Mar 22, 2013

Mathematics and Language 13

 Irene: I have a hypothesis -- just barely a hypothesis -- about "scientific" and "mathematical" language -- really, about any unfamiliar vocabulary. Maybe it's relevant to your case.

I believe -- tentatively, subject to refutation by empirical evidence -- that when we are learning a new subject, that it is advantageous if the vocabulary we use to speak about it is not new to us.

That is to say, that the words we will be learning are ones we have have often heard before, without having learned exactly what they mean. I believe that if you are having to become familiar with a new word at the same time that you are learning about the concept it stands for, your brain gets hung up on the new word and this prevents you from assimilating the concept. (This is not put very elegantly.)

An example: I believe that if you are going to study trigonometry at, say, 13, then at about 10 or 11, you should start hearing the words "sine", "cosine" and "tangent", in relation to triangles. You need not learn what they "mean" at this age, just that they are properties of an angle, in certain circumstances.

I believe that then, when you start learning what these words actually mean, how we may use them in mathematical problems, it will be significantly easier for you.

I know of no experimental evidence for this, and have never read of any research that addresses it (although there may well be some). It's just a hypothesis, based on my own (subjective, limited, selective) experience as both a tutor, and a learner. If it's true, I would expect that there is a neurological explanation, although perhaps beyond our current ability to isolate.

I believe that this is generally how we come to understand word meanings, as children. First we hear the word used, without understanding it. Then, as we hear examples of the word being used, we gradually come to that state we call "knowing the meaning". Very seldom is a word formally defined to us the first time we hear it, and almost never is it defined in the kind of emotional context that mathematical terms are usually presented in (which, for most children, is not a pleasant one).

I know also that mathematical terms are not necessarily like other words, in terms of there being a formal, precise quality to their proper use.

I would be very interested in others' comments on this, and in particular would appreciate hearing of any references to any relevant research (which does not necessarily have to be in the area of mathematics education).

@Doug: Just very small question:How you may define the meaning of BIG, SMALL, NEAR, FAR, LOW, HIGH, WIDE, MANY, FEW, NUMBER, PART, ...etc.

What is a number, that a man may know it, and a man, that he may know a number?

Look... a BIG tiger is coming NEAR. We had better get FAR away. (Except for those who believe reality is socially-constructed. They can remain.)

But I am stumped as to the definition of "etc".
And even if we all spoke Lojban, we would still have the "gavagai" problem.

When I was about 14 I discovered, like a host of similar teenagers before and after, the obnoxious pleasures to be found in asking people to "define their terms". When they tried to do so, they simply gave me targets for the Forward Observer's favorite command: "repeat!". Eventually I tired of this and went on to the delights of demanding a refutation of solipsism, or disproof of the no-free-will argument. Every child should do this!

But when we become adults, we should put aside childish things. If we wish to establish arithmetic on an axiomatic basis, a useful exercise, we will have to leave "zero", "successor", and "number" undefined, if we are not to have an infinite regress. A set is a collection and a collection is ....

This does not mean we should not seek to define, or at least clarify, our terms! Nor is it useless to follow arguments to their remorselessly 'logical' conclusions, beginning with Locke and ending with Hume.

By all means, awaken the dogmatic slumberers! But then we should agree with Hume that, although irrefutable, his scheme makes no practical difference in our daily lives.

While we teach children the truths of mathematics, we should also -- and not only in the mathematics classroom -- be teaching them how to think.

Does 1 + 1 = 2? Yes, in number bases higher than binary, and with a certain interpretation of the Hindu-Arabic numerals, but in binary 1 + 1 = 10. And of course, numbers are abstractions. In real life, we need to learn when to apply which set of abstractions. 55/10 is 5.5, but if I want to know how many ten-passenger mini-vans to hire to transport 55 pupils, 5.5 is not an answer. And 4/10 + 5/10 is 9/10 in some circumstances, and 9/20 in others.

Ideally, we should, in a sense, be teaching our children 'philosophy'. Their thinking should be flexible enough so that the mind-boggling results of assuming that the speed of light is a constant for all observers, will not boggle their minds, nor will the yet more mind-boggling results of our investigations of the world of electrons and photons.

If you can do that, well, te salud. For the time being, I'll settle for kids who know their times tables and can solve the sort of word problems that are routine for kids in Singapore. Perhaps a bit of Korzybski, so that they are not tangled up in metaphysical knots when faced with a question like "Is 4/3 a division, or is it a fraction?" (Hint: rephrase your question without using any form of the "to be" verb.)

But just as Donald Rumsfeld noted that "You go to war with the army you have," we have to teach mathematics with the teachers we have. We cannot jump over our own heads, and we cannot expect Mrs Smith who has been teaching 4th grade for the last 25 years, to take on board the sort of conceptual apparatus that would equip children for an easy acceptance of relativistic non-deterministic physics,

Some attention to our language, and even to the symbols we use for teaching mathematics, is, in my opinion, warranted, as is the encouragement of genuine critical thinking.

I think we could do more and better than we now do, in teaching children about what we have learned so far == so slowly and painfully! -- of the world and how it works.

