International Group for the Psychology of Mathematics Education, Vol. 5, pp. ... knowledge of âreasoning and provingâ that is important for teaching elementary.
CONTENT KNOWLEDGE FOR MATHEMATICS TEACHING: THE CASE OF REASONING AND PROVING1 Andreas J. Stylianides
Gabriel J. Stylianides
University of California-Berkeley
University of Pittsburgh
In this paper, we discuss issues of content knowledge that is important for mathematics teaching. Specifically, we present a framework for the content knowledge of “reasoning and proving” that is important for teaching elementary school mathematics, and we consider how this knowledge can be effectively promoted in mathematics courses for preservice elementary school teachers. We argue that these courses need to place emphasis on the use of a special category of tasks that we call “teaching-related mathematics tasks.” These are mathematics tasks that are connected to teaching, and have a dual purpose: (1) to foster teacher learning of mathematics that is important for teaching, and (2) to help teachers see how this mathematics relates to teaching. We discuss and illustrate the nature of these tasks. Over the past two decades an extensive body of research has focused on teachers’ content knowledge (Shulman, 1986), studying its relationship with effective teaching. It is now well documented in the literature that teachers’ ability to teach mathematics depends on their mathematics content knowledge (e.g., Ma, 1999). Nevertheless, it is still unclear what teachers need to know about each mathematical topic or activity they teach (Ball et al., 2001), and also how this knowledge can be effectively promoted in mathematics teacher preparation programs. In this paper, we take a step toward addressing this problem, focusing on the activity of reasoning and proving (RP). Specifically, we: (1) present a framework for the content knowledge of RP that is important for elementary mathematics teaching; and (2) consider how this knowledge can be effectively promoted in mathematics courses for preservice elementary school teachers, focusing on the nature of tasks used in these courses. BACKGROUND As background for the paper, we (1) elaborate on the importance of teachers’ content knowledge of RP, (2) present our conceptualization of RP, and (3) examine an episode from third grade to illustrate what RP might look like at the elementary school (grades K through 6) and to see what is important for elementary school teachers to know about RP in order to cultivate this activity in their classrooms. Importance of teachers’ content knowledge of reasoning and proving There is growing appreciation of the idea that doing and knowing mathematics is a sense-making activity, that is, activity characterized by meaningful learning. There is also appreciation of the intimate relation between sense making and the activity 1
The two authors had an equal contribution in writing this paper. We wish to thank Alan Schoenfeld for his useful comments on an earlier version of the paper.
2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 5, pp. 201-208. Prague: PME. 5 - 201
Stylianides & Stylianides of RP: a typical structure of students’ engagement in sense making is to first explore mathematical phenomena to identify patterns and make conjectures, and then investigate with arguments and proofs the truth of the conjectures to establish new knowledge (Boero et al., 1996; Mason et al., 1982; NCTM, 2000; Schoenfeld, 1983). Because students’ engagement in mathematics as a sense-making activity is a high-priority goal of school instruction, and because of the intimate relation between sense making and RP, many researchers and curriculum frameworks (especially in the USA) recommend that RP become central to all students’ mathematical experiences across all grades (e.g., Ball & Bass, 2003; NCTM, 2000; Yackel & Hanna, 2003). Yet, research shows that students of all levels face serious difficulties acquiring competency in RP (e.g., Healy & Hoyles, 2000; Reiss et al., 2002). Research shows also that teachers, especially elementary teachers, have weak content knowledge of RP (e.g., Martin & Harel, 1989; Simon & Blume, 1996), and that textbooks used in mathematics courses for preservice teachers do not systematically support rich opportunities for teacher learning of RP (McCrory et al., 2004). This situation is problematic: teachers’ ability to teach is a function of their content knowledge, so if teachers do not develop deep and robust content knowledge of RP, we cannot expect that they will be able to effectively promote RP in their classrooms. A conceptualization of reasoning and proving in school mathematics In school mathematics, the development of proofs is often treated as a formal process (in high school geometry), isolated from other mathematical activities. However, this treatment of proof is problematic. The work in which mathematicians themselves engage that culminates in a proof typically involves searching a mathematical phenomenon for patterns, making conjectures based on the patterns, and providing informal arguments demonstrating the viability of the conjectures (Schoenfeld, 1983). These activities aid any doer of mathematics in understanding the phenomenon under examination, building a foundation for the development of proofs (Boero et al., 1996; Mason et al., 1982; G. Stylianides, 2005). Thus, by viewing proof in isolation from the activities that support its development, we do not afford students the same level of scaffolding used by professional users of mathematics to make sense of and establish mathematical truth. Our conceptualization of RP situates the development of proofs in the set of activities involved in making sense of and establishing mathematical truth. Specifically, we use RP to describe the overarching activity that encompasses the set of activities associated with identifying patterns (general relations that fit given sets of data), making conjectures (reasoned hypotheses that are subject to testing), providing arguments (connected sequences of assertions) for or against the conjectures, and developing proofs (valid arguments from accepted truths that establish the truth or falsity of the conjectures) (G. Stylianides, 2005). This conceptualization of RP is not linked to any particular mathematical domain (e.g., geometry) or grade level. 5 - 202
