Or you could just cancel the twos, and subtract 1 from 3 to get your answer...
The "new" system here is fine, except that it requires a relatively large amount of mental overhead to keep all of the sub-results straight, it's almost as bad as doing it the traditional way. It's a good method to keep on the toolbelt. Most people don't realize it until they get to algebra, but most grade school (and college!) math is really just pattern matching to figure out the correct algorithm to use. Once you figure out what pattern the problem is in the rest is cake.
That said, this method is really helpful in cases where the traditional method is really difficult: When the individual digits on the subtractor (I've forgotten the proper term for this) are larger than the digits on the initial number and you have to do a lot of carries. And as it turns out, that pattern is extremely common in grade school math homework (usually almost every equation will follow that pattern because it's trickier and they want you to practice it more).
looks like the explanation is "Just because"
Hulk mad at new math! Hulk smash!
I don't have an opinion on the 'new math', but the sooner we get kids writing down arithmetic and then mathematics the better. Having a hell of a time getting my son to write stuff down because he does it in his head and the teachers can't comment on where he's gone wrong when he makes mistakes because they can't see his work.
I don't understand why all those steps are necessary, though. To use the example, if I buy a $4.30 coffee and only have a twenty, I round the $4.30 up to 5, which means I only have to do 20-15, then subtract $4.30 from 5 and I get 70 cents. Add 15 to .70 and I have $15.70. That's also four steps, but it seems easier to me than whatever is going on here.
edit: And I agree with the first comment, all you have to do is cancel out the twos (or mentally round them down to 30-10) and there you go.
This way may work for some. Not everyone wraps their head around it in the same way.
I am not brilliant at math or anything, but it helps for me to spatialize it in my mind based on units of 10 as a "block" (5 being half a "block").
The 2's match and cancel out, and the rest is just removing one "block" from a stack of 3 "blocks".
Going to 15 in the example seems unnecessary to me.
A formal application of this method would probably break down on power of 10, as most people have a pretty strong grasp on addition and subtraction with single digit numbers. So:
4622 - 1873
7 + 20 + 100 + 2000 + 600 + 20 + 2 or 2749.
One nice thing with this method is you're only adding two single digit numbers for each digit of the final answer. So it's 2, then 6 + 1, then 2 + 2, then finally 2 + 7. My example turned out to be a little extra easy because it didn't have any carries in the final addition, but that's because I chose an example that explicitly looks like something you would get 1,000 times as a fourth grader and would be annoying to do in the traditional system.
Funny how these things go in cycles, isn't it.
There was another period of time when people couldn't understand the new-fangled methods used to teach mathematics to children.
Of course, the funny thing is, the method parodied back then is exactly the same one that is being defended now as the "old" method.
This is an attempt to teach the children a concept called "number sense." People with number sense can rapidly calculate simple math and sometimes even complex math. A child who knows their way around the number line does not have to continually apply algorithms to solve math problems. I assume the adults reading this topic will understand what a valuable skill that is later in life.
Well, at least there's still numbers. Few educational stories terrify me quite so much as this 1998 description of "whole math", which sounds like what bureaucrats might come up with after reading Lockheart's Lament.
My sister teaches elementary school, and while I had an easy time with math, she had to really try to get math to stick in her mind. As a result she views math differently than I do, and she hated anything-math until she took a class on math pedagogy where she learned about the many ways to teach math. It's a very good thing for her career that math was tough for her, because she can explain one thing in several ways, helping a student find whichever way (or metaphor, or algorithm, or explanation) actually helps them. That way of doing it feels convoluted and strange to me, but that doesn't mean it wouldn't be the way to open a door for another person to get it.
Also, thank you for this:
Number sense helps in life a lot more than having a bunch of steps memorized.
Base eight is just like base ten, actually.
If you're missing two fingers.
At first glance the "new math" way looks confusing because the box is drawn over the + signs so that the first and third rows actually look like minus signs. So someone who doesn't understand the "counting on" concept that's happening sees "12-3=15" or "20-10=30'? That's crazy!
In fact, if you look closely, it's almost as if the box is drawn like this on purpose- it's not a single continuous line but kind of jags a few times to his some of those + signs right on the vertical. Someone call Dan Rather!
I usually approach this almost the same. To me, it makes more sense to computer 20-4 to get 16 and then drop one to 15 and do the same last step of adding (1.00-.30). I know it's basically the same, but I find it easier to subtract one later than to add one initially.
The reason giving change works is because you have markers--money--to keep track as you go along. The reason this is not like giving change is because you don't have those markers. As jandrese says, if you're going to do it, there's a lot of overhead involved in this method. Really, to do it the same as giving change, you'd have to take off your shoes and start cutting off toes and fingers. I prefer the old way, thanks.
Actually, the reverse method that I suspect most of us use makes a lot more sense: 32-10=22-2=20, or some variation of that
This is the exact reason why I would be a TERRIBLE math teacher. I never had any trouble with math growing up, so I am awful at trying to explain WHY something works the way it does.
I know we're talking about teaching children math, but the coffee example is one that likely applies to adults more often than children.
With most denominations of bills being multiples of 5, it's easy to use that as the base unit for the dollar bills and cents are of course out of 100.
$4.30 is 70 cents short of $5. $5 is $15 short of $20. The change is $15.70. No need to write anything down in order to remember 70 cents while calculating the $15. This isn't Murakami's Hard-Boiled Wonderland and the End of the World.
My suspicion is that something like this might be part of a lesson on counting by 5's.
Calculating the change without having the change in your hand is what I do. But then, I'm a math whiz who was noticed by the actuarial society in high school.
This is a rather poor example to use, because 32 and 12 have the same right digit. They should have used numbers where the right digit was off by 9 to make the point.