English as a Second
3 Findings and Discussion
According to the students’ answer sheets and the interview conducted toward all 65 subjects, those resulted indicator of conjecturing process in problem-solving of pat-tern generalizing as shown in the following table 1.
Table 1. Indicator of conjecturing process THE STAGES OF
CONJECTURING PROCESS
INDICATOR
Observing the case Observing and counting the number of objects construsting the pattern such as:
Observing and counting the number of the square at the down-left side up-right, the primary, the down-left and right without discer-ning the Black and White box .
Observing and counting the number of Black and White box separately
Observing and counting the number of the box without discer-ning the Black and theWhite box .
Organzing the case Implementing strategy to set the object constructing the pattern automatically, such as :
Writting the pattern of numbering series
Listing or labelling in order to link 1st number and 1st picture , 2nd number and 2nd picture , 3rd number and 3rd picture, and so forth.
Searching for and predicting the pattern
Observing certain objects both organized and non-organized ones, and considering the next objects which had not been recognized yet, such as:
Counting the différentiation among the 1st, 2nd, 3rd picture, and considering the next objects.
Counting the distinction among the White box of picture 1, 2, 3 and considering the next objects.
Writing down the symbols indicating the common pattern such as the underline, the round, or other forms, and counting the distinction of the 1st, 2nd, 3rd picture, also considering the next objects.
Formulating the con-jecture
Making a statement dealing with all possibilities of the case, based on the empical evidence, but had not been validated yet, such as:
Stating the pattern or fomulation of the-n picture which was valid
Stating the-n formulation of the Black boxes 2n + 1, the White boxes , 2n = 2, and the-n formulation of the Black and the Whi-te boxes :
Stating the general formulation or the number of the-n box
Validating the conjec-ture
Validating the conjecture by considering the conformity of particu-lar objects in order to determine the truth of the conjecture revea-led, such as:
Validating particular case in order to determin the truth of conjecture revealed, for instance, for the 1st, 2nd, 3rd picture, and …
Skhetching the next object representing the next pattern which might still be captured in order to determine the truth of the conjecture revealed, for instance, for the 4th, 5th, 6th picture, and so forth…
Generalizing the The changing conviction related to the conjection revealed that it
conjecture prevailed generally, such as:
Convincing the pattern or the formulation of the-n picture which pervailed in general.
Convincing the-n formulation of the Black square , White square and the-n formulation of the Black and White boxes which pervailed in general.
Convincing the general formulation or the number of the-n square which pervailed in general.
Justifying the genera-lization
Reasoning which explained the generalization of the conjecture that aimed to ensure people which its generalization was true, such as:
Justifying the generalization based on particular cases.
15 of 65 students who had accomplished TPGMP successfully constructed the con-jecture in symbolic manner. They could be classified into 3 categories based on the observation of the case, which was the basic of constructing the conjecture, as shown in the following table 2.
Table 2. Subject Classification based on Observing the case
OBSERVING THE CASE SUBJECT
Observing and counting the number of the square at the down-left side up-right, the primary, the left and right without discerning the Black and White boxes ..
S3, S4, S5
Observing and counting the number of Black and White boxes separately
S6, S7, S8, S9, S10
Observing and counting the number of the box without discerning the Black and theWhite boxes .
S1, S2, S3, S11, S12.
S13, S14, S15
Based on table 2, the conjecturing process dealing with the problem-solving of pat-tern generalization would be analyzed and discussed which were represented by S3, S6, and S1. The results of the analysis and the discussion were based on the stages including: observing cases, organizing cases, searching for and predicting patterns, formulating a conjecture, validating the conjecture, generalizing the conjecture, justi-fying the generalization, as follows.
3.1 Observing cases
In generalizing the pattern of Subject S3, S6, and S1, it had recognized that 1st, 2nd, and 3rd picture formed a pattern. In order to reveal the general formulation of counting the number of square in horizontal and vertical parts without discerning the Black and the White boxes of the 1st, 2nd, and 3rd picture. The following was a part of interview toward S3:
P 04 : What did the first thing come into your mind when you read this task?
S3 04 : I initially saw this (by pointing out the pattern of the boxes). The horizon-tal were 5 and the vertical were 2. Then, the one which lied under was 1 and which lied up was also 1. So, it must be only 2 standing.
Subject S6 viewed the case by observing and counting the number of the Black and the White boxes of the 1st, 2nd, and 3rd picture, separately. The following was a part of interview toward S6.
S607 I counted this separately. I counted the number of the Black and the White (by pointing out the pictured pattern)
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…
Subject S1 viewed the case by observing and counting the number of the box with-out discerning the black and the white ones of the 1st, 2nd, 3rd picture. The following was a part of the interview toward S1.
P 03 What did the first thing come into your mind when you read this task?
S103 You meant the difference, did you?
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 difference was 4, the next one was also 4, and so forth.
