[Maths-Education] Subtraction

[Neil A Pateman]

Ian wrote:
>Most books on teaching mathematics discuss 'the two aspects of subtraction:
>'take away' and 'difference'.
>
>Since the most efficient way to solve difference problems is usually by
>addition, and since 'difference' is commutative whereas 'take away' (and
>subtraction generally) is NOT, why do we persist in linking difference with
>subtraction?
>

Watching young students work at finding a difference with objects 
suggests that there are two methods they develop. In each case they 
initially make both collections ("these are yours, these are mine") 
then either:
(1) match up as much of the collections as possible and  push the 
matched ones aside (literally "taking away" the parts of the two 
collections that are equal), then counting the rest, or
(2) match up then find partners for those without (literally "adding 
on" to make the collections equal), then count the added ones.  So 
whether we like it or not "taking away" is connected to difference.

I have seen kids make the larger collection, then count off the 
smaller collection ("removing" them at least in their minds I 
suspect), say, "These are the same " and count the ones that "make 
the difference."

I think we have to think about all THREE situations as 
subtraction-like; (1) compare two collections, (2) take a smaller 
collection from a large collection, and (3) add to a small collection 
to make a larger collection.

Students learn that the solutions to all three  may be found by 
removing from  the larger collection, but using language appropriate 
to each situation--(1) "taking away" is obvious, (2) for the 
difference "these (the smaller collection) match the others, these 
(remaining collection) are the difference";    (3) for the addition 
to smaller "these (the smaller collection) are the ones I already had 
so these (the remaining collection) are the ones I need." Thus 
subtraction fits all three in the sense that taking away can be made 
to work for all three.

All this is of course premised on the use of real language situations 
for students to think about and to model throughout--rather than the 
presentation of symbols as beginning, middle, and end of story.

I am not at all averse though to kids making their own decisions 
about whether to add on, Ian's claimed most efficient method, or to 
subtract, based on personal preference. I suspect that the first is 
the preferred mental method and the second the preferred pencil and 
paper method. Of course using a calculator for adding on is rather 
awkward!

I enjoyed Judy's story and offer one from a colleague--a boy was 
struggling with "how many more" and continued to respond with the 
size of the larger collection when the boy beside him made the two 
collections with different colored Unifix cubes, laid the two 
collections side by side, counted up on the larger collection to 
where the two matched, snapped that many off the larger, gave our 
struggler the remainder and said, "These are the more ones!" "Oh!" 
was the response from the struggler who was completely satisfied with 
the explanation.

Aloha,
Neil