I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables.
Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi without first explaining how it's calculated? Where does the "four-thirds" part come from in calculating the volume of a sphere?
Not to mention that there's never enough history to go along with the mechanics - who discovered the quadratic formula? What was their life like? Why were they playing with quadratic equations in the first place? This make math seem less like magic incantations and more like something that was sort of cobbled together by flawed weirdos in order to solve real-life problems, and evolved over time.
that sounds utterly dreadful. computer science is often taught like this; to its folly.
imagine a kid learning to speak 'stop just memorizing words, you need to understand how language was derived before you learn how to speak it'.
its completely counter intuitive to how creatures learn. learn easy things, especially those that relate to problems we deal with, and then get deeper into the subject if necessary.
> Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi without first explaining how it's calculated?
If we're waiting until children are capable to derive those equations before we get them to use those a bit, then we're waiting for very long. And if you just keep telling them to add numbers and multiply numbers together for many years in a row without giving them any interesting problems to go with it, there'll be no one interested in math by the time they'd usually have the abstract maturity to deal with the more foundational modern math problems.
We don't ask Computer Science students to write an OS before using one, we don't ask carpentry students to build a hammer and cast nails before using them.
the beginning of the end of my math studies was eighth-grade algebra. i GOT algebra, it was easy, and i enjoyed it. ..until the quadratic equation was introduced.
"hey, that's really neat! how does it work?"
"oh, you'll learn that in calculus, which we won't allow you to take for another four years."
It's common not to explain where the quadratic formula comes from (which is silly, it's straightforward enough to show, but standard math education curricula are shot through with this blind formula memorization nonsense), but… do they really tell you you'll learn it in calculus? It's got nothing to do with calculus.
Not to mention, the quadratic formula per se is needlessly complicated. If you break it into two or three steps it makes a lot more sense:
Step 1: move pieces around and divide by the leading coefficient to put equation into the form x² + b = 2ax (or if you like, x² – 2ax + b = 0; or feel free to swap the sign of b if you prefer). The equation for the parabola is then x² + b = 2ax + y.
Step 2 (optional): rearrange that equation to get (x – a)² = a² – b
Step 3: x = a ± √(a² – b)
In this form, the “discriminant” is just a² – b, the x coordinate of the vertex is a and the y coordinate is b – a², Viète’s formulas tell us that the two roots satisfy ½(x₁ + x₂) = a and x₁x₂ = b. If the coefficients are real but the roots are complex, then we know each root has amplitude √b and phase arccos(a/√b). Etc.
Or even better in some contexts is the form x² + b² = 2ax, in which case we have a = ½(x₁ + x₂) [the arithmetic mean of the roots] and b = √x₁x₂ [the geometric mean of the roots], and the discriminant is a² – b², which has a nice symmetry.
> I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables.
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 was a proponent of "New Math" types of philosophies before I had a kid. It was only when I tried to teach concepts did I realize how important it was to be able to be able to arithmetic quickly off the top of your head. Its hard to explain, or even to grasp, how fundamental basic arithmetic is to almost everything in math. For us arithmetic is as simple as thinking a thought -- but only when I tried to actually explain concepts to a young kid did I realize how difficult it is for them to understand these concepts when arithmetic isn't ingrained in their brain.
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. :-)
I generally agree with your comment but have specific disagreements with two parts:
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.
It's way too early to judge "New Math" to be a failure. I think the idea of focusing on concepts over computation 100% in the right spot. The biggest problem to its acceptance is cultural. Parents don't feel comfortable with math concepts, and in my opinion, most of the negativity is coming from that insecurity. So they demand for things to be taught the "old way", even though that's produced a generation or two of mathphobic Americans.
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.
>I think the idea of focusing on concepts over computation 100% in the right spot.
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.
Not even wrong. You think New Math refers to the current modifications of common core. New Math actually refers to a series of cold war changes made in response to perceived russian scientific dominance.
yeah it is a bit rude..sorry! I just get very few opportunities to say it- and I was like yo- this is a good chance. Are you familiar with the history of the term? with pauli?
