The New Math
by Tom Temple
27 April 2005
How many times has the following happened to you?
You are in a restaurant with a bunch of people and it is time to compute the tip or divide the bill and people are intimidated about what to do. Inevitably the operation falls on you, the “math whiz”. You’re like, “What makes you guys think that I am any better at (multiplication/division) than any of you? I know someone has a cell phone with a calculator.” (Now to be honest, I am quick enough at number operations that I usually just do it for them. But if I think that the group is too math-intimidated, I say that as a public service.) The thing that bothers me is that my phone has both a calculator and a tip calculator. Why would there be both? You know what I’m getting at?
I have worse stories but some characters might be recognizable so I won’t share them.
I was sitting in class today faced with a terrifying mixture of calc, stats and linear. I couldn’t help but think that I was somewhat under-prepared. My first thought was to blame my previous teachers and the whole education system in general.
As the applications evolve, the education must be supple to keep up. People have been claiming for 40 years that mathematics education is to static and too rote. I can’t agree more. Perhaps it sounds funny (as in my favorite Tom Lehrer song), but “the important thing is to understand what you’re doing, rather than getting the right answer.”
Ask yourself where you spent the most time learning and where you spend the most time in application. If you’re like me, you spent 14 years learning how to compute things. But now I seldom actually add or multiply or divide actual numbers. I just write it out with the understanding that it could be done. I don’t often integrate stuff, I just write out what the integrals are. I don’t actually solve systems of equations, I just know when they can be solved.
When was the last time you used long division? Or even added 2 digit numbers?
The pinnacle of the standard mathematics education these days seems to be the integral. Don’t think that I am trying to downplay its importance but it is essentially long division again. Conceptually it is of critical importance but computationally it is a big waste of time.
When I tutored kids in vector calc I used this problem a lot.
d/dt(Something) =—divergence(Something)
I would say, “You already understand this equation. You’ve known it all your life. Give me examples of Somethings that makes this equation true.”
If we are over that hurdle, I don’t really care if they compute it wrong for some field or other. Alas, typically they pay me to increase their grade (as opposed to their smartness) and we end up spending most of our time computing it for this field and that field.
The nice part is that conceptually, most of math (the practical stuff, at least) is pretty intuitive. I’m convinced emphasizing the intuitive aspect would make learning easier. That, in turn, would free up time to cover more math. Maybe there’d be time to add the exponent and logarithm to the operations that the average American knows about. Maybe cover stats a little better and a little earlier. Maybe cover discrete math at all before sophomore Computer Science. Maybe make it to the convolution operation a little earlier. Maybe I wouldn’t have been hit up with quite so much new material today.
Since a number of you guys are or have been (or very nearly were) teachers, I am very interested in what you think about this.
Apr 28, 10:55 AM
This is a topic I think about frequently, and I have mixed feelings about it.
On one hand, I agree almost completely with your basic thesis, that the ability to grind through the mechanics of algebra is much less important than having a good intuition for its purpose and meaning. And by “algebra” I mean all the rote crap we have to do to solve real-world problems. It’s all just moving pebbles.
On the other hand, I believe the old potter’s saying applies: “If you’ve got no clay, the best hands in the world can’t make a bowl.” In math terms, that means if you haven’t ever thought about how to do arithmetic, and cranked through it, you’re damned if you’re ever going to grasp the intuition for more sophisticated abstractions.
Now, I am not claiming we should force everybody to spend their whole education doing stupid computational exercises. But the human mind is simply not designed to take in abstract ideas wholesale; we have to attach them to something concrete we understand. As Seymour Papert pointed out in the first chapter of his “Mindstorms” book, a lot of what makes someone “good” at picking up new abstractions is not that they have a better brain to begin with, but that they have a better set of concrete models to build upon. I think he’s right about that.
For those of us who are teachers, that means we can’t just tell our students to grind out a few word problems and expect them to just “get it.” As the Russians say, “Kazhdiy drochit, kak on xochet,” everybody strokes himself as he pleases. To get the “average American” to have a better grasp of mathematics is going to require an enormous commitment of time, creativity, and energy, not only by teachers, but by students.
If you ask me, that will never work in the American public education system, whose fundamental goal criterion is to give everybody a minimal education, not to get anybody a particularly good education.
