It seems that the prevailing wisdom in teaching mathematics is to make abstract concepts as concrete as possible. The thinking is that if math can be related to everyday concepts or pictures then it will be more palatable and understandable. I happen to disagree. I think part of what makes math fun is the abstractness of it. You make up some rules and follow them to their logical conclusions. This is also how I see children play. They like to invent make-believe worlds and then play within them according to the rules of the world.

Usually these attempts at concreteness seem harmless enough but I have recently come up with an example where making things concrete is much more confusing than just teaching a rule. The example is in division with remainders. My third grader was asked to “draw” 7 divided by 2 in terms of items and groups. Her instinct was to draw 2 groups with three balls each with one ball remaining, like this

(x x x)

(x x x)

x

She then was supposed to write this as a mixed number, which looking at the diagram she wrote 3 1/3. When she asked me if this is correct I asked her to multiply this by 2 and see if it gets back 7 and when she got 6 2/3, she was extremely confused as to why she didn’t get the right answer. I tried to explain to her that the way she grouped things, the remainder was in terms of the fraction of the number of groups, which is very unintuitive and almost impossible to explain. It would have been even worse if the example was 8 divided by 3.

I then tried to tell her that a better way to think of division is not to ask how many elements would you get if you divided 7 into 2 groups because this amounts to begging the question (phrase used the correct way), since you need to know the answer before you can do the operation. Rather, what you really want to ask is how many groups would you have if you divided 7 items into groups of size 2 (which is a local rule), whereupon the diagram would be

( x x)

(x x)

(x x)

x

Now if you write down the mixed number you get 3 1/2, which is the correct answer. She then argued vehemently with me that this is not what the teacher taught her, which may or may not be true.

I think even most adults would get confused by this example and maybe working through it would give them a new appreciation of division. However, if you wanted children to learn to divide correctly than teaching them the rule is better. To divide 7 by 2 you find the largest integer that multiplied by 2 fits into 7 and what’s left over is divided by 2. Even better, which introduces and motivates fractions, is that you write 7 divided by 2 as 7/2 and this then becomes 3 1/2. If you learn the rule, you will never end up with 3 1/3.

it took me awhile but i got the first one. i’m always confused—i saw michael freedman (fields medalworker) give a talk on his work on the poincare conjecture at gwu once—i misinterpreted since it involved the ‘undeidability of the word problem’, i thuight it was undecidable, but thats a different case. see also on math ed http://www.arxiv.org/abs/1407.1954v3 (from math under the microscope blog—-v i arnold and ‘a mathematcian’s lament’ by paul lockhart (published in AMS with an intro by Keith Devlin—and i looked at devlin’s ‘research’ page and was not too impressed, though again maybe thats an error) are more along my line.

LikeLike

ps ‘the conjunction fallacy’ (or ‘linda’s problem) is also a nice example. (i disagree with the standard solution, since its based on some assumptions). also a. a zenkin (rev. of modern logic, 2003-4) ‘logic of actualy infinity’ (he disproves cantor’s theorems; i even have a proof that the integers are uncountable though i’ve seen others have it and it relies on a few extra assumptions, though perhaps they add to -1/12).i think that’s einstein’s famous result on compound interest—if you save carefully, you’ll end up owing money.

LikeLike