Technology-supported learning of proof in mathematics

Cabri-Euclide, Luengo, 2005

Proof assistants, an automatic theorem proving research track, are reaching a maturity which suggests that is possible to the exploration of their use for the learning of proof in  mathematics first at the level of higher education and at tentatively the upper secondary school. 

It in the context of this pioneer research that is organised the PAT 2023 Thematic School which seeks to offer a broad spectrum of current research in the field of didactic of proof, the impact of the use of proof assistants in education, formalization of mathematics and user interfaces for theorem proving. I will give a lecture which will include  (1) a survey of the evolution of AI research on the learning of proof in mathematics, (2) lessons learned from the past focusing on the relations between reasoning-proving and knowledge representation, and on the problem of feedback, eventually (3) didactic analysis of the teaching of mathematical proof and its implications for the design of learning environments. The introduction will outline the history of the teaching of proof in mathematics, a short epilogue will raise epistemological issues.

Suggested readings:

Arzarello, F. (2007). The proof in the 20th century (From Hilbert to Automatic Theorem Proving Introduction). In P. Boero (Éd.), Theorems in School : From History, Epistemology and Cognition to Classroom Practice (p. 43‑63). BRILL.
Balacheff, N. (2023). Notes for a study of the didactic transposition of mathematical proof. Philosophy of Mathematics Education Journal, 2023 volume
Balacheff, N., & Boy de la Tour, T. (2019). Proof Technology and Learning in Mathematics : Common Issues and Perspectives. In G. Hanna, D. Reid, & M. de Villiers (Éds.), Proof Technology in Mathematics Research and Teaching. Springer.
Czocher, J. A., & Weber, K. (2020). Proof as a Cluster Category. Journal for Research in Mathematics Education, 51(1), 50‑74.
Hanna, G., & Xiaoheng, (Kitty) Yan. (2021). Opening a discussion on teaching proof with automated theorem provers. For the Learning of Mathematics, 41(3), 42‑46.
Luengo, V. (2005). Some didactical and Epistemological Considerations in the Design of Educational Software : The Cabri-euclide Example. International Journal of Computers for Mathematical Learning, 10(1), 1‑29.

Remarks on truth, the word and more

This post has been written in the context of the preparation of a text on the teaching and learning of mathematical proof. I noticed some years ago that we need to have a discussion on what we mean by "proof". I now realized that "truth" may deserve some attention as well, especially in the context of expressing in one language what we think in another one.


Proof and truth are inseparable concepts, yet discussions on what can count as proof in the mathematics classroom develop as if the meaning of truth were clear. The meaning and nature of truth seem to be an irrelevant issue in mathematics where true and false are just the two elements of a set where propositions or predicates take value, while the questioning of these meanings and their nature is one of the central subjects of philosophy. As it happens, Kleene’s first chapter to his classic book Mathematical logic (1967) does not spend even a sentence on defining true and false, even though the introduction is dedicated to fixing the formalism of his first chapter. But mathematical logic is not the logic of mathematics insofar as the activity of mathematicians is not reduced to carrying out a formalism. “Actually, the criterion of truth in mathematics is the success of its ideas in practice; mathematical knowledge is corrigible and not absolute; thus, it resembles empirical knowledge in many respects”, wrote Hilary Putman in a brief paper entitled What is mathematical truth? (1975 p. 529). There is something radical in Hilary Putman’s position paper, but it is not without relevance for proceeding with the objective I have in the text I am preparing. More than a science, in the mathematical classroom, mathematics is a practice.

The meaning of the words “true” and “truth” in the mathematics classroom borrows its substance from the vernacular culture and the culture emerging from the interactions among students and with the teacher, and from their management. But, if students in higher education as well as mathematicians maintain a difference between the mathematical meanings of true and truth from the meaning of these words in other scientific practices or everyday life, this is not the case for K-9 learners. Early mathematical conceptualizations develop based on a vernacular culture, and one can reasonably claim that this lasts until the middle of the junior high school when mathematical proof starts to appear as such. Hence, my papers, written in English but first contemplated in French deserve to question whether the translation of the French vrai by the English true carries any semantic significance.

