Experimental Approaches to Theoretical Thinking: Artefacts and Proofs

Abstract

This chapter discusses some strands of experimental mathematics from both an epistemological and a didactical point of view. We introduce some ancient and recent historical examples in Western and Eastern cultures in order to illustrate how the use of mathematical tools has driven the genesis of many abstract mathematical concepts. We show how the interaction between concrete tools and abstract ideas introduces an “experimental” dimension in mathematics and a dynamic tension between the empirical nature of the activities with the tools and the deductive nature of the discipline. We then discuss how the heavy use of the new technology in mathematics teaching gives new dynamism to this dialectic, specifically through students’ proving activities in digital electronic environments. Finally, we introduce some theoretical frameworks to examine and interpret students’ thoughts and actions whilst the students work in such environments to explore problematic situations, formulate conjectures and logically prove them. The chapter is followed by a response by Jonathan Borwein and Judy-anne Osborn.

Response to “Experimental Approaches to Theoretical Thinking: Artefacts and Proofs”

Jonathan M. Borwein, and Judy-anne Osborn

An overview of the chapter. The material we review focuses on the teaching of proof, in the light of the empirical and deductive aspects of mathematics. There is emphasis on the role of technology, not just as a pragmatic tool but also as a shaper of concepts. Technology is taken to include ancient as well as modern tools with their uses and users. Examples, both from teaching studies and historical, are presented and analysed. Language is introduced enabling elucidation of mutual relations between tool-use, human reasoning and formal proof. The article concludes by attempting to situate the material in a more general psychological theory.

We particularly enjoyed instances of the ‘student voice’ coming through, and would have welcomed the addition of the ‘teacher’s voice’ as this would have further contextualised the many descriptions the authors give of the importance of the role of the teacher. We are impressed by the accessibility of the low-tech examples, which include uses of straight-edge and compass technology, and commonly available software such as spreadsheets. Other more high-tech examples of computer geometry systems were instanced and it would of interest to know how widely available these technologies are to schools in various countries (examples in the text were primarily Italian with one school from Hong Kong) and how much time-investment is called upon by teachers to learn the tool before teaching with it. The general principles explicated by the authors apply equally to their low-tech and high-tech examples, and are thus applicable to a broad range of environments including both low and high-resourced schools.

The main theoretical content of the chapter is in the discussion of why and how tool-use can lighten cognitive load, making the transition from exploring to proving easier. On the one hand tool-use is discussed as it relates to the discovery of concepts, both in the practical sense of students coming to a personal understanding, and in the historical sense of how concepts make sense in the context of the existence of a given tool. On the other hand the kinds of reasoning used in the practice of mathematics are made explicit – deduction, induction and a third called ‘abduction’ – with their roles in the stages of mathematical discovery, as well as how tool-use can facilitate these kinds of reasoning and translation between them.

Frequent use of the term ‘artefact’ is made in the writing, thus it is pertinent to note that the word has different and opposing meanings in the educational and the science-research literature. In the educational context the word means a useful ­purposely human-created tool, so that a straight-edge with compass or computer-software is an artefact in the sense used within the article. In the science-research literature, an artefact is an accidental consequence of experimental design which is misleading until identified, so for instance a part of a graph which a computer gets wrong due to some internal rounding-error is an artefact in this opposite sense.

The structure of the chapter. There is an introduction: essentially a reminder that mathematics has its empirical side as well as the deductive face which we see in formal proofs. Then there are three parts. Part 1 begins with a discussion of the history of mathematics, with reference to the sometimes less-acknowledged aspect of empiricism. In its second half, Part 1 segues from history into modern teaching examples. Part 2 is the heart of the article. It deals with the kind of reasoning natural to conjecture-forming, ‘abduction’, the concept of ‘instrumentation’ and cognitive issues relating to ‘indirect proofs’; all through detailed examples and theory. Part 3 reads as though the authors are trying to express a large and fledgling theory in a small space. A general psychological paradigm called ‘Activity Theory’, is introduced, which deals with human activities and artefacts. An indication of digital technology as a means of translating between different ways of thinking is given in this context. We now discuss each Part in more detail.

Part 1. The historical half of Part 1 deals with geometric construction as a paradigmatic example, with the authors showing that since Euclid, tools have shaped concepts. For example, the straight-edge and compass is not just a practical technology, but helps define what a solution to a construction problem means. For instance cube duplication and angle trisection are impossible with straight-edge and compass (i.e. straight lines and circles) alone, but become solvable if the Nicomedes compass which draws a conchoid are admitted. A merely approximate graphic solution becomes a mechanical solution with the new tool. The moral is that changing the set of drawing tools changes the set of theoretically solvable problems, so that practical tools become theoretical tools. Another theme is the ambiguity noted in Descartes’ two methods of representing a curve, by either a continuous motion or an equation; and subsequent historical developments coming with Pasch, Peano, Hilbert and Weierstrass, in which the intuition of continuous motion is suppressed in favour of purely logical relations. The authors perceive that historical suppression as having a cost which is only beginning to be counted, and rejoice that the increased use of computers is accompanying a revived intuitive geometric perspective.

