____ THE ______ MATHEMATICS ___ _________ EDUCATOR _____ Volume 17 Number 1
Summer 2007
MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA
Editorial Staff
A Note from the Editor
Editor Kyle T. Schultz
Dear TME Reader,
Associate Editors Rachael Brown Kelly Edenfield Ryan Fox Na Young Kwon Eileen Murray Susan Sexton Catherine Ulrich
On behalf of our editorial staff and the Mathematics Education Student Association of the University of Georgia, I am pleased to present to you the first issue of Volume 17 of The Mathematics Educator. We have worked hard to bring to you a collection of articles that share new and interesting ideas in our field. We hope that you find them intriguing and that they may provide further fuel for your practice and research.
Advisor Dorothy Y. White
MESA Officers 2006-2007 President Rachael Brown Vice-President Filyet Asli Ersoz Secretary Eileen Murray Treasurer Jun-ichi Yamaguchi NCTM Representative Ginger A. Rhodes Undergraduate Representative Laine Bradshaw Rachel Stokely
In his guest editorial that leads off this issue, Tad Watanabe shares some ideas about what makes school mathematics curricula focused and coherent. Using his knowledge of Japanese school curricula and curriculum development, he provides some direction for improving U.S. curricula. Next, Rick Anderson shares his research of the development of mathematical identity within secondary-level students and provides practical advice for educators regarding how to foster positive mathematical identities within students. Shelly Sheats Harkness and Lisa Portwood, share their experiences in implementing a lesson with prospective and practicing teachers that focuses upon the mathematics of quilting. In addition, they provide a critical look at our notion of mathematical activity. Next, Zelha Tunç-Pekkan shares a unique look into the diverse practices of three professors who have taught a graduate-level mathematics curriculum course. In the final article, Jon Warwick reflects on his experiences in teaching undergraduate students mathematical modeling and provides suggestions for developing strong modeling skills. Concluding this issue, Ginny Powell reviews Mathematics Education Within the Postmodern, an “eye-opening and thought-provoking” book. This is my final issue as editor of TME. I would like to thank my staff of associate editors, previous editors and the Mathematics Education faculty and staff for their guidance and support, our reviewers who provide crucial feedback on each submitted manuscript, and my family and friends for their encouragement and patience. The journal will be in good hands with Kelly Edenfield and Ryan Fox assuming leadership of the journal as co-editors. I wish them success in the coming year.
Kyle T. Schultz 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124
tme@uga.edu www.coe.uga.edu/tme
About the Cover The Exploding Cube. Using Geometer’s Sketchpad, this image shows three stages of the partitioning of the cube (a + b)3 into components. These components correspond to the algebraic expansion of this expression.
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
Summer 2007
Volume 17 Number 1
Table of Contents 2 Guest Editorial… In Pursuit of a Focused and Coherent School Mathematics Curriculum TAD WATANABE 7 Being a Mathematics Learner: Four Faces of Identity RICK ANDERSON 15 A Quilting Lesson for Early Childhood Preservice and Regular Classroom Teachers: What Constitutes Mathematical Activity? SHELLY SHEATS HARKNESS & LISA PORTWOOD 24 Graduate Level Mathematics Curriculum Courses: How Are They Planned? ZELHA TUNÇ-PEKKAN 32 Some Reflections on the Teaching of Mathematical Modeling JON WARWICK 42 Book Review… The View from Here: Opening Up Postmodern Vistas GINNY POWELL 45 Upcoming conferences 46 Submissions information 47 Subscription form
© 2007 Mathematics Education Student Association All Rights Reserved
The Mathematics Educator 2007, Vol. 17, No. 1, 2–6
Guest Editorial… In Pursuit of a Focused and Coherent School Mathematics Curriculum Tad Watanabe
Most, if not all, readers are familiar with the criticism of a typical U.S. mathematics curriculum being “a mile wide and an inch deep” (Schmidt, McKnight, & Raizen, 1997). A recent analysis by the Center for the Study of Mathematics Curriculum (Reys, Dingman, Sutter, & Teuscher, 2005) reaffirms the crowdedness of most state mathematics standards. However, criticism of U.S. mathematics curricula is nothing new. In April 2006, the National Council of Teachers of Mathematics (NCTM) released Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. This document is an attempt by NCTM to initiate a discussion on what mathematical ideas are important enough to be considered as “focal points” at a particular grade level. But why is it so difficult to have a focused and cohesive school mathematics curriculum? Besides, what makes a curriculum focused and cohesive? In this paper, I would like to offer my opinions on what a focused and cohesive mathematics curriculum may look like and discuss some obstacles for producing such a curriculum. What makes a curriculum focused? Clearly, a crowded curriculum naturally tends to be unfocused. A major cause for the crowdedness of many U.S. textbook series seem to be the amount of “reviews,” topics that have been discussed at previous grade levels. Some amount of review is probably necessary and helpful. However, in many cases, the topics are redeveloped as if they have not been previously discussed. For example, in teaching linear measurement, most of today’s textbooks follow this general sequence of instruction: (a) direct comparison, (b) indirect comparison, (c) measuring with arbitrary or Tad Watanabe is an Associate Professor of Mathematics Education at Kennesaw State University. He received his PhD in Mathematics Education from Florida State University in 1991. His research interests include teaching and learning of multiplicative concepts and various mathematics education practices in Japan, including lesson study and curriculum materials.
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non-standard units, and (d) measuring with standard units. Often, the discussion of linear measurement in Grades K, 1, and 2 textbooks involve all four stages of measurement instruction at each grade level. In contrast, a Japanese elementary mathematics course of study (Takahashi, Watanabe, & Yoshida, 2004) discusses the first 3 stages in Grade 1, and the discussion in Grade 2 focuses on the introduction of standard units. Most Grade 2 textbooks, therefore, start their discussion of linear measurement by establishing the need for (and usefulness of) standard units through problem situations in which the use of arbitrary units is not sufficient. This redevelopment of the same topic in multiple grade levels may be both the symptom and the cause of a misinterpretation of the idea of a “spiral curriculum.” In the past few years, several elementary mathematics teachers who are using a “reform” curriculum told me that it is acceptable for children to not understand some ideas the first time (or even the second or third time) since they will see it again later. Such a view does not describe a spiral. Rather, it seems to be based on the belief that, by introducing a topic early and discussing it often, students will come to understand it. This view is incompatible with a focused curriculum. However, simply removing some topics from any given grade level does not necessarily result in a focused curriculum. If all items on a given grade level receive equal amount of attention, regardless of mathematical significance, then the curriculum lacks a focus. The Focal Points (NCTM, 2006) present three characteristics for a concept or a topic to be considered as a focal point: •
Is it mathematically important, both for further study in mathematics and for use in applications in and outside of school?
•
Does it “fit” with what is known about learning mathematics?
•
Does it connect logically with the mathematics in earlier and later grade levels? (p. 5) A Focused and Coherent Curriculum
Whether or not we agree with this particular set of characteristics, if a curriculum is to be focused, it must be based on a set of explicitly stated criteria for organizing its contents. What makes a curriculum coherent? It goes without saying that a coherent mathematics curriculum must have its contents sequenced in such a way that a new idea is built on previously developed ideas. Most agree that mathematics learning is like putting together many building blocks. Of course, there is typically more than one way to put together ideas. However, a cohesive curriculum and, ultimately, teachers must have a vision of how learners can build a new idea based on what has previously been discussed. This idea seems to be so obvious, but it is also very easy to overlook. Furthermore, I believe that textbook writers have the responsibility to make clear the potential learning paths they envision to support teachers who use their materials. This is where many U.S. mathematics textbooks seem to fall short. Too often, teachers’ manuals are filled with many suggestions without explicitly discussing how the target ideas may be developed from ideas previously discussed. Thus, teachers are left with an overwhelming amount of information without any guidance regarding how it can be organized and put to work. Another important factor that contributes to the coherence of a mathematics curriculum is how one part of a curriculum relates to another. For example, the Focal Points (NCTM, 2006) states that, in Grade 4, students are to “develop fluency with efficient procedures, including the standard algorithm, for multiplying whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems” (p. 16). However, in the “Connections to the Tad Watanabe
Focal Points”, the document also states, “Building on their work in grade 3, students extend their understanding of place value and ways of representing numbers to 100,000 in various contexts” (p. 16). Therefore, when students are developing fluency with multiplication procedures, the curriculum writers and teachers must pay attention to the products of the assigned problems to insure they will be in the appropriate range. As not all products of two 3-digit numbers will be less than 100,000, these two statements together suggest that the focus of a curriculum should be on helping students understand how and why their multiplication procedures work, rather than focusing solely on students’ proficiency with multiplying two 3-digit numbers. The coherence of a mathematics curriculum is also influenced by its mathematical thoroughness. For example, in many elementary and middle school mathematics curricula, students are asked to find the area of the parallelogram like the one shown in Figure 1. It is expected that most students will cut off a triangular section from one end and move it to the other side to form a rectangle, whose area they can calculate. This idea is discussed in Principles and Standards for School Mathematics (NCTM, 2000) as well. Based on this experience, most textbooks will then conclude that the formula for calculating the area of a parallelogram is base × height. However, this is an overgeneralization. For example, if this is the only experience students have, they will not be able to determine the area of the parallelogram shown in Figure 2, unless they already know the Pythagorean theorem. As a result, students cannot conclude that any side of a parallelogram may be used as the base to calculate its area.
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However, we will then need the Pythagorean theorem to determine the lengths of the base and the height. Therefore, for a curriculum to be cohesive, students should be provided with the opportunity to determine the area of the parallelogram like the one shown in Figure 2. Figure 3 shows some of the ways students may calculate its area. Some of these methods suggest that we could indeed use the horizontal side as the base if we consider the height to be the distance between the parallel lines containing the two horizontal sides. In addition to having a thorough sequence of mathematical ideas, the coherence of a curriculum may be enhanced by the selection of learning tasks and representations. For example, in a Japanese textbook series (Hironaka & Sugiyama, 2006), the following four problems were used in Grade 6 units on multiplication and division of fractions: •
With 1 dl of paint, you can paint
3 2 m of 5
boards. How many m2 can you paint with 2 dl of paint? •
•
! paint With 3 dl of paint, you can
4 2 m of 5
•
With
3 2 2 dl of paint, you can paint m of 4 5
boards. How many m2 can you paint with 1 dl?
! this particular By ! selecting the same problem context, textbook series hopes that students can identify these problem situations as multiplication or division situations, even though fractions are involved. We know from research (e.g., Bell, Fischbein, & Greer, 1984) that this decision is not trivial for students. Once the operations involved are identified, the series asks students to investigate how the computation can be carried out. A consistent use of the same or similar items across related mathematical ideas is not limited to the problem contexts. Another way the coherence may be enhanced is through the consistent use of representation. Figure 4 shows how Hironaka and Sugiyama (2006) use similar representations as they discuss multiplicative ideas across grade levels. In early grades, the representations are used primarily to represent the ways quantities are related to each other but, later on, students are expected to use the diagrams as tools to solve problems.
boards. How many m2 can you paint with 1 dl?
Why has it been so difficult to produce a focused and cohesive curriculum?
4 2 m of 5 2 boards. How many m2 can you paint with dl 3
We can probably list many different reasons to answer this question. For example, there is a general
With 1 dl of paint, you ! can paint
of paint?
! !
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A Focused and Coherent Curriculum
reluctance to remove any topic from an existing curriculum. Thus, today’s curricula include many ideas that probably were not included 50 years ago, yet virtually all topics from 50 years ago are still included in today’s curricula as well. However, I would like to discuss another idea that may be undermining our efforts to create a focused and cohesive curriculum: the lure of replacement units. The idea of replacement units, high quality materials used in place of a unit in a textbook series, may have started with a good intention. Some reform curriculum materials appear to be created so that parts of the curricula may be used as replacement units. Although many are indeed of very high quality, replacement units may have encouraged the compartmentalization and rearrangement of topics within a curriculum as necessary. Thus, a publisher may be able to “individualize” their textbook series to
match different state curriculum standards. If multiplication is introduced in Grade 2 in one state but in Grade 3 in others, there is no problem. One can simply package the introduction of multiplication unit in the appropriate grade level. However, it should be very clear that a focused and cohesive curriculum is much more than simply a sequence of mathematics topics that match the curriculum standards. In addition, as NCTM (2000) states, a curriculum is more than just a collection of problems and tasks (p. 14). One must pay close attention to the internal consistency and coherence of curriculum materials. A Japanese textbook series (Hosokawa, Sugioka, & Nohda, 1998) warned against teachers changing the order of units presented in the series. This is a stark contrast to a rather casual approach that some in this country seem to possess.
Figure 4. Consistent use of similar representations from Hironaka & Sugiyama (2006): (a) multiplication and division of whole numbers in Grade 3; (b) multiplication of a decimal number by a whole number in Grade 4; (c) multiplying and dividing by a decimal number in Grade 5; and (d) multiplying and dividing by a fraction.
Tad Watanabe
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What will it take to produce a focused and coherent curriculum? The most obvious response to this question is closer collaboration among teachers, researchers, and curricula producers. In Japan, such collaboration is achieved through lesson study. Although lesson study (e.g., Lewis, 2002; Stigler & Hiebert, 1999) is often considered to be a professional development activity, it also serves a very important role in curriculum development, implementation, and revision in Japan. At the beginning of a lesson study cycle, teachers engage in an intensive study of curriculum materials. The participating teachers ask questions such as, •
Why is this topic taught at this particular point in the curriculum?
•
What previously learned materials are related to the current topic?
•
How are students expected to use what they have learned previously to make sense of the current topic?
Bell, A., Fischbein, E., & Greer, B. (1984). Choice of operation in verbal arithmetic problems: The effect of number size, problem structure and context. Educational Studies in Mathematics, 15, 129-147.
•
How will the current topic be used in the future topics?
Hironaka, H. & Sugiyama, Y. (2006). Mathematics for Elementary School. Tokyo: Tokyo Shoseki. [English translation of New Mathematics for Elementary School 1, by Hironaka & Sugiyama, 2000.]
Is the sequence of topics presented in the textbooks the most optimal one for their students? During this process, teachers will read, among other things, existing research reports and often invite researchers to participate as consultants. After this intensive investigation of curriculum materials, the group develops a public lesson based on their findings. The public lesson is both their research report and a test of the hypothesis derived from their investigation. Through critical reflection on the observation of public lesson, the group produces their final written report. Japanese textbook publishers often support local lesson study groups, and the reports from those groups are carefully considered in the revision of their textbook series. Moreover, teachers examine the new curriculum ideas carefully through lesson study. Through this experience, teachers gain a deeper understanding of these new ideas, and they explore effective ways to teach them to their students. Because researchers, university-based mathematics educators, district •
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mathematics supervisors, and even the officials from the Ministry of Education regularly participate in lesson study open houses, lesson study serves as an important feedback mechanism for curriculum development, implementation, and revision. Lesson study is becoming more and more popular in the United States; however, the involvement by mathematics education researchers and curriculum developers is still rather limited. Moreover, the examination of curriculum materials is often limited as well. A closer collaboration between classroom teachers engaged in lesson study and mathematics education researchers and other university-based mathematics educators is critical if U.S. lesson study is to become a useful feedback mechanism to produce a more focused and coherent school mathematics curriculum. References
Hosokawa, T., Sugioka, T., & Nohda, N. (1998). Shintei Sansuu (Elementary school mathematics). Osaka: Keirinkan. (In Japanese). Lewis, C. (2002). Lesson study: Handbook of teacher-led instructional change. Philadelphia: Research for Better Schools. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author. Reys, B. J., Dingman, S., Sutter, A., & Teuscher, D. (2005). Development of state-level mathematics curriculum documents: Report of a survey. Columbia, Mo.: University of Missouri, Center for the Study of Mathematics Curriculum. Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas in the world’s teachers for improving education in the classroom. New York: Free Press. Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A splintered vision: An investigation of U.S. science and mathematics education. Dordrecht, The Netherlands: Kluwer. Takahashi, A., Watanabe, T., & Yoshida, M. (2004). Elementary school teaching guide for the Japanese Course of Study: Arithmetic (Grades 1-6). Madison, NJ: Global Education
A Focused and Coherent School Mathematics Curriculum
The Mathematics Educator 2007, Vol. 17, No. 1, 7–14
Being a Mathematics Learner: Four Faces of Identity Rick Anderson One dimension of mathematics learning is developing an identity as a mathematics learner. The social learning theories of Gee (2001) and Wenger (1998) serve as a basis for the discussion four “faces” of identity: engagement, imagination, alignment, and nature. A study conducted with 54 rural high school students, with half enrolled in a mathematics course, provides evidence for how these faces highlight different ways students develop their identity relative to their experiences with classroom mathematics. Using this identity framework several ways that student identities—relative to mathematics learning—can be developed, supported, and maintained by teachers are provided. This paper is based on dissertation research completed at Portland State University under the direction of Dr. Karen Marrongelle. The author wishes to thank Karen Marrongelle, Joyce Bishop, and the TME editors/reviewers for comments on earlier drafts of this paper.
