This Is What Common Core Looks Like
erickerickson.org
erickerickson.org
Edit: Here we go, Common Core math standards. Grade 2 starts on page 17. http://www.corestandards.org/wp-content/uploads/Math_Standar... Have a look, it's only 4 pages long.
To be specific, this is a third-grade textbook. Students learn subtraction in previous grades, so this is just a quick review before moving on to third-grade topics like multiplication. Notice the problems at the bottom don't require students to use this particular method. If your third-grader can't subtract, they were already behind before you switched to Common Core.
Also, is the method given really that hard to understand? It's probably not my favorite way to subtract, but it's kinda cool.
"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?
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 this point. If a kid didn't get it before, this curriculum throws a new method at them, hoping that this one will make sense to them. I'm not sure that I have a problem with the approach (though it wouldn't hurt tweak it so as to not confuse the kids who already know how to subtract).
The OP should take a minute to think about how the method of subtraction works, and then explain it to his kid.
I associate this panic reaction (OP: "this new method makes no freaking sense") with people who are math-phobic. The solution is to just settle down and think about it. (Common core, third grade objectives, number 1: "Make sense of problems and persevere in solving them.")
And, one thing I've learned about elementary educators is that they don't want to fight with parents. (Unsurprising!) So if you say (like OP did), "This common core math sucks," they are going to say, "yes, we don't like it either." If you said it was an interesting way to do subtraction, they'd agree with that too.
(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 did not know this technique. That was written in the early 1940s.
I remember being given change-making assignments in school, and being unable to complete them because I was not sure that the greedy algorithm (http://en.wikipedia.org/wiki/Change-making_problem#Greedy_me...) would always give the optimal result.
I was not able to articulate the reason for my uneasiness until years after I had failed to complete the work.
I use something similar to avoid getting too much change at a cashier, and to use up loose change (pennies, nickels, etc), except I usually count down and add in the difference. So, for example, if some bubble gum cost $1.28, I might give the cashier $2.03. Then, assuming the bewildered cashier doesn't try to give me the pennies back, I get three quarters for laundry, instead of two quarters, two dimes, and two pennies.
If I don't need laundry change, I'll count up or down searching for the amount that will maximally reduce the number of coins in my pocket (without avoiding bills, which would just be annoying to everybody). With practice you can do this in a split second, and it really blows the mind of some cashiers.
Isaac Asimov said something to the effect that smart people will tend to figure out these algorithms on their own. So it's kind of ironic that now somebody is complaining that it's being taught to kids. I suppose the only real legitimate criticism here is that, according to the link, there's only one example of this method, and then it's never mentioned again. If anything, it should be expounded upon.
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.
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 ;)
If you work through Kahn Academy math, you will certainly run through some of these areas where kids are pushed to do some kind of math problem in a highly stylized way that (i) they struggle with and (ii) don't see the relevance of. Elementary school teachers have told me the same thing, but it is not the end of the world.
There is "no one true way" to do arithmetic. When I was in the fourth grade I had the worst time with 3-by-3 digit multiplication. That didn't stop me from getting a B.S. in Physics and Math, with an A average in my majors, and also getting a PhD in theoretical physics. Recently I read that a lot of other people who were "good at math" struggled with the curriculum when I was a kid so nothing is going to be perfect.
First of all parents were not able to help their children with old methods either. Give someone a pen and paper and ask them how to add 17% to 58. You don't want an answer - just a working method. A lot of people can't do this on paper, but that's okay because we have computers to do that stuff. Lots of people can't add 17% to 58 using a computer. I find that a bit scary.
Going on to the example: I'm pretty stupid; I've often said that I am hopeless at math. I found that single example really easy to understand. I've not had any exposure to similar examples. I don't believe the author is actually baffled by the example. Perhaps a book might help?
http://www.robeastaway.com/books/maths-for-mums-and-dads
> What on earth are number bonds? What are partitioning and chunking? And why does my child look blank - or have a tantrum - when I demonstrate long multiplication? This book is for mums, dads and grandparents who want to help their primary school children or grandchildren with maths. To do so, many parents find they need to overcome their own rustiness and also to learn the strange new methods and terminology. Throughout the book are games, puzzles and examples of amusing ways in which kids ingeniously 'get it wrong'.
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 it because I had to start subtracting numbers in different bases for my MIPS assembly class.
> 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.
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.
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.
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 is older than common core.
Students learn math just like they learn to kick a ball. By practicing. Learn one way to subtract, and practice it really really well. Or learn two ways, and practice each way really, really well. Learn more ways if you want, but practice each way really really well. Most math curriculums assume that learning multiple ways is equivalent to (or better than) practicing one way. They don't require the repetitive practice.
As comments have pointed out, the common core standards don't explicitly require this multi-way approach. But the CC standards compound the problem by requiring students to explain concepts in order to demonstrate mastery. Completing lots of subtraction problems with a low rate of error doesn't count. You must explain, in order to understand. And if you really understand, shouldn't you be able to explain different approaches?
I would say no. Or at least, not yet. I don't want a third-grader explaining how to subtract unless they can also finish 50 subtraction problems in a row without pausing to struggle. And if they get the problems right? Good job, you get an A. Wonderful, let's skip grading the explanation because that is subjective as hell.
TL/DR: The article rightly points out a crap way to teach math. Common core is not totally to blame, but it makes a bad problem worse.
http://www.bbc.co.uk/programmes/b04dwbkt
Most relevant is probably this one: http://www.bbc.co.uk/programmes/b04gw6rh
There's no strong reason to glorify one particular subtraction algorithm over another, especially since the actual use case for it is relatively rare.