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Let me give an example of the sort of thing I believe you're saying. If we have a right triangle, then knowing one of the (non-right) angles is enough to totally tell you the shape of the triangle, because you can figure out the last angle by taking the complement. (1/1)

That doesn't tell us the length of each side of the triangle, because you could double all the sides and get a triangle the same shape as the first one, but with different side lengths. But that's the only way that two things of the same shape can differ, by similarity. (2/)

So the *ratio* of any two given side lengths in the triangle -- that is, how much longer or shorter one is than the other -- is determined by that one angle. And if you start with two lines whose lengths are in that ratio, and put them at that angle from each other, and connect them up... (3/)

I think I might be agreeing with you! I believe there is a stage of "getting a sense of" a concept which precedes and enables "making sense of" that concept. The transition is assisted by moving from an informal language of description to the technical language of definition. #UKMathsChat

I think at A level we often never move to that technical language of definition, (it's a bug bear of my sister who teaches at uni level that students don't come in with proper definitions). But I don't think students are helped by fuzzy language, which tends to be a problem with informal language.

Imo fuzzy language is often associated with fuzzy concept. An example of one of many irritants is: find the area of a circle.

It's tricky, because formal definitions really do require a certain level of mathematical sophistication that's beyond most pre-uni students (and is probably of no use to the vast majority of them). The interesting part is trying to get as *close* to formalism as you can!

Old post but pinned so i ll go with my opinion. I would say saying terminology is unnecessary for novice is, to make an anology, "demagogic". Yes of course, concept, but I'd say even more so, developping intuition is crucial. And, btw, intuition is the real language that you cannot speak.

Ah, but I'm not saying that terminology is unnecessary for novices. I'm saying that we should consider carefully when we introduce it and try to do so in a manner that aids learning, not inhibits it.

You'll be suggesting that I think about my teaching next...

But we need to exchange, explain to other, and explain to oneself. Mathematical terminology, there, is sometimes the most efficient, hence worthy to hear and use.

Not really an example of the above, but: A year 8 I teach, nurture set, with many students with educational needs and low prior attainments, are happy to use the word "cubed" when they see 4×4×4 and tell me their answer.

Fascinating. I hate absolutely everything about this! If "proper terminology" doesn't align well with "proper concepts", surely it's bad terminology!

There's a big difference between teaching using "proper" terminology, and teaching using language that makes sense, but might lack precision or nuance. It can be very easy to get caught up in "correcting" language, and forget that you don't actually need it to understand the concept.

Consider we have a sense of quantity but don't have numbers. We have stones that we can arrange in piles and patterns. A natural way to arrange stones is in rectangles *if you can*. I use this with 3-5 year olds and very quickly they are exploring and finding factorizations and identifying primes.

I am a *huge fan* of proper and consistent use of terminology. But I think there are contexts like this where it would be a huge mistake to offer the terminology without the need for it or until it is useful to support expanding understanding.

Is this an example of Andrew's point, though? Of course it's a good idea to get a sense of a concept before introducing terminology for it, but the point of learning the terminology is that it then becomes easier to talk about the concept.