Mathematics is a fairly formal business. But those who ‘get it’ realise that there is meaning behind the formalism, that it is not a matter of just shunting symbols around for its own sake. Terence Tao is not just a walking talking Mathematica program. Unfortunately, math teaching at one of the most prestigious schools in Melbourne are turning Maths into a silly test of whether you can follow instructions to the letter. Here are some examples from a recent Math Methods SAT* marking scheme.
You are given a density of the form kx(3-x) on 0<x<3. The task is to Show that k=2/9. The student wrote down the integral and equated it to one. Then he wrote down the indefinite integral and proceeded correctly to the answer. Result: 1 mark from a possible 2. The reason? He did not expand kx(3-x) into 3kx-kx2. He is allowed to assume that the indefinite integral of xk is (k+1)-1xk+1 but he is not allowed to assume basic expansion formulas.
You might think this is because of the word “show” in the question – that a high standard of pedantry is required. Not so. Later in the exam, we are asked to find the mean of the same distribution. We are again integrating a polynomial. A mark is again subtracted for not multiplying out the expression in the integrand. You can imagine that a student with a modicum of intelligence would, on receiving his marked script, be pretty angry and demotiviated with the subject of math methods.
But it gets way, way worse. Consider the discrete distribution bellow.
x 0 1 2 4 8 16
Pr(X=x) 0.1 0.1 0.05 0.2 0.4 0.15
You are asked to find the median. The student gives the answer 8. The definition of the median for a discrete distribution is an annoying exercise in pedantry itself, but the answer of 8 here is correct according to the provided definition. Result: 1 mark from a possible 2. The reason? The question asked that answers should be given to two decimal places. The required answer was 8.00. I kid you not. For those who think students should learn to follow instructions, why not ask the student to write all answers in red, or all even digits in green? We want to test mathematical understanding, not the ability to mindlessly follow instructions. They teach that in the cadet corp.
No wonder some students think that mathematics is a meaningless, mindlesss game that involves the slavish following of algebraic rules for their own sake.
* A SAT is a test adminstered within a school to determine within school rankings. These are combined with inter-school differences from the main exam.