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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???

Dominique
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amarius8312
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  • 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. – Henry Towsner Dec 31 '18 at 02:04
  • 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. – guest Dec 31 '18 at 04:00
  • @Henry...what's so difficult in algebra? That a smart kid couldn't grasp? Thevdifficulty of algebra is not about reasoning, in respect to reasoning, arithmetics may need more clever reasoning, jusr think about solving arithmetically a problem that can be solved algebraicallt by a system of equations, the arithmetic solution needs more creativity, more thinking, more steps. Where in algebra we write thevequations and then is a mechanical routine to solve them. But we teach arithmetics in such a way that creates a big gap towards algebra. – amarius8312 Dec 31 '18 at 05:32
  • Thebmain change when kids encounter algebra is that now we have unknown quantities, denoted by letters. If we wouldn't teach arithmetics with that " process-result" interpretation, this would be the only difference. It is this stupid "process" vuew that makes understanding and solving equations differenr and difficult. – amarius8312 Dec 31 '18 at 05:36
  • @guest. I don't agree with your viewpoint that our brains are not wired for math. They are. Math was not invented by some people who got bored, math simply is how the universe around us is functioning. I woukd agree with you if kids struggled with math from the beggining, from arithmetics. But i was talking about kids who ar good in arirhmetics, and thet love it, but when we progress to algebra they suddenly lose their interest and begin to hate math. And what does sex have to do here? Sex is just a basic instinct, wired in the reptilian brain. It has almost nothing to do withv he cortex. – amarius8312 Dec 31 '18 at 05:46
  • @ guest Math is all about logic and relations between "things", about reasoning and understanding. You can't say that the human brain is not wired for reasoning and understanding relations and connections between things. That is simply the purpose of the brain. So come on...even crows can " count" to 5 or 6. There's nothing that hard in math. And speaking about languages, the problem may be the symbolic language of mathematics and the way we teach it. – amarius8312 Dec 31 '18 at 05:54
  • There's nothing that hard in BASIC math, not talking about Galois theory or proof of Fermat last theorem here. Talkinf about x+1=2x+3 and such... And speaking about languages, the problem may be the symbolic language of mathematics and the way we teach it. – amarius8312 Dec 31 '18 at 06:02
  • Something doesn't have to be as complicated as the proof of Fermat's last theorem for it to be difficult. Understanding what's going on in algebra requires developing a more abstract understanding of numbers. That's genuinely difficult (particularly given the brain development going on in the children we teach it to). We still expect basically everyone to learn algebra. That's not because it's not difficult, it's because people are capable of doing difficult things. But we shouldn't be surprised that most people struggle on the way to doing difficult things. – Henry Towsner Dec 31 '18 at 20:25
  • The confusion students experience around the transition to algebra is the necessary process of them building their own model of how mathematics works. It's condescending and alienates students when we act like difficult subjects are easy merely because we've internalized them already. And it's naïve to think that we can shortcut that struggle by teaching things just the right way. – Henry Towsner Dec 31 '18 at 20:26
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    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. – Dan Christensen Jan 02 '19 at 02:49
  • @DanChristensen "strategies" work perfectly well for this task. For instance, a child might reason "I can fit 5 eights in giving me 40, so I have 16 left to go. Oh, that is just 2 more eights. So 7 eights all together!" – Steven Gubkin Jan 02 '19 at 16:15
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    @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. – Dan Christensen Jan 02 '19 at 16:52
  • @DanChristensen I am a professional mathematician, and if someone asked me to factor 56, I would not immediately recognize the factor of 8 or 7. I would probably repeatedly divide by 2. I do not think this has held me back as a mathematician. Also these "gymnastics" were not so laborious: in fact they were nearly identical to the thought process one uses for long division. – Steven Gubkin Jan 05 '19 at 14:18
  • @StevenGubkin Very strange. And not very helpful for a child struggling with introductory algebra, I would think. – Dan Christensen Jan 06 '19 at 13:22
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    @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. – Steven Gubkin Jan 07 '19 at 13:12
  • @StevenGubkin It has been a while,but I recall that algebra (and many other topics) were introduced with numbers that could be easily factored to start with. If students have not memorized the multiplication tables, this would not be an option for the teacher. I don't know if this is root of the current problem, but it seems reasonable to me. – Dan Christensen Jan 07 '19 at 15:21

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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!

amarius8312
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  • 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.
    • – Henry Towsner Jan 01 '19 at 04:15
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  • "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.)
    • – Henry Towsner Jan 01 '19 at 04:20
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  • 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.
    • – Henry Towsner Jan 01 '19 at 04:30