This was tried. It was called "New Math". Spectacular failure. Do you want to know what worked? Memorizing times tables.
There's just no way around the fact that drilling is key to early mathematical learning.
I hate to admit it, but I am now a believer that strong arithmetic skills are important, and drills get you there. I don't like to call it memorization, since I'm not sure it necessarily is simply memorization. But you do need the answers at a moments thought. It should be as natural as saying a word.
And its not to say that you don't teach concepts concurrently... but that the arithmetic is fundamental.
That said, I still believe that the long division algorithm taught isn't so useful. :-)
Its hard to explain, or even to grasp, how fundamental basic arithmetic is to almost everything in math.
I disagree. Not hard to explain at all. Point the person at a page of geometry problems and say, "Imagine struggling with 90 + 90 at the same time as you struggle with the concepts here." Point the person at a page of polynomials to factor and say, "Imagine trying to do these if you hadn't memorized basic multiplication facts." And so on.
I still believe that the long division algorithm taught isn't so useful.
At the risk of not knowing precisely which algorithm you're talking about, I can't imagine one that doesn't work by taking a large/hard division problem and breaking it down into small/easy division problems. And that's useful because it's a great example of what math does for your thinking.
To be specific, I think the important principle math teaches is that, when faced with a big, hard problem, break it down into smaller, easier problems. I would rather describe math as a "learn how to break down problems" discipline rather than use the vague and pretentious "learn how to think" description. All areas of education help your thinking.
If you ask me 9 * 7 I still do 10 * 7 = 70 - 7 = 63 in my head; but, I seem to handle spectral graph theory just fine.
EDIT: As pointed out, I mistook OPs invocation of New Math to be talking about the much maligned Common Core rethinking of math education. I was briefly a high school math teacher, but before the roll out of these changes, so I can't comment first-hand on what the new curriculum looks like in the actual teaching. But I do know how poorly prepared my students were for math beyond arithmetic. They were trained with similar curriculum that I had experienced growing up in the 90s, which I think is poorly thought out.
Apologies for the confusion caused by me not recognizing the term New Math.
BTW, "not even wrong" is a rather rude way to point out a misunderstanding.
Concepts of what? Without intuitive understanding of basic computation procedures, what concepts can anyone build out of nothing?
Math is concrete. It's from observation of actual quantitative phenomenon.
Modeling the world is a job that math the tool was created for, but it certainly isn't concrete just because the world that needed it was or is.
To split them, and think that, just because there is abstraction, and it's OK to develop the concept without the concrete substrates of actual experience is trying to make dream come true.
And I argue that no one should live in a dream. And it's obvious that only those dreams that have a strong connection with the real world have realistic chance of being made real.
Concepts over computation (or well, before computation) is probably right. But New math was about learning the abstract before the concrete. This was predictably an abject failure.
I think the best way to teach math is to follow the trajectory that humanity took when discovering it. The key that's missing is that math doesn't just come out of thin air, its just a systematization of precise quantitative thinking. If we motivate the concepts using real world examples, then explain how to abstract away the particulars into a general procedure, then these connections will get made that make math "real" and relevant.
I find it disheartening that I have to teach my kids basic fractions, ratios, and transformation cause the teachers don't or barely touch on it. Kids are supposed to "discover" and "explore" math, whatever that means. In my opinion it's all bullshit.
Math, in many ways, is like an engine, either it works meaning the answer is correct, or it doesn't.
The way people are teaching math in the US is baffling and weird, but Pearson's textbook design has little to do with the common core per se. Most common core-labelled material is from something else with a new shiny CC cover slapped on.
Many problems that people have with math seem to stem from not having internalized the most basic facts about addition and multiplication. If you don't know at least to 10 by 10, each tiny step of working through a problem will tend to be interrupted by counting. Fluency requires memorization.
Also I never completely learned my multiplication tables to the point where they were reflexive, such was my loathing for rote memorization. To this day I sometimes need to pause to mentally crunch something. This sucks for small numbers but it means I have the mental tools to grind out bigger ones that other people would need calculators for.
People who were taught by memorising go into outrage a new type of exercise is introduced. Suddenly thinking is needed and that is bad in their eyes. Not exactly success. Meanwhile, you can memorize time tables in later age if you decide it is useful (people rarely do).
Once people see what they need out of mathematics for success in life, they never choose to memorize multiplication tables. But they do often learn new problem solving techniques.
That should tell us something about which is more important.
(But of course, we don't have to choose between the two either.)
It sucked.
The next steps are to start doing things backwards, "how many times do we count by four to get to twenty?" And we start to introduce notation. Bingo, simple division. This leads directly to simple fractions. This opens up conversations on adding and subtracting fractions, then multiplying and dividing them.
My oldest kid, now in college, could add, subtract, multiply, and divide fractions by second grade and understood them. Oddly, she had a horrendous time learning decimals. Her mental model of numbers was fractions and decimals were "weird." She would have to change things like 5.045 to 5 45/1000 to understand it, and wanted to work with it as a fraction. It took a long time for her to get comfortable working with decimals.
