The significance of mathematical knowledge in teaching ... - SFU

Educ Stud Math (2011) 76:247–263 DOI 10.1007/s10649-010-9268-z

The significance of mathematical knowledge in teaching elementary methods courses: perspectives of mathematics teacher educators Rina Zazkis & Dov Zazkis

Published online: 16 September 2010 # Springer Science+Business Media B.V. 2010

Abstract Our study investigates perspectives of mathematics teacher educators related to the usage of their mathematical knowledge in teaching “Methods of Teaching Elementary Mathematics” courses. Five mathematics teacher educators, all with experience in teaching methods courses for prospective elementary school teachers, participated in this study. In a clinical interview setting, the participants described where and how, in their teaching of elementary methods courses, they had an opportunity to use their advanced mathematical knowledge and provided examples of such opportunities or situations. We outline five apparently different viewpoints and then turn to the similar concerns that were expressed by the participants. In conclusion, we connect the individual perspectives by situating them in the context of unifying themes, both theoretical and practical. Keywords Teacher education . Elementary methods course . Mathematics teacher educator . Teachers’ knowledge

1 Prologue and question Consider two candidates, Maria and Elena, who applied for the same position—instructor for a “Methods of Teaching Elementary Mathematics” course. Maria holds a Master’s Degree in Mathematics; she taught Mathematics for 5 years in secondary school and then for 3 years in college. She is a Ph.D. student in Mathematics Education. Elena, in contrast, holds Bachelor’s and Master’s Degrees in Education, and she has 15 years of teaching experience in elementary school at various grade levels. She is a Ph.D. student in Curriculum and Instruction. R. Zazkis (*) Faculty of Education, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada e-mail: [email protected] D. Zazkis Center for Research in Mathematics and Science Education, San Diego State University, 6475 Alvarado Road, Suite 206, San Diego, CA 92120-5013, USA e-mail: [email protected]

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R. Zazkis, D. Zazkis

Which one should be hired? To focus the comparison on mathematical background vs. teaching experience, we presented both candidates as female with similar academic standing. Further, let us assume that the candidates are of similar appearance, ethnicity, and age, and both have flattering recommendations from previous employers. Also, neither candidate has taught a methods course previously or worked with prospective teachers on issues related to teaching mathematics. Who is better suited to teach the course? A compelling argument can be presented for either candidate. Maria was hired. Disregarding external factors that may influence decision making, this decision implies preference towards the depth of subject matter knowledge to breadth of teaching experience at “relevant” (in this case elementary school) levels. (We are aware that some colleagues in mathematics education may disagree with this preference.) This hiring choice raises the following question: In what way is expertise in mathematics implemented in teaching methods for elementary school? Our study is aimed at investigating this issue.

2 Background The focus on mathematics teacher educators is a relatively new avenue in mathematics education research. Even (2008) suggests that “the recent focus on mathematics teacher education with lack of attention to teacher educators mirrors, to some degree, the early research in mathematics education, which centered on student learning, but lacked attention to teachers, teaching and teacher learning” (p. 59). A particular instance of this can be seen in the recent growth in the research on teachers’ mathematical knowledge but lack of attention to the knowledge that teacher educators bring to their practice. In providing background for the latter issue, which is the focus of this article, in what follows, we first attend to studies that focused on the work of teacher educators and then present a brief overview of research on teachers’ knowledge. 2.1 The practice of mathematics teacher educators According to Zaslavsky (2008), “teacher educators are ‘self made’”, that is, “they make their own transitions from their experiences as mathematics teachers and/or as researchers of mathematics learning and teaching” (p. 94). While there are no explicit accreditation requirements for this “developing professional” (Jaworski & Wood, 2008), researchers have described several personal traits that are desirable for mathematics teacher educators, such as being reflective, flexible, open-minded, and enthusiastic (e.g., Zaslavsky, 2008). Furthermore, Chapman (2008) summarized research on instructional practices developed and used by mathematics teacher educators and discussed usefulness of various instructional approaches when working with teachers. Zaslavsky and Leikin (2004) outlined the complexities of teacher educators’ work, as well as projects that promote their growth through practice. No one disputes the centrality of teachers’ knowledge of the subject matter. However, what constitutes “subject matter” for a teacher/instructor in a methods course cannot be outlined in a simple syllabus-style list of topics: it is knowledge of teaching mathematics, which is a complex combination of pedagogy, psychology, didactics, research, constructs and theories, curriculum, and, of course, mathematics itself. Zaslavsky (2008) describes the knowledge base of teacher educators as the knowledge for teaching mathematics together with the “knowledge on how to enhance teacher learning” (p. 112). In working with learners on teaching mathematics, some approaches are similar to those of working with

