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 and Eva, in the examples on the previous page, understand that their shortcuts work? It seems clear that they are solving these problems by using their reasoning – they are not thoughtlessly, mechanically, parroting a rehearsed procedure.

On the other hand, consider the following example.

Trixie collected 48 pieces of candy corn on Halloween. She wants to share them with her grandfather and her dog Punch. How many candies should they each get?

You may remember the following process for solving this problem. It is an example of what is called an "algorithm." This algorithm is sometimes called “short division.” It is one of several algorithms taught in school - for adding, subtracting, multiplying and dividing.

  • Make a “house.” Write the numeral 3 “outside the house” and the 48 “inside the house.”
  • Say to yourself, “3 goes into 4 once, with 1 left over.” Write the numeral 1 outside the house and above the 4. Write a little numeral “1” between the 4 and the 8 that are inside the house.
  • Say to yourself, “3 goes into 18 6 times.” Write the numeral 6 outside the house and above the 8.

Short division tells us that the answer is that Trixie, her grandfather and the dog Punch each get 16 candies. Short division is a complicated shortcut for solving the problem. It allows us to avoid making a physical model and counting. We don't have to count out 48 pieces of candy corn, we don't have to distribute them into 3 piles with the same number in each pile, and we don't have to count that there are 16 candy corns in each pile.

While you may remember how to "do" short 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? Most adults, even those who remember how to do short division, cannot explain why it works. They do not understand the process.

It takes a lot of practice, over a very long period of time, for children to become skillful in short division and in the other standard algorithms. Given the prevalence of calculators, I think that such practice is a waste of valuable time. But to the extent that these shortcuts are taught, they should at least be taught in such a way that children understand them. For the most part, that doesn't happen.