PROMOTING TEACHER LEARNING OF MATHEMATICS - CiteSeerX

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PROMOTING TEACHER LEARNING OF MATHEMATICS: THE USE OF “TEACHING-RELATED MATHEMATICS TASKS” IN TEACHER EDUCATION (1) Gabriel J. Stylianides University of Pittsburgh [email protected]

Andreas J. Stylianides University of California-Berkeley [email protected]

Research suggests that teachers need to have mathematics content knowledge that allows them to effectively deal with the particular mathematical issues that arise in their everyday practice. This implies the importance of providing teachers with learning opportunities that prepare them to both recognize situations in their practice where these mathematical issues arise and be able to apply their mathematical knowledge to successfully manage these situations. Yet, little research has focused on how such learning opportunities can be effectively promoted in teacher education. In this article we take a step toward addressing this limitation by discussing and exemplifying a special kind of tasks for use in teacher education which we call “teaching-related mathematics tasks.” These are mathematics tasks that are connected to teaching and can foster the development of teachers’ mathematics content knowledge that is important for teaching. In recent years, there has been increased research attention to the mathematics content knowledge that is important for teaching (e.g., Ball et al., 2001; Ball & Bass, 2000, 2003; Davis & Simmt, 2006; Ma, 1999; Shulman, 1986). A major idea advanced in this body of research, especially in the work of Ball and Bass, is that teachers need to have mathematics content knowledge that allows them to effectively deal with the mathematical issues that arise in their everyday practice, which are generally different from the mathematical issues that arise in the everyday practice of other professionals who use mathematics. For example, the work of an engineer does not (normally) necessitate that the engineer knows different methods for dividing fractions, or how these methods correspond to one another. A teacher, however, needs to be able to reason accurately and quickly about different methods – both valid and invalid – for dividing fractions in order to be able to function effectively in teaching situations where this mathematical issue arises. In the context of a classroom discussion of the standard “invert and multiply” method, a student may ask the teacher whether it is correct to use a different method such as “dividing numerators and denominators.” This situation raises for the teacher some critical questions: Is dividing numerators and denominators a valid method for dividing two fractions? If so, how does this method correspond to the standard invert and multiply method? The idea that teachers need to have mathematics content knowledge that allows them to effectively deal with the mathematical issues that arise in their everyday practice implies the importance that teacher education programs design opportunities for teachers to learn mathematics that are tailored for the particular needs of teaching. Specifically, it implies the importance that these programs provide teachers with learning opportunities that prepare them to both recognize situations in their practice where different mathematical issues arise (like the situation described in the previous paragraph) and be able to apply their mathematical knowledge to successfully manage these situations. Despite the importance of these learning opportunities for teachers, little research has focused on how such opportunities can be effectively promoted in teacher education (both teacher preparation and professional development programs). In this article, which is a continuation of Stylianides and Stylianides (2006), we take a step toward addressing this limitation by discussing a special kind of mathematics tasks – which we _____________________________ Alatorre, S., Cortina, J.L., Sáiz, M., and Méndez, A.(Eds) (2006). Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Mérida, México: Universidad Pedagógica Nacional.