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learners on mathematics, while others “alter subtly with the change of focus from the discipline of mathematics to the second-order discipline of teaching mathematics” (Mason, 2008, p. 8). However, in discussions regarding the practice of mathematics teacher educators, their mathematics knowledge is often taken for granted. While mathematics is positioned as the kernel in models of mathematics education research (Krainer, 2008) and at the top level in which mathematics teachers and mathematics teacher educators operate (Jaworski, 2001), it often remains in the background of the demands of this profession. While “relevant” or “core” mathematical knowledge is often acknowledged as a necessity, what is “relevant” for education of prospective elementary teachers (or teachers in general) is not spelled out. Our study attempts to address this gap by illustrating a variety of ways in which mathematical knowledge can be used, or is beneficial, when teaching prospective elementary school teachers about teaching mathematics. Since research in mathematics education has not yet paid significant attention to the mathematical knowledge of teacher educators, we turn to research that focused on mathematical knowledge of school teachers. 2.2 Teachers’ knowledge and awareness Teachers’ mathematical knowledge was the focus of a variety of recent studies. Hill, Ball, and Schilling (2008) further elaborated upon the classical notions of subject matter knowledge (SMK) and a pedagogical content knowledge (PCK) introduced by Shulman (1986) and proposed a model for mathematical knowledge for teaching (MKT). Specifically, SMK was seen as combined from common content knowledge (CCK), specialized content knowledge (SCK), and knowledge at the mathematical horizon. While CCK was described as shared among individuals who use mathematics, SCK was considered as the domain of teachers that allows them “to engage in particular teaching tasks” (ibid, p. 377). According to Ball, Thames, and Phelps (2008), “horizon knowledge is an awareness of how mathematical topics are related over the span of mathematics included in the curriculum” (p. 403). Furthermore, the PCK was seen as a combination of knowledge of content and students (KCS), knowledge of content and teaching (KCT), and knowledge of curriculum. Davis and Simmt (2006) argued against the traditional separation of content and pedagogy and claimed that “mathematics-for-teaching” can be considered as a distinct branch of the discipline of mathematics. Using complexity science, they offered several intertwining aspects of mathematics for teaching that included mathematical objects, curriculum structures, classroom collectivity, and subjective understanding. The different categories or aspects of knowledge were used to serve different purposes of researchers. Ball and colleagues focused on assessing separately different components of teachers’ knowledge. Davis and Simmt attempted to describe what mathematics teachers know or need to know, concluding that an “isolated focus on either questions of mathematics or questions of learning is inadequate in efforts to understand teachers’ mathematical knowledge” (p. 316). Other researchers attended to the uses of teachers’ knowledge (e.g., Adler & Ball, 2009), demonstrating how different kinds of knowledge are interrelated in practice (e.g., Zazkis & Mamolo, 2009). In contrast, Watson (2008b) avoided categorization and considered mathematical knowledge in teaching (MKiT) as “participation in mathematical practices in the classroom, and also during preparation for teaching” (p. 1). She argued that categorization of knowledge by identifying its types is unhelpful as it can veil the essential mathematical activity in which different kinds of knowledge relate and inform each other. However, what

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appears to be a unifying theme in different and seemingly opposing perspectives is the view that teachers’ mathematical knowledge is complex, and it has distinctive features that deserve research attention. These distinguishing features of teachers’ knowledge can be attributed to, following Mason (1998), particular “levels of awareness.” Awareness-in-action is the ability to act in the moment. This level of awareness is recognized when an individual corrects a mistake, suggests an answer to a task, but is unable to justify or explain his/her approach. Awareness-in-discipline is awareness of awareness in action. This awareness is essential in order to articulate awareness in action for others. According to Mason, the one important distinction between the two kinds of awareness involves the ability “to do” in contrast with the ability to instruct others. Teachers who possess awareness in discipline are able to articulate the choices they make in instructional situations and help learners develop awareness in action. Awareness-in-council is awareness of awareness in discipline. This awareness is essential in order to articulate awareness in discipline for others. It describes one’s sensitivity to what others require for building or enhancing their awareness. As such, awareness in council is what guides the practice of a teacher educator. We are interested in exploring the role of mathematics teacher educators’ mathematical knowledge in reaching this level of awareness. As a step in this direction, we focused on how experienced mathematics teacher educators describe the uses of their knowledge in their work with prospective elementary school teachers.

3 Method Five mathematics teacher educators, all with experience in teaching methods courses for prospective elementary school teachers, participated in this study. In a clinical interview setting, they were first presented with the Elena–Maria dilemma (mentioned in the beginning of this article) and asked to state their preference. All expressed an opinion that Maria was a preferred candidate, though this preference was subject to her having appropriate dispositions. This is hardly surprising, as all the interviewees were of Maria type, that is, all had extensive mathematical backgrounds (a Master’s degree in Mathematics or a Bachelor’s degree in Mathematics complemented by some graduate-level work in the subject). Since mathematical background was deemed important in the hiring choice, the participants were invited to discuss the value of their mathematical knowledge in relation to their teaching of an elementary mathematics methods course. More specifically, they were asked to consider where in their teaching of this course they had an opportunity to use, or to take advantage of, their advanced knowledge and provide examples of such opportunities or situations. The interviews were recorded and summarized by the authors. Each summary was presented to the respective interviewee with an invitation to comment on whether his or her ideas were presented as intended and correct if necessary. Two interviewees provided a minor extension or clarification; the other three participants approved the summary without a comment.

