# Science Math Primary/Secondary Education

### Edited by Marsigit

Doug Hainline:

I believe that the language (including the symbolism) that we use to teach, and do, mathematics, is a factor in how easy it is for chldren to learn the subject.

Both our language, and our symbolic notation, have "just growed", although our notation has been subject to a slightly more rational process of selection.

To take a small example: I would much rather discuss geometry with nine-year-olds, by focussing on the interesting and important features of, say, plane figures, than by having them memorize that a triangle with only two sides equal is an isoceles triangle, but one with three sides equal is an equilateral triangle.

I would rather just say, "Let's look at the smallest number of sides we need to make a pen that would hold in a horse.... see, you can't do it in fewer than three.... now let's look at what we can say about three-sided figures ... how they can differ among each other ... what they all have to have in common... " and then go on to discover that all three-siders can have no sides equal, or two sides equal, or three sides equal. And then to look at the angles that we can have with each of these kinds of three-siders.

I think there are many areas in mathematics where, if we were starting over, we could make things much easier for ourselves and our students, if we could choose our vocabulary and notation.

Two more things -- from a longer list - which irritate me as being unnecessary impediments in learning mathematics:

(1) Conflating identities and equations and relations.

(2) Making the exponentiation operator implicit instead of explicit.

I wonder if others have any thoughts on this issue?

I quite agree that the language of math needs as much attention as the process of math. I discovered that those students who can 'talk math' are the ones who succeed.

I compiled a list of words and phrases that have specific mathematical meanings for pre-algebra, beginning algebra, geometry and intermediate algebra (usually grades 7-10). There are over 2800 words and phrases that a teenager needs to know Vocabulary building activities are an essential part of any math instruction program that wishes to help all students succeed.

Two easy to implement ideas for vocabulary building are math cards and vocabulary sheets. Math cards are index card where students write the definitions of words, formulas to memorize, etc. The students are supposed to review the cards a few times a day. Vocabulary sheets contain words or phrases for which students are to find a definition. I have found that giving no credit for incomplete definitions, or sentence fragments increases the learning the students get from this activity.

One aid to math building activities is All Math Words Dictionary. For more information and discount rates, go to http://www.demcadams.com/allmathwordsdictionary.html

Helen Mason :

Doug, I have the same concern. Many math books use or assume a lot of vocabulary that is not necessary to the understanding of the mathematics. It would be useful to have people sit down and discuss how much vocabulary is necessary at what point. The current vocabulary-heavy topics can make it hard for students with learning difficulties or language processing issues to learn.

Marsigit Dr MA :

I also have the same concern; however, in the case of the role of language in learning math, I prefer to facilitate the students in order they are able to translate and to be translated, to produce and to be produced, to construct and to be constructed, to reflect and to be reflected, to evaluate and to be evaluated, to judge and to be judged.

Guangtian Zhu :

To provide another perspective from different culture, I would like to say that such concern would vanish in some other languages. For example, try this question
"how many sides are there in a

A.
B. C. D.
You don't even need to know Chinese to find out the answer to be C (same as the first character in

Helen Mason :

That would make things easier, Guangtian. Math and science terms in English tend to come from Latin or Greek roots. A century and half ago, educated people (who were upper class males only) learned those languages so might not have had the problems with the terms that students do today. Of course, there was more concentration on rote learning than on understanding back then.

Marsigit Dr MA :

So language has its own cultural context; the context in which the students are coming from. Therefore, I agree with the effort to employ language (mother tongue) to develop math educ. It leads to prove the fail of global software translator. In Indonesian we called Fraction as Bilangan Pecah. If we translate back into English using that translator, will be Broken Number. Ideot!. So language and mathematics can be contextual.

Helen Mason :

The same science and math terms have been employed for more than a century. The advantage of many of the current terms is that they are similar to those in other related European languages, making it easier to communicate between European cultures. For example:
English equilateral triangle
French triangle équilatéral
Italian triangolo equilatero
Spanish triángulo equilátero

As you can see, English, French, Italian, and Spanish all derive a lot of their words from Latin, which is no longer spoken.

David Reid :

It is not only the teachers who need to watch their vocabulary, and it is not just in the individual words that are used. The explanations in most mathematics textbooks for high school that I have used stink, not to put too fine a point on it. I often feel like doing like the English teacher in "Dead Poets' Society" and ripping out the gobbledygook --or at least what is gobbledygook for teenagers who have grown up with Tweets rather than sentences. Because of this language barrier, many mathematics teachers are tempted to just go for the numbers and equations, giving the students the erroneous idea that mathematics is only composed of numbers and equations. Reasoning seems to get lost in there somewhere.

Doug Hainline :

I thank all of you very much for these most interesting comments.

I have purchased your dictionary. I hope it does not contain the error made by a UK school dictionary of mathematics in my possession, which on one page defines a "fraction" as a "number smaller than one", and a few pages later defines pi as "a fraction approximately equal to ...."
I believe the names given to numbers and other mathematical concepts by different languages is a very interesting topic. The fact that the main Oriental languages are more logical than the European languages in their counting names -- and Guangtien has added a new bit of information on this subject -- has for about twenty years been adduced as one of the advantages that chldren learning to count in these languages have. ("ten-one, ten-two, ten-three ..." instead of "eleven, twelve, thirteen..." ) although this has recently been called into question, apparently.

Helen, thank you for your list of triangle-names in other European languages. And I don't think it's only kids out toward the end of the Special Needs spectrum who needlessly suffer because of clumsy, unnecessary vocabulary. I think we all do.

I wonder if anyone who knows other languages -- Arabic, or the languages of the Indian sub-continent -- might be persuaded to list some mathematical vocabulary, with literal translations if possible? What is an "equilateral triangle" [my "all-equal three-sider"] in Hindi? In Farsi? What are the names of the first three or four numbers after ten?