But this assumes that what we have learned so far is worth teaching.

Many children know the numbers without knowing that they are “numbers”; because in my language they are called “bilangan”. They do not need definition to know the concept of Number, Big, Near, etc. You did exactly the same with the younger learner to know them.

You memorize well your age 14; but I believe you cannot memorize your age 3. Still my questions how you may be able to know the concept of Big, Near, Far, Two, ..etc at your age 3? I hope my questions lead you to go deep into the endlessly state of knowing activities.

And, I am happy that you strived to go to your childish in order to remember how to know the very basic concept of mathematics i.e. concrete mathematics. Again, that is exactly the same with what the younger learner in striving to know mathematics.

If I continue this story, I believe I will not find the term “adult”; so there is no choice for you to go to younger world if you want to introduce your mathematics. And you did the simulation very well. If you do not mind I wish to call your simulation as “developing younger mathematics intuition”. I do agree to apply your statement “Every child should do this!” to this context.

The next is how to implement or follow up your simulation to your real world in which now you are as an adult who wish to interact with younger learner?

At the realm of infinite regress, more than your stumping of definition of “etc”; I challenge you or any other scientist to define “is”?

Still in the realm of infinite regress, when you become adults there will be new younger generations. It will be irresponsible behavior when then you pu aside the childish world. As an adults you should have your responsibility not to throw away undefined “zero”, “successor” and “number”; but to make them meaningful for your younger generation.

You seemed to be the victim of your older generation by repeating to fulfill their commands. And here, I do not agree with your statement that every child should do like you.

By the way, I spy that you are still confuse with the nature of Socio-constructivist. You seemed to mean this as thinking together. However, I will not elaborate it more due there are too many related references.

I wish to say that different perspectives have their different aspect of life. While I found that to some extent you still use your adults’ criteria to judge the younger life by trying to teach mathematical Truth to them. In my perspective, mathematics for the young learner is not about Body of Truth, Science of Truth or Structure of Truth, rather they are searching the pattern, relationship; solving the problems, investigating activities and doing communications.

If the Pope of Franciscus said “Protect all people especially the pure”; I will say “Protect all people especially the younger/powerless from the adults/powerful misbehavior”. Teaching cannot totally be compared with soldiers going to war. No, no way; because there is no enemy outside there but clearly the biggest enemy is inside the adults-powerful-self ego-pure mathematicians-determined teachers.

I prefer to compare teaching with the farmer’s growing the seeds. Critical thinking can only be performed if they are free to think and free to grow.


  1. Fatmawati
    PM.D 2016
    Dalam mempelajari matematika kita akan dipertemukan dengan berbagai simbol-simbol baru, contohnya seperti bilangan. Dalam mempelajari bilangan atau simbol-simbol tersebut bagi seorang siswa yang baru mengenal maka dia hanya perlu tahu arti dari simbol tersebut tidak perlu sampai ke definisi, karena siswa sekolah belum saatnya untuk memahami definisi dari simbol-simbol matematika. Oleh karena itu, menjadi perhatian penting bagaimana guru memahamkan siswa mengenai simbol-simbol yang ada di matematika.

    16709251056_PMC 2016
    Pendidikan Matematika-S2

    Mempelajari matematika dengan cara siswa akan memudahkan mereka dalam memahami matematika. Kita ingat kembali bahwa hakikat matematika sebenarnya adalah intuisi siswa itu sendiri. Maka sebaiknya, akan lebih tepat jika siswa dibiarkan berkembang sendiri melalui pikirannya untuk membangun sendiri konsep matematika dalam dirinya.

  3. Kunny Kunhertanti
    PPs Pendidikan Matematika kelas C 2016

    Sebenarnya penggunaan bahasa matematika sudah sering digunakan oleh anak-anak, namun anak-anak tidak menyadari hal tersebut. Bahkan mereka lebih cenderung merasa kesulitan jika harus menerjemahkan bahasa matematika. Disinilah tugas darp para orang dewasa atau guru, yaitu untuk membantu anak-anak kecil untuk menerjemahkan dalam bentuk matematika atau pun sebaliknya.

  4. Sehar Trihatun
    S2 Pend. Mat Kelas C – 2016

    Melakukan pembelajaran kepada siswa bukanlah mengajarkan siswa nya untuk dapat mmengikuti segala hal yang dapat dilakukan oleh gurunya. Bukan juga bertujuan agar siswa dapat meniru apa yang dapat dilakukan oleh gurunya. Karena hal itu tentu akan sulit bagi siswa yang memang kemampuannya tidak dapat disamakan dengan gurunya. Mereka memiliki keunikan tersendiri dalam memahami apa yang dia dapatkan dari hasil kegiatan belajarnya. Intinya siswa membangun pengetahuannya sendiri dengan cara yang berbeda-beda. Sehingga guru harus senantiasa memperhatikan bagaimana prosses siswa berpikir dan mengolah informasi yang diterimanya menjadi pengetahuan yang berguna baginya.

