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

The Arabic translation of the terms and expressions of the TEL Dictionary has been released

The Arabic translation of the entries of the TEL Dictionary has required a considerable effort which demonstrates once more that questioning the Technology Enhanced Learning vocabulary from a multiple perspective is needed. This time five expressions have not found a satisfactory translation: Constructionism, Pedagogical agent, Virtual pedagogical agent, Animated pedagogical agent and Programmable course. The case of "constructionism" is not a surprise, it is in fact in all languages not translated but transliterated. For the next three cases, it is the term "agent" which is resisting. In my opinion,  the last case should find a solution soon. Suggestions and contributions are welcome, for this purpose LinkedIn members are invited to join the LinkedIn "TEL dictionary initiative" group.

The terms documented by the TEL Dictionary are now available in...
English  • العربية • Български • Dansk • Español • Ελληνικά • Eesti • Français • Magyar • Italiano • 日本語 • 한국어 • Nederlands • Português • Русский • Slovenčina • Türkçe • Tiếng Việt • ‪中文(繁體)‬ • ‪中文(简体)‬

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.

cK¢, une introduction à l'occasion de la conférence PMENA 2013

J'aurai l'occasion cet automne de présenter les principes et objectifs du modèle cK¢ dans le cadre de la conférence annuelle de la section américaine du groupe international Psychology of Mathematics Education. Voici le résumé de mon exposé dont le texte est disponible sur HAL sous le titre "cK¢, a model to reason on learners' conceptions"

"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."

 La conférence PMENA a lieu du 14 au 17 novembre à Chicago.

Experiential learning, the new entry of the TEL Dictionary

What does exactly mean "experiential learning"? Is it learning serendipitously from an eventful life or is it learning from empirically from experiences one managed for you or that you managed yourself? How far is this concept from "inquiry learning" or "exploratory learning"?

The documented definitions of Experiential learning prepared by Vyara Dimitrova and Paul Kirschner for the TEL Dictionary clarify these issues and are excellent basis for further discussion.

One more question: would "experiential learning" from an educational perspective be a concept enhancing our tool box to understand the theoretical tensions between informal and formal learning?

#ocTEL MOOC (week 4 A42) Why would the student do or say this rather than that?

The second activity of this week on "Producing Engaging and Effective Learning Materials" is about the evaluation of resources in our area. So, it means, in my case, evaluating a resource for the learning of mathematics. However, I will start from a more general perspective. Whatever is the targeted learning, the first thing to check is the validity of the content the resource claims providing the learners with respect to the referent discipline. Then only, I will assess it from a learning perspective. Indeed, there are many issues to consider from accessibility to usability, motivation and autonomy. But, three questions have a hight priority in driving my evaluation:

Why would the student do or say this rather than that?
What must happen if she does it or doesn’t do it?
What meaning would the answer have if she had given it?

I borrow these formulations from the Theory of Didactical Situations (Brousseau 1997 p.65), but the questions are very pragmatic. The theory works here as a driver of our thinking; it is a tool to anticipate what could be the learning outcome, its likeliness, the possible limits and hence the needed intervention of a teacher. Depending on the responses, one may have to stage the use of the resource in one way or another.

Interaction and feedback are the main objects of the evaluation. The issue is not that students will do that or this, but why they do it,  because the constructed piece of knowledge must appear as the best adapted to the situation. Knowledge is something you reconstruct for yourself and appropriate because of its use value. The next issue is to verify, if the resource is interactive in some way, that it can feedback students so that they have a chance to realize that something went wrong and then react to that. If the resource is not interactive, then the issue is whether it is possible to figure out any thing about the activity (possibly, just reading) of students and find the appropriate support to bring. Eventually, the stake of this inquiry is the meaning possibly constructed by the student.

All this means that there is enough documentation about the resource, otherwise one has to guess or invent... just having a resource without information about its design, the intention of the designer and indications about its use, it is hardly possible to make a proper evaluation. This may be the reason why I couldn't do it for the proposed resource. But, anyway, I will make the exercise when achieving the third task of the week.

Some thoughts about Learning aware environments

Reference: Nicolas Balacheff, Learning aware environments, eAgenda 2006 European Forum, Castelldefels, Spain, 24 October 2006
 

Could we “introduce learning in every human activity”? From a non-English speaking perspective this question may sound strangely. Isn’t it the case that learning is present everywhere and at every moment in our life?  This is a matter of survival. Learning is a competence shared by all living organisms. Learning is life-long; it starts with our first breath and continues until the very last one. However there is something specific to human-beings, which is that not only do they learn to survive in their biosphere, but also they have to learn to survive in a noosphere that humanity is continuously building, renewing, transforming. The noosphere is made tangible by human artefacts, but essentially by language. Learning in the noosphere is so complex that specific strategies have been developed to support it, namely teaching (or education, instruction, training, coaching, etc.).

At this point it is interesting to come back to the origin of “learning” and “teaching” in the English language. Both words have a German origin, tracing back respectively to “læran” and “tæcan” in Old English. While the latter meant “to show” or “to persuade”, the former was preferred to mean “to teach” or “to guide”. Then, could we suggest that the English word learning has a teaching connotation, and that as a result the meaning of  the question is: “can we introduce læran in every human activity?”, what introduces the idea of environments with “teaching” capabilities.
 

Designing environments likely to stimulate and support learning outside formal education and training —or situations mimicking these—was in most cases out of reach until the emergence of the digital technology which bridges the biosphere, where our bodies and activities are developing, and the noosphere where minds and intellectual constructs are developing. While language and the related symbolic technology (writing and reading) were the privileged tools to support learning, digital technologies go beyond by producing highly interactive simulations and virtual worlds. But more significant is the development of augmented reality, the systematic embedding of sensors and system on ship in all artefacts which open the possibility of a “merge” of both spheres. Here is the challenge of ambient computing.

Just as the rest of our environment, modern digital technologies cannot support learning if they have not been designed on purpose by incorporating teaching (coaching, instructing, scaffolding, or else) features. This is the challenge of designing, implementing and understanding learning aware environments. They are environments which have the capacity to recognize and capture relevant events from observing the human activity, the ability to understand the learning needs and then to provide the adequate feedback in whatever form. This is a scientific and technological challenge for ambient computing and research on cognitive systems. This is also a political challenge because the full development of learning aware environment will not be possible without addressing ethical (protecting the individuals and the communities) and economical problems (accepting that knowing is a universal right).