On a related, perhaps deeper matter -- I think that words that sound familiar, which have everyday 'meanings', but which are also used in mathematics, can also be problematic. I am particularly thinking about the way we assign a word-sound to this sign: '='. Surely this sign -- " = " -- is used to refer to at least two, maybe three, rather different ideas. "2 + 1 = 3" and "X + X + X = 3X" has one meaning; in "X + 3 = 5" it has another; and in "y = 3X + 2" it has yet another. (Or perhaps not -- I'm not a mathematician. It just seems that way to me, and I find when I am tutoring, that it appears to help clarify things for my tutees if I point out that these are three different kinds of things.)

David Reid :

Doug-- in mathematics, there are primarily two possible meanings to equality (of course, you can make it mean a poodle if you want, but keeping a bit to convention....): intensional and extensional. The intensional meaning is basically that a=b if for all relations, you can substitute a for b or vice-versa, and the sentences retain the same respective truth values. The extensional one is when you treat your entities as sets (or as being in a one-one correspondence with sets which are elements in a structure isomorphic to your structure, etc.), and then the two entities are equal if they have the same elements (or if their sets with which they are in correspondence...you get the idea). (Then you have a separate definition for urelements if you want to include those.) In any case, no, the meaning of equality will be the same for all the examples you gave, just that in elementary mathematics we leave the quantifiers to be implicit. That is the difference between a sentence with variables and a sentence without them.

Doug Hainline :

David, thank you very much for your comment, which I don't really understand completely at first reading. But I'll think about it, as I find mathematical things are often this way. You have to come back and chew over the ideas.

Perhaps I am expressing myself clumsily, and blaming the innocent equals sign for a misdemeanor whose source is elsewhere.

Here's what I'm trying to say: that X^2 -Y^2 is the same thing as (X+Y) x (X-Y) is a fact that is true for all X and all Y provided they are real numbers. (I don't know about other kinds of numbers, like complex numbers. Perhaps it's true for them too.) You can, and should, learn this fact by heart (as well as knowing how to show that it's true.)

But X^2 -9X + 20 = 0, is only a fact for X = -5 and X = -4.

So perhaps it's not the equals sign, but something else, that I am trying to get at. In any case, it's my experience that young people learning mathematics are not taught that there is a difference between these two kinds of sentence (and the same when they come to functions), and I wonder if this is a barrier to learning to be proficient in using them.

David Reid :

Doug, sorry for being a bit short in my explanations. But your are right that
(x+y)(x-y)=x^2-y^2
(which is correct for complex numbers as well)
differs from the sentence
x^2 - 9x + 20 = 0
but not because of a difference in meaning of the equal sign in both. Rather, there is a difference in the quantifiers (such as "for all x" , "there exists an x such that...", ) that we suppress in schools. That is, the difference of squares reads, more fully,
For all x in D, for all y in D, (x+y)(x-y)=x^2-y^2
(where D is your domain: real numbers, complex numbers, whatever)
The second one is , if posed as a question,
There exists an x in D such that x^2 - 9x + 20 = 0. (The question is then to find all x which will satisfy this sentence.)
If it is posed as a fact as you expressed it, the sentence is
For all x, (x in {-5, -4} implies x^2 - 9x + 20 = 0)
So, that is the difference. Teachers in high school suppress these quantifiers in writing the equations because they are "understood" -- or at least that is the way the teacher sees it. The way the student sees it may be different. Teachers suppose, often incorrectly, that the pupils will understand this difference without having it explicitly explained to them.

Doug Hainline :

Got it! I've just been re-reading about universal and existential quantifiers -- oddly enough, in a little booklet on Language in Mathematics published in the early 1960s -- and I see that this makes perfect sense. I am going to experiment with explaining this to my tutees. (I have to scientific data on this, but my subjective impression is that ideas that seem far beyond school mathematics are actually not all that difficult, in their basic form, if introduced properly.)

But .. what about functions? y = 3X + 2 "looks like" one half of a simultaneous equation. Yet I teach my tutees that it expresses a relationship between X and Y, true for an infinite number of X's and Y's (assuming the domain of X is not restricted).

David Reid :

Doug: there is no problem here, both are true. y=3x+2, as a function, is simply
for all x, there exists a y such that y=3x+2.
[To emphasize that it is a function, you can put
for all x there exists a unique y such that y=3x+2, that is,
for all x there exists a y such that y=3x+2 & for all z (z=3x+2 implies z=y)].
Then, as far as simultaneous equations, in schools we are a bit sloppy: whereas we will use two letters to distinguish between the two functions, eg
f(x) = 3x+2 and g(x) = 4x-4
we then sloppily put
y = 3x+2 and y = 4x-4.
It should properly be
y = 3x+2 and z = 4x-4.
or, more fully, to talk about the two functions superimposed on a single set of axes (where really there are two vertical axes, y and z):
for all x in D (there exists a unique y such that y=3x+2 and there exists a unique z such that z=4x-4).
Then, "intersection" is "and", so putting them together,
for all x in D (there exists a unique y such that y=3x+2 and there exists a unique z such that z=4x-4 and y=z).
or more simply
There exists an (x,y) such that y = 3x+2 & y=4x-4.
Existence is not the same thing as finding it; for that one can use the iota notation, which is a "definite descriptor" (there is debate as to whether it can be called a quantifier, but it can be rewritten in terms of more conventional quantifiers)
(Unfortunately, this forum does not take Greek letters, so "iota" below stands for the Greek letter with that name; "iota x" is read as "the unique x such that....")
(x,y) such that y = 3x+2 & y=3x-4.
And yes, since English and other languages use quantifiers on a daily basis, there is no reason why a patient school child cannot learn them.

