Linking ck¢ and the Toulmin model

An original theoretical as well as methodological aspect of Bettina Pedemonte PhD work, was to use the Toulmin schema in relation with a knowledge model in the framework of the Theory of Didactical Situations. We have recently revisited this work and produced  a synthesis of this important outcome. This has led to a paper recently published in the Journal of Mathematical Behavior, in which we analyze students’ conceptions in geometrical problem-solving and their relations to proving. We show how students’ conceptions strongly impact the argumentation activity and the construction of a proof. This is illustrated by analyzing two pairs of students’ argumentations and proofs taken from a set of data collected from a teaching experiment. The use of the Toulmin's model enriched with the ck¢ model allows to elicit the complexity of a cognitive analysis of argumentation and proof that accounts for the students’ knowledge system. Toulmin's model is useful to select those elements in the argumentation that are part of students’ conceptions while ck¢ allows us to see the role they have inside the argumentation.
[click on the cover to get a free copy within the 50 coming days]

Calculus, a cK¢ perspective on learners understanding

On the occasion of the international meeting on learning and teaching calculus to be held in Mexico in September 2015, I will address the problem of understanding and modelling students conceptions in this domain.
I will introduce the keynote by a survey of the recent book from Gilbert Arsac about the birth of Uniform convergence and the role played by Cauchy. Then I develop the case of understanding learners conceptions of function. The objective of this talk is to present, taking Calculus as an example, the use of the cK¢ modelling framework and discuss its effectiveness. A workshop may be organized to discuss the relevance and the technicalities of the approach in the case of the research carried out by the audience.

(Long version of the) slides-show in support to the keynote

Bridging knowing and proving

The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory.

Talk at the conference "Explanation and Proof in Mathematics:
Philosophical and Educational Perspective"
held in Essen in November 2006

Balacheff N. (2010) Bridging knowing and proving in mathematics An essay from a didactical perspective. In Hanna G., Jahnke H. N., Pulte H. (Eds.) Explanation and Proof in Mathematics. pp.115-135. Springer.
Author preprint available from HAL and arXiv.

cK¢ takes up the challenge of modeling learners understanding (a response to Guershon Harel - continued)

Several of the questions Guershon Harel [*] asked after my talk at the PME-NA conference in Chicago concern the scope and objectives of the cK¢ modeling framework. In this post, I take each of these questions and give a short answer leaving for specific posts more elaborated answers when needed. So, here it is:

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

The answer is very simple: cK¢ provides a framework for modeling learning states. Indeed learning processes are of a paramount importance, but they are in my opinion more an object of study for psychology than for mathematics education. Indeed, I don't confuse "learning processes" which are of a mental and intellectual nature, and "problem solving processes" which correspond (to make it simple) to the activity the learner engage when he or she has to solve a problem in a given situation. We need to understand and model these processes, but even if they may inform us about learning processes they are only a dimension of them.

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?

cK¢ provides a framework to model learners' understanding (learning states, as just said) in mathematics from a situated perspective; situations may be set up within a classroom or in an other context. Actually, the objective is slightly larger, I would claim that cK¢ is a framework to model mathematical understanding taking into account the situational characteristics, not being restricted to learners. A key idea when I started the project was to find a way to model mathematical conceptions with the same tools, be it they conceptions of novices or experts, wrong or correct from whatever knowledgeable perspective.

6. What exactly are the challenging aspects of modeling learning relative to modeling content and pedagogy? 

Anyone will expect the content to be in some sense "correct" and explicit enough to be defined precisely as a content to be taught. Still, there are challenging aspects related to its nature; for example, to model algebra or geometry from an epistemic and teaching perspective is not of the same level of difficulty.
Concerning pedagogy, which is in the first place the product of a practice, the related knowledge is largely implicit. Practitioners can share and discuss their expertise within their community or with teacher students in an apprenticeship approach, but the communication register is largely based on a pragmatic co-construction of meaning referring to a shared practice; as a result it is difficult to model. The challenge is to make explicit what is largely knowledge in action. However, there are progresses as witnessed by research on tutoring and adaptive learning systems.
In line with what I wrote before, I will not consider learning but "the best conditions for learning" (best or optimal). So, I would consider the question: what are the challenging aspects of modeling (determining) the best conditions for learning compared to those of modeling content and pedagogy? I would say that these challenges are very close the one to the other. First, part of the challenge comes from the nature of the content at stake, in particular the role of representations and the complexity of validating the related mathematical statements. Arithmetic, algebra, geometry, calculus, probability, discrete mathematics raise their own specific learning challenges within mathematics. Then, determining the best conditions for learning requires knowing some critical things about the initial state of knowledge of the learners (as a matter of fact, we can only teach people who know); in other words it depends on our knowledge of the conceptions which have to evolve or to be rejected. Eventually, one can say that the challenge is of knowing the knowledge at stake from a learning perspective, understanding learners initial understanding, and being able to design situations likely to stimulate, support and validate the construction of new knowledge. Eventually, modeling pedagogy consists in stating principles for designing situations which implements the "best conditions for learning" under multiple constraints: curricula, institutional standard, cultural and economical environment, time and all material means to make the class working properly.