3.2 Organizing cases
Based on the number of 1st, 2nd, 3rd picture, S3 organized the case by making a list in order to link 1st number and 1st picture; 2nd number and 2nd picture; and 3rd number and 3rd picture. It was also emphasized by the students’ answer dealing with the given problem-solving task of pattern generalization as follows.
Fig. 2. The result of the task accomplished by S3
S6 organized the case and ordered the pattern of numbering series for the black boxes (3, 5, 7, 9) and the White ones (4, 6, 8, 10). It was also emphasized by the
stu-dents’ answers dealing with the given problem-solving task of pattern generalization as follows.
Fig. 3. The result of the task accomplished by S6
S1 organized the case by ordering the numbering series of the black and the white boxes which were 7, 11, 15. It was also emphasized by the result of the task done by S1as follows.
Fig. 4. The result of the task accomplished by S1
3.3 Searching for and predicting patterns
In searching for and predicting the pattern, S3 counted the difference between 2nd picture and 1st picture, 3rd picture and 2nd picture, and considered that the difference for the next picture was 4 boxes. It was also emphasized by the part of interview to-ward S307, S308, and the result of the task accomplished by S3 as follows.
S307 For instance, given 1st pictures had 5 boxes, or the horizontal were 5, the vertical were 2. So, if we counted them all, then it would be 7 boxes. Next, given 2nd picture, if we added these 5 horizontal boxes with 2 others, then it would also be 7. So, (while pointing out the boxes of the pictures) if this was added by this then it would also be 7. If we took a look the initial ones, there was 1 box for the first and 1 box for the second. So, there must be 2 boxes for the initial of each picture. In sum, 2 added 2 was 4, 7 added 4 was 11, its number were similar to the 3rd picture. Thus, the difference among each picture was 4 boxes.
P 08 What did you mean by 4 boxes?
S308 Let see, 1st picture had 7 boxes, 2nd one had 11, and 3rd one had 15, and so forth. Thus, the difference among each picture was 4 boxes, 11-7=15-11=4.
S6 searched for and predicted the pattern by counting the difference between 2nd picture and 1st one, the difference between 3rd picture and 2nd one. The difference between each black was 2 and the white one was 2. S6 also considered the next ob-ject, 4th picture. It was emphasized by the part of interview toward S6 as follows.
S610 The difference between 1st picture, 2nd picture and 3rd picture.
P11 What did you mean?
S611 As like the difference between the black boxes, I wondered about the number of its difference. If it was 2, tha it was also 2 for the white (by pointing out his answer)
S1 searched for and predicted the pattern by looking into the difference between 2nd picture and 1st picture, 3rd picture and 2nd picture, then considered that the next picture would be added by 4, and so forth. It was emphasized by the part of interview conducted toward S104 and the result of the task done by S1 in 4th picture.
3.4 Formulating a conjecture
In formulating a conjecture, S3 viewed the pattern of horizontal and vertical boxes in 1st, 2nd, 3rd picture. S3 formulated the conjecture in order to determine the number of boxes of the-n picture that equaled to (5 + n) + (2 + n). Then, S3 validated the con-jecture by viewing the conformity toward the number of the box in 9th picture. S3, furthermore, stated that the-n formulation was false. After considering that the conjec-ture formulated was false, S3 then tried to formulate the new one: (5+(n-1))+(2+(n-1)). Moreover, S3 made a validation by viewing the conformity of the formulation dealing with 2nd picture, then stated that it was also false. S3, then, realized that only one of the additions from both horizontal and vertical boxes (n-1) was proper. Since both them were joint, (n - 1) should be multiplied by 2. Thus, the general formulation was . S3 validated the conjecture revealed by counting 1st 2nd, and then viewed the conformity of its formulation toward 4th and 5th picture.
Fig. 5. The result of the task accomplished by S3
S6 formulated the conjecture by viewing the relation between 1st picture and the number of black boxes added by the white ones in 1st picture, 2nd picture and the number of the black boxes added by the white ones, and so forth. By looking into its relation, S6 formulated the conjecture in order to determine the number of the black boxes in the-n picture , for the-n of the white ones . Then, S1 vali-dated the conjecture by viewing the conformity of the total boxes in 1st, 2nd, and 3rd picture. S1 also counted the number of boxes in 4th, 5th, up to 100th picture. The fol-lowing was the result of the students’ task dealing with problem-solving of pattern generalization.
Fig. 6. The result of the task accomplished by S6
In order to formulate the conjecture, S1 viewed that the addition between 1st and 2nd picture was 4 boxes, and between 2nd and 3rd one was also 4. By realizing such addition, S1 formulated the-n formulation of the conjecture to determine the number of boxes for the-n picture = n + 4. S1, then, validated the conjecture by viewing the conformity toward the total boxes of 4th and 3rd picture, then stated that its n formula-tion was false. It could be seen from the following interview.
