Examining Interactions between Problem Posing and Problem Solving with Prospective Primary Teachers: A Case of Using Fractions

Abstract

Existing studies have quantitatively evidenced the relatedness between problem posing and problem solving, as well as the magnitude of this relationship. However, the nature and features of this relationship need further qualitative exploration. This paper focuses on exploring the interactions, i.e., mutual effects and supports, between problem posing and problem solving. More specifically, this paper analyzes the forms of interactions that happened between these two activities, the ways that those interactions supported prospective primary teachers’ conceptual understanding, and the difficulties that prospective teachers encountered while engaged in alternating problem-posing and problem-solving activities. The results indicate that problem posing contributes to problem-solving effectiveness while problem solving supports participants in posing more reasonable problems. Finally, multiple difficulties that demonstrate prospective primary teachers’ misunderstanding with fractions and their operations provide insight for teacher educators to design problem-posing tasks involving fractions.

Appendix

• Task 0

• Task 1

A task from class:

In an adult condominium complex, 2/3 of the men are married to 3/5 of the women. What part of the residents is married?

• 1. Solve this problem.

• 2. Pose as many mathematics problems as you can in terms of fractions.

• 3. Pose problems using either constraint manipulation strategy or goal manipulation strategy according to the problem you have solved, or you have posed, or any other ideas you have.

• 4. Describe a plan that you have for solving one of the problems that you just posed.

• Task 2

A situation from homework:

Adam put 1/2 of a cake in the freezer. Of the remaining half of the cake, Adam ate 1/5 and his dog ate the rest.

• 1. Pose problems using either constraint manipulation strategy or goal manipulation strategy, or any other ideas you have.

• 2. Solve one of the problems you have posed.

• 3. Could you pose any other problems after solving the problem? List your new problems.

• Task 3

A task from class work:

Q12. For each of the following, model the operation with your strip of paper, illustrate with a diagram the process you used, and shade your answer.

a. $3\times \frac{1}{4}$ b. $\frac{1}{4}\times 3$ c. $\frac{1}{3}\times \frac{1}{4}$ d. $\frac{1}{4}\times \frac{1}{3}$

• 1. Choose one of the operations and write a story problem representing it.

• 2. According to the story problem you wrote in problem #1, use either the constraint manipulation strategy or goal manipulation strategy to write new story problems for each of the following:

a. $\frac{3}{4}-\frac{1}{2}$ b. $\frac{1}{4}+\frac{1}{3}$ c. $\frac{2}{3}\div \frac{1}{2}$

• 3. Select the hardest problem you posed and solve it.

• Task 4

A real-life situation:

Four people share a house with four separate rooms in a city that experiences cold winters. They usually split the electric charge (including gas costs) equally. For a very cold month (30 days), they got a bill of $243.95. However, one of them left for home for 18 days, which means she only lived in the house for 12 days. How could they split the bill for this case in a way that seems reasonable?

• 1. Try to pose several easier mathematical problems based on the given situation using the constraint manipulation strategy or goal manipulation strategy.

• 2. Solve the initial problem. If you could not solve this problem, try to come up with a plan or some ideas you have.

• 3. Try to pose a more complex problem than the initial one.