topics arose in all groups, regardless of racial/ethnic or socioeconomic background. As mentioned above, these themes were related to instructional methods, curriculum, and peers. Regarding the instructional methods, 43% of the girls reported that their confidence was positively affected by the focus on cooperative group work. These girls explained that they appreciated the way that group work helped them see different approaches used by other students. One girl stated that this atmosphere helped her feel more confident because in math camp, we had hard questions and we had to try it out a lot of times ... and everyone did it a different way ... now I know there’s more ways to figure out one answer instead of just like giving up and just saying I can’t do it.
The girls also mentioned feeling that they shared common attributes and experiences with other girls in their groups, such as similar abilities and difficulties in mathematics, which contributed to their confidence. These feelings were also expressed by girls who had noticeably higher skill levels than many of those with whom they had worked, as implied in the following comment. I think that Math Camp built up my confidence because, at one point in time, I felt like I was the only person who didn’t understand math in a way, and then I met all these other girls who were having trouble or not having trouble in the same areas that I was and it was just really nice to know 36
Differences Between SES Categories Examining the comments by SES categories, some patterns in the girls’ views of the instructional methods and curriculum were noticeable, including specific teaching techniques and curriculum topics. Students who received free or reduced lunch were more likely (100% versus 50% of the other girls) to mention the benefit of revisiting topics they had learned previously, often explaining that they felt they developed a higher level of understanding or “refreshed their memory” in a helpful way. More of these girls (64% versus 13% of the other girls) valued the opportunity to present their group’s ideas in front of the class, explaining that this Math/Technology Camp Impact
also helped them increase their confidence, in part because it was another opportunity to see and share different approaches to the problems. One said she liked “the way you would have us in groups and then we’d share it with the class.” When the researcher asked, “Any idea why that made you more confident?” she replied, “Well, you get more answers and you see how other people have their point of view about the problem.” The frequency of these girls’ verbalized preference for making presentations to the group contrasted with some of the same girls’ discomfort in doing so at the beginning of the camp. According to the girls’ comments, this discomfort disappeared quickly, indicating that they found participation in the presentations to be a positive experience. Math Camp participation also appeared to affect the strategies used and the confidence of girls of low SES in confronting difficult problems when they returned to school. When asked how they handled difficulty with a problem after attending the camp, this group of girls was more likely than the others (36% versus 0%) to rely on their own strength and insight, rather than teachers or peers, as the first resource. They explained that they did not immediately ask for help, but first tried other approaches. One girl stated, I read over the problem to see if I understand it, kind of look at it in a different way, and that sometimes helps me understand the problem better. Math Camp affected me ... because it has made it so that I feel that I’m not so frustrated any more about the problems I don’t understand at first because I know that if I just keep looking over it, I’ll eventually understand it.
In her comment, this participant indicated that she changed from feeling frustrated by difficulty to believing she had the capacity to understand difficult problems if she just took more time. Thus, she believed the camp experience increased her optimism about and perseverance in mathematics. In contrast to the girls from low SES backgrounds, students in the higher SES group made fewer such comments. These students were more likely (50% versus 0%) to turn to the teacher as a first resource when they encountered difficulty, rather than using their own resources as discussed by the low SES group. Differences by Race/Ethnicity and SES Some trends by participants’ race/ethnicity were evident in the comments. These trends came from the girls of color who were also of low SES, rather than coming from all girls of color. For example, 2 of the 6 Janet Hart Frost & Lynda R. Wiest
girls in this group discussed, at length, the different mathematics teaching methods they had experienced at Math Camp and in class. They were explicit in explaining that they were much more interested in active, handson learning than in transmission models of instruction. One girl explained: Like our teacher ... he just like starts talking and talking and talking, and it’s all dark, like he has his little projector or whatever ... it makes you want to go to sleep. My last teacher, she will explain it to us and she’ll get things to show us and she had a lot of projects…. We’d actually do [a] survey, asking people … which was a lot easier to learn than just sitting there.
In terms of curriculum topics, 50% of the girls who were racial/ethnic minorities and receiving free or reduced lunch reported that they had difficulty with word problems in mathematics class. However, other girls in the same group commented that they were more confident doing word problems after extensive practice with them at Math Camp. Fewer similarities were evident among the White girls (of both SES groups) who attended the camp. Interestingly, in describing qualities of the camp that improved their confidence, this was the only group to mention learning to use graphing calculators (30%). Also, this group of students, like the group who were not low SES, commented that they were more likely to turn to the teacher as a resource when they had difficulty with problems. In summary, all girls valued cooperative group work, including the sense of commonality with their peers that they experienced in that setting as well as the study of geometry. The girls of color and low SES groups were more likely to talk about the value of participating in presentations, having the opportunity to review topics they had studied before, and learning through active, handson lessons. Each of these experiences at the camp was identified as helping them to improve their mathematical confidence. They also felt they had gained more confidence in their ability to be selfsufficient in resolving difficulties. In contrast, White girls and those who were not low SES said they were more likely to rely on the teacher for help, and White girls were more likely to express the idea that learning about graphing calculators increased their confidence. Discussion and Closing Thoughts Returning to the research questions, this study examined how and why the positive effects of Math Camp impacted girls’ confidence and whether there 37
were differences in the girls’ perspectives corresponding to race/ethnicity and SES. A secondary focus was placed on the implications that the girls’ perspectives had for the kinds of learning communities in which they felt most comfortable and validated. Although the small number of girls in this study and the unique characteristic of voluntarily participating in a summer mathematics and technology camp limit generalizations that apply to other populations, the results suggest many interesting considerations worthy of further exploration. In particular, when comparing the girls’ comments about factors that improved their mathematical confidence across the different demographic groups, both the common and differing responses made by the girls offer insight into the aspects of the camp experience they valued and perhaps found unique in the camp learning environment. Based on these comments we suggest directions for further study in order to understand why specific characteristics stood out to particular groups of girls. Several participants across all groups discussed the confidenceboosting impact of working with peers with whom they felt they had things in common. These comments suggest further investigation into middle grades girls’ perceptions of the characteristics they do or do not share with other members of their school classes and reasons why the camp experience may provide a greater sense of commonality with their peers than they had experienced in their classrooms. For example, did the prevalence of small group discussion as a means of generating conjectures and debates among different small groups at the Math Camp allow for beneficial sharing of perspectives that increased these girls’ sense of commonality? Boaler and Staples (2005) observed similar effective smallgroup work at a school with high proportions of students of color and low SES. Their work indicates that the girls’ positive experiences at the camp were not unique to that setting but can be created in classrooms by emphasizing students’ multidimensional skills and reliance on each other. In terms of curriculum, many students in all groups valued the opportunity to study geometry. One girl commented that she had never studied geometry before, and others made comments that suggested, that although they may have studied geometry in the past, they found the math camp instructional approach particularly enjoyable and memorable. This result suggests the benefit of further study of the way girls compare their school and camp experiences of learning geometry, particularly the amount of time they spent 38