Apr 28, 12:25 PM
I’m going to stir things up a bit. (i.e. do not mistake this for the sum total of my thoughts/response. I will gladly acknowledge that this is flamebait in its truest form.)
There is nothing as amusing as listening to amazingly smart people bemoaning the fact that not everyone is as smart as them:
“I mean really. What’s wrong with everyone else? div grad curl and all that makes perfect sense to me!”
Apr 28, 01:02 PM
Are you bad-mouthing Feynman? I just wont stand for that.
Firstly, I started this out with…bemoaning… But then, then I was talking about how I was having trouble with things. They weren’t even hard things, just in combinations I had never seen before. It would have been nice to have someone say, “here is what happens when you take the derivative of a covariance matrix with respect to a parameter vector.” before it is used as a vehicle for a harder concept.
In effect, I am complaining that my educators went too slow, something I’ve done before. That is, like Michael said, a lowest common denominator problem.
Second, I am was nicer to the kids that I tutored than I let on in that quote. They only get that question when I think they have divergence pretty well in hand.
Apr 28, 02:42 PM
I was not bad mouthing Feynman.
Again I’m going to preface this with the statement that I am deliberately picking out a relatively small aspect of what was posted by both Tom and Michael (last paragraph only) that I’m disagreeing with. Do not take these criticisms for the whole of my opinion or as personal attacks.
My point was that there is an inescapable element of elitism and arrogance in smart people complaining about an LCD problem in public schools.
Public schools are here to serve everyone.
(Astute readers with a careful knowledge of the history of American public schools will rightly be able to pillory this last statement. And I agree with them, per my caveat above. But the reality now is that public schools are here to serve everyone.)
Specifically, they are here to serve average kids. And to do that, I agree that the wishes and needs of very advanced and very remedial students are (somewhat) neglected.
But the fact that the very smart kids complain about this is (I think) inherently elitist and arrogant.
You complain, Tom, that your educators went too slow. That neglects the gifts you have. The rest of us look at you and marvel at how easily you seem to see into the heart of math/science concepts. When you complain that no one exposed you stats/discrete math/whatever until “too late”, I’m forced to ask:
1.) Even when you are exposed to things “too late”, you pick them up very quickly, relative to the rest of us. So what’s the big deal?
2.) What prevented you from checking a goddamn book out of the library and teaching yourself? You’re certainly smart enough.
Very smart kids inevitably reach some point in their education (college, grad school, retirement, whatever) when they realize that there are things that they need to know that were never taught to them.
My answer to this problem is that if you’re so smart, and you realize to need to know some stats in grad school, read a freakin book!
Instead, it sounds like you’re asking people to do that work for you: figure out everything I’m going to need to know in the future, and teach it to me!
Well, that’s impossible.
Indeed, I feel as though this style of intellectual inquiry is the essence of grad school. The presumption is that there will be things you need to know that you may not have learned. And the burden is on you to learn it.
The fact, Tom, that you can successfully absorb new math concepts in grad school is a testament to both
1.) Your natural ability
2.) The excellence of your education up till this point.
So that’s why it bugs me when smart people bitch about how no one ever taught them x or y before they got to college or grad school.
Finally, I’m going to retiterate that none of this is meant to label Tom or Michael(last paragraph only) as elitist or arrogant. Just a little unappreciative of how good their education really was.
Ok, I need to go running.
Apr 28, 03:20 PM
All I wanted to say was that math education should be tuned towards what math is being used for. What I am complaining about is that we spent nearly all of our time on the sort of things that I just don’t use. Michael brings up the point that you still need to know basic mechanics so you can build on it.
I would argue that we need far less mechanics than is being taught. I consider every single second of adding 3+ digit numbers a waste of time. By then we all knew what adding meant and we could all do it. At that point, I think they should’ve handed out calculators, showed us how to use them and moved on.
The problem, as I see it, is that math education hasn’t even adapted to the calculator yet let alone MATLAB, Mathmatica, etc.
I think it is immensely more valuable for a little kid to be able to say, “If I had N marbles and you took M, I would have N-M marbles.” As opposed to writing that out 210 times for every (N,M leq 20) with (M leq N). That is the sort of thing that I am talking about changing. I hear they tried it in the 70’s and the teachers rebelled.