Learning mathematics is a complex endeavor that involves developing new ideas while transforming one’s ways of doing, thinking, and being. Building skills, using algorithms, and following certain procedures characterizes one view of mathematics learning in schools. Another view focuses on students’ construction or acquisition of mathematical concepts. These views are evident in many state and national standards for school mathematics (e.g., National Council of Teachers of Mathematics [NCTM], 2000). A third view of learning mathematics in schools involves becoming a “certain type” of person with respect to the practices of a community. That is, students become particular types of people—those who view themselves and are recognized by others as a part of the community with some being more central to the practice and others situated on the periphery (Boaler, 2000; Lampert, 2001; Wenger, 1998). These three views of mathematics learning in schools, as listed above, correspond to Kirshner’s (2002) three metaphors of learning: habituation, conceptual construction, and enculturation. This paper focuses on the third view of learning mathematics. In this view, learning occurs through “social participation” (Wenger, 1998, p. 4). This participation includes not only thoughts and actions but also membership within social communities. In this sense, learning “changes who we are by changing our ability to participate, to belong, to negotiate meaning” (Wenger, 1998, p. 226). This article addresses how students’ practices within a mathematics classroom Rick Anderson is an assistant professor in the Department of Mathematics & Computer Science at Eastern Illinois University. He teaches mathematics content and methods courses for future elementary and secondary teachers.
Rick Anderson
community shape, and are shaped by, students’ sense of themselves, their identities. Learning mathematics involves the development of each student’s identity as a member of the mathematics classroom community. Through relationships and experiences with their peers, teachers, family, and community, students come to know who they are relative to mathematics. This article addresses the notion of identity, drawn from social theories of learning (e.g., Gee, 2001; Lave & Wenger, 1991; Wenger, 1998), as a way to view students as they develop as mathematics learners. Four “faces” of identity are discussed, illustrated with selected quotations from students attending a small, rural high school (approximately 225 students enrolled in grades 9–12) in the U.S. Pacific Northwest. Method The students in this study were participants in a larger study of students’ enrollment in advanced mathematics classes (Anderson, 2006). All students in the high school were invited to complete a survey and questionnaire. Of those invited, 24% responded. Fourteen students in grades 11 and 12 were selected for semi-structured interviews so that two groups were formed: students enrolled in Precalculus or Calculus (the most advanced elective mathematics courses offered in the school) and students not taking a mathematics course that year. These students represented the student body with respect to postsecondary intentions, as reported on the survey, and their interest and effort in mathematics classes, as reported by their teacher. All of the students had taken the two required and any elective high school mathematics in the same high school. One teacher taught most of these courses. When interviewed, this 7
teacher indicated the “traditional” nature of the curriculum and pedagogy: “We’ve always stayed pretty traditional. … We haven’t really changed it to the really ‘out there’ hands-on type of programs.” Participant observation and interviews with students corroborated this statement. Calvin, a high school senior, had enrolled in a mathematics class each year of high school and planned to study mathematics education in college. During an interview, he described a typical day: Just go in, have your work done. First the teacher explains how to do it. Like for the Pythagorean Theorem, for example, she tells you the steps for it. She shows you the right triangle, the leg, the hypotenuse, that sort of thing. She makes us write up notes so we can check back. And then after that she makes us do a couple [examples] and then if we all get it right, she shows us. She gives us time to work. Do it and after that she shows us the correct way to do it. If we got it right, then we know. She makes us move on and do an assignment.
Identity As used here, identity refers to the way we define ourselves and how others define us (Sfard & Prusak, 2005; Wenger, 1998). Our identity includes our perception of our experiences with others as well as our aspirations. In this way, our identity—who we are—is formed in relationships with others, extending from the past and stretching into the future. Identities are malleable and dynamic, an ongoing construction of who we are as a result of our participation with others in the experience of life (Wenger, 1998). As students move through school, they come to learn who they are as mathematics learners through their experiences in mathematics classrooms; in interactions with teachers, parents, and peers; and in relation to their anticipated futures. Mathematics has become a gatekeeper to many economic, educational, and political opportunities for adults (D’Ambrosio, 1990; Moses & Cobb, 2001; NCTM, 2000). Students must become mathematics learners—members of mathematical communities—if they are to have access to a full palette of future opportunities. As learners of mathematics, they will not only need to develop mathematical concepts and skills, but also the identity of a mathematics learner. That is, they must participate within mathematical communities in such a way as to see themselves and be viewed by others as valuable members of those communities.
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Identity as a Mathematics Learner: Four Faces The four faces of identity of mathematics learning are engagement, imagination, alignment, and nature. Gee’s (2001) four perspectives of identity (nature, discursive, affinity, institutional) and Wenger’s (1998) discussion of three modes of belonging (engagement, imagination, alignment) influenced the development of these faces. Each of the four faces of identity as a mathematics learner is described below. Engagement Engagement refers to our direct experience of the world and our active involvement with others (Wenger, 1998). Much of what students know about learning mathematics comes from their engagement in mathematics classrooms. Through varying degrees of engagement with the mathematics, their teachers, and their peers, each student sees her or himself, and is seen by others, as one who has or has not learned mathematics. Engaging in a particular mathematics learning environment aids students in their development of an identity as capable mathematics learners. Other students, however, may not identify with this environment and may come to see themselves as only marginally part of the mathematics learning community. In traditional mathematics classrooms where students work independently on short, singleanswer exercises and an emphasis is placed on getting right answers, students not only learn mathematics concepts and skills, but they also discover something about themselves as learners (Anderson, 2006; Boaler, 2000; Boaler & Greeno, 2000). Students may learn that they are capable of learning mathematics if they can fit together the small pieces of the “mathematics puzzle” delivered by the teacher. For example, Calvin stated, “Precalculus is easy. It’s like a jigsaw puzzle waiting to be solved. I like puzzles.” Additionally, when correct answers on short exercises are emphasized more than mathematical processes or strategies, students come to learn that doing mathematics competently means getting correct answers, often quickly. Students who adopt the practice of quickly getting correct answers may view themselves as capable mathematics learners. In contrast, students who may require more time to obtain correct answers may not see themselves as capable of doing mathematics, even though they may have developed effective strategies for solving mathematical problems.
Four Faces of Identity
One way students come to learn who they are relative to mathematics is through their engagement in the activities of the mathematics classroom: The thing I like about art is being able to be creative and make whatever I want… But in math there’s just kind of like procedures that you have to work through. (Abby, grade 11, Precalculus class, planning to attend college) Math is probably my least favorite subject… I just don’t like the process of it a lot— going through a lot of problems, going through each step. I just get dragged down. (Thomas, grade 12, Precalculus class, planning to attend college)
Students who are asked to follow procedures on repetitive exercises without being able to make meaning on their own may not see themselves as mathematics learners but rather as those who do not learn mathematics (Boaler & Greeno, 2000). A substantial portion of students’ direct experience with mathematics happens within the classroom, so the types of mathematical tasks and teaching and learning structures used in the classroom contribute significantly to the development of students’ mathematical identities. In the quotation above, Abby expressed her dislike of working through procedures that she did not find meaningful. In mathematics class, she was not able to exercise her creativity as she did in art class. As a result, she may not consider herself to be a capable mathematics learner. On one hand, when students are able to develop their own strategies and meanings for solving mathematics problems, they learn to view themselves as capable members of a community engaged in mathematics learning. When their ideas and explanations are accepted in a classroom discussion, others also recognize them as members of the community. On the other hand, students who do not have the opportunity to connect with mathematics on a personal level, or are not recognized as contributors to the mathematics classroom, may fail to see themselves as competent at learning mathematics (Boaler & Greeno, 2000; Wenger, 1998). Imagination The activities in which students choose to engage are often related to the way they envision those activities fitting into their broader lives. This is particularly true for high school students as they become more aware of their place in the world and begin to make decisions for their future. In addition to learning mathematical concepts and skills in school, students also learn how mathematics fits in with their Rick Anderson
other activities in the present and the future. Students who engage in a mathematical activity in a similar manner may have very different meanings for that activity (Wenger, 1998). Imagination is the second face of identity: the images we have of ourselves and of how mathematics fits into the broader experience of life (Wenger, 1998). For example, the images a student has of herself in relation to mathematics in everyday life, the place of mathematics in post-secondary education, and the use of mathematics in a future career all influence imagination. The ways students see mathematics in relation to the broader context can contribute either positively or negatively to their identity as mathematics learners. When asked to give reasons for their decisions regarding enrollment in advanced mathematics classes, students’ responses revealed a few of the ways they saw themselves in relation mathematics. For example, students had very different reason for taking advanced mathematics courses. One survey respondent stated, “I need math for everyday life,” while another claimed, “They will help prepare me for college classes.” These students see themselves as learners of mathematics and members of the community for mathematics learning because they need mathematics for their present or future lives. Others (e.g., Martin, 2000; Mendick, 2003; Sfard & Prusack, 2005) have similarly noted that students cite future education and careers as reasons for studying mathematics. Conversely, students’ images of the way mathematics fits into broader life can also cause students to view their learning of further mathematics as unnecessary. Student responses for why they chose not to enroll in advanced mathematics classes included “the career I am hoping for, I know all the math for it” and “I don’t think I will need to use a pre-cal math in my life.” Students who do not see themselves as needing or using mathematics outside of the immediate context of the mathematics classroom may develop an identity as one who is not a mathematics learner. If high school mathematics is promoted as something useful only as preparation for college, students who do not intend to enroll in college may come to see themselves as having no need to learn mathematics, especially advanced high school mathematics (Anderson, 2006). Students may pursue careers that are available in their geographical locale or similar to those of their parents or other community members. If these careers do not require a formal mathematics education beyond high school mathematics, these students may limit their 9
image of the mathematics needed for work to arithmetic and counting. In addition, due to the lack of formal mathematical training, those in the workplace may not be able to identify the complex mathematical thinking required for their work. For example, Smith (1999) noted the mathematical knowledge used by automobile production workers, knowledge not identified by the workers but nonetheless embedded in the tasks of the job. When students are not able to make connections between the mathematics they learn in school and its perceived utility in their lives, they may construct an identity that does not include the need for advanced mathematics courses in high school. The students cited in this paper lived in a rural logging community. Their high school mathematics teacher formally studied more mathematics than most in the community. Few students indicated personally knowing anyone for whom formal mathematics was an integral part of their work. As a result, careers requiring advanced mathematics were not part of the images most students had for themselves and their futures. Alignment A third face of identity is revealed when students align their energies within institutional boundaries and requirements. That is, students respond to the imagination face of identity (Nasir, 2002). For example, students who consider advanced mathematics necessary for post-secondary educational or occupational opportunities direct their energy toward studying the required high school mathematics. High school students must meet many requirements set by others—teachers, school districts, state education departments, colleges and universities, and professional organizations. By simply following requirements and participating in the required activities, students come to see themselves as certain “types of people” (Gee, 2001). For example a “collegeintending” student may take math classes required for admission to college. As before, students’ anonymous survey responses to the question of why they might choose to enroll in advanced mathematics classes provide a glimpse into what they have learned about mathematics requirements and how they respond to these requirements. Students were asked why they take advanced mathematics classes in high school. One student responded, “Colleges look for them on applications,” and another said, “Math plays a big part in mechanics.” Likewise, students provided reasons for why they did not take advanced mathematics courses 10
in high school, including “I have already taken two [required] math classes,” and “I might not take those classes if the career I choose doesn’t have the requirement.” While some students come to see themselves, and are recognized by others, as mathematics learners from the requirements they follow, the opposite is true for others. Students who follow the minimal mathematics requirements, such as those for graduation, may be less likely to see themselves, or be recognized by others, as students who are mathematics learners. The three faces of identity discussed to this point are not mutually exclusive but interact to form and maintain a student’s identity. When beginning high school, students are required to enroll in mathematics courses. This contributes to students’ identity through alignment. As they participate in mathematics classes, the activities may appeal to them, and their identity is further developed through engagement. Similarly, students—like the one mentioned above who is interested in mechanics—may envision their participation in high school mathematics class as preparation for a career. Mathematics is both a requirement for entrance into the career and necessary knowledge to pursue the career. Thus, identity in mathematics is maintained through both imagination and alignment. Nature Q: Why are some people good at math and some people aren’t good at math? A: I think it’s just in your makeup… genetic I guess. (Barbara, grade 12, Precalculus, planning to attend vocational training after high school)
The nature face of identity looks at who we are from what nature gave us at birth, those things over which we have no control (Gee, 2001). Typically, characteristics such as gender and skin color are viewed as part of our nature identity. The meanings we make of our natural characteristics are not independent of our relationships with others in personal and broader social settings. That is, these characteristics comprise only one part of the way we see ourselves and others see us. In Gee’s social theory of learning, the nature aspect of our identity must be maintained and reinforced through our engagement with others, in the images we hold, or institutionalized in the requirements we must follow in the environments where we interact. Mathematics teachers are in a unique position to hear students and parents report that their mathematics learning has been influenced by the presence or Four Faces of Identity
absence of a “math gene”, often crediting nature for not granting them the ability to learn mathematics. The claim of a lack of a math gene—and, therefore, the inability to do mathematics—contrasts with Devlin’s (2000b) belief that “everyone has the math gene” (p. 2) as well as with NCTM’s (2000) statement that “mathematics can and must be learned by all students” (p. 13). In fact, cognitive scientists report, “Mathematics is a natural part of being human. It arises from our bodies, our brains, and our everyday experiences in the world” (Lakoff & Núñez, 2000, p. 377). Mathematics has been created by the human brain and its capabilities and can be recreated and learned by other human brains. Yet, the fallacy persists for some students that learning mathematics requires special natural talents possessed by only a few: I’m good at math. (Interview with Barbara, grade 12, Precalculus class) I’m not a math guy. (Interview with Bill, grade 12, not enrolled in math, planning to join the military after high school) Math just doesn’t work for me. I can’t get it through my head. (Interview with Jackie, grade 12, not enrolled in math, planning to enroll in a vocational program after high school)
Although scientific evidence does not support the idea that mathematics learning is related to genetics, some students attribute their mathematics learning to nature. The high school student who says “I’m not a math guy” may feel that he is lacking a natural ability for mathematics. He is likely as capable as any other student but has come to the above conclusion based on his experience with mathematics and the way it was taught in his mathematics classes. Students who are not the quickest to get the correct answers may learn, albeit erroneously, that they are not capable of learning
mathematics. They do not engage in practices that are recognized, in this case, to be the accepted practices of the community. As a result, they view themselves, and are viewed by others, to be peripheral members of the community of mathematics learners. As shown by the provided responses from students, each of the four faces of identity exists as a way that students come to understand their practices and membership within the community of mathematics learners. I have chosen to represent these faces of identity as the four faces of a tetrahedron1 (Figure 1). If we rotate a particular face to the front, certain features of identity are highlighted while others are diminished. Each face suggests different ways to describe how we see ourselves as mathematics learners although they are all part of the one whole. This representation of identity maintains the idea that, as Gee (2001) wrote, “They are four strands that may very well all be present and woven together as a given person acts within a given context” (p. 101). When considering the four faces of identity as a mathematics learner, this context is a traditional high school mathematics classroom. While all four faces contribute to the formation of students’ identities as mathematics learners, the nature face provides the most unsound and unfounded explanations for students’ participation in the mathematics community. To allow for the development of all students to identify as mathematics learners, students and teachers must discount the nature face and build on the other three faces of identity. Developing an Identity as a Mathematics Learner To conclude this article, recommendations are offered to teachers for developing and supporting students’ positive identities as mathematics learners— members of a community that develops the practices of
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Figure 1. The four faces of identity Rick Anderson
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mathematics learning. The four faces of identity described here are used to understand how students see themselves as mathematics learners in relation to their experiences in the mathematics classroom and through the ways these experiences fit into broader life experience. Students’ experiences will not necessarily reflect just one of the four faces described (Gee, 2001). In fact, some experiences may be stretched over two or more faces. For example, learning advanced mathematics in high school can contribute to a students’ identity in two ways: (a) through imagination with the image of math as an important subject for entrance to higher education and (b) through alignment since advanced mathematics is required to attend some colleges. Taken together, however, we can see that a focus on a particular face of identity suggests particular experiences that can help to develop strong positive identities as a mathematics learner in all students. The engagement face of identity is developed through students’ experiences with mathematics and, for most high school students, their mathematics experiences occur in the mathematics classroom. Therefore, the most significant potential to influence students’ identities exists in the mathematics classroom. To develop students’ identities as mathematics learners through engagement, teachers should consider mathematical tasks and classroom structures where students are actively involved in the creation of mathematics while learning to be “people who study in school” (Lampert, 2001). That is, students must feel the mathematics classroom is their scholarly home and that the ideas they contribute are valued by the class (Wenger, 1998). As indicated earlier, teacher-led classrooms with students working independently on single-answer exercises can cause students to learn that mathematics is not a vibrant and useful subject to study. Boaler (2000), for example, identified monotony, lack of meaning, and isolation as themes that emerged from a study of students and their mathematics experiences. As a result many of these students were alienated from mathematics and learned that they are not valuable members of the mathematics community. Hence, mathematical tasks that engage students in doing mathematics, making meaning, and generating their own solutions to complex mathematical problems can be beneficial in engaging students and supporting their identity as a mathematics learner (NCTM, 2000). A good starting point is open-ended mathematical tasks, questions or projects that have multiple responses or one response with multiple solution paths (Kabiri & Smith, 2003). The mathematics classroom 12
can also be organized to encourage discussion, sharing, and collaboration (Boaler & Greeno, 2000). In this type of classroom setting, teachers “pull knowledge out” (Ladson-Billings, 1995, p. 479) of students and make the construction of knowledge part of the learning experience. With respect to imagination, the development of students’ identities as mathematics learners requires long-term effort on the part of teachers across disciplines. The various images students have of themselves and of mathematics extending outside the classroom—in the past, present, or future—may be contradictory and change over time. Teachers and others in schools can consistently reinforce that mathematics is an interesting body of knowledge worth studying, an intellectual tool for other disciplines, and an admission ticket for colleges and careers. Since students’ identity development through imagination extends beyond the classroom, teachers can provide students with opportunities to see themselves as mathematics learners away from the classroom. For example, working professionals from outside the school can be invited to discuss ways they use mathematics in their professional lives; many students may not be aware of the work of engineers, actuaries, or statisticians. Another suggestion is to require students to keep a log and record the ways in which they use mathematics in their daily lives in order to become aware of the usefulness of mathematics (Masingila, 2002). This activity could provide an opportunity for assessing students’ views of mathematics and discussing the connections between the mathematics taught in school and that used outside the classroom. Although many of students’ mathematical requirements are beyond the control of teachers and students, teachers can foster the alignment face of identity. Teachers can hold their students to high expectations so that these expectations become as strong as requirements. Also, knowledge of mathematics requirements for post-secondary education and careers can help students decide to enroll in other mathematics courses. Because students are known to cite post-secondary education and careers as reasons for studying mathematics (Anderson, 2006; Martin, 2000; Sfard & Prusak, 2005), teachers can facilitate this alignment face by keeping students abreast of the mathematics requirements for entrance to college and careers. Students may commonly reference the nature face of identity, but this face is the least useful—and potentially the most detrimental—for supporting Four Faces of Identity
students as they become mathematics learners. As mentioned earlier, the ability to learn mathematics is not determined by genetics or biology (Lakoff & Núñez, 2000). All students can become mathematics learners, identifying themselves and being recognized by others as capable of doing mathematics. Thinking about the tetrahedron model of identity, if the other faces are strong and at the fore, the nature face can be turned to the back As suggested above, the other three faces of identity can sustain mathematics learners’ identities—through engaging students with mathematics in the classroom, developing positive images of students and mathematics, and establishing high expectations and requirements—regardless of students’ beliefs in an innate mathematical ability. Gee (2001) points out that the nature face of identity will always collapse into other sorts of identities. … When people (and institutions) focus on them as “natural” or “biological,” they often do this as a way to “forget” or “hide” (often for ideological reasons) the institutional, social-interactional, or group work that is required to create and sustain them. (p. 102)
Teachers need to be aware of the four faces of identity of mathematics learners and of how their students see themselves as mathematics learners and doers. Detailed recommendations for developing students’ identities as mathematics learners are provided in Figure 2. The four faces of identity discussed in this article contribute to our understanding of how students come to be mathematics learners. Through consistent and sustained efforts by mathematics teachers to develop positive identities in their students, more students can come to study advanced mathematics and improve their identities as mathematics learners. As I have pointed out throughout this article, identities are developed in relationships with others, including their teachers, parents, and peers. We cannot assume that all students will develop positive identities if they have experiences that run to the contrary. We must take action so each face of identity mutually supports the others in developing all students’ identities as mathematics learners.