4 Results In presenting and analyzing results of this study, we first attend to the individual emphases related to uses of mathematical knowledge, which were explicitly attended to by each interviewee. Responses and views of each participant are summarized in Sections 4.1–4.5,


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occasionally illustrating the views with excerpts from the interviews. We next turn to the similar concerns, related to the teaching of elementary methods courses, that were expressed by the participants (Section 4.6). Then, in Section 5, we attempt to connect the individual perspectives by situating them in the context of unifying themes identified by the researchers. 4.1 Eric: mathematics vs. school math In describing his work with prospective teachers, Eric makes an explicit distinction between Mathematics and “school math.” His approach in his methods course is to introduce students to Mathematics. Eric: I always make a distinction between mathematics and school math, because they are coming in, and it takes me 3 months to undo what they think is mathematics, it’s the process of doing mathematics, it’s not the mathematics. Eric emphasized that his goal is to have his students1 “think mathematically.” When a specific example was requested, Eric mentioned that one of the problem solving activities he implements early in his course is the task of counting squares on a chessboard (that is, not only 64 small squares, but squares of all possible sizes). He mentioned, that to his repeated surprise, even after carrying out this task in four different offerings of the methods course, no student started by thinking of a “simpler but similar problem” (Polya, 1945/1988) in an attempt to develop a pattern or make a conjecture. But this is a kind of mathematical thinking in doing mathematics that Eric deems important, rather than the end result, and he tries to instill it in his students. In addition to developing mathematical thinking, Eric mentioned that his pedagogical approach attempts to put prospective teachers in students’ shoes, by involving them in work with bases other than 10. Eric: It’s the one exercise that puts them into position to see how their students will be reacting to these things. In addition to experiencing students’ challenges first hand, Eric sees the value of work with different bases in “reconstructing base 10.” He uses exponential notation in presenting the sequence of place values in different bases, emphasizing that the value of the basis changes, but exponents remain invariant. Eric sees the main usage of his mathematical knowledge in “knowing how to do math, not just school math, knowing what mathematics really is, and there is a huge distinction.” This invited a question, as to whether such “knowing” could be acquired without any formal study of mathematics at the University level. Eric believes it is quite possible if an individual engages in self-study outside of a school setting. However, this remains an extremely rare possibility. 4.2 Emily: problem solving For Emily, problem solving is the main focus of her course. She believes that when prospective teachers experience problem solving and are successful in it, then problem solving will feature in their teaching. Emily has an extensive repertoire of problems that she Given the context of this study, “students” refers to prospective elementary school teachers, who are students in the methods course.

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has collected over the years, and it is via problem solving that she gives her students “a taste of Mathematics.” When asked to provide several examples of problems that she uses with her students, she recalled the following problem from her most recent class: Barb has a box of candy with 25 pieces in it. She and some friends share the candy by passing the box around the table. Each friend, including Barb, takes one piece each time the box passes them. Barb always takes both the first piece and the last piece. How many friends could be sitting at the table? Emily mentioned that she liked several things about this problem: first, that there are multiple solutions (for example, 24, 8, and 6), a fact that her students find quite unusual; secondly, that the solutions can be found by trial and error; thirdly, when all the solutions are found, it is possible to see what they all have in common. Attending to this common feature makes it possible to vary the numbers in the problem. And finally, Emily likes the ambiguity of the wording of the problem as it creates an opportunity to engage students in a discussion of possible interpretations. For example, “always” is interpreted by students as either “every turn starts and ends with Barb taking a piece of candy” or “Barb starts and ends the process of sharing candy, however during the passing of the candy box she behaves exactly as her friends.” The interviewer mentioned that though this was a nice problem, the instructor did not need any “advanced” knowledge to engage students in the activity that Emily described. So she offered another example: Peter created a new dartboard for his sons. The board has two regions. The centre circle is valued at 9, and the outside one at 4. What is the largest number that cannot be achieved as a score in this game? Again, the solution can be found by trial and error, and Emily’s class starts the approach by generating a list of possible scores: 0, 4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, and 27. Once there are consecutive numbers on the list, there is an expected conjecture that all integers greater than or equal to 24 are on the list. While the proof of this conjecture can be carried out at different levels of mathematical sophistication, Emily recognizes immediately that the score (s) is a linear combination of two relatively prime numbers 4 and 9 that can span all integers (s ¼ 4x þ 9y). Though she has no intention to involve her students in a discussion of linear combinations and spans, this knowledge gives her confidence and immediate means to evaluate a solution found by students. Emily provided several additional examples; all involved problems either with multiple solutions or with multiple ways of finding a solution, some more sophisticated than others. Here is one additional example, brought to class by one of her students: You arrive at a hotel and have 3 sets of golden rings. The first set of rings has 4 rings all joined together, the second set has 2 rings and the third only has one ring. You cannot take these sets of rings apart or exchange them for a different form of currency, and the hotel clerk has no change. You want to stay at the hotel for 7 nights, and you have to pay one gold ring for each night that you stay. You cannot pay in advance, or all at once at the end of your stay. How do you pay for your 7 nights at the hotel? Emily mentioned that after some deliberation, all her students solved the problem having discovered that the rings paid previously can be used as “change.” However, they treated it


