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.

Future Directions

Can the same be said of school mathematics education? Maybe so, but my concern is how as classroom teachers you can have some small but real influence in shifting that future so that school mathematics education will be of relevance and value to a greater proportion of young people in schools; at the very least those in your classes. If any of what you have read strikes a chord and you feel compelled to reflect upon and change some parts of your classroom practice, then I want to encourage you in that direction. Get together with some like-minded colleagues and begin to explore how to develop mathematics learning that is built upon some of the principles in this book. Please remember that there are no quick fixes and the development of resources and teaching and learning styles takes time, especially if they are a departure from the things that we know best (Andrew Noyes, 2007, Rethinking School Mathematics, Paul Chapman Publishing, SAGE Publications Company, Page 118).

Mathematics and Citizenship

We have seen how mathematics can be used to develop ‘socio-political consciousness’ that goes beyond understanding political aspects of life with mathematics to using that knowledge to generate political engagement. This can happen at many levels and might be simply about having a say in aspects of school life. But at a different level, if young people in school, as a result of some analysis of data or modelling of some social issue, need to do something, for example, engage with teachers, businesses or politicians, then they should be supported in developing this sense of political engagement – even by mathematics teachers! This is Gutstein’s argument but is also, after all, what a useful citizenship curriculum might aim to do. These ideas of reading the world mathematically in a way that might lead to informed action has been described as critical mathematical literacy. Marilyn Frankenstein (2005: 19) outlines four goals for this form of literacy: (1) Understanding the mathematics, (2) Understanding the mathematics of political knowledge, (3) Understanding the politics of mathematical knowledge, (4) Understanding the politics of knowledge (Andrew Noyes, 2007, Rethinking School Mathematics, Paul Chapman Publishing, SAGE Publications Company, Page 104-105).

Mathematics and Society

Many mathematics educators recognize the importance of mathematical modelling in society and in the workplace. Such mathematics often requires that relatively low-level mathematical content be used in a fairly sophisticated way. It might well be that your own use of mathematics outside the classroom includes some kind of modelling processes. The National Curriculum Using and Applying mathematics (Ma1) strand should support the development of such modelling work in the classroom but many mathematics classrooms rarely offer opportunities for the kinds of modeling processes that learners might come across in the future. You would be forgiven for not noticing Ma1 in the NC as it is in the small print at the side of the main learning objectives. However, the processes and skills of using and applying mathematics, in this context for modelling, do not generally get learnt by accident. Well in some sense they might, but if we know that people use mathematics in these sorts of ways then surely it makes sense to spend time teaching and learning such skills. The NC does emphasize the problem-solving process when teaching Handling Data, but in reality most children’s experience is to acquire a set of data representation and interpretation tools and rarely to apply these to a problem of interest – to them! (Andrew Noyes, 2007, Rethinking School Mathematics, Paul Chapman Publishing, SAGE Publications Company, Page 91-92)

Mathematics and Cultures

Mathematics is a cultural human endeavour and this chapter considers how you might explore this through art, design, ethnomathematics, and so on.You will consider how history and culture might be integrated into classroom learning in an attempt to get away from simply seeing mathematics as a functional set of tools or tricks.This will only be a selection of examples with some indication of how they might be used in the classroom. There is much more that could go into this and other chapters but some ideas can only be mentioned in passing and you can explore their potential for classroom use with colleagues.

Teachers who want to develop an alternative mathematics curriculum have often experienced resistance owing to the culturally entrenched views of what mathematics classrooms should be like.Therefore it is important at the outset for you to look at your own classrooms and practices, and consider how they might develop to accommodate different types of mathematical activity. So, if you have not already read the introduction to this second part of the book you should go back and use the questions to think about your own classroom culture.

Much of Part I was looking at the politics of knowledge and I am aware that this is not a strong focus of this chapter. In fact, I bring much of my own cultural heritage and personal dispositions to bear on selecting examples and I am aware that they are not always representative. So, in reading these examples you might consider not only your own relevant knowledge but that of the young people in your mathematics classes. (Andrew Noyes, 2007, Rethinking School Mathematics, Paul Chapman Publishing, SAGE Publications Company, Page 73)

Building a School Culture of High Standards

Just about everyone in education talks about high standards–standards for student behavior, responsibility and thoughtfulness, standards for investment in work and dedication to work, and standards for the quality of work completed.
Educators everywhere share a goal of creating a learning environment which fosters, demands, and celebrates high standards. There is not, however, agreement on what this environment should look like, nor how it is best achieved. There is currently, and has always been, a great deal of debate among educators regarding what models of schooling best support high standards. For all this debate, there doesn’t seem to be much discussion or understanding of where standards originate. Every student in every classroom already carries around a notion of acceptable standards. Where this notion comes from is not clear. Because the origin of standards is poorly understood, I believe schools spend a great deal of time saying they want high standards while doing a great deal to undermine them.
As an elementary school teacher concerned with this issue, I’d like to offer a description of an approach to learning which I think is unusually effective in creating a school environment of high standards. I don’t contend that it’s the only effective approach, but simply one that seems to have worked very well in my classroom and in the public school in which I teach. I hope
through this description to shed light on some of the factors governing standards which I believe are not often considered or addressed in many school environments, or are addressed in a manner which tends to erode standards rather than cultivate them.
It is not an approach I invented; I take no credit for its premises or strategies. It is an approach I learned primarily from other educators, many of whom are fellow teachers at my school.
These teachers have served as models for me and as sources of inspiration over the past twelve years. I have seen this approach used effectively in a number of other schools, public and private, and all four of the public elementary schools in my small school district share much of its philosophy and orientation. Also, some well-known educational movements, such as Foxfire programs and the Process Writing movement, often exemplify its student directed and “student-owned” nature.
My personal contribution to this approach lies simply in my attempt here to present a portrait of a classroom in which it is used throughout the day, and in a particular passion I have in wedding this general approach to an effort to integrate arts into all aspects of learning. In my classroom, I’ve tried to build an environment where art is more than a decoration or supplement
for work, but rather a primary context in which most information is learned and shared. The infusion of arts has, I believe, had a profound effect on student understanding, investment, and standards.
I need to acknowledge that this classroom approach is not an easy one. It demands of teachers a willingness to abandon textbooks as much as possible, to gather and create resources themselves, and to work together. It demands of administrators a willingness to sanction and support teachers in doing this. It demands of everyone in the school the courage to trust children with a great deal of responsibility and autonomy. Many of the
specific strategies used in my classroom and school could not be easily transferred to other school environments, due to differences in size, support, orientation, and structure of the school day. This shouldn’t negate their value, as my goal is to share an approach, a way of thinking, rather than a blueprint for change. My hope is that aspects of this approach may be of some value in any school setting, and that they may spark interest in beginning to restructure classrooms or schools, even on a small scale.
Read more »

WordPress Themes