Developing and Supporting Students’ Identities as Mathematics Learners Engagement Use mathematical tasks that allow students to develop strategies for solving problems and meanings for mathematical tools.
•
Organize mathematics classrooms that allow students to express themselves creatively and communicate their meanings of mathematical concepts to their peers and teacher.
•
Focus on the process and explanations of problem solving rather than emphasize quick responses to single-answer exercises.
•
Imagination Make explicit the ways mathematics is part of students’ daily lives. That is, help students identify ways they create and use mathematics in their work and play.
•
Have working professionals discuss with high school students ways in which they use mathematics in their professional lives, emphasizing topics beyond arithmetic.
• •
Include mathematics topics in classes that relate to occupations, for example, geometric concepts that are part of factory work or carpentry (e.g., see Smith, 1999; Masingila, 1994).
Alignment • •
Maintain expectations that all students will enroll in mathematics courses every year of high school.
Take an active role in keeping students informed of mathematics requirements for careers and college and university entrance.
Figure 2. Recommendations
Rick Anderson
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References Anderson, R. (2006). Mathematics, meaning, and identity: A study of the practice of mathematics education in a rural high school. Unpublished doctoral dissertation, Portland State University, Oregon. Boaler, J. (2000). Mathematics from another world: Traditional communities and the alienation of learners. Journal of Mathematical Behavior, 18, 379–397. Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171– 200). Westport, CT: Ablex. Boaler, J., & Humphreys, C. (2005). Connecting mathematical ideas: Middle school video cases to support teaching and learning. Portsmouth, NH: Heinemann. D’Ambrosio, U. (1990). The role of mathematics education in building a democratic and just society. For the Learning of Mathematics, 10, 20–23. Devlin, K. (2000a). The four faces of mathematics. In M. J. Burke & F. R. Curcio (Eds.), Learning mathematics for a new century (pp. 16–27). Reston, VA: NCTM. Devlin, K. (2000b). The math gene: How mathematical thinking evolved and why numbers are like gossip. New York: Basic Books. Gee, J. P. (2001). Identity as an analytic lens for research in education. Review of Research in Education, 25, 99–125. Kabiri, M. S., & Smith, N. L. (2003). Turning traditional textbook problems into open-ended problems. Mathematics Teaching in the Middle School, 9, 186–192. Kirshner, D. (2002). Untangling teachers’ diverse aspirations for student learning: A crossdisciplinary strategy for relating psychological theory to pedagogical practice. Journal for Research in Mathematics Education, 33, 46–58. Ladson-Billings, G. (1995). Toward a theory of culturally relevant pedagogy. American Educational Research Journal, 32, 465– 491. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books. Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven, CT: Yale University Press.
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Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, UK: Cambridge University Press. Martin, D. B. (2000). Mathematics success and failure among African-American youth: The roles of sociohistorical context, community forces, school influences, and individual agency. Mahwah, NJ: Lawrence Erlbaum. Masingila, J. O. (1994). Mathematics practice in carpet laying. Anthropology & Education Quarterly, 25, 430–462. Masingila, J. O. (2002). Examining students’ perceptions of their everyday mathematics practice. In M. E. Brenner & J. N. Moschkovich (Eds.), Everyday and academic mathematics in the classroom (Journal for Research in Mathematics Education Monograph No. 11, pp. 30–39). Reston, VA: National Council of Teachers of Mathematics. Mendick, H. (2003). Choosing maths/doing gender: A look at why there are more boys than girls in advanced mathematics classes in England. In L. Burton (Ed.), Which way social justice in mathematics education? (pp. 169–187). Westport, CT: Praeger. Moses, R. P., & Cobb, C. E. (2001). Radical equations: Civil rights from Mississippi to the Algebra Project. Boston, MA: Beacon Press. Nasir, N. S. (2002). Identity, goals, and learning: Mathematics in cultural practice. Mathematical Thinking & Learning, 4, 213– 247. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34(4), 14–22. Smith, J. P. (1999). Tracking the mathematics of automobile production: Are schools failing to prepare students for work? American Educational Research Journal, 36, 835–878. Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge, UK: Cambridge University Press.
1
Others have used the idea of “faces” to convey the many interrelated aspects of a topic. For example, Devlin (2000a) describes “The Four Faces of Mathematics.”
Four Faces of Identity
The Mathematics Educator 2007, Vol. 17, No. 1, 15–23
A Quilting Lesson for Early Childhood Preservice and Regular Classroom Teachers: What Constitutes Mathematical Activity? Shelly Sheats Harkness Lisa Portwood In this narrative of teacher educator action research, the idea for and the context of the lesson emerged as a result of conversations between Shelly, a mathematics teacher educator, and Lisa, a quilter, about real-life mathematical problems related to Lisa’s work as she created the templates for a reproduction quilt. The lesson was used with early childhood preservice teachers in a mathematics methods course and with K-2 teachers who participated in a professional development workshop that focused on geometry and measurement content. The goal of the lesson was threefold: (a) to help the participants consider a nonstandard real-world contextual problem as mathematical activity, (b) to create an opportunity for participants to mathematize (Freudenthal, 1968), and (c) to unpack mathematical big ideas related to measurement and similarity. Participants’ strategies were analyzed, prompting conversations about these big ideas, as well as an unanticipated one.
What would happen if the activities we, mathematics teacher educators, use to model best practices and standards-based teaching in mathematics methods courses and professional development workshops honored mathematical activity that is nonstandard in the sense that it is sometimes not even considered mathematics? Mary Harris (1997) describes how she uses “nonstandard problems that are easily solved by any woman brought up to make her family’s clothes” (p. 215) in mathematics courses for both preservice and classroom teachers. To make a shirt, “all you need (apart from the technology and tools) is an understanding of right angles, parallel lines, the idea of area, some symmetry, some optimization and the ability to work from two-dimensional plans to threedimensional forms” (p. 215). Although none of these considerations are trivial, making a shirt is not typically considered mathematical activity. Harris raises questions about why this is true. Is it because the seamstress is a woman or because only school mathematics is valued by our society? And, more broadly, what constitutes mathematical activity? We contend that mathematical activity is both physical and mental. It requires the use of tools, such Shelly Sheats Harkness, Assistant Professor at the University of Cincinnati in the Secondary Education Program, is a mathematics educator. Her research interests include Ethnomathematics, mathematics and social justice issues, and the impact of listening and believing in mathematics classrooms. Lisa Portwood is a self-taught quiltmaker and quilt historian. She has been quilting for the past 21 years and is an active member of the American Quilt Study Group. She is also a financial secretary at Miami University in the Teacher Education Department.
Shelly Sheats Harkness & Lisa Portwood
as physical materials and oral and written languages that are used to think about mathematics (Heibert et al., 1997). In the process of doing mathematics, one thinks or reasons in logical, creative, and practical ways. According to Sternberg (1999), American schools have a closed system that consistently rewards students who are skilled in memory and analytical reasoning, whether in mathematics or other domains. This system, however, fails to reward, in the sense of grades, students’ creativity, practical skills, and thinking. In problem-solving situations, students should be encouraged to use both physical and mental activity to do mathematics in order to (a) identify the nature of the problem; (b) formulate a strategy; (c) mentally represent the problem; (d) allocate resources such as time, energy, outside help, and tools; and (e) monitor and evaluate the solution (Sternberg, 1999). Researchers who studied the consumer and vendor sides of mathematical reasoning found skills revealed by a practical test were not revealed on an abstractanalytical, or school-type, test (Lave, 1998; Nunes, 1994). Too often, mathematics is viewed as the mastery of bits and pieces of knowledge rather than as sense making or as sensible answers to sensible questions (Schoenfeld, 1994). Sensible questions arise from many nonstandard contexts. If we design problems that are based on those questions, model best practices, and elicit mathematical big ideas, our students might begin to see mathematics as a human endeavor. They may use logical, creative, and practical thinking to solve those problems.
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As sometimes happens, an idea for a nonstandard real-world lesson took root in an unexpected place. During lunchroom conversations, Lisa, a member of the American Quilt Study Group (AQSG), described her work on a reproduction quilt with Shelly, a mathematics teacher educator. AQSG members participate in efforts to preserve quilt heritage through various publications, an extensive research library, and a yearly seminar. At this seminar, the AQSG invites members to make smaller versions, or reproductions, of antique quilts from a specified time period so that many of these can be displayed in one area. Lisa and Shelly discussed the mathematical problems Lisa encountered as she designed the reproduction quilt. The square quilt she was
reproducing, one her neighbor owned, had a side length of about 88 inches; however, the display quilt could not be more than 200 inches in perimeter. She wanted to trace the templates for the design from her neighbor’s quilt (see Figure 1) and then use a copy machine to reduce these traced sketches (see Figure 2). She needed to decide which reduction factor to use and how much of each color fabric–white and blue–to purchase. She wanted to buy the least possible amount of fabric. These reproduction quilt problems became the context for the lesson that Shelly created and used with both early childhood education preservice teachers enrolled in a mathematics methods course and with K-2 classroom teachers in a professional development workshop.
Figure 1: Reproduction (left) and original quilt (right)
Figure 2: Templates 16
A Quilting Lesson
Literature Review of Lessons Related to Mathematics and Quilting Because of a desire to know more about the connections between mathematics and quilting, we began by searching for literature related to lessons for teachers. We found some rich resources that included a wide range of mathematical topics embedded in these lessons. Transformational geometry was the foundation for several lessons. Whitman (1991) provided activities for high school students related to Hawaiian quilting patterns with a focus on line symmetry and rotational symmetry. Ernie (1995) showed examples of how middle school students used modular arithmetic and transformational geometry to create quilt designs. Most recently, Anthony and Hackenberg (2005) described an activity for high school students that made “Southern” quilts by integrating an understanding of planar symmetries with wallpaper patterns. The patterns and sequences found in quilt designs provided a basis for mathematical topics in other lessons. Rubenstein (2001) wrote about several methods that high school students used to solve a mathematical problem related to quilting: finite differences, the formula for the sum of consecutive natural numbers, and a statistical-modeling approach using a graphing calculator. Westegaard (1998) described several quilting activities for students in grades 7-12 that reinforced coordinate geometry skills and concepts such as identifying coordinates, determining slope as positive or negative, finding intercepts, and writing equations for horizontal and vertical lines. Mann and Hartweg (2004) showcased third graders’ responses to an activity in which they covered two different quilt templates with pattern blocks and then determined which template had the greatest area. Reynolds, Cassell, and Lillard (2006) shared activities based on a book by Betsy Franco, Grandpa’s Quilt, which they incorporated into lessons for their second-graders. In these activities, students made connections to patterns, measurement, geometry, and a “lead-in” to multiplication. In a lesson for third graders, Smith (1995) described how she linked the mathematics of quilting–problem solving, finding patterns, and making conjectures–with social studies through the use of a children’s book, Jumping the Broom. Also with a connection to integration with social studies, Neumann (2005) focused on the significance of freedom quilts, the Underground Railroad, the book Sweet Clara and the Freedom Quilt Shelly Sheats Harkness & Lisa Portwood
(Hopkinson, 1993), and mathematics–the properties of squares, rectangles, and right triangles–in her lesson for upper elementary school children. In their book Mathematical Quilts: No Sewing Required!, Venters and Ellison (1999) included 51 activities for giving “pre- and post-geometry students practice in spatial reasoning” (p. x). These activities are situated within four chapters: The Golden Ratio Quilts, The Spiral Quilts, The Right Triangle Quilts, and The Tiling Quilts. The authors noted that the quilts that inspired their book were created when they were teaching mathematics and taking quilting classes in the mid-1980s: Because we had no patterns for our [mathematical] quilts, we had to draft the design and solve the many problems that arise in this process involving measurement, color, and the sewing skills needed for construction ... Taking on a project and working it through to completion provide invaluable experiences in problem solving. (p. ix)
These were the same challenges that Lisa faced when she created her reproduction quilt. What was missing from this extensive list of resources was any reference to using mathematics and quilting in lessons for preservice teachers or professional development workshops for teachers. We felt that Lisa’s real-world task would provide the opportunity for the preservice and classroom teachers to mathematize,1 recognize big mathematical ideas, and consider what constitutes mathematical activity. We thought the big ideas that would emerge from this task included: •
Measurement is a way to estimate and compare attributes.
•
A scale factor can be used to describe how two figures are similar.
Within this article, we briefly describe the preservice and classroom teachers who we worked with, the quilting lesson that we created–based upon the actual mathematical questions that Lisa faced as she created her reproduction quilt–and the mathematics that the preservice and classroom teachers used as they mathematized. We then summarize our follow-up conversations about participants’ general reactions to the reading, An Example of Traditional Women’s Work as a Mathematics Resource (Harris, 1997), and our question: What constitutes mathematical activity? Finally, we describe the emergence an unanticipated mathematical big idea, based on the ways that the preservice and classroom teachers approached the problem. 17
We piloted this lesson during the first semester of the 2004-2005 school year with a class of preservice teachers at a large public university in the Midwestern United States. We then obtained IRB approval to collect data in the form of work on chart paper that groups in a second class did the following semester. We used the lesson again during the summer of 2005 with a group of 20 classroom teachers, grades K-2, in a professional development workshop. The K-2 Preservice Teachers and Classroom Teachers Teaching Math: Early Childhood (TE300) was a 2credit methods course that preservice teachers enrolled in prior to student teaching. The course included a twoweek field experience in which the preservice teachers wrote one standards-based, “best practices” lesson plan, and then taught the lesson. During each class session throughout the semester, we focused on a chapter from Young Mathematicians at Work (Fosnot & Dolk, 2001) and one of the content or process standards from Principles and Standards for School Mathematics (PSSM) (National Council of Teachers of Mathematics [NCTM], 2000). The reading from PSSM for the week of the quilting activity focused on measurement. All but one of the sixty TE300 preservice teachers were female. In mathematical autobiographies written during the first week of the course, about two-thirds of these preservice teachers said they either disliked or had mixed feelings about their previous school mathematical experiences, K-12 and post-secondary. Some who reported dislike for mathematics described feeling physically sick before math class, helplessness, and lack of self-confidence. Those with mixed feelings wrote about grades of A’s and B’s as “good times” and grades of D’s and F’s as “bad times.” Many described board races and timed math tests over basic facts as dreaded experiences. Generally speaking, they hoped to help their own students experience the success that they did not enjoy in math classes. These mathematical experiences posed a special challenge for us because we felt that their beliefs about teaching and learning mathematics had to be addressed. Due to time constraints, we attempted to address them at the same time that we talked about best practices and standardsbased methods. The K-2 classroom teachers participated in a professional development workshop offered the summer after we implemented the lesson with the preservice teachers. Most opted for free tuition to earn graduate credit; each of them also received a $300 18
stipend to spend on math books, manipulatives, or other items. All but one of the 20 teachers were female. The majority of the classroom teachers described their own mathematical experiences as less than pleasant and their fear of mathematics was evident from the beginning. Similar to the preservice teachers, they were very bold about their dislike of mathematics. They openly discussed their views of mathematics as a set rules and procedures to be memorized. Although the focus of the workshop was on improving the teachers’ content knowledge in geometry and measurement, we felt that we also needed to address their beliefs about teaching and learning mathematics within that context. As a springboard for the semester and the workshop, we discussed the meaning of mathematics and mathematizing. To initiate discussion, we posed the following question: “Is mathematics a noun or a verb?” Some thought it was a noun because mathematics is a discipline or subject you study in school. Others argued that it was a verb because you “do” it. About half argued that it could be considered both. When both the preservice and classroom teachers read Young Mathematicians at Work (Fosnot & Dolk, 2001), we again negotiated the meaning of mathematics and mathematizing (Freudenthal, 1968), reaching a consensus consistent with Fosnot and Dolk’s interpretation: “When mathematics is understood as mathematizing one’s world— interpreting, organizing, inquiring about, and constructing meaning with a mathematical lens, it becomes creative and alive” (p. 12-13). These are all processes that “beg a verb form” (p. 13) because mathematizing centers around an investigation of a contextual problem. The Lesson According to Fosnot and Dolk (2001), situations that are likely to be mathematized by learners have at least three components: •
The potential to model the situation must be built in.
