Fascinating. You and I work and assign value/utility in totally antithetical ways! I agree, it’s not useful to say one is objectively superior … instead both are subjectively superior (for our respective selves). Fascinating!!

Ah, I remember you *think* like this from an old post – you use multipliers to move from side to side! Slightly related, I’ve taught myself to use cosecant, secant and cotangent … so I now *think* with them … I must say it’s liberating!! There’s an incredible feeling of fluency using all 6 lines.

Do you think algebraically? I mean, do you do the algebra in your head? (This is something I’m not very good at … I need pencil and paper. With a geometric approach for some reason I’m much happier doing it in my head.)

lol … a good friend of mine is exactly the same as you … he always takes the algebraic route and I always take the geometric. I feel more fluent working geometrically. I don’t understand why I don’t feel this with algebra. I feel algebra takes me out of the problem and geometry keeps me inside.

Thank you David for exploring this. I find with the sine rule that I *think* algebraically … but with a double application of right-angled-trig I think *geometrically* … and I prefer the latter.

Unintended actually ;-) (Just exploring why the sine rule is a thing … having thought about it I think I prefer a double application of right-angled-trig-similar-triangles … simply because I’m faster this way.)

Ah yes … the 32° / 116° was unintentional, lol. But thank you for reminding me it can be done like this (… did take me a moment to get it!). Again, I was exploring why the sine rule is a thing … I can’t not see this (now) as 9sin 32 cosec 116. I find it fairly immediate and less brain power. Hmm.

Yep … I posted it because I’m questioning (to myself) why the sine rule is a thing … I mean when I look at this I now can’t not see 4sin 56 cosec 39 … and it’s pretty immediate … for some reason I’m quicker this way than via the sine rule (for triangles like this).

Good point. Here’s where we need an expert to comment. I’m not sure where the boundary is between statistical inference and semantic inference. I’m assuming semantic inference is bolted on to the LLM but that the LLM is the driver of ‘decision making’. But I don’t actually know.

lol … the more the merrier ;-)

When you say ‘tiles’ do you mean tiles … or do you just mean the standard Khwarizmian-type area diagram? This is what confuses me … one doesn’t need ‘tiles’ for this … it’s just a traditional ‘surveyors’ area sketch.

I think the pertinent points (for me) which you’ve said are: 1) There’s a pedagogical responsibility to present multiple representations. 2) It’s okay to take the representation as only as far as works for you as the teacher.

Not the same but it reminds me of 14 million ÷ 2 million … and the surprising number of my pupils (including the head teacher actually) who said 7 million.

I find this happens a lot. They’re statistical … we shouldn’t forget that.

(− 1 + 20)(20 + − 2) … ;-) … but again I see your point.

It’s worth saying that in the 9th century Islamic mathematicians used geometry to validate their new algebra. I’m not advocating one does this today, just making the point that such practice has a long history.

Measuring is good! Always makes a good lesson. Sounds great :-)

Your phrase ‘to practice mentally manipulating shapes as well as numbers’ has stayed with me … and it’s true, we spend a lot of time teaching the latter and far less time teaching the former.

Yes and sorry if I sounded a bit dismissive. I will try and be more useful next time!

That is indeed true. Wise words. I maybe sounded a bit dismissive of their efforts!