This Is What Common Core Looks Like • Hacker News Board

Re: This Is What Common Core Looks Like

#21

Post by seanflyon » Mon, Oct 06, 2014, 11:21 PM UTC

blkhp19 wrote: post #3

While I agree that one example is not enough to grasp the concept for most people, I think that the "counting up" method is actually better because it shows kids what is going on in subtraction. The borrowing method that I learned growing up doesn't make much sense. You just follow the rules with no real reason. I'm in my third year of college and only just learned why the borrowing method works - and I only learned…

Perhaps the counting up method makes it easier for most kids to grasp the underlying concepts. The borrowing method can also teach kids the underlying concepts and follows nicely from addition as it is the inverse of "carrying the 1".

> You just follow the rules with no real reason.

This is the real problem here. We should not teach our kids to memorize a set of rules but to understand the concepts. When I was in school I had a rule that I would not memorize rules that I did not understand or formulas that I could not derive. I would probably be faster at arithmetic if I had memorized my multiplication table like I was supposed to, but I think that rule served me well.

Re: This Is What Common Core Looks Like

#22

Post by adamtj » Mon, Oct 06, 2014, 11:39 PM UTC

seanflyon wrote: post #21

blkhp19 wrote: post #3

While I agree that one example is not enough to grasp the concept for most people, I think that the "counting up" method is actually better because it shows kids what is going on in subtraction. The borrowing method that I learned growing up doesn't make much sense. You just follow the rules with no real reason. I'm in my third year of college and only just learned why the borrowing method works - and I only learned…

Perhaps the counting up method makes it easier for most kids to grasp the underlying concepts. The borrowing method can also teach kids the underlying concepts and follows nicely from addition as it is the inverse of "carrying the 1". > You just follow the rules with no real reason. This is the real problem here. We should not teach our kids to memorize a set of rules but to understand the concepts. When I was in sch…

When you're in 2nd or 3rd grade, it's more important to learn the mechanics of subtraction than to understand why it works. Kids at that age need to fill their brains with facts so that they have the raw materials for developing understanding when the are more mature and more able to understand.

In this case, borrowing is more compact and efficient. That makes it faster and easier, both on paper and in your head. If you write out the borrowing method as verbosely as this counting-up method, it's almost as easy to understand. However, that's not important in elementary school and shouldn't be done. The counting-up method has the disadvantage that you can't write it more compactly.

Students who learn the counting-up method will be hobbled in algebra: they'll be wasting their limited brainpower on the mechanics of subtracting the hard way when they could be using an easier method and devoting more brainpower to learning the concepts of algebra.

Of course it's important to understand how subtraction works, but by the time you get to algebra, that should be easy. Anybody who is uncomfortable in their ignorance can either just think about it, or ask a teacher. It's not hard. Just note that 325 = 300 + 20 + 5. Then line everything up and go. That understanding isn't worth a lifetime of pain.

Re: This Is What Common Core Looks Like

#23

Post by sp332 » Tue, Oct 07, 2014, 1:14 AM UTC

AnimalMuppet wrote: post #20

seanflyon wrote: post #6

Earlier quoted context omitted.

Just to be clear "But standardized tests, the SAT, and the ACT are all moving over to Common Core. So our child has to learn this insanity." which implies that standardized tests test for this particular "counting up" subtraction method. If I understand your statement you are saying that this is not the case. Right?

It's really hard for a standardized test to measure how you subtracted. (Yes, I am aware that it could be done.) But what it really measures is, can you subtract? Do you get the right answers? Personally, I don't like this approach, because to do one subtraction, you have to do four additions. That's kind of inefficient, in my view. But as sp332 pointed out, the kids should already know how to subtract numbers by thi…

I doubt the count-up method was intended to be a new method. I think the new curriculum had it in the second-grade book, and assumed that former second-graders had seen it there. It's not the textbook makers' fault that the school changed between years.