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as a puzzle, “solve, happy, done.” Emily suspected that the generality was missed in the particular strategy of ring exchanges. As such, she offered a different version: You arrive at a hotel and have a chain of 31 golden rings. You can take these rings apart, but the task is expensive, as you have to pay for each cut. You need to stay at the hotel for 31 nights, and you have to pay one gold ring for each night that you stay. You cannot pay in advance, or all at once at the end of your stay. What is the minimal number of cuts that you need? It is Emily’s mathematical knowledge that helped her in “seeing the general in the particular” (Mason & Pimm, 1984), recognizing the binary structure in the solution to the previous problem and extending the problem in such a way that its binary structure became essential for finding a solution. Emily: It is possible that more often than not my advanced knowledge of mathematics helps me teach elementary teachers, but my intimacy with mathematics helps me for sure. My years of problem solving experience, meaningful engagement with math, explorations are beneficial to my teaching. […] Choosing on the spot the appropriate counterexamples to disprove a conjecture or a valid method for proving it is a skill I doubt Elena possesses. It can only be achieved in advanced mathematics courses. There are easily accessible collections of mathematical problems for any level. So, the interviewer suggested provocatively, engaging students in problem solving should not be a problem. As expected, Emily disagreed. She believes that it is her years of studying mathematics and her personal experience with problem solving that guide her choice of appropriate examples, in evaluating the validity of students’ approaches, and, even more importantly, in providing a lens for meaningful “looking back” (a la Polya) at the problems that were solved. 4.3 Paul: NOT a rat in a maze Paul reinterpreted and rephrased the question about his usage of mathematical knowledge as follows: Paul: There are two ways to think about this: When do I use math beyond what I’m teaching to help teach what I’m teaching? And, when have I seen situations where it seems to me that what I’m working with would be easier understood if students had some knowledge that may come later. Another way to think of this is, why is it so easy to understand now and it was hard to understand then. As an example of the latter, Paul mentioned place value. He explained that when he is trying to instill a deeper understanding of place value in prospective teachers, he is limited by their lack of knowledge of exponential growth. He referred to this as a “time schism.” That is, the current curriculum introduces some concepts at an early stage (here: place value), where a deep understanding of these concepts may come only after some other concepts are learned at a much later stage (here: exponential notation). He further exemplified: Paul: Exponential growth is what allows us to see why place value is so misleading. Because it is so compact. A 4-digit number vs. a 5-digit number in linear measure, it’s only a difference of a digit, but you need to look exponentially to see how much

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bigger this is. […] The cardinality of a number grows exponentially in relation to the number of digits being used to represent it. Paul noted that his students understand the mechanics of place value, that is, they can identify correctly tens and hundreds and thousands, but “the sense of magnitude escapes them.” As an example that may help students get this sense of magnitude, Paul often uses the following task: Place 1,000 on a number line, where two points are already marked, 0 on one side and a million on another. Almost no one estimates the place of 1,000 correctly, usually the place marked by students is somewhere between 10,000 and 100,000. The mistake is in determining magnitude rather than a specific number. To develop the idea of time schism Paul provides other examples, radian measure and area–volume formulas: Paul: Radian measure, which is taught in high school, has absolutely no meaning until you actually understand how to take a derivative of a trigonometric function. […] Area–volume formulas in elementary school. Having a strong sense of dimension allows this to make more sense. Since the example of radians and derivatives was beyond elementary school, Paul was asked to elaborate on area and volume formulas. He explained that he often saw prospective teachers confuse formulas, having “one-too-many variables in an area formula, one-too-many dimensions, and not picking up to a fact that area formula should have only two dimensions,” or confuse formulas for area and circumference of a circle; seeing one as the derivative of another prevents such confusion. To illustrate further what knowledge helps in teaching, Paul mentioned dimensional analysis. When 8 apples are put in groups of 2 apples, “apples” serve as a unit that cancels out. While with “apples” it may be trivial, this view is most helpful when teaching division of fractions, where Paul starts with common denominator. Paul: When you are dividing 2 fractions with a common denominator, the denominator becomes the unit, in dimensional analysis when you divide 2 things with he same unit, the unit cancels out. He exemplifies this to his students by spelling out the denominator, that is, not using a fraction notation for 27 47, but actually writing “2 seventh ÷ 4 seventh”, to draw students’ attention to the fact that a “seventh” can be treated as a unit, similar to meters or centimeters. Then, Paul believes, when thinking of division of fractions by using common denominator the idea is better understood. Similar to other interviews, the interviewer probed in a rather provocative manner, which allowed Paul to provide his metaphoric description of the value of knowledge in teaching: Interviewer: So is it that for these few examples we prefer people with a mathematics background to teach elementary methods courses, or is there something else? Paul: There is something else. Call it the rat in a maze. I put it like this: If a teacher is just one lesson ahead of the students, they are basically the lead rat navigating through the maze, but the teacher has a really broad and deep understanding of mathematics, it allows them to see well beyond the horizon of what it is they are teaching, they are no longer the lead rat in a maze, they are outside of the maze, looking down, helping their students navigate through the maze. So this idea of having more mathematical knowledge allows, like it is always advantageous to see