I'd like to comment on suggestion of Helen Mason that we reduce the amount of math vocabulary that children have to learn. I say that, given the nature of math, that is possible only to a small degree. All of the vocabulary that exists in math exists for a reason: someone used a word or phrase, borrowed or invented, to express a specific concept. To eliminate a word or phrase would usually erase that concept from math.

The are a number of mathematical synonyms, which have little or no difference in denotation, at least in the middle school and high school settings. Some of these are Abelian and commutative; absolute magnitude and absolute value; accidental sample and convenience sample. Agreeing to use a particular word or phrase would reduce some of the confusions.

However, when one accounts for the multiple forms of English throughout the world, the phrase selection becomes more difficult. For example 'absolute magnitude' is preferred in the United Kingdom, and 'absolute value' is preferred in the U.S.

Even if these differences could be agreed upon, this method would only eliminate a few words or phrases from children's math vocabularies.

Guangtian Zhu's note suggests one way to accomplish this. Many of the words used in English math vocabulary have roots in Greek. To use Mr. Zhu's example, penta- means five. Replacement of the Greek roots would change pentagon to five-gon. One could go a step further and replace -gon with -sides. Then pentagon would become five-sides. And this brings to light a major problem with this methodology.

There are many things, both mathematical and not, with five sides that are not pentagons. Pentagon means specifically, a five sided object with straight sides where the two of the sides meet at each of five nodes. The use of -gon implies everything after the 'five'.

The use of ancient Greek has a specific advantage here. English students really don't care much how the root -gon was used in ancient or modern Greece. The just have to know what it means mathematically. And, it has a very specific, detailed mathematical meaning. So, I say it would be reasonable to rename pentagon into fivegon, but would be unreasonable to call it a fivesides.

Again, I say that reduction of math vocabulary for students in middle and high schools is possible on a small scale, attempting it on a large scale would create more confusion that before.

Doug Hainline :

Thank you very much. This is beginning to become clear to me now.

I have been trying to deal with my tutee's lack of clarity by talking about the equals sign, when actually, I should have been doing a bit of logical analysis. Your explanation of the meaning of functions, and especially of what we should be saying when we have more than one function, was very helpful.

I don't think the idea of universal and existential quantifiers is too difficult for kids doing algebra. I wonder why it's not taken up in mathematics curricula?

David Reid :

Doug, I am not sure which curricula you are referring to, but I will answer what I know of the US curriculum. (I have worked with other curricula as well, but I am less informed about the history of the curriculum development in those curricula.) As far as I understand it, the reason is historical: back in the 1960's, one decided to bring down some of the foundational work from the universities down to the schools, and set theory was introduced at all levels under the name "New Math". But teachers were not clear on how to teach it, the textbook writers were even worse (so of course the teachers followed the textbooks and exams were set accordingly) in that the ideas became downgraded to exercises that were just as much rote exercises as the ones that they were meant to be a conceptual support for. As a consequence, it didn't really form this support, and, time constraints being what they were, many of the basic skills never got learned, leading to a disastrous drop in skills in the 1970's. So the whole "New Math" was chucked overboard, and the baby was thrown out with the bathwater: now there is very little mention of set theory or logic in the US curriculum; what little there is is pitiful, and again the teachers still don't get taught it much, so how can they properly teach it?

David McAdams, I agree that tinkering with vocabulary in the schools is only possible on a limited scale. One reason that you did not bring up is that it is always a problem when a concept is named one thing at the school level and another at the university level: and there is no way the mathematics community is going to change their vocabulary to appease the school mathematics teachers.

Doug Hainline :

The big weakness of trying to make the mathematics curriculum more 'mathematical' is that the curriculum has to be delivered by the existing set of primary [elementary] school teachers, and in many countries, they are not sufficiently trained in mathematics to do anything like teach even elementary set theory. In fact, I am afraid a significant number of teachers at this level are not very well versed in mathematics, don't like it, even fear it, and this attitude is transmitted to their pupils.

I don't think anything can be done about this in short run. I do believe that the internet is going to make it possible for us to do a MUCH better job of teaching, even with weak teachers.

But it is interesting to speculate about what changes there ought to be, and also about what a more rational language for learning and thinking about mathematics would be.

With respect to language, I would certainly follow the Oriental languages and replace 'eleven' and 'twelve' with 'ten-one' and 'ten-two', as well as making the 'teens' into 'ten-three' , 'ten-four' and so on.

I would call plane figures by the number of the lines that made them up: three-liners, four-liners, five-liners, etc. (I wouldn't use the word 'side' because it's ambiguous: is a 'side' a line or a plane? I'd have 'lines', 'faces' and even 'corners' (instead of 'angles').

Of course this is all silly, because it's not going to happen. But at least we should be aware of the ridiculous, confusing hoops we're needlessly making children leap through, by making them have to learn that a ten-two-facer is a 'dodecahedron'. (And the screaming irony is, in Greek, it's almost as it should be: a two-ten-facer, or, rather, a two-ten-flat-thingie-you-might-sit-on.)

WIth respect to logic and set theory -- I don't believe that, at the elementary level, these subjects should be beyond the grasp of, say, 12-year olds.

At the moment, I think we almost teach anti-logic. I recall my granddaughter coming home with a handout informing her that there were four kinds of triangles: scalene, isosceles, equilateral, and right. After I finished banging my head on the kitchen table, I explained that this was like saying that there were four kinds of people at her school: teachers, boy pupils, girl pupils, and kids who played in the school orchestra.

