## Komunikasi Matematika

Seorang siswa pernah mempertanyakan ketidakpuasannya terhadap penjelasan guru matematikanya. “jadi, yang benar yang mana?” pernyataan tersebut dikemukakan oleh si siswa terhadap guru matematika lain yang saat itu menjadi observer guru model tersebut.

Hal tersebut menarik untuk dikaji, kenapa siswa tersebut merasa tidak puas terhadap penjelasan gurunya? Bila diperhatikan kembali, sepertinya ada dua kemungkinan yang menyebabkan siswa tersebut mengemukakan pernyataan tersebut. Kedua kemungkinan tersebut adalah: (1) Guru tidak menyelesaikan permasalahan hingga membuat siswa tersebut memahami apa yang dijelaskannya, (2) Guru tidak memanfaatkan keingintahuan siswa tersebut hingga berakhir pada solusi yang diperolehnya itu benar atau salah.

Sebuah tulisan menarik yang dikemukakan oleh Betsy McNeal (2001, Making Sense of Mathematics Teaching in Real Contexts)

**Example Lesson — January 15**

While half of the class was at physical education, Tilley explained to the remaining 10 students that they were going to do some “adding in their minds.” She worked with half of the group while her assistant teacher worked with the other half. Tilley asked the students to add the numbers she had written on the chalkboard (see Example 1). She wrote the numbers this way in order to allow students to use either horizontal or vertical alignment. Tilley then waited until all students had signaled that they had completed the problem.

Ms. Tilley: Who’s ready to give answers?

Will: 219.

(Ms. Tilley writes 219 on the board, other students signal agreement, except Adam.)

Ms. Tilley: Do you agree, Adam? Or did you get something else?

Adam: (Shrugs) I had 121.

Ms. Tilley: (Writes 121 on board next to 219) Okay, who wants to prove?

Will: (Says quietly to Adam who is next to him) [Did] you leave off the 132?

Adam: (to Ms. Tilley) I mean, *two *hundred twenty-one. (She erases 121 and writes 221.)

Tilley recorded all answers with the intention that the students would figure out the reason for the discrepancies as they “proved” their answers. The children are accustomed to this expectation and have learned that their answers will be accepted, so usually give their answers even when they may be wrong. She encourages her students to talk with each other about their work and so was pleased that Will addressed Adam directly as he tried to figure out the discrepancy between Adam’s answer of 121 and his answer of 219. The discussion continued:

Ms. Tilley: Okay, who wants to prove? (Hands up: Ruth, Will) Okay, Ruth.

Ruth: Well, 132…

Ms. Tilley: Let’s see, let me write it. (Writes vertically as shown in Example 2a)

Ruth: Plus 10, from the 12, is 142, is 144 with the 2, then add the 5 from 75, is 149, and then you add the 70 [inaudible].

Ms. Tilley: So you counted by tens? (Ruth nods) So you did plus 10 plus 10 plus 10?

Ruth: (Nods again) And I got 219.

Ms. Tilley: (Writes the tens as in Example 2b) So you got 219.

Will: Can I prove?

Ms. Tilley: (To the group) Okay. Everyone agree with *[Ruth’s] *proof?

(Students signal agreement)

Tilley’s questions indicated her desire to understand how Ruth added the 70. Asking if everyone agreed before considering another student’s “proof” was a reminder to Will of the expectation to check with the others.

The discussion continued with Will volunteering again:

Will: Can I prove it?

Ms. Tilley: You want to prove it, Will? Okay. I just want to go up by those tens, (counts the 7 tens in Example 2b before erasing) Okay, how did you do it, Will?

Will: Okay, what I did was I switched the 32 and the 75, so that’s 175, and then I took the 32.

Ms. Tilley: So you switched that, so it was 175 and 32?

Will: No. Then I took the 30 and put it with the 70 so it was 205.

Ms. Tilley: (Writes as in Example 3a) So you got 205 there, right? So you still had the 2 [in 132] and the 12.

Will: And then I added the 5, the 2, and the other 2 together.

Ms. Tilley: Now, wait, I’m confused.

Will: (Stands up and points) I didn’t use the 5 [in 75] yet.

Ms. Tilley: Oh! You didn’t use this 2 yet. (circles 2 in 132)

Will: I didn’t use that 2, that 2, or the 5 (pointing to 132, 12, and 75, respectively).

Ms. Tilley: And the 5.

