The transition from mathematical argumentation to mathematical proof, a learning and teaching challenge

Due to the development of the DOVID-19 pandemic, ICME-14 has been postponed by one year, probably until next summer 2021. Indeed, no body knows what the future will be like. So, I chose to share there the abstract of my lecture. Be life kind enough to allow me to attend ICME-14 whenever, wherever. 

Comments and questions on this post will be much appreciated. They will contribute to my reflexion which continues on the mathematical status of argumentation.

Mathematical proof is the backbone of mathematics as a scientific discipline. All along the 20th century, the meagre success of its teaching prompted most of the decision makers to postpone it until children have achieved a certain cognitive development. Research outcomes of the last decades suggest that the teaching challenge can be overcome, hence the nowadays wide consensus that mathematical proof ought to be part of curricula at whatever grade from kindergarten to university. To properly express this objective requires finding an adequate characterization of proof and the right words while one has been accustomed to using several different ones as mere synonymous.

First, I suggest to slightly change the didactical problem from learning proof to understanding how can be asserted the truth value of a statement in mathematics at different grades. This requires to tighten the links between problem-solving and proving, as well as between knowing and proving. I develop this position focusing on three terms: control, argumentation and proof. The choice of these terms intends to denote three regimes of validation whose respective weights change along the continuum from solving a problem to communicating its solution according to the mathematics standards in force at a given grade. 

Second, I shall shape the relations between argumentation and proof from an epistemological and didactical perspective. Doing this, I will pay attention to our linguistic, cultural and epistemological differences.

Although the historical roots of mathematical proof could give it legitimacy, the concept of mathematical argumentation will be a didactic concept and not the transposition of a mathematical one. The inherent social nature of argumentation would otherwise make a lasting impact on the understanding of mathematical proof. Although being the product of a human activity which certification is the outcome of a social process, a mathematical proof is independent of a particular person or group. The standardization of proof in mathematics, in addition to the institutional character of its theoretical reference, entails its depersonalization, decontextualization and atemporality. While argumentation is intrinsically dependent on an agent, individual or collective, and is “situated”.

Eventually, the characteristics of mathematical argumentation must not only distinguish it from other types of argumentation in order to manage its evolution to mathematical norms, but it must also be operational when it comes to arbitrating students' proposals in order to organize and capitalize on them in the classroom knowledge base. How, for example, can be arbitrated the case of the generic example that balances the general and the particular; a balance found at the end of a contradictory debate seeking an agreement which should be as little as possible a compromise?

笔译

Mathematical argumentation as a precursor of mathematical proof

I am delighted to discuss research on mathematical proof soon with Andreas Stylianides and the Cambridge Mathematics Education team.

Here is the seminar abstract:
Along history or across educational traditions, the space given to mathematical proof in compulsory school curricula varies from a quasi-absence to a formal obligation which for some has turned into an obstacle to mathematics learning. The contemporary evolution is to give to proof the space it deserves in the learning of mathematics. This is for example witnessed in different ways by The national curriculum in England (2014), the Common Core State Standards for Mathematics (2010) in the US or the recent Report on the teaching of mathematics (1918) commissioned by the French government; the latter asserts: The notion of proof is at the heart of mathematical activity, whatever the level (this assertion is valid from kindergarten to university). And, beyond mathematical theory, understanding what is a reasoned justification approach based on logic is an important aspect of citizen training. The seeds of this fundamentally mathematical approach are sown in the early grades. These are a few examples of the current worldwide consensus on the centrality proof should have in the compulsory school curricula. However, the institutional statements share difficulty to express this objective. The vocabulary includes words such as argument, justification and proof without clear reasons for such diversity: are these words mere synonymous or are there differences that we should pay attention to? What are the characteristics of the discourse these words may refer to in the mathematics classroom? Eventually, how can be addressed the problem of assessing the truth value of a mathematical statement at the different grades all along compulsory school? I shall explore these questions, starting from questioning the meaning of these words and its consequences. Then, I shall shape the relations between argumentation and proof from an epistemological and didactical perspective. In the end, the participants will be invited to a discussion on the benefit and relevance of shaping the notion of mathematical argumentation as a precursor of mathematical proof.

Monday 18th November 2019, 2.30-4.00pm
Faculty of Education, Donald McIntyre Building (room GS4)

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]

Explanation, proof and mathematical proof - A needed clarification

November 21st update: the final version of Gila Hanna paper “Reflections on proof as explanation” will no longer include the comment which justified this post. However, taking into account this comment was important and the following clarification of the misleading diagram is necessary. I thank Gila for the quality of our exchange and for giving me the opportunity of this clarification.

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 For about 30 years, I have used the Venn diagram reproduced here, without noticing how seriously it could be misleading once separated from its context. I realized that when reading recently Gila Hanna “Reflections on proof as explanation” (Hanna 2016). She  referred to this diagram in support to the claim “If one were to take the position that an explanation is simply a deductive argument, then all proofs would automatically be explanations”. This is the consequence of a quick reading of the paper where I used this diagram (Balacheff, 2010, p.130), but with a text making explicit the meaning of the three sets and the corresponding perspective.

In the said 2010 paper, entitled “Bridging proving and knowing in mathematics”, I postulated the following: “the explaining power of a text (or non-textual ‘discourse’) is directly related to the quality and density of its roots in the learner’s (or even mathematician’s) knowing.” I then added explicitly that such a text or discourse is “an “explanation” of the validity of a statement from the subject’s own perspective.” The following intended to position the three expressions: explanation, proof and mathematical proof.

“What is produced first is an “explanation” of the validity of a statement from the subject’s own perspective. This text can achieve the status of proof if it gets enough support from a community that accepts and values it as such. Finally, it can be claimed as mathematical proof if it meets the current standards of mathematical practice. So, the keystone of a problématique of proof in mathematics (and possibly any field) is the nature of the relation between the subject’s knowing and what is involved in the ‘proof’.” (my today emphasis)

Setting this framework was cautious enough not to restrict mathematical proof to logic in a narrow sense but to “the current standards of mathematical practice”. I must recognize that using such a diagram was a bit risky, and misleading for a quick reading.

I first developed this approach at the end of the 80s. Taking the perspective of the learner’s knowing, I chosen the word “explanation” instead of “argumentation” to account for the genuine effort of the learner to respond to the “why” a statement or a result is valid based on his or her “existing knowledge” – as Gila Hanna refers to it; the ground of the claim for validity being the functional organization and semantic value of the statements as opposed to what Duval called their epistemic value. Indeed, the ultimate aim of an explanation is to modify the epistemic value of the statement or result which initially is best qualified as a conjecture. Transforming one's own personal explanation of the validity of a statement into a proof (or a mathematical proof) is a complex process not always successful nor possible. When reading a proof, the reverse process necessary to get from it an explanation of the claimed validity is in itself an issue. It means constructing the links between the content and structure of the proposed proof and the reader’s own existing knowledge. It is in this manner that I understand the issue of the explaining power of a proof.

Hanna G. (2016) reflections on proof as explanation. In: 13th International Congress on Mathematical Education. Hamburg, 24-31 July 2016 [https://www.researchgate.net/publication/316975364]

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

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.