Presentation on theme: "Definitions of mathematical knowledge for teaching: Using these constructs in research on secondary and college mathematics teachers Natasha Speer The."— Presentation transcript:
1Definitions of mathematical knowledge for teaching: Using these constructs in research on secondary and college mathematics teachersNatasha SpeerThe University of MaineDepartment of Mathematics & StatisticsUMaine Research in STEM Education Center
2Consider 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?27 – 9 = ?2x2 – 16x = –30, what is x?y2 + 6x2 – 4 = 7y – 5x + 12, what is dx/dy ?How many solutions are there to x2 = 2x ? Provide a proof/justification for your answer.
3Some thoughtsWe 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?
4How 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 assistantsMathematiciansWe both were involved in curriculum development:Capstone courses for pre-service secondary teachers
5How I (and others) ended up thinking about these issues Looked at mathematicians teaching pre- service middle and high school teachers who were also math majorsHit various conceptual/linguistic obstaclesPerhaps: 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
6TodayThe education community’s descriptions of knowledge used in teachingOrigin/history of these descriptionsDesign of the inquiryThe findings are questionsCase/question 1Case/question 2Where does this leave us? Where might we go from here?
7Some things we have learned from research on teaching Content knowledge is not all that mattersTeachers’ 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)There is the suggestion in Rowan, Hill, and Ball that there is SCK (specialized content knowledge) that teachers need to know.7
8Subject matter knowledge Knowledge of content and studentsKnowledge of curriculumPedagogical content knowledgePedagogical knowledgeCommon content knowledgeKnowledge of content and teachingHorizon content knowledgeSpecialized content knowledge
9Horizon content knowledge Common content knowledgeHorizon content knowledgeSpecialized content knowledge
10Common 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)
11Specialized 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)
12Specialized content knowledge used to do the “mathematical work” of teachingto follow and understand students’ mathematical thinkingto evaluate the validity of student- generated strategiesshown to play a role in teachers’ practices and correlate with students’ learning (Ball & Bass, 2000; Hill et al 2004, 2005; Ma, 1999)12
13Horizon 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).
14Generally 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).
15Generally 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).
16But… Development involved elementary teachers Also: Grounded in practices of teachersFramework and item development in parallel
17So, do these constructs (e. g So, do these constructs (e.g., SCK, CCK, HCK) “work” at other grade levels?Reasons the answer might be YES:Intellectual work/tasks are similarCognitive processes appear to be similarReasons answer might be NO/IT DEPENDS:Typically, different amounts of content preparationTypes of knowledge are highly interconnected and influence one another (e.g., Even & Tirosh, 2002; Mewborn, 2003; Sherin, 2002)
18Why is this important? For practical reasons Design of math programs for teachersCapstone coursesFor research and theory- development reasonsHow 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?
19Beginnings 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, ormethods, ortheory
20From 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)
21Developments 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 frameworkAs a result, little occasion/need for theory examination/workMost focus on categorizing knowledge and are NOT tightly grounded in practice
22Developments at the tertiary level Much less is happeningAs of 2010, only 5 peer-review publications about college mathematics teachers’ practices (Speer, Smith & Horvath, 2010).Two of those were about knowledgeMany (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).
23Design of our inquiryAnalyze how constructs are defined in the literatureLook for illustrative examples in the (available to us) data corpusProducts of this work:QuestionsBeginnings of “road map” to places the community might look at more carefully
24We examined explicit definitions of CCK, SCK, and PCK operational definitions as found in the literature on elementary teachers’ knowledgethose 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
25Comparison & contrast work compared and contrasted characteristics of elementary school teachers with characteristics typical of secondary/post- secondary teachersexamined potential implications for the definitions of CCK, SCK, and PCK
26Data sources for illustrative cases Audio & video recordings of courses for pre-service secondary mathematics teachersTeaching methods courseCapstone course for mathematics majors enrolled in a secondary teacher certification programAudio 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)
27Case 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).
28With 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?
29Finding #1What 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 useTopics that recur again and again in increasingly sophisticated ways
30Case 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.
31A 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.
32The quiz: Quotient rule See handout. Find f’(x). Student does not use the quotient rule.
33With 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?
34In both situations the mathematician needs to: make sense of the mathematical ideas and reasoning presented by someone elsedetermine whether the reasoning is correct or incorrectformulate a response about the proposed solution
35Further thoughtsElementary 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?
36Finding #2What 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?
37Some 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.
38Concluding questionsCan 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?)
39So, what do we do?Be on the look out for “problematic” examples– they might help us understand and refine our definitionsSeek out other questions– they might focus our attention on aspects of theory/definitions that need some work
40AcknowledgementsThis 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.
41ReferencesBall, 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,
44An example: Even number definition Item to assess Specialized Content Knowledge1: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?1Drawn from examples given in talks by Deborah Ball and colleagues found at