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beyond the horizon of what it is you are trying to teach. It helps us see where things are going. In summary, Paul’s mathematics guides him in drawing students’ attention to what he perceives as important or beneficial for understanding mathematical concepts, formulas, or procedures. This “drawing of attention” may take different forms, such as unconventional representation (of fractions) or task design. 4.4 Nina: values and historical perspective We identified two themes in the interview with Nina: (1) historical perspective and humanizing mathematics and (2) enculturation and values. She mentioned that in her work with prospective teachers her knowledge of the history of mathematics helped her “give some perspective on a human front of where these ideas come from and who are the actual people behind it.” This was a significant focus of her course. Prospective teachers she works with may have thought that mathematics “comes from the sky” and they get excited when a historical perspective is incorporated in their course. Following the request for specific examples, Nina mentioned number systems, origins of symbols, and development of geometry—from Euclid to Klein, to transformational geometry. She also mentioned that connecting algebra to its ancient Greek origins and introducing algebra through geometry may help in teaching algebra with meaning. Nina: I show them actually how Euclid would have talked about x-square and compare it to how we see it coming out in the 16th century. The other theme, which Nina said was a “really big one” for her, is that of enculturation and values, that is, “the value system that drives mathematics.” Among what mathematicians value, and what she is trying to instill in her students, Nina mentioned precision, objectiveness, permanence, and being parsimonious. As an illustration, she described an activity she used in the beginning of her course, where each student gets a sticker with a shape on his back, and using only yes/no questions the goal is to find out what everyone’s shape is. She suggested that formulating questions drives the values of rigor, precision, and efficiency. Nina: why is it that mathematicians like to be so precise, it is not to make their lives miserable, but it is because it is one of the values. […] In other disciplines this may not be a value, but in mathematics it really is. She emphasized that prospective teachers studied different disciplines, in which different values may exist. For example, vagueness of descriptions may be valued in literature, but it is the opposite in mathematics. She further suggested that attention to symmetry in problem solving is related to values of efficiency as well as to esthetics. Nina: These things are important because it gives them a different level of why. Like why is it that someone would think that some way is a nice way of saying something, of convincing, and this is another thing that drives mathematicians—they like to find results that work out nicely. She further mentioned consistent choices and continuity, when moving from one system to another, as values to which she directs her students’ attention when, for example, introducing integers or fractions.

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In summary, Nina saw the importance of her mathematical knowledge in working with prospective teachers “in being able to identify examples in which a certain value is operating in a course of doing something.” However, to the question of whether a set of values, consistent with hers, can be developed without many years of learning mathematics, Nina responded: Nina: I think so, totally, but I don’t think it happens. {pause}It is nowhere in the curriculum that we explicitly talk about values of mathematics, they are implicit in everything that we do. […] We are not very good as teachers, and in the curriculum, being able to say why certain things are important. She further mentioned that while most important values of mathematics are not being developed until post-undergraduate study of mathematics, and at times only through teaching mathematics, there is no reason that some values cannot be developed and explicitly attended to in early elementary school. 4.5 Rachel: students’ questions Rachel may have interpreted the question of knowledge use more literally than others. She provided several examples where her mathematical knowledge acquired after high school was put in use. 4.5.1 Example 1: division by 0 Rachel indicated that she likes raising the question about division by zero with prospective elementary school teachers. Usually, there is confusion between division of zero by another number and division by zero. This is the first thing that needs to be clarified. Then, some students recall that division by zero is problematic. They describe it as “forbidden” or “nonsense.” Rachel described several explanations that may help students understand and remember that division by zero is undefined, either using a story (Zazkis & Liljedahl, 2009) or considering what division by zero can mean in terms of multiplication. One student suggested that the result of division by zero is “error” because that is what is shown on the calculator. To that Rachel responded that either the calculator programmer did not understand mathematics (as well as her class did), or he simply could not fit “undefined” on the calculator screen. This humorous comment served as a confidence boost for her students. However, knowledge use was exemplified in her response to another student, who suggested that the result of division by zero was “infinity.” This idea was supported by several other students, who recalled “learning this in the past.” Rachel immediately recognized the confusion between division by zero (arithmetic operation) and a limit of the expression a/x, where a is a constant (positive) value and x is approaching zero (from the right), in a notation from Calculus, limx>oþ ax ¼ 1. She did not want to dismiss the student’s suggestion; on the other hand, she did not want to immediately refer to Calculus. As such, she asked her students to complete a sequence of calculations: 3/2, 3/1, 3/0.5, 3/0.1 She asked students for a number smaller than 0.1, and then smaller than 0.01… And the presented sequence of tasks continued as 3/0.01, 3/0.001, … 3/0.0000000000001, etc. Students noticed that when the divisor gets smaller, the quotient gets larger. (This was yet another opportunity to revisit a popular belief of “division makes smaller.”) Or, as Rachel tried to mimic her students: “when you divide it by something really-really-small,


