A couple of weeks back, I ‘d have considered this concern meaningless. Less a genuine concern than a sort of unclear hand gesture.
But I simply checked out an excellent book that moved my thinking. [EDIT: I forgot to say which book! It’s The New Math: A Political History, by Christopher Phillips.] Now, instead of an unclear hand gesture, I see the hard/easy concern as a pointed and rather exact gesture. (No, not that gesture.) Particularly, the concern points towards another, larger concern:
What is mathematics class actually about?
One possibility: Is mathematics education for cultivating a basic quality of believed? For developing the capability to factor realistically? For honing the mind into an all-purpose tool?
In that case, it must most likely be as difficult as possible. The mind, like a pencil, requires friction to grow sharper. Not too much friction, or you’ll break the pencil (and the metaphor), however all else being equivalent, an excellent obstacle is a good idea.
Or, another possibility: is mathematics education for imparting a particular set of helpful abilities? Is it about information literacy, arithmetical knowledge, and algebraic fluency?
In that case, it must be as simple as possible. Simplify into itty-bitty jobs, and practice every one up until you have it cold. Mathematics ed should resemble motorists’ ed: so blindingly basic that any person can obtain these standard, required abilities.
A streamlined contrast, to be sure. For me, it’s an unexpected inversion of my normal thinking.
I most enjoy the difficulties of mathematics when I’m in “traditionalist” mode, when I feel genuine faith in the abilities and techniques that make up the historic discipline. (This isn’t my normal operating mode, however hey, I take pleasure in the chain guideline as much as anybody.)
But wait. If controling polynomials is really a human excellent, should not I wish to make it simple? Should not Iwant algebraic literacy to end up being like English literacy, with 99% of kids attaining it?
In the minutes when I’m most indulgent of “difficult,” I possibly should be pursuing “simple.”
Meanwhile, I’m wariest of mathematics sensation too hard when I’m in “progressive” mode, picturing a human-centered mathematics education. On nowadays, I tend to believe that mathematics is not about a particular ability, however broadening our intellectual toolkit and boosting our capability to factor.
But if this is my concern, why should I be troubled if mathematics is agonizing or undesirable? Sure, excessive obstacle will drive my trainees away. If my job is enhancing minds, then aren’t difficult days an excellent thing?
Of course, “difficult” is not one thing– its meaning differs trainee to trainee, job to job, minute to minute. There’s no “problem” dial I can turn. Still, these ideas leave me with a weird conclusion that I’m having a hard time to soak up.
When mathematics education has to do with the mathematics, I ought to be making every effort to make it simple. And when mathematics education has to do with the education, I should not avoid making it hard.