cK¢ , the Chicago talk and Guershon Harel questions

cK¢, a model to reason on learners' conceptions 
 

This presentation at PME-NA 2013 was followed by comments and questions from Guershon Harel on the invitation of the conference organisers.  I publish below with Guershon's permission his reaction on the talk, and will respond to his questions in coming posts on this blog.

Questions Inspired by or Generated from the cK¢ Model Presentation
Guershon Harel, University of California at San Diego

I would like to thank the program committee for inviting me to react to Nicolas Balacheff’s plenary talk. I have known Nicolas for many years, both professionally and personally. I feel honored to have the opportunity to react to his work.
A fundamental human nature is that not only do humans seek to resolve puzzles, but also they seek to be puzzled. Scholarly work, thus, is judged not only by the questions it answers but also by the questions it generates. Nicolas’ paper—of which the talk you have just heard is part—does exactly that: It addresses fundamental questions about learning and thinking and at the same time generates new questions.
A strong feature of Nicolas’ work, in general, and of this paper, in particular, is its attempt to define concepts and ideas rigorously. This puts the reader in a mood to follow suit, by asking questions of rigor as well.
What I will do in the time allocated to me is to share with you some of the questions Nicolas’ paper generated for me as I tried to build a coherent image of the cKc model. It is possible that the image I constructed is entirely idiosyncratic, not coinciding with the image—or better say conception—intended by Nicolas.
Whatever the case may be, I highlight that the sole purpose of the questions I present before you now, is to generate discussions, with the hope that they would further understanding, generate research studies, and advance effective classroom implementations of the cKc model. Balacheff’s paper is about a “[cognitive] model of a learner”. The adjective “cognitive” is important here to differentiate it from other types of models. So, following the rigorous style of the paper, the first question one might ask is:

1. What is a cognitive model and what are its purposes?

Briefly, and aggregately, the essential characteristics of “cognitive model”, as they appear in the literature include the following:

a. Cognitive models are approximation to processes of humans’ mental activities, such as attention, understanding, inferencing, decision making, etc.
b. They are derived from basic principles of cognition, such as a particular theory of learning.
c. They are based on rigorous methods of elicitation of cognition.
d. They are capable of explaining mental processes or interactions among them.
e. They are capable of generating testable predictions, both quantitative and qualitative.
f. They are described in formal, mathematical or computer, languages.
g. They aim at answering a specific question; for example: how do we learn to categorize perceptual objects? Such as:

i. How does a student learn to categorize problems according to their mathematical structure?
ii. How does a child transition from additive reasoning to multiplicative reasoning?
iii. How does one learn to categorize paintings according to the periods to which they belong?

h. They may target cognitive processes or cognitive states.

For example, the question, “What are humans’ categories of perceptual objects?” is a question about product rather than process. Likewise, the question “What are students’ proof schemes?” is a question about state, not process.
To illustrate the difference between these two types of models, I mention two examples of works many of you are familiar with. These are the seminal works of Marty Simon and Jere Confrey. What sets the research programs of Marty and Jere apart from many other works is their focus on the mechanisms that account for conceptual learning: namely, the transition from one conceptual state to another.
So relative to this background and characterizations, the questions one might ask about the cK¢ model are:

2. Is the cK¢ a cognitive model?

Or less rigidly,

3. To what extent is the cK¢ a cognitive model?

4. Is the cKc a model of learning processes or learning states?

Furthermore, given the unique nature of the mathematics discipline among the various disciplines, and given the complexity of the classroom setting, in general, and that of mathematics classroom, in particular,

5. Is the cK¢ a model of a learner (period), a model of a learner learning mathematics, or a model of a learner in a mathematics classroom setting?

As mathematics educators, we are most interested in the interactions among the three models outlined by Balacheff: the model of the learner, the model of the content to be learned, and the model of pedagogy. Nicolas indicates “For the last two [models], research has constantly been very active with some promising progress. On the contrary, modeling the learner proved to be a real challenge.” Two questions of interest, though they perhaps go beyond the scope of the paper, are:

6. What exactly are the challenging aspects of modeling learning relative to modeling content and pedagogy?
7. What are the interdependent relationships among these three models?
8. What is the efficacy of such models if they are constructed independently from each other? In particular, can models of content and pedagogy be viable without the presence of a learning model?

A more philosophically oriented, yet critical, question is

9. Are cognitive models of thinking possible?

This question is derived from the third characteristic of mental models I listed earlier; namely, a mental model is based on a rigorous method of elicitation of cognition. This characteristic is particularly problematic. Here is why. The cK¢ is a model of learning/thinking. As was pointed out by Colin Eden, “if we take seriously Karl Weick’s aphorism that we do not know what we think until we hear what we say, then the process of articulation—that is, the learner’s utterances and behaviors that constitute the data for the construction of the model—is a significant influence on present and future cognition. Since articulation and thinking interact, as is largely accepted, then an elicitation of cognition that depends upon articulation is always out of step with cognition before, during, and after the elicitation process.”

