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I think that the standard practice in the first grades when addition (or other operation) is taught as a "process" may be not so good. I always wondered why so many children lose interest in math immediately after arithmetics. And i think this may be one of the reasons. 1+2=3 viewed as: this is a process performed on two numbers to get a result. I think that, with this approach, we affect their future capability to think abstractly in mathematics and their understanding of algebra. Sometimes for good! I think instead we should view 1+2=3 as a relation between two numbers, on one side and another number on the other side. Relation, and not process. With this process idea they remain with the idea that "1+2" is not a true number, but only an "exercise to be performed " and "=" means just "compute". Which would be disastruous in algebra, where "=" can't have such meaning.

What do you think about it? Could this be the biggest reason why there is this big gap, children who are intelligent and bright in arithmetics suddenly get stuck in algebra?? Who in the world invented this "operation=process" idea???

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  • $\begingroup$ Surely the reason children get stuck in algebra is because algebra is difficult. There are better and worse ways to teach things, but no way of teaching is suddenly going to make algebra easy. $\endgroup$ – Henry Towsner Dec 31 '18 at 2:04
  • $\begingroup$ I think the human brain is not as well wired for math as it is for language, visual processing, sex, social relations, etc. Math is just hard for us walking apes. Look how easy it is to program a computer for math but how hard it is to program it for the topics I mentioned. We have a LOT of wiring with dedicated routines for the former topics. Not much for math. $\endgroup$ – guest Dec 31 '18 at 4:00
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    $\begingroup$ Perhaps the problem stems from the way multiplication (among other things) is taught in schools these days -- "strategies" instead of good old-fashioned memorization. Algebra at the introductory level elementary/high-school level is just working arithmetic backwards. The "strategies" for multiplication, for example, cannot be worked backwards. You can't do finger math to determine what number 8 times is 56. You just have to know that 7 times 8 is 56, without even thinking about it and wondering what it all means. $\endgroup$ – Dan Christensen Jan 2 at 2:49
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    $\begingroup$ @StevenGubkin Yikes! If a child must resort to such labourious mental gymnastics to determine what number times 8 equals 56, it is no wonder that they are finding algebra so difficult. $\endgroup$ – Dan Christensen Jan 2 at 16:52
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    $\begingroup$ @DanChristensen Would you think it "fair" to ask a student to factor 57? Your students have probably not memorized the x3 table up to 19, or the x19 table at all. So they will have to use "strategies" to factor this. In this case it is easy: you should notice that 57 is just 3 away from an "easy" factor of 3. Namely 57 = (60-3) = 3x(20-1) = 3x19. $\endgroup$ – Steven Gubkin Jan 7 at 13:12
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@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!

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    $\begingroup$ 1) Algebra requires treating equations as objects that can be observed and manipulated, and requires working with variables as entities in their own right (rather than just placeholders for numbers). This is a big jump in abstraction. $\endgroup$ – Henry Towsner Jan 1 at 4:15
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    $\begingroup$ 2) "It's normal to struggle" is not the same as "the teacher can't do anything". Good teaching lowers how much students struggle and helps students get through those struggles without hating the topic. Those differences are valuable, but they're incremental. It may well be that teaching arithmetic differently could make the transition to algebra easier, but it's not helpful to start that discussion with unrealistic expectations . (I have no opinion on the particular change you're proposing, incidentally.) $\endgroup$ – Henry Towsner Jan 1 at 4:20
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    $\begingroup$ 3) Having just watched an 8 month old who's learning to walk, it's particularly on my mind just how much struggling is a normal part of learning. $\endgroup$ – Henry Towsner Jan 1 at 4:30

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