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call “teaching-related mathematics tasks” – that can foster the development of teachers’ mathematics content knowledge that is important for teaching. Our goal is to further elaborate on the nature and illustrate the utility of this kind of tasks. To promote our goal, we connect the notion of teaching-related mathematics tasks with existing literature and we offer a critical reflective account of our own personal experiences in designing and implementing one such task in a mathematics course for preservice elementary teachers (grades K-6). To investigate more systematically the utility of the tasks in teacher education we are in the process of conducting a design experiment, the findings of which will appear in a future report. The Notion of “Teaching-related Mathematics Tasks” Consider the following two versions of a mathematics task that aims to promote teacher learning of proof in the domain of multiples of a number. The first version is a standard mathematics task typically used in mathematics courses for preservice elementary teachers, whereas the second version is a teaching-related mathematics task that we use in our mathematics course for preservice elementary teachers. Version 1 (standard mathematics task): Develop three proofs for the claim: “A multiple of 3 plus a multiple of 3 equals a multiple of 3.” One proof should use everyday language, another proof should use pictures, and the third proof should use algebra. Version 2 (teaching-related mathematics task): As the mathematics specialist in your school, you teach mathematics in all three fifth-grade classes: Class A, Class B, and Class C. The past week you have been working with all three classes on the notion of multiples of a number. The three classes developed appropriate definitions of multiples of a number, but in each class these definitions were represented in a different way. In Class A they were represented using everyday language, in Class B they were represented pictorially, and in Class C they were represented algebraically. In the next lesson, you plan to engage your students in the three classes in proving the claim: “A multiple of 3 plus a multiple of 3 equals a multiple of 3.” In preparation for this lesson, you want to take into account the fact that each class shares definitions that are represented in a different way. So, for Class A you want to prepare a proof that uses everyday language, for Class B you want to prepare a pictorial proof, and for Class C you want to prepare an algebraic proof. Write the three proofs. Although both tasks aim to promote teacher learning of proof in the domain of multiples of a number, they do so in substantially different ways. Contrary to Version 1 of the task, Version 2 facilitates connections between the intended learning and the work of teaching by situating the preservice teachers’ mathematical activity in an authentic teaching situation where this learning is crucial. In particular, Version 2 of the task helps preservice teachers make connections between learning equivalent proofs that utilize different representations and teaching situations where only certain representational tools are available in the shared knowledge of a particular classroom community. By so doing, Version 2 of the task helps preservice teachers appreciate what this task intends to teach them and, thus, makes it more likely than Version 1 of the task that preservice teachers will learn the mathematical ideas embedded in the task (Harel, 1998). According to Harel, “[s]tudents are most likely to learn when they see a need for what we intend to teach them, where by ‘need’ is meant intellectual need, as opposed to social or economic need” (p. 501). In the case of preservice teachers, “need” can be defined as their interest in

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developing mathematics content knowledge that is useful for their work and in becoming able to apply this knowledge to successfully manage teaching situations where this knowledge is called for. The consideration of preservice teachers’ need for learning is consistent with findings of research on the motivational aspects of cognition which suggest that a prerequisite for successful problem solving is that problem solving be situated in personally meaningful contexts (see, e.g., Mayer, 1998; Renninger et al., 1992; Weiner, 1986). According to the findings of this body of research, students learn better – that is, they think harder and process the material more deeply and with more likelihood of transfer – when they have an interest in the material. Given our analysis of versions 1 and 2 of the task above, we argue that mathematics courses for teachers (both preservice and inservice) need to place emphasis on the use of tasks like Version 2 that we call teaching-related mathematics tasks. These are mathematics tasks that are connected to teaching, and have a dual purpose: (1) to foster learning of mathematics that is important for teaching; and (2) to help teachers see how this mathematics relates to teaching, thereby increasing the possibility that they will learn it and use it in their work. By “task” we mean a sequence of related activities (e.g., engagement with a mathematical problem accompanied by reflection on the work to solve the problem) that focus on a particular idea and aim to promote a particular goal. Teachers’ engagement with teaching-related mathematics tasks can be thought of as a process of mathematizing teaching. Our use of the term “mathematizing” follows that of Freudenthal (1973, 1991). Freudenthal, used mathematizing to describe a notion of mathematics as an activity – as schematizing, structuring, and modeling the world mathematically, rather than as a discipline of structures to be transmitted, discovered, or even constructed. Also, we use mathematizing to include both horizontal and vertical mathematization, i.e., the process of converting a contextual (teaching) problem to a mathematical problem that can be solved mathematically and the process of taking the mathematical content to a higher (meta) level, respectively (Treffers, 1987). We elaborate on these ideas below. A teaching-related mathematics task engages teachers in studying (e.g., interpreting, analyzing) a teaching situation with a mathematical lens, and, in some cases, the task engages them additionally in studying their own mathematical activity in the task at a meta-level (e.g., reflecting on their own work in the task). Version 2 of the task we presented earlier provides an example of the former: it engages preservice teachers in studying mathematically the situation where a teacher needs to develop three proofs for a mathematical claim given the constraints in the representational tools available to different student populations. If the task additionally asked preservice teachers to reflect on their own mathematical activity in the task, this would facilitate studying the mathematical activity at a meta-level. Such a reflection could help preservice teachers identify their mathematical activity in this particular task as being part of the more general mathematical activity of producing equivalent ways to represent different mathematical ideas (e.g., arguments, concepts) within the constraints of given mathematical systems. Identification of this general mathematical activity would promote teachers’ understanding of where and how in teaching they would need to engage in a similar mathematical activity, thereby increasing the possibility of them transferring the mathematics content knowledge to be acquired from their engagement in this teaching-related mathematics task to new teaching situations. A Mathematics Course for Preservice Elementary Teachers Over the past three years, we have developed a set of teaching-related mathematics tasks on different content areas (e.g., algebra, geometry, number theory) that we have piloted – with promising results – in five enactments of a mathematics course for preservice elementary

