### What is mathematical literacy?

There is a wonderful mathematics joke, usually told about Von Neumann, but sometimes with other mathematicians swapped in, it features the following math problem

A train is traveling west at 70 miles per hour, and a train is traveling east at 80 miles an hour. They started 300 miles apart. A fly decided to challenge itself and, at 100 miles per hour, flew from the west-bound train train to the other, and then back, and then forward, and then back, until the two trains passed each other. (An exercise path that is somewhat akin to what we call “suicides” in high school swim practice.) Question: what is the total distance traveled by the fly?

There are two ways to do this problem.

1. The brute-force way. The fly and the east-bound train are moving toward each other at 180mph. Starting at 300 miles apart they would meet after 300/180 = 100/60 hours = 100 minutes, during which the fly would have traveled 500/3 miles, and the west bound train 350/3 miles. After the fly turned around, it will be 50 miles (= 500/3 – 350/3) from the west-bound train and they head toward each other at 170 miles per hour, so they would meet after 5/17 hours, during which the fly traveled 500/17 miles and the east-bound train 400/17. Now 100/17 miles apart the fly and the east-bound train head toward each other at 180mph and…

This eventually leads to an infinite series which one can potentially sum geometrically and arrive at a final answer. Or,

2. The “smart” way. The two trains move toward each other at 150 miles per hour. It takes 2 hours until they meet. The fly, traveling at 100mph, much have traveled a total of 200 miles during the trip.

The “joke” is usually told with some joker, say dear old Feynman, posing this mathematical problem to some more serious-minded mathematician, often renown for his computational abilities (say Von Neumann). The mathematician would reply instantly 200 miles. The joker would be disappointed and complain that the mathematician had heard this question before and somehow knew “the trick”, upon which the mathematician replies: “What trick? It wasn’t that hard to sum the series.”

I just came back from teaching an exercise course (in Princeton language, a precept), and the official answer to one of the exercises reminded me of the above problem. Here is the question:

You are given a biased coin which lands on tails with probability $p$. You flip the coin until you get heads, and record the number of tosses. (a) What is the probability that the number of tosses is exactly $n$? (b) What is the probability that the number of tosses is greater than or equal to $n$?

Part (a) is standard: for $n$ total tosses, the $n$th toss is heads and the rest are tails. So the probability is $p^{n-1}(1-p)$, representing $n-1$ successive tails followed by a single heads.

The official answer to part (b) is this: The probability of number of tosses is at least $n$ is the sum

$\displaystyle \sum_{k = n}^\infty P( X = k ) = \sum_{k = n}^{\infty} p^{k-1}(1-p)$

which after summing the geometric series we arrive at $p^{n-1}$.

Here’s what I thought after seeing the question: for at least $n$ total tosses, the first $n-1$ must be all tails. After which the tosses don’t matter: either one eventually ends up with heads at some finite time, in which case it is counted as some finite $k = N$ event, or one never hits heads, in which cases the probability is vanishingly small. So the probability is exactly the probability of getting exactly $n-1$ tails in a row, which is $p^{n-1}$.

Just like the fly problem, one of the solutions brute forces the answer, the other try to “reason” away the computational bits until one is left with a simple computation which can be done quickly in one’s head.

Which of these is mathematical literacy?

One may ask, “who cares? As long as the students can figure out the answer.” But teaching mathematics is always a large extent about teaching “how to solve problems”. It is not the answer, but how we get there that counts. We as a society have always been aware that many aspects of life demand reproducibility, and to be able to reproducibly provide solutions to mathematical problems, one needs to have some understanding of the methods involved, instead of just vague clairvoyance. And we grade students according to such. We ask students to explain, to show steps, to produce proofs, just so that we can see that they have a grasp of the material. But in the current climate where everything needs to be standardised and certified, we often also end up teaching students to a particular type of thinking: namely that of the person who produces the grading rubric.

And hence the question: what is the type of mathematics we should be teaching?

Each of the two methods of solving the question above exhibit a set of skills. To carry through the “brute force” technique, one needs to be computationally competent, and be sufficiently familiar with abstract thinking to keep track of multiple levels of recursion in one’s head. To even write down the infinite series requires some degree of pattern matching and symmetry-observing skills that form the basis of abstract reasoning in mathematics (something that is supposed to be taught in high-school level algebra). To sum the infinite series require some technical proficiency, at least in knowing how to some a geometric series. These are skills built up over years of mathematical education, where one is introduced, one at a time, to the various computational tools of mathematics, and shown when and how they should be used, and drilled until the tools become second nature.

