Introducing the Concept of Proof to Students

This week we saw four more interesting miniature lessons. By a good coincidence, three of them dealt with mathematical proof to differing extents. I want to speak about one lesson in particular, which covered the following geometry problem:

https://static.guim.co.uk/sys-images/Guardian/Pix/pictures/2014/10/16/1413457021371/da4d4757-3177-4772-88e4-e481c7cdc8ee-bestSizeAvailable.gif

The object is to determine the sum of the three angles indicated in the figure. I believe this puzzle originated with Martin Gardner (an excellent source for clever math problems and riddles) but I am not positive. We were given Geoboards and elastic bands as tools to help solve the problem. In addition, we we cut up a copy of the figure in order to see that the three angles should sum to a right angle. Although it's a very good problem, this was a rather difficult exercise (perhaps due the fact that we weren't expecting it). It made me wonder how it would be received by grade 10 students. However, if they had been working up to this exercise over the term by doing simpler geometric proofs, perhaps it would not catch them as off-guard. It is probably easier in the beginning to learn proof by proving the obvious, so whether this problem is at-level would depend on how many similar exercises the class had already seen. It is definitely the type of thing that needs to be seen by students more often, and early. If I had a choice, I would have done an algebraic proof, but being forced to do a geometric proof showed me how bad I really am at that kind of thing.

My first real encounter with proof was a second year university algebra course and it completely revolutionized my impressions of mathematics. It was not difficult, it taught me to think, and I really wished that I had seen something like that earlier. Another benefit of proof, particularly of the geometric kind, is that is does not use numbers. It is a purely logical/visual expression of math which I imagine many students may find refreshing, particularly those whose numeracy may be lagging behind their peers. Does the prevalence of calculation, data, word problems etc. perhaps create an implicit bias in the students' minds with respect to what mathematics actually is? Even geometry, when it is taught, is less about structure and more about numerical quantities of area, volume and so on. Which is more fundamental after all, shape or quantity? In schools, quantity seems to be hands-down winner of that debate (whether the debate ever took place I do not know). Perhaps it may be that the math class could do with fewer numbers.

Story Based Math Problems and Proportional Reasoning

Last week in our math course, we saw presentations of three condensed lessons. The topics covered were finance, graph transformations, and calculating volumes. Although all were good, I was to focus on the “volumes” lesson for this post. Nam (the presenter) based his lesson on the film adaption of the book “Holes”, which is a very common presence in the junior school English class. For those who haven’t read it, Holes is a young adult novel about a boy named Stanley who is wrongly convicted of theft and sent to a sort of juvenile disciplinary camp where the inmates are forced to dig a hole every day out in the desert. The depth of the hole and the diameter must be equal to the length of the inmate’s shovel. Nam started the class with a clip from the movie in which Stanley accidently gets in trouble with another inmate for grabbing a shovel slightly shorter than the rest of them. The question Nam posed to the class is, “does it really matter if the shovel is a bit shorter”. He then had us try and figure out how much less dirt we would have to dig over the course of a year if we used the shorter shovel (it turned out to be about a month’s worth of holes). I liked this set up for a lesson because it starts with familiar territory. Practically everyone in the class had read Holes so when we worked on this problem, we already had a context in mind for the problem. I also liked that I was surprised by the answer, I didn’t expect the difference in volumes would be so significant. All in all, it showed me the effectiveness of using stories to engage students with a problem (as opposed to the more generic types of word problems we often encounter). If I had one thing to improve upon, it would be the organization of the problem. The problem was posed in feet and inches and we were made to convert everything to metric units. This made the problem a bit numbers heavy. Additionally, it was a bit ambiguous what was meant by “how fewer holes …” since we had two different volumes. This is nitpicky of course, and easy to fix. An alternate approach could be to do away with inches and metres altogether and use the standard shovel length as your unit length. For example, if the shorter shovel was 7/8 of the standard shovel, the smaller hole has (7/8)³ the volume of a standard hole. This greatly simplifies the calculation and may be a useful way of incorporating proportional reasoning.