The etymology of true (n. truth), according to the Vocabulaire Européen des Philosophies (2004), goes back to a metaphorical origin in the word tree, which denotes firmness, steadfastness or faithfulness. Its evolution does not exclude this origin but includes other meanings among which are also certain, accurate, correct and mathematical meaning as well (i.e. logical necessity). The order of these meanings may vary from one dictionary to another, or from one edition of the same dictionary to another, but they are all still there. Today, the contemporary use of true/truth puts sincerity and reliability ahead of veracity (ibid. p.1350). Both notions underlie the first meaning proposed by the popular Oxford Advanced Learner’s Dictionary: “the truth [singular] the true facts about something, rather than the things that have been invented or guessed”. Then, a first example is given: “Do you think she's telling the truth?”

The etymology of vrai (n. verité) traces its origin back to the Latin word veritas whose paradigm is normative: it refers to the correctness and the validity of a rule; it is the legal truth that a legitimate institution locks and preserves (ibid. p.1342). The evolution of its meaning from the Middle ages to the contemporary epoch introduces the producer of the statements claimed true, with the assessment of their sincerity and correspondence – adequacy, conformity – to the thing. The latter dominates the former as witnessed by the popular French dictionary Le Robert which states: Vérité [n. f.] What the mind can and must give its assent to (as a result of a relationship of conformity with the object of thought, of an internal coherence of thought): sincerity is not absent but comes as the sixth and last meaning in the list.

I take the case of English and French, but in fact we must go beyond these cases by questioning the languages and background cultures of all research projects on the teaching and learning of mathematical proof. The epistemological differences silently shape research, while being aware of them should lead to a more sophisticated understanding of each other’s work and results. This is a general phenomenon known from research on comparative literature: “no philosophical argument or picture of the world can be divorced from the language, style, rhetoric, means of presentment and illustration in which it is stated” (Steiner, 1996, p. 157). In the case of mathematics, the publications and communications to which I have had access, do not suggest insurmountable conflicts or contradictions across the different cultures and history of truth: rather they present differences in the relative weights given to its various dimensions. We will benefit from being aware of the influence of these differences on the way we approach research.

Eventually, the investigation which started by noticing possible translation issues ends up inviting us to consider the vernacular epistemology as complementary to the attention classically paid to the role of the vernacular language. In other words, the tension between vernacular languages and the mathematical language – which we could pretend to be universal – should lead to questioning the culture that proof and truth carry with them, and despite which the socio-mathematical culture and norms must find their place in the school mathematics.

Confident in the wisdom of Donald Davidson’s (1996) warning that it is folly to try to define truth, I will not try to answer Hilary Putmann’s question, What is mathematical truth?, not even limiting its scope to the sole purposes of research in mathematics education. But the word truth and the concept it labels, do not stand alone; it is tightly related to the concept that the word proof refers. I am willing to take this relation into account, and I concur with Viviane Durand-Guerrier (2008, p. 373) when she asserts the “relevance of the distinction and relationship between truth and validity in mathematical proof for mathematics education”, taking on board the differences between common sense and mathematical logic, and generally “emphasizing the articulation between syntax and semantics” (ibid.). Her argumentation builds upon Alfred Tarski’s (1944) solution to “[the problem of] giving a satisfactory definition of [truth], i.e., a definition which is materially adequate and formally correct.” (ibid. p. 341). Tarski first defines satisfaction: “given objects satisfy a given [sentential] function if the latter becomes a true sentence when we replace in it free variables by names of given objects” (ibid. 353). Then comes the definition “of truth and falsehood simply by saying that a sentence is true if it is satisfied by all objects, and false otherwise” (ibid.) Tarski’s definition grounds the deduction theorem which bridges syntax and semantic, truth and validity. But, taking this perspective is not enough in the case of early learning since this definition requires that sentences are elements of “[a language] whose structure has been exactly specified.” (ibid. p. 347). Yet we know it is not the case for language at work at the K-10 level.

