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When I help in the classroom, I tend to get tasked with helping kids with math. It's like the teachers think that I must be the reason why my kids seem to intuit these things. So I see a lot of kids who are middleofthepack for their grade. This is a very high performing district where just about every kid is passing the state math exams at all grade levels tested (3+).
Here's the thing: Kids in first and second grade tend not to intuit this kind of thing. They can learn the rote rule for manipulating numbers, or count on their fingers (or number line or....) Teaching them to reshape their brains to do it differently takes work. For many children, it takes many, many exposures to a variety of different ways to do something to give them sufficient tools to identify the right (=fastest/easiest) solution. If they are not intuiting the short cut, they can't use it.
The fact of the matter is, though, that mental math strategies are an excellent life skill. I frequently add or subtract multiples of 7 in my head (figuring out dates without a calendar), multiply by 0.15 (tips), and multiply times 25 (figure out how much further I can go on that last gallon in the tank). If you can't multiply by 0.15 in your head, or multiply times 0.1 and add half again, then you need a piece of scratch paper just to calculate a tip. Or just tip inappropriately. Or get a crutch like an iphone app and spend $0.99 just to calculate a silly tip.
My own family's experience on two kids is that my daughter was taught this stuff in kindergarten and ran with it. She never looked back, and ±2 and similar strategies are just natural to her now that she was introduced to the idea. She's now 2 years accelerated in math (4th grader doing 6th grade math with ease), something that her school does once every 23 years, so for 1 kid in every 300 or so. DS was just accelerated 2 years in math (K>2), and will get a minimum of another year when he hits 4th grade, putting him 3 years accelerated 2 years years from now. The school curriculum coordinator just outlined for us a possible path that puts him in calculus as an 8th grader. The school district hasn't done this before, nor anything close. My son clearly came out of the womb with the intuition for a variety of mental math techniques, made possible by an innate understanding of what can be done with numbers. He did not need to be taught to do 2x+1, because he's already been using it since before he was 2.5. He amazes me daily when he verbalizes his reasoning for getting to an answer, most recently explaining to me that he thought that the distance around the Earth must be about 24000 miles, but he thought it might be a bit further. (He's under by 3%. He was disappointed he was that far off after I divided it out on my calculator.).
I think that for those of us here who come to this forum tend to watch math development in our kids with wonder. I don't think this is a common experience in the general population.
Tjej: The commutative property teaches kids that 3+4 is the same as 4+3. In kindergarten, this isn't taught by introducing the abstract property, but instead through repeated exposure to multiple examples. It's necessary to understand in the abstract when you get to algebra. Hopefully by the time a kid gets to algebra, their minds have been taught enough concrete examples that they are ready to move to the abstract. DD hit the terms describing the cummutative/associative/distributive properties in 4th grade ALEK, iirc, and in 6th grade math at school.
OK, so now I gotta ask  I went through enough of MM's history to see she's got a second grader. MM, can you tell us a little about your own experiences as well as the experiences of your daughter (or other kids?). I'm really curious what you both have experienced.
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captian optimism  IIRC, the commutative property is the one that says 3+4=7 and 4+3=7  position doesn't matter. I never really got the idea of this being a rule either. To me it was just stating the obvious of how #'s work. Like 3 buttons = 3. I don't think that needs a special button rule. :)
Tjej
Hm, interesting you said that.
It's true  it's obvious for integers. But there are mathematical objects where this isn't true  think about rotation in 3d. This is the kind of thing I have in mind when thinking/worrying perhaps kids don't get exposed to the bigger picture. If you're next to me now, I can show you why and chances are you'll get it right away because well you will see it  but somehow it's not taught in schools. The mathy kind will pick things up like this because they're interested, but everybody else can do too ... Anyway ...
The fact of the matter is, though, that mental math strategies are an excellent life skill. I frequently add or subtract multiples of 7 in my head (figuring out dates without a calendar), multiply by 0.15 (tips), and multiply times 25 (figure out how much further I can go on that last gallon in the tank). If you can't multiply by 0.15 in your head, or multiply times 0.1 and add half again, then you need a piece of scratch paper just to calculate a tip. Or just tip inappropriately. Or get a crutch like an iphone app and spend $0.99 just to calculate a silly tip.
...
OK, so now I gotta ask  I went through enough of MM's history to see she's got a second grader. MM, can you tell us a little about your own experiences as well as the experiences of your daughter (or other kids?). I'm really curious what you both have experienced.
I use those sorts of tricks all the times too ... they make life easier.
I'm still digesting this thread myself ... but anyway ...
I checked this thread because I don't have the slightest idea what kids do in regular school, mine is in some alternative program where they don't even do math as a separate subject, they take an integrated approach to all subjects. We've refused testing the last few years, dd is happily in a mixedage class and can explore other interests at school. She learns math mostly from us and a math club.
I don't know what to say in general about how we do math, never really think about this ...
Something specific perhaps ... with addition, we started with fingers/toes. Soon, that led to the idea of infinity, which led to the idea of mapping. We never did subtraction separately from addition  use negative numbers instead. Then, she wanted to know if you can add things that are not numbers  so she learnt that numbers are scalars ...