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teachers. The course places emphasis on the use of teaching-related mathematics tasks and aims to advance preservice teachers’ content knowledge that is important for teaching elementary school mathematics. A major goal of the course is to promote a notion of mathematics as a sense-making activity – that is, activity characterized by meaningful learning – and to create a community of mathematical discourse where ideas are validated based on mathematical argument. As a result the notion of proof holds a prominent place in the course. To facilitate connections between the mathematics learnt in the course and the work of teaching, the course makes extensive use of records of classroom practice (e.g., video or written reports of lessons, student artifacts). These records are derived from various sources, such as books (e.g., Carpenter et al., 2003), research reports (e.g., Zack, 1997), and a database of the Mathematics Teaching and Learning to Teach (MTLT) project at the University of Michigan that documents an entire year of the mathematics teaching of Deborah Ball in a United States public school third-grade class. An Example of A Teaching-related Mathematics Task From the Course In this section we present a teaching-related mathematics task we designed and implemented in our mathematics course for preservice elementary teachers. The presentation of the task is based on data (video records of lessons and student artifacts) we collected during the last enactment of the course in 2006; the instructor was the first author. Description This teaching-related mathematics task comes from the content area of number theory. In the class prior to the implementation of this task, the preservice teachers analyzed textbook definitions of even and odd numbers, and developed equivalent definitions of these concepts that are accurate mathematically and appropriate for use in the elementary grades. With this teachingrelated mathematics task, we aimed to help preservice teachers understand the utility of definitions in mathematical reasoning and, in particular, in proving true claims over infinite sets. The task had three parts: Individual work to prove the claim: An odd number plus an odd number equals an even number. Discussion of the mathematical issue raised in a videoclip from third grade where students try to prove the same claim. Individual and small group work to revisit their work in part A. Part B is particularly important, because it uses a teaching situation to help preservice teachers realize the limitations of empirical arguments (that predominate in their solutions of part A), understand the importance of definitions in the development of proofs, and motivate them to revisit their work in part A to produce a proof. Below we consider each part separately. Part A. The course was taken by 18 preservice teachers. Their written responses to part A of the task are classified as follows: two responses did not offer any real argument for the claim to be proved (e.g., they were faulty or irrelevant responses); nine were empirical arguments; two were unsuccessful attempts for a general argument; and five were general arguments (see Table 1 for illustrative examples). By “general arguments” we mean arguments that address appropriately the general case and either qualify as proofs or require minor refinements (e.g., better justification of a step in the argument) before they qualify as such. By “proofs” we mean valid arguments from accepted truths (e.g., definitions, axioms) for or against mathematical claims. In

