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Added explanations on the "behaviorist" and "socioconstructivist" part
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orion2112
orion2112
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To clarify, there is currently a shift in paradigm in teaching (well, it depends on the country but I think you'll understand). Essentially, it isn't about "how should we teach this", it's about "how should they learn this".

I'll try to explain it (grossly) here.

Before, the prerogative was that students were "entirely ignorant" and "wrong" until they got the same answer as the teacher. In other words, if a student does something that doesn't meet the teacher's standards, it's rubbish. So the teacher's job was simply to "punish" a student if he had different answers and "reward" if he had the same answers as the teacher. (By "punish", I mean things like saying "that is wrong", or marking 0, etc.) There is basically no regard to what goes on in the students mind, it's all about changing their behavior until they "learn" to do it right (like the teacher). This is, essentially, behaviorism.

Nowadays, this kind of approach is less and less popular. One of the reasons is that we know that kids are capable of being creative and telling them to meet very specific standards essentially inhibits that creativity. (Instead of learning the content, they learn how to do what teachers' want). So if you try to teach a 12 year-old how to do proofs and you tell him to start over until he's doing it just like you want (which, if I understood you correctly, sounds like the work of a much more advanced student), he'll end up completely lost, discouraged and unmotivated. At beast, he'd be able to copy a proof that you wrote on the board, but without understanding very much.

But an even better reason not to use this approach (behaviorism) is because there are better ones out there now, that implement what we now know about metacognition (knowing how we learn), with the advent of research in psychology, mainly.

Here's the picture:

Students arrive in class with their own conception of reality (or "mathematical truth" in our case). They learn new things (what the teacher tells them, what they read in books, what they hear from other students). From this new knowledge, they modify and improve their original conception of reality; they construct a new version of what they think reality is. It might not be perfect yet, but the idea is that this new version is (in general) better than the old one. In another course, or later in life, that same knowledge might be improved further as the student experiences more things. That's constructivism. So this is a pedagogical approach where knowledge is not fixed in place, but rather, is something that is constructed throughout one's life.

Socioconstructivism is essentially constructivism made through interactions with other people (peers, the teacher). Activities like group work, debates and such are characteristic of socioconstructivism. For example, you ask students a question. They answer it on their own. Then they trade their work with a neighbour. If they disagree, they have to argue with them to try and convince them that their answer is better. If they agree, they try to see if the neighbour's work had any interesting differences. By doing this, the students might glean some insight on the content that they wouldn't have had if they had only listened to the teacher.

In case you are worried that "everyone constructs their own reality" is a dangerous view in mathematics, that is where the teacher comes in. His/her role is (among other things) to make sure that what is assimilated (through constructivist activities) is not erroneous.

So if you have a 12 year old who "proves" in a very naive or intuitive way some statement, he's basically using all of his previous knowledge to try and come up with something convincing enough. To expect anything more abstract is a bit too much, since he/she doesn't have any background knowledge on how to do a proper proof. You can start doing proofs when you have some background knowledge about math. But when you first learn math, your background knowledge is just a bunch of intuitions and ideas that are probably littered with falsehoods; so that background knowledge ought to be fleshed out a bit.

For example, when you first learn that $1+1=2$, do you have to know that you are using a binary operation that is closed in some field? No. But if you want to make proofs involving fields, you ought to know how $+$ works in some intuitive way.

So to conclude with the "paradigm shift" I was talking about, instead of focusing on the content we want to pass on, we should focus on how to pass on the content first and socioconstructivist strategies are much richer than behaviorist ones, for the learners. I'm not entirely against teaching proofs, but the way you presented it goes against the new paradigm, so I'm hoping this answers your question.

To clarify, there is currently a shift in paradigm in teaching (well, it depends on the country but I think you'll understand). Essentially, it isn't about "how should we teach this", it's about "how should they learn this".

I'll try to explain it (grossly) here.

Before, the prerogative was that students were "entirely ignorant" and "wrong" until they got the same answer as the teacher. In other words, if a student does something that doesn't meet the teacher's standards, it's rubbish. So the teacher's job was simply to "punish" a student if he had different answers and "reward" if he had the same answers as the teacher. (By "punish", I mean things like saying "that is wrong", or marking 0, etc.) There is basically no regard to what goes on in the students mind, it's all about changing their behavior until they "learn" to do it right (like the teacher). This is, essentially, behaviorism.

