5-3
or
Five minus three
or
Five take away three
Lately, my friend PS and I have been discussing the wrongness of this idea of subtraction as "taking away". The conversation started with a TMBG video we show our kids about Zeroes and resurfaced recently when he fell asleep at a kindergarten curriculum night looking at the number line. Years ago I read a book called Where Mathematics Comes From which has really informed my thinking about math concepts. Out of these influences I have decided that we should throw out the old ways of "5 take away 3 leaves us with 2" to be replaced with the correct "marching 5 units to the right along the number line and then marching 3 units to the left leaves us exactly 2 units to the right of where we started".
(image from ehow.com)
The idea of subtraction as an act of "taking away" is sloppy thinking which most people get away with because they don't have to think things through. Not, that is, until they get to vectors in physics. What we find is that students think of negative acceleration as always meaning slowing down. This is because they think of negative meaning "taking away". The fact is if you are already heading in the negative direction, a negative acceleration will speed you up! (imagine going in reverse in your car and then stepping on the gas) Only now will the student begin to think of negative signs properly - as an indication of direction along the number line NOT as an act of taking away.
There is no subtraction. There are only positive and negative and these mean: "going to the right" and "going to the left" respectively.
"5 - 3" is short hand for "(+5) then (-3)"
If this were how students were taught to think of the minus sign then when we introduce them to negative accelerations they will think "that is an acceleration directed to the left as opposed to the right" and kinematics would be a lot easier for them (and for us to teach).
Learning is only hard when there is a lot of unlearning to do first.
I remember being taught the 5 - 3 is the same as 5 + (-3) in 8th grade, and I think it really tripped up a lot of people. Our teacher insisted that every time we saw a minus sign, we had to expand it out to + (-n) as we did our algebra homework(writing out the intermediate steps). I think overall, it did help in the high school years, but it was definitely a rough year of math for most of the class (including myself).
ReplyDelete