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sum, at the beginning of preservice teachers’ engagement with the task, only seven out of 18 made an effort to produce arguments that address the general case. Response Commentary An odd number multiplied by 2 always gives an This is a faulty response that does not odd number. offer any real argument for the claim to be proved. 3+7=10, 1+5=6, 7+9=16, 3+37=40, -47+3=-44, This is an empirical argument -3+1=-2, -3+(-1)=-4, -37+(-51)=-88, 573+697=1270. because the claim follows as a Any odd number added to another odd number generalization from the confirming would equal an even number. Any way that you do evidence offered by the examination of a it, it comes out to an even number. few particular cases (a proper subset of all the possible cases). Any odd number on a number line that is moved This is an unsuccessful attempt for a an odd number of spaces to the right (i.e., an odd general argument because, although it number is added to it), you always will land on an deals with the general case, it does not even number. produce a valid argument. An even number can be divided equally by two. This a general argument that qualifies An odd number can be divided by two with one left as a proof because it makes adequate use over. If you add together two odd numbers, you are of definitions of even and odd numbers to also adding the two leftovers, which will be able to deduce logically the truth of the claim be grouped by two without leftovers. “odd+odd=even.” Table 1. Illustrative examples of different types of responses to part A of the task Part B. After the preservice teachers completed their individual work on part A, and without discussing their solutions in the whole group, the instructor engaged them in watching and discussing a videoclip from the MTLT database. The videoclip shows the third graders in Deborah Ball’s class investigating the same mathematical claim as in part A of the task, namely, that the sum of any two odd numbers is even. Ofala, like many other students in the third-grade class, asserts that the claim is true because she verified it in a few particular cases (e.g., 1+5=6). Jeannie, however, begins to worry about what it really means to prove a claim that involves an infinite number of cases. She says: “Numbers go on and on forever and that means odd numbers and even numbers, um, go on for ever and, um, so you couldn’t prove that all of them work.” (See Ball and Bass [2003] for a more detailed description of the classroom episode in the videoclip.) The guiding question for the preservice teachers in watching the videoclip was the following: What was the mathematical issue raised in the videoclip and how could this issue be addressed? After some discussion in small groups and then in the whole group, the preservice teachers concluded that the main issue in the videoclip was that checking a few particular cases does not suffice to prove true mathematical claims like “odd+odd=even” that involve an infinite number of cases. Regarding the second part of the guiding question, one preservice teacher suggested that “there’s a consistency in how odd and even numbers behave and this consistency guarantees that what happens in particular cases happens in all cases.” The instructor highlighted this idea of consistency and asked the preservice teachers to think more about it. What exactly about even and odd numbers guarantees this consistency? Another preservice teacher then pointed out that the consistency resides in the definitions of even and odd numbers.

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Part C. After the discussion of the videoclip, the instructor introduced part C of the task: “Now that we’ve watched the clip and understood what the issue there was, I’d like you to think whether the proofs you gave [in part A] for the claim ‘odd+odd=even’ address this issue. If they don’t, how could you revise your proofs?” After the small group work, the instructor asked the different groups to present their proofs in the whole class. Out of five small groups, two tried unsuccessfully to produce a general argument and three produced general arguments. In other words, all groups made an explicit effort to avoid empirical arguments and to produce arguments that address the general case. Discussion A well-documented finding in the literature is that many students and teachers of mathematics demonstrate a reliance on the use of examples to verify the truth of general mathematical claims (e.g., Knuth et al., 2002; Martin & Harel, 1989; Simon & Blume, 1996). This finding is illustrated both in the work of our preservice teachers in part A of the task and in the episode used in part B of the task that showed the work of the third graders in Deborah Ball’s class. The teaching-related mathematics task helped us to engage our preservice teachers in thinking about the important mathematical issue of how one can prove true mathematical claims that involve an infinite number of cases. The innovative aspect of the task is that, instead of relying on the instructor and his/her authority to convince preservice teachers about the limitations of empirical arguments to prove general mathematical claims, it used a classroom episode from third grade that was raising the exact same issue. The set up of the task helped preservice teachers reconsider in productive ways their original approaches to proving the claim in part A of the task, resulting in significantly improved arguments in part C. Also, the preservice teachers appeared to be motivated to revisit their original approaches and to appreciate the value of what they were learning. The latter is encouraging, especially in the context of a task focusing on the notion of proof, because preservice teachers tend to consider proof as an advanced topic and, thus, they are often resistant to engage in activities that aim to develop their generally weak content knowledge in this important mathematical domain. Finally, we should note that part B of the task engaged preservice teachers in a process of mathematizing teaching, as it engaged them in interpreting and analyzing a teaching situation with a mathematical lens. Conclusion This article contributes to teacher educators’ understanding of how theoretical ideas on the mathematics content knowledge that is important for teaching can be used to design useful tasks that offer preservice teachers rich opportunities to learn mathematics in connection to the work of teaching. The reflective account of our own personal experiences in designing and implementing teaching-related mathematics tasks illuminates aspects of the role and nature of this kind of tasks, and suggests their promise in teacher education. Further research is needed to develop a comprehensive set of teaching-related mathematics tasks and investigate systematically their effectiveness by conducting experimental or quasi-experimental studies that will examine their impact on teachers’ learning of mathematics and on their teaching practice. Endnote 1. The two authors had an equal contribution in writing this article.

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