I disagree. I am not uncomfortable with math in general or the "new math". What I am uncomfortable with is the fact that my kids, and everyone else's kids, are essentially lab rats in a massive experiment. The result of which is 1+ generation of kids who can't calculate tax in their head or split a restaurant bill . They can't manage more than a few number without a calculator and are missing many of the basic blocks of math that are learned through rote.
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.
Disagree. If you read the common core standards, they are very sensible and make no curricular or pedagogical recommendations. "Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph." CCSS.MATH.CONTENT.HSF.IF.B.6.
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.
I think you misunderstand, "New Math"[1] was a thing from the 60's, he clearly took it as an adjective talking about common core, the current "new math", but not the same as the "New Math" bitwize was saying had been tried and was a failure.
I know New Math from the 1960s; the sibling comment to my post indicates that ghettoCoder is talking about Common Core today and I disagree with his characterization of Common Core. New Math as practiced in the 1960s is a fascinating and very different failure case, although it probably had similar problems in implementation (poor teacher training and shoddy repackaged curricula).
idk, I studied higher level math very seriously, and even there, practicing computations is pretty key to understanding the underlying concepts. Concepts and abstractions are important, but they don't stick without practice.
I agree that it's an important part of the process, but I think it works best when it's connected with actual application, and not just a worksheet full of disembodied computations. I saw way too much of that as a student and as a teacher.
You forcefully disintegrate concepts and physical world. All concepts have certain degree of abstractness for human understanding. It's not unique to Math. And it's not necessary that abstractions automatically alienate something from concreteness.
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.
Is it way too early, or has it produced a generation or two (or three!) of mathphobic Americans? How many generations of avoiding rote memorization would it take to judge an alternative, in your view?
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.
PhD physicist here. Struggled through the 'normal' public education math program, and didn't get good traction on stuff until college when my instructors starting focusing on more 'evidence based' math and physics.
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.
Memorizing time tables and similar rote learnign means that precisely kids who could be good at math hate it. Meanwhile, those who have good memory and sux at problem solving think they are good at it.
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).
I think this is a very astute observation. I know a lot of people who develop problem solving skills later in life after realizing that their mathematics education in high school was woefully inadequate. I don't know anyone who memorized a multiplication table by choice, and I know several professional mathematicians/engineers who haven't memorized their 10x10 or 12x12 multiplication table.
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.)
I never managed to memorize my times tables. I never managed to memorize other formulae either, which meant that in an exam without a cheatsheet I'd have to do things like draw a bunch of triangles, measure them (graph paper) then re-derive the Pythagorean formula that I vaguely remembered involving a square root.
Blind memorizing is a problem. Memorizing things that you have learned concepts behind makes a student faster. As an example, the way I've taught my kids multiplication starts with: let's count by 2s! Great, now you know that, let's do 3's! 4's, ... 12's. We got to the point where we could count fluently (speed + accuracy), and incorporated fingers to help. Then we started saying things like, "if we count by fours five times, what do we get?" and then we changed our words to "four times five." All my kids have done well so far with this method. There is some memorizing to become fluent, but it is fully backed by understanding.
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.
1) Did New Math actually try to do those things, or did you just bring it up to poison the well against any alternative pedagogies?
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.
Hah... glad to see I'm not the only one who does that. I never did truly memorize the times table, but I can work out most small multiplications easily enough.
I still feel a certain tinge of guilt though, over not memorizing that stuff.
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.
Also Kline spends a good part of his book shredding the standard math curriculum (especially rote memorization and mindless application of standard algorithms) and agreeing that it needs reform, at the end advocating constructivist alternatives, with word problems, use of physical manipulatives, and motivation via applications to other fields.
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.”
> Do you want to know what worked? Memorizing times tables.
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.
Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi without first explaining how it's calculated? Where does the "four-thirds" part come from in calculating the volume of a sphere?
Not to mention that there's never enough history to go along with the mechanics - who discovered the quadratic formula? What was their life like? Why were they playing with quadratic equations in the first place? This make math seem less like magic incantations and more like something that was sort of cobbled together by flawed weirdos in order to solve real-life problems, and evolved over time.