on the topic in their school classrooms and the nature of the lessons. Girls of low SES valued the opportunity to present their thinking in front of the class, in spite of the fact that some of the girls had initially been uncomfortable with this idea. The value they placed on these presentations suggests that further studies might examine how girls of varying SES backgrounds compare their experiences of presentations in front of their school classes with their experiences in Math Camp. Further research may reveal reasons why girls of low SES, more than other girls, were initially uncomfortable but later excited about the presentations. The low SES group found particular value in review opportunities. One hypothesis for this outcome would suggest that these girls did not feel they had fully mastered the ideas before this review. Therefore, future studies might examine reasons why review would be particularly helpful to girls of low SES–and more valued by them than by girls of higher SES. Was the review helpful because it was presented in a different way than the manner in which they had originally learned the material? Had they not mastered the ideas the first time they had been exposed to them? Or did they perceive reasons why it was especially important for them to master this material that other girls did not echo? The students of low SES were more likely (than the girls of higher SES) to describe themselves as more selfreliant in their mathematics work after attending Math Camp; the higher SES students continued to turn to their teachers for help. Additionally, the lower SES girls made comments that suggested they had increased confidence in their ability to understand difficult problems on their own after attending Math Camp. These results suggest the value of learning more about the resources students of low SES perceive are available–or not available–to them, as well as the importance they place on resolving issues on their own rather than relying on others and the reasons for this priority. Because girls of higher SES were more likely to make comments about turning to their teachers for help when they encountered difficult problems, future research might examine if there are differences in the ways students of varying SES experience their teachers’ availability for help. This research may also investigate whether there are other reasons why one group might be more likely to depend on their own perseverance while another turns to the teacher. Some girls of color and low SES described word problems as difficult in school. In contrast, other girls of color and low SES explained that they gained Math/Technology Camp Impact
confidence in their ability to problemsolve with word problems through their Math Camp experiences. Therefore, it would be valuable to pursue greater understanding of the differences between the experiences. Were the types of word problems different? Or was the environment in which they completed them different? Two girls of color and low SES provided extensive descriptions of classroom environments they found boring–namely, the situations in which the teacher stood at the overhead lecturing or demonstrating–in contrast to their experiences of conducting surveys and using these surveys as a foundation for their mathematics studies. Therefore, future studies might examine whether this perception is unique to girls of color and low SES or if other girls share the same perception. Are different types of instructional approaches more or less engaging for different groups of girls? Some studies suggest that girls are particularly likely to benefit from mathematics that they find engaging and meaningful (Boaler, 2002b; Boaler & Greeno, 2000). Are the benefits even greater for girls of color and low SES? Only the White girls who were not identified as low SES made comments about the value of learning how to use graphing calculators and that this learning contributed to their confidence. Therefore, it would be beneficial to understand more about the reasons why only this group named this aspect of the camp experience. Are there reasons why these girls would value the use of graphing calculators more than the girls of color or of low SES? Or are there reasons why the experience might have been more accessible to them? How much do girls of color or low SES see graphing calculators as an important part of their future school or life experience? Although this study was limited in the number of participants, the results of this study and the suggestions for future study echo the results of other similar studies of instructional approaches that have been successful with females, students of color, and students of low SES (Gavin & Reis, 2003; Gilbert, 2001; Gilbert & Gilbert, 2002; Goodell & Parker, 2001; Perez, 2000). Clearly, as described by advocates of situated theory, the interplay of student characteristics and the learning context has an important role in how students perceive and what they gain from their learning experiences. If students participate in an environment that promotes a sense of themselves as capable mathematicians and aligns with their sense of identity in such a way that they gain confidence in their abilities and interest in the subject Janet Hart Frost & Lynda R. Wiest
matter, they are likely to learn more and achieve success in their learning. For example, Boaler and Greeno (2000) described how students who experienced different instructional methods in their mathematics classroom were influenced to see mathematics (a) as abstract and demanding obedience and perseverance or (b) as both a creative and cooperative endeavor and a subject that was connected with their world. Students who viewed mathematics in the second way were much more likely to perceive mathematics as important in their lives. They valued mathematics and intended to continue their study of it because their view of mathematics aligned with their views of themselves and their futures. As described above, the comments made by the girls offer multiple directions for further study, particularly in terms of the ways that girls of different race/ethnicity and SES experience both the school classroom and a summer intervention program. Other factors also suggest further research agendas. For example, although this study combined girls of color into one group, Hispanic, African American, and Native American girls may have been impacted differently from attending the camp due to cultural or individual characteristics, and Asian girls–who were not part of this study–may have perspectives that differ from both White and minoritystatus groups. Similarly, English Language Learners may also have unique perspectives. Additionally, students with homes in rural or urban areas may have different perceptions and experiences. Each of these factors may affect students’ confidence levels and perceptions of their camp experience and are worthy of additional study. References Achinstein, B., Ogawa, R. T., & Speiglman, A. (2004). Are we creating separate and unequal tracks of teachers? The effects of state policy, local conditions, and teacher characteristics on new teacher socialization. American Educational Research Journal, 41, 557–603. Birenbaum, M., & Nasser, F. (2006). Ethnic and gender differences in mathematics achievement and in dispositions toward the study of mathematics. Learning and Instruction, 16(1), 26–40. Boaler, J. (2002a). The development of disciplinary relationships: Knowledge, practice, and identity in mathematics classrooms. For the Learning of Mathematics, 22(1), 42–47. Boaler, J. (2002b). Paying the price for “sugar and spice”: Shifting the analytical lens in equity research. Mathematical Thinking & Learning, 4, 127–144. Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematical worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171– 200). Westport, CT: Ablex Publishing.