Nowadays, this kind of approach is less and less popular. One of the reasons is that we know that kids are capable of being creative and telling them to meet very specific standards essentially inhibits that creativity. (Instead of learning the content, they learn how to do what teachers' want). So if you try to teach a 12 year-old how to do proofs and you tell him to start over until he's doing it just like you want (which, if I understood you correctly, sounds like the work of a much more advanced student), he'll end up completely lost, discouraged and unmotivated. At beast, he'd be able to copy a proof that you wrote on the board, but without understanding very much.

But an even better reason not to use this approach (behaviorism) is because there are better ones out there now, that implement what we now know about metacognition (knowing how we learn), with the advent of research in psychology, mainly.

Here's the picture:

Students arrive in class with their own conception of reality (or "mathematical truth" in our case). They learn new things (what the teacher tells them, what they read in books, what they hear from other students). From this new knowledge, they modify and improve their original conception of reality; they construct a new version of what they think reality is. It might not be perfect yet, but the idea is that this new version is (in general) better than the old one. In another course, or later in life, that same knowledge might be improved further as the student experiences more things. That's constructivism. So this is a pedagogical approach where knowledge is not fixed in place, but rather, is something that is constructed throughout one's life.

Socioconstructivism is essentially constructivism made through interactions with other people (peers, the teacher). Activities like group work, debates and such are characteristic of socioconstructivism. For example, you ask students a question. They answer it on their own. Then they trade their work with a neighbour. If they disagree, they have to argue with them to try and convince them that their answer is better. If they agree, they try to see if the neighbour's work had any interesting differences. By doing this, the students might glean some insight on the content that they wouldn't have had if they had only listened to the teacher.

In case you are worried that "everyone constructs their own reality" is a dangerous view in mathematics, that is where the teacher comes in. His/her role is (among other things) to make sure that what is assimilated (through constructivist activities) is not erroneous.

So if you have a 12 year old who "proves" in a very naive or intuitive way some statement, he's basically using all of his previous knowledge to try and come up with something convincing enough. To expect anything more abstract is a bit too much, since he/she doesn't have any background knowledge on how to do a proper proof. You can start doing proofs when you have some background knowledge about math. But when you first learn math, your background knowledge is just a bunch of intuitions and ideas that are probably littered with falsehoods; so that background knowledge ought to be fleshed out a bit.

For example, when you first learn that $1+1=2$, do you have to know that you are using a binary operation that is closed in some field? No. But if you want to make proofs involving fields, you ought to know how $+$ works in some intuitive way.

So to conclude with the "paradigm shift" I was talking about, instead of focusing on the content we want to pass on, we should focus on how to pass on the content first and socioconstructivist strategies are much richer than behaviorist ones, for the learners. I'm not entirely against teaching proofs, but the way you presented it goes against the new paradigm, so I'm hoping this answers your question.

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orion2112
orion2112
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I understand the question in the same way Tommi Brander did but I would like to give my own answer. Maybe I misunderstood the question or the age group you meant, but I assumed it to be children who are about 12-13 or under and who are beginners at learning fractions (for instance).

First off, I’m not sure where you get your premise that “learning something young means that that thing is more likely not to be learned properly”.

The article you linked mostly cites studies about Denmark kids. In particular, it says at the end that “In Denmark […] there is universal access to decent pre-kindergarten. Relatively few American cities and towns offer the same thing.” Put that way, of course you would expect kids who are put through better preparation would do better later in life.

The article cites one study about OECD countries, including data from Canada and the United States which says that kids who started school later are more likely to attend University.

I can’t conclude that “not attending University” means that I probably learned fractions wrong. So I’ll remain skeptical with the premise.

Second, even if we were to assume that the studies are right, the thing to do would be to postpone the age at which we start learning (at school), not replace the current content by other content. If kids are truly “getting fractions wrong” right now because they learn too soon, then they would learn number theory wrong instead…

Third, you seem to think that learning fractions is harder than doing number theory proofs. Imagine for one minute that you are a kid and you see this:

$$\forall x\in\mathbb{N}x+S(y)=S(x+y)$$

and this:

$$2-\frac{5}{6}$$

which do you think is easier to understand, or at least would take less time to understand properly?

You also suggest that kids who give “an intuitive reason” to “a certain statement that is obviously true” would be told to start again until “a formal proof” is written and that they can’t “collaborate with another student”. The whole group stays "locked" in that spot until everybody gets it “right”. This is so disgustingly behaviorist and so far off from modern socioconstructivism that I don’t think this would be applicable in the near future.

Children are not full-fledged mathematicians. Doing proofs requires some mathematical maturity that kids don’t have. It sounds like you want them to be able to do math the way a mathematician does, before they even start learning anything about math.