[quitit]

Where school kids tend to get stuck is that they'll hold contradictory views on how fractions can be represented.

First it'll be uncontroversial that ⅓ = 0.333... usually because it's familiar to them and they've seen it frequently with calculators.

However they'll then they'll get stuck with 0.999... and posit that it is not equal to 1/1, because there must "always be some infinitesimally small amount difference from one".

However here lies the contradiction, because on one hand they accept that 0.333... is equal to ⅓, and not some infinitesimally small amount away from ⅓, but on the other hand they won't extend that standard to 0.999...

Once you tackle the problem of "you have to be consistent in your rules for representing fractions", then you've usually cracked the block in their thinking.

Another way of thinking about it is to suggest that 0.999.. is indistinguishable from 1.

[Suppafly]

>However they'll then they'll get stuck with 0.999... and posit that it is not equal to 1/1, because there must "always be some infinitesimally small amount difference from one".

Honestly teachers are half of the problem because they seem to make a game out of pointing out these sorts of contradictions instead of teaching the idea that you need "to be consistent in your rules for representing fractions".

That and every next step in math classes is the teacher explaining that most of how you were taught to think about math in the previous step was incorrect and you really should think about it this way, only to be told that again the next year.