____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 19 Number 2

Winter 2009/2010

MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff

A Note from the Editors

Co-Editor Allyson Hallman Catherine Ulrich

Dear TME Readers, We are pleased to present you with this concluding issue of The Mathematics Educator’s Volume 19. This issue displays the variety of article types published in TME. Although all draw upon the work of researchers, the first is a position paper, the second a more traditional research report, the third a literature review focusing on theoretical frameworks of mathematics educators, and the last a research report analyzing issues of methodology. Although we did not intend for this issue to have a particular theme, as you read through the articles presented in Volume 19 Issue 2, you will see that the importance of the mathematical community comes through in each article. In the first article, mathematics is argued to be an inherently social endeavor. In the second article, the impact of student teachers’ interactions with mentors on their attitudes towards technology in the classroom is underlined. In the third article, the implications of viewing mathematics as a social construction are fleshed out. And in the fourth article, the effect of communication through letter writing on both secondary students and preservice teachers is explored.

Associate Editors Zandra deAraujo Eric Gold Erik D. Jacobson Laura Singletary Advisors Dorothy Y. White

MESA Officers 2009-20010 President Zandra deAraujo Vice-President Laura Singletary Secretary Laura Lowe Treasurer Anne Marie Marshall NCTM Representative Allyson Hallman Undergraduate Representative Hannah Channell Derek Reeves

In lieu of a guest editorial, we have decided to open this issue with a position paper by Thomas Ricks. He offers a challenge to return to the intellectual and social roots of mathematics activity in the classroom. His article would make an excellent reading for preservice or inservice teachers who do not see the importance of creativity and communication in school mathematics. The second article, by Asli ÖzgünKoca, Michael Meagher and Todd Edwards, is a report of research on the development of technological pedagogical and content knowledge (TPACK). TPACK is an area of pedagogical content knowledge that is sure to receive more attention in the coming years as increasingly sophisticated technology continues to offer new possibilities and challenges in the classroom. The third article, by Kimberly White-Fredette, brings a sense of the variety of theoretical frameworks used in mathematics education. An awareness, not only of one’s own perspective, but also of the perspectives of others, is certainly important for all mathematics educators. And, finally, Anderson Norton and Zachary Rutledge have written a follow-up article to “Preservice Teachers’ Mathematical Task Posing: An Opportunity for Coordination of Perspectives,” published in the first issue of Volume 18. Their original article focused on the theoretical perspectives of the researchers when conducting a study on preservice teachers’ letter writing exchanges with secondary students and on the results that these different perspectives helped them draw out of the data. In their followup article, they focus on the methodological obstacles when measuring student engagement. We would like to thank our associate editors and authors for all their hard work and dedication. We hope you enjoy reading this issue as much as we all have enjoyed working on it. Catherine Ulrich & Allyson Hallman TME Co-editors

105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

tme@uga.edu http://math.coe.uga.edu/TME/TMEonline.html

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

Winter 2009/2010

Volume 19 Number 2

Table of Contents 2 Mathematics Is Motivating THOMAS E. RICKS 10 Preservice Teachers’ Emerging TPACK in a Technology-Rich Methods Class S. ASLI ÖZGÜN-KOCA, MICHAEL MEAGHER, & MICHAEL TODD EDWARDS 21 Why Not Philosophy? Problematizing the Philosophy of Mathematics in a Time of Curriculum Reform KIMBERLY WHITE-FREDETTE 32 Measuring Task Posing Cycles: Mathematical Letter Writing Between Preservice Teachers and Algebra Students ANDERSON NORTON & ZACHARY RUTLEDGE 46 Upcoming conferences 47 Submissions information 48 Subscription form

© 2010 Mathematics Education Student Association All Rights Reserved

The Mathematics Educator 2009/2010, Vol. 19, No. 2, 2–9

Mathematics Is Motivating Thomas E. Ricks Mathematics is motivating; at least, it should be. I argue that mathematical activity is an inherently attractive enterprise for human beings because as intellectual organisms, we are naturally enticed by the intellectual stimulation of mathematizing, and, as social beings, we are drawn to the socializing aspects of mathematical activity. These two aspects make mathematics a motivating activity. Unfortunately, the subject that students often encounter in school mathematics classes does not resemble authentic mathematical activity. School mathematics is characterized by the memorization and regurgitation of rote procedures in isolation from peers. It comes as no surprise that many students have little motivation to continue mathematics study because it lacks intellectual and social appeal. I suggest several practical changes in school mathematics instruction that are drawn from the literature. These changes will lead to instruction that more readily engages students with the subject because they are rooted in the intellectually and socially appealing aspects of mathematics.

Mathematics is motivating. Or at least, it should be. Mathematical activity is an inherently attractive enterprise for human beings because of its intellectual and social aspects. This may be difficult to believe, especially when “so many people find mathematics impossibly hard" (Devlin, 2000, p. 1) and many openly admit strong dislike for the subject (Paulos, 1988). Certainly, critics might argue, a few gifted individuals might have a special inclination toward mathematical study. But, can mathematics be appealing for everyone? I claim that it can be; mathematics has the potential to be interesting for everyone because it is an intellectual and social endeavor. In the following sections, I detail what is meant by authentic mathematical activity, describing both its intellectual and social aspects. Comparing authentic mathematical activity to typical school mathematical activity, I suggest ways that teachers can draw on the intellectual and social aspects of mathematical activity to motivate and engage students in the study of mathematics. Mathematical Activity Because “mathematics is a woefully misunderstood subject” (Devlin, 2000, p. 3), for the purposes of this article I define authentic mathematical activity, or mathematical activity, to be what Dr. Thomas E. Ricks earned his PhD in Mathematics Education from the University of Georgia in 2007. He works as an Assistant Professor in the Elementary Education division of the Department of Educational Theory, Policy, and Practice at Louisiana State University. Currently, he is collaborating with LSU colleagues on a federal grant for the university’s preeminent undergraduate technology-assisted mathematics program.

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mathematicians do when they do mathematics. I use examples from the life of Stanislaw Ulam, a Polish mathematician, to describe authentic mathematical activity. Not as well-known as Euler, von Neumann or Einstein, Ulam was a rather ordinary mathematician. Ulam (1976) humbly stated, “after all these years, I still do not feel much like an accomplished professional mathematician” (p. 27). Much of Ulam’s descriptions of his work focused on the people he met, was inspired by, and worked with. Many of the experiences he reported illustrate intellectual and social aspects of mathematics. Intellectual Aspect of Mathematics Mathematical activity is the most intellectual endeavor of all the sciences. Mathematics is a creation of the brain... Mathematicians …work …without any of the equipment or props needed by other scientists…. mathematicians can work without chalk or pencil and paper, and they can continue to think while walking, eating, even talking. This may explain why so many mathematicians appear turned inward [which is] quite pronounced and quantitatively different from the behavior of scientists in other fields…. I have spent… on the average two to three hours a day thinking…. Sometimes… I would think about the same problem with incredible intensity for several hours without using paper and pencil. (Ulam, 1976, p. 53)

Since mathematics is “a thinking, flexible subject” (Boaler, 1999, p. 264), mathematical activity is

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characterized by a variety of mental methods, such as cognitive wrestle and creativity. Cognitive Wrestle Mathematicians wrestle with cognitively demanding problems that have no clear solution path. Such wrestle is invigorating, and “most mathematicians begin to worry when there are no more difficulties or obstacles” (Ulam, 1976, p. 54). Mathematicians rarely make progress at steady rates; rather, they struggle with little apparent progress, and then major strides are made suddenly. The mathematician Banach “worked in periods of great intensity separated by stretches of apparent inactivity” (Ulam, p. 33). Ulam described the importance of “‘subconscious brewing’ (or pondering) [which] produces better results than forced, systematic thinking” (p. 54), being “a discontinuous process. Nothing, nothing, at first, and suddenly one gets …it” (p. 70). Part of this cognitive wrestle involves exploring alternative possibilities through mental experimentation. I always preferred to try to imagine new possibilities rather than merely to follow specific lines of reasoning or make concrete calculations. …Forcing oneself to persist in a logical exploration becomes a habit after which it ceases to be forcing since it comes automatically. (Ulam, 1976, p. 54)

Creativity Mathematicians are creative by generating mathematical content; they do this by posing and solving problems. During this process they create the very fabric of mathematics, weaving a thread that connects to the work of others. For example, when only 25, Ulam (1976) “established some results in measure theory which soon became well known [by solving] certain set theoretical problems attacked earlier by Hausdorff, Banach, Kuratowski, and others” (p. 55). In turn, the Russian Besicovitch solved a problem posed many years earlier by Ulam. These events also illustrate that problem posing is another part of creative mathematical generation. Devlin (2000) claimed that all people, everywhere, have “a mind for mathematics" (p. 1), that every human being with a functioning brain has “an innate facility for mathematical thought” (p. xvi). The intellectual aspects of mathematics, such as cognitive wrestle and creative generation, are fundamental attributes of our species: This helps explain why all human cultures mathematize the world. The predisposition for patterned thinking is even seen in newborns (Dehaene, 1997).

Researchers know that “large parts of the brain are active when a person is doing mathematics" (Devlin, 2000, p. 12). Because we are intellectual beings, the intellectual appeal of mathematics makes it naturally enjoyable; people’s brains like doing mathematics. Mathematics is by far children's favorite subject in school, at least well into the fourth grade (National Council of Teachers of Mathematics [NCTM], 2000). Social Aspect of Mathematics Besides its intellectual character, doing mathematics is also highly social. Every mathematician works within a mathematical community. Mathematicians are social on both a local and global scale. Ulam, in particular, described communication and collaboration. Communicating Mathematicians are engaged in constant communication, in part to help them learn more mathematics. This may entail reading about mathematics, listening to lectures, or discussing mathematical ideas with knowledgeable others. Ulam was influenced by mathematicians’ books at an early age, such as Sierpinski’s Theory of Sets, Steinhaus’s What is and What is Not Mathematics, and Poincaré’s La Science et l’Hypothèse, La Science et la Méthode, La Valeur de la Science, and Dernières Pensées (1976, p. 21). Kuratowski was an early teacher of Ulam who made a formidable impression, and, in part, was responsible for starting him on a career in mathematics. Teachers and mentors play a significant role in mathematicians’ development. Many famous mathematicians bring to mind their mentors: Euler as a student of Bernoulli; Ramanujan as a student of Hardy; and Dedekind and Riemann as students of Gauss. Mathematicians recognize the benefit that flows from sharing and networking (Davis & Simmt, 2003). Ulam (1976) described how he would engage in mathematical discussions with friends and colleagues. During a break while attending an International Mathematical Congress, he got lost in the nearby woods, and bumped “into Paul Alexandroff and Emmy Noether [who were] walking together [among the trees] and discussing mathematics” (p. 46). The view of an isolated mathematician working long hours alone in the office with little interaction is almost everywhere false (Wiles’ work on the Fermat theorem being a notable exception). Much of the … historical development of mathematics has taken place in specific centers [or] a group in which mathematical activity flourished. 3

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Such a group possesses more than just a community of interests; it has a definite mood and character in both the choice of interests and the method of thought. Epistemologically this may appear strange, since mathematical achievement, whether a new definition or an involved proof of a problem, may seem to be an entirely individual effort, almost like a musical composition. However, the choice of certain areas of interest is frequently the result of a community of interests. Such choices are often influenced by the interplay of questions and answers, which evolves much more naturally from the interplay of several minds. (p. 38)

The work of a mathematician incorporates his or her surrounding influences. For example, Ulam (1976) also wrote that “most of my mathematical work was really started in conversations with Mazur and Banach” (p. 33, emphasis added). Even gatherings in a local café provided opportunities for sharing and discussing mathematics. A large notebook was permanently kept in the café and brought out by a waiter upon demand; it was the central repository of the group’s ideas. Ulam later translated the notebook and “distributed it to many mathematical friends in the United States and abroad” (pp. 49–51). Collaborating Beyond communicating about mathematics, active collaboration is also a natural part of mathematicians’ sociality. Ulam (1976) worked with many distinguished mathematicians during his career; “Collaboration [with Mazur and Banach] was on a scale and with an intensity I have never seen surpassed, equaled, or approximated anywhere—except perhaps at Los Alamos during the war years” (p. 34). Ulam said that upon his arrival in the United States, he and Borsuk “started collaborating from the first… my first publication in the United States ….was a joint paper with Borsuk” (p. 41). Later, he said, “a whole series of papers which we [Steinhaus and I] wrote jointly came from …collaboration” (p. 43). Ulam also worked with John von Neumann. The joint work of mathematicians results in mutually accepted definitions, terms, strategies, methods, and algorithms. From parking lots to offices, academic lunchrooms to conference halls, mathematicians scribble and get stuck, share questions and solution attempts, backtrack and refine, reattempt and debate; they question, raise counterexamples, reason, argue, collectively justify, and develop communal metaphors (Polya, 1945). As such, many believe mathematics, as a domain, transcends any 4

individual perspective (Boaler, 1999; Davis & Simmt, 2003; Devlin, 2000). It is not a static knowledge domain––an external thing to be internalized by a learner––but rather a socially created, culturally dependent, fallible domain (Ernest, 1990). Mathematics exists due to the collective actions of many people over thousands of years. It belongs to no one and yet is accessible to all; it is a constant, communal, and humanistic creation (Romberg, 1994). Great discoveries by many individuals and groups have woven the tapestry of current mathematical thought; people like the Pythagoreans, the Arab algebraists, Cardano’s band, and Newton and Liebniz have all contributed their threads. As Leopold Kronecker, a nineteenth century mathematician, remarked, “God made the integers, all else is the work of man" (quoted in Devlin, 2000, p. 15). All mathematics–– fundamental axioms, appropriate terminology, conventional representations, mathematically valid propositions––is socially driven, “a cultural product” (Ernest, 1990). From this perspective, mathematics is much more than numbers or computations; it emerges through correspondence, questions, and group deliberations. Inherent Mathematical Appeal I am not alone in believing that mathematics can be interesting for everyone. The authors of the NCTM Standards (2000) opine that mathematics is a meaningful, richly rewarding subject that all can learn and enjoy. Additionally, when given the opportunity to engage in meaningful mathematical tasks that maintain their cognitive integrity, students not only tolerate mathematical work, but report satisfaction and enjoyment (e.g., Boaler, 1999). These findings are not exclusive of any particular personality or culture. In addition, I have seen ample evidence to suggest that students, either high or low achieving, when allowed to engage in mathematics, are drawn to the activity (Ricks, 2007). There is something intrinsically motivating in the subject. School mathematics can share this attraction if students are able to engage in authentic mathematical activity. Unfortunately, “most people do not know what mathematics is” (Devlin, 2000, p. xvii), perhaps because they have not experienced authentic mathematical activity and are thus dissuaded from further mathematics study. School mathematics is characterized by the memorization and regurgitation of rote procedures in isolation from peers (Burton, 2004; Stigler & Hiebert, 1999). Therefore, it comes as no surprise that, devoid of its intellectual and social

Thomas E. Ricks

appeal, mathematics is not motivating for many students and that many do not continue formal mathematics study past high school. A corrective possibility is to harness the intellectual and social potential of mathematics activity; allowing students to engage in mathematical activity in their own classrooms affords simple, straightforward options to improving mathematics instruction by returning to root motivational aspects of the subject. The Lack of Intellectual and Social Appeal in School Mathematics School mathematics is characterized by learning definitions and practicing procedures (Stigler & Hiebert, 1999), activities that lead to intellectual boredom. The essential attributes of mathematical activity—cognitive struggle and creative generation— are absent. “The questions people [mathematicians] worried about and the struggles they went through trying to answer them almost never appear [in school mathematics]; instead we see the results of the struggles, neatly packaged into pieces of boxed text” (Cuoco, 2001, p. 169). School mathematics is, quite bluntly, an intellectual wasteland, a pseudomathematics. Richards (1991) describes the intellectually stagnating initiation-reply-evaluation sequence as the common form of classroom interaction. No wonder students are confused; no wonder they avoid further mathematics study! Intellectual Lack in School Mathematics Mathematics teachers often view their job as showing “a few standard facts and algorithms” to students, and, later, “supervis[ing] some drill and practice” (Romberg, 1994, p. 314). Students are expected only to memorize the various rules and procedures the teacher demonstrates (Boaler, 1999; Davis, 1994): Independent thought is not an expectation. The intellectual possibilities for “relatively sophisticated levels of mathematical reasoning, well beyond what is typically thought of as appropriate for primary school mathematics" (Yackel, 2000, p. 20) are rarely met. The current level of intellectual engagement in learning school mathematics pales in comparison to what could happen if children were allowed to think things through for themselves (Davis, 1994). School mathematics is also not viewed as a creative endeavor. The curriculum is set, the teacher and textbook are the authorities in classrooms. There is no room for questioning, no room for exploration, no

room for experimentation. “In many schools, mathematics is perceived as an established body of knowledge that is passed on from one generation to the next. Instead of seeing [theorems, formulas, and methods of mathematics] as the products of doing mathematics, these artifacts are seen as the mathematics” (Cuoco, 2001, p. 169). Said Burton (2004): It has long been my opinion that the mathematics experienced by students in formal education, and the ways in which it is encountered, offer explanations for [the] decline in interest. Public interest books about mathematics are readily bought so it cannot be that people have no wish to engage with mathematics…. once students make a choice to study mathematics, many of them report experiences that are not conducive to holding them in the discipline. The same pattern holds whether they are studying mathematics at school, as a preuniversity choice, at university as undergraduates or even at doctoral level. (p. 4, emphasis added)

Contrasting this intellectually diluted school mathematics with the work of mathematicians is enlightening: “as a result of such limited experiences, many students are prejudiced against the broader, more interesting aspects of mathematics” (Romberg, 1994, p. 290). Social Lack in School Mathematics Similarly, school mathematics deprives students of the natural socializing appeal of mathematical activity. Students are expected to sit quietly and listen to the teacher with little to no interaction with others (Davis, 1994). The necessary mathematical interactions needed for full mathematical activity are absent. This severely curtails children’s abilities to make “judgments about what is acceptable mathematically, for example, with respect to mathematical difference, mathematical sophistication, mathematical inefficiency, mathematical elegance, and mathematical explanation and justification,” and it deprives them of autonomous “mathematical power” (Yackel, 2000, p. 21). In school mathematics, students usually do problem sets alone, do homework alone, and take quizzes and tests in isolation. When do they have a chance to engage in social mathematical work? School mathematics perpetuates beliefs that heterogeneous class makeup is an “obstacle to effective teaching” and that “the tutoring situation is best, academically, because instruction can be tailored specifically for each student” (Stigler & Hiebert, 1999, p. 9). When they do work in groups, students usually do only superficial 5

