# A math teacher explains "new math"

The new system is a neat idea. However it is too easy to fail at. The normal algorithm is more intrinsically safe than this method. The traditional way will always give you the right answer. “New math” doesn’t have an algorithm, you’re just always starting from scratch.

Doing math “in your head” is a nice idea that some people can do well and some can’t. I am all for teaching this method, after the kids master an algorithm.

I’ve taught 4th graders math, and I will tell you that some of them do poorly because they just try to do it in their head (especially the “smart” kids) and get it wrong. This becomes rather acute when they start with algebra. The formalism of the classic methods feeds directly into the formalism you need when doing anything more advanced than arithmetic. A kid I taught who could do perfect arithmetic in his head (and thus never learned how to keep numbers organized on paper) was utterly lost when doing basic algebra because he wouldn’t work out his problems step by step. Less intelligent students did better because they already had the skills to organize their problems and implement reliable methods.

The idea of course is to get kids to “deeply understand” math. This is a good idea. Deep understanding of a subject comes about in two ways (in my experience):

1. A long period of more-or-less rote knowledge of a field builds to a level and then begins to synthesize together in an organic way more or less customized to your own mind. This is how organic chemistry is taught in college. You spend a year or so memorizing many many reactions while also learning theory. The theory and the reactions don’t usually seem to connect much. Eventually though (sometimes this is in a second year class) you start to understand the electronic theory which underlies the reactions. Retrosynthesis reactions remind you of things you’ve seen before and you have a bit of an epiphany. Some students never get to this point but what they can do is tell you about a lot of reactions. Students who get epiphanies go on to grad school in chemistry, students who don’t still get an A and go on to be happy people.

Musical education also works similarly. How do you become good at sight reading? You play everything until nothing surprises you. You practice a lot, deliberately, and it all synthesizes together.

1. Deep understanding can also come from “1st principles.” This is the way academics imagine it SHOULD work even though very few ever did it this way and those who did probably don’t bother with education as a field of interest. Ideally we would teach kids arithmetic, then algebra, then calculus, then physics, and then chemistry, molecular biology, organismal biology, ecology, etc. And each deep rich appreciation would build on the others.

The problem of course is that this approach is both deeply impractical and lacks any understanding of how the human mind works. You do not at all need to be good at calculus to be a good biologist. You need to understand what calculus is and what it can do, but you don’t need to be able to actually do it. Why? Because computers. A long time ago you did need to know calculus if you wanted to model anything, and you needed to be pretty good at it. Today though we have R and Matlab which can do the actual calculating. Heck, it wasn’t even 5 years ago before a working knowledge of Python was required to assemble your Illumina runs, now there’s an app for that (several even).

This “new math” is an attempted implementation of (2) which does not produce results because it’s based on a lot of misguided ideas about education and human minds.

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Jeez nobody knows (well very few people) know how to count back (Count up?) change anymore.
So yeah start with the change, 2 dimes for \$4.50, 2 quarters to make \$5, a 5 then bills to make the rest up to \$20 depending on what is in the till.
Making change is not hard and it is simple counting. Arithmetic jiggery pokery isn’t required.

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Exactly! Anyone who’s had to make change knows this. You start with the amount of the item and add money until you get the amount the customer handed to you. It comes naturally to add change amounts that bring the subtotals to round numbers (in terms of currency).

When it comes to teaching kids, there’s a tendency for the bureaucracy to want to latch onto a single idea. Unfortunately, kids learn differently. Some are visual, some verbal, some look for patterns and some need well defined rules, some need to write it down and others just cheat.

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The weird thing about math instruction is that math professionals seem utterly incapable of doing a rigorous statistical study on how to teach the simplest tasks.

For this reason alone, their idiosyncratic personal opinions should be filed with the Unabomber manifesto.

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'tis witchcraft

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You and me both, comrade. It was my experience in high school, of teaching adults how to make change on a register that wouldn’t do it for you, that disabused me of the notion that I might ever be a teacher. In my mind I was thinking "what do you mean you don’t get it, it’s just fucking counting!"while I was giving my best Customer Service Smile and saying, “let’s try again.”

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My father was a huge Tom Lehrer fan. His songs were the only thing we listened to on family roadtrips in the 1980’s. The New Math song always confused me for this very reason. The “new math” was exactly how I was taught to do math. I didn’t understand why the song ridiculed it as confusing. Moreover, the “old math” made no sense to me. It still really doesn’t. Granted, I understand it, but in the same sense as I know how to use my dad’s “reverse polish” calculator. I can do it (and I can see some advantages), but it seems very unnatural.

In other words, people prefer things to be the way they learnt them to be and do not like the unfamiliar. What is objectively “better” doesn’t matter.

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I am not sure where you get that. You can formalize the “new” approach just as much as the traditional one.