Marsigit Dr MA :

@ Dough: Still I am interested with your statement "The big weakness of trying to make the mathematics curriculum more 'mathematical' is that the curriculum has to be delivered by the existing set of primary [elementary] school teachers, and in many countries, they are not sufficiently trained in mathematics to do anything like teach even elementary set theory. In fact, I am afraid a significant number of teachers at this level are not very well versed in mathematics, don't like it, even fear it, and this attitude is transmitted to their pupils. "

However, my concerns are reflected, to some extend by reversing your notions, as the following "The big weakness of trying to make the mathematics curriculum more "humanized" are coming from the adults (teachers, educationist, and pure mathematicians). In fact, I am afraid a significant number of outsiders (pure mathematicians) behave inappropriately to intervene primary schools teachers, by employing their very simple logic that if their mathematics is okay, then everything (their teaching) will be okay.

No, teaching learning of mathematics is not just as simple as that idea. For me, to be a primary school mathematics teacher, she/he should not be a pure mathematician.

David Reid :

Dr. Marsigit, or someone who has better parsing skills than me, please clarify the sentence, which apparently contains a typographical error (but I am not sure what it is ): "I am afraid a significant number of outsiders (pure mathematicians) behave inappropriately to intervene primary schools teachers," ("intervene" is not a transitive verb, and if it is simply a case of a missing preposition, I cannot figure out which one would make the sentence make sense.)
By the way, there isn't much danger of a pure mathematician becoming a primary school teacher. In fact, the problem is usually in the other direction, as Doug pointed out, with primary school teachers knowing too little mathematics.

Marsigit Dr MA :

@ David Reid: Thanks for the response. Further I wish to know your ideas about the position of language in intuition or vice verse. Do you have any ideas about the role of language in developing mathematical intuitions? Thank.

David Reid :

Er, thanks for thanking me for the response, but you haven't answered the question I put there.
The role of language in developing mathematical intuitions is a very broad and complex subject, involving the question of the relations of the language centers in the brain with the other parts of the brain. Yes, my studies have included this theme, although I am no expert. In any case, any answer I would give in a short paragraph to such a complex question would be superficial. Perhaps you would like to make your question more specific?

Marsigit Dr MA :

@ David, firstly I wish to thank to Doug for his exposing the relationship between math and language. I think that the role of language in developing mathematical knowledge is a central; because mathematics itself is a language (Wittgenstain). Even now, I, you and others are exchanging the language. I wish to say that the language has its intentional and extensional dimensions ( I am not sure whether the terms I used are similar meaning with yours). By intentional dimension,I mean the deepness; and by extensional is the variety. I am aware that for a certain concept (not only in math) I and others have different terms to express. You have intensively posed and mixed the different level of language.

While, we understand, at a common-sense level, that the main purpose of using the language is for communications. The intentional dimension of communication can be translating and to be translated. If we extend this intentional dimension we may find that the role of language is for constructing the life. So I understand the small part of your life by understanding your notions. Further, this rule should meet to every single people. In sum, I wish to say that the language is the life itself. In fact, there is no similar people in the world; so theoretically, the number of language is the same with the number of human being ever life.

In the case of education, if you respect the existence of individual differences, you may aware that there may many peoples outside you who do not understand much of your notions you produced. Here, the role of language is not just as communication, but also for interaction in order to construct individual life. So the awareness and the study of language of a certain people from a certain context in which you may want to interact, is very-very important. That's why Theresia Nunes was very popular with her Street Mathematics in Brasil.This perspective lead to the study of ethnomathematics, supporting possibly by ethnography.

The last point will give us the space and time to do better for developing (students') mathematical intuitions. More than ninety percent of the younger mathematical knowledge are coming from their intuition. By developing the dimensions of the contextual language, ultimately we may conclude that the students are actually their languages. In the contemporary philosophy, even analytically the language can be define as this world.

So I always happy to read any language among mathematical contexts; not only to read but also to develop them. The implication for this kind of understanding may lead the adults (math teachers) to facilitate their students' learning of math in the perspective of their constructing of their life. In order to facilitate their needs, we as adults, to some extent do not need  always to indicate our determinations but to indicate passionate to wait the emerge of their mathematical intuitions. If you meet those criteria you may arrive at the stage of innovative mathematics teaching.

1. Mifta Tyas Laksita Sari
Pend. Matematika A 2013
13301241005

Komunikasi adalah kegiatan menyampaikan informasi melalui pertukaran pengalaman, pesan, atau informasi, dengan pidato, visual, sinyal, tulisan, atau perilaku. komunikasi hal yang sangat penting, salah satunya berkaitan dengan matematika. Matematika adalah pelajaran yang menggunakan gambar dan menerapkan rumus, tanpa komunikasi siswa tidak akan mengerti dengan guru yang dimaksudkan. Jadi antara matematika dan bahasa adalah hubungan yang tak terpisahkan karena tanpa hubungan di antara mereka, proses belajar tidak dapat dicapai.

2. Diah Nuraini Kartikasari
13301241006
Pend. Matematika A 2013

Bahasa merupakan sistem yang terdiri dari lambang-lambang, kata-kata, dan kalimat-kalimat yang disusun menurut aturan tertentu dan digunakan oleh orang untuk berkomunikasi. Sementara itu, matematika pun dapat dipandang sebagai bahasa karena dalam matematika terdapat sekumpulan lambang/simbol dan kata. Dalam pembelajaran matematika diperlukan bahasa dalam bentuk lambang/simbol untuk mengkomunikasikan istilah matematika. Bahasa dan matematika merupakan dua hal yang tidak dapat dipisahkan karena keduanya saling berhubungan. Dengan bahasa lambang/simbol, matematika dapat menyampaikan informasi dengan jelas dan singkat.