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Stylianides & Stylianides Reasoning and proving at the elementary school: An episode from third grade Episode: In a third-grade class, students are investigating what happens when they add any two odd numbers. They check several examples, identify the pattern that the sum in all the examined cases is an even number, and formulate – with their teacher’s help – the conjecture that the sum of any two odd numbers is even. The teacher then challenges the students to explain why their conjecture should be accepted. This provokes many different arguments. Jeannie argues that the class cannot prove the conjecture for all pairs of odd numbers, because “odd numbers and even numbers go on for ever and so one cannot prove that all of them work.” But other students disagree with her. For example, Ofala asserts that the conjecture is true because she verified it in 18 particular cases (e.g., 1+5=6). The lesson continues with students sharing their thoughts and the teacher pressing students to justify their thinking. The next lesson begins with the teacher reviewing the issue raised by Jeannie about why the class could not prove the conjecture. The teacher explains to the students that mathematicians would address this issue by trying to see what property of odd numbers makes the combination of two of them always an even number. She then helps the students remind themselves of their definitions of even and odd numbers, and challenges them to use these definitions to prove the conjecture. Betsy proposes the following proof for the conjecture: “All odd numbers if you circle them by twos there’s one left over. So, if you add two odd numbers, the two ones left over will group together and will make an even number.” (Ball & Bass, 2003)
What mathematics did the teacher in the episode above need to know to effectively manage her students’ engagement in RP? She needed to know at least three things: (1) The important idea that patterns can give rise to conjectures, which in turn motivate the development of arguments that may or may not qualify as proofs; (2) How to make sense of and evaluate mathematically different student arguments; and (3) The mathematical resources necessary for the development of a proof (notably, the definitions of even and odd numbers) in order to help her students acquire these resources. Thus, the episode illustrates the complexity and subtlety of the content knowledge of RP that is important for teaching elementary school mathematics, especially as it pertains to engaging students in RP. This raises a critical question: What content knowledge of RP is important for elementary school teachers to know? A FRAMEWORK FOR THE CONTENT KNOWLEDGE OF REASONING AND PROVING THAT IS IMPORTANT FOR ELEMENTARY SCHOOL MATHEMATICS TEACHING Using the findings of classroom-based research on elementary mathematics teaching that aimed to promote RP (Ball & Bass, 2003; A. Stylianides, 2005, in press), we developed a framework for the content knowledge of RP that is important for elementary mathematics teaching. The framework is structured around four ideas. Next we present the four ideas and illustrate them using the episode from third grade. Idea 1: PatternsÆConjecturesÆArguments (which may or may not qualify as Proofs) Description. Idea 1 derives directly from our conceptualization of RP and emphasizes the following important connection among patterns, conjectures, arguments, and proofs: patterns can give rise to conjectures, which in turn motivate the development of arguments that may or may not qualify as proofs (G. Stylianides, 2005). Teachers PME30 — 2006
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Stylianides & Stylianides need to know this connection: if they engage their students in the aforementioned sequence of activities, students are likely to share sufficient interest in knowing whether their conjectures are true and to recognize the need for a proof (Balacheff, 1990; Mason et al., 1982). An implication of Idea 1 is that teachers need to be able to distinguish between proofs and arguments that do not qualify as proofs, such as empirical arguments (Martin & Harel, 1989; Simon & Blume, 1996). Example. In the episode, the teacher demonstrated good knowledge of the connection described in Idea 1. She engaged her students in an activity that led to the formulation of a conjecture, which spurred debate over whether and how one could prove it. Thus, the search for a proof arose naturally from students’ activity. Idea 2: RP is Bounded by the Community’s Existing Knowledge Description. Idea 2 denotes that students’ engagement in RP is bounded by their existing knowledge (e.g., Ball & Bass, 2003; Simon & Blume, 1996; Yackel & Cobb, 1996). Accordingly, teachers need flexible content knowledge of RP that will allow them to identify mathematical resources that are crucial for students’ engagement in RP and organize their instruction accordingly. This flexible knowledge will help teachers make sense of and evaluate different student arguments, thereby supporting instruction that attends seriously to student thinking (Ma, 1999). Furthermore, teachers need to be able to produce multiple legitimate ways to define concepts, prove statements, etc., in order to accommodate the needs, and manage the constraints in the existing knowledge, of different student populations. Example. In the episode, the teacher recognized that knowledge of definitions of even and odd numbers was crucial for the development of a proof for the claim “odd+odd=even” (see Idea 3 for elaboration). For this reason, she focused her students’ attention on their definitions of these concepts. Idea 3: Mathematical Definitions are Central to RP Description. Idea 3 is that mathematical definitions are central to RP practices, in classrooms. Teachers need to know that definitions help the classroom community develop a shared understanding of mathematical concepts, and that definitions are the basis of arguments and proofs (e.g., Ball & Bass, 2003; Mariotti & Fischbein, 1997; Zaslavsky & Shir, 2005). An implication of Ideas 2 and 3 is that teachers need to develop an understanding of what makes a good definition and also the ability to tailor these definitions to their students’ existing knowledge. Example. Building on the example in Idea 2, Betsy’s proof in the episode was based on the following definition of odd numbers: “Odd numbers are the numbers that if you group them by twos there is one left over.” This definition expresses a property that is true for all odd numbers, thus supporting the development of an argument that covers the general case. Idea 4: Different Kinds of Tasks Can Offer Different Kinds of Opportunities for RP Description. Idea 4 is that different kinds of tasks can offer students different kinds of opportunities for RP. Specifically, there are two main mathematical characteristics of 5 - 204
