Secret Number

Analysis of the cycle 1 suggested little changes from what have been conducted in the cycle. There were two parts of this activity; the first one was playing and exploring a like guess-and-check game, and the second one is making tricks for the game. This activity concerned about the students‟ initial uses of algebraic elements, such as, symbols and equal signs for arithmetic purposes. Apart from it, a brief look at students‟ strategy to solve the problem perhaps also gives insights to strategies that students might have been able to perform.

In general, this strategy went differently from what was planned due to a slip up by the teacher when trying to explore the instructions provided for the game. In this case, she missed an important step (stopping and guessing students‟ secret numbers after the 4^th instruction; this was meant to emphasize the first important number), and swapped the last two steps; as such the final result (F) was the secret number itself, which unintentionally lost the stress for the 2^nd important number (check the lesson plan meeting 1 in the attachment; and the last paragraph of secret number cycle 1). As consequences, the task became less challenging and the emphasis of important numbers became weird to ask. As a solution, additional activity entitled learning to make secret number game was shortly conducted in the beginning of the upcoming meeting (see additional activity in meeting 2 in the attachment).

Students’ strategies

A strategy shown by the students during this activity was an inverse operation.

This strategy is observable in the students‟ typical way to make a secret number game trick (as shown in figure 5.16). In the figure, the students realized that to turn the

numbers back to its origin (after some operations), they need to inverse the operation they have performed. The students‟ arithmetic understandings probably have a great influence in the proposal of this strategy.

Figure 5.16 Students‟ typical strategy to create their own „secret number‟ trick

Working backwards, as appeared in the first cycle, was not shown during the meeting. This might be caused by the similar numbers appeared in the origin (secret number) and the final answer, which stimulated the students to think of the inverse operation.

Students’ representations

Despite the students‟ strategy, the researcher also puts concerns about the students‟ representations, which might support their understanding of algebraic representations, such as the use of equal signs and symbols.

The use of equal sign

Like what was found in the first cycle, all students, except one (in figure 5.17b), seemed to misuse the equal sign when recording the series of operations (see figure 5.17a). In the beginning of the math congress, they did not even realize that what they have written was indeed incorrect. A student argued that the equal sign should have led to the results of operations. In reaction to it, the teacher invited the students to pronounce the first two operations separated by the first equal sign carefully and

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Universitas Sriwijaya loudly, for instance, (from figure 5.17a). After a number of repetitions, some students began to feel strange and realized the mistake they have made. They then agreed that a correct way of stating it was, for example, a downward writing as shown in the figure 5.17b.

Figure 5.17 Students‟ records of arithmetic instructions

The fact that students understood equal sign as a sign of results of certain operations might come from their experience in working with arithmetic. This difficulty (according to Kieran, 2007) is common and tends to avoid students from performing the formal strategy in their initial algebra learning. The secret number activity helps students to treat the equal sign more as a relation rather than as an indication of results. Such an understanding might be helpful for students to later understand the existence of non-numbers element in an equation.

The use of symbols

Another task given in this session asked the students to create and explain their trick of secret number games. From this task, the students‟ ways of writing their secret number in their trick formula became a focus. To ease the analysis, examples of a group‟s answer is provided in the figure 5.18.

Figure 5.18 Students‟ uses of symbols in „secret number‟ activity 𝐴 𝐻 𝐻 𝐻 (Only 5 & 7 can be

For example: 𝐴 substituted with any

𝟕 𝟕 numbers)

Apart from the mistake in their uses of equal sign (in figure 5.18), it is observable that the students have decided to use letters to state their secret number. The „A‟ in their representation might stand for „Angka‟ which means „number‟, and the „H‟ stands for „Hasil‟ which means „result‟. In the upright corner, the students added explanation that perhaps indicates their awareness of generalized number. However, there was no further explanation about this representation during the class. Otherwise, the students probably just treat the number simply as a label.

Another evident that is probably more powerful to confirm the students‟

awareness of the existing of generalized numbers is shown in fragment 5.10. The conversation occurred in the focus group. Although they did not involve any algebraic symbols (or letters), they seemed to realize that the trick they have made is applicable for any numbers (line 84 and 90).

Fragment 5.10 Students realized the generalized numbers 1

2 3 4 5 6 7 8 9 10

Group 1 Amel Aldi Badri Badri Teacher Amel Badri Amel

(They tried their trick and found that it worked)

Would you try other numbers? Up to you. Any numbers.

(look convinced; she challenged her friends) How many instructions are there?

Four. 1, 2, 3, 4.

(Recheck with another number)

Would it be right if… if the secret number is not two?

Yes. Correct. Let‟s try.

(Try with 3)

Try zero. The easy way usually is 0.

Concluding remarks

As briefly explained in the beginning of this session, there was incident in the beginning of the session. This incident confused not only the students but also the researcher and the teacher of what to do. However, after some discussions, the class can be brought back normal.

However, in general, the students were still enthusiastic with the game and felt challenged to find out how the trick worked out. The goals of the activity were all observable, though with some differences with what we had in the cycle 1. Overall, the researcher is contented with the value of this activity as a good way to discuss the

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Universitas Sriwijaya students‟ misunderstandings of equal signs or to stimulate the students‟ uses of symbols for arithmetic purposes.