ANÁLISIS MARGINAL

The pairs EP-L and CP-L were clinically interviewed while working on both the pretest and the posttest. The researcher arrived to some inferences in relation to their

70 problem solving abilities. The analysis is presented based on the Qualitative Comparison table of the low and limited achieving groups (Appendix I) for the four assessed abilities.

4.2.2.1 Ability1: Understands the Problem

The qualitative analysis regarding the first ability revealed that EP-L showed more progress than CP-L. In general, EP-L’s ability to “understand the problem” improved. The group’s ability progressed from “unable to interpret enough given information to get started or make progress”, in the pre-equation problem, to “able to interpret all necessary information needed for a complete and correct solution”, in the post-equation problem as revealed in the figures 4 and 5 below.

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Figure 4: pre-equation EP-L Figure 5: post-equation EP-L

Figure 4 shows EP-L’s interpretation of the given information in the pre- equation problem. This problem requires students to find the distances travelled by each of four friends, when the total distance is given as well as the relations between different distances. When asked to justify their interpretation of the information, EP-L explained,

71 “It’s given that Mohammad’s distance is one half as Lara. So we can tell that it’s equal to half, and then we will double Lara’s distance.” Clearly, the group misunderstood the condition given on Mohammad’s distance to school, and did not realize that the distances need to be written in terms of one another. However, in the post-equation problem (Figure 5), where the students are given the number of cows given in every two adjacent pastures, and the task is to find the number of cows found in each pasture separately, the students in this group explained, “we know that the number of cows in the first cube, which is the pasture, and the second cube is 29”. When prompted to give further explanation about what this means, the students provided an example which demonstrated that EP-L completely understood the given situation. The following is an excerpt from the interview conducted with this group.

- for example, if in cube 1 we have 16, then we will have 13 in cube 2….but we also need to keep in our mind, that if cube 1 has 16, then cube 3 will have 25-16=9 cows.

- Interviewer: ok.. And can we have this?

- Yes, we can check… but no, 9 is less than 10…. And in the problem they specified that we need to have more than 10 cows.

On the other hand, in CP-L, the same ability regressed, where the members were “unable to interpret enough given information”, in the post-equation and post-pattern problems, after being able, in the parallel problems of the pretest, “to interpret most given information necessary to start the solving process. Figures 6 and 7 below shows this relapse which occurred between the pretest and the posttest. Figure 6 shows CP-L’s interpretation of the situation in the pre-equation problem, which is correct. The group explained, “Mohammad’s distance is one half Lara’s so whatever Lara’s distance is, Mohammad will be half. For example, if Lara’s distance is 20 Km, then Mohammad’s distance is 10 Km, and Sam will be 60 Km.” The group only overlooked the given about the total distance.

72 Figure 7 shows the interpretation of the given information in the post-equation problem, which is wrong. When asked why they had 5 pastures in their solution, the members of C P- L explained, “we have 5 numbers given, they must be the number of the cows in the pastures.

Figure 6: pre-equation CP-L

Figure 7: post-equation CP-L

EP-L also showed some progress in the post-pattern and post-reasoning problems by being “able to interpret some given information to solve part of the problem or to get part of the solution”. On the other hand, CP-L maintained the same level of being “able to interpret most given information” in the pre- and post-reasoning problems.

73 4.2.2.2 Ability 2: Uses information appropriately

Upon comparing the ability to “use information appropriately”, EP-L showed more improvement than CP-L, where the progress in this ability was reflected in two out of three problems with EP-L and in only one problem with CP-L.

EP-L was able to show good improvement, in the post-equation problem, where the group was able to “use all appropriate information correctly” after not being able to “use appropriate information correctly” in the pre-equation problem. Figure 8 shows EP-L’s use of information in the pre-equation problem. In the pre-equation, the only information that was used correctly was Sam’s distance, where members of EP-L decided to multiply Lara’s distance by 3 as specified by the problem; however, all other information was misused.

Figure 8: pre-equation EP-L

On the other hand, in the post-equation problem, EP-L was able to “use all appropriate information correctly” regarding the number of cows found in each two adjacent pastures while solving. Figure 9 shows how the students were using the information correctly.