Apr 28, 06:01 PM
Ok, Tom. I’ll respond to the actual meat of what you were saying now.
First,
“I consider every single second of adding 3+ digit numbers a waste of time. By then we all knew what adding meant and we could all do it.”
That’s because you’re very smart, Tom. This was/is not true for everyone.
Some people do lots of arithmetic problems, and never really truly grasp what’s going on. Some people eventually do, but it takes them lots of what you would consider rote and boring.
My point from before still applies. You’re treating math education from the perspective that everyone is just like you. They’re not. Some people are dumb.
Also:
“That is the sort of thing that I am talking about changing. I hear they tried it in the 70s and the teachers rebelled.”
It was more like the late 50’s and 60’s I think, though I’m not sure.
And there was a reason the teachers rebelled. “The New Math” sucked.
Now, I’m going to qualify that. The pattern in math education (in this country, in the past 100 years or so) has been to stumble across some miraculous new and different way to teach things every 20 years or so.
For the next decade or two, that becomes the only way to do things.
So the “New Math” that you are referencing (and that Lerher is singing about) literally involved zero rote practice and memorization.
The result? Lots of kids that knew a lot of set notation, but couldn’t actually add worth shit.
The fact of the matter, Tom, is that there is no “ideal” way to teach math (or anything else) that will work for everyone.
That is the great and beautiful challenge of teaching.
Instead, we are forced to develop methods that work pretty well for most people, and then augment them in individual cases, with individual students.
I would argue that (and I will later when I have more time) that calulators should be banished from schools until every student has more or less completely internalized arithmetic.
For you, that was 5 minutes after you learned it.
Not so for everyone else.
That’s the rub.
Apr 28, 07:03 PM
Joran wrote: “There is nothing as amusing as listening to amazingly smart people bemoaning the fact that not everyone is as smart as them.”
I know you’re trying to be inflammatory, but I can’t let this one completely slide by without a due response.
Since I’m not particularly smart, I feel I’m both safe and justified in making the following claim: The fact that the American school system is designed to serve the lowest common denominator has nothing to do with my intelligence or anybody else’s. Perhaps you do not like the phrase “lowest common denominator,” so let’s discard that for now.
The roots of our public education system arose out of the work of people like Horace Mann in the late 1700’s, who were struggling with the problem of how to make the newly-independent colonies stick together. Mann and Thomas Jefferson more or less created the idea of state-funded high-quality uniform education, in order to try to get traction against this problem.
However, despite the rhetoric of Mann and others to the contrary, “uniform” is as far as any centrally-run education system has ever gotten, and, in my opinion, as far as such a system ever will get. Inasmuch as there is natural variation in intelligence, any education system that wants to reach everybody is going to have to aim pretty low in order to hit that mark.
Apr 28, 07:31 PM
What a fantastic discussion we’ve kindled here. I shall, in the spirit of mathematics, number my points.
0) Joran is right that we have been unfair in considering the needs of those who find math (and learning) more difficult than ourselves. As I have diffuclty with, say, dancing and music.
1) Michael and Tom are right that our public schools do a mediocre job at the higher end. NCLB, for example, demands a lot of information on that lower end of the scale. Does it care at all if our young geniuses are languishing?
2) I agree with Joran that calculators should be abolished, but I disagree with extent. I think they should be abolished straight through college. I’m stick of typing numbers into them for ENGS problem sets that would be more interesting with just letters.
3) I agree with Tom that it is embarrassing how little we teach programing in general. We’re rapidly approaching an age in which inability to program will be tantamount to illiteracy.
4) There is clearly a balance to be reached between concepts and concrete skills. Even if you think algebra is better than arithmetic, and that letters are more important to math than numbers, you are still going to end up with some factors of 2 and 3, and the occasional 377.
5) Doing boring math problems over and over is important. It makes you good at math, it helps you recognize tricks, and it builds your confidence. Right now my confidence doesn’t last much more than a page of algebra, and that is a major shortcoming.
Apr 28, 08:15 PM
Even when I have actual numbers (and not letters) I write out the algebra without doing it. I would have said 13*29 instead. One of the things I tell kids when I am tutoring is don’t be too eager to do the arithmatic. They see 13s and 29s poping out of their equations and they are always so quick to consolidate them. Pretty soon they have a few huge (meaningless) coefficients that are more difficult to deal with.