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I recently made a purchase of \$19.something and paid with a \$20. The trainee at the cash register randomly grabbed a fistful of change made up of all denominations–an amount that by sheer bulk was clearly much more than a dollar–looked at her trainer and said, “Is this right?” This was with a fully functioning cash register, mind you.

That anecdote isn’t meant to be a “kids today!” grump or the like. But she was either tragically lazy or had been tragically failed by the U.S. education system. Possibly some of both.

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Cripes people, this isn’t about the “New Method” being taught as “the right way”. It’s just ANOTHER way of thinking about numbers and the operations. That’s ALWAYS A GOOD THING.

I just yesterday experienced the joy of my 4-year-old grasping the concept that two rows of 3 items is instantly recognizable as being “6”. Now he doesn’t need to count every item when he sees that pattern, he knows that there are 6. Does that mean I expect him to see two rows of 27 items and immediately recognize that there are 54? Of course not. It’s just another tool to use to make manipulating numbers in his head more intuitive and require less mechanical manipulation on paper.

That’s ALL the pictured example is meant to be. It’s not meant to replace standard arithmetic (and any teacher attempting to do so should be fired). It’s simply meant to demonstrate other ways to visualize and manipulate numbers. Those of us who are good at doing arithmetic in our heads do that kind of thing ALL THE TIME. That doesn’t mean I don’t use standard methods as well.

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Yes. This is a perfectly fine way to teach kids how to do math in their heads.

I certainly always do some version of this with harder numbers, but, because I have more experience, can usually just lump it all into one step.

What’s 100 - 42 (or, as I generally think of it, due to carpentry and what-not, the distance between 42 and 100)? 58. Because I just “see” the space between 42 and 50 as 8, and then another 50. The more of this kind of thing you do, the easier it becomes and the bigger jumps you make.

This is the kind of thing that people with good number sense take for granted. Other people gripe that kids aren’t being taught “the algorithm” or “the way they were taught” and fail to understand the point.

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This is not true in the least. There’s a formal method right there on the paper.

1. Working from the least significant (rightmost) digits of the subtractor, keep a mental tally of what you need to round it up to zero.
2. Once you run out of digits, add those numbers to the corresponding digit on the initial number.

Now you have your answer. It’s actually quite similar to the old method, except you do it in two steps with a stop at a midpoint and then add the results together in the end.

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We simply didn’t get credit for bare results. That annoyed me to no end when I was little.

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Yup. I’ve had this conversation with my youngest kid’s math teacher, who’s around my age. We joke that it’s the New Math of the 1960’s come home to roost. Our parents are laughing in their graves.

I was always told that if you didn’t show working but got the right answer, you could still get 100%, but if you got the wrong answer you’d get 0%. Show working and you could get points for method.

What you’re doing is essentially the same thing as what’s in the article. You start with \$4.30 and progressively step it up by round numbers until you have \$20. You’re basically doing this:

4.30 + 0.70 = 5.00
5.00 + 15.00 = 20.00

0.70 + 15.00 = 15.70

The article’s method would probably add an extra step, from 5 to 10 to 20, to break it down into smaller steps for the kids. Once you have a stronger handle on the concept you can start merging steps, as you did.

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Of course the best response to this was by legendary musical comedian and Harvard Mathematician, Tom Lehrer with his classic New Math

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“The normal algorithm is more intrinsically safe than this method. The traditional way will always give you the right answer.”

First, arithmetic is not always about the right answer. Number sense is often about understanding the magnitude of numbers, not about exactness to the last detail. (Scientists discard less significant digits in order to account for error.) You need to consider how effective this method is for estimation.

Second, you need to consider failure modes. When you make a mistake, how bad is it? The traditional method begins with and emphasizes the least significant digit at any given step. It is most likely to get the least important part of the calculation right. With a method like this (perhaps not exactly this), you can start with the most significant digit to get a grasp of magnitude, then progressively improve the precision of the result as you move to the right. While the fequency of errors may increase slightly, their magnitude is likely to decrease.

My approach is similar to this even though I was taught the “traditional” method. One reason, like others here, is making change. The other is that this is ideal for ballpark estimates, especially when the same principle is applied to multiplication. Quick, what’s 19x18? Start with 20x20 and you won’t be too far wrong. Then correct for errors in the original rounding. It seems to me the important part of this method is not the precise algorithm, but its emphasis on working towards easy-to-grasp ballpark magnitudes.

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Teach kids how to do Fermi Estimates and they’ll go far in life. At the very least, it’s an excellent way of detecting error, since if you get an answer that is way off from your estimate you have a pretty good idea that something went wrong somewhere.

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I went to school in the French system (at the Lycée Français de NY), and we actually had a section of every math class until the 5th grade for “Calcul Mental”, mental calculation. All I remember about it is that you got a good grade for getting the right answer and a bad grade (plus maybe a cutting remark from the teaching) for getting the wrong answer. I got pretty good at it, but I have no idea how.