You’re bringing subtraction into it too, when you use the fact that 87 - 7 = 80. It’s just simple enough for you that you’re able to treat it as a single operation in your mind.
I can see that 93 + 87 is 180 just by looking at it; why did you go to the trouble of breaking 87 into 80 and 7? Everybody can do some of these operations as single chunks in their mind, and for larger problems they have to break things down until they’re made up of those chunks.
Math is math, and addition is addition.
It ultimately doesn’t really matter if you add 53 and 37 the conventional way, by adding the ones, and carrying the tens, then adding the tens together, or if you decompose one or both of the numbers and then add them all back together (what the “new” math is doing).
Learning both ways teaches you valuable properties about numbers and adding- the conventional “carry the one” way teaches place value pretty solidly. The “new” math version really hammers home the commutative and associative properties of addition in ways that the conventional method doesn’t do.
Neither is right or wrong; a lot of the opposition is probably cranky old farts who think that there’s ONE way to teach math… i.e. the way they learned it, and anything else is new-fangled and dangerous nonsense.
Actually, I just came to ask what the 1960s-era New Math was, since I thought that was what the OP was talking about when I saw the thread title.
Cranky old white person here. When I hear 53+37, I automatically change it to (53-3)+(37+3)=50+40.
Once I personally witnessed a mathematics grad student take out a calculator to figure out 75*8, which I automatically reinterpreted as 3/4 of 800. When I asked my cranky old white wife (who is not a mathematician) about that, she did exactly the same thing I did.
If that’s all the new new math is, more power to them. The old new math was quite different. But the real problem is not cranky old white people, but undertrained teachers. If they are not with it, their pupils will not get it either. I saw that with my kids and the old new math.
According to the Staff Report, it has something to do with those newfangled Arabic numerals. 
Let the battle of the cranky old white persons begin! Ya done it all wrong! (Never mind that you got the right answer. We all agree that’s irrelevant, right?) 
Here’s what I did: Twice 75 is 150. [That much, I pretty much know by heart.] Double that to get 300 [now we have 754]. Double that again to get 600 [now we have 758].
Ha ha! My way rulez! You way suckz!
Us cranky old white persons just gotta crank. It’s in our blood!
(ETA: Slightly more thoughtful [I hope] post coming Real Soon Now, but first, it’s time to go pull the laundry out of the dryer and fold it all. Back soon!)
Why bring politics/insults into it? This is GQ, not GD. Just because YOU see the wisdom doesn’t mean everyone does, regardless of age or color (or sex, for that matter).
It’s all about the money. I am not surprised at all that Larson (or whoever profits from using the name Larson these days), went this way. Old textbooks are being replaced, and if Larson wouldn’t re-write his text to match what school districts are demanding, no more money for Larson!
As to this discussion, I think the “new way” is cumbersome, but that’s because I learned and have used the “old way” for decades. To me, it isn’t intuitive, but I suspect that children being taught the “new way” now will have the same trouble 30 years from now when their kids come home with an “old math” problem (which will be the “new old math”!)
I have no idea which way is officially “better”. I do know that I can add numbers in my head, and understand the “old” way because that’s the way I learned it, and through vast repetition over the years, it has become “the way” for me. If and when I have to learn the new method to help a student, I will. But I don’t have to like it.
I would also suspect that many teachers who are delivering the new method to their students don’t fully understand it either, and/or were not taught how to teach it properly.
On top of which, I don’t think there’s even any real consensus on “how to teach it properly”. I certainly have a notion of how I think is the best way to teach various math things, but I’ll bet it would not be widely accepted.
I remember that machine! It was a projector that displayed one line of text at a time, then scrolled to the next line, etc. It also had an option that would black out all of the line except for a gap in the blocking that would sweep from left to right (like the one you remember). I did not particularly like either one, since the reader had to keep up to prevent losing information, and I had not yet developed such discipline in my reading by that point (also 7th grade).