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the answer is really- really-really-huge, and the smaller this one gets, the larger your answer is.” This was an appropriate observation, following which Rachel offered the following to her students: Rachel: Our friends-mathematicians refer to this phenomenon as ‘limit’. It is not that division by zero is infinity, it is that when the divisor gets closer and closer to zero, we say approaches zero, the result approaches infinity. [italics indicate emphasis in voice] Rachel continued to say that maybe her taking four-course sequence of Calculus was not essential to be able to come up with this answer, but she could not imagine someone without exposure to Calculus providing a similar explanation. They would just either accept the student’s suggestion or reject it. 4.5.2 Example 2: exponent (−1) Another example that Rachel provided was that of interpreting the exponent of (−1). Again, she referred to a student’s question, which is captured as follows: Student: 3 to the negative 1 is one third [writes 3−1 =1/3], right? Rachel: Right. Student: So power “minus 1” means reciprocal, right? Rachel: Right. Student: But f-to-minus-1 [writes f−1] means the inverse function, this does not mean 1/f, right? Rachel: Right. Student: So they ran out of symbols, or what? Rachel asked this student how she understood the meaning of “inverse function,” and only after this student provided several examples and explained why g(x)=(1/2)x was the inverse of f(x)=2x (“because doing one after the other you get to where you started”), Rachel decided that the student was ready for a “mathematical” explanation. Rachel explained that in both cases the exponent of (−1) is a reference to “inverse,” and the inverse is determined with respect to an operation. So in the first case the operation is multiplication, whereas in the second the inverse is taken with respect to the composition of functions—connecting the term composition to what the student saw as “one after the other.” She also mentioned the idea of identity, which is f(x)=x with functions, and 1 in multiplication, and that is what we get when doing the operation of an element and its inverse. She offered several examples. Rachel shared that for a moment she was not sure whether her choice of explanation was appropriate, as she may have overwhelmed the student with her answer, especially with introducing some new terminology. However, when the student replied that it was “like 3 and (−3) make zero, right?” Rachel considered this as a confirmation that her approach was in some way understood. Rachel further wondered: Rachel: Do you really need to study some group theory for that? Maybe not. After all, without using the word ‘group’, the ideas of closure, closure with respect to an operation, and identity element are often introduced in a mathematics course for elementary teachers. But I guess it is my learning of group theory, where sometimes any operation is called ‘multiplication’, helped me to see things this way.

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Maybe, as Rachel noted, this question about exponent (−1) arises mainly with Englishspeaking students, where 1/3 is referred to as “reciprocal” of 3, rather than “inverse.” In other languages, the same term of “inverse” is used in both cases—for example, in Romanian the terms are “inversa funcţiei” and “inversă fracţiune,” for inverse function and inverse fraction, respectively—and this helps students make the connection. Of course it is possible to reply to this student’s question, as Elena type may suggest, that this is not the only case in mathematics where the same symbol is used for different things, and interpretation depends on the context. However, mathematical preparation helps in being equipped with an alternative answer, which can be used with students who are ready for it. In response to the interviewer’s invitation to provide more examples, Rachel mentioned that these were examples of “smart” mathematical questions. For most questions prospective teachers ask, one does not need advanced mathematics, but rather “a different take on the elementary mathematics.” These questions are usually phrased like “how would you explain to a student this or that.” In fact, what often hides behind such a question is a lack of personal understanding of a concept or procedure. As such, the conversation in Rachel’s class around different ways to explain a piece of mathematics for children, or around potential difficulties that children may have, is a way to combine mathematics and pedagogy and in doing so enhance the mathematical understanding of prospective teachers. 4.6 Common concern While all the participants viewed knowledge of mathematics beyond school as essential, there was also an agreement that, speaking mathematically, mathematical background was a necessary but not a sufficient condition for teaching the elementary methods course effectively. Consider the following excerpts: Nina: There are people who learned math at the undergraduate and graduate level that it makes them worse, there is like a period that makes them worse before they get better. And if you catch them at the bottom of that track, then you’re in trouble, as much trouble as if they don’t know any math. Eric: Mathematics is important, but it’s not the most important thing. If you could have Maria empathetic towards the experience these people have gone through, and if that could be the main component to how she interacts with these people, I would hire Maria in a heat beat. The issue that I have with Maria is that her experience might perpetuate more of the same for these teachers, pre-service teachers, and this is exactly the opposite of what these teachers want, need or are receptive to. Paul: So I would much rather have a teacher in general who has more mathematical knowledge, because then the stuff they’re teaching is situated within a larger context, it has purpose, even though that purpose may not be evident to the students at the time. The challenge is when we are moving to teacher education. Because sometimes people who have mathematics that goes well beyond the horizon don’t know enough mathematics up close to be able to help teachers navigate things. They can’t see mathematics through students’ eyes. In different words, these excerpts illustrate the same idea: Knowledge may be a barrier. It is important to note that similar reservations were voiced by all the participants in a conversation of their uses of their mathematical knowledge. That is, whereas the knowledge is only a part of the story, the other part should not be overlooked.