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Moses, R. P., & Cobb, C. E., Jr. (2001). Radical equations: Math literacy and civil rights. Boston, MA: Beacon Press. National Center for Fair and Open Testing. (2007). University testing. Retrieved April 20, 2007, from http://www.fairtest.org Orrill, R. (2001). Preface: Mathematics, numeracy, and democracy. In L. A. Steen (Ed.), Mathematics and democracy: The case for quantitative literacy (pp. xiiixx). Princeton, NJ: National Council on Education and the Disciplines. Parrott, L., Spatig, L., Kusimo, P. S., Carter, C. C., & Keyes, M. (2000). Troubled waters: Where multiple streams of inequality converge in the math and science experiences of nonprivileged girls. Journal of Women and Minorities in Science and Engineering, 6(1), 45–71. Perez, C. (2000). Equity in the Standardsbased elementary mathematics classroom. ENC Focus, 7(4), 28–31. Peterson, K. (2004, April). Advancing the agenda in gender equity for science, mathematics, engineering, and technology in Washington and Oregon: The Northwest Girls Collaborative Project. Paper presented at the Annual Meeting of the American Educational Research Association, San Diego, CA. Quinn, D. M., & Spencer, S. J. (2001). The interference of stereotype threat with women’s generation of mathematical problemsolving strategies. Journal of Social Issues, 57(1), 55–71. Rousseau, C., & Tate, W. F. (2003). No time like the present: Reflecting on equity in school mathematics. Theory Into Practice, 42, 210–216. Ryan, G. W., & Bernard, H. R. (2000). Data management and analysis methods. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 769–778). Thousand Oaks, CA: Sage. Schoenfeld, A. H. (2002). Making mathematics work for all children: Issues of standards, testing, and equity. Educational Researcher, 31(1), 13–25.
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The Mathematics Educator 2007, Vol. 17, No. 2, 41â€“51
An Analysis of the New Elementary Mathematics Teacher Education Curriculum in Turkey Mine Isiksal Yusuf Koc Safure Bulut Tulay AtayTurhan The purpose of this paper is to describe and reflect on the changes in the new elementary mathematics teacher education curriculum in Turkey. It is our goal to share the revised teacher education curriculum with the outside mathematics teacher education community. The paper is organized around four major sections: An overview of the teacher education system in Turkey, the characteristics of the previous mathematics teacher education curriculum, the need for the revisions, and characteristics of the revised curriculum.
All over the world, there has been increased attention on the professional education of teachers at all levels (Bishop, Clements, Keitel, Kilpatrick, & Leung, 2003; DarlingHammond & Bransford, 2005). Among other disciplines mathematics teacher education has been at the focus of reform in teacher education (Lin & Cooney, 2001). The curriculum for the education of mathematics teachers has also been considered a very crucial aspect of the mathematics teacher education enterprise (Grossman, Schoenfeld, & Lee, 2005). In Turkey, policy makers have recently revised the existing curriculum for the education of elementary school mathematics teachers as part of a larger scale revision of teacher education curricula (Higher Education Council [HEC], 2006). In this paper, our purpose is to describe and reflect on the Mine Isiksal is currently doing her postdoc at State University of New York. She received her Ph.D. in mathematics education in the Department of Secondary Science and Mathematics Education at the Middle East Technical University, Turkey. Yusuf Koc is currently a lecturer in mathematics education in the Department of Elementary Education at Middle East Technical University, Turkey. He received his Ph.D. in mathematics education from Indiana University. Safure Bulut is currently an associate professor in the Department of Secondary Science and Mathematics Education at Middle East Technical University, Turkey. She received her Ph.D. in mathematics education from Middle East Technical University. Tulay AtayTurhan worked at Middle East Technical University, Turkey for the past two years. She received her Ph.D. in Early Childhood Education from Florida State University. Mine Isiksal, Yusuf Koc, Safure Bulut, & Tulay AtayTurhan
changes in the new elementary mathematics teacher education curriculum in Turkey. We will discuss the history of mathematics teacher education in Turkey, the major motivations for revising the previous curriculum of elementary mathematics teacher education programs, and the development process and characteristics of the revised curriculum. History of Mathematics Teacher Education in Turkey Beginning from the foundation of the Republic of Turkey in 1923, Turkey has aimed to raise its standards in economical, social, political, and educational arenas to that of the developed countries through a wide range of reform efforts. Reforms in the field of education were among the most important changes in the 1920s (Cakiroglu & Cakiroglu, 2003). In 1924, the Turkish education system was centralized and all educational institutions were put under the control of the Ministry of National Education (MNE) (Binbasioglu, 1995). Changing the countryâ€™s teacher education system was another focus of reform. Before 1980, teachers were graduating from a variety of institutions with diverse experiences (see Cakiroglu & Cakiroglu, 2003 for details); however, in 1981, all teacher education institutions were placed under the authority of the HEC for a unified system of teacher training. Later, in 1989, the Council mandated that teacher candidates be educated in fouryear colleges in order to be eligible for a teaching certificate (Binbasioglu, 1995). The reform wave in Turkish teacher education was quite strong in the late 1990s. In 1998, with a support from the World Bank, the HEC restructured teacher education programs to enhance quality; this was part of 41
a fouryear project involving changes in all primary and secondary teacher education programs in colleges of education (Bulut, 1998). Prior to the 1998 changes, there were 26 departments or divisions of mathematics education awarding licenses for secondary mathematics teaching (grades 6 thru 11). As a result of these changes, the existing programs were closed and replaced by 28 elementary and 12 high school mathematics teacher education programs (Center for Student Selection and Placement, 1998). As part of the change, teacher education curricula, including the mathematics teacher education curriculum, were substantially revised. The 1998 curriculum was put in place starting from the fall semester of 1998; the details of the program are given in Appendix A. For additional details of the 1998 curriculum, see Bulutâ€™s 1998 article. The previous elementary mathematics teacher education curriculum was in use for eight years, from 1998 to 2006. During this time, in various academic and public platforms, including symposiums, panels, workshops and conferences, the qualifications of the preservice teachers in the previous curriculum were questioned. There was a consensus on the inadequacy of the subject matter knowledge, pedagogical content knowledge, and skills of the preservice teachers in the face of current societal and technological demands. It was concluded that the teacher education curriculum was partly responsible for such problems in the education of mathematics teachers. In order to find solutions to existing problems, the HEC collaborated with the faculty and deans of the colleges of education and decided to revise the existing curriculum. In this article, as a continuation of Bulutâ€™s (1998) discussions, the characteristics of the 2006 elementary mathematics teacher education curriculum are discussed and compared with the 1998 curriculum. Currently, in Turkey, elementary mathematics teachers are certified to teach 4th through 8th grade mathematics; yet, they mainly teach sixth, seventh and eighth grades. High school graduates are admitted to teacher education programs via the university entrance exam. Some high schools focus on careers in education; they follow the national high school curricula but offer professional education courses, e.g., introduction to education and educational psychology. Because graduates of these high schools are given extra points on the university entrance exams, most of the teacher education majors have graduated from such high schools. Preservice teachers take courses in three major areas: content courses, general education courses, and 42