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Stylianides & Stylianides the statement in a task that affect the RP activity in which students can engage: (1) whether the statement is true or false, and (2) whether the statement refers to a finite or an infinite number of cases (A. Stylianides, 2005). For students to develop an integrated understanding of RP, it is necessary that teachers know how to analyze the opportunities that different tasks can afford so as to implement effectively a balanced representation of different kinds of tasks in their classrooms. Example. In the episode, the statement “odd+odd=even” in the task is true and refers to an infinite number of cases. Students would get different experiences if they engaged in other kinds of tasks, such as the examination of the statement: “There are exactly 4 different ways for the sum of two dice to be 5.” This is a true statement that refers to a finite number of cases. Students can prove it by enumerating systematically all possible cases; systematic enumeration of each particular case cannot be used to prove statements that refer to infinitely many cases. PROMOTING THE CONTENT KNOWLEDGE FOR MATHEMATICS TEACHING: USING A SPECIAL CATEGORY OF MATHEMATICS TASKS It is well documented in the literature (e.g., Ball et al., 2001; Ma, 1999) that elementary teachers not only need to know the mathematics they teach (e.g., how to divide fractions), but they also need to know this mathematics in ways that are useful for teaching (e.g., they need to be able to respond to a student who might ask whether the “invert and multiply” procedure for dividing fractions is equivalent to the nonstandard procedure of “dividing numerators and denominators”). Thus, it is important that mathematics courses for preservice elementary teachers use mathematics tasks that provide preservice teachers with rich opportunities to learn mathematics in connection with the domain to which this learning will be used, namely, the work of mathematics teaching. Nevertheless, numerous anecdotal reports and some research (McCrory et al., 2004) suggest that existing mathematics courses for preservice teachers tend to use tasks that do not promote learning in the way described above. An example of a standard mathematics task used in mathematics courses for preservice elementary teachers is the following: “Prove that the sum of two odd numbers is even.” Although this task can promote teacher learning of proof, it is not designed to foster connections between the intended learning and situations in teaching where this learning can be useful. Such connections are crucial because: (1) they can help preservice teachers appreciate the need for the mathematics that teacher educators teach them, and thus, increase the possibility that preservice teachers will learn this mathematics; and (2) they make it more likely that preservice teachers will utilize this learning in their teaching. We argue that mathematics courses for teachers need to place emphasis on the use of a special category of tasks that we call teaching-related mathematics tasks. These are mathematics tasks that are connected to teaching, and have a dual purpose: (1) to foster teacher learning of mathematics that is important for teaching, and (2) to help teachers see how this mathematics relates to the work of teaching. The connections to teaching can take one or both of two forms. Next we outline and exemplify these PME30 — 2006
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Stylianides & Stylianides forms by using examples from our mathematics course for preservice elementary teachers. Our course uses heavily teaching-related mathematics tasks. Form 1: The task is embedded in a teaching context For example, in our course for preservice elementary teachers we show the video of the episode described above and we ask our preservice teachers to explain and evaluate mathematically the different student arguments in the clip. This task supports teacher learning related to Ideas 1 and 3 described above. Form 2: The task includes reflection of preservice teachers on themselves as teachers of mathematics For example, after our preservice teachers discuss the different student arguments in the episode described above, we ask them to reflect on whether knowing a single proof for the claim “odd+odd=even” would provide significant leverage for their work in managing classroom situations like the one in the video episode. Our preservice teachers bring up the idea that students who explore relations with these numbers and ultimately engage in the development of a proof for the statement “odd+odd=even” are likely to have significant variations in their existing knowledge, such as in their abilities to use different representations (pictures, algebraic notation, etc.). Thus, teachers need to know multiple ways to prove this statement to be able to evaluate different student arguments and develop a proof at the level of their students. This reflection is the first part of a teaching-related mathematics task that asks preservice teachers to develop, and explain the correspondences among, three different proofs for the statement “odd+odd=even” that may be accessible to different groups of students. This task supports teacher learning related to Idea 2 described above. Figure 1 summarizes our preservice teachers’ work in this task. Proof using everyday language:
Proof using algebra:
Proof using pictures:
Odd numbers are the numbers that if you group them by twos, there’s one left over.