•
The situation needs to allow learners to realize what they are doing. The Dutch used the term zich realizern, meaning to picture or imagine something concretely (van den Heuvel-Panhizen, 1996).
•
The situation prompts learners to ask questions, notice patterns, and wonder why or what if.
Guided by these components, we planned the lesson within Lisa’s quilting context. A Quilting Lesson
Throughout the semester and the workshop, Shelly read part of a picture book, Sweet Clara and the Freedom Quilt by Deborah Hopkinson (1993), in order to provide a context for the quilting problem. In this book, Sweet Clara is a slave on a large plantation. Her Aunt Rachel teaches her how to sew so that Clara can work in the Big House. There, she overhears other slaves’ talk of swamps, the Ohio River, the Underground Railroad, and Canada. Listening intently to these conversations, Clara visualizes the path to freedom and creates a quilt that is a secret map from the plantation to Canada. To launch the problem, Lisa told the groups about her work in the AQSG and explained why she wanted to produce a replica of a two-color quilt from the period 1800–1940. As it happened, her neighbor found a quilt in her basement and showed it to Lisa, who could hardly believe her luck! Not only did Lisa like the design, but she liked the two colors, blue and white, as well. She decided to use her neighbor’s quilt as the original for her reproduction. We then shared the parameters for the reproduction quilt, as given by the AQSG: •
Display: Each participant is limited to one quilt. Each quilt must be accompanied by a color image of the original and the story of why it was chosen.
•
Size: The maximum perimeter of the replica is 200 inches. This may require reducing the size of the original quilt. Size is limited to facilitate the display of many quilts.
•
Color: “Two-color” indicates a quilt with an overall strong impression of only two colors. A single color can include prints that contain other colors but read as a single color.
We also explained how the square-shaped original quilt had side lengths of 88 inches and showed them a photo of both the original and reproduction quilts (see Figure 1). In order to help both the preservice and classroom teachers immerse themselves in mathematizing and consider what constitutes mathematical activity related to measurement and similarity, we posed the following three questions that were actual questions that Lisa faced as she prepared to create the study quilt: 1.
By what percent did she need to reduce the original quilt to fit the 200 inches measured around all four sides (the perimeter)? The original was 88 inches on one side. (Lisa wanted to use the copy machine and a scale factor to reduce the pattern pieces she traced from the
Shelly Sheats Harkness & Lisa Portwood
original quilt to create the pattern pieces for the reproduction.) 2.
How much white fabric did she need to buy for the front and back of the quilt? (Please note: Fabric from bolts measures 44-45 inches wide.)
3.
How much blue fabric did she need to buy for the appliqués, borders, and binding around the edges? (Use the templates from the original quilt to determine your answer.)
Before they began to work, we also showed them an actual bolt of fabric and explained how fabric is sold from the bolt because we were not sure they would know what this meant (and many did not!). We gave each group original-sized copies of the templates used for the appliqué blue pieces on the original quilt (see figure 2) to use to answer the third question. We asked them to keep a record on chart paper of both the mathematics and mathematical thinking or processes they used to answer the questions so that they could share the results in a whole group discussion. Calculators, rulers, meter-yard sticks, tape measures, string, scissors, and tape were also available. We walked around, listened, and watched the groups work. Some had questions we had not expected: Is there white underneath the blue? Does the back have to be all one piece of fabric? Can we round our numbers? Should we allow for extra fabric? The students’ questions made us realize that, even though the three questions we posed might seem trivial for some quilters and mathematicians, they served as a springboard for the rich mathematical discussion that followed the small group work. The Mathematics We assessed the groups’ strategies while they worked to answer the three questions by listening to their discussions and analyzing the chart paper record of their strategies. We noticed that most groups, both in the class for preservice teachers and the workshop for classroom teachers, took the directions quite literally (i.e. that the reproduction perimeter must be exactly 200 inches) and used similar strategies. After the groups posted their chart paper on the walls, we began a whole-group discussion by posing the question: Are the two quilts, the reproduction and the original, mathematically similar? All agreed that they were but when asked why, their responses focused on the notion that they just looked similar. They knew the quilts were not congruent because they were different sizes. We told them that we would return to this question later so that we could negotiate a mathematical definition of similarity. 19
Figure 3: A solution to Question 1 Question 1 All but two groups thought of the perimeter parameter as exact and created scale factors to reduce the quilt so it would have a perimeter as near to 200 inches as possible. These groups said that the pattern should be reduced on the copy machine by either 57% or 43%; this led to an interesting conversation about how these were related and which one made more sense. Would we enter 57% or 43% into the copy machine? Which number makes the most sense based on what we know about copying machines and how they reduce images? The two groups that did not use the method adopted by the majority used the same scale factor as Lisa, approximately 67.7%. We asked these groups to share their thinking (see Figure 3) because it seemed like their mathematical calculations and reasoning were also valid. How could there be different answers? Instead of focusing on perimeter, these groups created a ratio of the total area of the reproduction quilt to the total area of the original quilt, 2500/7744 (assuming the quilt would measure 50 inches by 50 inches); the ratio was 32.3%. So, the area needed to be reduced by 67.7%. At first, we wondered why the percents differed when groups compared areas instead of perimeters for the same geometric figure. This led to the opportunity to discuss an unexpected big idea, something we had to think deeply about ourselves before we realized why the results for the groups were different: When the perimeter of a rectangle is reduced by a scale factor, the area is not reduced by the same scale factor. In fact, the ratio of the areas is the square of the ratio of the perimeters. In addition, this is true for any size of rectangular quilt or similar figures. We had to encourage the preservice and classroom teachers to think about why the two percents were different. When pressed, they realized that, because area is a square measure, taking the ratio of two areas resulted in a different value than taking the ratio of two 20
perimeters or side lengths. In fact, 32.3% was approximately the square of the ratio for the perimeter comparison, (50/88)2. This led us to rethink our questions about the copy machine. Does the word “reduction” lead to mathematical misconceptions? How does the reduction scale factor change the perimeter and the area? For instance, is the original image reduced by the selected percentage or does the machine create an image that is that percentage of the original? Experimentation with the copy machine reduction function helped us answer this question (we leave it to the reader to explore). Interestingly, even though most groups considered the perimeter parameter as strict, Lisa knew that the reproduction quilt could be no larger than 200 inches so she decided to use a 50% reduction—this made her study quilt 44 inches per side with a perimeter of 176 inches, which was “close enough.” Like Lisa, two groups decided that 50% was a reasonable and “friendly” number to use, making other calculations for the quilt less cumbersome. This led to a conversation about when close enough is sufficient for measurement and other uses of mathematics. We felt this was especially important because many of the preservice and classroom teachers experienced mathematics as problems with one exact answer. The idea that measurement can be precise but not exact was something they needed to think about. Questions 2 and 3 For the second and third questions, the chart paper revealed that the groups had a wide range of answers and some mathematical misconceptions. Most had answers close to 3 yards of white fabric and 1.5 yards of blue fabric. Some groups drew sketches of the fabric (see Figure 4). Groups that drew sketches or representations had the best estimates for conserving fabric. Even though one might think that determining
A Quilting Lesson
Figure 4: A solution for Question 2 the amount of white material would result in trivial mathematical conversations, we noticed that most groups tried various ways to overcome the fact that the width of the material (44-45”) posed a real contextual dilemma, as it was shorter than the width of the quilt. In other words, they had to consider both area and length in their attempt to minimize the amount of fabric needed. For the second question, one group decided that Lisa needed to buy 47 yards of white fabric. This group felt the sides of the quilt should measure 41 inches because the fabric was 44-45 inches wide (see Figure 5). This was similar to Lisa’s thinking and within the 200-inch parameter for total perimeter. However, we were shocked by their answer of 47 yards! They did not take into account the notion that
you must divide by 144 to convert square inches to square feet and by 9 to convert square feet to square yards. This mistake is one that could have been predicted with out-of-context problems, but Lisa had shown her reproduction quilt before they began working in their groups. What was most disturbing about this answer was the fact that 47 square yards made no sense given the size of one square yard. Similarity Returning to the definition of similarity, we again posed the question: Why are the two quilts mathematically similar? The preservice and classroom teachers negotiated a definition that made sense to them. They talked about “not the same size but the same shape” in terms of scale factors and created a
Figure 5: A solution for Question 2
Shelly Sheats Harkness & Lisa Portwood
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working definition: The scale factors or ratios of the corresponding sides of the two quilts are equal (or proportional) and the ratio of the areas is the square of the ratio of the side lengths. Generally speaking, we felt as though this task provided opportunities to talk about many mathematical notions related to measurement and similarity including exactness versus precision, estimation, ratio, proportion, percent, scale factor, perimeter, and area. We briefly discussed the kinds of symmetry—reflection (flip), rotation (turn), and translation (slide)—in the quilt but this was not a focus. The preservice and classroom teachers also noted that they used all five NCTM process standards (problem solving, communication, reasoning and proof, connections, and representation) as they worked in their groups and during our class discussion. What Constitutes Mathematical Activity? As a way to help the preservice and classroom teachers consider what constitutes mathematical activity, we gave them copies of An Example of Traditional Women’s Work as a Mathematics Resource (Harris, 1997) to read before our next class or professional development session. According to Harris, in mathematical activity, women are disadvantaged in two ways: (a) until very recently, female mathematicians were barely mentioned; and (b), in a world where women's intellectual work is not taken very seriously, the potential for receiving credit for thought in their practical work is severely limited. In her book, Harris showed her students a Turkish flat woven rug, called a kilim, and her students explored the mathematics involved in its construction. She also displayed a right cylindrical pipe created by an engineer and a sock knitted by a grandmother. She then posed the following questions: Why is it that the geometry in the kilim is not usually considered serious mathematics? Is it because the weaver has had no schooling, is illiterate, and is a girl? How do we know that the weaver is not thinking mathematically? Why is designing the pipe considered mathematical activity but knitting the heel of the sock is not? This reading helped create an opportunity for the K-2 preservice and classroom teachers, groups that are mostly female, to talk about their own beliefs regarding what constitutes mathematical activity in the context of women’s work. Many of them had considered school mathematics as the only kind of mathematics. After doing the quilting activity and discussing their reading
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of Harris (1997), however, they began to talk about doing mathematics within traditional women’s work such as measuring and hanging wallpaper, cooking, creating flower garden blueprints, playing musical instruments, and determining the number of gallons of paint needed to paint a room. We did not discuss the nature of mathematical activity as both physical and mental activity but, after thinking more deeply about it ourselves, we now realize that this was a missed opportunity. It seems that our conversation should also focus on the logical, creative, and practical ways in which we think and reason while doing mathematics. Concluding Remarks Through our collaborative effort to create a lesson with real-world applications and a reading related to what constitutes mathematical activity, the preservice and classroom teachers saw mathematics as something you do outside of school. They were mathematizing, organizing, and interpreting the world through a mathematical lens as they made conjectures about the same questions that Lisa faced when she created her reproduction study quilt. Analysis of the student strategies revealed opportunities to discuss big ideas related to measurement and similarity and what constitutes mathematical activity. It also prompted Lisa to take another picture of the two quilts, to illustrate the big idea that emerged from the groups’ sense-making: reducing the perimeter by 50% created a reproduction quilt with one-fourth the area of the original quilt (see Figure 6). Within this lesson, we modeled the kind of teaching we hope these preservice and classroom teachers will think about and use in their classrooms: helping students mathematize, make connections to big ideas and real-world mathematics, and question what constitutes mathematical activity. As Harris (1997) noted, the role of mathematics teachers should not be to teach some theory and then look for applications, but to analyze and elucidate the mathematics that grows out of the students' experience and activity. Using nonstandard contextual problems creates opportunities to honor school mathematics and mathematical activity that exists within the real world of everyday activity. By doing so, we also honor and respect our students’ logical, creative, and practical thinking. We give voice to their mathematics.
A Quilting Lesson
Figure 6: The reproduction quilt on top of the original quilt References Anthony, H. G., & Hackenberg, A. J. (2005). Making quilts without sewing: Investigating planar symmetries in Southern quilts. Mathematics Teacher 99(4), 270–276. Ernie, R. N. (1995). Mathematics and quilting. In P. A. House & A. F. Coxford (Eds.) Connecting mathematics across the curriculum (pp. 170–181). Reston, VA: National Council of Teachers of Mathematics. Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann. Freudenthal, H. (1968). Why to teach mathematics so as to be useful. Educational Studies in Mathematics 1, 3–8. Harris, M. (1997). An example of traditional women’s work as a mathematics resource. In A. B. Powell & M. Frankenstein (Eds.) Ethnomathematics: Challenging Eurocentrism in mathematics education (pp. 215–222). Albany, NY: State University of New York Press. Heibert, 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.
Nunes, T. (1994). Street intelligence. In R. J. Sternberg (Ed.) Encyclopedia of human intelligence, Vol. 2 (pp. 1045–1049). New York: Macmillan. Rubenstein, R. N. (2001). A quilting problem: The power of multiple solutions. Mathematics Teacher 94(3), 176. Reynolds, A., Cassel, D., & Lillard, E. (2006). A mathematical exploration of “Grandpa’s Quilt”. Teaching Children Mathematics 12(7), 340–345. Schoenfeld, A. H. (1994). What do we know about mathematics curricula? Journal of Mathematical Behavior 13, 55–80. Smith, J. (1995). Links to literature: A different angle for integrating mathematics. Teaching Children Mathematics 1(5), 288–293. Sternberg, R. J. (1999). The nature of mathematical reasoning. In L. V. Stiff & F.R. Curcio (Eds.), Developing mathematical reasoning in grades K-12 (pp. 37–44). Reston, VA: National Council of Teachers of Mathematics. van den Heuvel-Panhuizen (1996). Assessment and realistic mathematics education. Series on Research in Education, no.19. Utrecht, Netherlands: Utrecht University. Venters, D., & Ellison, E. K. (1999). Mathematical quilts: No sewing required! Berkley, CA: Key Curriculum Press.
Hopkinson, D. (1993). Sweet Clara and the freedom quilt. New York: Alfred A. Knopf, Inc.
Westegaard, S. K. (1998). Stitching quilts into coordinate geometry. Mathematics Teacher 91(7), 587–592.
Lave, J. (1988). Cognition in practice: Mind, mathematics, and culture in everyday life. New York: Cambridge University Press.
Whitman, N. (1991). Activities: Line and rotational symmetry. Mathematics Teacher 84(4), 296–302.
Mann, R., & Hartweg, K. (2004). Responses to the pattern-block quilts problem. Teaching Children Mathematics 11(1), 28–37. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Neumann, M. (2005). Freedom quilts: Mathematics on the Underground Railroad. Teaching Children Mathematics 11(6), 316–321.
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The term mathematize was coined by Freudenthal (1968) to describe the human activity of modeling reality with the use of mathematical tools.
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The Mathematics Educator 2007, Vol. 17, No. 1, 24–31
Graduate Level Mathematics Curriculum Courses: How Are They Planned? Zelha Tunç-Pekkan Even though there is much research related to the teaching and learning of K-12 mathematics, there are few studies in the literature related to university professors’ teaching. In this research report, I investigated how three professors of mathematics education structure their graduate level curriculum courses. The results show that three factors influence the ways that the professors design this course: (a) their view of mathematics curriculum, (b) their view of graduate students’ contributions to classroom atmosphere, and (c) their learning goals for the graduate students.
One of the goals in writing this article is to make the practice of teacher educators more public. This goal was motivated by Shulman’s (1998) exhortation to teacher educators: Now, think of your functioning as teachers. How much of what you do as a teacher—these great acts of creativeness, these judgments you make all the time as a teacher, the courses you design, the internships you tinker with, modify and strike gold with—how much of that ever becomes public? How much is susceptible to critical review by your colleagues or becomes a building block in the work of other members of the teacher education community throughout your own institution, much less the nation or the world? (p.18)
Preparing to teach a mathematics curriculum course requires more than deciding what mathematics topics to teach and in which order, as is the case for many content courses. It requires more than choosing K–12 mathematics classroom ideas to analyze with future teachers, as is the case in some methods courses. Preparing to teach graduate-level mathematics curriculum courses is a complex endeavor because these courses involve the integration of various aspects of mathematics education—curriculum, content and children’s mathematical learning—with the goals of educating graduate students. Such complexity seems to warrant the investigation of how professors design graduate-level mathematics curriculum courses. However, there are not many studies investigating how Zelha Tunç-Pekkan is a doctoral student in the Department of Mathematics and Science Education at the University of Georgia. Her research explores how 8th grade students use their fractional knowledge for the construction of algebraic knowledge including algebraic written notations. She received her bachelor’s degree from Middle East Technical University, Turkey and her master’s degree from Indiana University, Indianapolis.
