I was a teacher's assistant in high school, and I used to get impatient with students who would take a long time to make simple calculations. (Though I always tried to smile and keep my impatience from showing.) I asked a few of the students to explain their thought processes to me. I was rather surprised by the result. The students were (still) following calculation techniques step for step as they had been explained to them.
Q) 46 - 29 = ?
1) attempt 6-9. Oops, I need 10 extra.
2) 40 - 10 = 30
3) (30 + 16) - 29 = ?
4) 16 - 9 = 7 ones
5) 30 - 20 = 1 ten
6) 17
By contrast, when I was performing the calculations, I always looked for the simplest way to relate a problem to a math fact I had previously memorized, or which was exceedingly simple to compute.
Q) 46 - 29 = ?
1) 46 - 29 = (36 - 29) + (46 - 36)
2) 36 - 29 is equivalent to 16 - 9 = 7
3) 46 - 36 = 10
4) 10 + 7 = 17
In my head, it goes more like this:
What does it take to get to 36 from 29? 7
What does it take to get to 46 from 36? 10.
17.
I guess my questions is:
If these types of techniques make people faster, are teachers advocating them enough? No one ever suggested to me that I re-imagine problems as a series of simpler calculations (except for the standard model applicable to every single question of the same type). It just happened.
One of my favorite substitutions is the difference of 2 squares. For example, 14*18.
14*18 = (16-2)*(16+2) = 16^2 - 2^2 = 256 - 4 = 252.
Myfav, it's awesome that your boy is taking note of mathematical equivalents at a young age.
When I was young, I used to count and recount my coins. I didn't acquire money often, so often times I'd be re-counting the same quantity. I could count it in a lot of different ways, and naturally, some ways were more efficient than others. When the amount did change, sometimes the relative efficiency of the counting techniques would change too. Maybe one of the reasons I didn't end up blindly repeating the methods my teachers had taught me was that I had already come up with numerous techniques before they ever shared their technique with me.
When I was 5 and 6 I used to race the tellers (and their cash registers) in figuring out my mother's change when she purchased something. Now people use credit cards for everything.
Last edited by DAD22; 06/07/13 10:54 AM.