3. Devi Anggriyani
16701251023
S2 PEP B 2016

Matematika adalah bahasa. Itu adalah sebuah ungkapan yang menurut saya sangat tepat sebagai salah satu cara untuk menggambarkan seperti apakah matematika. Bahasa merupakan sistem yang terdiri dari lambang-lambang, kata-kata, dan kalimat-kalimat yang disusun menurut aturan tertentu dan digunakan oleh orang untuk berkomunikasi. Dalam tujuan pembelajaran matematika salah satunya adalah komunikasi. Komunikasi adalah kegiatan menyampaikan informasi melalui pertukaran pengalaman, pesan, atau informasi, dengan pidato, visual, sinyal, tulisan, atau perilaku. Dalam matematika siswa diharapkan mampu mengkomunikasikan matematika sehingga matematika menjadi lebih bermakna. Jika kita dalami, matematika itu adalah bahasa yang dibentuk dalam simbol-simbol. Dan simbol itu memudahkan dan menyederhanakan persoalan sehingga lebih mudah untuk diselesaikan.

4. Misnasanti
16709251011
PPs PMAT A

Sejumlah konsep matematika termuat dalam definisi. Menurut Wiggenstein konsep berperan membantu kita untuk memahami sesuatu. Ia berpendapat bahwa definisi merupakan aturan untuk menerjemahkan dari suatu bahasa ke bahasa lain dan setiap symbol yang benar harus dapat diterjemahkan ke dalam bahasa yang lain dengan suatu aturan. Definisi yang telah disepakati akan menjadi dasar komunikasi dalam suatu system matematika. Definisi sebagai dasar komnikasi dalam matematika merupakan unsur bahasa dan berfungsi sebagai alat untuk menghubungkan antara satu bahasa dengan bahasa lain.

5. Johanis Risambessy
16701251029
PPs PEP B 2016

Mengkomunikasikan matematika harus sesuai dengan bahasa yang mudah dipahami. Karena bahasa itu sebagai sarana menjelaskan konsep matematika yang abstrak kepada siswa. Dengan bahasa yang mudah, maka siswa dapat memahami setiap konsep yang dijelaskan oleh guru. Dengan mengkomunikasikan simbol-simbol yang ada menjadi lebih sederhana agar siswa dapat memahami setiap konsep yang ada. Dengan demikian, maka penggunaan bahasa anak haruslah dijadikan sebagai pengantar dalam mengkomunikasikan pendidikan matematika itu sendiri.

6. Fevi Rahmawati Suwanto
16709251005
PMat A / S2

Bahasa merupakan hasil kebudayaan yang mempunyai konteks sendiri. Matematika merupakan bidang ilmu yang dipelajari untuk mempersiapkan atau sebagai bekal anak pada kehidupan nyata, sesuai konteksnya. Maka untuk dapat mengajarkan matematika sebagai bekal kehidupan haruslah menggunakan bahasa-bahasa yang juga merupakan bahasa ibu bagi anak. Penggunaan ini ditujukan agar anak memperoleh pengertian dan pemahaman lebih mudah tentang pengetahuan matematika. Dengan pemahaman yang dimiliki, maka akan lebih mudah bagi anak untuk menerapkannya dalam kehidupan nyata.

7. Mega Puspita Sari
16709251035
PPs Pendidikan Matematika
Kelas B

Dari bacaan di atas dapat diperoleh bahwa peran bahasa dalam pembelajaran matematika adalah memfasilitasi siswa agar dapat menerjemah, membangun, mengevaluasi konsep dari matematika bukannya mempersulit siswa dalam belajar. Tidak sedikit dari buku-buku matematika memiliki bahasa yang sulit dimengerti oleh siswa dan menjadi masalah bagi siswa dalam belajar.

8. Kumala Kusuma Putri
13301241020
Pendidikan Matematika I 2013

Assalamualaikum Wr. Wb.
In my opinion, mathematics is really bounded with language. Not only mathematics, but all of life element are bounded with language. Language is really important thing in life. With language, we can communicate with each other with different language. Mathematics language, biology language, chemistry language are different language. They have speciality. Just like that. If we talk with child and adult, we should use different language. Like, if we want to explain mathematics to child, then we should use language that can be understood by child. Like Mr. Doug Hainline said. He would much rather discuss geometry with nine-year-olds, by focussing on the interesting and important features of, say, plane figures, than by having them memorize that a triangle with only two sides equal is an isoceles triangle, but one with three sides equal is an equilateral triangle. Mathematics and language are really bounded each other. Mathematics languange of adult and mathematics language of child are different language. I think that is enough. Thank you.

Wassalamualaikum Wr. Wb.

9. Rizqi Nefi Marlufi
13301241035
Pendidikan Matematika Internasional 2013

Knowing that Wittgenstain could give us a knowledge about Structuralism Mathematics in Language, it's so relief. Because, by this term we can made a mathematics just being a calculating mathematics, but it can goes into a mathematics language that can be understood by everyone easily.

10. Rizqi Nefi Marlufi
13301241035
Pendidikan Matematika Internasional 2013

By this understanding, it's easy to improve or expand mathematics as knowledge to implement them in a functional life. It's also made the system of mathematics learning process at school and at campus has a difference.

11. Rizqi Nefi Marlufi
13301241035
Pendidikan Matematika Internasional 2013

Make some level in mathematics learning process for the best step on the way of thought. This is a good culture that should be realized and be applied or produced for a better life.

12. MARTIN/RWANDA
PPS 2016 PEP B
I COMPLETELY stress on the use of mathematical language. as David said, those who pronounce math are the ones who succeed. this means that mathematics needs discussions. mathematics its self is meaningless but mathematics combined with language is meaningful and fruitful. i still aree with what Prof. Marsigit said that the curriculum should be designed basing on the needs of elementary and secondary schools.