I dissagree about calculators. The point is that there is an implicit understanding that your representation of the solution is tractable. I feel the same way you do about getting to the end of a problem and saying “42.656”. But when you write pi^(-1/2) in your answer that’s okay because it is a number that you could actually figure out if you had do. The thing that makes a symbolic answer okay is that fact that you could use it compute an actual answer if you had to.
In CS it happens all the time that I can write down a correct answer that is unnacceptable because it would take ages to compute. So at some level, people need to understand the tools they have at their disposal. Those tools include the calculator.
Apr 29, 05:00 AM
Tom wrote: “But when you write pi^(-1/2) in your answer that’s okay because it is a number that you could actually figure out if you had do.”
It’s acceptable because you can’t figure it out exactly. You can only approximate it. Unless you’re an engineer, of course. ;-)
I think there’s an excellent point in what you said here, that I’d like to draw out a bit, building upon what Jon said as well:
Teach people the rules of an algebra, and present them with a problem, and they will naturally try to apply all the rules they can, as eagerly as possible. One of the advantages of having them grind through boring problems over and over again (as Jon said) is that they will eventually build some intuition about when to reduce, and when to hold still. As Joran quite reasonably pointed out, this takes longer for some people than for others.
What Tom describes is a practical example of the difference between normal-order and applicative-order reduction in λ-calculus. If you do all the work up front, you will miss potential abstractions downstream (and, in the case of λ-expressions, may get into a cycle and miss the normal form). But you have to do some work, and knowing how to choose what to do is an art, not a science. A teacher cannot give you that intuition, only guide you toward it.
Apr 29, 07:22 AM
In order:
-Michael correctly takes me to task for my original (overly inflammatory) post. He is correct about the origins of our education system, but I want to point out one thing.
At the time (1700’s), the fact that our education system was uniform was not a problem. This was because it was actually not intended for everyone.
Only exceptional (and/or wealthy) kids were really expected to attend school for more than a few years, total. It wasn’t until the 20th century that it became expected that most kids would complete high school.
So I think that it’s not so much that the entire idea of a uniform education system is inherently flawed, it’s just that it doesn’t work so well when you force everyone to go through it for 13-17 years.
Jon: I agree with all five of your points (get it?!). In fact the only one that I have something to add on is #1.
I agree that public schools don’t provide exceptional students as much as they would like. As I was pointing out to Tom, I think I disagree with how large a problem this really is. In particular, I think that really smart people, being very close to this problem, tend to overestimate its importance.
Given that our public schools are charged with educating as many people as possible, for the aim of producing a more functional society, I place the following in descending order of importance:
a.) Providing lots of people with a great understanding of basic shit.
b.) Providing a small number of people with a tremendous understanding of really complicated shit.
My reasoning is basically what I was pointing out to Tom before. Super smart people are probably slowed down some by having to deal with the uniformity of our high schools. But these are people who are going to go to college and (likely) grad school anyway. Is it really that big of a catastrophe that they are forced to teach themselves some stuff when they get to college/grad school?
In short, producing talented people who have to play some form of catch-up in college or grad school is (in my opinion) harmless.
But pumping the masses out of high school who can’t do simple things like read, write or mental arithmetic is a serious problem for a functional society.
So I’m more inclined to worry about the mediocre, than the exceptional. (Obviously, I’m not advocating ignoring the talented. This is a matter of balance, as others have pointed out.)
Tom: I agree with your last post modulo calculators.
“The thing that makes a symbolic answer okay is that fact that you could use it compute an actual answer if you had to.”
My whole point is that for many people it takes a ton of time before they can “reliably compute an actual answer if they have to”. And doing this a lot gives people an intuition into the process (i.e. they get good at estimating). Let me give you an example:
My dad is a primary care physician who does a lot of teaching at the local hospital. These days, there is an enormous volume of pure data that physicians need to be able to recall very quickly.
Many medical schools have begun allowing students to (after a year or two) rely on PalmPilots and such to look shit up in.
So when my dad gets residents these days, they invariably have some PDA with them to look up drug interactions, dosages, diagnoses, etc.