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While Nina mentions that for some people knowledge “makes them worse,” Eric and Paul elaborate on what the trouble may be. For Eric, it is a possible lack of empathy, where he sees attentiveness to the needs of prospective teachers as the key in his teaching. This appreciation of attentiveness is constantly reinforced by an exercise that Eric implements early in his course. He refers to it as “spill this all out”—an invitation to prospective teachers to describe their experiences with mathematics. He is constantly shocked by the similarity of these experiences and the resulting negative emotions. As such, one of his goals is to create positive experiences in an attempt to rebuild confidence. Paul is concerned that some high school teachers, who may have “infinite fluidity with math,” cannot break this down to more basic steps and yet “we need more basic steps to help primary teachers and primary students.” His interpretation of “don’t know enough mathematics up close” refers to inability to, using Ball’s term, “unpack” mathematical knowledge (Ball, 2000; Ball & Bass, 2000). This also echoes Rachel’s view regarding the need for a “different take on elementary mathematics” in responding to students’ questions. As such, in addition to mathematical background, two components were viewed by participants as essential for successful teaching of an elementary mathematics methods course: ability to unpack elementary mathematics and ability to attend to students’ (here—prospective teachers’) thinking and sympathize with their difficulties. The ability to unpack is what distinguishes specialized content knowledge (SCK), which is a domain of teachers, from common content knowledge (CCK), which is accessible to others (Hill et al., 2008). This ability is also an indication of knowledge of students and teaching (KST), which demonstrates the interconnectedness of the knowledge components identified by Ball and colleagues. Sympathy and attention to a learner are a particular implementation of what is described in literature as “caring” or “developing caring relations” (Noddings, 2001; Sztajn, 2008). While these three components—mathematical knowledge, ability to unpack, and attention to a student—are needed for “good” teaching of any mathematics course at any level, a different combination of the three is essential for teaching an elementary methods course for prospective teachers. The mathematical knowledge needed in this recipe cannot be replaced or diluted by other ingredients.

5 Connecting the individual foci Though the five participants described apparently different perspectives regarding their personal knowledge and its usage, in what follows we reveal a strong connection among the expressed approaches and views. We observe this connection on two levels: the instructor’s theoretical perspective on the nature of mathematics and the instructor’s practical implications in task design. While the issues discussed in Section 4 were overtly referred to in the interviews, we note that the two themes discussed in Section 5 emerged in the subsequent analysis and often refer the tacit nature of participants’ knowledge, rather than to explicitly expressed ideas. 5.1 Different mathematics Existence of different mathematics was acknowledged in literature in several ways. For example, in his classical work that coined the terms instrumental and relational understanding, Skemp (1976) noted that “there are two effectively different subjects being taught under the same name, ‘mathematics’” (p. 91). Watson (2008a) articulated the difference between disciplinary mathematics and school mathematics. According to Watson, these have different sets of values, different core activities, purposes, and

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standards for accuracy. Moreover, they privilege different forms of reasoning as well as different methods of verification and establishment of validity. As such, “school mathematics is not, and perhaps never can be, a subset of the recognized discipline of mathematics” (p. 3). A related distinction is made by Lockhart (2009), who considers school mathematics to be “pseudo-mathematics.” In “pseudo-mathematics,” learners become “good at it” by following directions, whereas Lockhart claims that mathematics is about exploring new directions. When mathematics teacher educators discussed their usage of mathematical knowledge, they—either explicitly or implicitly—demonstrated attempts to introduce their students to ideas and approaches consistent with disciplinary mathematics. Eric explicitly mentioned that having students experience Mathematics, rather than “school math” was one of the goals of his course. Thinking mathematically—while engaged in a task—and applying efficient problem solving strategies indicated to Eric Mathematical (rather than school-mathy) experience. Nina’s efforts to instill in her students’ mathematical values can also be seen as an introduction to what is considered important in disciplinary mathematics. Rachel’s and Paul’s examples illustrate—and as such expose students to—unifying themes for a variety of mathematical ideas. This assists students in acquiring a view of mathematics as an interconnected web of knowledge, rather than a collection of unrelated facts and procedures. Emily’s focus on problem solving, especially “looking back” at a problem, brings her students closer towards authentic mathematical activity and gives her students a “taste of mathematics.” 5.2 Task design In responding to Watson (2008a), Zazkis (2008) suggested that, while school mathematics is “not a subset” of disciplinary mathematics, this should not be interpreted as seeing school mathematics and disciplinary mathematics as two disjoint sets; she focused on a possible intersection between the two sets. She considered task design in teacher education as a means for extending the intersection between school mathematics and disciplinary mathematics in order to bring the former closer to the latter. This idea is echoed to some degree in all the interviews. While tasks for prospective teachers serve many different purposes, both mathematical and pedagogical (Watson & Mason, 2007), it is the choice and design of tasks that provide the main avenue for the implementation of instructors’ mathematical knowledge. “Teaching is about directing learner attention” (Mason, 2008, p. 51), and the choice of tasks exemplified by mathematics teacher educators in this study serves exactly this purpose. For example, Nina focuses on values, and her way to instill mathematical values is the choice of tasks in which a certain value can be highlighted. Within Emily’s focus on problem solving, it is her mathematical background that allows her to choose problems and to extend problems in order to highlight specific methods and ideas. Eric engages students in certain tasks in order to develop their mathematical thinking via problem solving activity. While Rachel talks about students’ questions, rather than tasks, we recognize in her answers a careful task design that not only answers the question at hand but develops a mathematical idea. This ability to design tasks or extend problems in order to highlight certain aspects of mathematics or mathematical activity is what Paul communicates metaphorically as “not a rat in a maze,” that is, “looking at a maze from above,” which implies both the ability to navigate different pathways and also to foresee possible pitfalls and dead-ends. While the specific tasks described by the participants do not explicitly address the approaches of teaching the precise elementary school content, they address the issue implicitly, by extending prospective teachers’ ability to engage with mathematical activity.