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computational exercises, even though the group could enable individual students to overcome personal barriers in the problem-solving process. Rarely is mathematical understanding created by the group as a whole. School mathematics neglects the social aspects that make mathematics so appealing—the ability to participate in larger mathematizing collectives working toward shared meanings and common objectives. Classroom dialogue is characterized by univocal “number talk” rather than socially intertwined, mutually specified, dialogic functioning (Davis & Simmt, 2003; Richards, 1990, Wertsch & Toma, 1995). Such absences of mathematical activity in school mathematics classrooms led one researcher studying U.S. mathematics lessons to bemoan, “I have trouble finding the mathematics [in these lessons]” (quoted in Stigler & Hiebert, 1999, p. 26). The great ironic tragedy is that most students who claim to have little motivation to study mathematics have never really experienced authentic mathematics. To deal with a lack of motivation, non-mathematical strategies are often employed to maintain students’ attention in mathematics classes. However, such strategies do not work. Some examples are: (i) interrupting mathematics instruction to talk about a more interesting subject, (ii) using candy or prizes to excite students, (iii) presenting the lesson in the context of competitive games, or (iv) letting students work together on projects where the focus often shifts from mathematical ideas to creating attractive displays (Stigler & Hiebert, 1999). Making School Mathematics Authentic Mathematics learning does not require games, dramatic teacher presentations, external motivators, or even connections to real world activities, all common suggestions to motivate students in traditional mathematics classrooms. It requires instead a return to the core components of mathematics. The reason motivation is an issue at all is that current school mathematics is neither intellectual nor social; by focusing on “habitual, unreflective, arithmetic problems” (Richards, 1991, p. 16), school mathematics strips from the subject the very constituents that provide for meaningful mathematical experiences. Slight, subtle changes in the way mathematics is taught can significantly increase students’ motivation to learn mathematics. For example, teachers can engage students in (1) cognitively challenging (Stein, Smith, Henningsen, & Silver, 2000) and (2) socially oriented activities in 6

mathematics classrooms (Stein & Brown, 1990). Students can then be involved in the genesis of mathematical ideas in a group setting. These two components make mathematics a motivating activity: When a class follows an inquiry tradition of instruction, many of the ‘tasks’ that children engage in are tasks that they set for themselves as they attempt to reason about the dynamic interactions that occurs in small group interactions and in whole class discussions. In a real sense, by choosing what they reflect on, students individualize instruction for themselves in ways that only they can do. (Yackel, 2000, p. 20)

We can see how this understanding of what makes mathematics motivating is reflected in current trends in mathematics education. For example, the common recommendation to structure lessons around central challenging tasks (Stigler & Hiebert, 1999) would support the intellectual and social requirements of mathematical activity. A teacher’s ability to recognize, modify, or develop a central activity that is cognitively demanding, while, at the same time, maintaining the intellectual integrity of the task as students struggle, allows the mathematics to retain its intellectual vitality. Students’ mathematical experience would be less likely to degrade into mimicry, repetition, and boredom. Jointly working on a central task also provides for more robust whole-class discussions; the class shares the common foundations necessary to truly collaborate on mathematical work. The class can begin to emerge as a mathematical community through developing a common vocabulary and engaging in collective sense making. Ball & Bass (2003) write: Making mathematics reasonable is more than individual sense making. Making sense refers to making mathematical ideas sensible [and] comprises a set of practices and norms that are collective, not merely individual or idiosyncratic… That an idea makes sense to me is not the same as reasoning toward understandings that are shared by others with whom I discuss and critically examine that idea toward a shared conviction. (p. 29)

A second trend in mathematics education— relinquishing ‘mathematical authority’ (Cobb, Yackel, & Wood, 1992; Smith, 1996)—also respects the intellectual and social dimensions of mathematical activity. By purposely removing herself or himself as the source of mathematical truth, the teacher enables students to collectively develop mathematical knowledge. In fact, most current trends in mathematics education respect and enable the intellectual or social

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dimension of mathematics. For instance, establishing sociomathematical norms—establishing an environment for shared ways of mathematical sensemaking and making explicit appropriate means of questioning, justifying, and reasoning—enables the social aspect of professional mathematicians’ work in the classwork (Cobb, 1994; Cobb, Wood, & Yackel, 1990;Yackel, 2000; Yackel, Cobb, & Wood, 1999). The trend toward whole-class discussions, where the teacher orchestrates a respectful space for students to discuss and question each others’ thinking, (Cobb, Wood, & Yackel, 1990; Yackel, Cobb, & Wood, 1999); joint mathematizing or encouraging collaboration, where students combine their mathematical efforts (Grossman, et al., 2001; Ricks, 2007); and equalizing participation of students so particular students do not dominate class discussions (Noddings, 1989; Webb, 1995) are recommendations that attempt to catalyze the types of social interactions characteristic of mathematicians’ work. Eliciting students’ mathematical diversity, the teacher selection of different manners of student approaches and solutions to mathematical problems (Bennie, Olivier, & Linchevski, 1999; Borba, 1992; Linchevski, Kutscher, & Olivier, 1999; Linchevski, Kutscher, Olivier, & Bennie, 2000a, 2000b; Smith, 1992), helps students experience the range of creativity that is a hallmark of mathematical problem solving. Emphasizing dialogic functioning (Wertsch & Toma, 1995), when students think about each others’ thinking, also contributes to the intellectual work needed to make sense of others’ ways and means of operating mathematically. The ultimate goal of mathematics instruction should be for students to become “lifelong mathematics learners” (Cuoco, 2001, p. 169). What might this look like in a classroom? Although this article is not the place for delving into the specifics of intellectual and social rejuvenation of mathematics classes, I do offer three categories of examples at the levels of task, lesson, and overall curriculum to whet the reader’s appetite (e.g., similar to Usiskin, 1998). Mathematical Tasks The most obvious place for change is at the level of mathematical activities. Stein et al. (2000) detailed a task framework for measuring a task’s cognitive demand, ordered from lowest to highest: memorization, procedures without connections, procedures with connections, and doing mathematics. For them, doing mathematics is a cognitively demanding activity “requiring complex and non-

algorithmic thinking” where no rehearsed approach is used (p. 16), as opposed to memorization, defined as the recall or reproduction of previously learned material, or procedures, defined as emphasizing algorithms to produce correct answers with little explanation of thinking. They consider doing mathematics tasks to be the most beneficial student activities, and their study details several classroom case studies of teachers iteratively attempting to setup and implement doing mathematics tasks appropriately. Definitions can be developed through investigating, rather than having definitions presented by the teacher at the beginning of lessons as though they were axiomatic. For example, developing a definition for trapezoid could lead to intriguing intellectual and social possibilities. As there is no accepted standard definition for trapezoid (Wolfram, 2010). Are trapezoids quadrilaterals with at least one pair or only one pair of parallel sides? There are tantalizing ramifications of this choice, such as implications for trapezoidal classification as a subset or superset of parallelograms, how the trapezoid and parallelogram area formulas relate, etc. Students can explore how the taxonomy of other shapes change with similar definition modification. The teacher can mediate a class discussion about which definition is better and why, and the class can then adopt this specific definition in their future work. Mathematical Lessons Teachers can also structure their lessons to accentuate the intellectual and social aspects of mathematics. Instead of a lesson structured around teacher presentation, demonstration, and modeling of pre-packaged mathematical procedures (Cuoco, 2001), the teacher can pose challenging tasks, and then allow individual and/or group work followed by whole class discussions (Yackel, 2000). There are many examples of such lessons available for teachers, e.g., videos from the Annenberg collection (WGBH Boston, 1995). Another excellent example is the released 1995 Trends in Mathematics and Science Study (TIMSS) videos (NCES, 2003), including one of an eighth grade Japanese geometry lesson that revolved around a single task of dividing land equally. This lesson’s unified structure is particularly powerful when juxtaposed against the piecemeal problem review and teacher lecture of an eighth grade U.S. geometry lesson in another TIMSS video. Lessons from Deborah Ball’s 1989 third grade classroom offer further examples of rich intellectual and social mathematics lessons (Ball, 1991). In the 7

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lesson known as Shea’s Numbers, a student claims that the number six is both an even and an odd number. Rather than telling Shea that he is mistaken about the definitions (a typical teacher’s response), Ball instead allows him to fully explain his reasoning. Doing so allows others to value his argument, and to recognize that certain even numbers (2, 6, 10,…, 2 + 4n) have an odd number of twos; these the class calls Shea numbers. Mathematical Curriculum Mathematics teachers can also structure their units, courses, and curriculum to more accurately mimic the work of mathematicians. Examples of this abound, such as Moses’ (2001) Algebra Project where students learn algebra from real-life experiences, the Moore Method (Corry, 2007) of building mathematical structure from a small set of teacher-provided axioms, and Anderson’s (1990) mathematics courses “emphasizing that ordinary people create mathematical ideas and ‘do’ mathematics” (p. 354). Deborah Ball’s (1991) year-long third-grade class offers another glimpse at such curriculum innovation because she allows students to reason through their thinking in whole class discussions. Conclusion Current school mathematics strips from the subject the very constituents that provide for meaningful mathematical experiences. A solution to the crisis may be far easier than some think and this solution would not require more rigorous standards, more standardized testing, more funding for smaller classes, or more content training—only a return to the fundamental aspects that make mathematics so intriguing. The primary way to engage students in mathematics classrooms is to allow them to experience mathematical activity. Mathematical activity is a motivating activity because it connects the ubiquitous human capabilities of intellectualizing and socializing (Devlin, 2000). More specifically, mathematical activity welds together the intellectual and social dimensions of human beings as they collaboratively wrestle with and jointly create mathematical terrain in a process of social mathematizing. References Anderson, S. E. (1990). Worldmath curriculum: Fighting eurocentrism in mathematics. Journal of Negro Education, 59, 348–359. Ball, D. L. (1991). Shea’s numbers [A video package]. East Lansing: Michigan State University, Math Project.

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Ball, D. L. & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27–44). Reston, VA: NCTM. Bennie, K., Olivier, A. & Linchevski, L. (1999). Rethinking the notion of equity to address the fundamental assumptions and goals of Curriculum 2005. In M. J. Smit & A. S. Jordaan (Eds.), Proceedings of the National Subject Didactics Symposium (pp. 1-9). Stellenbosch, South Africa: University of Stellenbosch. Boaler, J. (1999). Participation, knowledge and beliefs: A community perspective on mathematics learning. Educational Studies in Mathematics, 40, 259–281. Borba, M. (1992). Teaching mathematics: Ethnomathematics, the voice of sociocultural groups. The Clearing House, 65(3), 134–135. Burton, L. (2004). Mathematicians as enquirers: Learning about learning mathematics. New York: Kluwer. Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7),13–19. Cobb, P., & Yackel, E. (1995). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. In D. T. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the Seventeenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3–29). Columbus, OH: PME-NA. (ERIC Document Reproduction Service No. SE 057 176) Cobb, P., Wood, T., & Yackel, E. (1990). Classrooms as learning environments for teachers and researchers. Journal for Research in Mathematics Education Monograph, 4, 125–146. Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2–33. Corry, L. (2007). A clash of mathematical titans in Austin: Harry S. Vandiver and Robert Lee Moore (1924-1974). The Mathematical Intelligencer, 29(4), 62–74. Cuoco, A. (2001). Mathematics for teaching. Notices of the American Mathematical Society (AMS), 48, 168–174. Davis, B. (1997). Listening for differences: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education, 28, 355–376. Davis, B., & Simmt, E. (2003). Understanding learning systems: Mathematics education and complexity theory. Journal for Research in Mathematics Education, 34, 137–167. Davis, R. B. (1994). Comments on Thomas Romberg’s chapter. In A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 312–321). Hillsdale, NJ: Erlbaum. Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press. Devlin, K. J. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. New York: Basic Books. Ernest, P. (1990). Social constructivism as a philosophy of mathematics: Radical constructivism rehabilitated. Retrieved July 14, 2007, from http://www.people.ex.ac.uk/PErnest/

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Fuchs, L. S., Fuchs, D., Kazdan, S., Karns, K., Calhoon, M. B., Hamlett, C. L., & Hewlett, S. (2000). Effects of workgroup structure and size on student productivity during collaborative work on complex tasks. The Elementary School Journal, 100(3), 183–212. Linchevski, L., Kutscher, B., & Olivier, A. (1999). Assessment in support of equity. Proceedings of the Fifth Annual Congress of the Association for Mathematics Education of South Africa (Vol. 1, pp. 155–167). Port Elizabeth, South Africa: Port Elizabeth Technikon. Linchevski, L., Kutscher, B., Olivier, A., & Bennie, K. (2000a, March). Diversity acknowledged and ignored: Achieving equity in school mathematics. Paper presented at the Second International Conference on Mathematics Education and Society, Lisbon, Portugal. Linchevski, L., Kutscher, B., Olivier, A., & Bennie, K. (2000b). Indispensable Mathematical Knowledge – IMK and Differential Mathematical Knowledge – DMK: Two Sides of the Equity Coin. In T. Nakahara & M. Koyama (Eds.), Proceedings of the Twenty-fourth International Conference for the Psychology of Mathematics Education (Vol. 3, pp.239– 246). Hiroshima, Japan: Hiroshima University, Department of Mathematics Education. Ma, L. (1999). Knowing and teaching elementary mathematics. Hillsdale, NJ: Erlbaum. Moses, R. P., & Cobb, C. E., Jr. (2001). Radical equations: Civil rights from Mississippi to the Algebra Project. Boston: Beacon. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. National Center for Educational Statistics (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. Washington, DC: Author. Noddings, N. (1989). Theoretical and practical concerns about small groups in mathematics. The Elementary School Journal, 89, 606–623. Paulos, J. A. (1988). Innumeracy: Mathematical literacy and its consequences. New York: Hill and Wang. Polya, G. A. (1945). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press. Richards, J. (1991). Mathematical discussions. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 14–15). Dordrecht, The Netherlands: Kluwer. Ricks, T. E. (2007). The mathematics class as a complex system. Unpublished doctoral dissertation, The University of Georgia. Romberg, T. A. (1994). Classroom instruction that fosters mathematical thinking and problem solving: Connections between theory and practice. In A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 287–304). Hillsdale, NJ: Erlbaum.

Smith, E. (1992, May). Constructivist mathematics as an ethical process. A paper presented at the Second International Conference on the History and Philosophy of Science in Science Education, Kingston, Ontario, Canada Smith, J. P., III (1996). Efficacy and teaching mathematics by telling: A challenge for reform. Journal for Research in Mathematics Education, 27, 387–402. Stein, M. K., & Brown, C. A. (1997). Teacher learning in a social context: Integrating collaborative and institutional processes with the study of teacher change. In E. Fennema & B. S. Nelson (Eds.), Mathematics teachers in transition (pp. 155– 191). Mahwah, NJ: Erlbaum. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College. Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York: Free Press. Ulam, S. M. (1976). Adventures of a mathematician. New York: Scribner. Usiskin, Z. (1999). Developments in school mathematics education around the world: Proceedings of the Fourth UCSMP International Conference on Mathematics Education, August 5–7, 1998. Reston, VA: NCTM. Webb, N. M. (1995). Group collaboration in assessment: Multiple objectives, processes, and outcomes. Educational Evaluation and Policy Analysis, 17, 239–261. Wertsch, J. V., & Toma, C. (1995). Discourse and learning in the classroom: A sociocultural approach. In L. P. Steffe & J. Gale (Eds.), Constructivism in education (pp. 3–15). Hillsdale, NJ: Erlbaum. WGBH Boston. (1995). Teaching math: A video library, K-4. South Burlington, VT: The Annenberg/CPB Math and Science Collection. Wolfram (2010). Retrieved January, 29, 2010 from http://mathworld.wolfram.com/ Trapezoid.html Yackel, E. (2000, July). Creating a mathematics classroom environment that fosters the development of mathematical argumentation. Paper prepared for Working Group #1: Mathematics Education In Pre and Primary School of the Ninth International Congress of Mathematical Education, Tokyo/Nakuhari, Japan. Yackel, E., Cobb, P., & Wood, T. (1999). The interactive constitution of mathematical meaning in one second grade classroom: An illustrative example. Journal of Mathematical Behavior, 17, 469–488.

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The Mathematics Educator 2009/2010, Vol. 19, No. 2, 10–20

Preservice Teachers’ Emerging TPACK in a Technology-Rich Methods Class S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards There is a dearth of research on the mechanisms for preservice teachers' development of the pedagogical knowledge necessary for effective use of such technologies. We explored the emergent Technological Pedagogical and Content Knowledge (TPACK) (Niess 2005, 2006, 2007) of a group of secondary mathematics preservice teachers in a methods course as they designed and implemented technology-rich teaching materials in field settings. Participant surveys and collected assignments were analyzed through the lens of the TPACK framework. The data were also analyzed to examine the trajectory of the participants’ beliefs about the appropriate role of advanced digital technologies in mathematics. The results indicate that the participants’ understanding of technology shifted from viewing technology as a tool for reinforcement into viewing technology as a tool for developing student understanding. Collected data supports the notion that preservice teacher TPACK development is closely related to a shift in identity from learners of mathematics to teachers of mathematics. In a class where advanced digital technologies were used extensively as a catalyst for promoting inquiry-based learning, preservice teachers retained a great deal of skepticism about the appropriateness of using technology in concept development roles, despite their confidence that they can incorporate technology into their future teaching. To me, it's [the use of calculators in mathematics instruction] more about where kids are at developmentally. The methods are influenced by this. When kids are younger and inexperienced, they need to be taught the basics using direct instruction like I was. Now that I know some things, I can use the calculator to learn more. But I have a good foundation in the basics FIRST.