Re: This Is What Common Core Looks Like

#24

Post by seanflyon » Tue, Oct 07, 2014, 1:39 AM UTC

adamtj wrote: post #22

seanflyon wrote: post #21

Earlier quoted context omitted.

Perhaps the counting up method makes it easier for most kids to grasp the underlying concepts. The borrowing method can also teach kids the underlying concepts and follows nicely from addition as it is the inverse of "carrying the 1". > You just follow the rules with no real reason. This is the real problem here. We should not teach our kids to memorize a set of rules but to understand the concepts. When I was in sch…

When you're in 2nd or 3rd grade, it's more important to learn the mechanics of subtraction than to understand why it works. Kids at that age need to fill their brains with facts so that they have the raw materials for developing understanding when the are more mature and more able to understand. In this case, borrowing is more compact and efficient. That makes it faster and easier, both on paper and in your head. If…

I think that understanding basic concepts is foundational. It not only gives you an understanding you can build off of, but also teaches you how to think, how to approach a new problem. You say that kids need to fill their heads with facts, but I think they should fill their heads with concepts that they actually understand. Simple concepts that they understand, not facts, are the raw material for developing more complex understanding when they are more mature.

I realize that I cannot prove what I just said, so perhaps I stated it too strongly. It's possible even, that what I claim is true for some children and what you claim is true for others. I know that an emphasis on understanding has served me well, and that an emphasis on rote memorization has worked poorly for several people I know.

Re: This Is What Common Core Looks Like

#25

Post by daltonlp » Tue, Oct 07, 2014, 1:53 AM UTC

scarmig wrote: post #19

daltonlp wrote: post #15

The first two paragraphs illustrate a huge problem with US math education, which pre-dates common core, but is reinforced by it. That problem is the false belief that numeracy is improved by learning different ways of looking at number operations. (i.e. four ways to subtract). Our child's second-grade teacher (a very good teacher) has a poster illustrating 7 ways to subtract. Seven ways, no exaggeration. This poster…

Which programmer would you rather hire: the one who memorized a particular algorithm but has no idea how it works, or someone who can develop it from first principles and enlightened judgment? There's no strong reason to glorify one particular subtraction algorithm over another, especially since the actual use case for it is relatively rare.

Full agreement - any subtraction algorithm is fine, as long as it's well-learned (which takes a non-trivial amount of practice).

Re: This Is What Common Core Looks Like

#26

Post by daltonlp » Tue, Oct 07, 2014, 1:56 AM UTC

DanBC wrote: post #16

daltonlp wrote: post #15

The first two paragraphs illustrate a huge problem with US math education, which pre-dates common core, but is reinforced by it. That problem is the false belief that numeracy is improved by learning different ways of looking at number operations. (i.e. four ways to subtract). Our child's second-grade teacher (a very good teacher) has a poster illustrating 7 ways to subtract. Seven ways, no exaggeration. This poster…

Children learn by getting things wrong. A child doing the sum and getting the wrong answer may be learning as much as a child doing the sum and getting the right answer. http://www.bbc.co.uk/programmes/b04dwbkt Most relevant is probably this one: http://www.bbc.co.uk/programmes/b04gw6rh

Yes exactly. That's the reason so much practice is necessary. When you learn subtraction, you need to see lots and lots of number combinations, many of which you'll initially get wrong, but soon learn to do correctly, then recognize instantly, and finally use as building blocks.

Re: This Is What Common Core Looks Like

#27

Post by hyperair » Tue, Oct 07, 2014, 2:20 AM UTC

I wouldn't fault Common Core. I'd fault the textbook. That little page on counting-up subtraction didn't make sense to me either, so I went to Wikipedia, read this shorter snippet[1], and understood it immediately. That textbook's just crappy.

