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Definitions of mathematical knowledge for teaching: Using these constructs in research on secondary and college mathematics teachers Natasha Speer The University of Maine Department of Mathematics & Statistics UMaine Research in STEM Education Center

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Consider this: For each item below, without having to actually do any work on it, do you know that you can find the solution? Would a middle school student’s response to the above question differ from yours? What about a post-doc in mathematics? 1) 27 – 9 = ? 2) 2x 2 – 16x = –30, what is x? 3) y 2 + 6x 2 – 4 = 7y – 5x + 12, what is dx / dy ? 4) How many solutions are there to x 2 = 2 x ? Provide a proof/justification for your answer.

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Some thoughts We call the same mathematical task an “exercise” or a “problem” depending on who is solving it. Is it possible that a similar framing might be needed for teacher knowledge? In particular, is the designation of something as “specialized content knowledge” a function of who the person is who holds that knowledge?

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How I (and others) ended up thinking about these issues Collaborators: Karen King, NCTM; Heather Howell, ETS. I had data. Karen needed data. I had been doing research involving: ◦ Graduate student teaching assistants ◦ Mathematicians We both were involved in curriculum development: ◦ Capstone courses for pre-service secondary teachers

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How I (and others) ended up thinking about these issues Looked at mathematicians teaching pre- service middle and high school teachers who were also math majors Hit various conceptual/linguistic obstacles Perhaps: The type of knowledge something is might be a function of the (characteristics) of the person possessing that knowledge? Decided: Let’s use these situations to examine (and refine?) the constructs

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Today The education community’s descriptions of knowledge used in teaching Origin/history of these descriptions Design of the inquiry The findings are questions ◦ Case/question 1 ◦ Case/question 2 Where does this leave us? Where might we go from here?

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Some things we have learned from research on teaching Content knowledge is not all that matters ◦ Teachers’ having more courses in content is not strongly correlated with higher achievement for their students (Begle, 1979; Monk, 1994) “The conclusions of the few studies in this area are especially provocative because they undermine the certainty often expressed about the strong link between college study of a subject matter and teacher quality” (Wilson, Floden, & Ferrini-Mundy, 2002, p. 191)

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Horizon content knowledge Specialized content knowledge Common content knowledge Knowledge of content and students Knowledge of curriculum Knowledge of content and teaching Pedagogical content knowledge Subject matter knowledge Pedagogical knowledge

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Horizon content knowledge Specialized content knowledge Common content knowledge

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Common Content Knowledge mathematical “knowledge of a kind used in a wide variety of settings – in other words not unique to teaching”; these are not specialized understandings but are questions that typically would be answerable by others who know mathematics” (Ball, Hoover Thames, & Phelps, 2008, p. 399)

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Specialized Content Knowledge “the mathematical knowledge ‘entailed by teaching’ – in other words, mathematical knowledge needed to perform the recurrent tasks of teaching mathematics to students” (Ball, Hoover Thames, & Phelps, 2008, p. 399)

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Specialized content knowledge used to do the “mathematical work” of teaching to follow and understand students’ mathematical thinking to evaluate the validity of student- generated strategies shown to play a role in teachers’ practices and correlate with students’ learning (Ball & Bass, 2000; Hill et al 2004, 2005; Ma, 1999)

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Horizon Content Knowledge “an awareness of how mathematical topics are related over the span of mathematics included in the curriculum” (Ball, Hoover Thames, & Phelps, 2008, p. 403).

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Generally accepted that Mathematical understanding required to teach (elementary school) is different from that needed of non-teacher adults (Ball et al, 2007) This type of understanding not likely to be acquired via traditional college course work expected of (elementary school) teachers (Ball, 2005).

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Generally accepted that Mathematical understanding required to teach (elementary school) is different from that needed of non-teacher adults (Ball et al, 2007) This type of understanding not likely to be acquired via traditional college course work expected of (elementary school) teachers (Ball, 2005).