  5. Nama: Ilma Rizki Nur Afifah
    NIM: 17709251020
    Kelas: S2 Pendidikan Matematika A

    I am very attracted to discussion above that the mathematics is not something to be memorized and studied without an understanding. In mathematics, the concept is needed. Mathematical concepts need to be introduced early in students. Not just tell them to memorize formulas or make a quick in answering the questions. The basic concepts are introduced early mathematics aims to enable students easy to solve problems in mathematics further. Indeed, it is not easy for a teacher to explain mathematical concepts to students especially for those who teach students under 10 years old. But remember, it is extremely important concepts. Perhaps teachers need to teach the concept with more innovative ways so that students easily understand checkers implemented on the questions.

  6. Dimas Candra Saputra, S.Pd.
    PPs PMA 2017

    Assalamualaikum prof,
    Dalam proses belajar maupun mengenali dunianya, anak-anak kecil tidak memerlukan definisi dan pengetahuan mereka tidak diawali dengan definisi. Seiring perkembangan proses belajar, pengalaman dan interaksinya dengan lingkungan, mereka bisa membuat suatu definisi sendiri. Tidak mungkin anak kecil bisa bersepeda yang dimulai dengan definisi dan teori bersepeda, proses belajarnya akan lebih baik ketika mereka berlatih langsung melalui interaksinya dengan sepeda. Demikian halnya dalam pembelajaran matematika, mengajarkan matematika yang dimulai dengan definisi tidak akan mudah dipahami oleh siswa. Siswa akan lebih mudah memahaminya bila mereka diberikan kesempatan untuk berinteraksi dengan objek maupun dengan orang disekitarnya. Di samping itu, guru juga tidak cukup hanya berhenti pada membangun konsep, guru juga harus menciptakan pembelajaran yang bermakna dan juga membuat siswa mampu menggunakan pengetahuannya secara otomatis. Dengan demikian siswa menggunakan pengetahuannya dalam aktivitas yang lebih tinggi.

  7. thank you very much prof, from the above article we can find an idea that the topic of mathematics language may be used as an interesting research topic because it is still very rarely studied.

  8. Auliaul Fitrah Samsuddin
    PPs P.Mat A 2017

    Thank you for sharing this, Prof. I really love your comparison between learning process and growing the seeds. We need to create a learning atmosphere where students can feel free to construct their concept, idea and logic.

  9. I Nyoman Indhi Wiradika
    PEP B

    Terima kasih Prof, telah memberikan saya pengetahuan baru melalui artikel menarik ini. Anak-anak ketika memasuki lingkungan orag dewasa kerap terperangkap dengan definisi-definisi yang membingungkan. Hal ini hampir sama seperti anak yang tidak pernah sekolah, namun sangat paham melakukan transaksi jual-beli. Tentu, anak yang tidak bersekolah memiliki peluang yang sangat tinggi untuk tidak mengenal definisi matematika, namun secara intuisi, anak tersebut dapat dikatakan memiliki kemampuan matematika, walaupun tidak paham dengan definisi-definisi membingungkan yang diciptakan oleh para ilmuwan murni.

  10. Nama : Habibullah
    NIM : 17709251030
    Kelas : PM B (S2)

    Assalamualaikum wr.wb

    Konsep matematika merupakan sebuah objek, pola pikir yang murni, tidak dapat didengar dan tidak dapat dilihat. Karena tidak ada cara untuk mengamati secara langsung pikiran seseorang, harus menggunakan alat-alat yang dapat didengar atau dilihat seperti mengucapkan kata atau bunyi lainnya, penulisan kata atau tanda lainnya yang ditulis di atas kertas. Melihat pentingnya bahasa sebagai alat kumunikasi dalam pembelajaran matematika maka guru harus selalu melatih kemampuan public speakingnya agar apa yang ingin disampaikan dapat diserap dan dipahami siswa, sehinga akan terjadi respon timbal balik. Dengan begitu guru dan siswa besama-sama dapat menguasai kompetensi yang diharapkan.

  11. Nama: Hendrawansyah
    NIM: 17701251030
    S2 PEP 2017 Kelas B

    Assalamualaikum wr wb

    Situasi dan keadaan anak rasanya telah dikuasai oleh matematika dewasa.Generasi muda menjadi kebingungan.Mereka bagaikan karang yang terhempas di dasar laut oleh riuhnya gelombang.Pikirannya telah terkontaminasi dengan aturan konsep yang masih mengambang.Kebebasan berpikir adalah haknya.Maka biarkan ia melambungkan pikirannya untuk menyesuaikan apa yang anak-anak dapat dengan yang ada di pikirannya. Itulah yang dapat saya ungkapkan melalui postingan ini.

  12. Junianto
    PM C

    Belajar matematika bagi siswa masih menjadi sesuatu yang sangat menyulitkan siswa. Hal ini salah satunya disebabkan oleh adanya campur aduk antara matematika sekolah dan matematika murni. Untuk siswa sekolah dasar dan menengah, matematika yang mereka pelajari adalah matematika sekolah dan matematika murni adalah untuk mahasiswa tingkat Universitas. Selain itu, pemilihan metode dan media pembelajaran juga harus tepat sesuai dengan topik bahasan yang akan diberikan kepada siswa.