Let’s take dosages. Prescribing the correct amount of medicine is tricky. There are a lot of variables (weight, age, other medicines etc.), and so people have devised formulas that calculate the correct dosage given a handful of values.
Many of the residents my dad teaches probably spent a semester in med school calculating with those actual formulas, but have long since been told that they aren’t worth memorizing and to just use a PDA.
This happens to my dad regularly:
Dad: “How much Drug X should we prescribe?”
Resident: (tapping away at the PDA) “100mg every 5 hours”
Dad: (my dad blinks) “That seems a little high.”
Resident: “That’s what the formula says.”
Now, my dad has had to calculate these things for common medicines a ton. He probably has no more idea what the actual formula is off the top of his head than the resident.
But he does know that the residents answer is off by a factor of ten! That could be fatal!
Somewhere the resident typed in .0001 instead of .001. Honest mistake! But they had no clue that the answer they got was potentially fatal!
So what’s the moral? Forcing people to do arithmetic by hand at least through high school is hugely beneficial for lots of people, and isn’t that large a burden on everyone else.
Apr 29, 08:56 AM
I remember listening to the CEO of Intel talking about how math education in this country is like walking down the blade of a knife. To get to the level where the skills are needed (implicitly by the end of school), you get no mulligans, there simply isn’t enough time to catch up. Of course, he’s wrong to the extent that really smart people will be able to catch up by teaching themselves. But he was more concerned with more average smart kids, the ones who could fall either way. He wasn’t talking about the top research positions, he was talking about engineering in general.
So we want to lead the world in engineering and we also want the general population to be educated adequately. As much as this conflicts with everything I think about social issues, it seems we need two sets of schools.
Jon and I went to pretty fantastic public schools. Let me tell you a little about my HS. There were about 600 kids in my class. In all of high school, I took any classes (besides gym) with maybe only a quarter of them. I took any math or science with less than an eighth of them. It was very segregated. There were lousy teachers but they didn’t teach any of my classes. There really were at least two distinct schools there. I had some ideas about how to better integrate them that I already linked and I wont rehash them here.
You could call that problematic, but I think it is probably worth it. Once you get to the college level, noone seems to have a problem with splitting up the kids. How is that so different?
I think your PDA example is flawed. I don’t think that your dad was better off because he had worked it out by hand. By hand I am far more likely to move a decimal point from time to time. Now the important thing is that you understand what is going on and you hace some intuition for it. Indeed actually solving the half-life equation is pretty illuminating but at the point when you are just using formulas (like the resident) I don’t see any problem with just tapping it out on the PDA, he should just be smart enough to check his freakin answer rather than try to impress your father with his speed. Your dad caught it because he knows that 100mg is too much. He probably knows that because he’s dosed it before, not because he is more intimate with the math (as you admitted, he probably isn’t).
Apr 29, 04:04 PM
Well, things got a little dramatic today and I’m sorry to have missed the real-time posting.
Having never set foot inside a public school classroom that has a class going on in my life, but having been inside many private school classrooms (as a student and teacher) perhaps I can offer a few observations. If my memory serves me correctly I first got accelerated in math in first grade when I got the second grade book, same thing happened in second grade. Then in third grade it became too complicated for me to have my own math curriculum (we’re talking a class of about 8-10 remember) and so I did third grade again in third grade. I stayed on normal pace until 8th grade when our class (now 16) was split into two groups. Obviously in high school no one had a problem sending kids from the same grade into different math classes. Sophomore year I skipped from Algebra II to Pre-Calc halfway through the year, so that I could do two years of Calc in high school. I feel like I was pretty lucky with my education. I had teachers who recognized that I was smarter (at math anyways) than most of the kids around me (fortunately no Jon’s or Tom’s growing up with me in Putney) and they helped me stay challenged. I gave this history to point out that I am not the norm.