In the above quote, a preservice teacher shares his views on the use of technology to teach mathematics. Based on his strong views on this issue, we might not expect him to use a great deal of advanced digital technologies in his classroom nor to employ discovery activities (using technology or otherwise). Many researchers have highlighted the important influence of S. Asli Ozgun-Koca teaches mathematics and secondary mathematics education courses at Wayne State University, Detroit, MI. Her research interests focus on the use of technology in mathematics instruction and understanding mathematics teachers' views about and knowledge on the technology use in teaching and learning of mathematics. Michael Meagher is an Assistant Professor of Mathematics Education at Brooklyn College - CUNY. His research interests include the use of advanced digital technologies in teaching and learning mathematics, and the role alternative certification programs in training mathematics teachers for teaching in urban schools. M. Todd Edwards teaches mathematics and secondary mathematics education courses at Miami University, Oxford, OH. His research interests include the use of technology in the learning and teaching of school mathematics. In his spare time, he enjoys exploring the joys of dynamic geometry software with his three children, Cassady, Ian, and Dylan. 10

teachers' beliefs and views on instructional decisionmaking and classroom practice (Ball, Lubienski, & Mewborn, 2001; Borko & Putnam, 1996; National Council of Teachers of Mathematics [NCTM], 1991; Richardson, 1996; Stipek, Givvin, Salmon, & MacGyvers, 2001; Thompson, 1984, 1992). Furthermore, Peressini, Borko, Romagnano, Knuth, and Willis-Yorker (2004) argue that none of the experiences (mathematics and teacher preparation courses, preservice field experiences, and employment) in learning to teach are independent of one another, which ensures a complicated collection of influences on a prospective teacher's learning trajectory. A growing body of research indicates that digital technologies, including graphing and Computer Algebra System (CAS)-enabled calculators, can enhance young students' conceptual and procedural knowledge of mathematics (Dunham, 2000; Thompson & Senk, 2001). As teachers decide whether and how to use advanced digital technologies in their teaching, they need to consider the mathematical content that they will teach, the technology that they will use, and the pedagogical methods that they will employ. Moreover, they need to reflect on the critical relationships between these concepts: content, technology, and pedagogy. Drawing on a series of case studies, Zbiek (2002) suggests some direction for the development of a model of effective teaching using CAS. This model stresses the importance of many of the influences discussed by Peressini et al. (2004), including conceptions of school mathematics and how available curriculum materials intersect with

S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards

technology. Zbiek concludes that such a model could be a useful analytic tool for describing and facilitating teachers' evolution in teaching with CAS. As will be discussed in more detail in the next section, Niess (2005, 2006, 2007) developed the Technological Pedagogical and Content Knowledge (TPACK) framework to provide such a tool for using advanced digital technologies in teaching in general. However, empirical studies of preservice teachers' emerging TPACK remain in short supply. Theoretical Context Shulman (1986) provided an analysis of teachers' knowledge as a complex structure including content knowledge, pedagogical knowledge, and pedagogical content knowledge (PCK). Ensuing research on teacher knowledge is grounded in his framework. With Mishra and Koehler's (2006; Koehler & Mishra, 2005) and Niess' (2005, 2006, 2007) conception of TPACK, the field has additionally gained “an analytic lens for studying the development of teacher knowledge about educational technology” (Mishra & Koehler, 2006, p. 1041). TPACK involves the content knowledge (CK), pedagogical knowledge (PK), and technological knowledge (TK) required to teach in technology-rich environments (see Figure 1). In our case, content knowledge is high school mathematics. Pedagogical knowledge includes learning theories and instructional methods. Technological knowledge includes the knowledge of how to operate technology-oriented tools (such as Geometer's Sketchpad or TI-Nspire) and the ability to adapt to ever-changing, novel technologies.

Figure 1. Re-creation of Mishra and Koehler's TPACK model.

Shulman's (1986) discussion of PCK focuses on the two-way relationship between content and pedagogy, for instance, how particular pedagogical methods might help (or hinder) students' learning of specific content. Niess's (2005, 2006, 2007) TPACK model extends this relationship to include relationships with other constructs, including technological content knowledge (TCK) and technological pedagogical knowledge (TPK). TCK is viewed as the intersection of the technology and the content wherein a wholly different perspective on content may arise. For example, technology can be used to explore the fact that a quadratic with integer coefficients is highly unlikely to be factorable, drawing attention to and questioning the traditional content of school mathematics. With respect to TCK, Mishra & Koehler (2006) say, “teachers need to know not just the subject matter they teach but also the manner in which the subject matter can be changed by the application of technology” (p.1028). On the other hand, “technological pedagogical knowledge (TPK) is knowledge of the existence, components, and capabilities of various technologies as they are used in teaching and learning settings, and conversely, knowing how teaching might change as the result of using particular technologies” (p. 1028). Research Question There is much to consider when studying pre- and inservice teachers' knowledge, views, beliefs, attitudes, and decisions about the use of technology in their classrooms. Whereas Niess (2006, 2007) discussed how teachers' beliefs and views about teaching mathematics with technology play a crucial role in the development of TPACK, our research question was: How does preservice teachers' TPACK emerge during their methods classes and field placement? Therefore, in a methods course intended specifically for preservice secondary mathematics teachers, we examined teachers' emerging TPACK (Niess 2005, 2006, 2007) as manifested in their use of advanced digital technologies in the design and implementation of technology-rich teaching materials in field placements. Moreover, through written responses regarding the use of the TI-Nspire (Texas Instruments, 2007) and other advanced digital technologies, we studied their views about the use of technology to teach mathematics. Data Collection We studied a group of 20 preservice teachers enrolled in a first-semester mathematics teaching 11

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methods course at a small Midwestern university. The sample is one of convenience (Lodico, Spaulding, & Voegtle, 2006). The participants were students in one of the researchers’ classes, which was designed to introduce participants to inquiry-based learning with open-ended questioning. In past research, we have found that technologies such as the TI-Nspire calculator, virtual manipulatives, and dynamic geometry software (DGS) open up new possibilities for teachers to promote connections between representations, encourage students to explore dynamic mathematics environments, develop students' skills of inquiry, and support students’ construction of knowledge (Özgün-Koca, Meagher, & Edwards, in press). Based on this result, the instructor placed considerable emphasis on the use of such technologies in the teaching and learning of mathematics, with particularly extensive use of the TI-Nspire. The TINspire is a handheld device that incorporates functionalities such as graphing, manipulating algebraic expressions, and constructing geometric figures and analyzing data in a dynamic environment, while dynamically linking all of these representations. Activities in the course focused primarily on pedagogical tasks (e.g. constructing lesson plans and grading rubrics, creating technology-oriented math activities) and content-related activities (solving mathematics problems, analyzing mathematical accuracy of student work). For example, participants completed problem sets designed to give them the opportunity to explore (and extend) content and pedagogical knowledge of secondary school mathematics. As part of their field experience, participants completed two reports in which they researched, developed, and implemented mathematics lessons. In addition, they submitted five secondarylevel mathematics activities constructed and/or modified for use with the TI-Nspire. They were encouraged to use these materials in their field teaching whenever possible. Finally, participants conducted original research dealing with the teaching of a secondary mathematics problem (or set of related problems) using the TI-Nspire. The field experiences varied in the extent to which technology was used, from almost none in some classrooms to extensive and skilled implementation in others. At both the beginning and end of the course, the participants completed a mathematics technology attitudes survey (MTAS), which included questions rated on the Likert scale. Additionally, they participated in three short surveys administered electronically in weeks 4, 8, and 13 of the course. Each 12

of these surveys consisted of a combination of Likert scale and open-ended items. Finally, participants completed an open-ended exit survey at the end of the course with questions that were more general than those asked in the earlier surveys. Likert scale questions from the MTAS and short surveys included: 1. Graphing calculators help me understand mathematics. 2. Graphing calculators are a useful support for discovering algebraic rules. 3. Students shouldn't use calculators until they have thoroughly mastered the required skills by-hand. 4. Graphing calculators help people who have difficulties with algebra to still be able to do mathematics. 5. I have been thinking and working a lot on the technology of the course we are designing. 6. Our group has been considering how course pedagogy and technology influence one another. Some of the examples of open-ended questions were: 1. What kind of technology skills that you can use later in your profession are you learning? Describe how you intend to use those skills in your future teaching. 2. Discuss the extent to which you have been thinking and working with pedagogical issues in the student activities you have been designing in our class. While recently observing classroom instruction in a local high school, a mathematics teacher made the following comment to me: "Content and pedagogy influence one another, especially when I use technology with kids in my classroom." Discuss your thoughts regarding this statement. 3. Similarly, a student in the aforementioned classroom noted that "technology changes the way our teacher teaches mathematics and the way I learn mathematics." Discuss your thoughts regarding this statement 4. Lastly, the classroom teacher noted that "technology changes the mathematics content that I teach." Discuss your thoughts regarding this statement.

S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards

Data Analysis Our analysis focused on the data collected through the five secondary-level mathematics activities, field experience reports, and surveys. The first level of analysis focused on survey responses. We used descriptive statistics for quantitative data and searched for emerging codes and themes in the qualitative data. The initial themes arising from the analysis were (i) a shift in thinking of technology as a reinforcement tool to thinking of technology as a tool for developing mathematical concepts, and (ii) a change in relationship to technology predicated on a shift of the participants’ own identity from learner of mathematics to teacher of mathematics. Once these themes had been identified, we re-analyzed changes between pre- and post-survey data and found the themes to validly reflect key characteristics of the data. We analyzed the activities and field experience reports through this lens and found further evidence to support the conclusions. We used the TPACK framework to guide the qualitative data analysis for the open-ended survey questions, the secondary-level mathematics activity write-ups, and the field experience reports. The first level of analysis involved coding the data for instantiations of the participants' attitudes towards, skill in using, and deployment of, TK, CK, and PCK. For instance, if a participant were to say that calculators should not be used until students master the skills by hand, or if a participant were to discuss how using technology in their activity write-ups affected instructional planning, then we would code this as TPK. While analyzing the activity write-ups, we focused on three key issues: implementation of technology, implementation of inquiry-based methods, and quality of problem solving. Our interest in the interactions among the various domains of the model fueled our second level of analysis. We developed codes for each of the possible interactions between TK, CK, and PCK. We then analyzed the data for important aspects of these interactions. For example, a participant’s statement suggesting that the use of calculators means that certain topics should be deemphasized would be coded as how technological knowledge influences content knowledge. We feel that the multiple data sources and use of different lenses in the analysis provided sufficient data and researcher triangulation to ensure trustworthiness of our findings in this study (Miles & Huberman, 1994). Results and Discussion Two major themes emerged from the data analysis. Firstly, the participants’ understanding of technology

showed perceptible shifts and mutations from thinking of technology as a reinforcement tool to a tool for developing mathematical concepts. Glimpses of this evolving relationship to technology, which we see as a positive development in their TPACK, are reflected in candidate comments throughout the semester. Secondly, we saw an interesting change in participants’ relationship to technology as they shifted their identity from being a learner of mathematics to being a teacher of mathematics. This also represents a positive step in developing TPACK, specifically in TPK. The course was the first methods course for these teacher participants and, therefore, their first opportunity to give serious thought to the use of technology from a teacher's perspective. Reinforce or Develop: The Use of Technology in Lesson Plan Development The development, or lack thereof, of TPACK in teacher participants is reflected in the learning activities they design for students. Developing good tasks that incorporate technology presents a challenge for the preservice teachers since they have to mix CK of the topic they wish to address, TK of the technology they choose to use, and PK in designing an inquirybased task for their students. Their intersection, TPACK, proved particularly interesting in this challenge. In the quotes below, we see two participants’ reflections on how they started to think about technology as an instructional tool to build conceptual understanding: I am using technology because we are required to do so. However, the second activity write-up used the TI-Nspire extensively because I thought it would be really neat to see if I could use it for my idea. At first, the activity seemed to me that we had to use and had to incorporate technology in our activity. Now is seems that technology is more of a tool to help us design a really good hands on, visual activity.

TPACK was evident in the content-specific ways that preservice teachers took advantage of the functionalities and affordances of the technology to engage students in inquiry-based tasks. The examples discussed below illustrate that participants moved beyond a naïve use of the technology into a more sophisticated incorporation of technology into the mathematics of their tasks. When developing activities and creating lesson plans using technology, preservice teachers often incorporated technology into lessons through a 13

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superficial use of the available tools rather than taking advantage of those capabilities that were specific to the technology at hand (e.g. drawing shapes within a DGS environment while ignoring dynamic construction capabilities of the software, calculating simple results or using graphing functions while ignoring linkages between different representations), thus showing some lack of TK. This also showed an underdeveloped sense of TPK since the technology was not being used in a sophisticated way to help provide inquiry-based experiences for the students to develop understanding. In many of the participants’ lessons, technology use was not tied to acquisition of the mathematics content—technology and content were envisaged as separate constructs rather than as intertwined entities. Therefore, although we found little evidence of TPK, there was an increasing trend throughout the study. For example, early work samples of student-generated activities revealed naïve understandings of various TINspire tools and had low cognitive demand (see Figure 2). The activity in Figure 2 does not use the dynamic capabilities of TI-Nspire, focusing strictly on its drawing and measurement capabilities. Utilized in this manner, the technology contributed few, if any, advantages over use of traditional paper and pencil.

answer the question, determining the length of each side of the triangle and verifying that the square of the side opposite theta was strictly less than the sum of the squares of the other two side lengths. While theta is clearly an acute angle when viewed statically (as is the case on the printed page), when vertices of the triangle are dragged, theta may also assume values larger than 90 degrees. Hence, viewed dynamically, it is impossible to determine if the triangle is right or not. Therefore, in addition to being a very low-level identification task (identify the type of angle denoted by theta), the prompt makes no sense in a dynamic context. Below we can see a participant’s first activity (see Figure 3) in which he used a real-world problem and both the tabular and graphing capabilities of the technology to find a point of intersection of two graphs. Here the use of technology was helpful, but not essential; the problem could have been solved just as easily with pencil and paper. Figure 3. Participant 3’s Activity, highlighting weak

use of technology

Figure 2. An example of a student-generated activity of low cognitive demand. In this activity, asking the student to “say whether this triangle is a right triangle or not” implies that the angle theta remains fixed. Although it is not clear from the prompt alone, the participant intended the student to use the converse of the Pythagorean Theorem to 14

However, when we analyzed the second activity that he created later in the semester (see Figure 4), we saw that he constructed the cross-sections of various polyhedra using CABRI 3D to determine that crosssections of a cone can form an ellipse, a circle, and parabola, and that cross-sections of a regular tetrahedron can form a scalene triangle, an isosceles triangle, an equilateral triangle, an isosceles trapezoid, and a quadrilateral. Without the technology, such constructions are impractical and not readily available to teachers or students. In this second example, the software is arguably an instructional necessity, indicating a more mature utilization of technology.

S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards

Figure 4. Participant 3’s Second Activity with Effective Use of Technology While there was a general improvement in the quality of the activities and lesson plans written by participants as the semester progressed, the activities written by those students with field placements in technology-rich environments showed more sophistication, not just in the use of technology, but also in terms of implementing inquiry-based and openended instructional approaches. Not only did both their technological and pedagogical knowledge develop, but the intersection of these two constructs, their TPK, also developed during those experiences. When participants did not have a rich technology experience in the field, they typically indicated that technology was not an instructional must: I didn't really see any technology in schools when I had field, and I'm not convinced that kids need it. They need the basics first in order to actually UNDERSTAND what the calculator is doing. This is like reading. Kids need to know letters of the alphabet first before they can read.

Change in Identity: Technology for Them and Technology for Their Students On the pre-and post-surveys, a large percentage of participants agreed or strongly agreed that graphing calculators helped them better understand mathematics. In addition, a clear majority of the subjects agreed or strongly agreed that graphing calculators increased their desire to do mathematics (73% in the pre-survey and 65% in the post-survey). Based on these observations, it seems that the participants had a wellestablished understanding of how they themselves can use advanced technologies in doing mathematics. On the other hand, when it came to the issue of teaching with calculators, the participants had mixed views. In the pre-survey, 82% agreed that calculators help people who have difficulties with algebra to still be able to do mathematics. However, this percentage decreased to 70 in the post-survey. We, therefore, conclude that their perspective changed somewhat when putting themselves in the position of teachers of mathematics.

This could be an indication that the participants, based on their experiences in their field placements, remained attached to the idea that students' can be overly dependent on calculators and that calculators can interfere with students' learning of basic concepts. In addition, during the fourth week, two of the preservice teachers discussed their concern that they were learning about technology to which they were unlikely to have access as classroom teachers. Again, this shows that they had started to reflect on the issues as prospective teachers. That week the preservice teachers also discussed SMARTTM Boards (n=8), websites (n=4), and Geometer’s Sketchpad (n=1) as possible tools to use in their future teaching. However, by the eighth week, after some field experience, no one discussed TI-Nspire calculators, although many discussed the limited access that they had to advanced technologies in actual school settings: “After going out in the field, I believe more than I did before that the technology I am learning to base my lessons off of, though, is far too advanced.” After their field experiences, many participants reported that they found internet-based resources, such as interactive web applets,more practical—both in terms of their accessibility in classroom situations (e.g. most classrooms were equipped with one demonstration computer with internet access) and their low cost (unlike the TI-Nspire, most applets were freely available). The TPACK Model and Advanced Digital Technologies In this section, we discuss how the TPACK model helped us to reach the two main conclusions discussed above. Technological knowledge. Participants mentioned a variety of technologies when discussing which of the technological skills they were learning would be useful in their future teaching. In the fourth week, eight preservice teachers mentioned that they liked TINspire calculators. Some mentioned the technological skills that they were learning, such as how to operate the TI-Nspire: I've not had a lot of practice in using calculators besides the TI-84 and with that only the basic functions. The technological advanced TI-Nspire on the other hand, as I'm learning, is very user friendly, with menus you can go to find out more about what is available.

On the other hand, some preservice teachers had technical difficulties learning how to use some of the technology: “I found the TI-Nspire to be too 15

Preservice Teachers’ Emerging TPACK

complicated and not worth the hassle figuring it all out. I spent more time trying to figure out how to use it than I did learning about math.” Another mentioned that, “one of the issues I've struggled with is the extent [to which] we would use technology. A number of cases in using technology have required extensive knowledge/experience with the technology.” Clearly a lack of TK for a particular technology could be an important factor in preservice teachers’ consideration of whether to use that technology in their future classrooms. Content and pedagogical content knowledge. The CK required by these teachers is, minimally, the high school mathematics content they will teach. Participants in the methods course mostly agreed that both their university class and field placement required them to work extensively with high school mathematics content as they designed teaching activities. Even though many said that they were not learning about mathematics content as they designed activities, quite a few mentioned they were “remembering” and that the activities were “refreshing us” on the high school mathematics while looking at content from a teacher's perspective: So far, we have covered many of the content areas including algebra and geometry. These were important for my growth because I was unaware of the severity of my 'rustiness' when it came to basic algebraic and geometric principles. Quite a few of the activities we have done in class have made use of mathematics that I have not used since high school. These activities have reminded me how to do a number of problems. Overall, the course is requiring me to look at mathematics from a teachers’ perspective and not a students’ view of question and answer.