Odd numbers are the numbers of the form 2n + 1, where n is an integer.
Odd numbers are of the form:
Even numbers are the numbers that if you group them by twos, there’s none left over.
Even numbers are the numbers of the form 2n, where n is an integer.
Even numbers are of the form:
If you add two odd numbers, the two ones that are left over will make another group of two.
If you add two odd numbers, you get: (2k + 1) + (2m + 1) = (2k + 2m) + (1 + 1) = 2 • (k + m + 1)
The resulting number can be grouped The resulting number is of the form 2n by twos with none left over and, thus, is and, thus, is an even number. an even number.
odd number
... ...
=
... ... ... ... odd number
+ ... ...
... ...
=
... ... even number
Figure 1. Three proofs for the statement “odd+odd=even” and their correspondences. The examples presented above illustrate how we transformed the standard mathematics task “Prove that the sum of two odd numbers is even” to teachingrelated mathematics tasks that support teacher learning related to Ideas 1, 2, and 3. In working on these tasks, our preservice teachers engaged meaningfully with important mathematical ideas in connection with their future work, and they appeared to 5 - 206
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Stylianides & Stylianides appreciate the value of what they were learning. The latter is encouraging, especially in the context of RP, for teachers tend to consider RP as an advanced topic (Knuth, 2002) and, thus, they are often resistant to engage in activities that aim to foster the development of their generally weak content knowledge in this area. CONTRIBUTIONS TO THEORY AND PRACTICE This paper offers insights into the often-problematic relationship between theory on teachers’ knowledge and the practical work of mathematics teaching. Specifically, it 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 with the work of teaching. The reflective account of our own personal experiences in designing and implementing teaching-related mathematics tasks illuminates aspects of their role and nature, and suggests their promise for mathematics teacher education. Research is needed to develop a comprehensive set of teaching-related mathematics tasks and explore further their effectiveness through design of experimental or quasi-experimental studies that will examine the implementation of these tasks in mathematics teacher preparation programs. References Balacheff, N. (1990). Towards a problématique for research on mathematics teaching. Journal for Research in Mathematics Education, 21(4), 258-272. 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: National Council of Teachers of Mathematics. Ball, D. L., Lubienski, S., & Mewborn, D. (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, NY: Macmillan. Boero, P., Garuti, R., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems: a hypothesis about the cognitive unity of theorems. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 113-120). Valencia, Spain. Healy, L., & Hoyles, C. (2000). Proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428. Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum. Mariotti, M. A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics, 34, 219-248. Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20, 41-51. PME30 — 2006
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Stylianides & Stylianides Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. London: AddisonWesley. McCrory, R., Siedel, H., & Stylianides, A. J. (2004). Mathematics textbooks for prospective elementary and middle school teachers. Symposium in the Research Presession of the Annual Meeting of the National Council of Teachers of Mathematics. Philadelphia, PA. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. Reiss, K., Hellmich, F., & Reiss, M. (2002). Reasoning and proof in geometry: Prerequisites of knowledge acquisition in secondary school students. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, Volume 4 (pp. 113-120). Norwich, Great Britain: University. Schoenfeld, A. H. (1983). Problem solving in the mathematics curriculum: A report, recommendations, and an annotated bibliography. The Mathematical Association of America, MAA Notes, No. 1. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3-31. Stylianides, A. J. (2005). Proof and proving in school mathematics instruction: Making the elementary grades part of the equation. Unpublished doctoral dissertation, University of Michigan, Ann Arbor. Stylianides, A. J. (in press). Proof and proving in school mathematics. Journal for Research in Mathematics Education. Stylianides, G. J. (2005). Investigating students’ opportunities to develop proficiency in reasoning and proving: A curricular perspective. Unpublished doctoral dissertation, University of Michigan, Ann Arbor. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458-477. Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 227-236). Reston, VA: National Council of Teachers of Mathematics. Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317-346.
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