7. What are the interdependent relationships among these three models? 

Indeed, as the above answers suggests it, the three models are tightly related. In particular, from an educational perspective modeling knowledge is under learning constraints because what we need is not a "knowledge model" for itself but a model of the intended learning outcomes. This knowledge (intended learning outcomes) must be learnable (accessible to learners) and teachable (manageable by teachers); actually, the objective of the didactical transposition is exactly to produce this knowledge which is always at a distance from a knowledge of reference which to some extend justify it. 

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?

Be it explicitly the case or not, any pedagogical model includes a learning model; I mean a model of the (claimed) best conditions for learning. 

Then, a more philosophically oriented, yet critical, question is

9. Are cognitive models of thinking possible?

Once we have agreed on what means "cognitive", "model" and "thinking" my answer would be: yes... but a discussion of this answer may go far beyond my field of expertise and beyond the scope of this blog as well.

cK¢ is not a cognitive model (a response to Guershon Harel)

The first question Guershon Harel [*] asked about cK¢ is

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

Actually, this question comes after a more general one: (1) "What is a cognitive model and what are its purposes?" and a more direct one (2) "Is the cK¢ a cognitive model?
My response is very simple and direct:

cK¢ does not propose a framework to construct cognitive models. It does not pretend to model an "approximation to processes of humans’ mental activities" and do not ambition to be "capable of explaining mental processes or interactions among them", eventually it does not aim at answering a specific question such as "how do we learn to categorize perceptual objects?"

Yet, cK¢ has a very strong relation to the learner by being focused on his or her interaction with a learning environment (more precisely the "milieu"). Indeed, cK¢ could contribute to a cognitive approach, but it is not its objective in the first place.

Based on the evidence we can get from the learner's activity, the objective is to characterize it in terms intelligible from a mathematical perspective and which can serve as inputs to take teaching decisions. Two types of evidence are easy to get: representations manipulated by the learner and operators he or she uses in order to achieve a task or to solve a problem. Actually, these operators are not always explicit but it is not impossible to have an interpretation of the learner's behaviours which makes sense from a mathematical perspective (this corresponds to the Vergnaud coup de force when he coined the concept of "theorem in action"). It is then reasonable to claim that we have a picture of the learner understanding when these representations and operators are stable within a problem-space. This has to be completed by a description of the means the learner uses to take a decision about the validity of his or her activity and the related outcomes. It is the idea of the control structure. Once we have a characterization along these four dimensions, we can conjecture a mathematical meaning, but this does not tell what are the related mental activities or cognitive structures as psychology or neuroscience would understand them. It is very likely that different learning theories would shade different lights on these characterizations. However, my claim is that such characterizations are sufficient to assess the so-called mathematical understanding, and to take teaching decisions or  to design learning environments.

For the rest, cK¢ shares many of the scientific characteristics of "cognitive models": it is based on "rigorous methods", it is "capable of generating testable predictions" and of generating descriptions in "formal, mathematical or computer, languages". It does not describe processes but  nothing prevents it a priori to contribute to such descriptions, this is something to explore.

Eventually, it is important to realize that cK¢ does not ambition to construct models to respond to the question "How does a child transition from additive reasoning to multiplicative reasoning?" but to the question "What are the optimal conditions to initiate and support the child transition from additive reasoning to multiplicative reasoning?"

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

cK¢, an introductory talk on the occasion of the PMENA annual conference

Next fall, on the invitation of PMENA (the Psychology of Mathematics Education North American Chapter), I will have the occasion to present an introductory talk to the cK¢ model. The text of the talk entitled "cK¢, a model to reason on learners' conceptions" is now available on the arXiv.org. Here is a summary:

"Understanding learners' understanding is a key requirement for an efficient design of teaching situations and learning environments, be they digital or not. This keynote outlines the modeling framework cK¢ (conception, knowing, concept) created with the objective to respond to this requirement, with the additional ambition to build a bridge between research in mathematics education and research in educational technology. After an introduction of the rationale of cK¢, some illustrations are presented. Then follow comments on cK¢ and learning. The conclusion evokes key research issues raised by the use of this modeling framework."

 The PMENA 2013 conference is held in Chicago from the 14th to the17th of November.