pedagogy courses (Appendixes A & B). Mathematics courses, science courses, and mathematics teaching methods courses constitute the content area courses. General education courses, referred to in the United States as a core curriculum, provide preservice teachers with necessary background in technology, social, cultural and historical topics such as computer literacy, foreign language, and Turkish History and Language. Preservice elementary mathematics teachers are also required to take a number of pedagogy courses. These are mainly devoted to topics in general pedagogy, including educational psychology, classroom management and counseling. Preservice elementary mathematics teachers also engage in field experiences and student teaching activities as part of their education. These students take 146 credit hours of courses to graduate from the program. It is always hard to characterize an entire education system, but we will try to portray what happens in these three categories of courses. Mathematics and physics are usually taught in a very traditional way, through lecture. There are generally two midterms and one final examination to assess student performance in such courses. The mathematics teaching methods courses are expected be taught in a nontraditional format where theory and practice are blended to help the preservice teachers understand applications of theories and earn firsthand experience through various activities. General education courses are taught by faculty with diverse backgrounds; as a result, while some of them prefer lecturing, others promote more student participation. In addition to regular paperpencil tests, projects are used to assess student performance in those courses. The pedagogy courses are taught in various ways. While lecture is common, student presentations, video presentations, small group work, whole group discussions and other nontraditional methods are used to deliver pedagogy courses. The entire teacher education program takes four years, equivalent to a regular undergraduate degree. The two academic semesters, fall and spring, each last about 14 weeks. Teaching certificates awarded by universities are permanent and valid throughout the country. Certified teachers are required to pass the Government Staff Selection Exam (KPSS) in order to work in public schools. Only 2% of all elementary schools (757 schools out of 34,656) are operated by private organizations. They recruit teachers based on their own criteria.
Teacher Education Curriculum
Motivations for the Revision of the Teacher Education Curriculum We will discuss four major motivations for the revision of the teacher education curriculum. These are (a) preparation for integrating with the European Union, (b) the changes in the elementary school mathematics curriculum, (c) the need for better qualified teachers, and (d) problems with the previous curriculum (HEC, 2006). Integration With the European Union Turkey, in general, could benefit in many ways from European Union (EU) membership. In particular, we expect to become more integrated, more prosperous, more autonomous, and more democratic as an EU member. In addition to reorganizations in political, economical and social areas, education is considered a critical component of the integration process with the EU. Turkeyâ€™s longstanding wish to be a member of the EU motivated teacher educators and curriculum developers to revise teacher education programs and improve the level of teaching standards (HEC, 2006). This reform is not limited to the K12 curriculum. Changes in the Elementary School Mathematics Curriculum In 2003, the Turkish Ministry of National Education organized a curriculum development team to revise the existing elementary school mathematics curriculum. The new curriculum was designed as part of a largerscale curriculum reform initiative that included five content areas: mathematics, science, social sciences, life science and Turkish language (Koc, Isiksal, & Bulut, 2007). One of the major objectives of this curriculum reform was to promote teaching and learning environments in which students can share their ideas and actively participate (MNE, 2006). The new curriculum placed a heavy emphasis on childrenâ€™s cognitive development, emotions, attitudes, interests, selfconfidence, beliefs, anxiety, selfregulation, psychomotor development and social skills. Additionally, the curriculum promoted student discussion, inquiry, and curiosity about what is going on in their families, schools, and society (MNE, 2004). Researchers suggest replacing rote memorization with learning for understanding (Hiebert, Carpenter, Franke, et al., 1997). In order to address this and to increase student participation, teachers were encouraged to set up studentcentered classroom environments. Furthermore, it was within the goals of the curriculum to encourage students to work collaboratively, to Mine Isiksal, Yusuf Koc, Safure Bulut, & Tulay AtayTurhan
communicate effectively about their ideas and to reflect on their learning. Research suggests that this will give students a chance to express their ideas and increase selfconfidence (Bandura, 1986). Problem solving, a critical aspect of understanding (Polya, 1957), was introduced as an integral aspect of all subject areas in the new curriculum. For meaningful student learning, the curriculum encouraged teachers to consider the outside contextual elements, such as lifestyle and geographical factors, while designing classroom tasks. Finally, assessment was regarded as an essential part of the classroom instruction (Irish National Council for Curriculum and Assessment, 1999; United Kingdom Qualifications and Curriculum Authority, 1999; National Council of Teachers of Mathematics [NCTM], 1995, 2000; Romberg, 2004). Expected teacher dispositions in the new school curriculum. Teachers are expected to exhibit a variety of skills and characteristics in order to effectively implement the new elementary school curriculum. First of all, they should believe that all students can learn mathematics. This particular characteristic is essential in promoting equity in mathematics learning. Aligned with this, teachers are to respect and follow all aspects of human rights and ethical values in mathematics classrooms. Teachers are also expected to work toward helping students develop positive attitudes about mathematics. Additionally, teachers need to guide and motivate students in learning mathematics. They should also motivate students to ask questions, engage in critical thinking, state and support ideas, and inquire about the subject matter. To accomplish these goals, teachers must know their students, parents, and the community in which they live. Additionally, it is explicitly stated in the new curriculum that mathematics teachers should enjoy teaching mathematics. Teaching responsibilities of elementary school mathematics teachers. In elementary schools, mathematics teachers are required to teach 15 class hours per week. They are paid for every extra hour above 15 hours. Additionally, in some schools, mathematics teachers are assigned a class of advisees. Like all other teachers, mathematics teachers are also responsible for organizing and leading social activities such as sporting events, national ceremonies, and extracurricular activities. There are at least two department meetings and two general faculty meetings in elementary schools. Additionally, some teachers organize after school and weekend courses for students willing to receive extra mathematics instruction for a minimal fee. 43
