For proofs, just click “play”

July 18, 2012, 9:44 am

This week I am adding to the playlist of screencasts for the inverted intro-to-proofs class I first mentioned here. There are seven chapters in the textbook we are using and my goal is to complete the screencasts for the first three of those chapters prior to the start of the semester (August 27). Yesterday I added four more videos and I am hoping to make four more tomorrow, which will get us through Chapter 1.

The four new ones focus on conditional (“if-then”) statements.  I made this video as the second video in the series as a prelude to proofs, which are coming in Section 1.2 and which will remain the focus of the course throughout. Generally speaking, students coming into this course have had absolutely no exposure to proof in their background with the exception of geometry and maybe trigonometry, in which they hated proofs. Watch a part of this and see if you can figure out my emphases:

In talking to students who claim to hate proofs, the problem seems to be that their proofs in geometry and trig were either totally mechanistic — like some of the mind-numbing proofs of trig identities — or the steps in those proofs just came out of nowhere, with no discussion of how an ordinary person might have thought of the next step. Or both. There’s very little humanity in the process of proof as it is often taught to middle- and high school students (if it’s taught at all), which is a huge loss for students that compounds over time. Proof is the culmination of an intellectual activity that seems unique to mathematics — it’s the running-in-the-dark thrill of figuring things out followed by the climb-your-way-out struggle to express your discoveries clearly in print. It’s a uniquely human process and it needs to be presented as such. (And “presenting” it is not enough. Students have to be involved because they are human too.)

So my main goal in this short video is to frame the whole enterprise of proof without mentioning “proofs” at all (I think I said that word once or twice) but rather as a process of problem-solving that requires a certain kind of solution. And the organizing principle of this sort of problem-solving — of just about any sort of problem-solving — is play. To understand whether the statement “If \( p \) is prime, then \( 2^p – 1 \) is prime”, you cannot just start writing the proof out. Because there is no proof. The problem is that we have a statement that appears to be true but it’s unclear at the outset whether it is really true. So you have to play with the problem, whereupon you run into \( p = 11 \) and find out that the statement is not true after all.

The “director’s cut” of this video had a second example on the Twin Prime Conjecture. I asked viewers to notice that there are a lot of prime numbers that are “right next to each other”, like 5 and 7 or 11 and 13, and as you go higher in the list of primes you continue to find these “twin prime pairs” but less and less frequently. So it leads naturally to the question, does the list of twin prime pairs go on and on? Or do we eventually reach a point where there are literally no more? Playing with the problem helps you understand why this is a problem: you feel the sparsity of the twin prime pairs increase, and yet every now and then you still find one. You don’t hit an obvious counterexample as you do with the \( 2^p – 1 \) question. So it’s not a story that resolves itself easily, and of course as of today, nobody has either proven or disproven the Twin Prime Conjecture, although it has been verified as of 2011 up to \( 3756801695685 \cdot 2^{666669} \pm 1 \). (I cut this example to keep the video length down to under 7 minutes, but I’ll probably reintroduce it as an in-class “homework” exercise.)

This last point, too, is an important thing for students in this class to see — that there are unsolved problems, or unresolved mysteries, in mathematics and the process of mathematical investigation is alive and well, indeed so dynamic in some cases that it’s very hard to keep up with the new discoveries being made. Students are so used to liking mathematics because “it always came easy for me” and “there’s only one right answer” that it’s a healthy system shock to see that it’s not always easy, but there is always an accessible entry point to understanding any problem; and that not only is there not always one right answer, some problems currently don’t even have answers.

When we meet the class for the very first time, I’m planning on giving students a similar problem to play with. Many of you will recognize it, but I doubt the students will. You can try it too if you want. Suppose \( n \) is a positive integer and define the function

\[ T(n) = \left\{ \begin{array}{cl} \frac{3n+1}{2} & n \ \textrm{odd} \\ \frac{n}{2} & n \ \textrm{even} \end{array} \right. \]

Directions: Pick a value of \( n \) at random and plug it into \( T \). Then take the result and plug that into \( T \). Repeat. What eventually happens?

Then I will give them a clicker question that says:

The sequence of numbers generated by plugging repeatedly into \( T\) always terminates in 1.
(a) True
(b) False
(c) I don’t know

The correct answer is “I don’t know”, because we cannot say whether this statement is true based on a finite list of examples. We believe, in our hearts, that it is true — but this is different than knowing. And in fact, this is another famous unsolved problem that has a rich literature of work in the recent past and currently ongoing. (If somebody happens to come up with a proof or counterexample in the course of answering the clicker question, I’ll thank them and then quickly co-opt it and publish it under my name.)

The point here is that, with this little video, I want to set the stage for mathematical work as being something that appeals to their better natures — the natures they share with kids of exploration, skepticism, and play — and which sadly get planed down throughout the years of repetitive school math too often.

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