One of the questions asked by Liping Ma and Deborah Ball of Chinese and U.S. mathematics teachers was the following:
- Divide 1 3/4 by 1/2;
- Explain how you did the calculation;
- Make up a story problem for 1 3/4 divided by 1/2.
Ball’s study of U.S. elementary school teachers showed that many of them had trouble doing and explaining calculations with fractions, and more had difficulty in making up story problems. Ma’s study of Chinese elementary school teachers showed that all of them were able to do the calculation and explain how it was done, and over 90 percent made up appropriate story problems.
One of the Chinese teachers said she preferred not to use dividing by _ to illustrate the meaning of division of fractions because one can easily see the answer without really doing the division. She then proposed asking how long a jump_rope is if 4/5th of it is 1 3/4 meters.
Another of the Chinese teachers responded with three different story problems, all dealing with sugar:
- If you have 1 3/4 kg of sugar and want to wrap it into packs of _ kg each, how many packs can you wrap?
- If you have 1 3/4 kg of white sugar and _ kg of brown sugar, how many times more is the weight of white sugar than that of brown sugar?
- If you have 1 3/4 kg of sugar on the table and that is half of what you have at home, how much sugar do you have at home?
This teacher would put all three problems on the board and have the students compare the different meanings they represent. Then the students would be asked to make up their own story problems to represent different models of division by fractions.
One telling illustration of the Chinese teachers’ deeper understanding of elementary mathematics was that most of the story problems proposed by the U.S. teachers dealt with round food, like pizza, or money, while the Chinese examples were from many different areas.