The curriculum states that teachers should develop and implement instructional activities that promote mathematical understanding, regularly monitor and evaluate student learning, effectively manage instructional time, and encourage students to evaluate their own and their peers’ progress. Teachers are also expected to use assessment and evaluation results to improve the quality of instruction. Furthermore, mathematics teachers should collaborate with parents, other school personnel, and the outside community to improve the quality of schooling. Given the above expectations for the teachers, it is definitely important that teachers develop selfconfidence and selfregulation skills. Finally, they need to continuously improve their professional knowledge and experiences through a variety of activities, including following scientific research literature and conducting smallscale research projects. The Need for BetterQualified Teachers One of the key factors in improving instruction and student understanding in the mathematics classroom is the role of the teacher (Hiebert et al., 1997). The above discussions indicate that there is a need for qualified teachers to be able to implement the new elementary mathematics school curriculum effectively. Teaching mathematics effectively is a complex endeavor, and there are no easy recipes for success. Effective teachers must know and understand the mathematics they are teaching, and they must flexibly draw on that knowledge (Hill, Schilling, & Ball, 2004). While challenging and supporting students, teachers need to understand the gap between what their students know and what they need to learn (NCTM, 2000). NCTM (1991) emphasizes that “teachers must help every student develop conceptual and procedural understandings of numbers, operations, geometry, measurement, statistics, probability, functions, and algebra and the connections among ideas” (p. 21). Thus, in order to develop the conceptual and procedural understanding of students, teachers should understand the content on both of these levels. Mathematics teachers not only need to have sufficient content knowledge of mathematics, but also pedagogical content knowledge (Even, 1990). Teachers need to know why mathematical statements are true, how to represent mathematical ideas in multiple ways, what constitutes an appropriate definition of a term or concept, and methods for appraising and evaluating mathematical methods, representations, or solutions (Hill, Schilling, & Ball, 2004). Subject matter knowledge and pedagogical content knowledge are 44
essential in effective mathematics teaching and in the preparation of mathematics teachers (NCTM, 2000). Tirosh (2000) states that a major goal in teacher education programs should be to contribute to the development of preservice teachers’ knowledge of common ways children think about school mathematics topics. She conjectures that the experience acquired in the course of teaching is the main, but not the only, source of teachers’ knowledge of students’ common conceptions and misconceptions. Preservice teachers’ own experiences as learners, together with their familiarity with relevant developmental and cognitive research, could be used in teacher education programs to enhance their knowledge of common ways of thinking among children. In summary, mathematics content and pedagogical content knowledge are critical factors in the effectiveness of mathematics teachers. The new school curriculum requires teachers to expand their theoretical knowledge and studentcentered teaching experiences. These needs motivated the Turkish Higher Education Council to revise the teacher education programs. The HEC aimed to increase the quantity and quality of the courses in the teacher education curriculum to help preservice teachers increase their professional knowledge and skills to teach elementary school mathematics. In Turkey, teacher education programs experience a number of challenges in achieving these goals, including a limited number of faculty members specialized in teacher education, inadequate universityschool partnerships, lack of enough field and student teaching experiences, and issues with the quantity and quality of the teaching methods courses. Problems With the Previous Curriculum The developers of the new teacher education curriculum identified major handicaps of the content and implementation of the curriculum that had been in place during the previous eight years. Teacher education programs experienced a number of challenges in achieving these goals, including a limited number of faculty members specialized in teacher education, inadequate universityschool partnerships, lack of enough field and student teaching experiences, and issues with the quantity and quality of the teaching methods courses. Here, we mention the major problems and how they were overcome in the revised curriculum. These problems fit into two main categories: (a) content and pedagogy and (b) policy.
Teacher Education Curriculum
In the previous curriculum, there were 27 content and pedagogy courses, including mathematics, science and teaching methods courses (Appendix A). Among these 27 courses, only 13 of them were mathematics courses. Preservice elementary mathematics teachers were also required to take 9 content courses outside their major area, such as biology, physics, and chemistry, in order to build interdisciplinary connections between science and mathematics. These students were also expected to reach the same level of proficiency in science as that required of science teachers. In fact, preservice elementary mathematics teachers were required to obtain a supplementary teaching certificate in elementary science. Interestingly, they were only required to complete 13 courses in their major area, mathematics. This imbalance between the number of science and mathematics courses became one of the major concerns of teacher educators and other specialists as they revised the teacher education curriculum. Another concern regarding the content of the teacher education curriculum was the limited number of general education courses. In the previous program of study, there were ten courses designed for general education of the preservice teachers, including history, Turkish language, foreign language, and the principles of Kemal Ataturk, the founder of the republic (Appendix A). Such courses aimed to increase teacher candidates’ awareness of social, cultural, and historical issues (HEC, 2006). However, there was no specific course designed for the preservice teachers to develop awareness of social, cultural, and historical issues in their local communities, which could be accomplished through community services and universitycommunity partnerships. Also missing were opportunities for the preservice teachers to learn more about the history of Turkish education and the cultural and philosophical roots of mathematics teaching. Thus, during the curriculum revision process, teacher educators and specialists agreed on increasing the number of general education courses to fill these needs. In the old teacher education curriculum, there were two teaching methods courses (Appendix A). The first one was devoted to general teaching methodologies and philosophies, in which the preservice teachers were involved in more theoretical aspects of pedagogy. In the second course, the students were exposed to more specific and practical applications of the teaching methodologies in mathematics teaching. They were provided experiences in constructing relationships among mathematical concepts, representations, and processes. Thus, at the end of these courses, preservice Mine Isiksal, Yusuf Koc, Safure Bulut, & Tulay AtayTurhan
