New Math: An Analysis of its Validity

[DiracPool]

I always remember my dear departed grandmother referring to something called "New Math" when I was a youngster, but at the time I wasn't interested enough in whatever it was to look into it. However, I just read the Wiki page on it:

http://en.wikipedia.org/wiki/New_Math

And it looks like something that got phased out shortly before I entered grade school. From the face of it, it looks like a crazy, overoptimistic project, but I wasn't really there for it. Is there anybody that actually experienced this and was taught this in school, or someone who parented a child that encountered this movement? How did it work out? Is there some validity in this manner of early instruction in children "in an ideal society," at least? Or was it just misplaced to begin with--driven by cold war fears?

 

[ImaLooser]

I always remember my dear departed grandmother referring to something called "New Math" when I was a youngster, but at the time I wasn't interested enough in whatever it was to look into it. However, I just read the Wiki page on it:

http://en.wikipedia.org/wiki/New_Math

And it looks like something that got phased out shortly before I entered grade school. From the face of it, it looks like a crazy, overoptimistic project, but I wasn't really there for it. Is there anybody that actually experienced this and was taught this in school, or someone who parented a child that encountered this movement? How did it work out? Is there some validity in this manner of early instruction in children "in an ideal society," at least? Or was it just misplaced to begin with--driven by cold war fears?

Yes, I learned the New Math. And I got a master's degree in mathematics, and can tell you what it is all about.

The story begins in about 1800 with Carl Gauss, possibly the best mathematician that ever lived. He wrote a text in Latin that put all of mathematics on a new intellectual basis we call "abstract algebra." It's the basis of group theory, which led to Riemannian geometry (Riemann was one of Gauss' graduate students) which is in turn the basis of much if not most of the mathematics used in physics. It also led to the axiomization of mathematics.

The New Math is a very simplified version of all this. The trouble is, it is so simplified it's boring and useless. The real applications of it are way too abstract for high school students. So that was a flop.

 

[DiracPool]

Yes, I learned the New Math. And I got a master's degree in mathematics, and can tell you what it is all about.

The story begins in about 1800 with Carl Gauss, possibly the best mathematician that ever lived. He wrote a text in Latin that put all of mathematics on a new intellectual basis we call "abstract algebra." It's the basis of group theory, which led to Riemannian geometry (Riemann was one of Gauss' graduate students) which is in turn the basis of much if not most of the mathematics used in physics. It also led to the axiomization of mathematics.

The New Math is a very simplified version of all this. The trouble is, it is so simplified it's boring and useless. The real applications of it are way too abstract for high school students. So that was a flop.

Interesting, although I'm sensing almost a contradiction. It was a flop but you learned it and went on to become a mathemetician. Is it that it was an approach that worked only for the "mathemically inclined," and that helped you in particular develop important skills which helped later, while it confused everyone else? Or was it that you would have gone into mathematics in either case and the New Math was just a flop in general, for everyone involved?

 

[Danger]

It is widely known here on PF that I have a grade 9 math education because of assaulting a math teacher in grade 10. The fact is, I flunked math every grade after 4, but we progressed based on overall grades rather than specific classes until high-school. New math was one of the main causes, along with the absolute worst teaching staff on the face of the planet.
Bearing in mind that I was 10 years old when this crap was inflicted upon me, it was only the arithmetic component that I was subjected to. I loved the subject up until then. All of a sudden, I was told that I was no longer allowed to do division the proper way. I had to do what they called "short division" and we were penalized if we fell back into "long division". I never learned how, and by the time I was old enough to tell them what to do with it, I had forgotten how to do it properly. I would gladly throttle every one of the MF's who were involved in my Ontario education. (Oh, I forgot to mention that I was forcibly relocated to southern Ontario from my beloved here in Alberta in December of '65 when I was in grade 4.)
I moved back here in '78, and began tending bar a couple of years later. After a few more years, one of the local high-school jocks graduated and came to work with us. Something looked familiar to me while he was doing a cash-out, so I took a closer look and there he was doing long division. Apparently, the Alberta school system hadn't been stupid enough to adopt that new crap. I got him to show me how to do it again, but unfortunately the lesson didn't take. I can still contact him, so maybe I should ask again. (By the bye, the last straw in public school was when my teacher, who was also the school's math teacher, yelled at me and whapped me with a ruler for winding my watch in class.) My only consolation is that surely all of them are dead by now.
I can't help thinking that if those *****'s had left me alone I might have graduated high-school, maybe gone to university (I so much wanted an PhD in astrophysics), and had some sort of decent career. Fie on them all.