13. Azwar Anwar
16709251038
Pendidikan Matematika S2 Kelas B 2016

Dalam pembelajaran matematika perlunya komunikasi atau interaksi yang baik anatara guru dan siswa. Salah satunya dengan bahasa, matematika adalah bahasa yang melambangkan serangkaian makna dari pernyataan yang ingin kita sampaikan. Simbol-simbol matematika bersifat artifisial yang baru memiliki arti setelah sebuah makna diberikan. Jadi bahasa berusaha untuk menghilangkan sifat kabur atau samar-samar di dalam siswa untuk memahami matematika.

14. Fatya Azizah
16709251039
Pendidikan Matematika B PPS UNY 2016

penggunaan bahasa matematika selain menarik dan memiliki keindahan tersendiri dikarenakan setiap wilayah tentunya memiliki bahsa matematikanya masing masing dan juga bersifat umum untuk seluruh dunia, selain itu juga berguna bagi kehidupan dan bagi pengajaran kepada siswa.

15. Erni Anitasari
16709251007
S2 Pend. Matematika Kelas A

Sarana belajar siswa sangat menunjang proses pembelajaran, salah satunya yaitu buku pendamping siswa. Penggunaan kosa-kata dalam buku pendamping siswa ternyata memiliki pengaruh besar. Apabila kata yang digunakan di dalam buku terlalu sulit untuk dimengerti oleh siswa, itu juga akan menjadi kendala untuk siswa memahami materi.

16. Rospala Hanisah Yukti Sari
16790251016
S2 Pendidikan Matematika Kelas A Tahun 2016

Assalamu’alaikum warohmatullahi wabarokatuh.

Pembelajaran matematika kontekstual memang baik jika diterapkan dalam pendidikan sekolah. Misalnya, SD dan SMP. Karena dalam jenjang tersebut, siswa masih memerlukan visualiasi dan contoh konkrit dalam kehidupan mereka yang kemudian dituangkan dalam pembelajaran matematika kepada mereka. Dengan begitu, pembelajaran matematika menjadi menyenangkan.

Wassalamu’alaikum warohmatullahi wabarokatuh.

17. Nuha Fazlussalam
13301244023
s1 pendidikan matematika c 2013

mengajarkan matematika harus sempurna daam matematika murni, kita tidak bisa belajar dengan kesalahan dong, padahal salah satu yang paling kontribusi dan diingat adalah belajar dari kesalahan, sehinga mereka belajar. itu sama halnya dengan orang baik yang ingin mengajarkan kebaikan, ada kebaiokan murni ada pendidikan kebaikan, kebaikan murni itu dasar- kebaikan, kebaikan orang dewasa, pendidikan mkebaikan adalah kebaikan yang bisa dpikir dan dipahami oleh anak-anak, tidak segampang itu mengajar kan kebaikan, tetapi harus memberikan contoh real, dimana contoh real penuh dengan kontradiksi, perlu media, perlu mengetahui psoikologi siswa sehingga siswa dapatmenerima ilmu kebaikan tersebut. analoginyas perti itu, jadi kesimpulannya, penddiiajn matemtika perlu belajar dan berusaha menjadui matematika murni, namun tidak harus sejajar dengan matematika murni, karena mereka perlu memahami siswa, matematika murni tidak wajib mengetahui siswa, pndidikan matematika wajib.

18. Yurizka Melia Sari
16701261003
PPs PEP A 2016

The language of which the teacher use in delivering the mathematics is mostly differ with what students use in their learning. This condition is what van Hiele consider as the missmatch condition. in fact, this condition always limits the process of teaching and learning that makr the students unable to understand at all. Therefore, the students tend to perform the lowest level in learning, which is mrmorizing facts. Memorizing mathematical facts is, honestly, a rudimentary work. This idea is inline with what Doug Hainline had claimed above.

19. Yurizka Melia Sari
16701261003
PPs PEP A 2016

"language has its own cultural context; the context in which the students are coming from." (Marsigit)

I agree with this statement. The fact that the language may support or hinder the learning is inevitably. In Indonesia, we have bahasa and javanese language whe have disagreement in defining, for example, multiplication. 2x3 is read dua kali tiga in bahasa. This has the meaning that we already have three and multiply the quantity by two folds. Another representation on it in bahasa can be stated as 3+3. Nevertheless, in javanese language, 2x3 is read as loro ping telu. This representation, in short, can be stated as 2+2+2 which is, clearly, differ from the bahasa meaning.

Another example is about the method of completing square as one of the methods in solving quadratics equation. In bahasa, we read the method as "kuadrat sempurna" or if we translate back to english, it may be read as perfect square. Regardless the algebraic structure of the method, if we may look at the history of algebra, we would notice that actually the completing square method resulted from the process of making algebraic form into geometrical object. In details, the method, then, operates in forming a real complete square figure by taking and adding a certain area. I believe that this idea, if we implement in the class, may produce significant different understanding than showing the students the algebra only of the method.

20. Yurizka Melia Sari
16701261003
PPs PEP A 2016

Most mathematicians would proud on how mathematics are their language. The abstraction on the formalized mathematical objects may be enjoyed by only a few of them. They don't care if the beauty of mathematics could not be understood by others especially the young people. However, some mathematician, like Freudhental, worry that in some point people won't aware on the beauty of mathematics. Furthermore, the existence of mathematician would be dissappear from this world. That concerns especially for mathematics at school. some would agree that the language of mathematics delivered in the classroom tend to dictate the students and left no space for their own creativity. One important point suggested by Freudhental is the idea that mathematics is a human activity. Mathematics should be able to be thaught, performed, expressed, and imagined by the students. This point a view should be implemented in the language of mathematics delivered in the class, unless the students would consider that mathematics is only a collection of remembered formula.