teachers should have had appropriate experiences aimed at improving their understanding of the content and pedagogy of mathematics. However, two courses devoted to the content and pedagogy of elementary school mathematics were not sufficient for addressing all elementary school mathematics concepts. Thus, a suggestion was made to increase the number of mathematics teaching methods courses for preservice teachers. Another issue concerning teacher preparation was the limited emphasis on instructional planning and assessment. Previously, there was only one course covering both planning and assessment; however, addressing all objectives of the course in one semester was difficult. For example, instructors could not spend enough time on developing classroom tests. Finally, preservice teachers were not provided enough experience in working with students from diverse populations, including students with special needs. Related with this issue, teacher education courses were not designed to help preservice teachers teach the subject matter for all students. There were also issues related to the policy of double licensure. As previously mentioned, preservice elementary mathematics teachers were also certified to teach science in elementary school. Officials from the teacher education department of the Ministry of National Education decided that, due to the adequate supply of elementary school science teachers and the need for elementary mathematics teachers, the double licensure system should be rethought during the revision process. Development Process of the New Curriculum At the beginning of the reform process, the HEC asked some of the colleges of education to review the 1998 program, identify weaknesses and strengths of the program, and give suggestions for improvement. Then, a program revision group was formed with 25 faculty members from various teacher education programs. This group met for a week in March 2006 to form the blueprint of the revised curriculum. The blueprint was distributed to colleges of education around the country for feedback. Based on the received feedback, the final version of the curriculum was approved by the Higher Education Council in July 2006. What’s “New” in the New Mathematics Teacher Education Curriculum? The new mathematics teacher education curriculum changes are detailed in Appendixes B and 45
C. First of all, as seen in Table 1, the number and percentages of courses in the content area, pedagogy courses, and general education courses are different. Table 1 Number and Percentages of the Course Types in the Previous and Current Curricula Type of courses
Content & Content Teaching Methods (C) Pedagogy (P) General Education (GE) Total
Number and percentages of the courses in the previous curriculum 27 (56%)
Number and percentages of the courses in the current curriculum 22 (44%)
11 (23%) 10 (21%)
13 (26%) 15 (30%)
48 (100%)
50 (100%)
As indicated in Table 1, the total number of courses in the elementary mathematics teacher education curriculum increased from 48 to 50. In particular, while the percentages of content and content teaching methods courses decreased (from 56% to 44%), the percentages of pedagogy courses and general education courses increased (P: from 23% to 26%; GE: from 21% to 30%). In sum, the number of mathematics, mathematics teaching methods, general education and pedagogy courses increased; whereas, the number of science and science teaching methods courses decreased from nine to two in the revised curriculum (see Appendixes). For instance, the 13 mathematics (C) courses in the previous curriculum increased to 16 in the revised curriculum. Also, the number of general education (GE) courses increased from 10 to 15, and the number of pedagogy (P) courses increased from 11 to 13. In addition, the dual licensure requirement was removed from the teacher education programs due the ineffectiveness of the process. One of the significant changes in the new curriculum is the emphasis given to the general education courses. Curriculum developers expected future mathematics teachers to learn more about the Turkish culture, history of the Turkish educational system, philosophy of science, and history and philosophy of mathematics. Teachers with sufficient background and skills in general studies and information technologies, sufficient experiences in performing research, and a multidimensional perspective may be more effective in their classroom practices. Yet, these history and philosophy courses are 46
only recommended, not required, for the teacher education programs because there are not enough faculty members to teach the courses in some universities. A new teaching profession course, Community Service Practice, helps teacher candidates become more aware of current social problems and develops universitycommunity partnerships. As mentioned earlier, preservice teachers were not given enough opportunities to work on communityrelated projects in the previous curriculum. In this particular course, they are expected to work with governmental and nongovernmental organizations to engage in a wide range of projects, e.g., helping people in poverty, assisting local libraries, and working with students in rural areas. The goal of this course is to motivate preservice teachers to participate in volunteer opportunities and increase their sense of empathy and awareness to social issues. They will be encouraged to participate in professional activities, such as panels, conferences, symposiums and workshops throughout the Community Service Practice course. The new teacher education curriculum requires completion of a research methods course. The research course will provide learning opportunities for the preservice teachers to improve their research skills and practices. In particular, the course will be the main vehicle for promoting teachersâ€™ studying of their own teaching via scientific research methods, as in action research. This idea comes from Harrison, Dunn, and Coombe (2006) who argue that classroom research will be more effective if classroom teachers, active practitioners of teaching, are involved in conducting the research. Another addition to the elementary mathematics teacher education curriculum is the Turkish Educational System and School Management course in which preservice teachers will learn about the structure and philosophy of the Turkish educational system. Also, students will have a chance to see how the school administration contributes to quality instruction. The HEC developed the blueprint of the curriculum for the teacher education programs, but the schools of educations have flexibility in utilizing the curriculum. That is, they can remove, add, and revise the name and content of up to 30% of the total courses; however, they are not allowed to remove any pedagogy (P) courses (HEC, 2006). This flexibility will give the colleges of education a chance to organize their own teacher education programs based on their needs and capabilities.
