Bob is an elementary school student. He is learning to calculate. He just learned about addition and multiplication but there are some things that he doesn’t understand. For example, how come 1+3 = 3 + 1? How can it be the same thought Bob? Every morning I have 1 piece of bread for breakfast while Dad has 3 pieces. If I have 3 pieces while Dad has 1 piece, I will be too full and Dad will be hungry?

When they added three numbers, Bob did not understand (1+2) + 1 = 1 + (2+1). Usually I like to drink 1 cup of coffee with 2 spoons of milk then afterwards have a piece of bread. I would not feel well if I first drink a cup of coffee then afterwards drink 2 spoons of milk while having 1 piece of bread. How come they are the same, thought Bob.

The most confusing part was after the lesson on fraction. Bob learned that 1/2 = 2/4. So when he got back home he tried to share 6 apples with his sister Linda. He divided the 6 apples into two groups – 2 apples in one group and 4 apples in another group.

From the set of two apples he gave 1 to Linda. That’s 1/2. From the set of four apples, he took 2, that’s 2/4. It is equal he said. But Linda did not agree with him because she got 1 apple less that he. Bob thought, how can this be? Why would 1/2 = 2/4 not work for apples!

The next day, the teacher asked Bob to add 1/2 and 2/4? Bob wrote 1/2 + 2/4 = 3/6 because taking 1 apple from 2 apples then 2 apples from 4 apples, he must have taken a total of 3 apples from 6 apples!

This story is adapted from *A Framework of Mathematical Knowledge for Teaching* by J. Li, X. Fan, and Y Zhui at the EARCOME5 2010 conference.

Point for reflection:

What has Bob missed about the meaning of addition of natural numbers? the meaning of fraction?

You may want to read the following posts about math knowledge for teaching:

- Math knowledge for teaching fractions and decimals
- Math knowledge for teaching – counting cubes
- Math knowledge for teaching tangent to a curve
- Two Plus Two or Why Indians Flunk