Will: So 5, 2, and 2, that’s 9 …

Ms. Tilley: Okay. (Writes as in Example 3b)

Will: And then I knew I had 10 left, so I took that so it was 219. (Ms. Tilley hesitates, not writing anything) Because 10 plus 9 equals 19. (Adam, Elizabeth, Jessica, Ruth give agree signal; Elizabeth’s hand goes up.)

Ms. Tilley: Oh, that 10 there [in 12] you mean! Ah! Okay, oh no the 5 was gone (Fixes Example 3a then completes Example 3b.) Okay.

Adam: [I probably miscounted.]

Ms. Tilley: Did you? You did it the same way, but you lost track of something, do you think? Okay.

Tilley believed students should decide the correctness of answers on the basis of examination of their solution processes, rather than on teacher evaluation. She was pleased that Will felt free to “teach” her by showing her the 5 he did not use. This allowed her to admit being confused herself, signaling to the students that uncertainty and errors are a natural part of learning. When Tilley hesitated, students appeared to take this as an indication that she did not understand what they were saying, rather than that their answer was wrong: They offered clarification for their thinking, rather than immediately changing their answers. Tilley’s response to Adam shows her belief that thinking together can help one recognize one’s error and understand what might have caused the error. Adam’s comment, “I probably miscounted,” shows that he realized his answer was wrong and, listening to the others, speculated as to the reason on his own. Tilley’s response showed she values this thinking.

Tilley then called on two other students to prove. After the group had signaled agreement with each of their solution methods, she turned for a final comment to Adam.

Ms. Tilley: I’m wondering if you left out one of the twos because . . . look, yours is just 2 less than, 2 more than the others (pointing to the 219 and 221). Or if you added one of the twos twice? You think maybe that’s what happened? (small nod from Adam, then Ms. Tilley turns to the others) What do you think, do you think maybe thafs what he did? (small nods from the others).

By returning to analysis of Adam’s answer and inviting group participati Tilley indicated that sorting out errors is part of what is to be figured out by a mathematical community.

As well as illustrating how Tilley’s teaching practice fits with her stated philosophy, these excerpts show how the children were thinking about addition of two- and three-digit numbers prior to instruction in the standard algorithm. Their methods did not begin with adding the ones’ digits neither did their strategies always move sequentially from hundreds, to tens, to ones. Although Ruth decomposed the 12 and 75 into tens and ones, she began with the 132 in its entirety. Will, like Ruth, decomposed the addends into hundreds, tens, and ones, but pieced these back together by first using the 100, then some of the tens (70 + 30) to make another hundred, then the ones, and then another ten. Throughout the year, all but one of the children in the class provided proofs of their calculations that combined hundreds, tens and ones in ways that showed a clear understanding of place value. All strategies also showed the children seeking relationships among the addends that would aid them in adding efficiently and keeping track of their total.

The next day, one child, Jane, used a strategy that Tilley thought might provide a useful connection to the standard algorithm. An example of Jane’s strategy is shown in Example 4. Jane added 358 + 471 by first adding 300 + 400 and writing a 7 in the hundreds’ column, then added 50 + 70, erased the 7 in the hundreds’ column and wrote an 8 followed by a 2 in the tens’ column, then finally added 8 + 1 for a 9 in the ones’ column. Jane’s “erasing strategy” appeared to be very close to the standard addition procedure.

Dari petikan proses pembelajaran yang berlangsung antara Tilley dan siswa-siswanya, komunikasi matematika terjadi pada siswa yang berbeda pendapat dengan siswa lainnya. Perbedaan tersebut terinisiasi dari perbedaan jawaban oleh beberapa siswa, momen menarik pun muncul ketika Guru memanfaatkan antusiasme siswa untuk membuktikan ide mereka dengan memperjuangkan solusi yang diperolehnya itu benar atau salah. Tindakan guru pun membuahkan hasil, tampak ketika Adam menyadari kalau ia telah melakukan kesalahan komputasi terhadap bilangan-bilangan tersebut. Kesadaran Adam adalah dampak dari suatu pembelajaran, komunikasi matematika telah membuat mereka memahami suatu konsep dan memberikan pembelajaran bagaimana suatu masalah dipecahkan bersama. Artikel ini menginspirasi pembelajaran melalui melalui diskusi, mengeksplorasi pengetahuan siswa, dan memberikan kebebasan kepada siswa untuk memperdebatkan perbedaan mereka untuk mendapatkan jawaban yang benar, adalah aktivitas dalam komunikasi matematika.