To clarify the consequence of having at stake such a teaching context – but even though it may still hold beyond this – I adopt the distinction made by John Langshaw Austin between statement and sentence:

“A statement is made and its making is an historic event, the utterance by a certain speaker or writer of certain words (a sentence) to an audience with reference to an historic situation, event or what not.
A sentence is made up of words, a statement is made in words. A sentence is not English or not good English, a statement is not in English or not in good English. Statements are made, words or sentences are used. We talk of my statement, but of the English sentence (if a sentence is mine, I coined it, but I do not coin statements).” (Austin, 1950, p. 3)

The utterance of a statement requires words and a good command of linguistic rules to produce a sentence true to the communication objective which underpins it; this objective includes semantic adequacy and formal correctness. But, paraphrasing Donald Davidson, we know that most students may not speak the language for which mathematical truth has been defined (ibid. 1996, p. 277). They have to learn the language of mathematicians, becoming aware of the way it deals with mathematical objects, properties and relationships, which are involved throughout the mathematical activity. Acquiring this linguistic competence is key for the success of learning. Moreover, John Langshaw Austin’s introduces a speaker and an audience, in other words the intentional character of the speech act uttering truth, and its social dimension. Hence, aside from coherence and correspondence, the hypothesis – more often than not implicit – of sincerity and steadfastness of the speaker and of the audience must be included.

Although this discussion may be somewhat limited, it sheds light on the difficulty of comprehending the meaning of the words true/truth when taking a step beyond mathematical logic while remaining within the mathematical territory – of which learning mathematics is part. While mathematics as a scientific discipline is universal, mathematical activity is diverse. It embraces the cultural, historical and contextual characteristics of the society in which it develops. This is even more so for its learning and teaching, which are situated mathematical activities framed by institutions and political projects of a society.

I will not dare a definition but I propose four conditions on a sentence to be held true: 

• to be ethically minded (sincerity, reliability)
• to be linguistically appropriate (statement vs sentence)
• to be semantically adequate (correspondence)
• to be formally correct (coherence)

Based on what research has shown so far, we can tell that these conditions will not have the same importance within the transition from argumentation at the earliest learning stages to mathematical proof. Nevertheless, these should be considered and assessed against the level of students’ acculturation to the mathematical practice. I suggest that we ought to take on such epistemological and didactical perspectives to revisit the classical issue of defining mathematical proof in order to fit the needs of mathematics learning and teaching. 

Aknowledgements: many thanks for their valuable feedback  to Viviane Durand-Guerrier, Patricio Herbst, Joel Hillel, and Richard Noss.

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.

---

 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]

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]

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

An hommage to Juliana Szendrei

Tomorrow, in Budapest, researchers and teachers from Hungary and abroad will meet to pay hommage to Juliana Szendrei who passed away early January this year. I could not join the conference, so it is with this short post that I will participate and share this moment in memory of Juliana.

Beyond conferences and readings, I came to know Juliana Szendrei from a collaboration within the framework of a Tempus project, thanks to the complicity of Paolo Boero who introduced me. I remember our first meeting in Budapest in the mid-90s. It was in the beginning of the winter, the weather was cold and cloudy, the material conditions a bit limited and the use of the technology somewhat uncertain. But Juliana was there. She was so enthusiastic and eager to facilitate everything that very soon I forgot all these difficulties and enjoyed contributing to her project to enhance teacher training and mathematics learning.

As a research leader in mathematics education, Juliana Szendrei was committed to the international movement to improve the research area and to set firm theoretical foundations, including on a topic which I am specially interested in, the learning of mathematical proof. Actually, she was not only a researcher in mathematics education but also a good mathematician, this shades a very special light on her work. In particular, she was aware of the evolution of her own understanding of what a proof is. She shared this view during one of our working sessions. It has then been published in a book on proof edited by Paolo (see below). She views this evolution as a series of steps, from step 1 to a step 7... at step 1, as a "conformist learner", she saw mathematical proof as a "ceremony" the rules of which she was quite able to follow. The rewards of the teacher led her to step 2: the feeling of being part of a community, something like a community of mathematicians. But, Juliana was concerned by the fact that this could result in a split between this community and the rest of the world. So, she found herself better when teaching probability and coming to the belief that "mathematics is about the theory, not about the real coin". I will not describe here all the steps she told she went through, but notice that her understanding of mathematical proof as a mathematics educator was rooted in this awareness of the role of mathematics as a modelling tool, and the role of proof in making this tool so robust and efficient. This understanding that the meaning of a theory rests in the dialectical relationship between the theory and the concrete world is also a mark of her view on research in mathematics education.

Juliana Szendrei primary objective was the concrete enhancement of mathematics teaching and learning in schools as they were, with the curricula as they were at that time. This pragmatic view of her responsibility as a researcher guided her action. Sure she would smile if I took Tomas Varga words to sum up the lesson I learned from her: teaching and learning problems "cannot be settled without further research and deeper insight into the learning process. But we cannot wait until they are".

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.