One time that I think it paid off. I told her that 0.999... is equal to 1. She said false. I said, no, it is true. Can you tell me why? At this time, she was in algebra, and I was expecting to show her how to prove it using algebra. She had a much better way of looking at it. In about a couple of seconds, she said, "well, 1/9 is 0.111... and 9/9 would be 0.999... and that is also 1." Her answer was much better than mine. :)
An example of when memorizing is bad (ie, when the underlying knowledge is skipped) was her 7th grade algebra teacher. In teaching the laws of exponents, he said "anything to the 1st power it itself and anything to the 0th power is 1. We don't know why, it is just one of those math things." Teaching like this is why we have students who, later in high school, can't do x^0 or x^1 because they think, "it is either 1 or 0 or itself, I don't remember." As opposed to applying mental models and patters to see that 3^3 -> 3^2 -> 3^1 -> 3^0 is just dividing by 3 each time. These students know 3^2 is 9. So they should know that the next is 9/3 = 3 and that the next is 3/3 => 1.
2) Did New Math fail because of poor results or because of popular revolt? (Would New Coke have failed if there was never any such thing as the original Coke?)
2a) If New Math did actually have poor results, was it because it hewed too closely to the goals I brought up, or because of other issues?
3) Memorizing times tables clearly didn't work, or we wouldn't be having this discussion
4) I never said that drilling times tables isn't important, so I'm not sure what your last sentence is in response to
I was told to memorize the times table, and tried but never managed to succeed. Instead, I found that I got along just as well by memoizing them instead; that is, I would compute the parts I needed on the fly in the margins of the paper. (Example: Say I need to find 37. I happen to know that 33=9, which I can double to get 36=18, plus 3 to get 37=21. These figures would be written down, so when I later needed 47 I could easily add another 7 to get 28. I had similar tricks for various other numbers, and could generally get the figure I needed--if it wasn't already written down--in a few hops.)
These contortions don't seem to have significantly affected my mathematical development, but they did* improve my logic and reasoning skills (or possibly merely showed that I had them). Particularly now that nearly everyone I know carries a calculator in their pocket, I don't see why we would continue to focus on rote learning over actually understanding how the underlying principles work.
I still feel a certain tinge of guilt though, over not memorizing that stuff.
No it wasn't, at least writ large in the USA.
And that's not what New Math was. Explaining what multiplication is doesn't demand an introduction to set theory.
If we read Feynman's CRITICISM of New Math, we actually find that he ADVOCATES for exactly what your parent is suggesting ("cobbled together.. in order to solve real-life problems"). So clearly, your parent isn't describing "New Math". Or perhaps Feynman is just a raving lunatic.
So I'm no advocate for "New Math", but I do oppose the argument you're making here, in which "New Math" is taken to mean "anything other than memorizing times tables" and is then denigrated on face. Without regard to the fact that the most vocal opponents of "New Math" were in fact advocating for exactly what your parent post is suggesting.
> Spectacular failure
So brief was the new math intervention that, to this day and despite all of the hoopla, we don't have a good empirical basis for claiming new math worked or did not work.
New Math was barely attempted, and its primary opponents were mathematically illiterate parents and teachers. This is just true, even if there were mathematically literate opponents to New Math, e.g., Kline or Feynman.
(But also note Meder’s reading of Kline. It's also worth noting that Mathematicians are maybe not the ultimate authority when discussing secondary pedagogy, especially in the mid 20th century. I have no basis for this belief, but IMO lots of mathematicians who weighed in on New Math were very possibly waging a sort of proxy battle as part of a larger war over the future of their own field -- pure vs applied.)
> Do you want to know what worked? Memorizing times tables.
Is this satire (honest question)? For all the things we don't know about math ed, we know that this doesn't work. Students who memorize times tables are routinely incapable of multiplying 12 by 13 or 55 by 55.
> There's just no way around the fact that drilling is key to early mathematical learning.
No, there isn't. But there's also no way around the fact that drilling without understanding is why a whole bunch of students who are "good at math" can't get through even the most dumbed-down versions of proof-based courses in college, or in some cases can't even get through a full calculus sequence. But they're "good at math" because they can rattle off 12*7 real fast!
New Math advocates (and their opponents!) were all at least correct about one thing: we REALLY SHOULD seriously ask what good is learning "math" if the student does not become a better problem solver. It's not 1417 anymore -- problem solving is important, but human calculators don't pull down living wages.
Probably the answer is that we should all be equal opportunity critics: memorization without understanding is intellectually lazy and limits growth potential, while understanding without practice is for most learners a contradiction in terms.
His beef with the New Math is with an emphasis on axioms, deductive reasoning, rigorous abstract logic, linguistic purity, and symbol manipulation, rather than with teaching conceptually or letting students think for themselves. He also doesn’t like the specific content of the New Math (set theory, inequalities, alternate number bases, boolean algebra, modular arithmetic). [I haven’t studied the New Math curriculum enough for myself to know how fair these arguments are.]
His key criticism: “Psychologically the teaching of abstractions first is all wrong. Indeed, a thorough understanding of the concrete must precede the abstract. Abstract concepts are meaningless unless one has many and diverse concrete interpretations well in mind. Premature abstractions fall on deaf ears.”
That obviously hasn't worked as most people are quite awful at math and society at large hates it. Memorizing times tables has been a spectacular failure as has memorizing formulas.
Math is about problem solving, not memorizing answers to common things; it's the focus on memorization that's made so many people bad at math to begin with. Common core is an attempt to address this by focusing on how the problem is solved rather than what the answer is, it's freaking parents out, but it is a better approach if you're actually trying to teach math.