21. Yurizka Melia Sari
16701261003
PPs PEP A 2016

The language of which the teacher use in delivering the mathematics is mostly differ with what students use in their learning. This condition is what van Hiele consider as the missmatch condition. in fact, this condition always limits the process of teaching and learning that makr the students unable to understand at all. Therefore, the students tend to perform the lowest level in learning, which is mrmorizing facts. Memorizing mathematical facts is, honestly, a rudimentary work. This idea is inline with what Doug Hainline had claimed above.

22. RAHMANITA SYAHDAN
16709251013
PPs Pmat A 2016

Bismillahirrahmanirrahim
Menurut Galileo Galilei (1564-1642), seorang ahli matematika dan astronomi dari Italia, “Alam semesta itu bagaikan sebuah buku raksasa yang hanya dapat dibaca kalau orang mengerti bahasanya dan akrab dengan lambang dan huruf yang digunakan di dalamnya.”
Dan bahasa alam tersebut tidak lain adalah matematika. Matematika adalah bahasa yang melambangkan serangkaian makna dari pernyataan yang ingin kita sampaikan. Simbol-simbol matematika bersifat “artifisial” yang baru memiliki arti setelah sebuah makna diberikan kepadanya. Tanpa itu, maka matematika hanya merupakan kumpulan simbol dan rumus yang kering akan makna. Berkaitan dengan hal ini, tidak jarang kita jumpai dalam kehidupan, banyak orang yang berkata bahwa X, Y, Z itu sama sekali tidak memiliki arti.
Bahasa matematika memiliki makna yang tunggal sehingga suatu kalimat matematika tidak dapat ditafsirkan bermacam-macam. Bahasa matematika adalah bahasa yang berusaha untuk menghilangkan sifat kabur, majemuk, dan emosional dari bahasa verbal. Lambang-lambang dari matematika dibuat secara artifisial dan individual yang merupakan perjanjian yang berlaku khusus suatu permalahan yang sedang dikaji. Suatu obyek yang sedang dikaji dapat disimbolkan dengan apa saja sesuai dengan kesepakatan kita (antara pengirim dan penerima pesan).
Menurut Wittegenstein, matematika merupakan metode berpikir yang logis.

23. Wan Denny Pramana Putra
16709251010
PPs Pendidikan Matematika A

Bahasa yang digunakan dalam pendidikan matematika sebaiknya menggunakan bahasa yang mudah dipahami oleh peserta didik. Apalagi mengajarkan matematika pada sekolah dasar maka kita sebagai guru dituntut untuk menggunakan bahasa kontekstual sehingga anak-anak menjadi lebih mudah menerima pelajaran matematika.

24. Sumbaji Putranto
16709251028
Pend. Matematika S2 Kelas B

Bahasa menjadi bagian penting dalam setiap ilmu pengetahuan. Tak terkecuali dalam matematika. Kemudahan dalam memahami matematika juga dipengaruhi oleh bahasa yang digunakan. Menjadi sangat perlu untuk dikoreksi ketika sekarang ini terdapat banyak buku matematika yang menggunakan banyak kosakata yang sulit dan rumit. Kosakata ini justru akan membuat siswa mengalami kesulitan belajar karena siswa harus memproses kosakata tersebut, apalagi untuk siswa yang memang mengalami kesulitan belajar maka kesulitan tidak semata-mata karena kurangnya terhadap konsep namun bisa jadi karena kendala bahasa. Sebaiknya guru maupun penulis dapat memilih kosakata yang tepat agar tidak menyulitkan anaka dalam belajar matematika.

25. RISKA AYU ARDANI
16709251021
PMAT KELAS B PPS UNY 2016

Bahasa matematika berusaha dan berhasil menghindari kerancuan arti, karena setiap kalimat (istilah/variabel) dalam matematika sudah memiliki arti yang tertentu. Ketunggalan arti itu mungkin karena kesepakatan matematikawan atau ditentukan sendiri oleh penulis di awal tulisannya. Orang lain bebas menggunakan istilah/variabel matematika yang mengandung arti berlainan. Namun, ia harus menjelaskan terlebih dahulu di awal pembicaraannya atau tulisannya bagaimana tafsiran yang ia inginkan tentang istilah matematika tersebut. Selanjutnya, ia harus taat dan tunduk menafsirkannya seperti itu selama pembicaraan atau tulisan tersebut.
Sehingga dalam proses pembelajaran matematika, maka guru perlu memilih cara yang baik bagaimana komunikasi kepada siswanya , mentransformasikan sifat matematika kedunia siswa. Kemudian harapannya siswa dapat menerima dan memahami dunia matematika tersebut.

26. Erlinda Rahma Dewi
16709251006
S2 PPs Pendidikan Matematika A 2016

Komunikasi dalam matematika dan pembelajaran matematika menjadi sesuatu yang dibutuhkan seperti yang diungkapkan oleh Lindquist (1996), jika kita setuju bahwa matematika adalah bahasa dan bahasa sebagai pembahasan yang terbaik di masyarakat, maka mudah untuk memahami bahwa komunikasi adalah esensi dan mengajar, belajar, dan mengakses matematika. Komunikasi merupakan bagian yang sangat penting dalam matematika dan matematika pendidikan. Komunikasi adalah cara untuk berbagi ide dan memperjelas pemahaman. Melalui komunikasi ide dapat tercermin, ditingkatkan, dibahas dan dikembangkan. Proses komunikasi juga membantu membangun makna dan ide-ide mempermanenkan dan proses komunikasi juga dapat mempublikasikan ide-ide.

27. Kartika Nur Oktaviani
16709251032
Pendidikan Matematika S2 UNY kelas B

Assalamu'alaikum wr wb.
Bahasa matematika menurut saua adalah bahasa yang paling teratur karena sebelum kita melanjutkan ke langkah ini, kita harus melewati langkah itu.
Bahasa matematika juga penuh dengan bahasa yang simbolis dan misterius.
Wassalamu'alaikum wr wb.

