I saw a poignant comic strip in the newspaper today, and thought it was illustrative to mention here:
Paige has no clue what to do with and or how to solve that set of equations. Her brother then asks her “if two shirts and a sweater costs $60, and a shirt and two sweaters cost $75, what does each item cost?”, which she can solve without difficulty. In the last panel, she is still desperate about how to solve and . Of course, this is only a comic strip. But it touches on something I suspect all math teachers have encountered in their classes: (1) the total disconnect between the perceived world of math class and the outside world; and, almost in spite of that, (2) glimpses of real mathematical thinking from students who don’t do well in class.
In many a traditional math class, those glimpses of real mathematical thinking from students are irrelevant. They are a distraction, they are a hurdle on the way to having children open their mouths and swallow. For those children, all this (to mathematicians) beautiful machinery we adduce to help them solve problems in real life, is simply one of a set of bitter pills to be swallowed so they can finally get cured from a disease called school.