This revival also offers the prospect of teachers who better understand mathematics in its historical context. Ideally, their students will gain a better appreciation of the lustrous history of mathematics. It is not unreasonable that students find hard concepts which took the best minds in Europe decades or centuries to understand and capture.

Part 1 is completed by examples from three educational studies followed by a discussion of the importance of the role of the teacher. Each example uses a tool to explore some mathematical phenomenon, with the teaching aim being that students develop a theoretical perspective. The first study involved over 2,000 students in various year-groups and 80 teachers, setting straight-edge and compass in the wider context of mathematical “machines”. The second study, of Year 10 students, sits in the context of a particular DGS (Dynamical Geometry System), specifically the software called ‘Cabri’. Students first revised physical straight-edge and compass work, then worked in the virtual Cabri world, in which their drawings become what are termed ‘Evocative Computational Objects’ | no longer just shapes but shapes with associated Cabri commands and the capacity to be ‘dragged’ in interesting ways whose stability relates to the in-built hierarchical structure of the object. Interesting assertions made by the authors are that drag-ability relates to prove-ability, and that the original pencil drawings become signs for the richer Cabri objects. As in the previous study, a central aspect was student group discussion and comparison of solutions. The third study was of the use Year 9 students made of a CAS (Computer Algebra System) to explore the behaviour of functions. The ­students initially made numerical explorations, from which they formulated conjectures. Then, largely guided by a suggestion from their teacher, they substituted letters for numbers, at which point the path to a proof became evident.

The way in which the role of the teacher is crucial, in all three studies, is described with reference to a model expressed in Fig. 5.4. On the left of the diagram, activities and tasks chosen by the teacher sit above and relate to mathematics as a general entity within human culture. On the right side of the diagram, student’s productions and discoveries from carrying out the tasks sit above and relate to the mathematical knowledge required by the school curriculum. The artefact (purpose-created tool) sits in the middle. Reading the picture clockwise in an arc from bottom left to bottom right neatly captures that teachers need to choose suitable tasks, students carry them out, and teachers help the students turn their discovered personal meanings into commonly understood mathematics. It is pointed out that as students discuss their use of artefacts, teachers get an insight into students’ thought-processes.

Part 2. In Part 2, we get to the core of the article’s discussion of proving as the mental process of transitioning between the exploratory phase of understanding a mathematical problem to the formal stage of writing down a deductive proof. The central claim is that this transition is assisted by tools such as Dynamic Geometry System (DGS) softwares and Computer Assisted Algebra (CAS) softwares, provided these tools are used within a careful educational design. The concept of abduction is central to the authors’ conceptual framework. The term is used many times before it is defined – a forward reference to the definition in about the tenth paragraph of Part 2 would have been useful to us. It is worth quoting the definition (due to Peirce) verbatim:

The so-called syllogistic abduction (C.P.2.623), according to which a Case is drawn from a Rule and a Result. There is a well-known Peirce example about beans:

Rule: All beans from this bag are white

Result: These beans are white Case:

These beans are from this bag

Clearly this kind of reasoning is not deduction. The conclusion doesn’t necessarily hold. But it might hold. It acts as a potentially useful conjecture. Nor, as the authors note, is this kind of reasoning induction, which requires one case and many results from which to suppose a rule.

The authors’ naming and valuing of abduction sits within their broader recognition and valuing of the exploratory and conjecture-making aspects of mathematics which can be hidden in final-form deductive proofs. Their purpose is to show how appropriate abductive thinking arises in experimentation and leads to deductive proofs, when the process is appropriately supported.

The role of abductive reasoning in problem solving strikes these reviewers as a very useful thing to bring to educator’s conscious attention. One of us personally recalls observing an academic chastise a student for reasoning which the academic saw as incorrect use of deduction, but which we now see as correct use of abduction in the early part of attempting to find a proof.

An example of 10th grade students faced with a problem about distances between houses and armed with a software called TI-Nspire is presented in detail in this section. The empirical aspect of mathematical discovery is described in an analogy with a protocol for an experiment in the natural sciences. We note that this example could be usefully adapted to a non-computerised environment. A point which the authors make, specific to the use of the computer in this context, is that the software encourages/requires useful behaviour such as variable-naming; which can then assist students in internalising these fundamental mathematical practices as psychological tools.

A teaching/learning example regarding a problem of finding and proving an observation about quadrangles, presented to 11th and 12th grade students is given. The authors give a summary of the steps most students used to solve the problem, and then interpret the steps in terms of the production of an abduction followed by a proof. The authors write

In producing a proof, (Phase 5) the students write a proof that exhibits a strong continuity with their discussion during their previous explorations; more precisely, they write it through linguistic eliminations and transformations of those aforementioned utterances.

This statement is in the spirit of a claim at the start of Part 2 that empirical behaviour using software appropriately in mathematics leads to abductive arguments which supports cognitive unity in the transition to proofs.