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But … Development involved elementary teachers Also: ◦ Grounded in practices of teachers ◦ Framework and item development in parallel

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So, do these constructs (e.g., SCK, CCK, HCK) “work” at other grade levels? Reasons the answer might be YES: ◦ Intellectual work/tasks are similar ◦ Cognitive processes appear to be similar Reasons answer might be NO/IT DEPENDS: ◦ Typically, different amounts of content preparation ◦ Types of knowledge are highly interconnected and influence one another (e.g., Even & Tirosh, 2002; Mewborn, 2003; Sherin, 2002)

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Why is this important? For practical reasons ◦ Design of math programs for teachers ◦ Capstone courses For research and theory- development reasons ◦ How generalizable are the constructs? ◦ What do we need to consider when creating assessment items/tests of these kinds of knowledge? ◦ How should we advance theory?

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Beginnings of a discussion Others have noted the possible influence the elementary context has had on the development of these constructs (e.g., Ruthven & Rowlands, 2010) But it tends only to be NOTED, not analyzed/examined for implications for: ◦ findings, or ◦ methods, or ◦ theory

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From last paragraph of 291-page book: …we have already noted how research has only started to map out the details of mathematical knowledge in teaching. There is scope for a more comprehensive research programme to extend scrutiny beyond the particular phases, systems and topics that have received most attention to date: to examine mathematical knowledge in secondary and tertiary teaching as much as primary, beyond a small group of anglophone cultures, and in relation to areas and aspects of mathematics other than arithmetic.” (Ruthven & Rowlands, 2010, p. 291, emphasis added)

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Developments at secondary level Work is occurring! (e.g., Knowledge of Algebra for Teaching project (KAT); Teacher Education and Development Study in Mathematics (TEDS-M); High School Mathematics from an Advanced Standpoint Project (HSMFAS)) Many take the framework as a “given” Some develop their own, but appear influenced by existing framework As a result, little occasion/need for theory examination/work Most focus on categorizing knowledge and are NOT tightly grounded in practice

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Developments at the tertiary level Developments at the tertiary level Much less is happening As of 2010, only 5 peer-review publications about college mathematics teachers’ practices (Speer, Smith & Horvath, 2010). Two of those were about knowledge Many (myself included) take/took framework as a “given” One difference: This work has been (mostly) grounded in practices of graduate students and faculty (e.g., Wagner, Speer & Rossa, 1007; Speer & Wagner, 2009).

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Design of our inquiry Analyze how constructs are defined in the literature Look for illustrative examples in the (available to us) data corpus Products of this work: ◦ Questions ◦ Beginnings of “road map” to places the community might look at more carefully

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We examined explicit definitions of CCK, SCK, and PCK operational definitions as found in the literature on elementary teachers’ knowledge those definitions and their relationships to typical characteristics of elementary teachers (e.g., their level of content preparation, their experiences doing mathematics) implicit in those definitions

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Comparison & contrast work compared and contrasted characteristics of elementary school teachers with characteristics typical of secondary/post- secondary teachers examined potential implications for the definitions of CCK, SCK, and PCK

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Data sources for illustrative cases Audio & video recordings of courses for pre-service secondary mathematics teachers ◦ Teaching methods course ◦ Capstone course for mathematics majors enrolled in a secondary teacher certification program Audio of interviews with course instructors (mathematics educators and research mathematicians) Video of upper division mathematics courses and audio of post-instruction interviews with the instructors (research mathematicians)

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Case 1: Case 1: Defining “Common” and “Specialized” (see handout) A teacher is using the following problem with her students: Suppose that a staircase comprises ten steps and that you can climb the stairs one or two steps at a time. In how many different ways can you climb these ten steps? (Rubel & Zolkower, 2007/2008).

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With a neighbor (or two): Do the problem. Examine the student-generated solution. Is this sequence really the Fibonacci sequence? Why? How does this solution connect to the combinatorial solution? Is this an important mathematical connection to make? To follow/make sense of the solution did you use your common content knowledge, specialized content knowledge, pedagogical content knowledge, or something else? How could you tell which it was?

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Finding #1 What is the relationship of CCK to SCK for those holding a bachelors degree or higher in mathematics? Places to investigate: ◦ Topics where we expect proof or sophisticated definition use ◦ Topics that recur again and again in increasingly sophisticated ways

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Case 2: Case 2: The Nature of Mathematicians’ Work Consider the teaching tasks of examining, evaluating and formulating a response to a student-generated solution. This is a type of work that researchers of elementary teachers assert necessitates (and enables the development of) specialized content knowledge.

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A day in the life of a mathematician In the morning, she’s grading Calculus 1 quizzes. In the afternoon, her colleague stops by and asks her to look over a proof he’s been working on.