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6 Conclusion “The demands on teacher educators, in terms of knowledge and qualities, are enormous and multifaceted” (Zaslavsky, 2008, p. 94). However, according to Simon (2008), there is an insufficient research-based knowledge base “for teacher education efforts that promote envisioned mathematics education reforms” (p. 18), and this is “a serious impediment in the development of mathematics teacher educators” (p. 26). Even (2008) endorses this observation by noting “the literature offers only limited empirical information about the practice of mathematics teacher educators” (p. 60). Our study is a contribution to the knowledge base related to the practice of mathematics teacher educators that highlights one aspect of their work: how mathematical knowledge is used in their teaching and how it shapes their work with prospective elementary school teachers. The focus on the usage of mathematical knowledge in teaching is a relatively new avenue in mathematics education research (Adler & Ball, 2009) that complements previous attention on what teachers know or need to know. Our contribution is in extending a conversation about the usage of subject matter knowledge, focusing on teaching of prospective teachers by mathematics teacher educators. Though there are examples in mathematics education literature of researchers/teacher educators using their personal profound knowledge of mathematics in their work with elementary school teachers (e.g. Simon, 2006), the educators’ knowledge has not been the focus of these studies. In his interview Eric claimed that his usage of mathematical knowledge is “not so much about the content.” But this is, in our view, an extended experience with the content that contributes to acquiring what Cory (2001) alludes to as ‘image of mathematics’, that is, “knowledge about that discipline” (p. 168, italics in original), in addition to the “body of mathematics.” Returning to Skemp (1976), “what constitutes mathematics is not the subject matter, but a particular kind of knowledge about it” (p. 95, authors’ italics). As mentioned above, our participants, while responding to questions about their mathematical knowledge usage, described other essential elements in the work of a mathematics teacher educator. Our emphasis on the mathematical knowledge and its usage is based on the assumption that other important components—such as empathy with students’ difficulties and ability to unpack—are much more likely than mathematics to be acquired or reinforced during one’s practice. We further acknowledge that lack of explicit attention in this paper to the pedagogy and methods of teaching elementary school mathematics is the result of the particular focus of this study, rather than lack of attention to these issues in the courses taught by the participants. According to Mason (2008), “the effective teacher educator aims to direct attention so that participants’ attention is drawn out of the actions of doing mathematics and also out of the actions of teaching mathematics, so that awarenesses become explicit” (p. 50). Such awareness of a teacher educator is what Mason (1998) referred to as “awareness-in-council,” that is, awareness that is needed to articulate awareness for others. Either explicitly or implicitly, the teacher educators in our study exemplified how their mathematical knowledge contributed to their awareness in council—in task design, in choice of problem solving activities, and in responses to students’ questions. We suspect that the themes identified in this study provide only a partial view on the usage of instructors’ mathematics knowledge, their observations and examples illustrate only knowledge which they are consciously aware of using and that they recalled at the time of the interview. Their tacit knowledge of mathematics most likely affects them in deeper and subconscious ways. For example, guiding discussions in classrooms on teaching elementary concepts, as well as choices of assignments for students, may be influenced by

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knowledge of more advanced but related concepts. Further research is needed to explore whether this influence exists and what aspects of teaching it affects. The mathematics teacher educators interviewed in this study imply that their mathematical knowledge—as used in their work with prospective elementary school teachers—is not the kernel but the means: the means for developing values, appreciating problem solving, and getting “outside the maze.” In other words, the mathematical background is the means towards developing a “third level awareness”, or “awareness-in-council” (Mason, 1998). The question that remains is whether achieving this higher level of awareness, or a “particular kind of knowledge about [mathematics]” (Skemp, 1976), is possible without formal exposure to advanced content. Ramanujans and Marjories Rice provide an existence proof for such a possibility. For all others, we believe, a University degree in Mathematics (preferably graduate, possible undergraduate) is a more common route.

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