The next main idea in Part 2 after ‘abduction’ is that of ‘instrumentation’. The special kind of ‘dragging’ which has been referred to during discussions of DGS softwares is recognised as maintaining dragging (MD), where what is being maintained during the dragging is some kind of visible mathematical invariant. Furthermore the curve that is traced out during dragging is key to conjecture-formation and potentially proof.

The third main concept dealt with in Part 2 is that of ‘indirect proof’ and the difficulties that students often have with it. The authors usefully describe how software-mediated abductive reasoning may help, which they support with two plausible arguments. First, the authors note that indirect proofs can be broken up into direct proofs of a related claim (they use the term ‘ground level’) together with a proof of the relationship between the two claims (they use the term ‘meta level’). Thus an argument for software-mediated reasoning says that use of software helps students keep track of the two levels of argument.

Second, the authors argue that abduction is useful to students partly because of what happens to formal (mathematical) claims when they are negated. The authors state that in some sense the cognitive distance between the conjecture and the proof is decreased in the negation step. They give two examples. The first is a study of a Year 9 student presented with a delightful problem about functions and their derivatives and anti-derivatives. In this case, refutation of an argument by abduction turns out to coincide with refutation of an argument by deduction. The second example is of a study of two students from Years 9 and 10 in Hong Kong given a problem about cyclic quadrilaterals in a Cabri environment. In this case, ‘dragging’ behaviour led the students to an argument which collapsed to a formal ‘reductio ad absurdum’.

To summarise one stream of thought from Part 2 relating to the practice of learning and teaching: (a) tool-use facilitates exploration, especially visual exploration; (b) exploration (in a well-designed context) leads to conjecture-making; (c) practical tool-use forces certain helpful behaviours such as variable-naming; (d) this ‘instrumentation’ can lead to internalising tools psychologically; (e) for indirect proofs, the way negation works helps bridge the distance between kinds of reasoning used in conjecture-making and proof.

In the closing section of Part 2, the authors go beyond the claim that abduction supports proof and become more speculative. They quote Lopez-Real and Leung to claim that deduction and abduction are parallel processes in a pair of ‘parallel systems’, Formal Axiomatic Euclidean Geometry on the one hand, and Geometry realised in a Dynamic Geometry Environment (DGE) on the other hand, and that interaction between the two ways of knowing and storing information could be productive in ways not yet fully elucidated. It would be fascinating to see these ideas fleshed out.

Part 3. Part 3 introduces ‘Activity Theory’, a general framework concerned with the know-how that relates to artefacts, and attempts to situate the discussions of Parts 1 and 2 in this context, however as readers we found it difficult to gain insight from this formulation lacking as we do previous detailed knowledge of Activity Theory. The attempted translation between languages is scanty, although there are some illuminating examples, for instance an interesting use of ‘turtle geometry’ to explore hyperbolic geometry is presented in this section, where the turtle geometry is regarded as an artefact within Activity Theory. This part also expands upon the idea of instrumentation, linking ideas about concept-development, Gestalt perception and embodiment. There is much here which could be further developed; and which assuredly will be.

Conclusions. We first highlight a notion which is implicit throughout the chapter, which is the valuing of the teaching of proof in schools. Proof is a central component of mathematics however the valuing of the teaching of proof is not always taken for granted. For instance in the Australian context we know of instances of stark contrast, where the current state-based curricula does not emphasise proof (it is mentioned in the context of upper level advanced classes only), although we know of cases in which teacher-training does emphasise it. In short, we believe that there is often not enough teaching of proof in schools and that the chapter under review may help by providing a conceptual and practical bridge for students and their teachers between the activities of exploring mathematics and of creating and understanding proofs.

We also highlight the authors’ own advisements about implementation in practice of the theory they have articulated. The authors emphasise that the role of the teacher is crucial both in lesson design and classroom interaction; as is neatly captured by Fig. 5.4 near the end of Part 1. They observe, for instance in their discussion of “maintaining dragging” in Part 2, that desired student understandings and behaviours often do not arise spontaneously. Further, they warn early in Part 1 (quoting Schoenfeld) that counterproductive student behaviour can arise as unintended by-products of teaching. At the end of Part 1 the authors give references to studies in which the kinds of useful interventions that teachers repeatedly make are analysed. It is helpful to have the centrality of the mathematics teacher made so clear. The importance of design and interaction are emphasised in quotations such as “The teacher not only selects suitable tasks to be solved through constructions and visual, numerical or symbolic explorations, but also orchestrates the complex transition from practical actions to theoretic arguments”; and “The teacher, as an expert representative of mathematical culture, participates in the classroom discourse to help it proceed towards sense-making in mathematics” (our emphasis).

In summary, this work repays the effort to read it. The historical perspective at the beginning brings the duality between empiricism and deductive reasoning usefully to mind. The examples, language and theory developed in Part 2 are likely to be clarifying and inspiring to both educators and theorists. The more speculative aspects at the end of Part 2 and in Part 3 call for further elucidation and development to which we look forward.