Moreover, one preservice teacher mentioned that he or she was focusing on the “why” question more: I feel as though I am not learning new mathematics content, but instead, I am thinking of what I already know in a different light. The class has caused me to think more about the why than the how, and to me, that is, the most important element of being a mathematics teacher.

We see here the participants’ transition from thinking of themselves as learners of mathematics to thinking of themselves as teachers of mathematics. The participants' CK provided a basis for the development of their PCK. Another participant mentioned that: Both the problem sets and our activities have required us to investigate mathematics content. The 16

Folded Paper problem is a perfect example. This problem could be solved with a table, via a graph or using calculus. Exploring each of these is valuable in understanding the content of the math and understanding multiple representations.

Preservice teachers started thinking about pedagogical issues together with content. Moreover, another preservice teacher discussed how designing lesson plans helped him learn, not only about the content, but also about incorporating technology: I have been trying to figure out how to make lessons based on certain content. This class has been helping me to identify how to design lessons based on content which is something I had no experience with. I now have a better understanding of how to incorporate things like technology into the lesson as well.

Pedagogical knowledge. PK includes teaching strategies appropriate for student learning. When preservice teachers were asked to discuss pedagogical issues as they designed activities during the fourth week, only two mentioned the use of technology. Specifically, one mentioned TI-Nspire and one mentioned websites. Other than that, participants focused their discussion primarily on the use of manipulatives, inquiry, problem solving, differentiation, and other pedagogical issues. The class has made me realize the importance of manipulatives and hands-on activities in the classroom. These types of activities help students to become active learners, and thus, cause them to retain more of the material.

In the quote above, the candidate did not make an explicit connection between the use of technology and the use of manipulatives in an active/hands-on learning style. I feel in my activity write-ups, I am constantly considering pedagogical issues. The one main issue is using an inquiry method for problem solving. I try to have my students explore a topic, such as finding the length of the diagonal of a square through investigation rather than lecture.

Once again, the candidate does not explicitly link the use of digital technologies to the use of an inquirybased pedagogical approach. The interactions among pedagogy, content and technology. In a survey given in the thirteenth week, three open-ended questions asked the participants to discuss the relationships between content and pedagogy, content and technology, and technology and pedagogy. Figure 5 illustrates the interactions between

S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards

content, technology, and pedagogy reflected in the data. The direction and thickness of the arrows represent the relationships articulated by the preservice teachers. If many preservice teachers articulated a relationship, the arrow representing that relationship is bolded. Dashed arrows signify relationships that were not mentioned or are non-existent.

Even though no one mentioned how the content influenced the technology directly, several preservice teachers mentioned “appropriate uses” of technology when discussing the relationship between content and pedagogy. In these cases, they were not focusing on technological skills, but on the use of technology as one of many possible teaching strategies: Some things that we taught kids were naturally studied with the Nspire. For instance graphing functions was a natural with the “Graphs and Geometry” application. The math you are teaching totally influences the way you teach it. Some topics are well suited for use with technology. Like when you are learning about transformations, then Sketchpad is a natural tool to use. When you're studying graphs, then calculators are a natural learning tool. Other topics, such as trigonometric proofs and identities, aren't as obviously hands-on. I feel more limited teaching these topics using methods other than lecture.

Figure 5. Interactions between the content, the technology and the pedagogy. Many preservice teachers mentioned that they could see the influence of content on pedagogy. For instance, they acknowledged that the complexity of the high school content affects the teaching method to be employed: I totally think that content influences how topics are taught. I'm not sure if pedagogy should influence content. The methods you use depend on the kids you teach and the topics you are teaching. The content you teach is related to the way you teach it. When I taught kids about area and perimeter in the field, we used plastic tiles to study the topics in a hands-on manner. But we didn't use hands-on materials when we studied cross multiplying. There just wasn't a good way to do this hands-on.

Even though a few said that they were not sure if pedagogy should influence the content, the examples that they gave showed that they were considering the pedagogy to content direction in Figure 5. One mentioned, “in my field placement, I saw how dynamic geometry software influences what content is taught and how it is taught.”

Thus, in Figure 5, we connected content and technology through pedagogy with one arrow. When discussing the influences of technology on content, participants focused on the capabilities of technology. Due to the capabilities of advanced technologies, the content (curriculum) might have been affected or some contents might have become more accessible: I have been using more technology now with the problems than I ever thought I would. I remember from problem set two that using technology made doing the problem much easier to do. Technology does change the content. There are things that I studied in school that don't seem relevant in algebra class anymore. For instance, factoring. We did an assignment on the Nspire that showed that 99% (or more) of quadratics aren't factorable. So why do we spend all of this time factoring? I think technology changes the content sometimes, but I don't think it should. I think math should influence the technology, not the other way around.

In regard to the relationship between technology and pedagogy, some preservice teachers discussed technology as a pedagogical tool as opposed to focusing on needed technical skills. Therefore technology was embedded in pedagogy. They noted how the use of technology might impact how a task develops, which in turn could influence student learning. In these instances, the participants are thinking through how technology is deployed rather 17

Preservice Teachers’ Emerging TPACK

than, as discussed earlier, which topics might naturally lend themselves to technology use: Technology can make it easier to test conjectures. For instance, with sketchpad, we can test countless conjectures much more quickly than possible when using pencil and paper. When we observe behaviors in sketchpad (or with the TI-Nspire), students are more motivated to ask “does this always happen?”

I have been able to incorporate things such as the TI Nspire and GSP into my activity write-ups, and I think that incorporating these types of technology into lessons helps to make them (the activities) more multifaceted and thus easier for a larger percentage of the students in a classroom to understand.

Not all preservice teachers thought that technology influenced students’ learning. In particular, one candidate noted: Technology doesn't change the way kids learn math. They have to learn it the way I learned it, by repetition and practice. It's like learning how to read. You have to do some memorizing and repetition before you can get to the good stuff.

Other preservice teachers noted the importance of first using paper and pencil experiences in students’ learning. In these cases, it appeared that their beliefs about how learning occurs affected the extent to which they would use technology in their teaching. During the eighth week, after having some field experiences, one preservice teacher felt that students were dependent on calculators for computation: “The main issue that I dealt with in my classroom (i.e. methods field experience) was the severe dependency on calculators that students seemed to have.” In the exit survey, approximately 70% of the preservice teachers agreed that they would specifically like to use the Nspire when they become a full-time teacher. However, in the fourth week, on a more general question about the place of technology in the mathematics classroom, approximately 68% of the preservice teachers agreed that students should not use calculators until they have thoroughly mastered the required skills by hand. This percentage decreased to 44% at the end of the study. We should point out that only 9 participants completed the survey in the thirteenth week, as opposed to the 20 that completed the survey in the fourth-week survey.

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One of the preservice teachers’ main messages was that content should be a teacher’s first priority. As one preservice teacher put it: Good teachers think about content first and ways to better deliver content to students. Putting pedagogy first (and even WORSE, putting technology first) is irresponsible. We should always be thinking about WHAT we [want] our kids to know MATHEMATICALLY . . . then figure out how (or if) technology or various teaching methods support THAT . . . not the other way around.

Eventually, they started to look at the content from a teacher's perspective, thinking about issues related to teaching and learning Later, technology came into play as a pedagogical tool with novel capabilities that paper and pencil (or chalk and blackboard) cannot provide. At the end of the course, participants thought themselves better prepared to use technology in their teaching. At the beginning of the course, only 40% of them considered themselves at least fairly prepared to have students use technology to explore new concepts. This percentage increased to 84% at the end of the course. Similarly, 64% of them felt fairly or very well prepared to have students use appropriate educational technology to learn mathematics at the beginning of the year. This percentage also increased to 84% in the exit survey. Conclusions This study sought to examine teachers’ emerging TPACK as manifested in their use of advanced technologies in the design and implementation of technology-rich activities in their student teaching. We did this through an examination of their views on the use of advanced technologies, such as the TI-Nspire. Our major conclusions are that (1) preservice teachers’ development of TPACK is related to their shift in identity from being learners of mathematics to teachers of mathematics; and that, even in a class where advanced digital technologies are used extensively as a catalyst for promoting inquiry-based learning, (2) preservice teachers retain a great deal of skepticism about the role of technology in mathematics education even though they felt much better prepared to incorporate technology into their teaching. We have shown above that, often, a preservice teacher’s first use of advanced technologies is naïve and incorporates technology superficially. We believe this results from a combined intial lack of PCK and TK. These two deficits, in tandem, make it hard to

S. Asli Özgün-Koca, Michael Meagher, & Michael Todd Edwards

design tasks that allow students to explore mathematical concepts. The data show that, initially, preservice teachers’ mathematical focus is on content and their own ability to solve problems. By and large, preservice teachers have been successful in doing mathematics in traditional environments. As they make the shift to being mathematics teachers, they begin to develop pedagogical knowledge and become interested in hands-on activities and inquiry-based learning. However, for many, there may remain a feeling that advanced technologies “do too much.” The data shows that they do not see advanced technologies as part of an inquiry-based approach. Preservice teachers can certainly develop their technological, pedagogical, and content knowledge separately, but integrating these types of knowledge through the development of their TPK, TCK and TPACK gives them a more holistic view of their teaching and helps them transition from learners of mathematics to teachers of mathematics. Our data show that close attention must be paid to the relationship between the university classroom and the field placement; ideally, every preservice teacher would see that what they learn in the university classroom has an impact on their work in the field. Field placements are where preservice teachers face the reality of a classroom and experience first-hand that how they design tasks affects student learning. Our conclusions suggest several directions for further research. Perhaps the most obvious of these is the need for further investigation of what happens when participants complete their preservice training and become full time teachers: What are the crucial influences on the development of TPACK? Our past research (Özgün-Koca, Meagher, & Edwards, in press) suggests that experiencing success in the classroom and reflection, through journal writing or interviews, are vital elements in continuing the development of TPACK. Other potential influences to consider include access to technology, availability of materials to support inquiry-based instruction, and the existence of a supportive professional environment. Another area for further research is studying the effect, on preservice teachers’ attitudes and practices, of seeing exemplary inquiry-based instruction in a technology-rich environment. There is not enough data in this study to support strong claims, but our data does suggest that students found it difficult to appreciate the possibilities of advanced technologies in instruction without experiencing exemplary use in an authentic classroom situation. Such experience is highly dependent on field placement, although use of remote

video technology could be employed to make an exemplary experience available to an entire class. Using advanced technologies in methods classes puts preservice teachers in the position of being learners. This allows them to pay explicit attention to developing their TCK, which in turn encourages them to reflect on their PCK and CK. Thinking about, and engaging with, advanced technologies gives preservice teachers a vantage point from which to examine their beliefs about, and attitudes towards, what it means for their students to be successful. References Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp. 433-456). New York: Macmillan. Borko, H., & Putnam, R. T. (1996). Learning to teach. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 673-708). New York: Macmillan Dunham, P. H. (2000). Hand-held calculators in mathematics education: A research perspective. In E. Laughbaum (Ed.), Hand-held technology in mathematics and science education: A collection of papers (pp. 39-47). Columbus, OH: The Ohio State University. Koehler, M. J., & Mishra, P. (2005). What happens when teachers design educational technology? The development of technological pedagogical content knowledge. Journal of Educational Computing Research, 32, 131–152. Lodico, M. G., Spaulding, D. T., & Voegtle, K. H. (2006). Methods in educational research: From theory to practice. San Francisco: Jossey-Bass. Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Thousand Oaks, CA: Sage. Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers College Record, 108, 1017–1054. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author. Niess, M. L. (2005). Preparing teachers to teach science and mathematics with technology: Developing a technology pedagogical content knowledge. Teaching and Teacher Education, 21, 509–523. Niess, M. L. (2006). Guest Editorial: Preparing teachers to teach mathematics with technology. Contemporary Issues in Technology and Teacher Education, 6, 195–203. Niess, M. L. (2007, January). Professional development that supports and follows mathematics teachers in teaching with spreadsheets. Paper presented at the meeting of the Association of Mathematics Teacher Educators (AMTE) Eleventh Annual Conference, Irvine, CA. Özgün-Koca, A., Meagher, M., & Edwards, M. T. (In Press). A teacher's journey with a new generation handheld: Decisions, struggles, and accomplishments. School, Science and Mathematics.

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Peressini, D., Borko, H., Romagnano, L., Knuth, E., & WillisYorker, C. (2004). A conceptual framework for learning to teach secondary mathematics: A situative perspective. Educational Studies in Mathematics, 56, 67–96.

Thompson, A. G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127-146). New York: Macmillan.

Richardson, V. (1996). The role of attitudes and beliefs in learning to teach. In J. Sikula, T. J. Buttery, & E. Guyton (Eds.), Handbook of research on teacher education (pp. 102–119). New York: Macmillan.

Thompson, D. & Senk, S. (2001). The effects on curriculum on achievement in second year algebra: The example of the University of Chicago School Mathematics Project. Journal for Research in Mathematics Education, 32, 58–84.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

Zbiek, R. M. (2002). Influences on mathematics teachers’ transitional journeys in teaching with CAS. The International Journal of Computer Algebra in Mathematics Education, 9, 129–137.

Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers’ beliefs and practices related to mathematics instruction. Teaching and Teacher Education, 17, 213–226. Thompson, A. G. (1984). The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105–127.

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The Mathematics Educator 2009/2010, Vol. 19, No. 2, 21–31

Why Not Philosophy? Problematizing the Philosophy of Mathematics in a Time of Curriculum Reform Kimberly White-Fredette This article argues that, as teachers struggle to implement curriculum reform in mathematics, an explicit discussion of philosophy of mathematics is missing from the conversation. Building on the work of Ernest (1988, 1991, 1994, 1998, 1999, 2004), Lerman (1990, 1998, 1999), the National Council of Teachers of Mathematics (1989, 1991, 2000), Davis and Hersh (1981), Hersh (1997), Lakatos (1945/1976), Kitcher (1984), and others, the author draws parallels between social constructivism and a humanism philosophy of mathematics. While practicing mathematicians may be entrenched in a traditional, Platonic philosophy of mathematics, and mathematics education researchers have embraced the fallibilist, humanist philosophy of mathematics (Sfard, 1998), the teachers of school mathematics are caught somewhere in the middle. Mathematics teachers too often hold true to the traditional view of mathematics as an absolute truth independent of human subjectivity. At the same time, they are pushed to teach mathematics as a social construction, an activity that makes sense only through its usefulness. Given these dichotomous views of mathematics, without an explicit conversation about and exploration of the philosophy of mathematics, reform in the teaching and learning of mathematics may be certain to fail.

The teaching and learning of mathematics is going through tremendous changes. The National Council of Teachers of Mathematics’ (NCTM, 2000) Principles and Standards for School Mathematics calls for reforms to both curriculum and classroom instruction. Constructivist learning, student-centered classrooms, worthwhile tasks, and reflective teaching are all a part of NCTM’s vision of school mathematics in the 21st century. Along with calls for changes in how mathematics is taught, there are numerous calls for changes in who engages in higher level mathematics courses. NCTM’s Equity Principle calls for high expectations, challenging curriculum, and high-quality instructional practices for all students. In addition, recent publications from the National Research Council (NRC, 2001, 2005) have, in many ways, redefined the teaching and learning of mathematics. These documents call for a move away from the teaching of isolated skills and procedures towards a more problem solving, sense-making instructional mode. This changing vision of school mathematics—student-centered pedagogy, constructivist learning in our classrooms, focus on the problem-solving aspects of mathematics, and mathematics success for all— cannot come about Kimberly White-Fredette has taught elementary, middle, and high school mathematics, and is currently a K-12 math consultant with the Griffin Regional Educational Service Agency. She recently completed her doctorate at Georgia State University. Her work focuses on supporting teachers as they implement statewide curriculum reform.

without a radical change in instructional practices and an equally radical change in teachers’ views of mathematics teaching and learning, as well as the discipline of mathematics itself. As state curricula, assessment practices, and teaching expectations are revamped, a discernable theoretical framework is essential to the reform process (Brown, 1998). This theoretical framework must include a re-examination of teachers’ views of mathematics as a subject of learning. What are teachers’ beliefs about mathematics as a field of knowledge? Do teachers believe in mathematics as a problem-solving discipline with an emphasis on reasoning and critical thinking, or as a discipline of procedures and rules? Do teachers believe mathematics should be accessible for all students or is mathematics only meant for the privileged few? Recent studies examining teacher beliefs and mathematical reform have primarily focused on teachers’ views of mathematics instruction (see e.g., Bibby, 1999; Cooney, Shealy, & Arvold, 1998; Hart, 2002; Mewborn, 2002; Sztajn, 2003). Further research is required to understand how teachers view not only mathematics teaching and learning, but mathematics itself. Unlike many previous studies, this research should examine teachers’ philosophies, not simply their beliefs, regarding mathematics. Philosophy and beliefs, although similar, are not identical. Beswick (2007) asserted that there is no agreed upon definition of the term beliefs, but that it can refer to “anything that an individual regards as