 

[ImaLooser]

Interesting, although I'm sensing almost a contradiction. It was a flop but you learned it and went on to become a mathemetician. Is it that it was an approach that worked only for the "mathemically inclined," and that helped you in particular develop important skills which helped later, while it confused everyone else? Or was it that you would have gone into mathematics in either case and the New Math was just a flop in general, for everyone involved?

It was a flop because students and parents hated it as useless. It has no application in everyday life. Clock arithmetic. Who cares?

I didn't like it either. I was not interested in mathematics at all in high school. It bored the hell out of me, and I usually took a nap in class. I got into math much later. Group theory IS interesting, but the interesting parts are way beyond what can be done in an ordinary high school.

 

[ImaLooser]

It is widely known here on PF that I have a grade 9 math education because of assaulting a math teacher in grade 10. The fact is, I flunked math every grade after 4, but we progressed based on overall grades rather than specific classes until high-school. New math was one of the main causes, along with the absolute worst teaching staff on the face of the planet.
Bearing in mind that I was 10 years old when this crap was inflicted upon me, it was only the arithmetic component that I was subjected to. I loved the subject up until then. All of a sudden, I was told that I was no longer allowed to do division the proper way. I had to do what they called "short division" and we were penalized if we fell back into "long division". I never learned how, and by the time I was old enough to tell them what to do with it, I had forgotten how to do it properly. I would gladly throttle every one of the MF's who were involved in my Ontario education. (Oh, I forgot to mention that I was forcibly relocated to southern Ontario from my beloved here in Alberta in December of '65 when I was in grade 4.)
I moved back here in 78, and began tending bar a couple of years later. After a few more years, one of the local high-school jocks graduated and came to work with us. Something looked familiar to me while he was doing a cash-out, so I took a closer look and there he was doing long division. Apparently, the Alberta school system hadn't been stupid enough to adopt that new crap. I got him to show me how to do it again, but unfortunately the lesson didn't take. I can still contact him, so maybe I should ask again. (By the bye, the last straw in public school was when my teacher, who was also the school's math teacher, yelled at me and whapped me with a ruler for winding my watch in class.) My only consolation is that surely all of them are dead by now.
I can't help thinking that if those *****'s had left me alone I might have graduated high-school, maybe gone to university (I so much wanted an PhD in astrophysics), and had some sort of decent career. Fie on them all.

What the heck is short division?

 

[Danger]

What the heck is short division?

New math. You have a thing that looks like an "L" but rotated 90° clockwise. It's like a long division frame, but without the line down the right side. The denominator goes under the crossbar, the numerator goes on the left of the vertical bar, and the quotient goes on top of the crossbar. There are a bunch of subtracted things underneath the denominator.

 

[DiracPool]

Thanks for everyone's input. Looks as though the consensus so far is that this was an experiment gone wrong, very wrong in some circumstances

 

[L4xord]

New math. You have a thing that looks like an "L" but rotated 90° clockwise. It's like a long division frame, but without the line down the right side. The denominator goes under the crossbar, the numerator goes on the left of the vertical bar, and the quotient goes on top of the crossbar. There are a bunch of subtracted things underneath the denominator.

It still seems to be taught at school. I and all my friends have learned the short division method. In fact, a couple of years ago when we were introduced to long division, most people could not do it. To this day (as far as I know) a fair amount of people can not do it and do know care for it.

I don't know what to think anymore…

 

[Danger]

Thanks for everyone's input. Looks as though the consensus so far is that this was an experiment gone wrong, very wrong in some circumstances

Yeah. The Ontario school board, in about 1976 (2 years after I should have graduated grade 12 and perhaps started 13), publicly admitted that they had made a horrible mistake and reverted to normal math. Far too late for some of us.

edit: Hi, L4xord. I wasn't ignoring you; you slipped in while I was composing. Sorry to hear that they're still teaching that crap.

