Mathematics as a silly exercise in pedantry
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.
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November 22nd, 2010 at 4:37 pm
This is precisely what I found frustrating about Maths Methods 15 years ago, when SATs were CATs and so on and so forth. I solved the problem in the major project in general terms, and then applied the general solution to the specific question, and stubbornly stuck to my solution in face of advice from my teachers that my answer, while correct, would not get me a high score. The result? B+ on that project, while I got straight A+ on every other bit of maths assessment I did, including the allegedly harder Specialist Maths. Very silly.
November 24th, 2010 at 1:06 pm
Marking Schemes must be darn difficult to do right. Assignments and exams are hard enough, I’ve set some and found that they measured things badly.
The overall goal is generally to find out whether the student understands the concepts and can apply the techniques. In my teaching experiences (mostly physics and maths) I have often found a student approaching a problem from a different angle - sometimes a truly inspired one! On the rare occasions I’ve worked with a marking scheme I’ve been able to apply discretion where discretion is due. But it would be truly difficult to build in discretion at all the right places while designing a marking scheme critical to university entry for thousands of students …
Now if the rubbish Chris describes is only relevant to a single school, it should be open to correction by the teachers using it.
As a student I remember looking out for the instruction “Show all working”, and still struggling because several steps would flow into one before they reached my conscious mind. How trivial should we go? As a teacher the problem is different - trying to rule out students having copied from one another. For exams, you should be able to trust your exam conditions; for assignments, the “show all working” instruction is more useful.
I do want to comment on the last example, though. Since the questions have been transcribed and not reproduced, I’m not sure if this criticism will work for the original. However, my training says that in general answers should have comparable accuracy to the input data. (Naturally there are exceptions.) In this case, x is supplied as an integer - without two decimal places - and supplying an answer to two decimal places would violate the rule; it would imply the accuracy of the median is quite different to the original data (which is either perfectly precise or has about a +-0.5 uncertainty.)
I suspect I picked up this rule from my physics side, but I’m pretty sure my maths training never contradicted it.
Anyway. If the stupidity I describe is indeed present in the original exam, then the logical deduction is that the teacher who wrote the exam & marking scheme is not aware of the potential meanings behind the formal symbolism of mathematics; in other words, they don’t “get it” - and that’s sad. It’s sad any time a teacher doesn’t “get” their subject matter, and more so if they have somehow become senior and/or influential.