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true” (p. 96). Pajares (1992) affirmed the importance of researching teacher beliefs, although he acknowledged that “defining beliefs is at best a game of player’s choice” (p. 309). Not only is any definition of beliefs tenuous, but distinguishing beliefs from knowledge is also a difficult process (Pajares, 1992). I argue that a study of philosophy moves beyond the tenuousness of beliefs, in that philosophy is a creative process. “Philosophy is not a simple art of forming, inventing, or fabricating concepts, because concepts are not necessarily forms, discoveries, or products. More rigorously, philosophy is the discipline that involves creating concepts” (Deleuze & Guattari, 1991/1994, p. 5). Why philosophy? Current calls for reform in mathematics education are not without controversy (Schoenfeld, 2004). This controversy, and the reluctance towards change, may well be rooted in philosophical considerations (Davis & Mitchell, 2008). Webster’s Dictionary (2003) defines philosophy as “the critical study of the basic principles and concepts of a particular branch of knowledge, especially with a view to improving or reconstituting them” (p. 1455). A study examining philosophy, therefore, seeks to better understand those basic principles and concepts that teachers’ hold regarding the field of mathematics. Philosophy, not just philosophy of mathematics teaching and learning, but the philosophy of mathematics, is rarely examined explicitly: “Is it possible that teachers’ conceptions of mathematics need to undergo significant revisions before the teaching of mathematics can be revised?” (Davis & Mitchell, p. 146). That is a question not yet answered by the current research on teacher change and mathematics education. Researchers seldom ask teachers to explore their philosophies of the mathematics they teach. But such a study is in keeping with the writings of Davis and Hersch (1981) and Kitcher (1984) who sought to problematize the concept of mathematics. If we are to change the nature of mathematics teaching and learning, we have to look beyond the traditional view of mathematics as a fixed subject of absolute truths, what Ernest (1991) and Lerman (1990) termed an absolutist view. Constructivist teaching and inquirybased learning demand a new view of mathematics, the fallibilist view that envisions “mathematical knowledge [as a] library of accumulated experience, to be drawn upon and used by those who have access to it” (Lerman, p. 56) and “focuses attention on the 22

context and meaning of mathematics for the individual, and on problem-solving processes” (Lerman, p. 56). Sfard (1998) argued that mathematicians are entrenched in an absolutist view of mathematics while researchers in mathematics education are deeply immersed in the fallibilist view: On the one hand, there is the paradigm of mathematics itself where there are simple, unquestionable criteria for distinguishing right from wrong and correct from false. On the other hand, there is the paradigm of social sciences where there is no absolute truth any longer; where the idea of objectivity is replaced with the concept of intersubjectivity, and where the question about correctness is replaced by the concern for usefulness. (p. 491) The teachers of school mathematics are caught between these two opposing groups, yet are rarely asked to explore their philosophies of mathematics. The very existence of philosophies of mathematics is often unknown to them. Yet the question, what is mathematics, is as important to the work of K–12 mathematics teachers as it is to the mathematics education researcher and the mathematician. In the following sections, I will outline the recent explorations that researchers and mathematicians have undertaken in the areas of philosophy of mathematics and mathematics education. Unfortunately, few researchers have engaged teachers of mathematics in this important discussion. Changing Views of Education and Mathematics An investigation of philosophy of mathematics is rooted in three areas. Postmodern views of mathematics, 20th century explorations in the philosophy of mathematics, and social constructivism have contributed to discussions regarding the philosophy of mathematics. I begin by describing the emergence of social constructivism, which in many ways is the driving force behind mathematical reform in the United States and other nations (Forman, 2003). Social Constructivism Forman and others (e.g., Restivo & Bauchspies, 2006; Toumasis, 1997) have argued that NCTM’s Professional Standards for Teaching Mathematics (1991) and the later Principles and Standards for School Mathematics (2000) clearly build upon a social constructivist model of learning. But Ernest

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(1991, 1994, 1998, 1999) argues that social constructivism is more than just a learning theory applicable to the teaching and learning of mathematics. According to Ernest, social constructivism is a philosophy of mathematics that views mathematics as a social construction. Social constructivism focuses on the community of the mathematics classroom and the communication that takes place there (Noddings, 1990), and grew out of Vygotsky’s (1978) work in social learning theory. It has been further developed in mathematics teaching and learning through the work of Confrey (1990), Lerman (1990, 1998, 1999), and Damarin (1999). This theory is in keeping with NCTM’s (2000) emphasis on the social interplay in mathematics instruction: Students’ understanding of mathematical ideas can be built throughout their school years if they actively engage in tasks and experiences designed to deepen and connect their knowledge. Learning with understanding can be further enhanced by classroom interactions, as students propose mathematical ideas and conjectures, learn to evaluate their own thinking and that of others, and develop mathematical reasoning skills. Classroom discourse and social interaction can be used to promote the recognition of connections among ideas and the reorganization of knowledge. (p. 21) Overall, social constructivists advocate that educators form a view of mathematical learning as something people do rather than as something people gain (Forman, 2003). It is upon this foundation that Ernest (1998) built his theory of social constructivism as a philosophy of mathematics. He argued that the teaching and learning of mathematics is indelibly linked to the philosophy of mathematics: Thus the role of the philosophy of mathematics is to reflect on, and give an account of, the nature of mathematics. From a philosophical perspective, the nature of mathematical knowledge is perhaps the central feature which the philosophy of mathematics needs to account for and reflect on. (p. 50) Without that link, Ernest argued, we cannot truly understand the aims of mathematics education. Ernest (2004) emphasized the need for researchers,

educators, and curriculum planners to ask “what is the purpose of teaching and learning mathematics?” (p. 1). But, in order to answer that, both mathematics and its role and purpose in society must be explored. Dossey (1992) also placed an emphasis on the philosophy of mathematics: “Perceptions of the nature and role of mathematics held by our society have a major influence on the development of school mathematics curriculum, instruction, and research” (p. 39). Yet, in the educational sphere, there is a lack of conversation about and exploration of philosophy that “has serious ramifications for both the practice and teaching of mathematics” (Dossey, p. 39). Without a direct focus on philosophy, the consequences of differing views of mathematics are not being explored. Ernest (1991, 1998) described two dichotomist philosophical views of mathematics—the absolutist and the fallibilist. The Platonist and formalist philosophies both stem from an absolutist view of mathematics as a divine gift or a consistent, formalized language without error or contradiction. Both of these schools of thought believe mathematics to be infallible, due either to its existence beyond humanity, waiting to be discovered (the Platonist school), or to its creation as a logical, closed set of rules and procedures (the formalist school). The fallibilist philosophy, what Hersh (1997) termed a philosophy of humanism, views mathematics as a human construction and, therefore, fallible and corrigible. One important implication of the fallibilist philosophy of mathematics is that if mathematics is a human construct then so must be the learning of mathematics. In the fallibist philosophy, mathematics is no longer knowledge that is simply memorized in a rote fashion. It is societal knowledge that must be interpreted in a manner that holds meaning for the individual. The constructivist approach to learning, therefore, aligns well with the fallibilist philosophy of mathematics. Ernest (1991, 1998) characterized a cycle of subjective and objective knowledge to support his view of the social constructivist foundations of mathematical knowledge. In this cycle, new knowledge begins as subjective knowledge, the mathematical thoughts of an individual. This new thought becomes objective knowledge, knowledge that may appear to exist independent of humanity, through a social vetting process. This objective knowledge then enters the public domain where individuals test, reformulate, and refine the knowledge. The individuals then internalize and 23

Why Not Philosophy?

interpret the objective knowledge, once again transforming it to subjective knowledge. The social process of learning mathematics is intricately linked to society’s ideas of what is and is not mathematics. Thus, Ernest was able to connect a learning theory, social constructivism, with a philosophy of mathematics. A Postmodern View of Mathematics During the past 50 years, there has been a growing discussion of the historical and philosophical foundations of mathematics. What was once seen as objective is now viewed by some as a historical and social construction, changing and malleable, as subjective as any social creation. Aligned with these changing views of mathematics are new ideas about mathematics instruction. The absolutist view of mathematics is associated with a behaviorist approach, utilizing drill and practice of discrete skills, individual activity, and an emphasis on procedures. The fallibilist view of mathematics aligns itself with pedagogy consistent with constructivist theories, utilizing problem-based learning, real world application, collaborative learning, and an emphasis on process (Threlfall, 1996). Although there have been numerous calls to change and adapt the teaching of mathematics through the embracing of a constructivist epistemology, little has been done to challenge teachers’ conceptions of mathematics. The push towards student-centered instructional practices and the current challenges to traditional views of mathematics teaching have been brought together through Ernest’s work over the past 20 years: “Teaching reforms cannot take place unless teachers’ deeply held beliefs about mathematics and its teaching and learning change” (Ernest, 1988). Ernest’s (2004) more recent work advocates a postmodern view of mathematics. He seeks to break down the influence of what he terms the “narratives of certainty” that have resulted in “popular understandings of mathematics as an unquestionable certain body of knowledge” (Ernest, 2004, p. 16). Certainly, this understanding still predominates in mathematics classrooms today (see e.g., Bishop, 2002; Brown, Jones, & Bibby, 2004; Davison & Mitchell, 2008; Handel & Herrington, 2003). However, Ernest draws upon postmodern philosophers such as Lyotard, Wittgenstein, Foucault, Lacan, and Derrida, to challenge traditional views of mathematics and mathematics education. He embraces the postmodern view because it rejects 24

the certainty of Cartesian thought and places mathematics in the social realm, a human activity influenced by time and place. Others have joined Ernest in exploring mathematics and mathematics instruction through the postmodern perspective (see e.g. Brown, 1994; Walkerdine, 1994; Walshaw, 2004). Neyland (2004) calls for a postmodern perspective in mathematics education to “address mathematics as something that is enchanting, worthy of our esteem, and evocative of wonder” (p. 69). In so doing, Neyland hopes for a movement away from mathematics instruction emphasizing procedural compliance and onto a more ethical relationship between teacher and student, one that stresses not just enchantment in mathematics education but complexity as well. Walshaw (2004) ties sociocultural theories of learning to postmodern ideas of knowledge and power, drawing, as Ernest does, on the writings of Foucault and Lacan: “Knowledge, in postmodern thinking, is not neutral or politically innocent” (p. 4). For example, issues of equity in mathematics can be seen in ways other than who can and cannot do mathematics. Indeed, societal issues of power and reproduction must be considered. A postmodern analysis forces a questioning of mathematics as value-free, objective, and apolitical (Walshaw, 2002). Why are the privileged mathematical experiences of the few held up as the needed (but never attained) mathematical experiences of all? Furthering a postmodern view of mathematics, Fleener (2004) draws on Deleuze and Guattari’s idea of the rhizome1 in order to question the role of mathematics as lending order to our world: “By pursuing the bumps and irregularities, rather than ignoring them or ‘smoothing them out,’ introducing complexity, challenging status quo, and questioning assumptions, the smoothness of mathematics is disrupted” (p. 209). The traditional view of mathematics has ignored the bumps and irregularities, forcing a vision of mathematics as smooth, neat, and orderly. Another postmodern view is that our representations of mathematics cannot be divorced from the language we use to describe those representations: Any act of mathematics can be seen as an act of construction where I simultaneously construct in language mathematics notions and the world around me. Meaning is produced as I get to know my relationships to these things. This process is the source of

Kimberly White-Fredette

the post-structuralist notion of the human subject being constructed in language. (Brown, 1994, p. 156) Brown used Derrida’s ideas on deconstructing language to examine how the social necessity of mathematical learning means that mathematics is always, in some way, constructed. And, in examining new mathematical ideas, the learners cannot help but bring their entire mathematical and personal history to the process. Brown’s (1994) postmodern view of mathematics strengthens Ernest’s (1998) own contention of the philosophical basis of social constructivism. Mathematical ideas begin as social constructions but “become so embedded within the fabric of our culture that it is hard for us to see them as anything other than givens” (Brown, p. 154). Thus, the establishment of mathematical metanarratives2 camouflages the social/cultural roots of mathematical knowledge. As a result, mathematics continues to be viewed primarily as something discovered, not constructed. Siegel and Borasi (1994) described the pervasive cultural myths that continue to represent mathematics as the discipline of certainty. In order to confront this idealized certainty, they state a need for an inquiry epistemology that “challenges popular myths about the truth of mathematical results and the way in which they are achieved, and suggests, instead, that: mathematical knowledge is fallible… (and) mathematical knowledge is a social process that occurs within a community of practice” (p. 205). This demystifying process is necessary, argued Siegel and Borasi, if teachers are to engage students in doing mathematics, not simply memorizing rote procedures and discrete skills. New Ideas in the Philosophy of Mathematics A world of ideas exists, created by human beings, existing in their shared consciousness. These ideas have objective properties, in the same sense that material objects have objective properties. The construction of proof and counterexample is the method of discovering the properties of these ideas. This branch of knowledge is called mathematics. (Hersh, 1997, p. 19) A reawakening of the philosophy of mathematics occurred during the last part of the 20th century (Hersh, 1997). In their landmark book The Mathematical Experience, Davis and Hersh (1981)

explored ideas of mathematics as a human invention, a fallibilist construct. Davis and Hersh described several schools of philosophical thought regarding mathematics—including Platonism and formalism. In the Platonist view, mathematics “has evolved precisely as a symbolic counterpart of the universe. It is no wonder, then, that mathematics works; that is exactly its reason for existence. The universe has imposed mathematics upon humanity” (p. 68). The Platonist not only accepts, but embraces, God’s place in mathematics. For what is mathematics but God’s gift to us mortals? (Plato, trans. 1956). The Platonist, forever linking God and mathematics, sees the perfection of mathematics. If there are errors made in our mathematical discoveries (and, of course, they are discoveries not inventions because they come from a higher power), then the errors are ours as flawed humanity, not inherent to the mathematics. And because mathematics is this higher knowledge it follows that some will succeed at mathematics while many others fail. Mathematics, in the Platonic view, becomes a proving ground, a place where those who are specially blessed can understand mathematics’ truths (and perhaps even discover further truths) while the vast numbers are left behind. Euclid’s Elements was (and still is) the bible of belief for mathematical Platonists (Hersh, 1997). As Davis and Hersh (1981) pointed out, “the appearance a century and a half ago of non-Euclidean geometries was accompanied by considerable shock and disbelief” (p. 217). The creation of non-Euclidean geometries—systems in which Euclid’s fifth postulate (commonly known as the parallel postulate) no longer held true—momentarily shook the very foundations of mathematical knowledge. The loss of certainty in geometry was philosophically intolerable, because it implied the loss of all certainty in human knowledge. Geometry had served, from the time of Plato, as the supreme exemplar of the possibility of certainty in human knowledge. (Davis & Hersh, p. 331) A result of the uncertainty brought on by the formation of non-Euclidean geometries was the development of formalism. In formalism, mathematics is the science of rigorous proofs, a language for other sciences (Davis & Hersh, 1981). “The formalist says mathematics isn’t about anything, it just is” (Hersh, 1997, p. 212). In the early part of the 20th century, Frege, Russell, and Hilbert, among others, each attempted to formalize 25

Why Not Philosophy?

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all of mathematics through the use of the symbols of logic and set theory. Russell and Whitehead’s “unreadable masterpiece” (Davis & Hersh, p. 138), Principia Mathematica, attempted the complete logical formalization of mathematics. But the attempts to complete the logical formalization of mathematics were doomed to failure as demonstrated by Gödel’s Incompleteness Theorem that proved any formal system of mathematics would remain incomplete, not provable within its own system (Goldstein, 2005). Proofs and Refutations: The Logic of Mathematical Discovery, a beautifully written exploration of the philosophy of mathematics penned by Imre Lakatos (1976), offered an alternative philosophy of mathematics to those of Platonism and formalism, termed the humanist philosophy. In Proofs and Refutations, Lakatos used the history of mathematics, as well as the structure of an inquirybased mathematics classroom, to explore ideas about proof. Through a lively Socratic discussion between a fictional teacher and students, Euler’s formula ( V − E + F = 2 ) is dissected, investigated, built upon, improved, and, finally, made nearly unrecognizable. Lakatos used the classroom dialogue to challenge accepted ideas about proof. He forced the reader to question if proofs are ever complete or if mathematicians simply agree to ignore the nonexamples, which Lakatos’s students termed monsters, that contradict the proof. Through this analogy, Lakatos demonstrated that in mathematics there are many monsters, most of which are ignored, as though the mathematical community has made a tacit agreement to turn away from that which makes it uncomfortable. Ernest built much of his philosophy of mathematics and mathematics education on the writings of Lakatos. Like Lakatos, Ernest (1998) saw mathematics as indubitably tied to its creator— humankind: “Both the creation and justification of mathematical knowledge, including the scrutiny of mathematical warrants and proofs, are bound to their human and historical context” (p. 44). Hersh (1997), in his book, What is Mathematics, Really?, included both Lakatos and Ernest on his list of “mavericks”— thinkers who see mathematics as a human activity and, in so doing, influenced the philosophy of mathematics. Others are included as well: philosophers Charles Sanders Peirce (Siegel & Borasi, 1994) and Ludwig Wittgenstein (Ernest, 1991, 1998b), psychologists Jean Piaget and Lev

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Vygotsky (Confrey, 1990, and Lerman, 1994), and mathematicians George Polya and Philip Kitcher. Polya’s (1945/1973) classic, How to Solve It: A New Aspect of Mathematical Method, revived the study of the methods and rules of problem solving— called heuristics—in mathematics. Although he eschewed philosophy, Polya saw mathematics as a human endeavor. He described the messiness of the mathematician’s work: Mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies; you have to try and try again. (Polya, 1954/1998, pp. 99) Both Polya and Lakatos led mathematicians into new areas that questioned the very basis of mathematical knowledge. Their combined impact on the philosophy of mathematics was as important as the development of non-Euclidean geometries and Gödel’s Incompleteness Theorem (Davis & Hersh, 1981). By defining mathematics as a social construct, they opened up the field to new interpretations. Polya’s heuristic emphasized the accessibility of problem solving. Lakatos, by using dialogue to trace the evolving knowledge of mathematics—the proofs and refutations—stressed the social aspects of mathematical learning as well as the fallibility of mathematical knowledge, and defined mathematics as quasi-empirical. No longer was mathematics a subject for the elite. Ernest (1998) credited Lakatos with a synthesis of epistemology, history, and methodology in his philosophy of mathematics—a synthesis that influenced the sociological, psychological, and educational practices of mathematics. Ernest (1998) and Hersh (1997) also referred to Kitcher as a maverick, in that he stressed the importance of both the history of mathematics and the philosophy of mathematics. Kitcher (1983/1998) underscored the concept of change in mathematics: “Why do mathematicians propound different statements at different times? Why do certain questions wax and wane in importance? Why are standards and styles of proof modified?” (p. 217). His conclusion was that mathematics changes in practice, not just in theory. Kitcher identified five components of mathematical practice—language,