2nd edit: Just out of curiosity, what educational jurisdiction do you live in?

 

[WannabeNewton]

I took a look at the topics covered by it, in the wiki article. My high school still teaches many of those things in that same manner. For example in my sophomore honors algebra 2 class we started off immediately with groups and fields (we never did rings or modules though).

 

[jedishrfu]

My recollection of short division was look at the problem , see the answer and write it down vs long division look at the problem, figure out the answer digit by digit with lots of intermediate steps. and write it down.

More recently, we look at the problem, type it into Google and Google figures it out for harder problems ask PF.

http://en.wikipedia.org/wiki/Short_division

http://en.wikipedia.org/wiki/Long_division

Personally, I liked the Trachtenberg system but nobody was interested in teaching it:

http://en.wikipedia.org/wiki/Trachtenberg_system

 

[lisab]

Well I love new math :!), and I'm sorry they don't teach it anymore. I feel incredibly lucky that I was taught math that way - it laid the foundation for the rest of my life, really.

 

[jedishrfu]

Well I love new math :!), and I'm sorry they don't teach it anymore. I feel incredibly lucky that I was taught math that way - it laid the foundation for the rest of my life, really.

That would mean you are a Power Set.

 

[PrincePhoenix]

Are the wikipedia articles about long and short division accurate? There isn't much different in the two methods they show.

 

[jedishrfu]

Are the wikipedia articles about long and short division accurate? There isn't much different in the two methods they show.

The difference is in the writing of the intermediate results on down the page (long) vs little superscript numbers (which I don't remember writing) as you work from left to right.

 

[collinsmark]

 

[zoobyshoe]

I'm 58 and recall very well the nuns who taught my grade school telling us we were learning "New Math" which would be different from what our parents learned.

Strangely, though, I don't recognize anything that is called "New Math" in that Wiki article. I do believe we were introduced to the concept of sets at some point but I don't recall it being so abstract as to be confusing, nor was it center of everything else.

 

[jim hardy]

I went through grade school about a year ahead of "New Math". It appeared in Miami ca 1959 to best of memory. I was pretty good at math and remember a very pretty eighth grade girl (Sharon D.) who came to me for some help with her 'new math'. I was a ninth grader . We'd meet at the town library to work our respective math homeworks.

I remember it this way -
Old math taught us the mechanics of solving math problems. We learned "what" , and if a kid was interested enough he could figure out they "why" by playing with numbers just like you play with bicycle gears to understand them. Ever take apart a Sturmey Archer 3 speed bicycle transmission? Planetaries...

Anyhow -

Sharon's book tried to explain "Why" by introducing meaningless(to me) terms and rules to memorize. The drill questions in her book were not repetitively working problems that demonstrate arithmetic concepts like carry and factoring, instead they were memory tests- recite the rules.
Took me an hour or two to figure out what were the terms in her book referring to.
I showed her how to work some problems and then we went back and studied how the rules applied and from whence came the terms. Sharon was very bright and picked up right away. She went on to make very good grades in high school. I think she worked backward from the observations to the rules, like I do.

That summer my Aunt from Seattle asked me about new math - her daughters were struggling with it too. Their Dad was an engineer and he taught them the same method i'd shown Sharon.

So - in my opinion "New Math" is not for the masses.
"Why" before "What" is contrary Descarte's admonition :
"never to accept anything for true which I did not clearly know to be such".There are kids bright enough to learn 'new math' as evidenced by Lisa's experience. They are the fortunate few.

For the gifted, 'New Math' is just an appetizer.

Myself I blame 'new math' for this country's dearth of engineers and scientists. It discouraged most kids early on.

You see, we plodders learn by doing and we figure out the rules empirically.
We need to keep our "Descartes before the Hors ( d'ouvres)."

That's my opinion and I bet you're sorry you asked.

old jim

 

[Evo]

I'm 58 and recall very well the nuns who taught my grade school telling us we were learning "New Math" which would be different from what our parents learned.