28. Rhomiy Handican
16709251031
PPs Pendidikan Matematika B 2016

Bahasa dan matematika sangat erat kaitannya ketika bahasa digunakan dalam komunikasi matematika. matematika dengan penyampaian komunikasi yang baik akan melahirkan penganut matematika yang menyatakan matematika itu indah melalui bahasa. walaupun bahasa matematika yang banyak menggunakan bahasa simbolis namun dalam komunikasi matematika bahasa akan dapat di terjemahkan dengan baik. dan menjadi guru tantangan kita adalah untuk membuat komunikasi matematika yang dapat diterima dan dipercaya.

29. Arifta Nurjanah
16709251030
PPs P Mat B

Bahasa yang digunakan dalam proses belajar mengajar dan juga dalam melakukan matematika merupakan faktor yang penting dalam menentukan kemudahan siswa belajar. Bahasa akan membantu siswa terjemah dan diterjemahkan, membangun dan dibangun, serta mengevaluasi konsep dari matematika bukannya mempersulit siswa dalam belajar. Maka penting bagi guru untuk memperhatikan penggunaan bahasa dalam mengkomunikasikan matematika pada siswa. Penggunaan bahasa juga penting bagi interaksi dalam mengkonstruk kkehidupan siswa. Bahasa dan matematika yang digunakan sebaiknya yang kontekstual. Jika kita dapat memilih kosakata dan notasi lambang yang cocok maka akan sangat menentukan keberhasilan siswa.

30. Muh. Faathir Husain M.
16701251030
PPs PEP B 2016

Terkadang memang antara simbol matematika dan bahasa memiliki gap yang jauh, ambil contoh frase "jika dan hanya jika", sebagian orang akan memandang itu pemborosan kata, padahal dalam matematika itu menunjukkan bikondisional artinya, semua statemen harus emilki status yang sama, baik subyek maupun predikatnya. Selain itu ada pula simbol matematika yang sampai sekarang beberapa teman guru masih menyamakan konteks tak hingga dan tak terdefinisi karena samanya simbol matematika yang digunakan.

31. RAIZAL REZKY
16709251029
S2 P.MAT B 2016

Matematika dan bahasa merupakan dua bdang keilmuan yang tidak dapat dipisahkan, dikarenakan dalam matematika terdapat bahasa yang menjelaskan persoalan-persoalan dalam matematika, bahkan dalam penggambaran simbol-simbol matematika mempunyai bahasa tersendiri yang mengartikan simbol-simbol itu sendiri.

32. Faqih Mu'tashimbillah
12313244030
Pend Matematika Internasional

Assalamualaikum.wr.wb
Saya setuju dengan pendapat Helen Mason.
Banyak buku matematika yang menggunakan banyak kosakata yang sulit dan rumit.
Kosakata ini justru akan membuat siswa mengalami kesulitan belajar karena siswa harus memproses kosakata tersebut, apalgi untuk siswa yang memang mengalami kesulitan belajar.
Sebaiknya guru maupun penulis dapat memilih kosakata yang tepat agar tidak menyulitkan anaka dalam belajar matematika.

33. Ummi Santria
16709251008
S2 Pend. Mat Kelas A – 2016

Bahasa adalah kunci perkembangan kognitif karena bahasa merupakan alat komunikasi antara manusia. Untuk memahami konsep-konsep yang ada diperlukan bahasa. Bahasa diperlukan untuk mengkomunikasikan suatu konsep. Bahasa yang merupakan suatu sistem terdiri dari lambang-lambang, kata-kata dan kalimat-kalimat yang disusun menjadi hal yang penting bagaimana tidak hanya siswa, tetapi guru dan lainnya juga membutuhkannya dalam mengomunikasikan sebuah konsep. Bagaimana siswa menjelaskan lagi apa yang dipahami dan apa yang dilihatnya dari sebuah konsep ada hubungannya dengan pemakaian bahasa dalam matematika.

34. Bismillah
Ratih Kartika
16701251005
PPS PEP B 2016

Assalamualaikumwarahmatulahiwabarrakatuh¬
Bahasa merupakan sistem yang terdiri dari lambang-lambang, kata-kata, dan kalimat-kalimat yang disusun menurut aturan tertentu dan digunakan oleh orang untuk berkomunikasi. Komunikasi adalah kegiatan menyampaikan informasi melalui pertukaran pengalaman, pesan, atau informasi, dengan pidato, visual, sinyal, tulisan, atau perilaku. Dalam KBM, guru dan siswa berkomunikasi dari sini diharapkan guru mentransfer matematika dengan baik dan siswa memahami matematika dengan lebih bermakna.
Wassalamualaikumwarahmatulahiwabarakatuh

35. Siska Nur Rahmawati
16701251028
PEP-B 2016

Matematika memiliki bahasanya sendiri yang digunakan untuk memahami materi matematika. Bahasa matematika digunakan untuk mereduksi pikiran agar dapat memahami makna. Guru perlu menjelaskan pentingnya menggunakan bahasa matematika yang baik dan benar agar siswa tidak salah dalam memahami konsep matematika.

36. Asma' Khiyarunnisa'
16709251036
PPs PM B 2016

Bahasa akan mempermudah siswa dalam mempelajari matematika. Bahasa yang dimaksud disini adalah bahasa matematika yang berupa simbol, notasi, dan lain-lain. Hal tersebut diberikan kepada siswa beserta dengan makna nya sehingga dapat membantu siswa memahami pernyataan matematika.

marsigitina@yahoo.com, marsigitina@gmail.com, marsigit@uny.ac.id