DISCUSIÓN Y CONCLUSIONES

students with originally different levels of problem solving abilities?

The researcher relied on both qualitative and quantitative analyses to answer this supplementary question.

5.1.3.1 Conclusions based on the qualitative analysis

To qualitatively determine which groups showed more improvement in each of the achievement groups, the researcher counted the number of problems in which each group showed improvement, for a particular ability. The conclusion is presented in the following sections based on each of the four abilities

The ability to “understand the problem”

Out of three problems, the low achieving pair of the experimental group improved in all problems, while low achieving pair of the control group did not show improvement; the average achieving pair of the experimental group improved in 2, while the average achieving pair of the control group improved in none; both high achieving pair of the experimental and control groups did not face problems in this ability from the beginning of the study, so there was no noticeable change.

99 Thus, in general, it can be determined that the low and limited achieving and the average achieving students of the experimental group benefitted from the problem posing activities.

The ability to “Use information appropriately”

Out of 3 problems, the low achieving pair of the experimental group improved in 2, while the corresponding control pair reflected no improvement; the average achieving pair of the experimental group improved in 3, while the corresponding control pair improved in 2 problems; the high achieving pair of the experimental group regressed in 1, but at the same time the corresponding control pair regressed in 3 problems.

The researcher inferred that the problem posing strategy was more beneficial to the low and limited achieving and the average achieving students of the experimental group.

The ability to “Apply appropriate procedures/representations”

Out of 3 problems, the low achieving pair of the experimental group improved in 2 while the corresponding control pair improved in only 1; the average achieving pair of the experimental group improved in 3, but the corresponding control pair improved in only 2; the high achieving pair of the experimental group regressed in 1 problem and improved in 1 problem, but the corresponding control pair regressed in 2.

The researcher decided that problem posing activities were beneficial to students of different achievement levels of the experimental group with respect to this ability.

100

The ability to “answer a problem”

Out of 3 problems, the low achieving pair of the experimental group improved in 2 while the corresponding control pair did not show improvement; the average achieving pair of the experimental group improved in 3, whereas the corresponding control pair improved in 2; the high achieving pair of the experimental group improved in 1 problem, while the corresponding control pair did not show improvement in any of the problems.

The researcher deduced that problem posing activities were more effective with the limited and low achieving students and to a lesser effect with the average and high achieving students of the experimental groups.

In conclusion, the qualitative analysis which focused on the work of the sample chosen for the clinical interviews revealed the following inferences:

1- Regarding the four assessed abilities, problem posing activities are found to be effective with students of all achievement categories.

2- The effect of problem posing on students’ ability to “understand a problem” and to “use information appropriately” is more evident in the low and limited achieving and average achieving groups.

3- Problem posing actitvities seem to assist high achieving students with their use of procedures and representaions, as well as providing a correct answer to a problem.

101 5.1.3.2 Conclusions based on the quantitative analysis

The stratified quantitative analysis, which assessed the performance of different achievement categories in each of the two groups, is discussed in this section per ability, after which a general conclusion is presented.

“Understands the question”

The low, average and high achieving experimental groups were more successful than the corresponding control groups, with the analysis revealing a significant association between the high achieving experimental and control groups (p-value=0.002), where the students’ ability to “understand a problem improved by 35% in the high achieving experimental group, while those in the control group showed no improvement.

“Uses information appropriately”

The low, average and high achieving experimental groups showed more improvement than the corresponding control groups. It has to be noted that the average and high achieving students improved slightly more than the limited and low achieving students.

“Applies appropriate procedures/representations”

The analysis revealed a slight improvement for the average and high achieving experimental groups over the corresponding control groups.

“Answers a problem”

The analysis reveals a slight improvement for the average and high achieving experimental groups over the corresponding control groups. The analysis also shows a

102 regression for the low achieving experimental group; however, the low achieving control group showed more regression than the corresponding experimental group.

According to the stratified quantitative analysis done on the achievement groups, the researcher reached the following conclusions:

1- Regarding the four assessed abilities, problem posing activities seem to have a positive effect on students of different achievement groups.

2- The high and average achieving students showed more improvement in the abilitiy to “understand a problem” than the low and limited achieving students. 3- In general, problem posing activities had more effect on the average and high achievers of the experimental group than the low and limited achievers. The analysis revealed a more significant statistical association for the improvement of the average achieving experimental group over the correstponding control group (p-value=0.19) than the association for the improvement of high achieving experimental group over the corresponding high achieving control group (p-valule=0.27). However, the researcher decided that the improvement of the high achieving experimental group is educationally significant due to the smaller sample size of the high achievers in comparison to the average achievers.

After crossing the results of the quantitative and qualitative analyses, the study revealed the following conclusions:

Implementing problem posing activities, during the learning/teaching of problem solving, has positive effects on the problem solving abilities of students with different achievement levels, as is reflected by the qualitative and quantitative analyses. However,

103 quantitative analysis revealed that the improvement in the ability to “understand a problem” is more evident with average and high achieving students. On the other hand, the qualitative analysis revealed that this improvement was more evident with limited and low achieving and average achieving students. Since the sample sizes for the qualitative analysis was small, and that for the quantitative was relatively higher, then in the current study, the researcher feels that the quantitative results were more reflective of the effect of problem posing on students’ problem solving abilities. Thus, the researcher will adopt the results of the quantitative analayis to determine which achievement group was affected most by the problem posing approach. In general, the high and average achieving students reflected more improvement in the posttest. Statistically, the average group showed a more significant improvement than the high achieving group; however, due to the considerable difference in the sample sizes of these two groups, the improvement made by the high achieving group is also taken into account.

After taking the limitations that were present in this study into consideration, the researcher concludes that problem posing has a positive effect on students’ problem solving abilities, specially in terms of prompting understanding, flexibility, and creativity.