Strangely, though, I don't recognize anything that is called "New Math" in that Wiki article. I do believe we were introduced to the concept of sets at some point but I don't recall it being so abstract as to be confusing, nor was it center of everything else.

I'm glad to hear you say that because I also didn't recognize anything in that wiki article, nor do I have any memories of it being strange or confusing. I was beginning to think all of my childhood memories of math were false.

I think I had a bit of both, luckily old math came first.

I thought this bit from The Straight Dope was funny.

The following examples may help to clarify the difference between the new and old math.

1960: A logger sells a truckload of lumber for $100. His cost of production is 4/5 of this price. What is his profit?

1970 (Traditional math): A logger sells a truckload of lumber for $100. His cost of production is $80. What is his profit?

1975 (New Math): A logger exchanges a set L of lumber for a set M of money. The cardinality of set M is 100 and each element is worth $1.

(a) make 100 dots representing the elements of the set M

(b) The set C representing costs of production contains 20 fewer points than set M. Represent the set C as a subset of the set M.

(c) What is the cardinality of the set P of profits?

1990 (Dumbed-down math): A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20. Underline the number 20.

1997 (Whole Math): By cutting down a forest full of beautiful trees, a logger makes $20.

(a) What do you think of this way of making money?

(b) How did the forest birds and squirrels feel?

(c) Draw a picture of the forest as you'd like it to look.

— Ian, Jill, and Dex

http://www.straightdope.com/columns/read/1529/what-exactly-was-the-new-math

 

[collinsmark]

 

[WannabeNewton]

People should learn abstract mathematics instead of being mindless formulaic zombies. I was very lucky my high school went with the former. The beauty of math and the challenge of math is in the abstractions; anyone can plug stuff into formulas and crank out results from equations with little to no thought. Where's the skill in that?

 

[jedishrfu]

My recollections of elementary school were struggling to memorize the multiplication tables, reciting them to my Mom in the hospital while she was delivering my younger brother. I always had trouble with 8x7 vs 9x6 which I finally resolved when I learned that things times 9 add up to 9 from I concluded that 54 was right for 9x6 and 56 was for 8x7.

Later I remember doing stuff with Venn diagrams and diagramming sentences in English... and that would have been around 1957-1964. I do remember my cousin teaching me some Algebra when I was in 5th or 6th grade which picqued my curiosity and learning about Atomic energy in 6th grade which picqued my science curiosity.

So perhaps the New Math was phased in and didn't hit my class.

Anway how would I know new from old it was all gnu to me.

 

[jim hardy]

Evo that was a great link-- Thanks!

People should learn abstract mathematics instead of being mindless formulaic zombies. I was very lucky my high school went with the former. The beauty of math and the challenge of math is in the abstractions; anyone can plug stuff into formulas and crank out results from equations with little to no thought. Where's the skill in that?

I dunno, WBN - I never thought much of "Abstract Art" . Boston's "Museum of Bad Art" is full of it.

This is the mechanized age.

I respect those who think on higher planes . They advance science. And I wish I could understand my little book on 'irrational numbers'.
But 99% of us are more involved in maintaining the wheels and cogs of civilization.
Einstein was not a skillful handyman. Among his most prized possessions was a small reflector telescope some students built as a gift for him.

"A Mathematician's Apology" G H Hardy, no relation...

Beauty is the first test: there is no permanent place in the world for ugly mathematics.

Is there a museum of ugly mathematics ? I'd suggest that, so far as being a tool for producing math capable kids, 1960's "New Math" belongs there.

 

[jedishrfu]

And yet Einstein was an inventor and held a patent on a Refridgerator:

http://en.wikipedia.org/wiki/Einstein_refrigerator

and in a similar vein Hollywood actress Hedy Lamarr invented on of the most significant ideas of the 20th Century used in most wireless communication devices:

http://en.wikipedia.org/wiki/Hedy_Lamarr

http://en.wikipedia.org/wiki/Frequency-hopping_spread_spectrum

now back to new math...

 

[jim hardy]

If I'm correct in my understanding,,, Einstein started out with Euclidean geometry which builds from observations of constructs...What before why... rules evolving from observations...

And I wasn't demeaning him at all. He was a thinker.

