The world of mathematics is populated by strange creatures like bumblebees who while normally flying in straight lines at constant velocity, are able to change or even reverse direction instantaneously. We have men in rowboats who proceed along a river at constant velocities and drop their hats in the water when they pass under a bridge. And recently I wrote about a spider capable of instantly calculating the optimum trajectory along the walls and ceilings between two arbitrary points in a room.
I do not object to these magical creatures even though their abilities and behavior are strange or even absurd when compared to real spiders and bumblebees. Why then do I object to a woman who decides to randomly "sample" her shoe collection when packing her suitcase for a business trip?
In mathematics, we use spiders and bumblebees as shorthand symbols to represent certain ideal behavior which is graphically suggested by the creatures we use as stand-ins. There is a certain charm to a math puzzle which is composed using such symbols, and there is little doubt as to what is intended. The homework problem we looked at the other day is an entirely different kettle of fish. It represents everything that is wrong with math teaching in today's schools, be they high schools or universities.
The school system pays lip service to the idea that they want students to learn to think and understand, but in practise the system demands that the student memorize instructions and follow algorithms. The homework problem I discussed the other day shows both of these hypocrisies in their fullest form: first, the lip service to the idea that the student should think about what is about what is going on...namely, a woman randomly choosing two pairs of shoes for a trip...and then, the turnaround where the student is told what formula to use for the calculation...."sampling with replacement", taking into account the order of selection.
I have already discussed the absurdity of calculating the answer based on which order the shoes are selected. If you pick the pumps first and the boots second or vice versa, you still end up with the same two pairs of shoes in your suitcase. If you want to create a word problem where the student is required to take into account the order of selection, then there has to be a practical reason within the word problem why it should matter. You don't write a problem where the order doesn't matter, and then tell the student to use the formula where it does.
And then there is the point of "sampling with replacement". Sampling with replacement means you put the first pair of shoes back in the closet before you choose the second pair. Now, how are you going to pack your suitcase if you do that? It just doesn't make sense. She picks a pair of shoes, puts it right back in the closet, shuffles the boxes around so they are again totally randomized and then picks another pair. How does she end up with two pairs of shoes in her suitcase? Maybe their are women who pack their bags that way, but I don't know how you can do any kind of mathematical calculation.
If I thought I was picking on one example of a bad teacher who made a silly mistake in a homework assignment, I wouldn't be writing this article. I've singled it out because this goes on all the time, and it's typical of everything that's wrong with math teaching.
(Oh, and I forgot to mention the orange trees.)