Goals - Part 4

My goals in helping children with arithmetic are simple. I want them

  • to be good at solving all different types of word problems,
  • to like solving word problems,
  • to understand the problem solving methods (shortcuts) that they use and
  • to be able to communicate their reasoning to others.

4. What Does It Mean To Understand A Shortcut?

Understanding a shortcut means understanding why it works – that is, understanding why it gives the same answer as modeling does. Can there be any doubt that Rose, in the example on the previous page, understands the shortcut that she uses? It seems clear that she solved the problem by using her reasoning – she is not thoughtlessly, mechanically, parroting a rehearsed procedure.

On the other hand, consider the following problem.

I have 962 candies and I want to put them into 37 party bags with the same number in each bag. How many candies should I put into each bag?

Perhaps you remember how to solve this problem using “long division.” Long division is a complicated shortcut for solving certain kinds of problems. If we use long division we don’t have to count out 962 objects, we don’t have to distribute them into 37 piles, and we don’t have to count how many candies are in each pile.

While you may remember how to "do" long division you will very likely have a hard time explaining why it works. In the example above, are you even certain that it does work - that if you actually dealt out the candies that you would get the same answer as the one that long division provides? Most adults, even those who remember how to do division, cannot explain why it works. They do not understand the shortcut - they are merely mechanically using it.