What seems to be a focus on understanding in reform math is really all about going more slowly and providing more (and repeated) introductory explanations. It's not about an abstract mathematical understanding that comes from mastery and application of the basic math identities.
This is something I think about frequently - or, rather, something I (hope I) experience frequently.
It gets back to Saxon's & Neumann's observations about math being different or being something you get used to.
A couple of years ago, I stopped looking for "math for poets"-type books -- books that explain math to the non-mathematical.
I'm not criticizing these books; I'm sure they have value; I'm glad someone's taking the time to write them.
But at some point - and this was when I was still working my way through K-5 arithmetic, I realized I wasn't really learning math from them. At some point, with math (and correct me if I'm wrong...) you have to become airborne.
And "airborne," in math, means you have to start climbing a purely abstract ladder. You've left the world of words behind.
I think.
I agree with Steve. Everyday Math is like a bunch of introductory lessons strung together. When my kids first started with the program, I kept expecting some follow-up to the exercises. I thought many of the homelinks were good jumping off points for exploring the material in-depth.
ReplyDeleteBut those follow-up lessons never came. There's no depth to the program. The emperor has no clothes.
Robyn
"And 'airborne,' in math, means you have to start climbing a purely abstract ladder."
ReplyDeleteI would call it plateaus and there are a number of them. It takes a certain amount of climbing to get there. You could also look at it as achieving some sort of critical mass where all of the pieces fit.
I mentioned before that it took me until my junior year in high school before I felt that I really understood algebra. This meant that I could manipulate equations any which way without confusion. Up until then, there was always something that caught me off-guard; something that challenged my understanding. In my junior year, all of that stopped.
Trig provides another, perhaps smaller plateau, and calculus provides a larger one, especially when you have to use it to solve real problems.
Then, there are always other gaps. It took me until college to fully understand (appreciate?) and use the different forms of equations, like explict, implicit, and parametric.
Reaching all of these mathematical pleateaus might be helped with pictures of pies or marbles, or using manipulables, but you won't really get there unless you understand it all on the abstract, definition, or identity level.
When I work with my son, I make sure that he always relates his steps to the basic identities, no matter how stupid it seems. He wants to talk about canceling terms, but I make sure he knows that canceling means
a/a = 1
and then
b*1 = b
because he says things like "it goes away" or worse, "it's zero". Vague understandings will get you into big trouble later on. Marbles or tiles won't carry you very far.
Everyday Math in sixth grade talks about general and specific rules, but then applies it to all sorts of equations. This doesn't make it clear that there are only 14 basic math identities that you have to know and apply.
i'll go so far as to say
ReplyDeletethat the symbols on the page
become the subject matter ...
but you *never* leave the world
of words behind! in fact,
as i see it, you have to take
words more seriously than is common
elsewhere: definitions in mathematics
are *really* definitions
(those things in the dictionaries
are just handwaving *suggestions*).
vlorbik