Our teacher asked the class which method we perferred; I liked the whole line better since I could read for context that way, but the majority of the class liked the “sliding window” better than the whole line.
Regarding the learning of addition, I recall learning some parts of number theory, in that we were taught to add with the aid of flash cards: two apples on this card, three apples on the other card, how many apples all together? This teaches the children to count the number of objects on one card, then start from that point while counting up the objects on the other card.
I think the idea here was to demonstrate that 1 + 1 is NOT an arbitrarily 2 – you can count and prove it for yourself! But I feel sure that many kids were learning how to count the objects, and those kids were now at a disadvantage when they started to get to numbers over ten.
I do not care how you slice it, though, eventually you either have to memorize the addition table or get a calculator.
Okay, that is one pack of gun for 50 cents and $3.95 for the jerky. That’ll be $25.07, please.
Okay, an attempt at making some serious commentary here.
I was in grades 1- 6 from 1957 to 1963; junior high (7-9) 1963-1966; high school 1966-1969. Class of 1969 rulz! That put me right about at the beginning of the “first” wave of New Math. I learned basic arithmetic (grades 1-6) with methods that I believe would be described as pre-New-Math. Five Fundamental Laws in grade 4, IIRC.
Started Algebra I in 9th grade, with what I believed (then and now) was a New Math approach. At least as I saw it, I thought then that it was dumb and in retrospect I still think so. It has potential, but I got the sense that neither the teachers nor the textbooks were clear on the concept of what they were trying to accomplish, or how to do it. This was the math of which Tom Lehrer spoke “The important thing is to understand what you’re doing, rather than get the right answer” except that I don’t think they really did a good job teaching us what we were doing either, and they did mark us down when we got answers wrong.
I was hot in math through all those years, and I got it very easily. But I don’t think it was the textbooks nor teachers that helped me do that, and I don’t think a lot of other students “got it”. My sense was that it was all designed by mathematicians who knew their stuff, much of it in an abstract sense, but had little clue how to teach it to beginners. (Remember, I’m talking at the beginning Algebra I level.)
The book focused too much on abstract concepts, too early.
Set theory was big. The first chapter was all about sets. Okay, fine. But after that, pretty much all we used sets for was to write the answers to problems as “solution sets”. You couldn’t just solve a problem and write x = 7 for the answer.
Oh, no, No, NO! It had to be: x ∈ { 7 }
Nobody knew why that was so important, and I didn’t think it was such a big deal. But by George, you had to write your answers that way. We also did a lot with intervals (open, closed, half-open) and set-builder notation to describe them, like: { x : x > 7 and x < 10 }
I remember to this day our first “official” introduction to negative numbers. If I hadn’t already learned about them on my own, I don’t think I would have grokked this:
To each real number a we assume there corresponds another real number, denoted by -a and called the additive inverse of a, having the property that:
a + (-a) = (-a) + a = 0.
Got that? Totally abstract, the boiled-down and condensed wisdom of 400 years of thought about negative numbers, and doesn’t even mention the word “negative”, sounding like it was written by a lawyer. In common everyday thought, we would say these numbers are “less than zero”, but even that isn’t obvious from the above. You have to figure that out (with the help of a few definitions yet to come.) Then, from the above, we worked through some algebraic mumbo-jumbo to derive some of the properties of these mysterious additive inverses and how to work with them. But no mention of why we might actually find numbers less than zero useful (even when we eventually manage to intuit that they are just talking about numbers less than zero), except to solve algebra problems like x + 5 = 3 that had no solutions before.
I thought that, then and now, that that was a bass-ackwards to teach, especially at the beginner level. Purely abstract stuff like that works better, pedagogically, at more advanced levels (like learning, if differential equations, what an “analytic function” is).