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university professors conceptualize the courses they teach, especially graduate-level mathematics curriculum courses. This study had two purposes: (a) to gain insight into university teachers’ decision-making processes in planning a graduate-level curriculum course and (b) to make these insights public so that educators who teach or plan to teach similar courses will have a stronger base of information to guide their decision-making processes. Literature Review Shuell (1993) stated that teaching and learning at all grade levels are dynamic and reciprocal processes and that research should attempt to account for the complex and simultaneous effects of developmental, affective, and motivational influences, as well as cognitive factors. Many investigations of K–12 mathematics teachers’ practices have been conducted to explore motivational influences and cognitive factors that affect these complex reflexive processes (e.g., research on teachers’ knowledge, beliefs, and motivation). However, there is a paucity of research on university professors’ motivation or cognitive processes when they reflect upon their practice as teachers of graduate level mathematics education courses. Looking beyond mathematics, there are a few general studies of university professors’ beliefs about students’ learning, and how they perceive their practice as course instructors. For example, considering professors’ teaching practices, Kugel (1993) theorized that professors undergo three stages of development as teachers: self, subject, and student. During the first stage of their career, professors primarily focus upon their own role in the classroom and how they feel about their own abilities, i.e., self. In the second stage, professors focus on teaching the subject, which Graduate Level Mathematics Curriculum Courses
includes the subject matter and the materials they use when teaching. In the final stage, professors focus on students and how they learn. This stage is the least stressful part of the development, according to Kugel, because at this stage professors have already mastered the previous two, and are better prepared to focus on their students. Kugel admitted that not necessarily all professors experience these stages, but he claimed that they were a commonality in the experiences of many. Stark and Lattuca (1997) made an additional generalization about professors and wrote about their unsystematic way of thinking and acting in their teaching practices:
professors’ practices. In addition, Jackson (1994) claimed that context and values play a fundamental role in professors’ designing processes at the undergraduate level. These general studies offer some insight into investigating how mathematics education professors conceive their practices. However, there is still a need for studies that investigate professors’ design processes for graduate-level courses, where many students are mature adults coming to school to learn more about their own profession. Thus, this study focuses on the following research question: How do university professors decide what to teach in graduate level mathematics curriculum courses?
Instructional methods are chosen more often according to personal preferences or trial and error rather than through systematic attention to [the] nature of the expected learning, the nature of the student group or audience, and many varied practical constraints, such as size of the class (p. 288).
Research Design & Methods
In contrast to these authors, I believe, most professors use some trial and error in their classroom in a systematic way. They revise their practice considering the learning goals for their courses and the nature of the student group. Therefore, the definition of trial and error and how it affects teaching needs more investigation at the graduate level. Hence, there is a need to understand how professors take into consideration the nature of the material they want to teach, the group of students they teach, and their expected learning goals when making decisions on how to teach the course differently. With this study, I hoped to gain insight in these issues. Jackson (1994) observed: "We do not know what teachers in higher education think about their teaching and we do not know the cognitive processes in which they engage when they develop curriculum" (p. 2). In her study, she interviewed 11 university professors from different disciplines to understand their conception of curriculum design. In particular, her principal questions were “How do university teachers see themselves as curriculum makers; how do they think and make decisions about their teaching; how do they interpret their experiences and give meaning to their work?” (p. 2). In her findings, Jackson indicated that professors’ decisions about the curriculum design of a course are based upon the context of the course as well as their individual and disciplinary values. As a summary, Kugel (1993) indicated what professors might focus on when they reflect on their practice, whereas Stark and Lattuca (1997) suggested that unsystematic trial and error might be part of Zelha Tunç-Pekkan
The study consisted of three cases. Patton (2002) said, "Cases are units of analysis. What constitutes a case, or a unit of analysis, is usually determined during the design stage and becomes the basis for purposeful sampling in qualitative inquiry" (p. 447). In this vein, I selected three professors who approach teaching from different theoretical frameworks as my units of analysis. These professors, Martin, Rafaela, and Adam1, all have taught mathematics education courses at the undergraduate and graduate levels. To a certain extent, this study was an intrinsic case study (Stake, 2000) because I had a personal interest in trying to better understand the selected cases. As both a graduate student and a future instructor of curriculum courses, I was interested in how those particular professors decided what to teach in their curriculum courses and how they perceived graduate students. I used interviews and course artifacts to get detailed information about the cases and interviewed each participant for one hour. An overall interview guide (see Appendix) was developed for this semi-structured interview. The use of the interview guide followed Patton‘s (2002) suggestions: The interview guide provides topics or subject areas within which the interviewer is free to explore, probe, and ask questions that will elucidate and illuminate that particular subject. Thus, the interviewer remains free to build a conversation within a particular subject area, to word questions spontaneously, and to establish a conversational style but within the focus on a particular subject that has been predetermined. (p. 343)
The interview guide consisted of ten open-ended questions. The foci of the guide were the professors’ understanding of curriculum, their goals for the curriculum course, and the difference between teaching 25
graduate level mathematics education curriculum courses and teaching undergraduate mathematics content and method courses. The interviews were audiotaped, transcribed, and analyzed. Participants were asked to answer follow-up questions based on the initial data analysis. I also collected course syllabi, selected books, organized readers, and other artifacts used in these professors’ courses to help with the analysis of the interview data. Context and Participants This study focused on a graduate-level curriculum course taught in a large university in the southeastern region of the United States. The course was listed as a three credit hour graduate course in the university graduate catalog, and had the following description: “Mathematics curriculum of the secondary schools, with emphasis on current issues and trends.” It was a required course for graduate students in the master’s program. In addition, several doctoral students chose to take the course as preparation for doctoral-level advanced curriculum studies if they had not taken a similar course in their master’s program. One female and two male professors participated in this study. One of the participants, Martin, had taught this course more than 20 times. The last time he taught the course, Martin used videotapes from the Third International Mathematics and Science Study (TIMSS) to discuss curriculum and analyze current curriculum issues. He also used recent publications from the National Council of Teachers of Mathematics (NCTM) and the state’s K–12 mathematics standards. In addition, Martin drew upon his own articles and experiences related to curriculum development. Rafaela had taught this course three times at this institution. Every semester, she began this course with a book called The Saber-Tooth Curriculum by Peddiwell (1939) to create a discussion about the nature and purpose of curriculum. She also valued NCTM’s (2000) Principals and Standards for School Mathematics and thought of NCTM’s standards for the K–2 grade band as the basis for how she conceptualized K–12 mathematics education. Rafaela divided each course session into two parts. In the first part, she conducted a more theoretical discussion of selected articles regarding current issues in mathematics education curriculum (e.g., “math wars” or equity). In the second part, she incorporated activities and investigations related to mathematics education curriculum. For example, she recently used innovative curriculum materials that were funded by the National Science Foundation. She asked her 26
students to consider what their classrooms would look like if they taught with these curricula. The third participant, Adam, had taught this course five times. He did not ask students to read articles or have discussions about curriculum in his course; rather, he preferred to work on their mathematical knowledge by engaging them in mathematical investigations. Adam’s aim was to make mathematics teachers creators of curriculum by strengthening their mathematical knowledge. Data Analysis After analyzing the cases individually, I searched for themes that cut across all three cases. Three themes emerged: professors’ views of curriculum, their views of graduate students’ contributions to the course, and their learning goals for the graduate students. In the remainder of this article, I focus on data related to these three themes. During the analysis, I realized that the three participants did not share similar conceptions of what I considered to be basic terminology, such as mathematics curriculum. For example, Martin’s focus was on curriculum as instantiated by textbooks, state standards and other documents. Rafaela and Adam, on the other hand, deeply questioned what curriculum is. However, Rafaela and Adam’s ways of questioning curriculum also differed due to their backgrounds and roots in mathematics education. Views of Curriculum What is curriculum? Clements (2002) summarized a few classic definitions of curriculum in the United States as follows: the ideal curriculum is what experts propound; the available curriculum is the textbooks and teaching materials; the adopted curriculum is the one that is adopted by authorities; the implemented curriculum is what teachers teach in the class; the achieved curriculum is what students have learned; and “the tested curriculum is determined by the spectrum of credibility tests” (p.601). The three different perceptions of curriculum that emerged in this study are related to different aspects of curriculum as defined by Clements. Martin’s view of curriculum was closer to the idea of the available curriculum. For him, mathematics curriculum is represented in textbooks; curriculum is how teaching materials are organized. Because of the importance he gave to this view of curriculum, Martin used many different curriculum materials in his course. He discussed how these materials are organized, and why certain things are added to or omitted from school Graduate Level Mathematics Curriculum Courses
mathematics curriculum. In doing so, he followed the history of the available mathematics curriculum. Interviewer: What are your goals for your students in this course? Martin: Well, I want to get them [the grad students] to analyze, first of all the structure of the curriculum. How the curriculum is [organized], that is really, where we start usually. The structure of the curriculum, how the U.S. curriculum is organized, and why it is organized the way it is, and various attempts to change the organization. How new topics have come in, how other topics have gone away, how certain things have been emphasized in different times. So when we look at old textbooks, we look at the kinds of problems that are posed, we look at the organization of topics, we look at how, and what definitions are offered for certain things. To compare them with each other and to see, for example, how certain definitions have changed, or how the kinds of activities in the books have been changed.
Curriculum for Rafaela included a wide spectrum of issues and, when talking about curriculum, she utilized many of the aspects mentioned by Clements (2002). To her, curriculum is not just textbooks or that which is taught by mathematics teachers: curriculum is a complex political and theoretical concept. She encouraged her graduate students to discuss what curriculum is and how it affects students and teachers in K-12 schools. Even though Rafaela incorporated some discussion on curriculum theory into her class, she believed there were underlying expectations at her university about what to teach in this curriculum course, e.g., history of curriculum and NCTM Standards. When asked why she did not include more theoretical discussions of curriculum in her course, her answer provided insight into her ideas about curriculum.
Adam included both teachers’ and students’ actions in his definition. This definition combines Clements’ notions of implemented and achieved curriculum. Adam’s view of curriculum was not related to what is in textbooks or how they are organized. Thus, he created curricula based upon the actions of his students as he taught these courses. He modeled his view of curriculum for his students by using mathematical investigations. Based on his knowledge about what is essential middle- and high school-aged children’s mathematics, Adam formed a possible curriculum, what it should include implicitly, and then further developed the investigations for his students as he interacted with them throughout the course. Interviewer: How do you perceive curriculum? Adam: … You can view curriculum like books on the shelf: it is already in place. It serves me to teach and it is objective. That is one view to curriculum that is a normal view people in mathematics education take. … It is already there in place and already there before the teacher. And the teacher just implements. My view of curriculum is quite different: my view of curriculum is, it is done by the teacher and by the student, it is a dynamic growing, evolving thing, defined by the participants in the classroom.
Views of Graduate Students When talking about the graduate students in his course, Martin mainly focused on their teaching experiences. He saw that the classroom discussions changed immensely depending on whether there were many graduate students with teaching experience. Hence the teaching experiences of the students enabled him to conduct the undergraduate- and graduate-level curriculum courses differently.
Interviewer: What do you think why you don’t talk about it [the theory]?
Interviewer: So, was summertime different than how you taught it [the graduate course] in spring or fall semester?
Rafaela: The way I read what it says in the description of the course, this is not really a theory of curriculum course. It is math ed. curriculum in schools … so I understand it more as a discussion of curricula that is out there. Because I have taught curriculum theory classes, I try to include some of the theory in my class. So, in my class we talk about things like different types of curriculum. I try to get students to think about what is curriculum: is it your textbook, is it the politicians? … We talk about hidden curriculum: the things you teach but you don’t even know you are teaching, like values … so I bring all that in …
Martin: Well, summertime is, of course, shorter. It makes the course a little bit different, but the course is basically the same. What changed the course the most is whether most of the students have done teaching. I had classes where almost everybody had teaching experience, if not everybody, so these were experienced teachers. On the other hand, I’ve taught [when] almost nobody had done any teaching except possibly, student teaching, and that makes the course very different. That makes more of a difference, I think, than when the course is given, whether it is given in the summer …
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Interviewer: So you think that there are differences between those graduate students because … Martin: If they have never taught, then there are some of these issues [that] don’t occur to them or are not realistic for them. Or they have trouble seeing some of the issues. Let me give one example. I have in recent years—since the TIMSS video studies came out—I have occasionally used the TIMMS videos. I have shown it in the course to discuss curriculum issues. I have shown it in [another graduate course] also, but occasionally shown it in [the curriculum graduate course] so we can talk about some … curriculum questions … We were concentrating in that course only [on] American teachers [in the videos] … and I’ve noticed that they [graduate students with teaching experience] see different things; they notice different things about the topics that are being taught. … They have different reflections on the video.
Martin could do more in the graduate-level course by using different curriculum materials and readings about the history of school mathematics because curriculum is more real to graduate students who have taught and experienced it in their professions. Interviewer: Are there undergraduate curriculum?
any
courses
for
Martin: Yes. Mostly in that course we remind them [undergraduate students] what the curriculum looks like and there is very little history. …There is not very much analysis of new materials—some innovative materials. There is a little bit of look at new textbooks, … but we try to balance that because we recognize that most of these people would not be using these new materials right away... so part of what we try to do is familiarize them with the most common materials out there. And again, since none of them have taught … in that course, it is really a very different orientation because usually in [the graduate curriculum course] you can expect some people have done some teaching, and so they can talk about some of these curriculum ideas from their own perspective: this is what we had in our school, these are the materials, this is what we like, this is what we didn’t like, and so forth. The undergraduate course doesn’t have that kind of discussion.
Rafaela also indicated that as a professor she could try different things with her graduate students compared to her undergraduate students. On the other hand, her vision of graduate students not only differed in their teaching experiences but also in their willingness to try NCTM materials. Rafaela believed that most graduate students who live in the academic 28
environment are familiar with NCTM (1989, 2000) standards and already believe they can teach with these standards. On the other hand, she observed differences among the graduate students who are currently practicing teachers. These graduate students had varied views of NCTM’s standards and other reform-oriented curriculum materials. She thought that these practicing teachers especially needed to be exposed to NCTM standards in order to analyze their own teaching practice and observe similarities and differences between their practice and Standards-based teaching. Therefore, to accommodate these practicing teachers, Rafaela planned student demonstrations of teaching a topic from non-traditional textbooks as an important part of her curriculum class. Rafaela: I don’t think it is my goal to convince them [about NCTM standards]. I think my goal is to help them analyze what they believe. They can be critical and write a paper about why they don’t agree with that [NCTM standards]. So, I think, especially teachers, they finish the course thinking that it is a good idea, but you can’t really implement it. Some of it, it is hard to convince them that they can do it. And in the methods course, I am more interested in convincing them what they can do. I don’t do as much of that in the curriculum course. But I try to give them a vision of what it would look like if they were to try it. And I have changed the materials used in the course over the years. Last year we had Connected Mathematics Project materials. … Two or three of the students would be teachers from that class and we were the students. Because I started noticing that some of the students didn’t have a vision: “What would it look like if I were to do what the standards say? If I wanted to do that in the classroom, what does it look like?” … So I started giving them more of an idea, well, this is a different thing. Some of them liked it … I am just talking about the classroom teachers who come back, not the regular students who are in this environment that talk about NCTM and change. … Two years ago, I had one classroom teacher who came to me and said, “I am very lost. You really took the carpet from my feet” … for him things could be different … and I have had other students saying that “I have been doing that but I did not know how to call this.” But I also have had teachers, come and leave thinking that I am a dreamer. You know, anything I said is not possible.
Similarly to Martin, Adam mainly talked about how he viewed the graduate students and their contributions to his teaching by comparing them to his undergraduate students. Graduate students’ teaching experience was an important component of how he Graduate Level Mathematics Curriculum Courses
viewed graduate students and their contributions. Teaching experience provided the possibility for graduate students to be involved in secondary school students’ mathematical thinking. For Adam, having previously engaged in students’ mathematical thinking made graduate students more able to appreciate the importance of the basic mathematical concepts and operations they investigated in his course. These graduate students were able to establish meanings of basic ideas in mathematics from the point of view of a teacher, not just a student. Adam: It is very difficult for them [undergraduate students] because they are struggling. They struggle for the actual thinking that is involved. … There is a qualitative distinction between the natures of the students in the two courses. Interviewer: Nature of the students? Adam: The way students view themselves, the way they view what those courses should be about. … [Undergraduates] are not as mature as graduate students in actually working with students. They just did not have a chance to become involved with others people’s thinking. So, they don’t appreciate how important their thinking is in trying to understand the thinking of other students. So, their basic orientation in [the undergraduate course] is not to understand the thinking of the students. It is more, “what do I have to do when I go teach the topic that is already given?” That is their orientation. Interviewer: But don’t some students have that kind of orientation in [the graduate] class? Adam: Oh yes. By all means, they had that orientation. But I think they are more mature and probably little bit willing to consider the possibilities. OK. But [for the] most part few students that went through the course always knew what we were doing and quite appreciated what we are doing. The distinction between [the] two classes is quite profound in the maturity of the students and appreciation for investigating basic mathematical concepts and operations [and the] meaning of basic ideas in mathematics from the point of view: How do I make these things? How do I make meaning for them? How do I formulate a constructive itinerary of mathematics and the relationships and the connections to mathematics? I think [the graduate course] students are much more able to deal with that than the [undergraduate course] students.