For the computational parts, arithmetics I guess, her teacher thinks she's a few years ahead. What we do quite differently at home is the conceptual parts, more abstract math. We recently talked about equivalence relation and modulo, remainders, and all that. We've also started talking about prime numbers, which leads to fundamental theorem of arithmetics. She also learnt about boolean logic too, so she can get some taste of what proofs look like.
OK mamas, I'm talked out, I don't really think about this stuff ... Math, yes, but math education no  I was just surprised when I first read this thread.
I'm not talked out. I find math education and development of mathematical understanding fascinating.
MM, it sounds like your family would fit right in with ours, where dinner table discussions find us flittering from fractals to music and octaves to imaginary numbers.
Cool, Geofizz!
Yes, right, that's what I thought the commutative property was! We (my kid and I) read the first chapter of an algebra textbook together and it was all about these rules that are just names slapped on stuff that's really obvious. But isn't this rule under discussion sort of like that, too? It's like, uh,
(X+1) + X= 2X+1
right? Isn't that also the commutative property, or do we have to call it something else if there is multiplication?
Anyway, I learned a lot of math trying to help him because of his asynchronous development. What I mean is, he couldn't read the books that had the stuff in them he wanted to know, so I wound up reading a lot of books to him about math for a popular audience. I do not remember any of this stuff being any fun until I did it with him. I sort of knew what fractals were and how music related to math and so on, but now I'm responsible for his experience of this, and I need it to be good. He has a good experience of math at school, don't get me wrong, but it's the experience of praise and approval for knowing how to do it all, not the experience of learning new things.
captian optimism  IIRC, the commutative property is the one that says 3+4=7 and 4+3=7  position doesn't matter. I never really got the idea of this being a rule either. To me it was just stating the obvious of how #'s work. Like 3 buttons = 3. I don't think that needs a special button rule. :)
Tjej
Divorced mom of one awesome boy born 232003.
Hm, interesting you said that.
It's true  it's obvious for integers. But there are mathematical objects where this isn't true  think about rotation in 3d. This is the kind of thing I have in mind when thinking/worrying perhaps kids don't get exposed to the bigger picture. If you're next to me now, I can show you why and chances are you'll get it right away because well you will see it  but somehow it's not taught in schools. The mathy kind will pick things up like this because they're interested, but everybody else can do too ... Anyway ...
Admittedly, you've either outmathed me on this one (which is very possible), or I am suffering from newborn brain again. I don't understand how that general concept doesn't apply to 3d objects. If I rotate a shovel in the air, it is still going to be the same shape and size. The length of the shovel never changes. Its reach in any particular direction might change, but the shovel itself is always the same shovel. But from my understanding, the commutative property rule itself isn't created to describe 3d objects. But again, I am easy to outmath.
Tjej
Yes, right, that's what I thought the commutative property was! We (my kid and I) read the first chapter of an algebra textbook together and it was all about these rules that are just names slapped on stuff that's really obvious. But isn't this rule under discussion sort of like that, too? It's like, uh,
(X+1) + X= 2X+1
right? Isn't that also the commutative property, or do we have to call it something else if there is multiplication?
I'm not sure. I think it is just the concept that a things value doesn't change. It holds true in the equation you gave, but it doesn't really communicate the idea of simplifying math by changing things into easier math and then modifying to get the right answer. If that makes sense.
AoPS Pre Algebra does a really good job of presenting the commutative and associative properties in very simple terms, and then very elegantly revealing how they work as justifications for various algebra steps. I really like the analogy to the three buttonsI had thought the same thing, why isn't this just obvious? When DS was doing AoPS this fall, it was an aha moment for me of seeing the properties as useful scaffolding.
Really interesting to hear so many mathy kids here coming up with doubles... I think I remember DS first playing around with nondouble additionthree peas in one pod half and four in another.
I also remember DS being really delighted by the idea of adding 1.99 and 1.99 in a store.
Geofizz, it was also so interesting to hear your experience working with children in the classroom and the usefulness of working on different strategies and ways of understanding. DS really wanted to find his own strategy and algorithm, so our experience also with him was more facilitating than teaching a checklist of strategies.
Heather
If I rotate a shovel in the air, it is still going to be the same shape and size. The length of the shovel never changes. Its reach in any particular direction might change, but the shovel itself is always the same shovel. But from my understanding, the commutative property rule itself isn't created to describe 3d objects. But again, I am easy to outmath.
It's about 3dspace  not 3d objects. Yup, you're right, the shovel stays the same shape, but its final orientation depends on the order you do rotations.
I'm not sure. I think it is just the concept that a things value doesn't change. It holds true in the equation you gave, but it doesn't really communicate the idea of simplifying math by changing things into easier math and then modifying to get the right answer. If that makes sense.
But that is the doubling trick. Right? Isn't the trick:
42+43=2(42) +1?
Which is the same as X+(X+1)=2X+1
It's not changing things to easier math to get the right answer. It's expressing the same equation in two different ways. First you have to recognize that 43=42+1, though. (Which, obviously, the OP's child did.)
I don't think this precludes learning, "when you see two twodigit numbers, add up the ones and then add up the tens column."
Divorced mom of one awesome boy born 232003.

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