For the students in Level 3, both addition and multiplication were used in solving the proportional problems. For instance, student 3A was able to write multiplicative expressions but calculated them using addition, which was perceived as easy and safe leading to the correct answer. It is noteworthy that the students in this level understood the inverse proportional situation by saying that if one quantity was increased, then the other was decreased. However, they were not successful in solving the problems.
The students in Level 4 completed successfully almost all the proportional problems in the questionnaire. They were the only students who could recognize the invariable quantities and found the missing values in the inverse proportional context. The following transcript illustrates a solution of the student 4B.
Interviewer: Can you explain how you solved this problem?
S-4B: To complete the task, 4 people need to work for 6 days. Multiplying 4 and 6 is 24 and dividing 24 by 8 is 3. So 3 days are needed if 8 people can work.
Interviewer: What does the number 24 mean? S-4B: Completing the given task.
However, even the students in Level 4 often relied on drawing pictures or reasoning additively. These strategies were sometimes helpful to understand or solve the problems, but in other times they were a barrier to reason proportionally.