Conceptual (knowledge) confusion: Some deliberatively provocative remarks Jon R. Star Harvard Graduate School of Education

“These [three] truths are self-evident” A key learning outcome in mathematics is the development of CK PK is also important, but optimally PK should be connected with CK In the absence of connections to CK, PK is likely be known only by rote Wed April 9NCTM 2008 Salt Lake City2

Conceptual confusion Chapter 1: In the beginning... –A story from my past, where my faith in the power of these three truths is shaken Chapter 2: Help me out! –Where I describe several areas of confusion with the construct of CK/PK framework and the 3 truths Chapter 3: Anticipated responses –Where I speculate about possible responses to my areas of confusion (and respond to these responses) Wed April 9NCTM 2008 Salt Lake City3

Chapter 1: Beginnings Star as novice high school teacher –About 20 years ago Mentored by reform-oriented dept. chair –Active nationally and regionally in NCTM Teaching absolute value equations Wed April 9NCTM 2008 Salt Lake City4

Focus on conceptual knowledge What are the key concepts that I want students to understand in working with absolute value equations? –Absolute value as distance Goal is for students to really understand what they are doing –Knowing about absolute value as distance will help the procedure make more sense Wed April 9NCTM 2008 Salt Lake City5

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What happened? Absolute value as distance was not easy to connect or integrate into the procedure In students’ minds, understanding this concept did not help them solve these problems CK was peripheral and separate for students Faded quickly, unless I brought it up continually and with great emphasis! Wed April 9NCTM 2008 Salt Lake City7

Over time... Students remembered the procedure and could implement it successfully on a wide range of problems, seemingly without accessing CK As students forgot CK, I did not see any differences in their ability to solve this and other similar (and even harder) problems Wed April 9NCTM 2008 Salt Lake City8

(But Jon, what about problems like this?) Some students did this, but they then checked their answers and found neither solution worked –Justified “no solution” by noting that absolute values can’t be negative – no mention of distance Wed April 9NCTM 2008 Salt Lake City9

Crisis of confidence? Was I wrong in thinking that the purpose of teaching CK was that it helped students become better problem solvers? –In this case, CK did not seem to impact students’ ability to solve a wide range of easy and hard problems Why do we teach CK? –Am I teaching CK because CK is important to know? (Maybe I just wasn’t a good teacher?) Wed April 9NCTM 2008 Salt Lake City10

Chapter 2: I dive into CK/PK What are CK and PK? How do we think they are related? Why and how do we teach CK and PK? In spite of my firm conviction that we need to teach mathematics for understanding, I became quite confused about the PK/CK framework, particularly about how math education as a field talked about and studied CK/PK Wed April 9NCTM 2008 Salt Lake City11

Areas of confusion My current list of areas of confusion about (the way that I perceive many in our field talking about and studying) PK/CK (Perhaps my confusions are more about the ways that the PK/CK framework is used to describe and justify the 3 truths, rather than the truths themselves...) Wed April 9NCTM 2008 Salt Lake City12

1. Confusion of type and quality From Star, 2005: Wed April 9NCTM 2008 Salt Lake City13

Double entendre? Does CK mean “knowledge of concepts”? Or does CK mean “that which is known deeply”? For many, it seems that CK means both –I see these are two different meanings PK can mean “knowledge of procedures” or “that which is known superficially” –I see these as two different meanings Wed April 9NCTM 2008 Salt Lake City14

2. Confusing definitions? Wed April 9NCTM 2008 Salt Lake City15 Conceptual knowledge is knowledge that is rich in relationships Procedural knowledge – less well connected; relationships sequential or to other procedures When procedural knowledge is connected to conceptual knowledge, what do we call this knowledge?

3. Limited operationalization Hiebert 1986 book and Baroody 2003 books –Elementary math topics –Baroody and Battista comments to follow – ditto? It seems odd that we assume we can easily generalize from elementary to secondary school math about CK, PK, and the relationship between them Why isn’t there more work using this framework in high school math topics, for example? Wed April 9NCTM 2008 Salt Lake City16

4. Absence of good assessments How do we assess CK? Are there any/many reliable and relatively efficient ways to assess CK about a given math topic, for use in a study of several hundred students? –The “I know it when I see it” problem Some disagree on the premise that CK can be assessed with a multiple choice test Wed April 9NCTM 2008 Salt Lake City17

Chapter 3: Answers? How might my colleagues respond to my confusion? Let me speculate, and also Provide further questions and responses Wed April 9NCTM 2008 Salt Lake City18

Possible responses There is clear research evidence that CK helps problem solving and aids transfer Yes, these terms are not precisely defined, but we really do know it when we see it The best way to assess CK is by interviewing a student; why would we need other types of assessment? It’s all about relationships between CK and PK anyway Wed April 9NCTM 2008 Salt Lake City19

Sounds good, but... Wed April 9NCTM 2008 Salt Lake City20 Evidence Assessment Definition/Theory/Operationalization

Not there yet - Evidence Evidence is not clear or robust enough that critics are convinced If you show me a study, I’ll want to know how you assessed CK and how you assessed problem solving outcomes Difficult to produce convincing evidence without good assessments Wed April 9NCTM 2008 Salt Lake City21

Not there yet - Assessments There are no widely-used, “standard” assessments for CK of particular math topics Despite the value of interviews to assess CK, we also need other instruments that can be used in large-scale quantitative studies Difficult to produce good assessments without clear definitions and operationalization Wed April 9NCTM 2008 Salt Lake City22

Not there yet - Definitions Agreeing that the focus is on relationships doesn’t eliminate the need to develop good theory and definitions of CK and PK Every curriculum and professional development program claims to foster CK in teachers and students We have no way to evaluate or refute such claims with good definitions and operationalizations Wed April 9NCTM 2008 Salt Lake City23

In sum... Arguing about the meaning of CK and PK, how these words are used, and what the theory says about the developments or and relationships between these types of knowledge is not merely an esoteric issue but is something of great importance to our field CK/PK is primarily an ideological framework and not an empirical one, which is problematic Wed April 9NCTM 2008 Salt Lake City24

Thanks?! Jon R. Star This presentation and other related papers and presentations are available at: Wed April 9NCTM 2008 Salt Lake City25 I’m Jon Star, and I approved this message.