My experience (at least in private schools) is that nobody has a problem separating kids out for different math classes once they reach the pre-Algebra/Algebra stage. This usually happens in 7th or 8th grade. I think this is a fine age to start the separation. Well, I guess there are two ways to look at it. Up until middle school I think it’s more important to keep kids with their peers for social development, and the math you’re learning (or not learning if you’re Tom) is very basic, but you’re not going to really get behind by going slow. I guess if you had a big enough class that you could separate it into an accelerated and normal paced math group starting earlier that would be fine, but I don’t like the idea of throwing a 4th grader into 6th grade math just because he’s smart enough. (There was a kid my last year at Putney who was in Pre-Calc as a freshman. He was not socially ready for high school – at all. Whether his social development was hampered by his accelerated math is something I’ll never know, kids that smart tend to be socially awkward anyways, but I don’t think he was necessarily served all that well. That’s my argument against having separate classes at a young age, but on to the more interesting stuff.
0) No calculators. I could not agree more with the latest consensus on calculators. I got kids last year in Geometry at SMS who pulled out the calculator every time they had to add two numbers totalling less than 50. With numbers that small I think we’d all agree that everyone should be able to add those numbers in their head. That they can’t is because they’ve been given calculators since they were old enough to press the buttons and they don’t actually know the basics.
1) Repetition. I was tutoring a few kids (who needed remedial education) who would do the same algebra problems different every time. All I had to do was change the numbers or letters and they’d go from perfectly doing the problem to having no clue. It’s going to take a lot longer for these kids (who we don’t want to leave behind) to be able to understand how to do some basic algebra than any of us posting on this.
1.b) Even the “smart” kids need practice. After half a year of Geometry (which is probably mostly useless in it’s current high school form – but that’s a different post) most of them needed a reminder on how to solve an algebra problem that is more complicated than ax = b. They remember faster than the average kids, but they need practice.
I’ve rambled a bit, but here is what I’m trying to get at. I think it’s great to push the kids. As a teacher, I much preferred the younger kids who were being pushed a little out of their element (and were much more focused on it) than the ones who were bored because it was too easy.
However, you run into a lot of problems when you push people. You can never teach to the fastest kids. Regardless of how many brilliant students you have, some are going to be more brilliant. They are going to be bored and either need to go find a book, have an exceptional teacher who will challenge them independently, or just sit tight. If you do teach to the fast kids, you leave too much of the class behind and you turn off a lot of smart people. (That would have been 11th grade where Julian and I thought calculus was great and everyone else thought the teacher went about 35% faster than he should have.)
In an ideal world we could have math classes of about 7-10 kids and everyone could be in a class moving at roughly the right speed for them. Until that happens I think we’re better off slowing things up a little and risking boring some child prodigies than we are teaching to their level. The kids who I had in 10th grade who hadn’t succeeded at math for a few years were pretty much hopeless. It wasn’t that they were incapable of doing it, rather they truly believed they were incapable of doing it. I broke a few kids out of this hopelessness, but only for a few weeks at a time. As soon as they fell behind they pretty much gave up.
I think it makes a lot of sense to try and get everyone through basic algebra as solidly as possible. That means with their peers and with the confidence that they can write down something like a set of simultaneous equations, see that they’re solvable, and be happy with that … knowing they could solve it if they had to. But until people are at that point, they shouldn’t get calculators.
All right. Time to eat.
Apr 30, 07:10 AM
I’d like to add another voice in agreement with the general sentiment against calculating machines in the classroom. However, I would also like to tie in something Jon said earlier:
“I agree with Tom that it is embarrassing how little we teach programing in general. We’re rapidly approaching an age in which inability to program will be tantamount to illiteracy.”
This is absolutely true, particularly in the U.S. and Europe, and it’s happening practically under our noses. I have a longer rant on this topic elsewhere, but I think the quoted sentence above is a pretty good synopsis of the real situation.
So how do we get kids to be literate in the use of computing machinery, without letting them use it? The answer, I think, is to separate the classroom from the laboratory: No calculators for adding, subtracting, multiplying, dividing, or basic algebra. But, if you want to solve a complex equation and you can write a program to do so, why, I think that’s just grand. In fact, I think that this kind of work should become part of the core mathematical curriculum. People can learn to drive word processors, drawing programs, and games on their own time—but code is a subtle algebra all its own.
Apr 30, 03:50 PM
Well said Michael.
If you can write a program to solve a complicated problem there’s a pretty good chance you understand how to solve the problem without the calculator. You probably would have to understand how to solve the general form of the equation to be able to write a useful program. I’m all for that.