Teacher Education Curriculum
Discussions and Recommendations Reform efforts in the Turkish education system focused on increasing the quality of education from kindergarten to university in all content areas (Binbasioglu, 1995). The change in the elementary mathematics teacher education curriculum is part of this larger scale reform agenda. There were four major factors that motivated policy makers, teacher educators and other specialists to revise the previous elementary mathematics teacher education curriculum: integration with the European Union, changes in the elementary school mathematics curriculum, the need for more qualified teachers, and dissatisfaction with the previous curriculum. Turkey’s goal of becoming a permanent member of the European Union (EU) catalyzed their efforts to meet European Union countries’ educational standards. It is believed that with the successful implementation of the new elementary school curriculum and the teacher education programs, Turkey will reach the educational levels of other European Union countries. To implement the new elementary school curriculum, teachers need to be equipped with appropriate knowledge, skills, and experiences. The new teacher education curriculum places a considerable degree of emphasis on a successful utilization of the elementary school curriculum. This includes motivating students to ask questions, engaging students in critical thinking, and using mathematical inquiry in their mathematics classroom practices. The new teacher education curriculum is also concerned with overcoming problems in the previous curriculum. These problems were based on content, pedagogy and policyrelated issues. In the new mathematics teacher education curriculum, the number of mathematics and mathematics teaching methods, general education, and pedagogy courses increased in order to provide preservice elementary mathematics teachers with previously lacking learning opportunities. Additionally, with the increased number of mathematics teaching methods courses, the preservice teachers are expected to have a more indepth understanding of mathematical relationships and procedures and pedagogical content knowledge. Furthermore, the introduction of more general education courses will provide preservice teachers with more experiences to increase their awareness of social, cultural, and historical issues. The new teacher education curriculum has been implemented nationwide since Fall 2006. The transition from the previous curriculum to the new one
Mine Isiksal, Yusuf Koc, Safure Bulut, & Tulay AtayTurhan
has been challenging for the programs because the students in the same cohort do not progress at the same pace; although most of the students are able to follow their program successfully, there are others who repeat courses or fall behind their peers. Curriculum developers aimed to increase the quality of mathematics teacher education to that of the international standards; however, revising the curriculum is not sufficient to reach the desired level. Implementation of the new curriculum program will be monitored and continuously evaluated in order to enhance the quality of teachers to the highest level. The implementation of the curriculum will be monitored by the Turkish Higher Education Council through continuous feedback from teacher educators, preservice teachers and K12 institutions. Further monitoring attempts may be carried out by individual teacher education programs to explore the strengths and weaknesses of the revised curriculum. References Bandura, A. (1986). Social foundations of thought and action: A social cognitive. Englewood Cliffs, NJ: Prentice Hall. Binbasioglu, C. (1995). Türkiyé de Eğitim Bilimleri Tarihi [History of Educational Sciences in Turkey]. Ankara, Turkey: Ministry of National Education. Bishop, A. J., Clements, M. A., Keitel, C., Kilpatrick, J., & Leung, F. K. S. (2003). Second international handbook of mathematics education. Dordrecht, Netherlands: Kluwer Academic Publishers. Bulut, S. (1998). Changes in mathematics teacher education programs in Turkey. The Mathematics Educator, 9(2), 30–33. Cakiroglu, E., & Cakiroglu, J. (2003). Reflections on teacher education in Turkey. European Journal of Teacher Education, 26, 253–264. Center for Student Selection and Placement. (1998). Öğrenci seçme ve yerleştirme sınavı ikinci basamak kılavuzu [Student selection and placement examination manual for the second level]. Ankara, Turkey: Author. DarlingHammond, L., & Bransford, J. (2005). Preparing teachers for a changing world: What teachers should learn and be able to do. San Francisco: JosseyBass. Even, R. (1990). Subjectmatter knowledge for teaching and the case of functions. International Journal of Mathematics Education in Science and Technology, 14, 293–305. Grossman, P. L., Schoenfeld, A., & Lee, C. D. (2005). Teaching subject matter. In L. DarlingHammond & J. Bransford (Eds.), Preparing teachers for a changing world: What teachers should learn and be able to do (pp. 201–231). San Francisco: Jossey Bass. Harrison, L. J., Dunn, M., & Coombe, M. (2006). Making research relevant in preservice early childhood teacher education. Journal of Early Childhood Teacher Education, 27, 217–229.
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Higher Education Council. (2006). Yeni Programlar Hakkında Açıklama [Description of the new teacher education curricula]. Retrieved December 22, 2006, from http://www.yok.gov.tr/egitim/ogretmen/aciklama_program.do c Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann. Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. The Elementary School Journal, 105(1), 11–30. Irish National Council for Curriculum and Assessment. (1999). Primary school mathematics curriculum. Dublin, Ireland: Author. Johnson, D. W., & Johnson, R. (1990). Cooperative learning and achievement. In S. Sharan (Ed.), Cooperative learning: Theory and research (pp. 23–37). New York: Praeger. Koc, Y., Isiksal, M., & Bulut, S. (2007). The new elementary school curriculum in Turkey. International Education Journal, 8(1), 30–39. Lin, F. L., & Cooney, T. J. (2001). Making sense of mathematics teacher education. Dordrecht, Netherlands: Kluwer Academic Publishers.
Ministry of National Education. (2005). PISA 2003 Projesi ulusal nihai raporu [Final national report of PISA 2003]. Ankara, Turkey: Egitim Arastirmalari ve Gelistirme Dairesi. Ministry of National Education. (2006). İlköğretim matematik dersi (15 sınıflar) öğretim programı [Elementary school mathematics curriculum (grades 15)]. Ankara, Turkey: Author. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Polya, G. (1957). How to solve it (2nd ed.). Princeton, NJ: Princeton University Press. Romberg, T. A. (2004). Standardsbased mathematics assessment in middle school: Rethinking classroom practice. New York: Teachers College Press. Tirosh, D. (2000). Enhancing prospective teacher’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31, 5–25. United Kingdom Qualifications and Curriculum Authority. (1999). The national curriculum. London: Author.
Ministry of National Education. (2004). Müfredat geliştirme süreci: Program geliştirme modeli çerçevesinde yapılan çalışmalar [Curriculum development process: Activities conducted around the curriculum development model]. Ankara, Turkey: Author.
