I've been purposefully drowning myself in math for quite some time now, after having done no formal math at all for about 18 years. This has given me a rather strange, distanced point of view on how math appears to be presented, or taught.

There are two things that seem to be taught: proofs (sometimes, if you're lucky), and steps for solving a problem. There is an emphasis on the latter, and an expectation that the student will learn those steps and use them. "If you follow 1., 2., 3., you will have the solution." This isn't just how things are being presented in the college calculus classes I'm currently taking, but I also recall it rather poignantly from my grade school and high school experiences years ago (most of which were frightful, despite the fact that I love math and understand it easily, b/c I usually need to invent my own solution steps).

It is curious to me, especially now that I am older and have a clearer sense for what works for me and what does not, that a key concept recognized in other disciplines is not recognized in mathematics--at least it's not being recognized at the academic level I'm currently at. That key concept is this: *if a person has deep, real understanding and ownership of an idea, they can use that idea creatively. Conversely, creative application of an idea leads to deeper understanding*. One is expected to make creative use of learned concepts in the arts, right? And even the graduate science course I am taking expects (and *grades on*) creative problem exploration. Why is the expectation in math that one will simply plug in values to a memorized series of steps that were invented by someone else and be happy with that? That that's *enough*. Why isn't there an emphasis on using concepts creatively to find an individually optimized series of steps for solving a problem, or using concepts creativity to get at the core of what is wrong or right with methods of solution. If a concept is understood enough to be used creatively, then that implies a much deeper understanding of the concept than rote regurgitation.

I wonder if it is because we are taught in grade school that there is always "one right answer" to a math problem that we get bogged down into thinking that mathematics is purely formulaic. The closer I get to more advanced mathematical concepts, the less I find this paradigm useful (not that I've ever found it particularly useful). What would be different if from the start children were taught mathematical concepts instead of someone else's steps, and were told to find optimal methods of solution? What if from the start children were encouraged to play with their math, to discover for themselves how each jeweled facet of it works?

Eh, maybe things have changed in the way math is taught in primary education in the 20-30 years since I was a kid, but still, I've been thinking about this a good deal. I'm not entirely certain where I'm going with it yet, but I thought the ramble had relevance to this community :-)

hyperfl0wApril 3 2005, 02:02:14 UTC 11 years ago

It really was 'backwards' i think but I can't remember anymore. I was in the math olympics and over time, I was assimilated to use the standard conventions.

My solution "worked" in the sense that I could answer the problems given to me. I am willing to accept that the solutions didn't 'scale well', considering it might well have required some memory which would break down when doing say 4 digits X 3 digits... Too much to compute in a mind without loosing some precision.

Oops, did

youjust find a flaw in the system?I think you did!

Is it always appropriate to use a Btree?

How about storing 10 numbers out of sequence?

Without challenging the ways that structure is formed, we discourage more optimal solutions, even if those solutions are for the individual.

Nice Post.

## Good question, wrong perspective...

heyoka_hoka_heyJune 23 2005, 01:06:35 UTC 11 years ago

....Kern....

lerAugust 28 2005, 14:25:14 UTC 11 years ago

djsinOctober 13 2005, 06:09:34 UTC 11 years ago

I was debating along the same lines.

Also growing up, I hated math. I never met a math teacher I liked until college. My thing is if math were taught differently maybe more people would excel in it. People who are good with words might like math more if they learned the history and concepts behind math to essentially help them connect better with it.

My 0.2 cents

raeven_beakAugust 12 2006, 01:43:03 UTC 10 years ago

djsinAugust 12 2006, 23:18:08 UTC 10 years ago