Kimberly White-Fredette

metamathematical views, accepted questions, accepted statements, and accepted reasoning—that are developmentally compatible: As one component changes, others must change as well (Hersh, 1997). Kitcher’s five components emphasized the social aspect of mathematics as a community activity with agreed upon norms and practices. Kitcher’s view of mathematics mirrors Ernest’s cycle of subjective knowledge → objective knowledge → subjective knowledge and Lakatos’ idea of proofs and refutations in that each generation simultaneously critiques, internalizes, and builds upon the mathematics of the previous generation (Hersh, 1997). Conclusion Few studies have addressed the issue of teachers’ philosophies of mathematics. Too often, those that have relied on surveys and questionnaires to define the complexity that is a teacher’s philosophy (see e.g., Ambrose, 2004; Szydlik, Szydlik, & Benson, 2003; Wilkins & Brand, 2004). Although studies conducted by Lerman (1990), Wiersma and Weinstein (2001), and Lloyd (2005) briefly examined their participants’ expressed perceptions of mathematics, none of these studies specifically examined the results of an exploration of philosophy. What remains to be investigated is what happens when teachers are presented with non-traditional views of mathematics and explore philosophical writings about mathematics. At a university in Greece, Toumasis (1993) developed a course for preservice secondary school mathematics teachers that centered on readings about the history and philosophy of Western mathematics, as well as “discussion and an exchange of views” (p. 248). The purpose of the course was to develop a reflective mathematics teacher because: To be a mathematics teacher requires that one know what mathematics is. This means knowing what its history, its social context and its philosophical problems and issues are. . . . The goal is to humanize mathematics, to teach tolerance and understanding of the ideas and opinions of others, and thus to learn something of our own heritage of ideas, how we came to think the way we do (p. 255). According to Toumasis, teacher preparation programs continue to shortchange mathematics teachers by focusing only on coursework in higher

level mathematics, e.g., Linear Algebra, Discrete Mathematics, and Analysis. Knowledge of mathematics, especially if one is to teach mathematics, must include a reflexive study of mathematics. Toumasis (1997) argued that the philosophical and epistemological beliefs about the nature of mathematics are intrinsically bound with the pedagogy of mathematics. In his examination of the philosophical underpinnings of NCTM’s Curriculum and Evaluation Standards for School Mathematics (1989), Toumasis identified a clear fallibilist point of view; mathematics is “a dialogue between people tackling mathematical problems” (p. 320). Yet in current attempts to reform mathematics based on both the 1989 Standards and the later Principles and Standards for School Mathematics (NCTM, 2000), an investigation of philosophy is rarely undertaken. The teaching and learning of mathematics is a politically charged arena. Strong feelings exist in the debate on how “best” to teach mathematics in K–12 schools, feelings that are linked to varying perceptions about the nature of mathematics (Dossey, 1992). Is mathematics an abstract body of ideal knowledge, existing independently of human activity, or is it a human-construct, fallible and everchanging? These perceptions of mathematics then drive beliefs about the appropriateness of instructional practices in mathematics. Is mathematics a body of knowledge that must be memorized and unquestionably mastered, or do we engage the learners of mathematics in personal sense-making, in constructing their own mathematical knowledge? That we are still in the midst of “math wars” is indisputable (Schoenfeld, 2004). What it means to teach mathematics and the very nature of mathematics is at the center of these wars: Traditionalists or back-to-basics proponents argue that the aim of mathematics education should be mastery of a set body of mathematical knowledge and skills. The philosophical complement to this version of the teaching and learning of mathematics is mathematical absolutism. Reform-oriented mathematics educators, on the other hand, tend to see understanding as a primary aim of school mathematics. Constructivism is often the philosophical foundation for those espousing this version of mathematics education. (Stemhagen, 2008, p. 63) 27

Why Not Philosophy?

I agree with Schoenfeld (2004), Greer and Mukhopadhyay (2003), and others (e.g., Davison & Mitchell, 2008) that the math wars are based on philosophical differences. It has therefore been my intent to inject philosophy into the discussion of mathematics educational reform and research. Research is needed that focuses on what we teach as mathematics and, more importantly, how teachers view the mathematics that they teach. Is mathematics transcendental and pure, something that exists outside of humanity, or is it a social activity, a social construction whose rules and procedures are defined by humanity (Restivo & Bauchspies, 2006)? An extensive review of the literature found no studies that led teachers to explore their philosophies of mathematics. Yet Restivo and Bauchspies recognized the need to push teachers’ understanding of mathematics beyond the debate of mathematics as a social construction. To understand mathematics (and thus to teach mathematics) is to understand the social, cultural, and historical worlds of mathematics (Restivo & Bauchspies). Should we not then explore mathematics in a philosophical sense, its “basic principles and concepts…with a view to improving or reconstituting them” (Webster’s Dictionary, 2003, p. 1455)? Change in classroom practices may not be possible without first “improving or reconstituting” teachers’ philosophies of mathematics. Many studies have addressed the need to engage both preservice and inservice teachers in constructivist learning in order to change their instructional practices (see, e.g., Hart, 2002; Mewborn, 2003; Thompson, 1992). Yet little has been done to engage teachers in a philosophical discussion of mathematics: “Teachers, as well, should be encouraged to develop professionally through philosophical discourse with their peers” (Davison & Mitchell, 2008, p. 151). Philosophy and mathematics have a long-standing connection, going back to the ancient Greeks (Davis & Hersh, 1981). Mathematics teachers are seldom asked to explore philosophy beyond an introductory Philosophy of Education course. If one is going to teach mathematics, one should ask “Why?” What is the purpose of teaching mathematics? What is the purpose of mathematics in society at large? Should not mathematics’ purpose be tied to how we then teach it? These questions come back to teachers’ perception of mathematics, and more specifically, their philosophies of mathematics.

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I contend that discussions of philosophy, particularly philosophy of mathematics, should be brought to the forefront of mathematics education reform. If teachers are never asked to explore the philosophical basis of their perceptions of mathematics, then they will continue to resist change, to teach the way they were taught. The growing philosophical investigations of mathematics (see, e.g., Davis & Hersh, 1981; Hersh, 1997; Restivo, Van Bendegen, & Fischer, 1993; Tymoczko, 1998) in the past 30 years have not often been addressed in mathematics education research. We seem afraid to raise issues of philosophy as we implement curriculum reform and study teacher change. But philosophy too often lies hidden, an unspoken obstacle in the attempt to change mathematics education (Ernest, 2004). Researchers can bring the hidden obstacle to light, should engage both policymakers and educators in a conversation about philosophy, not with the intent of enforcing the “right” philosophy but with the acknowledgement that, without a continued dialogue about philosophy, the curriculum reform they research may continue to fall short. I end this article by revisiting a definition of philosophy of mathematics: “The philosophy of mathematics is basically concerned with systematic reflection about the nature of mathematics, its methodological problems, its relations to reality, and its applicability” (Rav, 1993, p. 81). If our goal in mathematics education reform is to make mathematics more accessible and more applicable to real-world learning, we should then help guide today’s teachers of mathematics to delve into this realm of systematic reflection and to ask themselves, “What is mathematics?” 1

Building from the botanical definition of rhizome, Deleuze and Guattari (1980/1987) used the analogy of the rhizome to represent the chaotic, non-linear, postmodern world. Like the tubers of a canna or the burrows of a mole, rhizomes lead us in many directions simultaneously. Deleuze and Guattari described the rhizome as having no beginning or end; it is always in the middle. 2 A meta-narrative, wrote Kincheloe and Steinberg (1996), “analyzes the body of ideas and insights of social theories that attempt to understand a complex diversity of phenomena and their interrelations” (p. 171). It is, in other words, a story about a story; a meta-narrative seeks to provide a unified certainty of knowledge and experience, removed from its historic or personalized significance.

Kimberly White-Fredette

References Ambrose, R. (2004). Initiating change in prospective elementary school teachers’ orientations to mathematics teaching by building on beliefs. Journal of Mathematics Teacher Education, 7, 91–119. Beswick, K. (2007). Teachers’ beliefs that matter in secondary mathematics classrooms. Educational Studies in Mathematics, 65(1), 95–120. Bibby, T. (1999). Subject knowledge, personal history, and professional change. Teacher Development, 3(2), 219–232. Bishop, A. J. (2002). Critical challenges in researching cultural issues in mathematics education. Journal of Intercultural Studies, 23(2), 119–131. Brown, M. (1998). The paradigm of modeling by iterative conceptualization in mathematics education research. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (pp. 263–276). Norwell, MA: Kluwer Academic. Brown, T. (1994). Describing the mathematics you are part of: A post-structuralist account of mathematics learning. In P. Ernest (Ed.), Mathematics, education and philosophy: An international perspective (pp. 154–162). London: The Falmer Press. Brown, T., Jones, L., & Bibby, T. (2004). Identifying with mathematics in initial teacher training. In M. Walshaw (Ed.), Mathematics education within the postmodern (pp. 161–179). Greenwich, CT: IAP Information Age. Confrey, J. (1990). What constructivism implies for teaching. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on teaching and learning of mathematics, Journal for Research in Mathematics Mducation, Journal for Research in Mathematics Education Monograph no. 4 (pp. 107–124). Reston, VA: National Council of Teachers of Mathematics. Cooney, T., Shealy, B., & Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. Journal for Research in Mathematics Education, 29, 306–333. Damarin, S. (1999). Social construction and mathematics education: The relevance of theory. In L. Burton (Ed.), Learning mathematics: From hierarchies to networks (pp. 141–150). London: Routledge Falmer. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston: Birkhauser. Davison, D. M., & Mitchell, J. E. (2008). How is mathematics education philosophy reflected in the math wars? The Montana Mathematics Enthusiast, 5(1), 143–154. Deleuze, G., & Guattari, F. (1987). A thousand plateaus: Capitalism and schizophrenia (B. Massumi, Trans.). Minneapolis, MN: University of Minnesota Press. (Original work published 1980) Deleuze, G., & Guattari, F. (1994). What is philosophy? (H. Tomlinson & G. Burchell, Trans.) New York: Columbia University Press. (Original work published 1991) Dossey, J. (1992). The nature of mathematics: Its role and its influence. In D. A. Grouws (Ed.), Handbook of research in mathematics teaching and learning (pp. 39–48). New York: Macmillan.

Ernest, P. (1988). The impact of beliefs on the teaching of mathematics. [Electronic Version]. Retrieved May 2, 2006, from http://www.people.ex.ac.uk/PErnest/impact.htm Ernest, P. (1991). The philosophy of mathematics education. New York: Falmer Press. Ernest, P. (1994). Social constructivism and the psychology of mathematics education. In P. Ernest (Ed.), Constructing mathematical knowledge: Epistemology and mathematics education (pp. 62–72). London: The Falmer Press. Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany, NY: State University of New York Press. Ernest, P. (1999). Forms of knowledge in mathematics and mathematics education: Philosophical and rhetorical perspectives. Educational Studies in Mathematics, 38, 67–83. Ernest, P. (2004). What is the philosophy of mathematics education? [Electronic Version]. Philosophy of Mathematics Education Journal, 18. Retrieved January 4, 2006, from http://www.people.ex.ac.uk/PErnest/pome18/PhoM_%20for_I CME_04.htm Fleener, J. M. (2004). Why mathematics? Insights from poststructural topologies. In M. Walshaw (Ed.), Mathematics education within the postmodern (pp. 201–218). Greenwich, CT: IAP Information Age. Forman, E. A. (2003). A sociocultural approach to mathematics reform: Speaking, inscribing, and doing mathematics within communities of practice. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.). A research companion to Principles and Standards for School Mathematics (pp. 333–352). Reston, VA: National Council of Teachers of Mathematics. Goldstein, R. (2005). Incompleteness: The proof and paradox of Kurt Gödel. New York: W. W. Norton & Company. Handel, B., & Herrington, A. (2003). Mathematics teachers’ beliefs and curriculum reform. Mathematics Education Research Journal, 15(1), 59–69. Hart, L. (2002). A four year follow-up study of teachers’ beliefs after participating in a teacher enhancement project. In G. Leder, E. Pehkonen, & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education (pp. 161–176). Norwell, MA: Kluwer Academic. Hersh, R. (1997). What is mathematics, really? New York: Oxford University Press. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Mathematics Learning Study Committee. Washington, DC: National Academies Press. Kincheloe, J. L., & Steinberg, S. R. (1996). A tentative description of post-formal thinking: The critical confrontation with cognitive theory. In P. Leistyna, A. Woodrum, & S. A. Sherblom (Eds.), Breaking free: The transformative power of critical pedagogy (pp. 167–195). Cambridge, MA: Harvard Educational Review. Kitcher, P. (1984). The nature of mathematical knowledge. New York: Oxford University Press. Kitcher, P. (1998). Mathematical change and scientific change. In T. Tymoczko (Ed.), New directions in the philosophy of mathematics (pp. 215–242). Princeton, NJ: Princeton University Press. (Original work published in 1983)

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Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press. (Original work published in 1945) Lerman, S. (1990). Alternative perspective of the nature of mathematics. British Educational Research Journal, 16(1), 53–61. Lerman, S. (1998). Research on socio-cultural perspectives of mathematics teaching and learning. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (pp. 333–350). Norwell, MA: Kluwer Academic. Lerman, S. (1999). Culturally situated knowledge and the problem of transfer in the learning of mathematics. In L. Burton (Ed.), Learning mathematics: From hierarchies to networks (pp. 93– 107). London: Routledge Falmer. Lloyd, G. (2005). Beliefs about the teacher’s role in the mathematics classroom: One student teacher’s explorations in fiction and in practice. Journal of Mathematics Teacher Education, 8, 441–467. Mewborn, D. (2002). Examining mathematics teachers’ beliefs through multiple lenses. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA, April 1–5, 2002. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academies Press. National Research Council. (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press. Neyland, J. (2004). Toward a postmodern ethics of mathematics education. In M. Walshaw (Ed.), Mathematics education within the postmodern (pp. 55–73). Greenwich, CT: IAP Information Age. Noddings, N. (1990). Constructivism in mathematics education. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on teaching and learning of mathematics, Journal for Research in Mathematics Education Monograph no. 4 (pp. 7–18). Reston, VA: National Council of Teachers of Mathematics. Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307–332. Polya, G. (1973). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press. (Original work published in 1945) Polya, G. (1998). From the preface of Induction and Analogy in Mathematics. In T. Tymoczko, (Ed.), New directions in the philosophy of mathematics (pp. 99–101). Princeton, NJ: Princeton University Press. (Original work published in 1954)

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Rav, Y. (1993). Philosophical problems of mathematics in the light of evolutionary epistemology. In S. Restivo, J. P. Van Bendegem, & R. Fischer (Eds.), Math worlds: Philosophical and social studies of mathematics and mathematics education (pp.80–109). Albany, NY: State University of New York Press. Restivo, S., & Bauchspies, W. K. (2006). The will to mathematics. Foundations of Science, 11, 197–215. Restivo, S., Van Bendegen, J. P., & Fischer, R. (Eds.). (1993). Math worlds: Philosophical and social studies of mathematics and mathematics education. Albany, NY: State University of New York Press. Schoenfeld, A. (2004). The math wars. Educational Policy, 18 (1), 253–286. Sfard, A. (1998). The many faces of mathematics: Do mathematicians and researchers in mathematics education speak about the same thing. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (pp. 491–511). Norwell, MA: Kluwer Academic. Siegel, M., & Borasi, R. (1994). Demystifying mathematics education through inquiry. In P. Ernest (Ed.), Constructing mathematical knowledge: Epistemology and mathematics education (pp. 201–214). London: The Falmer Press. Sztajn, P. (2003). Adapting reform ideas in different mathematics classrooms: Beliefs beyond mathematics. Journal of Mathematics Teacher Education, 6, 53–75. Szydlik, J. E., Szydlik, S. D., & Bensen, S. R. (2003). Exploring changes in pre-service elementary teachers’ mathematical beliefs. Journal of Mathematics Teacher Education, 6, 253– 279. Threlfall, J. (1996). Absolutism or not absolutism: What difference does it make? [Electronic Version]. Philosophy of Mathematics Education Journal, 9. Retrieved February 23, 2006, from http://www.people.ex.ac.uk/PErnest/pome/pome9.htm Toumasis, C. (1993). Ideas and processes in mathematics: A course in history and philosophy of mathematics. Studies in Philosophy and Education, 12, 245–256. Toumasis, C. (1997). The NCTM Standards and its philosophy of mathematics. Studies in Philosophy and Education, 16, 317– 330. Vogotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press. Walkerdine, V. (1994). Reasoning in a post-modern age. In P. Ernest (Ed.), Mathematics, education and philosophy: An international perspective (pp. 61–75). London: The Falmer Press. Walshaw, M. (2002). Epistemic terrains and epistemic responsibility [Electronic Version]. Philosophy of mathematics education, 16. Retrieved February 25, 2006, from http://www.people.ex.ac.uk/PErnest/pome16/epistemic.htm Walshaw, M. (2004). The pedagogical relation in postmodern times: Learning with Lacan. In M. Walshaw (Ed.), Mathematics education within the postmodern (pp. 121–139). Greenwich, CT: IAP Information Age. Webster’s New Universal Unabridged Dictionary. (2003). New York: Barnes & Nobles.

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Wiersma, L., & Weinstein, G. L. (2001). Mathematical sophistication and educational philosophies among novice mathematics teachers [Electronic Version]. Philosophy of Mathematics Education Journal, 14. Retrieved February 23, 2006, from http://www.people.ex.ac.uk/PErnest/pome14/wiersma.htm

Wilkins, J. L. M., & Brand, B. R. (2004). Change in preservice teachersâ€™ beliefs: An evaluation of a mathematics methods course. School Science and Mathematics, 104(5), 226â€“232.

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The Mathematics Educator 2006, Vol. 19, No. 1, 32–45

Measuring Task Posing Cycles: Mathematical Letter Writing Between Algebra Students and Preservice Teachers Anderson Norton Zachary Rutledge In a secondary school mathematics teaching methods course, a research team engaged 22 preservice secondary teachers (PSTs) in designing and posing tasks to algebra students through weekly letter writing. The goal of the tasks was for PSTs to elicit responses that would indicate student engagement in the mathematical processes described by NCTM (2000) and Bloom’s taxonomy (Bloom, Englehart, Furst, Hill, & Krathwohl, 1956), as well as student engagement in the highest levels of cognitive activity described by Stein, Smith, Henningsen, and Silver (2000). This paper describes our efforts to design reliable measures that assess student engagement in those processes as a product of the evolving relationship within letter-writing pairs. Results indicate that some processes are easier to elicit and assess than others, but that the letter-writing pairs demonstrated significant growth in terms of elicited processes. Although it is impossible to disentangle student factors from teacher factors that contributed to that growth, we find value in the authenticity of assessing PSTs’ tasks in terms of student engagement rather than student-independent task analysis.