The “smart” technique requires, on the other hand, a different level of abstract thinking. This is a level of observation which “transforms” the problem to be solved. It requires also some pattern-matching and symmetry-observing, but not at the level of manipulation of symbols, but at the level of abstract algorithms or “mathematical situations”. It requires an understanding that two mathematical problems can be analogous without the immediately obvious methods of solution be even remotely similar. But it also does not require technical proficiency. The “smart” in the label is not entirely complimentary: this method advocates finding short cuts and analogies, and doesn’t always work in real life. Not every problem one encounters is carefully crafted to admit a beautiful simplification.

In short, the first method focuses on “technique”. The second on “insight”. As any practicing mathematician will tell you, clear mathematical thinking requires big doses of both. One of the tragedies of the genius Ramanujan was that he was full of insight but lacking in technique, whereas the United States high school regularly churn out individuals with plenty of technique (much of which programmed into their graphing calculators) but with no insight.

Insight, however, is hard to teach, and even harder to grade.

One sense of insight is the ability to draw analogies between apparently disparate situations. It is, by definition, the ability to see what is hard to see. And as such, a new insight is almost always difficult to explain. One of the central points in William Thurston’s “Proof and progress” essay is precisely the importance of trying to explain this, the unexplainable: it is only when the insights become routine and well-understood, that it becomes another tool in our toolbox and mathematics as a whole progresses.

From the point of view of the teacher, how do we explain insight? The Zen buddhists had it easy: they just kept hitting the disciples over the head with a stick. I suspect that corporal punishment and unanswerable riddles are unlikely to be endorsed by the US Department of Education. But how do we make someone, whose brain is not yet tuned to “see”, see? I have no answer to that. For teaching the “technical” aspects of mathematics, I often like to employ the method of Socrates. But in those cases the tool is already there. I just wait for the students to grasp on to the correct tool, prompting them with occasional questions. Rhetoric is great for pointing out what went wrong; and for the technical aspects the student can learn even through a process of elimination. But their effectiveness in exploring new mental territories is limited.

How did I learn my insights? Through problem solving. Doing lots and lots of math problems, some for school work, some for contests, the random firings of my synapses sometimes lead to unexpected connections. These build and grew as I learned mathematics, and once the first few hurdles are past, the brain is very susceptible to new ideas. But not every student has the luxury of discovering the joys of mathematics in his own time. There are university course work set on a schedule, and practical examinations or certifications that one must obtain. This reminds me of martial arts novels and mangas: the inner force (ki) is supposed to be built up bit by bit through diligence in daily practice. Yet the heroes (or villains) often are individuals who through some miraculous coincidences managed to acquire through entirely unorthodox methods a large amount of ki. How do we, as teachers, reliably do the unreliable, and stimulate the students into obtaining those insights? In the martial arts novels, the downside to unorthodox training methods is often that it easily leads to insanity or death even when the most minute things go wrong. In our teaching, applying too much pressure to the students lead them to feel stagnant and unproductive, eventually declaring themselves “unable to do” or “hating” mathematics.

And on the flip side, how do we as educators reward the students for insights? Thinking back to my own experiences, insights are often extremely difficult to explain, especially when one has not yet been taught the proper language to describe them. When I first discovered algebra for myself, it was with great trepidation that I wrote on my homework exercises a system of equations like

The number of guests / the number of dishes + 4 times the number of dogs = the age of the host

Up until then I’ve only been taught that one can only perform arithmetic on numbers, so I didn’t even know whether what I wrote was “legal”. Furthermore, I didn’t know that the usual rules of computation can apply to symbols. So I couldn’t explain how I arrived at the final answer. It was “obvious” to me that my answer is the unique solution to the system of equations, but I hadn’t the technical facilities to justify that obvious fact! Talking to some students today, I wonder how often do student nip their own insights in the bud, just because “it is something difficult to explain, and requires explaining in words instead of formulae, and therefore cannot possibly be the correct answer that this exercise/exam is looking for.”