Proof & TOK #1
Saturday 12 January 2013
Along with teaching Math SL and Math HL, I've also been involved in teaching Theory of Knowledge (TOK) for the past 12 years. One of the reasons I enjoy teaching TOK is for the opportunity to explore different subject areas without the constraints of a detailed content list to be covered and the pressure of external assessments on this material. Don't get me wrong - a detailed syllabus and end-of-course assessments are absolutely necessary for any high-quality academic program that is preparing students for university. However, there is a lot of interesting and engaging ideas in all subject areas that students are not exposed to while in secondary school - ideas that give a more wide-ranging (and I would argue more accurate) view of the particular subject (area of knowledge). This is true for all six areas of knowledge that are to be explored in a TOK class: Ethics, Mathematics, Natural Sciences, Human Sciences, History, The Arts.
There are so many interesting areas of mathematics - all within the intellectual grasp of a typical teenager - that most secondary school students do not encounter because of practical constraints due to course syllabi, teaching time, textbook limitations, etc. TOK lessons are a great opportunity for students to explore some of these 'non-syllabus' mathematical topics - such as non-euclidean geometries (especially 2-dimensional elliptic geometry, i.e. spherical geometry), topology (e.g. demonstrating properties of a mobius strip), recreational mathematics and puzzles (e.g. Rubik's cube and card tricks), and applications of mathematics to other subjects such as music and art.
At its heart, TOK is a course in critical thinking that gives students an opportunity to examine, discuss, and personally reflect on weighty ideas and issues connected with the pursuit of knowledge in different human pursuits (mathematics, history, etc). Perhaps the most important 'big' idea or aspect of mathematics - and that distinguishes it fundamentally from other areas of knowledge - is proof. Proof in mathematics specifically refers to a rigorous deductive proof.
I like to start discussions about proof in a TOK class by demonstrating a couple examples of deductive proof - both of which are very brief and ones that I'm quite confident most students would not have seen before.
The first proof I demonstrate concerns a problem often referred to as the mutilated chessboard problem. I like it because it is very visual and hands-on (give chessboards and dominoes to groups of students) and involves no equations or symbolic math - but definitely deductive reasoning. Given a standard chessboard with 64 squares - 32 black and 32 white - and a sufficient number of dominoes each of which is the size of two chessboard squares, all the squares of the chessboard can be completely covered ("tiled", or "tessellated"). Consider if the chessboard is 'mutilated' such that two opposite corners (they will be the same color) are removed. Can such a chessboard (62 squares) be completely covered with the dominoes? No - it is not possible. With enough prodding, a student in the class can usually provide a logical explanation why it is not possible. Here is one explanation with a nice visual display of the problem.
A second proof that I like to share with students is proving that 1 = 0.99999... (or expressed as ).
Start with the conjecture that 1 = 0.99999... Let x = 0.99999...
10x = 9.99999...
x = 0.99999...
9x = 9.0 then x = 1; therefore 1 = 0.99999... Q.E.D. (quod erat demonstrandum)
After these brief examples of deductive proof and getting students to discuss, express their own ideas, and ask questions about the nature of proof in mathematics, I show them the excellent BBC documentary made by Simon Singh on Fermat's Last Theorem (see below) - the very human story about Andrew Wiles finally putting together a proof of the theorem (although it should have been called Fermat's Last Conjecture) after first being proposed (and claimed to have been proven) by Pierre de Fermat more than 350 years earlier. The documentary is just under 50 minutes long and I usually have students view the documentary during two different class meetings allowing some time for discussion. It is a very well made documentary and succeeds at getting a good balance between sufficiently describing the high-level math involved but not too much to make the documentary too dry or obsure for the non-mathematician. It especially does a great job showing the human (and very emotional) side of Andrew Wiles' seven-year journey in composing a correct proof. It also has a very nice portion early in the documentary where several leading mathematicians offer words to describe what proof is in mathematics - and why "that's what mathematics is about." Just an excellent documentary for TOK students to watch and dicuss.