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.

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".

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.

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.

@Doug:

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.

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.

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