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The quiz: Quotient rule See handout. Find f’(x). Student does not use the quotient rule.

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With a neighbor (or two): Do the problem. Examine the student-generated solution. Is the student’s solution correct? What was the student thinking as he generated the solution? What will you tell the student about his solution method? Why would or wouldn’t you encourage the use of this method? To follow/make sense of the solution did you use your common content knowledge, specialized content knowledge, pedagogical content knowledge, or something else? How could you tell which it was? Are the same things at play when the mathematician looks over her colleague’s (written) proof and then discusses it with him?

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In both situations the mathematician needs to: make sense of the mathematical ideas and reasoning presented by someone else determine whether the reasoning is correct or incorrect formulate a response about the proposed solution

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Further thoughts Elementary and secondary teachers generally do not (typically) examine the mathematical work of their peers. Does that mean that the knowledge used while checking the validity of student- generated solutions is common content knowledge for mathematicians but specialized content knowledge for others? Does it matter that the responses will be for different people (student versus colleague)? Is the knowledge used to do that work the same in both contexts?

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Finding #2 What is the relationship between the type of work mathematicians do in their research and while teaching mathematics? Issue for further consideration: ◦ How much do the source of the work and audience for the response matter? ◦ Is examining/validating student work just “messier” than looking at a colleague’s? ◦ Do mathematicians see/feel the similarities?

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Some evidence that mathematicians feel a difference “Where does that come from? Where does that lead? … I just don’t understand and haven’t thought enough about differential equations as a subject to be taught so that I feel any flexibility at all.” But other examples where they report it is quite similar.

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Concluding questions Can something be SCK for one person but CCK for someone else? If so, what does that mean for theories of knowledge? What does it mean for the practical task of designing assessment items? Where are (and how do we find) the boundaries? Does the source of the mathematical work (e.g., student versus colleague) influence the type of knowledge used to make sense of and validate that work? If so, what causes that differentiation? Is there some boundary between the two? (What happens when a mathematician looks at the work of their graduate student or post-doc?)

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So, what do we do? Be on the look out for “problematic” examples– they might help us understand and refine our definitions Seek out other questions– they might focus our attention on aspects of theory/definitions that need some work

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Acknowledgements This material is based upon work supported by the National Science Foundation under Grants No. DRL and DUE Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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References Ball, D., Hoover Thames, M., & Phelps, G. (2008). Content knowledge for teaching: What makes It special? Journal of Teacher Education, 59(5), Even, R., & Tirosh, D. (2002). Teacher knowledge and understanding of students’ mathematical learning. In L. English (Ed.), Handbook of international research in mathematics education (pp ). Mahwah, NJ: Laurence Erlbaum. Hill, H., Rowan, B., & Ball, D. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), Hill, H., Sleep, L., Lewis, J., Ball, D. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts? In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp ). Reston, VA: National Council of Teachers of Mathematics. Mewborn, D. S. (2003). Teaching, teachers’ knowledge, and their professional development. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp ). Reston, VA: National Council of Teachers of Mathematics. Ruthven, K., & Rowlands, T. (Eds.). (2010). Mathematical knowledge in teaching. Dordrecht: Springer. Sherin, M. (2002). When teaching becomes learning. Cognition and Instruction, 20(2), Speer, N., & Wagner, J. (2009). Knowledge Needed by a Teacher to Provide Analytic Scaffolding During Undergraduate Mathematics Classroom Discussions. Journal for Research in Mathematics Education, 40(5), Wagner, J., Speer, N., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry-oriented differential equations course. Journal of Mathematical Behavior, 26,

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An example: Even number definition Item to assess Specialized Content Knowledge 1 : Ms. Lin looks up the definition of “even number” in several textbooks and reference books. Which of the following definitions is both mathematically correct and accessible below the middle school level? a) A number that can be divided in two equal parts with nothing left over is even. b) A whole number is even if it can be divided into groups of 2 with nothing left over. c) A number with 0, 2, 4, 6, or 8 in the ones place is even. But would we consider this “specialized” for a mathematics major? A mathematics graduate student? 1 Drawn from examples given in talks by Deborah Ball and colleagues found at

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Case 1: Case 1: Defining “Common” and “Specialized” Recognizing the accuracy of a definition is SCK for an elementary school teacher, but (might it be considered) CCK for a mathematics major?

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