Better approach: First, teach what positive integers (and zero) are good for: Counting things. Then what positive real numbes are good for: Measuring things. They are tools for working on certain kinds of problems. Challenge students to consider: What in the world would a number less than zero be good for? Zero is the least you can have of anything, isn’t it? Then start pointing out some kinds of real-world problems for which numbers less than zero make sense: (1) Problems where numbers have a direction to them. (Did that airplane change altitude upward or downward? Did John owe Jill some money, or did Jill owe John? If positive numbers describe the progress of a train from Chicago to Denver, how would you use numbers to describe the progress of a train from Denver to Chicago?) (2) Number scales that have an arbitrary rather than absolute zero-point: Temperature scales being the obvious example.
Get those real-world ideas out on the table, with concrete examples that you can show. THEN give the abstract formulation, and describe how it works, and how it can be used to develop the “rules” for working with these new numbers less than zero.
That’s the kind of development that I thought was conspicuously missing from the whole New Math curriculum, as I saw it in the mid-to-late 1960’s.
There is an absolutely vast amount of complete shit out there being called “Common Core,” from crap various teachers have thrown together as their interpretation of what parts the curriculum mean, to the same old ed-mill junk sold to the trade for years, now stamped COMMON CORE on the cover.
Mrs. B. brought home a number of howlers last year, collected from various teachers she encounters, most bearing a sort of officialish “Common Core” slug or stamp or title (or just URL including .CommonCore.) - and every one of them traced to bogus sources.
Whatever value CC has (and I’d say little), this rush of educators and ed-predators to smother their audiences and markets with appalling relabled shit is doing its acceptance no good at all.
If you want to reason “Alright, it’s $3.95, plus two nickels and four dimes to make the 50 cents, so we get 4 bucks using one of those nickels and then 45 more cents from the rest”, in the vein of the OP’s illustration, rather than running some traditional addition algorithm, you’ll be alright.
If you want to reason “Alright, it’s about 4 bucks, plus 50 cents, so it’s about $4.50”, you’ll also be alright, mind you. If you later want to reason carefully, but slowly, through the to-the-penny total, almost anyone who understands what addition means (and I do agree that conceptual understanding of addition is a vital skill children must be inculcated with), whether or not they’ve ever been drilled on quick, accurate mental or pen-and-paper decimal notation math, could do it with time.
If you even want to pull out your phone, you’ll be alright. Really!
And, yes, if you want to use a traditional decimal notation addition algorithm rotely applied, you’ll be alright too. It’s just not the only way.
Keep in mind: people got by buying and selling things in an age before standardized math education. Hell, they got by for thousands of years doing so before decimal notation! Which isn’t to deny that decimal notation addition algorithms have value (they were invented for a reason). It’s just not vital that everyone be drilled in running them manually in lieu of use of non-algorithmic reasoning or calculators (which were also invented for a reason).
To make it easier. In fact we can break it down a bit more to make it even easier. 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=37
That’s a cheap gun. I’ll take 10 gun please.
No, you will not be alright.
Customers don’t want to get stiffed, not ever, not even for a nickel, and every boss these children are ever going to work for will take a dim view of this sort kind of “it’s about” attitude.
Every time I’ve ever gone grocery shopping, I’ve not kept a to-the-penny running total in my head, and I’ve been alright.
I, of course, can do a to-the-penny calculation if needed. And when it is needed, I have no qualms busting out a calculator, because, why not?
But, really, my concern isn’t so much with manual arithmetic calculation as it is with the desire that this be instilled via some standardized algorithm. Any idiot can follow any symbol-shuffling algorithm with the rules in front of them and weak time constraints, and most can be trained to furthermore memorize the rules and carry them out quickly without error. But why should they have to, if they can get what matters to them another way just as well? (I worry that the heavy focus in early math classes on making computers out of humans ends up killing for many students wider possibilities of discovering understanding and joy in mathematics.)
Sounds kinda like Lockhart’s Lament condensed to a single sentence.
The customer is not supposed to have to keep an accurate to-the-penny total anywhere. The business is.
The business does so with cash registers and computers.