The complexity of the epistemological genesis of mathematical proof

Travelling through Tokyo and Singapore, it is a great pleasure to make a stop and meet colleagues and friends, hence one talk and two seminars. First at the Joetsu Seminar of Research on Mathematics Education in Tokyo on September the 13th, then in Singapore for a seminar at the Mathematics and Mathematics Education (MME) laboratory of the National Institute of Education (NIE) on September the 18th.

Abstract
Early learning of mathematics is first rooted in pragmatic evidences or learners’ confidence in the facts and procedures taught. Nonetheless, learners develop a true knowledge which works as a tool in significant problem situations, and which is accessible to falsification and argumentation. As teachers know, they could validate what they claim to be true, but based on means in general not conforming to mathematical standards. Teaching these standards requires an evolution of their understanding of what can count as a proof in the mathematical classroom, as well as an evolution of their mathematical knowing. This claim is discussed from the perspective of modelling the learners ways of knowing (the model cK¢), within the framework of the theory of didactical situations, bridging the semiotic system they use, the type of actions they perform and the controls they implement either to construct or to validate the solutions they propose to a problem.

Although this presentation is self-content, it could be interesting to complement it with the CINVESTAV talk which focused more on the didactical situations of validation [ppt], one of the specific situations of the Theory of didactical situations [ppt]

What would the Theory of Didactical Situations mean to my research? A workshop

Math Ed. Doctoral Colloquium at CINVESTAV ... in relation to my lecture, I will run a workshop, interactive and collaborative, on the possible contribution of the Theory of didactical situation (TSD) to the design and implementation of a research project in mathematics education.   Questions or comments on this post are welcome (in Spanish, French or English), I will consider them during the workshop. Here is the presentation of the workshop:

Choosing a theoretical framework to address a research question in mathematics education is one of the difficult decision PhD students must take. This workshop, as a follow up of Nicolas Balacheff lecture, will offer an opportunity to present and discuss PhD research projects from a theoretical perspective. The TSD has several integrated dimensions which allows to build bridges with other frameworks such as constructivism, epistemology, situated learning, collaborative learning and educational technology as well. The discussion will allow to deepen the theoretical issues and understand how the TSD can contribute to the shaping of a research project. 

A suggested format is : two minutes presentation of the PhD topic, then five minutes to present an issue which could be either theoretical, methodological or related to the identification and presentation of results. Five to six different projects could be presented  within the 90mn workshops.

The complexity of the epistemological and didactical genesis of mathematical proof (2)

Math Ed. Doctoral Colloquium at CINVESTAV ... hereafter an advanced version of the slides in support to my talk (see the post below for a summary). Questions or comments are welcome (in Spanish, French or English), I will consider them for the talk.

Notes from a research journey on learning proof for
the CINVESTAV doctoral colloquium 2015

The complexity of the epistemological and didactical genesis of mathematical proof

To come soon: an invited lecture at the Math Ed. Doctoral Colloquium at CINVESTAV, ... hereafter a summary of issues I will address:

Students’ mathematical knowledge is first rooted in pragmatic evidences and in the effort to make sense of the content and procedures taught. They develop a true knowledge which works as a tool in problem situations, and is accessible to falsification and argumentation. They can validate what they claim to be true, but based on means which may not conform to current mathematical standards. The theory of didactical situations (TSD) is based on the recognition of the existence of this true knowledge and the analysis of the specific complexity of the teaching situations from an epistemological perspective. It is in this framework that I propose to address the problems raised by the teaching and learning of mathematical proof. The main issue which I will discuss is that the evolution of the students understanding of what count as proof in mathematics implies – and is constitutive of – an evolution of their knowing of mathematical concepts. This discussion will support the claim that the “situation of validation” conceptualized by the TSD must be the starting point of any didactical engineering.

To prepare your participation, here some outlines of the TSD

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¢ 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.

A didactical view on authenticity

Retrieved from the TEL opinion blog, August the 27th, 2008

The search for authenticity of learning situations is a concern for most designers of TEL environments. Most of them realise soon that this is a desperate project since any environment is a representation of some kind of a reference, often called "reality", which keeps staying at a distance. To be as close as possible to reality does not mean much, unless we can qualify or quantify the closeness. Indeed, this is a challenge and we are not be well equipped today to take it up. A solution might be to find a theoretical framework within which we can formulate the problem, and then search for a solution within this framework. This first step will put limits on this solution, but it will make it much more tangible and so accessible to further progress. Currently we too much lack definitions and references to ensure that we can seriously discuss the issue. But, let's try something...

 First, we may agree that a learning environment becomes such if it is embedded in a situation which can contextualise the learner activity and hopefully stimulate, support and validate his or her successful learning. Be they formal or informal, these situations have an objective which can be made explicit in learning terms at least from the point of view of their designers; the fact that this objective is explicit for learners is another story. Following Brousseau(*), let's call "didactical" these situations. Didactical situations can be distinguished from other situations by their explicit intention to "teach". And here is the problem! As soon as the learner identifies this intention and bases on it his or her activity, it is very likely that the learning outcome will not have the expected "authenticity". All the search for authenticity of learning situations (and learning environments) is dedicated to the overcoming of this difficulty.

