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by housecarpenter
729 days ago
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The way I see it, the distinction between = and ≡ isn't really to do with equality having more than one meaning. An "identity" like sin^2 x + cos^2 x ≡ 1 is really just shorthand for "for every real number x, we have sin^2 x + cos^2 x = 1". The equals sign here has the same meaning as in a statement such as "there is a real number x such that sin^2 x + cos^2 x = 1"; the difference is in the surrounding language, and what meaning it assigns to the variable x. So perhaps rather than just emphasizing the difference between = and ≡ more, it would be better to go further and emphasize the difference between universal and existential quantification more. Quantifiers can be confusing, but I think people also find having two different equals signs confusing; and it wouldn't be necessary to give a full account of predicate logic to high schoolers, I'm thinking more of just informally describing the difference between "for all" and "there exists" and reminding them that a bare variable has no meaning if you don't know what set it ranges over and how it's quantified. This is just my speculation, I have no experience with mathematical education whatsoever. |
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Right, I've uni math also formal logic so I've knowledge of universal and existential quantification, etc. thus an understanding of the issues.
You're right, that stuff's a bit too heavy for highschoolers. Perhaps all that's needed is to be told 'that at times these appear the same but later on you'll need to understand there's a mathematical distinction' then emphasize the difference aspect to drive the point home.
Even though it was a long time ago I mostly recall what the teacher said and whilst he gave a few examples he never emphasized that there was a mathematical difference and that it was an important fact to know. Matters became more ambiguous from our science courses, the use of 'equivalent' was very loose.
I reckon the same or similar should apply to other topics, calculus for example.