Yeah that's a good approach. The problem remains though that if you use the + operator on ratios you're still overloading it to mean something different in a way that doesn't retain its meaning when you start expressing things as fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just spitballing here. There's definitely enough options in Unicode that an existing operator should be suitable for this purpose: https://en.wikipedia.org/wiki/Mathematical_operators_and_sym...
And + is overloaded in a bunch of ways students encounter in high school, and much time is spent talking about when you're allowed to add and when you're not and which rules apply when. Examples:
1 + 2 - fine.
1/2 + 1/2 - one set of rules.
1/2 + 1/3 - a subtly different set of rules.
1:20 + 0:45 - yet another set of rules. Modular.
30° + 350° - fine? But maybe modular.
It would be lovely if mathematics were taught as a strongly typed language without overloaded operators, alas all our corpus is in the language it's in.
Indeed. And if they're cartesian vectors, you're good. But if that second number is an angle measured in radians, you use yet a different set of rules for the addition.
The conversation with 16 year olds when you explain that their previous teacher who told them that you couldn't add points wasn't lying, but was, perhaps oversimplifying things to make their life easier, is a fun and fraught one.
I've had to reason kids through the fact that a^2 + b^2 is not equal to (ab)^2 or even (a+b)^2 more times than I can count. What's particularly difficult is that, confronted with the fact that 25 and 49 are manifestly different numbers, many still cling to the rule that a^2 + b^2 = (a+b)^2, because of the "law of distribution", which they haven't learned as the "law of distribution of multiplication of monomials over addition, and only that".
How did I go all of my life so far without hearing the super useful word "monomial"? It's such an obvious concept to have a meaningful name, and yet I don't recall anyone ever having said that word.
I also didn't encounter it until I was relatively older. Our school has been pushing to introduce more rigorous language and definition in our 7-12th math program. For some students, it really seems to help. For others, it's really hard.
I mean, the general case is certainly trouble for kids. But 15% and 20% look very similar to 15x and 20x. That the two (sometimes!) operate using different rules causes confusion for many students. The implicitness of the fractions conceals something important that's explicit if you write out the fractions. For example, multiplying percents does not do what most students intuitively think it does.
15% of 8 times 20% of 10 isn't 35% of any nice arithmetic combination of 8 and 10. That's hard to communicate to many students.
The main thing I find myself explaining with percentages (IANA educator) is that 'grossing up' != adding the same percentage back, e.g. 80% * 1.2 != 100%, which looks pretty obvious like that, but it's a common mistake among adults talking about real life percentages like taxes.
Another common, and I suppose related, one (but that I don't bother correcting often) is 'percent' != 'percentage point'. Talk of 47% of something being '3% less than' half of it really winds me up - and it's stupidly common among journalists - but it's too common to bother pointing out IMO. Live and let get wrong.