Goals for Graduate Students Martin’s overall goal was to make students aware of current curriculum issues. For this goal, discussing NCTM’s (2000) new standards was important for him. Interviewer: What is the purpose when you are using NCTM standards and why do you want to use those? Martin: Well, to acquaint them [the students] with some of the issues in the field. These current publications reflect efforts in the profession to change, in the case of the curriculum standards, to change the curriculum. So, I think it is important for them to know what people are advocating. … I usually add in some critiques of this, or if we don’t read a critique we actually make a critique ourselves, … especially if they are experienced teachers, they don’t necessarily agree with all of the things that are in these documents so we discuss them. … So, my purpose is to get them thinking about current issues… As it says here [pointing to his course syllabus], I wanted them to … “gain some skill in analyzing issues and trends.” Because these people, whatever they end up doing in [their] profession, they are going to be using, or at least knowing about, curricula and they are going to know, I hope, that [there] will be issues out there.
Martin felt that graduate students needed to look at mathematical topics locally (for a grade) and globally (across grades) when discussing curriculum issues. For example, he discussed the emphasis on proof in NCTM’s 2000 standards as opposed to the earlier Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) and added that there would always be debate on certain curriculum issues, such as the inclusion of real life applications and technology. Hence, his goal was to make graduate students aware of those issues and enhance their skills in critiquing those issues. Martin: For example, how much emphasis should be put on proof? … now we have PSSM and a stronger emphasis on proof, … but … the 89 Standards didn’t emphasize it.…the way that 2000 Principles and Standards is structured, it raises the question of what is to be done about proof in early grades and what is to be done about proof at the later grades. And this raises questions of how the curriculum is organized across the grades. Even though the focus is on the secondary curriculum, there is always a question, “How does it build on the elementary curriculum?”
Rafaela’s overall goal was related to her conceptualization of the curriculum. Similarly to Zelha Tunç-Pekkan
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Martin, she wanted graduate students to think about and reflect on curriculum. However, her goal was to make an implicit change in graduate students’ teaching practice. She wanted them to think about how curriculum played a role in their own teaching and the effects of their use of curricula on their teaching. Rafaela believed in the existence of a hidden curriculum that teachers implemented but were not aware of. Therefore, her goal was to help graduate students clarify their own teaching goals. Interviewer: What are your goals? Is it the little course description? Rafaela: To think about, “What is curriculum?” is my goal, probably because…I come from this curriculum studies perspective. … You have to decide, what do you want to teach? As a teacher, what are your goals? And those are things, I can’t help anyone to decide but I can help them to think about it. So, my overall goals are to bring the class … to think … “Yes, there is a hidden curricula that I teach and never thought about …Why am I teaching this? What kind of people am I trying to educate? What [are] my goals as a teacher for my students?” … That is what I want them to reflect on. Inside that there is my view that … we want to create thinkers. … I think the NCTM standards … are a good venue for helping create thinkers who reflect mathematically … so I do present it from that perspective. …Who decides all those things? Who decides the curriculum? Who decides [the state standards]? Do we have to follow? What kind of people are we going to create by following that?
With this course, Adam also wanted to make a change in his graduate students’ educational experiences. His main goal was to reorient graduate students to think about the basics of school mathematics. In order to understand and value K–12 students’ mathematical activities, he believed that teachers need to have mathematical experiences such as understanding and formulating mathematical rules they use everyday in their teaching. Therefore, he provided learning opportunities in mathematical investigations and hoped graduate students would develop meaningful itineraries for some mathematical topics. Adam: How the teacher thinks is totally critical. … How students think is totally critical. So, my view of curriculum is manifested in how I acted in the [graduate] course. I involved … the participants deeply in doing basic mathematical activities in a way that they probably haven’t thought about before. … Investigate the basic ways of reasoning in mathematics, the basic meaning of … linear 30
functions. … Where they come from, what is the constructive itinerary for that? So, I want the participants to become aware how they think mathematically. OK. I want them to be aware of what they are doing mathematically … For example, addition of fractions: half plus a third is viewed as a procedure, as an algorithm. … I want them to go back to very basic ways of reasoning … How would I formulate that for the sum, if I don’t know already those rules? What do those rules mean? … I think that attitude is very essential for teachers because they have to respect … productive thought and creativity, and potential creativity, of the students. So, they are not just giving the mathematics procedurally to students, but the students are constructing it meaningfully.
Final Comments Depending on the professor, the learning experiences graduate students have in this curriculum course may differ immensely. The professors’ views of curriculum (e.g., static as in textbooks or already given as in the school standards versus dynamic views), views of graduate students, and their goals for the course influenced what kinds of materials they chose and how they used these materials. Martin, Rafaela, and Adam all believed this curriculum course should make graduate students better thinkers and better analyzers. Whereas Martin and Rafaela focused on discussing existing curriculum materials when talking about their learning goals, Rafaela was also concerned about changes in her students’ teaching practice. Adam, on the other hand, focused on helping graduate students become better at analyzing their own and their K-12 students’ mathematical activities. The professors’ learning goals were closely connected to their views of curriculum and their views of graduate students. For example, because Martin regarded school mathematics curriculum as textbooks, written documents, and the evolution of mathematical topics in those documents over time, he took these components into consideration in his planning. Martin focused on the organization of the materials with his graduate students and used a variety of current and historical curriculum materials for that purpose. He aimed to help his students better analyze current issues. In addition, he viewed graduate students’ teaching experiences as the factor that most affected the quality of discussions. Rafaela also used reading materials, but she concentrated on the discussions of how graduate students conceptualize curriculum, what NCTM standards mean in terms of teaching and learning, and Graduate Level Mathematics Curriculum Courses
who is making curriculum. For Rafaela, curriculum meant a theoretical discussion of teaching practice, so using NCTM materials as an orientation was a good venue for that purpose. She believed that some of the graduate students, mostly the currently practicing teachers, were hesitant to think about curriculum differently. Therefore, NCTM and other reform materials provided a context for this discussion. Using this context, she could expose teachers to new ideas that they could try in their practice. For Adam, curriculum was a dynamic phenomenon that is formed by teachers and students in the classroom. He thought graduate students should be creators of curriculum, like him, with their own students inside the classroom. In his classes, he tried to provide a model of this view by dynamically creating a curriculum with his graduate students. Teachers’ mathematical knowledge, as well as their teaching experiences, played an important role in that creation. He interacted with graduate students using a mathematical domain as the medium. His aim was to provide opportunities to graduate students to rethink mathematics curriculum in schools by engaging them with the basics of mathematics. This investigation of a graduate-level curriculum course reveals that various factors affect the ways in which professors design graduate-level courses. However, further research is needed to investigate the learning experiences of graduate students and how professors’ ideas about teaching curriculum are compatible with their practices in the classroom. References Clements, D. H. (2002). Linking research and curriculum development. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 599– 630). Mahwah, New Jersey: Lawrence Erlbaum. Jackson, S. (1994, April). Deliberation on teaching and curriculum in higher education. Paper presented at the Annual Meeting of American Educational Research Association, New Orleans. Patton, M. Q. (2002). Qualitative research and evaluation methods (3rd edition ed.). Thousand Oaks, CA: Sage.
Shulman, L. S. (1998, February). Teaching and teacher education among the professions. Paper presented at the American Association of Colleges for Teachers Education 50th Annual Meeting, New Orleans, Louisiana. Stake, R. (2000). The case study method in social inquiry. In R. Gomm, M. Hammersley, & P. Foster (Eds.), Case study method. London: Sage. Stark, J. S., & Lattuca, L. R. (1997). Shaping the college curriculum: Academic plans in action. Boston: Allyn and Bacon.
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Appendix: Interview Protocol 1.
How many times have you taught this course? Have you taught similar courses in different institutions?
2.
How is this course different from any mathematics education content courses or method courses you taught before? How does curriculum have a special or different emphasis in your design of the course?
3.
What are your goals for the course? How do these goals affect your decisions when you are designing the course?
4.
Since this course is for graduate level students, how do you take this audience into consideration (graduate students might be in-service teachers) when you design the course?
5.
How do you know your graduate students understood the curriculum ideas emphasized in the course? How do you check it?
6.
What components of the K-12 mathematics curriculum are important in your design of this curriculum course? How do you know you have emphasized them enough when teaching this course?
7.
How do you revise the content of the course or the way you teach the course each time? What factors do you take into account? (colleagues, recent related research, students’ success or responses, the departmental needs, etc.) and How?
8.
What would you like to gain as a teacher when teaching this course and how does this affect your design of the course?
9.
How does your research affect your teaching of graduate level mathematics curriculum courses? Or vice versa?
10.
In which ways do you think your [graduate curriculum] class is similar/different from the [graduate curriculum class] taught by other instructors?
Kugel, P. (1993). How do professors develop as teachers? Studies in Higher Education, 18, 315–328. National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principals and Standards of School Mathematics. Reston, VA: Author. Peddiwell, J. A. (1939). The saber-tooth curriculum. New York: McGraw-Hill. Shuell, T. J. (1993). Toward an integrated theory of teaching and learning. Educational Psychologist, 28, 291–311.
Zelha Tunç-Pekkan
All names used in this article are pseudonyms.
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The Mathematics Educator 2007, Vol. 17, No. 1, 32–41
Some Reflections on the Teaching of Mathematical Modeling Jon Warwick This paper offers some reflections on the difficulties of teaching mathematical modeling to students taking higher education courses in which modeling plays a significant role. In the author’s experience, other aspects of the model development process often cause problems rather than the use of mathematics. Since these other aspects involve students in learning about and understanding complex problem situations the author conjectures that problems arise because insufficient time within mathematical modeling modules is spent reflecting on student work and enabling “learning to learn” about problem situations. Some suggestions for the content and delivery of mathematical modeling modules are given.
Over the last 20 or so years of teaching in higher education, I have had the pleasure of teaching various aspects of the mathematical sciences to students at levels ranging from pre-degree to master levels. Although each module1 that one teaches presents challenges, the one subject that has been the most challenging to my students and myself has been that of mathematical modeling. In this article, I reflect on the mathematical modeling process and how it has influenced the way I teach modeling. My own experience of modeling has been acquired within the management science domain. This domain is concerned not only with modeling physical processes but can also include considerations of systems and organizational culture. Although this may give my views a different slant than those of someone working as a modeler in the pure sciences, the issues discussed apply across many modeling domains. By mathematical modeling I mean the “pencil and paper” type of modeling characterized by written assumptions, equations, and so on, as opposed to computer-based simulation models that can be built using graphical interfaces. Students usually enjoy the latter since the medium is interesting. These situations often divert attention from tough modeling considerations and the need to see the dynamic equations! This, however, is another story and I wish Jon Warwick completed his first degree in Mathematics and Computing at South Bank Polytechnic in 1979 and was awarded a PhD in Operations Research in 1984. He has many years of experience in teaching mathematics, mathematical modeling, and operations research in the Higher Education sector and is currently Professor of Educational Development in Mathematical Sciences at London South Bank University. He is also the Faculty Director of Learning and Teaching.
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to restrict my discussion to mathematical models derived without the use of software. Examples of these pencil-and-paper models are often presented in management science or operational research texts and would include some standard models relating to inventory control, waiting line models, and mathematical programs. These models can be written in terms of equations that give optimal order quantities, average waiting times, and so on for differing sets of conditions. For these standard models the underlying assumptions are well known. Students taking my modules at the undergraduate level are encouraged to develop their own models which may be based on a standard form but must be described using mathematical notation and with pencil and paper. In practice I have often used academic library management as a contextual area where, over the last forty years, mathematical modeling has been applied to good effect, producing a wealth of accessible literature and different types of models (Kraft and Boyce, 1991). By way of example, I shall describe some experiences from an introductory modeling module given to undergraduate students studying mathematics related to management. Having first spent some time with the students studying examples of a number of the standard model forms found in management science (stochastic, deterministic, simulation, etc.), the students are given a simple situation to start the modeling process. Briefly, this involves the students working in groups to develop a model that can be used for determining the effect of changing the loan period of a single title (multiple copy) text appearing on a class reading list. My students must develop a model (and if possible solve it) using pencil and paper only. A crude measure of
The Teaching of Mathematical Modeling
Investigation and Problem Identification
Mathematical Formulation of the Model
Collect Data and Obtain a Mathematical Solution to the Model
Interpret the Solution Compare with Reality Implement the Solution and Report Writing Figure 1. Stages in the modeling process. library user satisfaction is the likelihood of finding a book on the shelf when desired. Students are asked to find loan periods that provide certain satisfaction levels for differing numbers of copies and class size. The idea is, at this stage, to encourage simplicity in modeling and highlight the importance of assumptions. Working in groups is also important as group discussion facilitates the model development process. The Modeling Process Examination of textbooks dealing with undergraduate mathematical modeling (or any of the related fields, such as management science) will normally yield a description of the modeling process in general terms incorporating the stages as outlined in Figure 1. There are many variations on this theme from both specialist texts on mathematical modeling (see Edwards & Hamson, 2001) or texts on more general quantitative analysis (see Lawrence & Pasternack, 2002), but the basic structure of the process is usually similar to that shown. There are two things to notice about the process. First, as described in Figure 1, it is essentially a looping process. Second, it is a process that students Jon Warwick
generally find difficult to undertake, despite the fact that the process is fairly simple to state, the steps are logical, and the language fairly non-technical. The Art of Modeling As a student of mathematical modeling, I was introduced many years ago to an article that dealt with the process of mathematical modeling and attempted to give some hints and tips as to how the novice modeler might proceed (Morris, 1967). It is a paper I often recommend to my students as it recognizes the difficulties that many of them are facing. Morris makes the valid point that, when students read about the development of mathematical models and look at examples of models that have been developed by others, the writing is nearly always in the spirit of justification rather than the spirit of inquiry. By this we mean the writing justifies the final product and comments on the results obtained, the validity of the model, etc. However, it does not dwell on the frustrations and problems that may have been encountered on the way to the final model, the models that were discarded, or false trails that were followed. Adopting the latter style of writing, describing the ups and downs of the 33
inquiry process that eventually produced the final model, would be far more illuminating to students than just a description of the final model. In addition, Morris (1967) gives a nice description of the art of modeling and notes that the model development process has a looping structure with two major loops. The first looping process is developing a working model from a set of assumptions and continually testing the model against real data until it may be regarded as acceptable within the limits set by the realism of its assumptions. The second involves changing the assumptions either by relaxing those that seem unrealistic or by imposing new assumptions if the model is becoming too complex. These two looping processes are often in operation at the same time as the modeler strives to balance model tractability with performance. Model tractability here means the ease manipulating and solving the model. Morris refers to the looping process through which model assumptions are relaxed and the model enhanced as enrichment and elaboration. In addition to these two primary looping processes, Morris (1967) gives a checklist of hints and tips that he suggests will help the novice modeler. These may be summarized as:
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•
Try to establish the purpose of the model to give clues to model form and perhaps the level of detail necessary.
•
Break the problem down into manageable parts so these smaller pieces can be solved before being reincorporated into the larger whole.
•
If possible, use past experiences or other similar problems already solved to give clues as to the solution required by the current model. This is the process of seeking analogies and is a powerful weapon in the modeler’s armory (see for example Warwick, 1992).
•
Consider specific numerical examples. This may give clues as to where assumptions might be needed or how the problem situation is structured.
•
Establish some notation as soon as possible and begin building relationships in the form of mathematical equations.
•
Write down the obvious!
These hints and tips together with an appreciation of the general looping processes involved in model development are the core activities that students need to master in order to build models. They are easily learned, or perhaps memorized, and yet still students find model building difficult. There are at least two learning processes with which students are required to engage in order to become proficient in modeling. Each makes quite distinct demands of the student. The first learning process requires the student to become conversant with the tools of the trade such as mathematical symbolism, algebraic manipulation, the stages involved in model building, the looping processes, and archetypical model forms. These elements, often as not, form the core content of mathematical modeling modules. In terms of the type of learning that is being undertaken, we can refer to Bloom’s (1956) taxonomy of learning in the cognitive domain that describes different categories of learning arranged sequentially. The learning required to become proficient in the mechanics of model building is primarily within the three lowest categories–knowledge, comprehension, and application–and my students seem to have few problems here. Problems begin to surface when we consider the second learning process, which is not explicit in Figure 1. In this second process the modeler is coming to terms with the intricacies of the problem being modeled, the subtleties of the situation being studied, and the implications these will have for the model being developed. No model can be developed successfully unless the modeler has a clear understanding of what is to be modeled and this learning will need to take place as the modeling proceeds. Yet there is nothing in the modeling process model that helps the modeler with this. In other words, there is a requirement for the skills of learning to learn to be appreciated by students as every modeling situation they meet will be different and often complex. Learning to Learn What do we mean by learning to learn? Reference to the literature allows three general observations. First, this idea has been the subject of research for more than 30 years with researchers considering learning-to-learn issues at the K-12 (Greany and Rodd, 2003) and university levels (Wright, 1982), as well as within the work environment (Ortenblad 2004). Learning-to-learn The Teaching of Mathematical Modeling
Table 1 Considerations for learning to learn and comparison with Morris (1967). Achieving Learning to Learn Begin with the past;
Some Key Considerations for the Individual It is important to look back and consider what was your previous experience about how you learn, how was learning structured before and what worked well in similar circumstances.