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Teacher Education Curriculum
Appendix A: Previous Elementary Mathematics Teacher Education Curriculum Semester I Course
T
Pr
Cr
C Analysis C Abstract Mathematics C General Biology GE Principles of Kemal Ataturk I GE Turkish I: Written Expression GE Foreign Language I P Introduction to Teaching Profession Total hours and credits
4 3 3 2 2 3 3 20
2 0 2 0 0 0 0 4
5 3 4 2 2 3 3 22
Semester III Course C Analysis III C Linear Algebra I C Physics I C General Chemistry I P Developmental Psychology and Learning GE Computer Total hours and credits
T
Pr
Cr
4 3 4 3 2
0 0 2 2 2
4 3 5 4 3
3 19
0 6
3 22
Semester V Course Statistics and Probability I C Introduction to Algebra C Laboratory Applications in C Science I Analytic Geometry C Instructional Technology and P Material Development GE Elective I Total hours and credits
T
Pr
Cr
2 3 2
2 0 2
3 3 3
C C
3 2
0 2
3 3
P C
3 15
0 6
3 18
C Elective II Total hours and credits
Semester VII Course Special Teaching Methods II
T
Pr
Cr
2
2
3
C
Semester II Course C C
Analysis II Geometry General Biology II GE Principles of Kemal Ataturk II GE Turkish II: Oral Expression GE Foreign Language II P School Experience I Total hours and credits
C C C C P
Semester IV Course Analysis IV Linear Algebra II Physics II General Chemistry II Instructional Planning Evaluation
Total hours and credits
C
P
Semester VI Course Statistics and Probability II Elementary Number Theory Laboratory Applications in Science II Classroom Management Special Teaching Methods I
Semester VIII Course Evaluation of Subject Matter Course Books Elective V
Computer Assisted Instruction in 3 0 3 C Mathematics Education Teaching Science 2 2 3 GE Elective VI C School Experience II 1 4 3 P Counseling P 3 0 3 P Student Teaching GE Elective III Elective IV 3 0 3 C Total hours and credits 14 8 18 Total hours and credits C: Content and mathematics teaching methods courses, P: Pedagogy courses, GE: General Education courses T: Number of hours for the theoretical part Pr: Number of hours for the practical part Cr: Number of credits
P
and
T
Pr
Cr
4 3 3 2 2 3 1 18
3 0 2 0 0 0 4 9
5 3 4 2 2 3 3 22
T
Pr
Cr
4 3 4 3 3
0 0 2 2 2
4 3 5 4 4
17
6
20
T
Pr
Cr
2 3 2
2 0 2
3 3 3
2 2
2 2
3 3
3 14
0 8
3 18
T
Pr
Cr
2
2
3
3
0
3
3 3 2
0 0 6
3 3 5
13
8
17
49
Appendix B: Revised Elementary Mathematics Teacher Education Curriculum Semester I Course
T
Pr
Cr
General Mathematics Turkish I: Written Expression Principles of Kemal Ataturk I Computer I Foreign Language I Introduction to Education Science
4 2 2 2 3 3
2 0 0 2 0 0
5 2 2 3 3 3
16
4
18
T
Pr
Cr
4 3 4 2 2
2 0 0 0 0
5 3 4 2 2
P Instructional Principles and Methods Total hours and credits
3 18
0 2
3 19
Semester V Course C Analysis III C Analytic Geometry I C Statistics and Probability I Methods of Teaching C Mathematics I C Introduction to Algebra P Elective III GE History of Science * Total hours and credits
T
Pr
Cr
3 3 2 2
0 0 2 2
3 3 3 3
3 2 2 17
0 0 0 4
3 2 2 19
T
Pr
Cr
3 3 2 2
0 0 0 0
3 3 2 2
C GE GE GE GE P
Total hours and credits
C C C C GE
C C P GE
Semester III Course Analysis I Linear Algebra I Physics I Elective I Scientific Research Methods
Semester VII Course Elementary Number Theory* Elective IV Special Education * History of Mathematics*
Semester II Course
T
Pr
Cr
3 3 2 2 3 2 3 18
0 0 0 0 0 2 0 2
3 3 2 2 3 3 3 19
T
Pr
Cr
4 3 4 3 2
2 0 0 0 2
5 3 4 3 3
Total hours and credits
16
4
18
Semester VI Course C Differential Equations C Analytic Geometry II C Statistics and Probability II C Methods of Teaching Mathematics II GE History of Turkish Education * GE Community Service Practices** P Measurement and Evaluation Total hours and credits
T
Pr
Cr
4 3 2 2
0 0 2 2
4 3 3 3
2 1 3 17
0 2 0 6
2 2 3 20
T
Pr
Cr
2 3 3 2
0 0 0 0
2 3 3 2
2
6
5
12
6
15
C Abstract Mathematics C Geometry GE Turkish I: Oral Expression GE Principles of Kemal Ataturk II GE Foreign Language II GE Computer II P Educational Psychology Total hours and credits
C C C GE P
C GE P P
Semester IV Course Analysis II Linear Algebra II Physics II Elective II Instructional Technologies Material Development
Semester VIII Course Philosophy of Mathematics * Elective V Elective VI Turkish Education System School Management Student Teaching
P Counseling 3 0 3 P P Field Experience 1 4 3 P Classroom Management 2 0 2 Total hours and credits 16 4 18 Total hours and credits C: Content and mathematics teaching methods courses, P: Pedagogy courses, GE: General Education courses T: Number of hours for the theoretical part Pr: Number of hours for the practical part Cr: Number of credits
50
and
and
Appendix C: Critical Changes C C P C C C C C P
Courses removed from the program General Biology I General Biology II School Experience I General Chemistry I General Chemistry II Analysis IV Laboratory Applications in Science I Laboratory Applications in Science II Computer Assisted Instruction in Mathematics Education
T 3 3 1 3 3 4 2 2 3
Pr 2 2 4 2 2 0 2 2 0
Cr 4 4 3 4 4 4 3 3 3
C GE GE GE C C GE GE P C P
Courses added to the program General Mathematics Computer II Scientific Research Methods History of Science * Differential Equations Analytic Geometry II History of Turkish Education * Community Service Practices** Special Education *
Philosophy of Mathematics * Turkish Education System and School Management C: Content and mathematics teaching methods courses, P: Pedagogy courses, GE: General Education courses T: Number of hours for the theoretical part Pr: Number of hours for the practical part Cr: Number of credits
T 4 2 2 2 4 3 2 1 2
Pr 2 2 0 0 0 0 0 2 0
Cr 5 3 2 2 4 3 2 2 2
2 2
0 0
2 2
51
The Mathematics Educator (ISSN 10629017) is a semiannual publication of the Mathematics Education Student Association (MESA) at the University of Georgia. The purpose of the journal is to promote the interchange of ideas among the mathematics education community locally, nationally, and internationally. The Mathematics Educator presents a variety of viewpoints on a broad spectrum of issues related to mathematics education. The Mathematics Educator is abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on Mathematical Education). The Mathematics Educator encourages the submission of a variety of types of manuscripts from students and other professionals in mathematics education including: • • • • • • • •
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