S108 It was added by 4 boxes, then, it was added again by 4 boxes, and so forth.
But, if we took a look one more time, I preferred that the-n was 4 respective-ly, thus it could 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.
After realizing that the conjecture formulated was false, S1 tried a new strategy in order to formulate the conjecture. The strategy was by searching for the initial number before it was added by 4 (for the patter 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 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. Furthermore, S1 applied the new strategy by searching for the initial number before it was added by 4, multiplied by n. for 2nd picture, 2 , so x = 3, for 3rd picture, , so x = 3. Thus, the initial number before it was added by 4 was 3.
After finding out the number, S1 formulated the conjecture of general formulation:
, and validated the conjecture based on the number of boxes which had been found. This was shown by some parts of interview and the following task ac-complished by the students.
Fig. 7. The result of the task accomplished by S1
3.5 Validating the conjecture
In validating the conjecture, S3, S6, and S1 validated it with particular case in or-der to determine the truth of the revealed conjecture, by viewing the conformity of the formulation toward 1st, 2nd, and 3rd picture. They also sketched the next object which represented the next pattern that might still be reachable to determine the truth of the revealed conjecture, such as for 4th, 5th, and 6th picture.
3.6 Generalizing the conjecture
After validating the conjecture, S3, S6, and S1 believed that the-n pattern of the black boxes: , the-n pattern of white box: , and the-n formulation of both black and White boxes: , were generally valid.
3.7 Justifying the generalization
S3 justified the generalization with intention to ensure people that the conjecture revealed was true by demonstrating particular example as what had been demonstrat-ed while validating the conjecture. It could be shown by the following part of inter-view.
P 15 So, how to explain to others that this formulation revealed was true?
S315 I would show the examples, Sir P 16 How could it be?
S316 For instance, given 1st picture: . It was true. Given 2nd picture: it was true. Given 3rd picture: , it was true,given 4th picture: , it was true and so forth.
S6 justified the generalization with intention to ensure people that the conjecture revealed was true by explaining how to reveal the formulation and counting the total boxes as what had been conducted while validating the conjecture. It could be shown by the following part of interview.
P 19 So, how to explain to others that this formulation revealed was true?
S619 I would explain how I revealed and counted the total number of the black and the white boxes (while pointing out his answer).
S1 justified the generalization with intention to others that the conjecture revealed was true by particular examples. S1 counted the total boxes of 4th picture:
, and 3rd picture had 19 as well. According to such examples, S1 justi-fied the generalization revealed. This could be shown by the following part of inter-view:
P 15 Why could you be so sure with the answer?
S115 Because if it was taken into account to search for 4th picture, then it must have been appropriate: . 3rd picture also had 19 for the result. Thus, it was true for adding 4 boxes within each picture.
In the stage of observing the case, there were 3 possibilities which would be con-ducted by the subject: observing and counting the number of boxes within the part of down-left, up-right, the primary, the left and right without discerning the black and the white boxes; observing and counting the number of the black and the white boxes separately; observing and counting the number of boxes without discerning the black and the white boxes. This was appropriate to Gestlat’s theory dealing with observa-tion: the closeness order, the closed-ness order, and the similarity order (Wertheimer, 1923). Furthermore, during the stage of organizing the case, the subject listed and labeled to link 1st number with 1st, picture 2nd number and 2nd picture, 3rd number and 3rd picture, and so forth. They also wrote the pattern of numbering series which re-ferred to activities that might ease them to find out the pattern from the given prob-lem.
Next, during the stage of searching for and predicting the pattern and formulating the conjecture, each subject gave a representation in various ways but meaningful for them along with their initial insight. This was appropriate to what were suggested by Steinbrig and Yerushalmy (2008) that the symbol of mathematic was the tools for codifying and describing the insight, and also communicating the mathematic insight.
The subject successfully accomplished such stages.
At the stage of validating the conjecture, the subject validated the conjecture by particular object that had be recognized, and then sketched the next object which rep-resented the next pattern which had not been recognized yet, but still reachable in order to determine the truth of the conjecture revealed. During this stage, there some-what that was called as internal validation with the object which had already been recognized, and external validation with the object which had not been recognized yet.
Next, in generalizing the conjecture, there were 50 subjects who generalized the conjecture with non-symbolic manner. Generalizing the pattern was not adequate merely launched the general rule and the pattern order. It was also necessary to launched the general rules with symbol as well. This was in accordance to Caraher and Martinez (2008) who argued that the students did not only used notation or sym-bol, but they also represented and made certain mathematical reasoning, made a summary and generalization by their own way.
The last stage was justifying the generalization. In this stage, the subjects justified the generalization with intention to ensure people that the conjecture revealed was true by particular example whether it had already been recognized or not. In justifying the generalization, the subjects did similar activities as what they did in validating the conjecture.