Designing and posing tasks plays a central role for mathematics teaching (Krainer, 1993; NCTM, 2000). However, research indicates that preservice teachers lack ability to pose appropriately challenging mathematical tasks for students (e.g., Silver, MamonaDowns, Leung, & Kenney, 1996). This article addresses the development of such ability by engaging preservice secondary teachers (PSTs) in posing mathematical tasks to high school algebra students through mathematical letter writing. We consider our approach an extension of the kind of letter-writing study performed by Crespo (2000; 2003). In a previous article (Rutledge & Norton, 2008), we reported results from this project related to the letter-writing interactions between PSTs and students. That article focused on comparing cognitive constructivist and socio-cultural lenses for examining the interactions. The purpose of this article is to investigate the mathematical processes that PSTs’ tasks elicited from students. Crespo (2003) engaged preservice elementary school teachers in posing mathematical tasks to fourthgrade students through letter writing. The purpose of her study was to elicit and assess students’ mathematical thinking. She found the tasks preservice Anderson Norton is an Assistant Professor in the Department of Mathematics at Virginia Tech. He teaches math courses for future secondary school teachers and conducts research on students' mathematical development. Zachary Rutledge is completing a PhD in Mathematics Education at Indiana University. He works as an actuary in Salem, Oregon.

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teachers wrote became more open-ended and cognitively complex over the weeks of letter writing. This result affirmed her key hypothesis that the preservice teachers’ extended and reflective interactions with an “authentic audience” (p. 243) would provide opportunities for them to learn how to pose appropriately challenging tasks. Crespo’s work informed our approach to studying the development of task-posing ability among PSTs, and we too used letter writing with PSTs to foster such development. Rather than focusing on the tasks PSTs posed, as Crespo did, we specifically examined elicited student responses as a product of the evolving relationship within letterwriting pairs. During a secondary methods course, 22 PSTs were paired with high school algebra students; the PSTs posed tasks to their student partners and assessed the responses. As researchers, we independently examined the responses from the algebra students to make inferences about their cognitive activity. Considering this study to be an extension of Crespo’s work, we introduce a method for measuring PSTs’ progress in learning to design and pose individualized mathematical tasks through letter writing. We measured the effectiveness of PSTs’ tasks by assessing the cognitive activities those tasks elicited from students (as indicated by student responses), and we hypothesized that such measurements would demonstrate growth over the course of letter-writing exchanges between the pairs. We report on our design of measurements for the effectiveness of the letter-writing pairs, in addition to

Anderson Norton & Zachary Rutledge

the results of applying that design. In particular, we relied on descriptions of cognitive activities described in three main sources: Bloom’s taxonomy (Bloom et al., 1956; Kastberg, 2003), Principles and Standards of School Mathematics (NCTM, 2000), and a chapter on “cognitively complex tasks” by Stein, Smith, Henningsen, and Silver (2000). We chose these sources because they are common readings in the PSTs’ methods courses, and they provide potential metrics for assessing the quality of tasks. We collectively modified them to form a comprehensive and complementary framework for assessing students’ responses to the tasks. In the following section we summarize the original authors’ descriptions of these processes. We then describe how we operationalized the processes to assess the cognitive activities indicated by each student response. In the final two sections, we report on the reliability of our measures and the evolution of cognitive activity elicited by the PSTs’ tasks over the course of 12 weeks. Findings from this study inform the following research questions: How can we reliably measure the effectiveness of the letter-writing exchanges in terms of elicited cognitive activity from the high school students? And, using the measurements we develop, in what ways do the PSTs demonstrate progress in designing and posing appropriately challenging mathematical tasks for students? We wanted PSTs to learn to pose more engaging mathematical tasks and to assess students’ thinking based on their written responses. We hypothesized that over the 12 weeks the PSTs’ tasks would elicit more of NCTM’s Process Standards (2000) and the four highest levels of reasoning in Bloom’s taxonomy (i.e. application, analysis, synthesis, and evaluation). We also expected a general progression toward responses that indicated students were using Procedures with Connections and Doing Mathematics, moving away from responses reliant upon Memorization or Procedures without Connections (Stein et al., 2000). Theoretical Orientation Task Posing Since Brown and Walter’s (1990) seminal work on problem posing (read as “task posing”), many subsequent publications focused on teachers engaging students in posing problems (e.g., Gonzales, 1996; Goldenberg, 2003). Whereas these publications have implications for teacher education, they do not examine teachers’ abilities to design appropriately challenging tasks for their students. Research investigating teachers’ abilities to design such tasks has

typically focused on student-independent attributes of the tasks, such as whether they introduce new implicit assumptions, initial conditions, or goals (Silver et al., 1996). Similarly, Prestage and Perks (2007) engaged PSTs in modifying givens and analyzing mathematical demand of tasks in order for these future teachers to develop fluency in creating ad hoc tasks in the classroom. However, Prestage and Perks noted, “the analysis of the mathematics within a task can only offer a description of potential for learning” (p. 385). Understanding the actual cognitive demand of a task depends upon the learner. “Today, there is general agreement that problem difficulty is not so much a function of various task variables, as it is of characteristics of the problem solver” (Lester & Kehle, 2003, p. 507). One such characteristic that has received insufficient attention during task posing is the students’ understanding of mathematics content (NCTM, 2000, p. 5). As Crespo (2000; 2003) demonstrated, letter writing can provide a rich context for PSTs to develop task-posing ability through mathematical interactions with students and help PSTs better attend to student understanding of content. Liljedhal, Chernoff, and Zazkis (2007) described another important component of task-posing for PSTs: “predicting the affordances that the task may access” (p. 241) as PSTs attempt to elicit particular mathematical concepts or processes from students. However, within letter writing such analyses no longer determine whether the task is ‘good’ because PSTs can rely on students’ actual responses for making that determination. Crespo (2003) described letter writing as an opportunity for “an authentic experience in that it paralleled and simulated three important aspects of mathematics teaching practice: posing tasks, analyzing pupils’ work, and responding to pupils’ ideas” (p. 246). The authenticity of PST-student interactions is highly desirable because the PSTs can assess the effectiveness of their tasks without relying on the authority of a teacher educator. The benefits of this kind of authenticity might be analogous to students’ experiences when they view their own mathematical reasoning as an authority, rather than relying on the text or a teacher for validation. The PSTs’ problem becomes one of “witnessing the development of the activities provoked by the task, and comparing it to the ones they predicted and to the initial task” (Horoks & Robert, 2007, p. 285). This development allows PSTs to use these comparisons as they modify their initial tasks and design new tasks.

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Measuring Task Posing Cycles

Cognitive Measures (in Theory) In order to measure PSTs’ progress in eliciting mathematical activity from students through task posing, we looked to three sources: Bloom’s taxonomy (Bloom et al., 1956), the NCTM Process Standards (2000), and the levels of cognitive demand designed by Stein et al. (2000). PSTs’ familiarity with these sources was important to us for the following reason: Often these sources (or others that describe a hierarchy for analyzing student thinking) are introduced to PSTs as useful ideas to adapt into their future teaching. Teacher educators should move beyond introduction of these sources and instead facilitate opportunities for PSTs to investigate ways that they prove beneficial in working with students. Therefore, we asked PSTs to assess their students’ responses to tasks using these sources. This mirrors the way that we used them in this study to assess the PST’s task-posing ability.

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Table 1 presents the measures created for this study based on these frameworks. The short definition provides a summary of the different measures as described by the original authors. The first four measures come from Bloom’s taxonomy of educational objectives (Bloom et al., 1956), the next five measures come from NCTM’s Process Standards (2000), and the last four measures come from Stein et al. (2000). It is important to note that we chose to disregard the first two levels of Bloom’s taxonomy, Knowledge and Comprehension, because we felt that these measures were too low-level and would likely be elicited with great frequency. On the other hand, we kept all four of Stein’s levels of cognitive demand because they provide a necessary hierarchy for ranking tasks and measuring growth, as we describe in the following section.

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Methodology Setting The 22 PSTs who participated in this study were enrolled in the first of two mathematics methods courses that precede student teaching at a large midwestern university. Mrs. Rae, a local high school mathematics teacher, was interested in finding ways to challenge her students by individualizing instruction. When the PSTs’ methods instructor (first author) approached her about task-posing through letter writing, Mrs. Rae agreed that such an activity would serve the educational interests of her students, as well as the PSTs. Each PST was assigned to one student from Mrs. Rae’s Algebra I class, and wrote letters back and forth to her or his assigned student, once per week for seven weeks. The PSTs were given no guidelines on the type of problems to pose; instead they were instructed to focus on building students’ mathematical engagement. The high school term (trimester) ended after the seventh week and students were assigned to new classes, so the PSTs began writing letters to a new group of students in the eighth week. They wrote to Mrs. Rae’s Algebra II students the final five weeks of the project. Each week, the methods instructor and Mrs. Rae collected the letters and responses, respectively, and exchanged them. In the title of this article, we use the term cycle to refer to the PSTs’ iterative task design. After posing an initial task, we expected PSTs to use student responses to construct models of students’ mathematical thinking. That is, we expected PSTs to try to “understand the way children build up their mathematical reality and the operations by means of which they try to move within that reality” (von Glasersfeld & Steffe, 1991, p. 92). Using this knowledge, the PSTs could design tasks more attuned with their students’ mathematics—the students’ particular mental actions and ways of applying those actions to problem-solving situations. In turn, we hypothesized that the well-designed tasks would presumably increase student engagement and cognitive activity. By focusing PSTs’ attention on the cognitive activities described by NCTM, Bloom et al., and Stein et al., we hoped to provide a framework for PSTs to begin building models. Whereas PSTs’ goals for student learning often revert to mastery of procedural knowledge (Eisenhart et al., 1993), we promoted goals for conceptual learning among the PSTs through class readings and discussions. We encouraged PSTs to use open-ended tasks (i.e. tasks that invite more than one particular response) so student responses would be rich enough

for PSTs to make inferences about the students’ thinking. We hoped the opportunity to make inferences about the students’ mathematical thinking would lead the PSTs to construct models of students’ mathematics. We also encouraged the PSTs to rely on their models to imagine how students’ mathematics might be reorganized in order to become more powerful, allowing the students to engage with a broader range of mathematical situations. Data Analysis Data consisted of PSTs’ letters and students’ responses. PSTs complied these documents into their notebooks, and we collected them at the end of their methods course. After removing 31 letters that were not matched with task responses, 233 tasks/response pairs remained to be analyzed. Data analysis had four phases: (a) operationalization of our cognitive measures, (b) the raters’ individual coding, (c) reconciliation of our individual coding, and (d) interpretation of the final codes. The operationalization concerns the way in which we transformed the theoretical processes given in the previous section into heuristics that allowed us to identify cognitive activity. Individual coding relied on this operationalization while continuing to inform further operationalization of the cognitive measures. As not to distort inter-rater reliability scores, in the interim we met only to discuss clarifications of the cognitive activities, without sharing notes or discussing particular responses. At the end of the letter-writing project, we computed the inter-rater reliability of our coding for the cognitive measures. Following this analysis, we reconciled our codes by arguing points of view regarding scoring differences until we reached consensus. Finally, we could interpret the reconciled codes, graphically and statistically. Graphs of the relative frequency of each cognitive activity, as measured week-by-week, provide an indication of growth among the PST-student pairs. We use the graphs to describe patterns in elicited activity over time. Although the two different groups of students involved in letter writing (Algebra I students in the first seven weeks, and Algebra II students in the final five weeks) render a 12-week longitudinal analysis untenable, data from the two groups do provide opportunity for us to consider differences in PSTs’ success in working across the groups. Finally, we performed linear regressions on aggregate results to provide a statistical analysis of progress.

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Cognitive Measures (in Practice) The operationalization occurred mostly during the individual coding phase with minor adjustments required during the reconciliation phase. That is to say, we essentially transformed the 13 measures into a system allowing a researcher or a practitioner to categorize his or her inferences of studentsâ€™ cognitive activity. To achieve this transformation, we began with the previously discussed definitions for the various measures and then made adjustments throughout the individual coding phase. Whenever one of the raters (authors) encountered difficulty in assessing a student response, he would approach the other to discuss the difficulty, in a general way, without referring to a particular student response. This interaction would allow the raters to decide how to resolve the difficulty and individually reassess previous ratings to ensure consistent use of the newly operationalized measure. Table 2 describes the most fundamental changes that we made to the measures. The adjustments are the results of the following two goals: (1) to ensure that measures could be consistently applied from task to task and (2) to ensure that no two measures were redundant. With regard to redundancy, we had to differentiate Connections, Procedures with Connections, and

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Application. We used Connections to refer to connections among disparate mathematical concepts; we reserved Procedures with Connections to describe connections between mathematical procedures and concepts; and we reserved Application to describe connections among mathematical concepts and other domains. With regard to our ability to consistently apply a measure, we had difficulty with Problem Solving and Doing Mathematics. As defined in Table 1, Problem Solving requires a struggle toward a novel solution, rather than the application of an existing procedure or concept. Lester and Kehle (2003) further characterized problem solving as â€œan activity requiring the individual to engage in a variety of cognitive actionsâ€? (p. 510). Lester and Kehle (2003) described a tension between what is known and what is unknown: Successful problem solving involves coordinating previous experiences, knowledge, familiar representations and patterns of inference, and intuition in an effort to generate new representations and patterns of inference that resolve the tension or ambiguity (i.e., lack of meaningful representations and supporting inferential moves) that prompted the original problem solving activity. (p. 510)

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Other cognitive activities, such as representation and inference (possibly involving reasoning and proof), may support a resolution to this tension. For us to infer that a student engaged in Problem Solving, we needed to identify indications of this perceived tension, and we needed to infer a new construction through a coordination of cognitive actions. This meant responses labeled as involving Problem Solving were often labeled as involving other cognitive activities as well, such as Analysis and Reasoning & Proof. While Stein et al.’s (2000) definition of Doing Mathematics provided some orientation for our work, we found it to be too vague for our purposes. So, we relied on Schifter’s (1996) definition for further clarification; she defined Doing Mathematics as conjecturing. To infer a student had engaged in Doing Mathematics, we needed to infer the student had engaged in making and testing conjectures. For example, if we inferred from student work that the student designed a mathematical formula to describe a situation and then appropriately tested this formula, then we would consider this to be Doing Mathematics. We recognize that our restriction precludes assessment of other activities that Stein et al. (2000) would consider to be “doing mathematics,” but this restriction provided a workable resolution to assessing students’ written responses. As a final and general modification of the original

cognitive measures, we required that each process (other than the lowest two levels of cognitive demand) produce a novelty. For example, assessing Synthesis required some indication that the student had produced a new whole from existing constituent parts. Furthermore, we needed indication that the student generated the cognitive activity as part of their reasoning. If a PST explicitly asked for a bar graph, the students’ production of it would not constitute Representation, because it would not indicate reasoning. Such a response would probably indicate a Procedure without Connections. Results A Letter-Writing Exchange To illustrate the exchanges between letter-writing pairs, and to clarify the manner of our assessments, we provide the following sample exchange. A complete record of the exchange can be found in Rutledge and Norton (2008). Figure 1 shows the task posed by a PST, Ellen, in her initial letter to her student partner, Jacques. The task is similar to other introductory tasks posed by PSTs and seems to fit the kinds of tasks to which they had become accustomed from their own experiences as students. However, there is evidence (i.e. the “why” questions at the task’s end) that Ellen, like fellow PSTs, attempted to engage the student in responding with more than a computational answer.

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Jacques’ response (Figure 2) indicates that he did not meaningfully engage in the task of finding equations for lines meeting the specified geometric conditions. However, he was able to assimilate (make sense of) the situation as one involving solutions to systems of equations. From his activity of manipulating two linear equations and their graph, we inferred that the task elicited only procedural knowledge from Jacques. It is possible that Jacques may have had a more connected understanding of the concepts underlying the procedure, but there was no clear indication from his response that allowed us to infer this. Therefore, we coded the elicited activity as Procedures without Connections.

provoked Jacques to struggle through finding equations for the lines. In addition, Ellen seemed to detect an overall trend that Jacques engaged more readily with familiar procedures. She adapted to the trend and began to frame future tasks around a procedure with which she felt Jacques was likely familiar. This kind of adaptation to the student indicates that Ellen began to model the student’s thinking.

Other codes assigned to Jacques’ response included Application and Communication. The former was based on our inference that Jacques used existing ideas in a novel situation. He effectively applied an algebraic procedure to a new domain when he applied his knowledge of systems of equations to a situation involving finding equations of intersecting lines, When coding for Communication, we inferred that Jacques’ written language intended to convey a mathematical idea involving the use of systems of equations to find points of intersection. Subsequent tasks and responses indicate Ellen began to model Jacques’ mathematical thinking. Using these models, she designed tasks that successfully engaged Jacques in additional cognitive activities, such as problem solving. For example, Ellen asked questions to focus Jacques’ attention on the angles formed in the drawing on Figure 1. Her questions

reconcile discrepancies. To understand inter-rater reliability, we considered three measures as shown in Tables 4 and 5. These measures were Cohen’s Kappa, Percent Agreement, and Effective Percent Agreement. As Table 4 indicates, percent agreement was high on all four measures from Bloom’s taxonomy, but this result is because of the rarity of either rater identifying the measures. Effective percent agreement provides further confirmation of this outcome by considering agreement among those items positively identified by at least one of the raters. Moving from left to right, the Kappa scores in Table 3 show decreasing inter-rater reliability as we progress to higher levels of Bloom’s taxonomy. Sim and Wright (2005) cite Landis and Koch who suggest the following delineations for interpreting Kappa scores: less than or equal to 0 poor, 0.01-0.20 slight, 0.21-0.40 fair, 0.41-0.60 moderate, 0.61-0.80

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Inter-Rater Reliability Results When we finished individually coding all of the task responses for the letter-writing pairs, we met to compile the results into a spreadsheet, for this process allowed us to measure inter-rater reliability and

Anderson Norton & Zachary Rutledge

substantial, and 0.81-1 almost perfect. With this in mind, we see Application has a substantial agreement, Analysis has moderate agreement, Synthesis has slight agreement, and Evaluation did not have agreement distinguishable from random. The high percent agreement in combination with a low effective percent agreement for Synthesis and Evaluation highlight the fact the raters rarely identified these two constructs. In the few times one rater identified such a construct when the other did not, this resulted in a low Kappa score. For further support of this assessment, we note that the 95% confidence interval for these two measures includes 0; thus, there is no support for inter-rater reliability with these two measures. Table 4 displays the data for the NCTM Process Standards. In a similar analysis, Communication and Problem Solving have moderate agreement. Representation is on the border between fair and moderate; Reasoning & Proof is on the border between slight and fair; and Connections has a poor inter-rater reliability. Connections and Reasoning & Proof were quite rare, hence the high levels of percent agreement and the lower level of Kappa, as was the case with

Synthesis and Evaluation. In fact, the 95% confidence interval for each of these measures includes 0, indicating no reliability on these measures. Table 5 displays the ratersâ€™ responses for the levels of cognitive demand described by Stein et al. (2000). The Kappa was .55â€”described as moderately reliableâ€”with a 95% confidence interval of .47 to .64 (note that we report only one Kappa because we could choose only one categorization for each task response). We see that although 164 of the 233 items are on the main diagonal (showing agreement), there is definite spread away from the diagonal as well. We determined this was partly attributable to a shift in how conservatively the raters interpreted student responses. Specifically, one rater tended to identify items as eliciting lower cognitive demand than the other rater. This is seen in the total column and row, where one rater identified 178 items as either Memorization or Procedures without Connections and only 45 items as Procedures with Connections. Alternately, the other rater found 157 items to be either Memorization or Procedures without Connections and 70 items to be Procedures with Connections.

