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Proof

by Jake Tawney

 

The mathematician is a professional “proof writer.” In this way, Euclid’s presentation of geometry is the archetype of professional mathematics: a formal system that builds upon first principles (“postulates” or “axioms”) and definitions. From there, the mathematician makes conjectures about mathematical truths and proceeds to offer rigorous logical proofs for these conjectures. The final proofs are the “product” of the art of mathematics. The investigation, conjecture, and writing of proofs is the very doing of mathematics. Therefore, proofs should be present at all levels of mathematics education, though the level of detail and formality will vary depending on the student’s mathematical development.

A proof is a movement of logic that unfolds as a narrative. It is incumbent upon the mathematician not only to know the truth, but also to communicate the truth to an audience. Therefore, the very act of proof writing is also an act of rhetoric.

The rigor of this rhetoric will vary depending on the mathematical experience of the student. At the earliest ages, the proof consists simply of a rough sense of why the thing is true. “The area of a triangle is half a box because I can draw a picture showing it to be so.” For an elementary student, this may be sufficient. For a more mature student, there are details to fill in. How do we know that the triangle is exactly half of the rectangle? How do we know that the corresponding angles in each triangle are congruent? How do we know that the lines are parallel? Several other questions will arise as we press into the issue. As we ask more questions, we will produce a more “rigorous” proof. In Euclid, as in other examples of mathematics, we encounter the “rigorous proof,” one in which all of the questions have been answered. Even the rigorous proof, though, still contains the essence of the elementary student’s insightful narrative: the triangle is half the box.

Seeing proof writing from this “top down” perspective, wherein we first grasp the basic narrative (the basic why of the proof and the process of filling in the details by asking, “How do we know that?” at each step) is a very different method than the more common “Start with the ‘given,’ end with the ‘proof,’ and make logic steps along the way.” The problem with this latter pedagogy is that a student may end up with a correct proof, and may even understand every step, but there is no guarantee that the student understands why the theorem is true, only that it is true. It risks lacking the moment of insight by proceeding from the details to the whole rather than the other way around. The top down process of emphasizing the narrative of the proof also squares (pun intended) very nicely with the Socratic method. Once a student has demonstrated why a truth holds, we then press into their reasoning by asking for various sub-claims to be clarified.

Above all, from Kindergarten through Calculus, students are embarking on a fascinating journey through the pure forms of the heavens, and the art of proof allows them to discover objective truth in immaterial and eternal realities: those things that were true before they were born and will be true long after they are gone. Whether or not Edna St. Vincent Millay was correct that “Euclid alone has looked on Beauty bare” may be up for discussion, but it seems at least clear that those who strive for mathematical truths—be they of arithmetic, geometry, algebra, or calculus—will have encountered the intersection of logic and beauty and in the case of the latter will “have heard her massive sandal set on stone.”


Jake Tawney is a seasoned secondary teacher of mathematics and a much sought-after teacher trainer (who frequently includes a five-pin juggling demonstration for effect). He currently serves as the Vice President of Curriculum at Great Hearts Academies.

In the modern era, mathematics has arisen to become the international lingua franca of science, enabling empirical researchers and scientists to symbolically represent all types of patterns.

Doing mathematics properly requires one to think in a certain way about a certain sort of thing (mathematics), and eventually one becomes familiar with it and how it tends to go.

A proof is a movement of logic that unfolds as a narrative. It is incumbent upon the mathematician not only to know the truth, but also to communicate the truth to an audience. Therefore, the very act of proof writing is also an act of rhetoric.

Edna St. Vincent Millay (1892–1950) was one of America’s most talented 20th century poets, whose brilliance as a sonneteer has secured her place in numerous collections of children’s poetry.

This Socratic approach equips students to take more responsibility for their learning, increasing their confidence and motivating them to undertake the further exploration of mathematical reality, just as Socrates did with the young boy in Plato’s Meno.

Practical application often motivates such beautiful abstract mathematics; while, at the same time, abstract thinking often motivates practical mathematics. It’s time to share that side of the coin and how both sides work together.

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