conceptual (knowledge) confusion: some deliberatively provocative remarks

Conceptual (knowledge) confusion: Some deliberatively provocative remarks

Jon R. StarHarvard 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 9 NCTM 2008 Salt Lake City 2

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)


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


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



x −3 =4x=7

x −3 =−4x=−1

x −3 =4

3 7-1

4 4

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!


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


(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


x −6 =−2

x −6 =−2x=4

x −6 =2x=8

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?)


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


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...)


1. Confusion of type and quality

• From Star, 2005:


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


2. Confusing definitions?


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?


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


Chapter 3: Answers?

• How might my colleagues respond to my confusion?

• Let me speculate, and also• Provide further questions and responses


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


Sounds good, but...


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


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


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


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


Thanks?!

Jon R. [email protected]

This presentation and other related papers and presentations are available at:

http://gseacademic.harvard.edu/~starjo/


I’m Jon Star, and I approved this message.

conceptual (knowledge) confusion: some deliberatively provocative remarks

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