Metacognitive Activities in a Conjecturing Process toward Problem Solving of the Pattern Generalization
3 Results And Discussion
Based on the students answer sheet dealing with the problem-solving of patterns generalization and the interview conducted toward 6 subjects which resulted in sym-bolic manner, we acquired some data that metacognitive activities existed in conjec-turing process while the students were accomplishing the task, including awareness regulation, and evaluation. The type of metacognitive activities in such conjecturing process could be seen in the following table 1.
Table 2. Metacognitive Activity in Conjecturing Process Subject Metacognitive activities in the stages of conjecturing process
1 2 3 4 5 6 7
S1 A - A A, R, E E - -
S2 A A A, R, E E - -
S3 A - A A, R, E E - -
S4 A - A A, R, E E - -
S5 A - A A, R, E E - -
S6 A - A A, R, E E - -
Note:
The stages of conjecturing process were: (1) Observing case; (2) organizing case; (3) searching for and predicting patterns; (4) formulating the conjecture;
(5) validating the conjecture; (6) generalizing the conjecture; (7) justifying the generalization
Metacognitive activities: A=Awareness, R=Regulation, E=Evaluation
Among 6 subjects of this study, 2 subjects were described: S1 and S6. The descrip-tion was related to metacognitive activities in conjecturing process toward the prob-lem-solving of patterns generalization as follows.
3.1 Subject S1
In generalizing patterns, S1 had recognized that 1st, 2nd, and 3rd picture formed a pattern. For finding out the general formulation about the number of boxes in the-n picture, S1 initially observed and counted the total boxes without discerning the black and the white ones. The following was a part of interview conducted towaed S1 and the metacognitive activities done by the subject.
P 03 What did the first thing come into your mind when you read this task?
S103 The difference, Sir
P 04 What did you mean by the difference?
S104 I meant the difference of the pictures displayed, Sir. The first one was 7, the second was 11, and the third was 15 (by pointing out the boxes). So, the dif ference was 4, the next one was also 4, and so forth. (Awareness)
Based on the number of 1st, 2nd, 3rd picture, S1 organized the case by ordering a se-ries of numbering respectively. Then, S1 searched for and predicted patterns by view-ing the difference between 2nd and 1st picture, 3rd and 2nd picture; and then thought that the next picture should be added by 4, respectively. This was emphasized by a part of interview toward S104 and the students’ problem-solving task as follows.
Fig. 2. The result of the task accomplished by S1
For formulating the conjecture, S1 viewed the addition between 1st and 2nd picture was 4, as well as between 2nd and 3rd picture. Viewing such addition, S1 formulated the conjecture to determine the number of boxes within the-n picture . S1, then, validated the conjecture by viewing the conformity of the number of boxes with-in 4th and 3rd picture, and mentioned that formulation of its-n was false. The following was a part of interview conducted toward S1.
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P 07 Ow, so what did you mean by this? (Pointing out the result) S107 It was false. It was mistyped.
P 08 what did you mean?
S108 this should be added by 4, added by 4 again, added by 4 again, and so forth.
But, when we took a look one more time, it seemed to be false (Awareness).
Itended to see the-n as 4, in respective manner, so it would be 8. It would be
different if we took the-n as 3 then added by 4, which resulted in 7, thus it became false (Evaluation)
After realizing that the conjecture formulated was false, S1 tried to apply a new strategy to formulate the conjecture, which was searching for the initial number be-fore it was added by 4 (as the pattern was always added by 4). S1 searched for the initial number by reducing the number of boxes in 1st, 2nd, and 3rd picture. It was 4 boxes reducing for each picture. The result was 3, 7, 11 respectively. S1 realized that the initial number sought had not been appropriate yet since it was still different. It could be shown by a part of interview conducted toward the students in accomplish-ing such problem-solvaccomplish-ing task related to patterns generalization as follows.
P 09 ow… I see. If you knew that it was false, (while pointing out the result), then, what did you immediately think after that?
S109 I attempted to find out which initial number before it was added by 4 (Regu-lation)
P 10 What did you mean?
S110 for instance, this 2nd picture had 11 boxes, it was 7 before it was added by 4, thus it was fail (Evaluation)
P 11 What did you mean by fail?
S111 the initial number for each picture was 3, 7, 11. It indicated that it was false (by pointing out his result)
Fig. 3. The result of the task accomplished by S1
S1, then, applied the new strategy, searching for the initial number before it was added by 4 times n. for 2nd picture, so, , for 3rd picture, , so became the initial number before it was added by 4 times n was 3. After finding out the initial number, S1 formulated the conjecture of general formulation: and validated the conjecture based on the number of boxes that had already recognized. This could be shown from a part of interview conducted by the students and their task accomplishment as follows.
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P 11 So, what was your next step?
S111 I searched the initial number before it was added by 4 times n. 2nd picture was equal to , so . And 3rd picture was equal to , so . Hence the initial number before it was added by 4 times n was 3. That is, the general formulation was (evaluation) P 12 4 times n, what did you mean by that?
S112 let’s see, the difference between the initial number and 7 was 4, the differ-ence between 7 and 11 was 4, the differdiffer-ence between 7 and 15 was 8. That is, the range from 7 up to 15 was the two-ply of 4, the difference between the in-itial number and 15 was the triple of 4. Thus, it was the third of n.
Fig. 4. The result of the task accomplished by S1
S1 justified the generalization with intention to ensure people that the conjecture revealed was true by particular examples. S1 counted the number of boxes within 4th picture: and 3rd picture also had similar result after it was added by 4. Based on the example, S1 justified the generalization revealed. This could be shown by the following part of interview:
P 15 Ok. So, how could you explain to others that this formulation revealed was true?