Re new math -I stand by my assertion : "what before why" produces larger numbers of math capable kids.
I arrived at that from my own observations in early 1960's.
Are there any new studies that disagree?

 

[AlephZero]

I did some "new math" in the 1960s at school in the UK. I don't think it did me any lasting damage.

This thread reminds me of a story from one of my Univ. lecturers. He had decided his own kids ought to learn math concepts as early as possible. So one day he was walking along the riverside with his 4 year old kid, watching some rowing eights training. He pointed out to the kid that there were the same number of oars as the number of rowers, and this was called a one-to-one correspondence.

A few days later they were on the river bank again and he asked his kid if he could remember what a 1-to-1 correspondence was. With great confidence the kid answered, "It's a special sort of boat".

 

[WannabeNewton]

I dunno, WBN - I never thought much of "Abstract Art" . Boston's "Museum of Bad Art" is full of it.

This is the mechanized age.

I respect those who think on higher planes . They advance science. And I wish I could understand my little book on 'irrational numbers'.
But 99% of us are more involved in maintaining the wheels and cogs of civilization.
Einstein was not a skillful handyman. Among his most prized possessions was a small reflector telescope some students built as a gift for him.

The Boston thing made me laugh quite a bit haha. I'm not saying applied math is useless; by all means I mean the opposite because it drives our world :D. What I was talking about was more along the lines of the kind of teaching where you are thrown some formulas and have to memorize them and plug and chug come exam time. I don't see any utility in that kind of teaching. By abstractness I meant more along the lines of focus more on concepts rather than actual numbers.

Perhaps as an example, explaining the concept(s) and diagrams behind the Fourier transform of an impulse-response system into the frequency domain would probably be more helpful in solidifying the theory behind why it is useful and why it works as opposed to just giving say a mass-spring system with some numerical parameters and saying "here, go find the Fourier transform and give me the amplitude of the response in the frequency domain!"

 

[Turion]

Perhaps as an example, explaining the concept(s) and diagrams behind the Fourier transform of an impulse-response system into the frequency domain would probably be more helpful in solidifying the theory behind why it is useful and why it works as opposed to just giving say a mass-spring system with some numerical parameters and saying "here, go find the Fourier transform and give me the amplitude of the response in the frequency domain!"

Pretty sure we're talking about grade school here.

 

[WannabeNewton]

Pretty sure we're talking about grade school here.

Pretty sure Jim was talking about engineers and inventors.

 

[Turion]

Pretty sure Jim was talking about engineers and inventors.

Good point. Yeah, understanding is important for engineers and inventors.

 

[MarneMath]

My father is an Engineer and I had a conservation with him regarding new math many years ago during my high school time. From what I gathered, he felt completely discouraged in his mathematical abilities for a good part of his education, because he never understood the purpose behind of stating axioms while solving algebraic equations or looking at problems in terms of sets. If it wasn't for the fact that a family friend, who worked at NASA explicitly told him that 'while there is a use for this kind of thought, at the end of the day, as an engineer, it's your ability to use math to solve real problems, not made up ones, that matter.' that got him to stay and eventually obtain an engineering degree instead of working the family avocado farm.

Looking back at my own educational experience, I can say with a high degree of confidence I would've felt the same. I disliked most of abstract algebra, I disliked most of my theory based probability, and I wrote my thesis with the sole intent of taking an abstract idea and making it concrete for myself. In the end, the mathematics has always just been a tool for me to analysis real world data. I think education needs to accept that for a lot of people, mathematics is just that, if not less.

 

[jim hardy]

In the end, the mathematics has always just been a tool for me to analysis real world data. I think education needs to accept that for a lot of people, mathematics is just that, if not less.

Thanks all - that was exactly my line of thought.

 

[zoobyshoe]

I'm glad to hear you say that because I also didn't recognize anything in that wiki article, nor do I have any memories of it being strange or confusing. I was beginning to think all of my childhood memories of math were false.

Maybe you and I both happened to go to schools that were administered the placebo.

 

[dlgoff]

The beauty of math and the challenge of math is in the abstractions ...

It's been too long to really appreciate your quote, but not so long as to remember I began feeling exactly the same somewhere around differential equations.