Definitions of mathematical knowledge for teaching: Using these constructs in

research on secondary and college mathematics

teachers

Natasha SpeerThe University of MaineDepartment of Mathematics & StatisticsUMaine Research in STEM Education Center

Consider this: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?

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.

Some thoughtsSome 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?

How I (and others) ended up How I (and others) ended up thinking about these issuesthinking 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

How I (and others) ended up How I (and others) ended up thinking about these issuesthinking about these issuesLooked 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

TodayTodayThe education community’s

descriptions of knowledge used in teaching

Origin/history of these descriptionsDesign of the inquiryThe findings are questions

◦Case/question 1◦Case/question 2

Where does this leave us? Where might we go from here?

Some things we have learned Some things we have learned from research on teachingfrom 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)

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

Horizon content knowledge

Specialized content

knowledge

Common content

knowledge

Common Content Common Content KnowledgeKnowledgemathematical “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)

Specialized Content Specialized Content KnowledgeKnowledge“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)

Specialized content Specialized content knowledgeknowledgeused 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)

Horizon Content Horizon Content KnowledgeKnowledge“an awareness of how

mathematical topics are related over the span of mathematics included in the curriculum” (Ball, Hoover Thames, & Phelps, 2008, p. 403).

Generally accepted thatGenerally accepted thatMathematical 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).

Generally accepted thatGenerally accepted thatMathematical 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).

ButBut……Development involved

elementary teachers

Also:◦Grounded in practices of teachers◦Framework and item development

in parallel

So, do these constructs (e.g., SCK, So, do these constructs (e.g., SCK, CCK, HCK) “work” at other grade CCK, HCK) “work” at other grade levels?levels?Reasons the answer might be YES:

◦Intellectual work/tasks are similar◦Cognitive processes appear to be

similarReasons 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)

Why is this important?Why is this important?For practical reasons

◦Design of math programs for teachers

◦Capstone coursesFor 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?

Beginnings of a discussionBeginnings of a discussionOthers 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

From last paragraph of 291-From last paragraph of 291-page book: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)

Developments at Developments at secondary levelsecondary levelWork 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

Developments at the Developments at the tertiary leveltertiary levelMuch 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).

Design of our inquiryDesign of our inquiryAnalyze how constructs are

defined in the literatureLook 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

We examinedWe examinedexplicit definitions of CCK, SCK, and

PCKoperational 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

Comparison & contrast Comparison & contrast workworkcompared 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

Data sources for illustrative Data sources for illustrative casescasesAudio & 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)

Case 1Case 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).

With a neighbor (or two):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?

Finding #1Finding #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 use ◦Topics that recur again and again in

increasingly sophisticated ways

Case 2:Case 2: The Nature of

Mathematicians’ WorkConsider 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.

A day in the life of a A day in the life of a mathematicianmathematicianIn 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.

The quiz: Quotient ruleThe quiz: Quotient rule

With a neighbor (or two):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?

In both situations the In both situations the mathematician needs to: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

Further thoughtsFurther 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?

Finding #2Finding #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?

Some evidence that Some evidence that mathematicians feel a mathematicians feel a differencedifference“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.

Concluding questionsConcluding 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?)

So, what do we do?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

AcknowledgementsAcknowledgementsThis material is based upon work supported

by the National Science Foundation under Grants No. DRL-0629266 and DUE - 0536231. 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.

ReferencesReferences Ball, D., Hoover Thames, M., & Phelps, G. (2008). Content knowledge for teaching:

What makes It special? Journal of Teacher Education, 59(5), 389-407. 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. 219-240). 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), 371-406.

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. 111-156). 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. 45-52). 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), 119-150.

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), 530-562.

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, 247-266.

An example: Even number An example: Even number definitiondefinitionItem 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 http://www-personal.umich.edu/~dball/presentations/index.html

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?