Considerations from Morris (1967)
Proceed to the present;
There needs to be a clear reason for doing what you are doing! Which parts are important? Which should be tackled first? What is controllable and what is not and which bits are already learned to form a basis for further learning?
Try and establish early on the purpose of the model so that this will give clues to model form and perhaps the level of detail necessary for the model.
Consider the process ‌
What is the structure of the work to be learned? Get a feel for the general theme, the main points, key words. Are they understood?
Write down the obvious! Establish some notation as soon as possible and begin building relationships i.e. writing equations. Break the problem down into manageable parts so that these smaller pieces can be solved before combination back into the larger whole.
‌ and the subject matter;
How much of this subject is known about already? How much is known about related subjects and what is the link? What resources are available and are they accessible now? Decisions need to be made about how quickly to proceed through material, when to attempt questions, when to seek guidance etc.
Seek analogies and associations with other, related, modeling problems. Consider specific numerical examples— this may give clues as to where assumptions might be needed or how the problem situation is structured.
Build in review;
Decide here what went well and what did not and how this might affect further learning attempted.
This is a key area that Morris describes as lacking in modeling articles and reports. In practice, I have found it useful for students to keep a log or workbook that includes reflections on the various models built during the course of a taught module.
research also spans academic disciplines with examples from such diverse subject areas as history (Knight, 1997), physical education (Howarth, 1997), and science (Hamming, 1997; Elby, 2001). Little has been written in the context of mathematical modeling. Second, the recent interest in learning to learn has coincided with the development of research in cognitive and metacognitive strategies (Waeytens, Lens, and Vandenberghe, 2002) and the expansion of higher education. As a result, many universities now recruit students from a variety of backgrounds and consequently with a range of abilities and previous Jon Warwick
If possible, try and use past experiences or other similar problems already solved to give clues as to the solution required.
educational experiences. Third, there is little agreement about the definition of learning to learn or how it should be taught. Some researchers have a narrow view in which learning to learn involves essentially study skills, hints, and tips, whereas others take the broader view that students should be able to apply skills in critical analysis, goal setting, personal planning, and so on (Rawson, 2000). Regarding how learning to learn should be taught, there has been debate as to whether it is appropriate to approach it as a separate, isolated module or whether it should be embedded into other 35
regular study modules. These days, conventional wisdom suggests it must be taught within regular modules and not as an isolated subject (Waeytens et al., 2002). In my view learning to learn incorporates a broad set of skills including reflective and critical thinking and it should be approached within the context of a module. In fact, it is crucial in developing effective modeling skills. Because learning to learn is now becoming a key part of many university learning and teaching strategies, one way of approaching it is to consider the key elements as shown in Table 1 (amended from Landsberger, 2005). These considerations apply as much to the learning of mathematical modeling as they do to any other subject. The hints and tips given by Morris (1967) do, in fact, sit quite well within this framework, as seen in Table 1. In other words, Morris seems to be tacitly addressing the learning-tolearn difficulties associated with modeling through his practical advice. We can further strengthen this idea that modeling is as much about learning as it is about applying mathematics. To accomplish this, we must reconsider the classic process model of mathematical modeling (see Figure 1) and re-formulate it to emphasize the learning processes that are truly going on. True to the spirit of Morris (1967), we can do this by seeking analogies with other models of the learning process. A particularly useful representation has been developed within the field of organizational learning. Organizational Learning: Single and Double Loop Learning According to Senge (1990), learning enables us to do things we were never able to do, change our perception of the world and our relationship to it, and extend our capacity to create. In this context, learning organization is an “organization that is continually expanding its capacity to create its future” (Senge, 1990, p.14). In their classic work on organizational learning, Argyris and Schon (1978) define learning as occurring under two conditions: (a) when there is a match between an expected or desired outcome and the actuality or real outcome, and (b) when there is a mismatch between expected or desired outcomes and reality that is identified and corrected so that the mismatch becomes a match. Argyris and Schon (1978) describe two types of learning response that can occur when a mismatch is detected, single loop learning and double loop learning. Single loop learning is described as focusing on the status quo by narrowing the gap 36
between desired and actual conditions (University of Luton, 2006). It is a simple feedback loop where the learner’s actions are changed to accommodate mismatches between expected or desired and observed results in the perceived real world. Single loop learning has also been described as an errorcorrecting or fine-tuning process. There are, however, a number of limitations to single loop learning (Peschl, 2005): •
It is an essentially conservative process that seeks to retain the existing knowledge structures rather than exploring new alternatives.
•
There is very little chance that new insights will be gained or that anything new or innovative will be learned.
It is a process that lacks any form of reflection. Double loop learning (or reflective learning), on the other hand, tries to overcome these limitations by first examining and altering the current mental model and then the actions. It is single loop learning with an extension, or second feedback loop, that allows for the possibility of change in assumptions, premises, mental models, etc. As Peschi (2005) states: •
In double loop learning a second feedback loop introduces a completely new dynamic in the whole process of learning: each modification in the set of premises or in the framework of reference causes a radical change in the structure, dimensions, dynamics, etc. of the space of knowledge. By that process, entirely new and different knowledge, theories, interpretation patterns, etc. about reality become possible. (p. 92)
This allows us to adapt our mental models in the light of experience and information. An example of the structure of single and double loop learning is shown in Figure 2. To illustrate the difference between these models, consider a fall in enrollment numbers on a previously popular course. In single loop learning (i.e. identifying a mismatch between desired and actual outcome), faculty members may respond by increasing efforts to publicize the course in the media, with feeder schools, and with colleges, as well as working more closely with the local community. Fundamental beliefs are unchallenged but actions are amended to address the mismatch. An alternative response characterized by double loop learning would be to re-examine beliefs about the The Teaching of Mathematical Modeling
course, such as the suitability of its curriculum, the attractiveness of the subject area to potential students, and whether its current state is “fit for purpose�. This double-loop-learning response may result in radical change to the course offering. Organizations as well as individuals derive and amend their mental models through experience, observing, and interpreting the outcomes of their actions and decisions (Argyris and Schon, 1978; Bartunek, 1984; Levitt and March, 1988). In this sense, double loop learning requires the generation of new knowledge, insights, and intuitions by modifying existing models. Double Loop Learning and the Modeling Process We now can see how the mathematical modeling process can be placed within the framework of double loop learning. When we develop a mathematical model, there are two aspects to be considered. First, when we develop a model based on a set of assumptions derived from our current understanding of the problem situation, we effectively engage in single loop learning. The assumptions we have made determine the formulation of the model, the data requirements, and so on. Once the data has been collected, we solve the model and interpret this solution within the context
of the problem situation. This leads to model validation and verification considerations. The validation and verification process may indicate problems with the model, a mismatch between our expectations and real situation dynamics. In this case, it may be that the model has not been formulated correctly in terms of the assumptions, that the model contains errors in its formulation, or that the data used is unreliable or inappropriate. In any event, the model needs to be amended. Within the limitations of our current set of active assumptions about the problem situation, we seek to find a model that does not deviate from our expectations. This is a single loop learning process. When the model is decided to be valid, then we can begin the process of enrichment and elaboration, extending and developing the model by broadening our understanding of the problem situation, in terms of both the breadth and sophistication. This produces an amended set of working assumptions for the model requiring further development. This second looping process is double loop learning. It requires from the student not just the technical mathematical and statistical skills, but also the learning to learn skills that were described above. The mathematical modeling process is outlined in Figure 3.
Information Feedback
Comparison with the Real World
Single Loop Learning
Expectations, Desires, Decisions
Double Loop Learning
Personal Mental Models
Worldview, Strategy
Figure 2. Single and double loop learning – adapted from Sterman (2000).
Jon Warwick
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Assessment of verification and validation results.
Find and interpret the solution to the model.
Collect the appropriate data
Enrichment and Elaboration By Learning to Learn
Single Loop Learning
Appreciation of the problem situation, structure and complexity
Double Loop Learning Derive/amend or correct the mathematical formulation of the model
Assumptions made to determine the model structure, boundary etc.
Figure 3. The amended modeling process. To paraphrase Dooley (1999), the single loop learning phase can be described as “building the model right” whilst the double loop learning phase relates to “building the right model” (p. 13). This mathematical modeling process model (Figure 3) is richer than the conventional process model used with students. It allows the discussion of mathematical modeling to be extended to include elements related to the double loop learning aspect of the process. These are the difficult elements of the modeling process for both teachers and students. Yet these are just the skills that enable effective modeling and engage the students in the higher levels of learning as described by Bloom’s taxonomy (i.e. synthesis and evaluation). Now, returning to our example drawn from library management, my teaching experience suggests that students will initially adopt a variety of model forms often using analogy as recommended by Morris (1967). Common themes here are the conceptualization as either one of an inventory control problem or as a waiting line (queuing) problem. In the case of the inventory model, the copies of the title on the shelf are the stock being demanded (borrowed) by students and then immediately re-ordered. The average inventory level is a measure of the satisfaction level and lead times are assumed fixed initially, corresponding to an assumption that all books are kept for the full loan period and then returned promptly. 38
For the waiting line model, the service mechanism represents the copies of the title (one server for each copy) and average service time equates to the loan period. Actual borrowing times are assumed to be random in the basic waiting line model. The queue itself might represent reservations having been made for the title if it is not immediately available. In this case, the satisfaction level is related to the probability of finding idle servers. Calculations can also be made of average waiting times to get the book depending on the number of copies, the loan period, and the class size. Students are provided with some sample data, and then test their models. If necessary, they refine and correct any faults until they are satisfied with the results. This is iteration around the single-loop learning phase. The complexity of the situation is then increased gradually so that students will, at first, try to adopt single loop learning in order to accommodate any new information within their existing models. Eventually, they must consider broader and more complex issues that may require radically changed assumptions, significant new modeling and understanding, and, in the extreme, adopting a completely new model formulation. For example, I might begin by asking students to relax their assumption about borrowing times by allowing users to return their copy early or late according to some probability distribution. This modification can be built in to both models described above relatively easily but requires the students to The Teaching of Mathematical Modeling
research how to do this. As a result, their models become more complex, moving away from standard inventory or waiting line models into more specialized versions. A higher level of complexity is introduced by allowing feedback into the system. Students are asked to consider that demands for the title will not be regular but depend on the perceived likelihood of obtaining a copy in reasonable time. If many copies are available in the library (high satisfaction levels), then this encourages use of the library, increasing demand and eventually reducing satisfaction levels. Otherwise, if copies are never available, potential borrowers might go elsewhere (or buy it for themselves), lowering demand. Dealing with these new complexities requires students to engage with aspects of double loop learning. For example, they need to explore their existing model, ask further questions of the system, and revisit their assumptions and their understanding of the situation to incorporate these new factors. At this stage, students often get stuck dealing with the additional complexity and need help moving forward with double loop learning. I have been able to help students with this by using structured discussion. The Importance of Advocacy, Inquiry, and Reflection In this paper, I have argued that the skills that are most difficult for students to master are those related to the double loop-learning cycle in the modeling process. We have borrowed the notion of double loop learning from the field of organizational learning and, in completing this analogy, we can shed some light on how this sort of learning can be fostered in students. Senge (1990) argues that, in helping organizations undertake double loop learning, members of the organization should be able to combine advocacy and inquiry. Advocacy refers to the ability to solve problems by taking a particular view, making the appropriate decisions, and then gathering whatever support and resources are necessary to make things happen. Inquiry, on the other hand, is being open to questions, asking questions of others, inquiring into the reasoning of others, and expressing one’s own reasoning. Senge states: When both advocacy and inquiry are high, we are open to disconfirming data as well as confirming data–because we are genuinely interested in finding flaws in our views. Likewise, we expose our reasoning and look for Jon Warwick
flaws in it, and we try to understand others reasoning. (p. 200)
Thus, creative outcomes are far more likely as a consequence of using advocacy and inquiry. When working with a group of students modeling a complex situation, they should be encouraged to use advocacy and inquiry to challenge and explore modeling ideas. There are a number of guidelines proposed by Senge (1990) that, when used as prompts, can encourage students to explore the problem situation. For example, when advocating personal views the guidelines may be summarized as: •
Make your reasoning explicit.
•
Encourage others to explore your views.
•
Encourage others to provide different views.
Actively inquire into others’ views that differ from yours. Or, when inquiring into others’ views, try to: •
•
State any assumptions you are making about the views of others.
•
State the data on which your assumptions are based.
•
Ask what data or logic might change their view.
If there are disagreements, design an experiment or collect data that might provide new information. Discussion among the groups of students can be structured using these types of prompts, resulting in creative thinking about the way modeling should proceed (I have rarely seen aspects of creative thinking mentioned as part of mathematical modeling module descriptions!). In practice, students find this sort of debate and discussion difficult. They often need prompting from the teacher when group discussion has reached a dead end. Eventually, a greater understanding of the new problem is achieved. This usually leads to the amendment of the existing model, incorporating new assumptions and factors. As a result, students investigate stock control models with variable demand patterns or waiting line models with nonindependent arrival patterns. In this way, students develop further mathematical knowledge and research skills as well as engage in a cycle of learning. In extreme cases of paradigm shift, students will reject the existing model completely in favor of a •
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new formulation. This was the case with one group who rejected a simple inventory control model as too restrictive in favor of a model built using simple differential equations linking the number of copies available on the shelf with the number of potential borrowers. If the number of copies available on the shelf is low, then frustration will reduce the number of potential borrowers. In time, this causes the number of copies available to increase (reduced demand), leading to an increase in potential borrowers, and so on. These students found stable solutions to their model and investigated its sensitivity to changes in the loan period. Finally, we turn to reflection. Having looked at one way to encourage double loop learning, we need to then give students the skills to reflect individually on their performance, their learning, and how they can further improve their modeling skills. Thus, it is important to get students into the habit of reflecting on their work and the work of others. As King (2002) states, when undertaking reflection, “a variety of outcomes can be expected, for example, development of a theory, the formulation of a plan of action, or a decision or resolution of some uncertainty” (p. 2). Furthermore, “reflection might well provide material for further reflection, and most importantly, lead to learning and, perhaps, reflection on the process of learning.” (King, 2002, p. 2) Morris (1967) pointed out that reflective writing is sadly lacking in the professional literature. However, recent educational research has addressed reflective writing (see for example Moon, 2000) and how skills in reflection and reflective writing can be developed. King (2002), for example, suggests a model of the reflective process as having seven stages: Purpose, Basic Observation, Additional Information, Revisiting, Standing Back, Moving On, and either Resolution or More Reflection. Although UK Higher Education courses are expected to promote reflective thinking in many aspects of student’s work (Southern England Consortium for Credit Accumulation and Transfer, 2003), I would argue that it is a particularly crucial aspect of the mathematical modeler’s toolkit. Some General Conclusions Reflecting on my own teaching of mathematical modeling over the years has led to a number of changes to the way modules are designed, delivered, and assessed. When working with students (whether undergraduate or graduate) the following has been
40
useful in meeting some of the issues referred to in this article: •
Ensure that the content of the module includes some mathematical and statistical theory (as required by the particular program) but also sessions on creative thinking, learning to learn, and reflective writing.
•
Although some smaller models are used for the purposes of example, students are encouraged to work on a progressively more complex problem during the course of the module. This gives the opportunity for the development of successive models through enrichment and elaboration. Furthermore, it is helpful if the problem at hand can be modeled using a variety of approaches. This enables students to identify alternatives and to reflect upon the criteria for selection.
•
For longer projects, allow students to work in groups. Group meetings are held during class time so that the instructor can observe the discussion and try to move the students towards double loop learning as they seek to enrich their models. Questioning each other using the prompts discussed earlier can help here. Students take minutes of their meetings so that there is a record of the inquiry process.
Assessment is based upon the models students produce as a group as well as students’ individual reflection, both on the model development process and on their own learning. Each student each keeps a reflective log of his or her work during the module, commenting on the skills and lessons learned and identifying the skills needing further development. It is difficult to say whether students who complete a mathematical modeling unit with this type of structure are better modelers at the end. What I can say, from my experience, is that this structure engages students more readily than modeling taught as a more technically-oriented and solitary experience. The basic skills required of a mathematical modeler are probably little different now from when Morris (1967) originally wrote his guide. Technology, of course, has advanced enormously, but the individual’s ability to learn •
The Teaching of Mathematical Modeling
about, understand, and unpack a complex problem remains at the heart of modeling. References Argyris, C., & Schon, D. A. (1978). Organisational learning: a theory of action perspective. Reading, MA: AddisonWesley.
Levitt, B., & March, J. G. (1988). Organisational learning. Annual Review of Sociology, 14, 319–340. Moon, J. A. (2000). Reflection in learning and professional development. Abingdon, United Kingdom: Routledge Falmer. Morris, W. (1967). On the art of modelling. Management Science, 13(12), 707–717.
Bartunek, J. M. (1984). Changing interpretive schemes and organisational restructuring. Administrative Science Quarterly, 29, 355–372.
Ortenblad, A. (2004). The learning organisation: Towards an integrated model. The Learning Organisation, 11(2), 129– 144.
Bloom, B. S. (1956). Taxonomy of educational objectives, Handbook I: The cognitive domain. New York: David McKay.