Even if we overcome this philosophical hurdle, an empirical question emerges:

10. To what extent can a general learning model be viable, given human diversity of character, culture, and circumstances?

The fourth component of the cK¢ model is control. Balacheff characterizes control under the general umbrella of metacognitive behaviors. The control component is crucial, and is Balacheff’s significant addition to Vergnaud’s model. It is crucial because it is the place where issues of the learner’s understandings are to be revealed. The set of four examples Balacheff discusses to illustrate the cK¢ models are illuminating, but I still found myself wanting to better understand the cK¢’s definitions and treatment for crucial control constructs such as understanding, meaning, and ways of thinking.

These are crucial constructs with various instantiations. For example, when we talk about “understanding” and “meaning”, we—researchers and teachers—want and need to distinguish, for example, between “understanding in the moment” and “stable understanding”, and between “meaning in the moment” and “stable meaning”. Likewise, we want and need to observe ways of thinking, or habitual anticipations of meanings, both desirable and undesirable. Thus, it is natural to ask:

11. What are “understanding,” “meaning,” and “way of thinking” for the cKc model, and what is a reliable methodology to elicit them?

12. What is “Problem” for the cK¢ model?

Recall that Balacheff’s definition of “conception” is a quadruplet (P, R, L, Σ). Balacheff recognizes that the first component, Problems, is problematic; namely, he faced the question as to how to characterize the set of the problems for a particular conception. After considering two possible characterizations, one by Vergnaud and one by Brousseau, Balacheff describes P as a set of problems prototypical to the field to which the conception belongs. This characterization raises theoretical, methodological, and instructional questions.

Specifically, the cK¢ model postulates that problems are the source and the criteria of learning and knowing. And following Vergnaud and Brousseau, problems are also held as the engine of the teaching process. A consequence of these largely agreed upon positions is that the cK¢  hinges upon the school prototypical problems one chooses to elicit conceptions.

The difficulty that arises here is that many of these prototypical problems are alien, not intrinsic, to the students. The students might be able to solve them, but the kinds of perturbations they engender with the students are didactical, aimed solely at satisfying the will of the teacher. Thus:

13. If the problem is alien to the learner, what meaning can a researcher give to the operations and control components of the model?
14. How is to be determined by researchers, and more importantly by teachers, whether the problem posed to elicit conception is intrinsic or alien to the learner, and how does this determination effect the observer’s conception of the learner’s conception?

The Problem component is also a crucial factor in Balacheff’s definition of generality. Generality is one of the factors in the cK¢ model shaping relations between conceptions. As such, it is crucially important, for the simple reason that it provides a criterion for conceptual development; namely, how one conception is more general than other.

Balacheff defines generality as follows:

C=(P, R, L, Σ) is more general than C’= (P’, R’, L’, Σ’) if there exists a function of representation ƒ: L’→L so that ∀p ∈P’, ƒ(p)∈P.

The examples of relative generality discussed in the paper work nicely according to this definition. Balacheff’s definition also worked well with many of the examples I tested. For some
cases, where P=P’, the definition may need further refinement. Consider the following example:

A 13-year-old girl, Tami, and an 8-year-old boy, Dan, were interviewed in pair.

Interviewer: One pound of candy cost $7. How much would 3 pounds of candy cost?
Tami: Three times seven, 21.
Dan: I agree, three times seven.
Interviewer: What if I changed the 3 into 0.31? What if the problem were: One pound of candy cost $7; how much would 0.31 of a pound cost?
Tami: The same. It is the same problem, you have just changed the number, 0.31 times 7.
Dan: No way! It isn’t the same. Can’t be [angrily]. It isn’t times. Why did you [speaking to the interviewer] agree with her?
Interviewer: I didn’t agree with her, I’m just listening to both of you. How would you solve the problem?
Dan: You take 1 and you divide by 0.31. You take that number, whatever that number is, and you divide 7 by that number.

Indeed:

On the one hand, the set of problems belonging to Tami’s conception is identical to set of problems belonging to Dan’s, and it seems that there is always a translation between the corresponding L and L’ satisfying Balacheff definition of generality. Hence, the two conceptions seem to be equivalent. On the other hand, intuitively, I want to attribute a greater generality to Tami’s conception, with all the great admiration I have for Dan’s conception.

In closing,

Three of Balacheff’s goals for introducing the cK¢ can be summarized as follows:

a. Make more efficient our own research.
b. Clarify concepts and their relationships.
c. Contribute to better understanding of learners’ understanding, so as to support decision making for teachers and learners.

It is against these goals that I chose the questions I have just presented.

Thank you