@Doug, Wonderful examples. I loved the "bad_writing" site, and the Sokal
hoax has been one of my favourite jokes for years. It reminds me of the
lines from Woody Allen's "Love and Death":

Sonja: "Judgment of any system, or a priori relationship or phenomenon exists in an irrational, or metaphysical, or at least epistemological contradiction to an abstract empirical concept such as being, or to be, or to occur in the thing itself, or of the thing itself. "

Boris: "Yes, I've said that many times. "

(Woody Allen actually lifted Sonja's line from a Russian author, who had been serious when he wrote it.)

Yes, there is a strong connection between Mathematics and Language: mathematical thinking (which is more important than learning to integrate, unless you're an engineer I guess) leads to clear language.

In general, very good points.

@Marsigit: Although your contention that there is more to reality than mathematics is debatable (if one has a wide enough definition of mathematics), nonetheless we are talking about teaching mathematics here. Yes, fuzzy poetry and Zen koans are OK with contradictions, but if we want to facilitate (to use your word) the students to learn mathematical thinking, the first thing they need to learn is to avoid contradiction, as this is at the heart of all mathematical thinking. If they invent something else, that may be fine for Art or Literature classes, or for some religion, but it is not mathematics. In any case, my remark in my previous post was not concerned with the issue whether one should be too pedantic about avoiding contradiction while teaching (yes, Holism has a certain place in thinking, but so does Reductionism: one without the other gives nothing of value), but was asking you to clear up a direct contradiction that you had made about teaching. You are evading the question.

Sonja: "Judgment of any system, or a priori relationship or phenomenon exists in an irrational, or metaphysical, or at least epistemological contradiction to an abstract empirical concept such as being, or to be, or to occur in the thing itself, or of the thing itself. "

Boris: "Yes, I've said that many times. "

(Woody Allen actually lifted Sonja's line from a Russian author, who had been serious when he wrote it.)

Yes, there is a strong connection between Mathematics and Language: mathematical thinking (which is more important than learning to integrate, unless you're an engineer I guess) leads to clear language.

In general, very good points.

@Marsigit: Although your contention that there is more to reality than mathematics is debatable (if one has a wide enough definition of mathematics), nonetheless we are talking about teaching mathematics here. Yes, fuzzy poetry and Zen koans are OK with contradictions, but if we want to facilitate (to use your word) the students to learn mathematical thinking, the first thing they need to learn is to avoid contradiction, as this is at the heart of all mathematical thinking. If they invent something else, that may be fine for Art or Literature classes, or for some religion, but it is not mathematics. In any case, my remark in my previous post was not concerned with the issue whether one should be too pedantic about avoiding contradiction while teaching (yes, Holism has a certain place in thinking, but so does Reductionism: one without the other gives nothing of value), but was asking you to clear up a direct contradiction that you had made about teaching. You are evading the question.

@David: We are adults people, so our utterances is in the mode of
adults. It will be very inappropriate to bring every single adult's
utterances to the younger. Again, your argument have proved that you are
still a pure math minded to behave in front of your younger learner.
You will face the difficulties or you force it to them then you have no
problem. In the case of extensive dimension of life 1+1 = 2 can be
contradictory i.e.if we are concern with the time and space. But this is
only and only for adults. See my last response to Doug.

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).

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.

@David: My exchange with you is about dimension of communication. For
me, there was no contradiction of my previous statement. You are still
in the stage of formal thinking (math way) to translate language
phenomena. If you wish to understand the life better by translating
language of life, so you need to improve (lift up) your dimension of
communication i.e. go into the normative stage. It was very clear, in my
perspective, from normative dimension, that technology is the product
of adults, however we should be wise to use them to facilitate the
younger learner. In the case of interaction with the adults' product,
the younger should have their own determination, effort, creativity in
the scheme of constructing their life (math). Finish.

Further, still about the term 'contradiction', I will say that as other phenomena, this also has its dimension. The mathematical contradiction is different with the normative contradiction. The contradiction in mathematics means 'inconsistency'; but in the normative dimension, it means not in accord with the Identity rule.

Further, still about the term 'contradiction', I will say that as other phenomena, this also has its dimension. The mathematical contradiction is different with the normative contradiction. The contradiction in mathematics means 'inconsistency'; but in the normative dimension, it means not in accord with the Identity rule.

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ReplyDeletePEP B S2

Memamng berbeda matematika dewasa dengan matematika pada dunia anak. Perbedaan dapat dimukai dari segi kebahasaan. Pada orang dewasa tentunya akan sangat berbeda dengan pemahaman bahasa anak anak. Pada orang dewasa lebih komplek, sedangkan sebaliknya pada anak anak tentunya masih sangat sederhana. Kontradiksi dalam matematika sendiro juga terkait dengan bahasa. Karena dengan bahasa semua fapat berinterakso dan berkomunikasi.

Jika dalam anak anak matematika bukanlah hanya sebatas dimensi formal saja, namun lebih dengan pendekatan kobstruktivis dengan bahasa yabg dapat dinengerti, sehingga konsep yang diingkan akan dapat ditransfer dan diaplikasikan dengan baik oleh subjek itu sendiri