[1] https://en.wikipedia.org/wiki/Subtraction#Counting_up

Re: This Is What Common Core Looks Like

#28

Post by deciplex » Tue, Oct 07, 2014, 2:51 AM UTC

wahern wrote: post #17

cafard wrote: post #9

Back in the days before computerized cash registers, this was typically how cashiers made change. Hand one $5 for a $1.34 purchase, and the cashier counts it back out as (penny) $1.35 (nickel) $1.50 (quarters) $1.75, $2 (dollar bills) $3, $4, $5 Some still do it that way. In Teacher in America , chapter "Let x Equal ...", Jacques Barzun mentioned as an example of widespread innumeracy the half-trained cashiers who di…

It's actually a very useful method for quickly finding the difference between two numbers. And I would think that any computer programmer would appreciate the algorithm. The algorithm is a classic divide-and-conquer technique that is perfect for quickly performing arithmetic in ones head. And it would also be a good entre into teaching things like associativity and commutativity. I use something similar to avoid gett…

It's a good method for making change, but for doing subtraction in general it is (in my opinion) not so great, the reason being that you have to keep track of more figures in your short term memory, when compared to left-to-right.

To do subtraction by counting up you need to remember: the minuend, how far you've already counted up, and the number you've already counted up to. So, three figures. Subtraction left-to-right (or right-to-left) only requires you remember two numbers: what remains of the minuend, and what remains of the subtrahend. It's pretty easy even for primary school kids once they practice a bit.

Of course, when you're making change, you don't actually care how far you've counted up, because you're not really after the difference exactly, you're only trying to arrive at the correct number of coins and bills. So you can forget that part, or rather you can leave it to the coins in your hand to remember for you, and now you're back to two numbers. But I'd bet that if I asked you to tell me on the spot what the difference was without looking at the change in your hand you'd struggle to tell me.

Making change by subtraction left-to-right is cumbersome because, while you still only need to remember two numbers, after you calculate the difference in your head you must now count out the change, whereas with counting up you're already done.

Different tools for different purposes. I agree with Asimov.

Re: This Is What Common Core Looks Like

#29

Post by wahern » Tue, Oct 07, 2014, 5:47 PM UTC

deciplex wrote: post #28

wahern wrote: post #17

Earlier quoted context omitted.

It's actually a very useful method for quickly finding the difference between two numbers. And I would think that any computer programmer would appreciate the algorithm. The algorithm is a classic divide-and-conquer technique that is perfect for quickly performing arithmetic in ones head. And it would also be a good entre into teaching things like associativity and commutativity. I use something similar to avoid gett…

It's a good method for making change, but for doing subtraction in general it is (in my opinion) not so great, the reason being that you have to keep track of more figures in your short term memory, when compared to left-to-right. To do subtraction by counting up you need to remember: the minuend, how far you've already counted up, and the number you've already counted up to. So, three figures. Subtraction left-to-ri…

In practice you're not doing every step consciously.

Take, for example, subtracting 188 from 500. Count up to 200 to get 12. The difference between 500 and 200, however, is automatic. It takes no thought at all. So you arrive at 312.

Sometimes you can count down, too. Take 512 - 333. That's just 200 - 21. You can find 21 by doing it left-to-right if you want (it's how I just did it).

I'm not saying it's some kind of magical way to handle arithmetic, and it's not something you use to the exclusion of other techniques. Rather, what you're doing is converting the problem to something that is easier to solve. Left-to-right is a very useful, general, and systematic approach to do that, but there are faster ways depending on context. You can analogize it to programming in that you can trade CPU (left-to-right) for memory (cached tables of differences) to arrive at an optimal algorithm for the problem at hand.

So I would just re-iterate that learning these techniques isn't simply for the sake of doing arithmetic faster in your head. Rather, understanding how and why they work, and how to select the methods, improves numeracy in general.

And FWIW, I never look at the change in my hand. I don't literally count up like a cashier might when handing you change. To be honest, until _just_ right now I never really fully comprehended exactly what they were doing. I knew it was similar, but it only now just clicked that they were counting up in a fashion similar to what I've always just done in my mind's eye by breaking up and rounding numbers. (Not that I think I'm particularly good at this--please don't walk up to me on the street and drill me ;)