@Henry Townsner:
Ok, so let's be specific: what exactly are the difficulties students experience when transitioning to algebra and why? Specific, not general statements as "more abstract understanding of numbers" or "building their own model of how math works", that is too general.
I just wonder WHY CLEVER students begin to hate math when they encounter algebra. In my opinion it HAS something to do with teaching. We cannot just lay there and just say "algebra is difficult, so it is normal to struggle " and watch them struyggle and stryggle and hating math more and more, while they once loved it. We should try to find what exactly are the difficulties, why they are happening and what we can do to make students overcome those difficulties.
And speaking about "building their own way of how math works", even if this is a too general, non specific idea, i think the problem is just here, we teach arithmetical operations in the "process-result" manner , a manner which is unsuitable for understanding algebra.
And this is not an original idea of mine, there is a ton of research on the subject, see for example articles by Anna Sfard.
That's why they are so confused when starting to learn algebra, because the old process -result view of operations is no more suitable and they are just confused ...and it is quite normal that they start to hate math.
I don;t "Act" like algebra is simple or difficult, but let's be realistic, what's the main difference between arithmetics and algebra? It is simply the presence of variables and unknowns, aka the "letters".
And i honestly don't think that the letters denoting variables and unknowns are the problem. And i say that because studnets solve equations in arithmetic too. It is just that instead of the "intimidating" letters we have blank spaces or question marks or small circles, etc...
Arithmetic is full of equations like 2*?+3=17 so students perform excellently at equations of the form ax+b=c.
The ones of the type ax+b=cx+d are the problem. Because in ax+b=c they still use the old arithmetic interpretation of "=" as meaning "do the computation" but in ax+b=cx+d "=" cannot have the same meanning. So here they encounter a major confusion! Produced exactly by the "process-result" way we teach them arithmetics!
What i want to say is that we should make the best effort from the start to avoid them seeing "=" as "compute" and "a+b" like a process with the result c. And "=" as compute!
We should from the start make them see "2+3" as a DESCRITPTION OF A NUMBER, and not a process, and "=" in 2+3=5 to mean "is the same number(with different "descriptions") and NOT "compute" or "evaluate" or other crap.
even in the practical way we teach them counting with real objects, the second you put 2 coins next to three coins you there have an exact number of 5 coins, there is no further "process" to perform and no "result".
So what i think is that someone invented this ridiculous "process-result" interpretation of addition just to make arithmetic simpler to teach. But with the price that when advancing to algebra students will be confused and have more difficulties.
So we make arithmetic "simpler" (even if that might not be necessary) but with the price of compromising the tyransition to algebra for most students... How silly is that??
We just made them struggle with algebra, and know we just lay there and say "algebra is difficult, iot is normal to struggle bla bla, tacher can't do anything, just tell them math is hard and they should struggle bla bla". How wrong is that?
No, IT IS NOT NORMAL that kids have to struggle, in a well designed teaching system! They should not struggle! If they struggle it means the teaching is not optimal!
