Issues and Challenges in Instrumental Proof

Abstract

Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof, reasoning and epistemic necessity, taking advantage of the possibilities offered by the development geneses and fibrations in an instrumented perspective. The coordination of the semiotic, discursive and instrumental geneses of the working space founds discursive-graphic proofs, mechanical proofs and algorithmic proofs that are activated at school in the subject-milieu interactions. We end with a discussion on some consequences of the computer-assisted modelling of the learning conditions of mathematics, and we conclude on a necessary reconciliation of heuristics and validation.

Appendices

Appendix 1: Area Partition Activity

In an activity of dynamic geometry, de Villiers (2018)¹⁴ proposes the study of situations which allow us to show that “to the working mathematician, proof is not merely a means of verifying an already discovered result, but often also a means of exploring, analyzing, discovering and inventing new results”. It begins by proposing an interactive version of the proof without words presented in Sect. 2.4, automatically providing the measures on both sides of equality:

This is a proof that is both mechanical and discursive-graphic, as is our considerations around Fig. 9. At the interface of the situation, the author adds a help button that participates in the devolution of the problem (hint) and a generalization button that allows the user to move to the next situation (pentagon). In fact, the user can always move to a more general situation (pentagon or to another (k+1)-gon, until an octagon) without even having solved the previous problem (parallelogram or k-gon):

This dynamicity of the activity makes it possible to identify invariants in the equality of areas of figures and to reinvest them in a set of proof problems:

Carefully reflect on your proof, and consider how this same proof can also apply to a certain type of pentagon, hexagon, etc. Make generalizations and check your generalized conjectures by clicking on the Link buttons on the right to go to pentagons, hexagons, etc. with a similar area partition property. (extract from the activity instructions)

It would then be a particularly rich activity of proof which also makes it possible to engage an algorithmic proof.

Appendix 2: Transcription of Text from Dense Figures

This is the sequential transcription of Trahan’s approach structured in Fig. 6.

Step 1

Theorem

(not surprising)

If you cut a polygon into a finite number of pieces to reform another, then both polygons will have the same area.

Theorem

(more surprising)

Should any two polygons have the same area, it is possible to cut the first into a finite number of pieces to then form the second.

Step 2

Propositions

(program)

• 1. Any polygon can be cut into a finite number of triangles.

• 2. Any triangle can be cut to form a rectangle.

• 3. Any rectangle can be cut to form a square.

• 4. Any pair of squares can be cut to form a square.

Resolution

(breakdown by case)

• 1. Demonstration in two cases: convex polygon and concave polygon.

• 2. Check the four corners of the rectangle and check the «joints» (alignment of points).

• 3. In a rectangle L by ℓ: case ℓ < L ≤ 4 ℓ, checking of the angles, of the «joints» then sides; case L > 4 ℓ, check also lengths and sides.

• 4. Verification of angles, “joints” then sides.

Conclusion

Any polygon can be cut into a finite number of pieces to then form a square.

Step 3

Deriving proposition

If two polygons of the same surface can be cut into a finite number of pieces to each form a square, then there is a split of the first polygon to form the second.

Solution

Superposition of the two cuttings.

Can we do better?

Yes, the demonstration makes it possible to obtain a division between two polygons, but this is not optimal.

Verification

with interesting cuttings (demonstrated in other research work)

• Square in equilateral triangle.

• Square in nonagon.

• Some cutting of the pentagon.

Step 4

New situations

(to pose and solve)

• Generalization: curved surface?

• Generalizations in 3 dimensions

• (…)

Hilbert’s 3rd problem: given two polyhedra of equal volume, can we cut the first polyhedron into polyhedra and bring them together to form the second polyhedron?

• No: Dehn found an invariant (Dehn invariant) that is preserved during a cut; the cube and the tetrahedron do not have the same invariant.

• Some possible cuttings…