Second, let's consider the limit case of a didactical situation which didactical intention is completely transparent. If learning occurs in such a situation, we could ascertain that its outcome has the expected "authenticity": it does not owe the didactical intention (in other words the reasons for the activity of the learners are in the knowledge at stake not due to any guessing of the teacher or trainer expectations) These are "adidactical" situations, let's quote Brousseau:

"[The student must know that] this knowledge is entirely justified by the internal logic of the situation and that she can construct it without appealing to didactical reasoning. Not only can she do it, but she must do it because she will have truly acquired this knowledge only when she is able to put it to use by herself in situtations which she will come across outside any teaching context and in the absence of any intentional direction. Such situation is called an adidactical situation. Each item of knowledge can be characterized by a (or some) adidactical situation(s) which preserve(s) meaning; we shall call this a fundamental situation." (Brousseau p.30)

These three concepts: didactical situation, adidactical situation and fundamental situation will allow us to locate our problem of authenticity, and to formulate it.

So, designing an authentic learning situation depends on our capacity to characterize the related fundamental situation in relation to the piece of knowledge which learning is at stake. The problem is then not the closeness to reality, but the fact that the situation has the epistemic properties specific to this piece of knowledge. The specification of a so-called authentic environment requires first the expression of these epistemic properties and of the way they can be "translated" in the tangible world. However, once we have such a situation, there may be still a long way to designing an adidactical situation likely to be made available to learners. The design of this adidactical situation and the related environment is the challenge of designers of authentic TEL environments. After that, there is still one issue for the teacher: bring to life this adidactical situation in the classroom without damaging its "authenticity", what is the didactical challenge!

A quick example to conclude this post: the concept of "angle" in mathematics finds its full meaning when linear measurement is not possible or too "expensive" (for example when sailing on the Atlantic). Let say that the fundamental situation for "angle" is the problem of locating a point in the macro-space. We can realize that the classroom can hardly host a macro-space, we have then to find a situation which has the characteristics of the macrospace (linear measurement being impossible or too "expensive") and can be implemented in a classroom. If the problem were presented in the frame of a piece of paper (micro-space), the situation may appear quite artificial in the students eyes (a ruler is enough), then the corresponding didactical situation would be delicate to negotiate and in the end fragile. Technology can offer a solution, opening the window of the computer on the macro-space...

Brousseau G. (1997) Theory of Didactical Situations in Mathematics . Springer (Kluwer Acad. Pub.)

Teaching counts

  Retrieved from the TEL opinion blog, December the 22th, 2005

The reasons why the learner, either a child or an adult, needs "teaching inputs" are very often hidden as a corollary of the emphasis on—and possibly the misunderstanding of—the constructivist principles of design of learning environments. I would like to suggest here that these needs are especially important in the case of modern environments which are largely distributed and provide a potential access to a huge amount of knowledge and information. The following questions illustrate some of the issues that learners may have to face when left on their own in the wild web of digital resources: "How to look for something you don't know? ", "How to know that what you have found is what you were looking for? ", "How to know that you have learned?". Here are some of the issues that a teaching assistant should help to address. Another crucial question is: "How will others know that you know?"

It is not enough that learners have solved problems for them to understand that they have learned. Creative problem-solving which is at the core of the constructivist approach is so rich in new intellectual constructs that it is even a problem for the learner to realise what is worth remembering. Here again is a specific task for a teaching assistant. There is no general teaching model which could be implemented to equip a learning environment with the corresponding functionalities.

The nature of complex knowledge (as opposed to basic skills) is another reason to seriously refocus the design of learning environments on teaching issues. One of the main characteristics of such knowledge is, first that to master it requires to master several different pieces of knowledge organised in the form of a system, and second that its use depends on methods which are not mere algorithms. Such knowledge cannot be constructed spontaneously even when learners are provided with an adequate problem-situation, and actually in some cases such situations are even still unknown (e.g. linear algebra). As a result, complex knowledge requires specific learning environments and content specific teaching strategies. The complexity of such knowledge also comes from the fact that the corresponding learners' conceptions (i.e. learners' cognitive constructs), can be very different the one from the other and rather complex to understand and to model. The current research on students' understanding of the concept of "function" in mathematics or of the concept of "energy" in physics witnesses this complexity. The development of technological tools aiming at supporting the use of these knowledge (formal computation, simulation, etc.) even increases the difficulty by modifying within a kind of systemic loop the nature of the users' conceptions.

We cannot expect one single universal agent to be able to handle the complexity of supporting the learning process in the case of complex knowledge. On the contrary, there is a need for specialised agents, either artificial or human, able to cooperate and to coordinate their actions in order to provide the best support to the learner—indeed, one could remark at this point that the situation might not be so different for the so called "basic skills"…

My claim is that: the educating function of a system is an emerging property of the interactions organised between its components, and not a functionality of one of its parts.