Peschl, M. F. (2005). Acquiring basic cognitive and intellectual skills for informatics: Facilitating understanding and abstraction in a virtual cooperative learning environment. In P. Micheuz, P. Antonitsch, & R. Mittermeir (Eds.), Innovative concepts for teaching informatics (pp. 86–101). Vienna: Carl Ueberreuter.
Dooley, J. (1999). Problem solving as a double loop learning system. Retrieved on January 12, 2006, from http://www.well.com/user/dooley/Problem-solving.pdf Edwards, D., & Hamson, M. (2001). Guide to mathematical modelling. Basingstoke, United Kingdom: Palgrave Macmillan. Elby, A. (2001). Helping physics students learn how to learn. American Journal of Physics, 69(7), 54–64. Greany, T., & Rodd, J. (2003). Creating a learning to learn school. London: Network Educational Press. Hamming, R. W. (1997). The art of doing science and engineering: Learning to learn. Amsterdam: Gordon and Breach Science. Howarth, K. (1997, March). The teaching of thinking skills in physical education: Perceptions of three middle school teachers. Paper presented at the annual meeting of the American Educational Research Association, Chicago. King, T. (2002). Development of student skills in reflective writing. In A. Goody & D. Ingram (Eds.), Spheres of Influence: Ventures and Visions in Educational Development. Proceedings of the 4th World Conference of the International Consortium for Educational Development. Perth: The University of Western Australia. Knight, P. T. (1997, May). Learning How to Learn in High School History. Paper presented at ORD-Congress (Onderwijsresearchdagen), Leuven, Belgium. Kraft, D. H., & Boyce, B. R. (1991). Operations research for libraries and information agencies: Techniques for the evaluation of management decision alternatives. San Diego: Academic Press. Landsberger, J. (2005). Learning to learn. Retrieved on January 12, 2006, from http://www.studygs.net/metacognition.htm Lawrence, J. A., & Pasternack, B. A. (2002). Applied management science. New York: John Wiley and Sons.
Jon Warwick
Rawson, M. (2000). Learning to learn: More than a skill set. Studies in Higher Education, 25(2), 225–238. Senge, P. M. (1990). The fifth discipline: The art and practice of the learning organisation. New York: Doubleday. Southern England Consortium for Credit Accumulation and Transfer. (2003). Credit level descriptors for further and higher education. Retrieved on January 17, 2006, from http://www.seec-office.org.uk/SEEC%20FE-HECLDsmar03def-1.doc Sterman, J. (2000). Business dynamics: systems thinking and modelling for a complex world. Boston: McGraw-Hill Irwin. University of Luton. (2006). Effecting change in higher education. Retrieved on January 13, 2006, from http://www.effectingchange.luton.ac.uk/approaches_to_cha nge/ index.php?content=ol Waeytens, K., Lens, W., & Vandenberghe, R. (2002). Learning to learn: Teachers’ conceptions of their supporting role. Learning and Instruction, 12, 305–322. Warwick, J. (1992). Modelling by analogy: An example from library management. Teaching Mathematics and its Applications, 11(3), 128–133. Wright, J. (1982). Learning to learn in higher education. London: Croom Helm.
1
My intention is to use 'module' to mean part of a course of study so that a student studies several modules per year.
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The Mathematics Educator 2007, Vol. 17, No. 1, 42–44
Book Review… The View from Here: Opening Up Postmodern Vistas Ginny Powell Walshaw, M. (Ed.). (2004). Mathematics education within the postmodern. Greenwich, CT: Information Age. 254 pp. ISBN 1-59311-130-4 (pb). $34.95 The term “postmodern” has been used in many different ways by many different people. And that’s just fine with postmodernists. Those given credit for the creation of postmodernism cared little for the name, and today’s evangelizers feel no need to pin it down to one meaning. That is the point, after all. Postmodernism was born as a reaction against the “modernist project” of finding the one final answer to every question. The real world, postmodernists would say, is much more complex than that. But whether you consider yourself a postmodernist or not, you will find Mathematics Education within the Postmodern (edited by Margaret Walshaw, $34.95) an eye-opening and thought-provoking book. As postmodern pioneer Valerie Walkerdine says in the preface, the purpose of this volume is to “challenge accepted wisdoms” (p. vii) about mathematics, mathematics education, and mathematics educators. Up to now, “the post” has led to few insights into mathematics education, even as its contribution to other disciplines has grown.1 A volume such as this is a sign that this interesting perspective is growing in popularity and recognition, pulling up a seat at the table, and joining the fray. For those new to postmodernism, editor Walshaw provides a nice introduction in the first chapter, along with an explanation of modernism, for those who are not aware they are embedded in it. Accurately, though disturbingly, she describes the postmodern approach as “unsettling” and about “exploring tentativeness” (p. 3). There is an unapologetic lack of answers here: no “tips for teachers” (p. 222), as Cotton says in the final chapter. This book, like this approach, is about opening up new ways of thinking about matters we did not realize we needed to think about, things we thought Ginny Powell is an Instructor of Mathematics at Georgia Perimeter College, and a doctoral student in Mathematics Education at Georgia State University. She is interested in the teacher-student dynamic, especially in cross-cultural and tertiary settings, and in the fairness, or lack thereof, of standardized assessment.
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were fixed and decided, the “unthinkable.” I like to picture postmodernism as pointing out a new path I never noticed before, which invites endless exploration, but also carries possible dangers. This book is the fourth volume in the International Perspectives in Mathematics Education 2 series, and international certainly describes it; contributors hail from, or have worked in, Colombia, Brazil, the United States, Australia, Kiribati, Denmark, the United Kingdom, and New Zealand, though only the last two countries have multiple representatives. As a mathematics teacher in the United States, I found nothing that seemed lost in the translation across cultures. It was all relevant and recognizable, sometimes troublingly so. Organization of the Book After the introductory chapter, the book is divided into three roughly equal parts. The three chapters in “Part I: Thinking Otherwise for Mathematics Education” treat the broad subjects of how postmodernism might lead to new ways of thinking about mathematics itself, research in mathematics education, and the practice of mathematics education. “Part II: Postmodernism within Classroom Practices” includes four chapters attempting to show, with varying success, how a postmodern attitude has changed or could change the classroom. The final part, “Part III: Postmodernism within the Structures of Mathematics Education,” takes on teacher training, curriculum design, and assessment from the postmodern perspective. This structure seems arbitrary. Chapters from the first and last parts could have easily been put together, as all treat large issues in mathematics education. Then again, some of them treat classroom practices, and might have fit better in the second part. The chosen organization seems to echo that of the previous volume in the series rather than any logical arrangement. That Walshaw’s volume has one less chapter and is one Opening Up Postmodern Vistas
hundred pages shorter than the previous volume leads one to wonder if some of the contributions would not have been accepted if there had been more submissions. Perhaps in the future there will be more researchers willing and able to contribute to future mathematics education publications in the postmodern vein. The Big Picture Paul Ernest starts off the first part by taking mathematics to task for its failure to respond meaningfully to fundamental issues, or rather for responding by gobbling up each new paradox and going on as though nothing had happened. “Gödel’s Theorem did not even cause mathematics to break its stride as it stepped over this and other limitative results” (p. 17). Some would see that as a strength, but Ernest, consistent with the postmodern emphasis on deconstruction, would rather explore where those issues might take us. Also in this chapter, he discusses several postmodernists and pre-postmodernists, such as Lyotard, Foucault, and Lakatos, illuminating the origins of some of the basic ideas of postmodernism. Ernest gently points out how we sometimes have to unlearn what we thought we knew in order to learn something new, whether it’s “addition makes a bigger number” or “there is one best way to teach.” In the next chapter, Valero takes us into a classroom in Colombia to explore what we think we know about mathematics students. Adopting a postmodern attitude, she offers insight into the unreality of the “laboratory children” (p. 43) mathematics education researchers claim to have knowledge about. Real children are much more complicated, of course, and she discusses how we might better approach them. Even if some children are willing to make it easy for us by playing by the rules of mathematics and the classroom, she asks if that is really all we want for them. Finishing up Part I of the book, Neyland tackles ethics and what postmodern ethics might mean. He charges current, “modern” ethics with being “undesirable and illusory” (p. 56), leading to educational “reforms” like national curricula and standardized tests that seem to him flawed from the ground up. His goal is not to replace this state of affairs with a new set of unquestionable standards—an “objectively founded and universal ethical code is impossible to obtain” (p. 60)—but to explore other possibilities based on the individual self, possibilities that might lead to a “re-enchantment” (p. 60) with mathematics. Ginny Powell
Hands On The next section, on classroom practices, should be the most enjoyable for the neophyte postmodernist reader, if only for its concreteness. Those set afloat by the endless questioning of the first part will find something to hold onto here, as we see teachers and students interacting in recognizable ways. But soon we will be led to question what we thought we knew about such a familiar setting, as the authors point out the obvious-once-you-hear-them, shocking undercurrents of teacher-student relationships. Unfortunately, the first chapter in this section seems completely out of place in this volume. In it, Macmillan discusses interactions in preschool mathematics groups in Australia. Her occasional use of a word from the official postmodern lexicon (e.g., agency, discourse) cannot hide her essentially constructivist approach. She painstakingly systematizes everything and explicitly and unquestioningly accepts the conventions of the current modernist classroom. One wishes she had read and learned from Valero’s chapter above, or Hardy’s below. The rest of the second section, however, is filled with questions and new perspectives. In her piece, Hardy vividly dissects an “exemplar” teacher training video. Drawing on Foucault, she discusses the normalizing effect on students and teachers, asking how we might “choose to do otherwise” (p. 116). Speaking as though for the entire book, she hopes that “by working through alternatives, by exploiting the lack of stability of many of our professional notions, we might open up spaces from which we can counter ill-posed problems and look for sites of resistance” (p. 117). Editor Walshaw contributes a chapter to this section as well, bringing Lacanian psychoanalysis into the fray. Once again she acts as helper to the reader, defining and explaining constructivist and sociocultural theories of knowing and outlining Lacan for us, before bringing him to bear on a single student and that student’s interactions with her mathematics teacher. What she has to say about the idea of a “model” pupil will have you deconstructing all your notions about your relationship to your students as a teacher and your relationship with your teachers as a student. Cabral continues the Lacanian analysis by describing a classroom where postmodern ideas have already influenced practice. The result is, as promised, unsettling. She coins a new phrase, “pedagogical transference” (p. 142), to explain her ideas about how learning is affected by the unconscious, by feelings and moods. But even without adopting her terms, the 43
reader can come to the, by now familiar, space of constant questioning as they experience this new vision. The result is not necessarily the urge to run out and replicate her classroom, but the reader becomes aware of yet another space for change in her or his own teaching and learning.
way. His vision is clear, but the example he gives of his attempt to actually use his new assessment shows just how difficult change can be, as ten-year-olds demonstrate how embedded they already are in the world that standardized testing has wrought.
More Big Issues
This book is not an exhaustive treatise on mathematics education or on postmodernism. It is only a beginning, a step towards possible change. While the authors come at postmodernism from different angles and through the work of different thinkers (Lacan, Foucault, Lyotard, Deleuze, etc.), they all share one goal: to make us think about mathematics education differently. They remind us that there is no one “right” way to teach or learn mathematics. Through their own examples, they inspire us to seek new insights of our own. As Walshaw says in the introductory chapter, “Ultimately it is the hope of all the authors that the ongoing engagement will mark a fruitful and productive convergence between mathematics education and postmodernism” (p. 11). This book would be an interesting and useful read for anyone involved in the teaching or learning of mathematics at any level, kindergarten through college, and for administrators and policymakers who are in a position to make broad decisions about mathematics education. It can be read all at once, or the reader can use the index or Walshaw’s excellent introductions in Chapter 1 to find something applicable to their own situation. Whether or not anyone embraces postmodernism as a result of reading this book is irrelevant. As long as new questions are asked, progress has been made toward a more flexible system of mathematics education.
The last part of the book returns to confrontation with current large issues in mathematics education. In their article, Brown, Jones, and Bibby search for insight into the thinking of elementary school teacher trainees. They find nearly universal fear and lack of facility with mathematics. This chapter isn’t “teacher bashing,” nor is it a panacea, but rather it exposes a reality that needs to be addressed. Brown, Jones, and Bibby ask how a teacher’s identity and feelings about her- or himself as a mathematics learner and teacher affect future students, and how we might change those feelings for the betterment of all. Meaney, in the next chapter, takes us back a step to look at curriculum design. Specifically, she looks at how she, as an outside consultant brought in to facilitate the development of a mathematics curriculum among the Mäori, negotiated the many power relations inherent in the situation, and how she might do it differently next time. There is much here about crosscultural pitfalls, but also about the dangers of top-down decision making. Once again, she asks only that we begin to think about how and why things are done, and what alternatives are possible. “Do you ever think about what you don’t think about?” (p. 202) Fleener asks provocatively at the beginning of her chapter. The sole American contributor, she draws on Deleuze and Guattari, as well as popular movies, to question the most basic structures of mathematics education. “Why do we teach division after multiplication? …Why is mathematical aptitude considered evidence of intelligence? … Why do we teach 400-year-old algebra and calculus and 2500-year-old geometry?” (p. 202). Echoing Ernest, she castigates the mathematical community for ignoring the possibilities that foundational problems create, expressing a wish that we “celebrate rather than bemoan the loss of certainty and structure” (p. 203) and help our students “fall in love” (p. 205) with mathematics, not just regurgitate it. In the final chapter, Cotton challenges current assessment practices and how they can take on a selfperpetuating life of their own. Drawing on Lyotard, he deconstructs assessment as it is currently practiced in the UK and then sets forth his own criteria for a better 44
Putting It All Together
References Linn, R. (1996). A teacher's introduction to postmodernism. Urbana, IL: National Council of Teachers of English. Walshaw, M. (Ed.). (2004). Mathematics education within the postmodern. Greenwich, CT: Information Age. 1
For example, in 1996 the National Council of Teachers of English (NCTE), in the NCTE Teacher's Introduction Series, published the book A Teacher's Introduction to Postmodernism (Linn, 1996). 2 Volume 1 was Multiple Perspectives on Mathematics Teaching and Learning (2000), edited by Jo Boaler; Volume 2 was Researching Mathematics Classrooms: A Critical Examination of Methodology (2002), edited by Simon Goodchild and Lyn English; and Volume 3 was Which Way Social Justice in Mathematics Education? (2003), edited by Leone Burton, who is also series editor.
Opening Up Postmodern Vistas
CONFERENCES 2007, 2008‌
AMESA 13th Annual National Congress
Mpumalanga, South Africa
July 2-6, 2007
Seoul, South Korea
July 8-13, 2007
Salt Lake City, UT
July 29-August 2, 2007
First Joint Meeting with the Polish Mathematical Society http://www.ams.org
Warsaw, Poland
July 31-August 3, 2007
GCTM Georgia Council of Teachers of Mathematics Annual Conference
Rock Eagle, GA
October 17-19, 2007
Indianapolis, IN
November 1517, 2007
Lake Tahoe, NV
October 25-28, 2007
San Diego, CA
January 6-9, 2008
Tulsa, OK
January 24-26, 2008
Oklahoma City, OK
March 6-8, 2008
Salt Lake City, UT
April 7-9, 2008
Salt Lake City, UT
April 7-12, 2008
New York, NY
March 24-28, 2008
http://www.amesa.org.za/AMESA2007/ PME-31 International Group for the Psychology of Mathematics Education http://pme31.org JSM of the ASA Joint Statistical Meetings of the American Statistical Association http://www.amstat.org/meetings/jsm/2007/
http://www.gctm.org/ SSMA School Science and Mathematics Association http://www.ssma.org PME-NA North American chapter International Group for the Psychology of Mathematics Education http://pmena.org MAA-AMS Joint Meeting of the Mathematical Association of America and the American Mathematical Society http://www.ams.org AMTE Association of Mathematics Teacher Educators http://amte.net RCML Research Council on Mathematics Learning http://www.unlv.edu/RCML/conference2007.html NCSM National Council of Supervisors of Mathematics http://www.ncsmonline.org/ NCTM National Council of Teachers of Mathematics http://www.nctm.org AERA American Education Research Association http://www.aera.net
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The Mathematics Educator (ISSN 1062-9017) 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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The Mathematics Educator strives to provide a forum for collaboration of mathematics educators with varying levels of professional experience. The work presented should be well conceptualized; should be theoretically grounded; and should promote the interchange of stimulating, exploratory, and innovative ideas among learners, teachers, and researchers.
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An electronic copy is required. (A hard copy should be available upon request.) The electronic copy may be in Word, Rich Text, or PDF format. The electronic copy should be submitted via an email attachment to tme@uga.edu. Author name, work address, telephone number, fax, and email address must appear on the cover sheet. The editors of The Mathematics Educator use a blind review process therefore no author identification should appear on the manuscript after the cover sheet. Also note on the cover sheet if the manuscript is based on dissertation research, a funded project, or a paper presented at a professional meeting.
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In this Issue, Guest Editorial… In Pursuit of a Focused and Coherent School Mathematics Curriculum TAD WATANABE Being a Mathematics Learner: Four Faces of Identity RICK ANDERSON A Quilting Lesson for Early Childhood Preservice and Regular Classroom Teachers: What Constitutes Mathematical Activity? SHELLY SHEATS HARKNESS & LISA PORTWOOD Graduate Level Mathematics Curriculum Courses: How Are They Taught? ZELHA TUNÇ-PEKKAN Some Reflections on the Teaching of Mathematical Modeling JON WARWICK Book Review… The View from Here: Opening Up Postmodern Vistas GINNY POWELL