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design in that these measures were more commonly identified in the coding. Elicited-Response Results

To summarize, Evaluation was the measure with the weakest reliability, for it had a negative Kappa associated with it. Although having a positive Kappa, Synthesis, Connections, and Reasoning & Proof had confidence intervals that contained 0; this result indicates we cannot be certain whether it was above 0 randomly. It is also important to note these measures were some of the least-often identified. Conversely, the most reliable measures were Communication, Problem Solving, and Application. In addition, our assessments of levels of cognitive demand were reliable to a similar degree. Again, this conclusion is supported by our

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After measuring inter-rater reliability, we reconciled our scores by arguing for or against each discrepant score. For example, the second author assessed Task 1 (Figure 1 and Figure 2) as having elicited Connections. However, the first author successfully argued that the evidence was stronger for connections to concrete situations, and according to our operationalization that should be coded as Application. We report on the reconciled scores in Figures 3, 4, and 5. Each of those figures illustrates the percentages of responses that satisfied our negotiated measures over the course of the twelve weeks. We excluded missing responses from all calculations. Although we included letters from week 1, we note many of the introductory letters did not include tasks, presumably because the PSTs were becoming familiar with the students and the format of the activity. We also note the first seven letters were written between PSTs and Algebra I students, whereas the final five letters were written between PSTs and Algebra II students. Once again, the letters written in week 8 were introductory letters, though these included many more tasks. In Figures 3, 4, and 5, the dark vertical line

Anderson Norton & Zachary Rutledge

between weeks 7 and 8 marks the transition from Algebra I letters to Algebra II letters. Ignoring the introductory letters from week 1, the general trends illustrated in Figure 3 indicate the frequency of PSTs eliciting Application from students decreased over the duration of letter writing, even across correspondences with Algebra I and Algebra II students. Analysis was elicited more frequently, with a pronounced spike among correspondence between PSTs and Algebra II students in the final weeks. As previously mentioned in the reliability results, we found both Synthesis and Evaluation were rarely elicited in correspondence with either group of students. These patterns indicate the levels of cognition described by Bloom’s taxonomy—at least in our operationalization of them—were heavily dependent on the PSTs and the tasks they posed. Interestingly, these patterns show little apparent dependence on the groups of students (i.e. Algebra I and Algebra II). This outcome may be because many of the posed tasks inherently required application and analysis to resolve them, with PSTs gaining a greater appreciation for students’ use of analysis over the course of the semester. Application tasks tended to describe new situations where the PSTs inferred, often correctly, that the students could use existing knowledge. For

example, we characterized Jacques’ response in Figure 2 as indicating an Application, for he applied his knowledge about solving linear equations to a concrete situation that required finding the point of intersection of two lines. Analysis tasks often involved equations whose components needed to be examined. For example, in a subsequent exchange with Ellen, Jacques broke down the triangle (Figure 1) into three lines and correctly identified the sign of the slopes of these lines. Using this knowledge, he attempted to formulate the equations of these lines. It seems that either PSTs were less familiar with the kinds of tasks that might elicit Synthesis and Evaluation, or students did not readily engage in such activity. In Figure 4, we begin to see some differences between the elicited responses of the two groups of students. Whereas Connections, Representation, and Reasoning & Proof were rarely elicited from either group of students, there is a pronounced increase in Problem Solving among correspondence with the Algebra II students as compared to the Algebra I students. Communication also increased among correspondences with Algebra II students, but seemed to be elicited in a pattern that was similar across the two groups. We see mathematical communication from both groups of students increased to a peak during the

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middle weeks, and then decreased toward the end of the correspondence between letter writing pairs. This trend may be due to the PSTs’ initial interest in the students’ thinking, which was replaced by more goaldirected tasks, once the PSTs determined a particular trajectory along which to direct the students. We found Communication dropped among both groups of students after their fourth week. This could be due to a lack of enthusiasm among the students once the novelty of letter writing had faded. In fact, Mrs. Rae noticed the Algebra I students began to tire of writing responses and wrote less in later weeks. Figure 5 illustrates a general trend away from tasks eliciting Memorization. It seems the PSTs used students’ recall of facts in order to gauge where the students were developmentally, both at the beginning and end of their correspondence with the students. Procedures without Connections dominated the elicited responses from students, whereas Procedures with Connections seemed to play a significantly lesser role. It is also interesting to note that the few instances identified as Doing Mathematics occurred among correspondence with Algebra II students. Along with the previous observation about Problem Solving (namely, that problem solving was elicited much more with Algebra II students), these results lead us to one

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of two conclusions: (1) the PSTs held higher expectations for Algebra II students (in terms of cognitive activity, and not just content) and were, therefore, more inclined to challenge them with higherlevel tasks, or (2) the Algebra II students were better prepared (either from previous learning or accepted social norms) to engage in these higher levels of cognitive activity. A Statistical Analysis In addition to considering the measures individually, we performed a linear regression on aggregate results over time. The first column in Table 6 lists the average number of processes elicited weekby-week, among the five NCTM Process Standards and the four highest levels of Bloom’s taxonomy. For example, the PSTs elicited, on average, two of the nine processes during Week 10. The second column in Table 6 lists the average ranking of the levels of cognitive demand elicited week-by-week. We ranked Memorization as 0, Procedures without Connections as 1, Procedures with Connections as 2, and Doing Mathematics as 3. If we accept this simple form of ranking, the student responses from Week 7 indicated, on average, as a 1, Procedures without Connections.

Anderson Norton & Zachary Rutledge

measurements. We intend to improve the measurements in terms of their reliability and their value as assessments of professional growth. First, we recognize areas of weakness in reliable uses of the measures, as well as areas of weakness in elicited responses. These areas coincide because cognitive activities that were least assessed were assessed least reliably; they include Synthesis, Evaluation, Connections, Reasoning & Proof, and Representation. Reliability of Measures as Operationalized

Table 7 reports the slopes and r-squared values for each column over each of the following time periods: the first seven weeks (interactions with Algebra I students), the final five weeks (interactions with Algebra II students), and the entire 12 weeks (across the two groups of students). In addition, Table 7 includes the corresponding p-values to indicate whether the slopes are statistically significant. We calculated these values using rank coefficients. The slopes provide indications of the group’s growth from week to week to the degree that the r-squared values approach 1 and p-values approach 0.05. It is interesting to note that the slope, r-squared value, and p-value for the final five weeks of letter writing suggest considerable growth in the level of mathematical engagement during interactions between PSTs and Algebra II students. Discussion of Findings and Implications Having operationalized measures of cognitive activity and having applied them to a cohort of letterwriting pairs, we are now prepared to evaluate the

When we reconciled our independent assessments of task responses, common themes emerged concerning the least assessed cognitive activities. Some of these involved highly subjective judgments, such as the novelty of the activity for the student and the student’s familiarity with particular concepts and procedures. This subjectivity highlights the need for us to make our assessments based on inferences about the student’s mathematical activity, just as we asked the PSTs to design their tasks based on such inferences. For example, one student assimilated information from a story problem in order to produce a simple linear equation. During coding, this response would typically be evaluated as Synthesis; however, one of the raters inferred that the student was so familiar with the mathematical material that her actions indicated a procedural exercise. For our reconciliations of all measures, we agreed each cognitive activity needed to produce a mathematical novelty, such as a tabular representation a student produced to organize data in resolving the task. If the PST were to request the table, then the student’s production of it would not be considered novel. Therefore, this response would not be labeled as a Representation. Likewise, we decided Connections should be used to refer to a novel connection between two mathematical concepts, such as a connection a student might make between his or her concepts of function and reflection in resolving a task involving transformational geometry. The need to make inferences about the novelty of students’ activities introduced ambiguity in assessing student responses,

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particularly because we assessed responses week-byweek without considering the history of each student’s responses. We also realized we introduced some reliability issues through our selection of cognitive activities. Whereas we were pleased with the diversity of measures offered by the three categorizations (Bloom’s taxonomy, NCTM’s Process Standards, and Stein’s levels of cognitive demand), these are not mutually exclusive. Despite our efforts to operationalize the measures in a way that would reduce overlap, we realized, for example, Connections would always implicate Procedures with Connections or Doing Mathematics. Additionally, we recognize that Doing Mathematics would always implicate Problem Solving, and Reasoning & Proof would always implicate Communication. Recognizing such implications might reduce ambiguity and increase reliability by eliminating some of the perceived need for raters to choose one measure over another. Finally, because frequently elicited cognitive activities were measured reliably, we conclude that supporting PSTs’ attempts to elicit all cognitive activities can increase the reliability of each measurement. This support would also promote our goals for PSTs to design more engaging tasks. Eliciting Cognitive Activity We originally hypothesized that PSTs’ tasks would elicit more cognitive processes over time, and we anticipated a general progression toward the highest levels of cognitive demand. Such findings would indicate growth in the evolving problemposing/problem-solving relationships between PSTs and students. Our hypothesis is confirmed to the degree that r-squared values indicate the positive slopes reported in Table 7. Those values indicate that the relationships were particularly productive between PSTs and Algebra II students. There are many reasons students’ content level might have influenced the relationship, and we cannot discern the main contributors. Possible contributors include the following: (1) PSTs wrote to the Algebra II students second and for a shorter duration so the students’ remained motivated throughout the project; (2) greater content knowledge of Algebra II students contributed to greater process knowledge as well by allowing the students to engage in more problem solving or make more connections; (3) PSTs were more familiar with the content knowledge of Algebra II students so the PSTs were better prepared to design more challenging tasks; (4) social norms in the two classes differed and affected students’ levels of engagement. In any case,

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our findings do indicate that PSTs—as a whole and over the course of the entire twelve weeks—became more successful in eliciting cognitive activity through their letter writing relationships. Our findings also indicate which cognitive activities seem most difficult to elicit through letter writing, and we have suggested classroom social norms play a role in students’ reluctance to engage in some of those activities, such as Problem Solving and Doing Mathematics. However, we also found that PSTs were able to engage students in some cognitive activities, such as Application and Communication, which affirms, “prospective teachers have some personal capacity for mathematical problem posing” (Silver et al., 1996, p. 293). Moreover, PSTs demonstrated increased proficiency at engaging their student partners in additional higher-level cognitive activities, such as Analysis. Silver et al. found, “the frequency of inadequately stated problems is quite disappointing” (1996, p. 305). Although, like Silver et al., our expectations for our PSTs’ task design were not met, we found students accepted nearly all of the tasks as personally meaningful and engaged in some kind of mathematical activity as a result. The disparity of this finding with that of Silver et al. (1996) might be attributed to our disparate approaches in studying problem posing. Most notably, the PSTs in our study designed tasks with particular students in mind and used student responses to assess the effectiveness of those tasks and to model students’ thinking. We believe such experiences are essential to making methods courses personally meaningful to future teachers. References Bloom, B., Englehart, M. Furst, E., Hill, W., & Krathwohl, D. (1956). Taxonomy of educational objectives: The classification of educational goals. Handbook I: Cognitive domain. New York: Longmans. Brown, S. I., & Walter, M. I. (1990). The Art of Problem Posing. Hillsdale, NJ: Erlbaum. Crespo, S. (2000). Seeing more than right and wrong answers: Prospective teachers’ interpretations of students’ mathematical work. Journal of Mathematics Teacher Education, 3, 155– 181. Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in preservice teachers' practices. Educational Studies in Mathematics, 52, 243–270. Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993). Conceptual knowledge falls through the cracks: Complexities of learning to teach mathematics for understanding. Journal for Research in Mathematics Education, 24, 8–40.

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Goldenberg, E. P. (2003). Problem posing as a tool for teaching mathematics. In H. L. Schoen (Ed.), Teaching mathematics through problem solving: Grades 6-12 (pp. 69–84). Reston, VA: National Council of Teachers of Mathematics. Gonzales, N. A. (1996). A blueprint for problem posing. School Science and Mathematics, 98, 448–456. Horoks, J., & Robert, A. (2007). Tasks designed to highlight taskactivity relationships. Journal of Mathematics Teacher Education, 10, 279–287. Kastberg, S. (2003). Using Bloom’s taxonomy as a framework for classroom assessment. The Mathematics Teacher, 96, 402– 405. Krainer, K. (1993). Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction. Educational Studies in Mathematics, 24, 65–93. Lester, F. K., & Kehle, P. E. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism (pp. 501–518). Mahwah, NJ: Lawrence Erlbaum. Liljedahl, P., Chernoff, E., & Zazkis, R. (2007). Interweaving mathematics and pedagogy in task design: A tale of one task. Journal of Mathematics Teacher Education, 10, 239–249.

Prestage, S., & Perks, P. (2007). Developing teacher knowledge using a tool for creating tasks for the classroom. Journal of Mathematics Teacher Education, 10, 381–390. Rutledge, Z., & Norton, A. (2008). Preservice teachers' mathematical task posing: An opportunity for coordination of perspectives. The Mathematics Educator, 18(1), 31–40. Schifter, D. (1996). A constructivist perspective on teaching and learning mathematics. In C. T. Fosnot (Ed.), Constructivism: Theory, perspectives, and practice (pp. 73–80). New York,: Teachers College Press. Silver, E. A., Mamona-Downs, J., Leung, S. S., & Kenney, P. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27, 293–309. Sim, J., & Wright, C. C. (2005). The Kappa Statistic in reliability studies: Use, interpretation, and sample size requirements. Physical Therapy, 85, 257–268. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press. von Glasersfeld, E., & Steffe, L. P. (1991). Conceptual models in educational research and practice. The Journal of Educational Thought, 25(2), 9–103.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

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Manuscript Submission Guidelines

The Mathematics Educator (ISSN 1062-9017) strives to provide a forum for collaboration of mathematics educators at varying levels of professional experience throughout the field. The Mathematics Educator is abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on Mathematical Education). 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. The Mathematics Educator encourages the submission of a variety of types of manuscripts from students and other professionals in mathematics education including: • reports of research (including experiments, case studies, surveys, and historical studies); • descriptions of curriculum projects or classroom experiences; • literature reviews; • theoretical analyses; • critiques of general articles, research reports, books, or software; • commentaries on research methods in mathematics education; • commentaries on public policies in mathematics education. The work must not be previously published except in the case of: • translations of articles previously published in other languages; • abstracts of or entire articles that have been published in journals or proceedings that may not be easily available. Manuscripts submitted to TME are put through blind review before they are accepted into the journal. The blind review process typically takes four months. Guidelines for Manuscript Specifications • Manuscripts should be double-spaced with one-inch margins, 12-point Times New Roman font, and a maximum of 25 pages (including references and footnotes). An abstract should be included and references should be listed at the end of the manuscript. The manuscript, abstract, and references should conform to American Psychological Association Publication Manual, Sixth Edition (APA 6th). • An electronic copy is required. The electronic copy must be in Word format and 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 TME 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. • Original figures, tables, and graphs should appear embedded in the document and conform to APA 6th. To Become a Reviewer Contact the Editors at the postal or email address below. Please indicate if you have special interests in reviewing articles that address certain topics such as curriculum change, student learning, teacher education, or technology. Allyson Hallman & Catherine Ulrich Postal Address: The Mathematics Educator 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

Electronic address: tme@uga.edu

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TME Subscriptions TME is published both online and in print form. The current issue as well as back issues are available online at http://math.coe.uga.edu/TME/tmeonline.html. A paid subscription is required to receive the printed version of The Mathematics Educator. Subscribe now for Volume 20 Issues 1 & 2. You may subscribe online by visiting http://math.coe.uga.edu/Mesa/MESA.html and emailing your shipping address to tme@uga.edu. Alternatively, send a copy of this form, along with the requested information and the subscription fee to The Mathematics Educator 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

___ I wish to subscribe to The Mathematics Educator for Volume 20 (Numbers 1 & 2). ___ I would like a previous issue of TME sent. Please indicate Volume and issue number(s): ___________________

Name

Amount Enclosed ________________ subscription: $10/domestic; $15/international

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In this Issue, Mathematics Is Motivating THOMAS E. RICKS Preservice Teachers’ Emerging TPACK in a Technology-Rich Methods Class S. ASLI ÖZGÜN-KOCA, MICHAEL MEAGHER, & MICHAEL TODD EDWARDS Why Not Philosophy? Problematizing the Philosophy of Mathematics in a Time of Curriculum Reform KIMBERLY FREDETTE-WHITE Measuring Task Posing Cycles: Mathematical Letter Writing Between Preservice Teachers and Algebra Students ANDERSON NORTON & ZACHARY RUTLEDGE

The Mathematics Education Student Association is an official affiliate of the National Council of Teachers of Mathematics. MESA is an integral part of The University of Georgia’s mathematics education community and is dedicated to serving all students. Membership is open to all UGA students, as well as other members of the mathematics education community.

Visit MESA online at http://www.coe.uga.edu/mesa