S115 I would show the conformity among 1st, 2nd, and 3rd picture, and since it had been correct if it was used for counting the 4th picture:
. 3rd picture would have similar result when it was added by 4. It indicated that the formulation was true.
3.2 Subject S6
In generalizing patterns, S6 had realized that 1st, 2nd, and 3rd picture fomed a pat-tern. In order to reveal the general formulation of the total boxes in the-n picture, S6 firstly observed and counted the number of black boxes and white boxes separately.
The following was the part of interview toward S6.
S607 I counted this one by one. So, how many the black ones were, how many the white ones were, how manya both the black and the white were (by pointing out the pictured patterns) (Awareness)
P 08 What did you mean?
S608 Let’s take a look; the 1st picture had 3 black boxes, and 4 white boxes. The 2nd picture had 5 black boxes and 6 white boxes. The 4th picture had 5 black
boxes and 8 white boxes, and so forth…
Based on the total boxes within 1st, 2nd, and 3rd picture, S6 organized the case by ordering the pattern of numbering series. S6, then, searched for and predicted patterns by viewing the difference between 2nd and 1st picture, 3rd and 2nd picture. The differ-ence was 2 for each black and white box. S6 also considered the next object, 4th pic-ture. This was emphasized by the interview conducted by S608 and the students result of the given task as follows.
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Fig. 5. The result of the task accomplished by S6
In order to formulate the conjecture, S6 viewed the relation between 1st picture and the number of black boxes added by the white ones within the picture, 2nd picture and the number of black boxes added by the white ones within the picture, and so forth.by viewing such relation, S6 formulated the conjecture to determine the number of black boxes within the-n picture n , for the white ones, . S6, then vali-dated the conjecture by viewing the conformity of the number of boxes within 1st, 2nd, and 3rd picture, and counted 4th, 5th,… up to 100th picture. The following was a past of interview conducted toward S6 and the students’ task accomplishment.
S614 Hemppp..what I meant is…we needed to find out the-n pattern, don’t we?.here, we needed to count the furthest pattern as well, such as tens, hun-dreds, thousands, don’t we? (awareness). Thus, we needed to see the rela-tion among each picturethat’s why, I concluded this , 6+1=7.
, 6+2=7 so, here, if 2nd picture is , 4+1=5, this would be , 4+2=6 (Regulation)
P 15 Then, how was the next?
S615 Let’s try to find out 4th picture.
P 18 Why were you so sure with your answer?
S618 Because I had counted 4th and 5th picture. Then, I fitted to 1st, 2nd, nd 3rd pictures. I did also count the furthest as well, like 100th and the result was true (Evaluation).
Fig. 6. The result of the task accomplished by S6
S6 justified the generalization with an intention to ensure people that the conjecture revealed was true by explaining how to reveal the formulation and to count the num-ber of boxes as what had been done while validating the conjecture. This could be shown by this following interview.
P 19 Ok. So, how did you explain to people that the formulation revealed was true?
S619 I would explain how I got this and counted the number of boxes, both black and white, like this (by pointing out his result).
P 19 Ok. Terus bagaimana cara nya adik menjelaskan pada orang lain bahwa rumus yang dihasilkan ini benar.
S619 Saya akan menjelaskan bagaimana cara saya mendapatkan dan menghitung jumlah persegi hitam dan putih seperti ini (sambal menunjuk hasil pekerjaannya).
At the phase of pattern generalization, S1 and S6 had realized that 1st, 2nd, and 3rd pictures constructed a pattern which they got its materials of the pattern when they were 7th graders. This finding was in a line with the Wilson and Clarke’s (2002) description of metacognitive awareness that awareness involved (1) What individual knew (knowledge for particular task, relevant mathematics insight, personal problem-solving strategy; (2) certain condition Where individual were in the process of prob-lem-solving; (3) What to do in need, what have done, or what could do. Furthermore, in the phase of formulating the conjecture, both S1 and S6 conducted metacognitive regulation. S1 tried new strategy to formulate the conjecture: finding out the initial number before it was added by 4, thus, he/she reveal the general formulation: 3 + (4 x n). S6 formulated the conjecture by viewing the relation between the number of black boxes and white boxes. By considering such relationship, S6 formulated the conjec-ture to determine the number of black boxes in the-n picconjec-ture: n= 2n + 1, and for the white boxes: n = 2n + 2. The strategy employed by S1 and S6 were in accordance to Wilson and Clarke (2002) that metacognitive regulation referred to individual’s in-sight of selecting and employing the strategy, including how and why they used par-ticular strategy. Magiera and Zawojewski (2011) also suggested that indicator of met-acognitive regulation included: (a) making a plan, (b) planning a strategy, (c) select-ing particular problem-solvselect-ing strategy, (d) alterselect-ing the way which had already been conducted.
Going to the next stage, validating the conjecture, S1 and S6 conducted met-acognitive activity that was evaluation. In this phase, S1 viewed the conformity of the total boxes at 3rd and 4th picture. After conducting such evaluation, S1 concluded that the formulation of the-n was false. S6 validated the conjecture by viewing the con-formity of the total boxes at 1st, 2nd, and 3rd pictures, and then counted the total boxes at 4th, 5th …up to 100th picture. Validating the conjecture conducted by S1 and S6 was intended to see the conformity of particular objects in order to determine the truth of the conjecture revealed. Magiera and zawojewski (2011) also suggested that the indi-cator of metacognitive evaluation included: (a) the problem-solver checked the an-swer. (b) Assessed the result, (c) assessing whether the result was true, (d) assessing whether it had been well-conducted. Validating the conjecture was conducted by viewing the conformity of particular objects in order to determine the truth of the conjecture revealed.