74 Figure 9: post-equation EP-L

The following is an excerpt from the interview conducted with EP-L while working on the post-equation problem.

- Interviewer: Can you explain to me what these calculations are?

- We are trying to find the number of cows in each of these cubes, so we are picking a random number, and checking if it works with the other numbers.

- Interviewer: Can you tell me how this is working with you?

- We are going to start with 15 cows for the cube here (pointing to the top left corner)… we do 29-15=14 in this cube (pointing to the top right), also 25-15=10 in this cube (pointing to the bottom left), then the 28-10=18 in this cube (pointing to the bottom right).

- Interviewer: Are you done?

- We can use these to check too… for example here (pointing to the top right with the bottom right) we have 32 in both. We can check by doing 32-18 from the one before to get 14, which is what we got before. I think we are done.

The group did not notice that they made a calculation mistake while subtracting 10 from 26.

The same ability regressed in both groups in the post-pattern problem, but improved slightly in the post-reasoning problem.

4.2.2.3 Ability 3: Applies appropriate representations/procedures

The qualitative analysis showed progress in EP-L’s ability to “apply appropriate representations/procedures” to solve problems. The group was able to “apply completely appropriate representations/procedures” in the post-equation problem, whereas it was

75 unable to do so in the pre-equation problem. The members “applied inappropriate procedures” to solve the post-pattern problem. In the post-pattern problem, the task is to find out the minimum number of tunnels needed to connect each of 30 cells with every other cell. The group was unable to devise an appropriate plan for solving, but they refrained from using the same wrong procedure they followed in the pre-pattern problem. The group members explained, “We can’t divide by 2, even if every two cells are connected, because this is wrong. It means that not all are connected, but we don’t know what else to do.” The group faced a similar task in the pre-pattern problem where the students has to find out the number of games to be played, if six teams are competing such that every team plays with every other team only once. In that problem, members of EP-L were confused between two inappropriate procedures: 1-dividing 6 teams into groups of 2 to get 3 games, and 2- raising 2 to the power of 3.

The group’s work in the post-reasoning problem also revealed some improvement in their ability to “use appropriate representations/procedures”, though only slightly in this case. In the post-reasoning problem, students are given some clues to try and find out the number of dogs found in different kennels. EP-L made a diagram (Figure 10) which represented the different colored kennels. The group tried to represent differently colored kennels, where the students actually used different colors to keep track of some of the given information; however, they did not use all information appropriately as the group believed that one of the kennels is red and blue, instead of having two separate kennels for the mentioned colors.

76 Figure 10: post-reasoning EP-L

In comparison, CP-L was not able to “use appropriate procedures” in the post- equation and post-pattern problems, where in the post-pattern problem the group divided the 30 cells by 2 to get 15 tunnels. However, the post-reasoning problem revealed a slight improvement in the group’s ability to “use appropriate procedures”, where the group used a table to fill in the number of dogs found in each kennel.

It should also be noted that, after the intervention, EP-L developed an ability to rely more on representations while solving. The group used some representations in the post- equation and post-reasoning problems to assist them while solving, while representations were poor in their pretest. Figure 11 shows a sample of the representations used in EP-L’s posttest while working on the post-equation problem.

77 CP-L, on the other hand, did use some representations in both tests; however, the representations were not useful while solving. For example, Figure 12 shows CP-L’s representation while working on the pre-reasoning problem, where the students had to figure out a way to fill 6 liters of water in a drinking tank using only a 5L and an 8L container. The representation is poor and did not help the group while solving.

Figure 12: pre-reasoning CP-L 4.2.2.4 Ability 4: answers the problem

The improvement in the ability to “answer the problem” was more noticeable in EP-L’s work. The group’s ability progressed from submitting a “wrong answer”, in the pre-equation and pre-reasoning problems, to giving “a correct solution with a minor technical error”, and to “a partial answer to the problem”, in the post-equation and post- reasoning problems respectively. CP-L’s ability did not show any progress at all.