More math education
From over at Mathematics Weblog (a paragon of naming wit), I find a few videos on YouTube dealing with math education. The first one shows me why Washington State has to cripple its graduation requirement exams. Math Education: An Inconvenient Truth illustrates some of the “reform math” texts for fourth and fifth graders, and shows the incredibly intricate — yet somehow called simpler — algorithms for basic place-value multiplication and division. The New Math overshot students’ capacity to learn abstract concepts, but these methods undercut their ability to learn at all, and at every step they come back to their all-powerful god: calculator.
I think it’s well summed up in this quote from the teachers’ reference for one such program:
The authors of Everyday Mathematics do not believe it is worth students’ time and effort to fully develop highly efficient paper-and-pencil algorithms for all possible whole-number, fraction, and decimal division problems. Mastery of the intricacies of such algorithms is a huge endeavor, one that experience tells us is doomed to failure for many students. It is simply counter-productive to invest many hours of precious class time on such algorithms. The mathematical payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator.
Okay, here’s why this is a complete load: the extensibility of the algorithm. The standard long division algorithm leads to polynomial division, and even power series division without any modification at all! If a student has mastered place-value long division, they are prepared for these later necessary algorithms. And students today simply are not. In fact, I just demonstrated power series division to my second-semester calculus classes the other day, and I’m pretty sure that I smelled some of the students wetting themselves when I said, “It just goes like long division.” These are ivy-league college students reduced to gibbering terror over a basic method that I remember being taught in elementary school less than twenty years ago. I can only imagine what it’s like at other universities.
Fine, I’ll grant you that most people will never have to long-divide even a polynomial in their “everyday” lives. But without this ability, and the comfort with mathematical algorithms it acts as a marker of, students will never achieve in mathematics, science, or engineering. What the authors are really saying in the above quote is, “we don’t think most of your kids are ever going to do much of anything in science or engineering, so we’re going to cripple all of them — even those who might otherwise have done.”
And here’s another point: you don’t always have a calculator. To indulge a bit of hyperbole, there’s a reason that a naval submarine carries paper copies of its own schematics rather than a CD-ROM of them. Jim Lovell didn’t have a calculator when things went wrong on Apollo 13. Pencils and paper never get shorted out, never run out of batteries, never have hardware glitches. In a mission-critical situation, having the ability to do without a calculator or computer is essential. Yes, most people will never be slingshotted around the dark side of the moon in a tin can, but the authors of these curricula are saying they have no problem with preventing that being your child on the first manned spaceflight to Mars.
The second video, Math Education: A University View, shows the ultimate effect of these “reform” math curricula on the quality of incoming undergraduates, particularly at the University of Washington. It also illustrates the general stylistic trends of such curricula, like the emphasis on students “discovering” mathematical facts on their own rather than being instructed by a teacher.
I knot that what I’m about to say is horribly politically incorrect (”counter-revolutionary”?), but the most basic mathematical methods must be taught — they cannot be discovered without mastery of the basic algorithms. I know I’m more interested in “why” than in “how” in higher mathematics, but this is simply not how people lay the foundations of familiarity with mathematics. It’s just not how human beings learn to think mathematically.
You want to know how people learn mathematics? There’s actually a movie about it, though you might not have recognized it as such at the time: The Karate Kid. Mr. Miyagi starts by making Daniel wax his car, paint his fence, sand his deck, and so on. Daniel doesn’t understand why he’s supposed to go through these motions, but Mr. Miyagi knows that the understanding will come only after mastery of the basic techniques, not before.
I have not met anyone who understands why the place-value algorithm for multiplication works without having learned how to do it first. After the student is forced to master the formal manipulations, authoritarian as it might seem, the underlying reasons may be taught or discovered. Besides, there’s simply too much depth to any of these manipulations for any student to understand why before how. I only just recently found out (via Michi’s blog) that place-value addition “really” arises from group cohomology. Did my not knowing this impede my doing addition problems? Of course not. And neither did my not understanding the ring-theoretical underpinnings of place-value algorithms before I learned them, nor my not understanding the set-theoretical underpinnings of natural number arithmetic before I learned them.
The algorithm comes first, and understanding comes later. Mathematics simply is. It cannot be negotiated. Mathematics education as realized in the NCTM standards has been taken over by sociologists, or even Critical Theorists. They are vehemently opposed to the seemingly-authoritarian rote method and saying “just do it like this and don’t ask why”. Never mind the fact that in this case “why” comes naturally after “how”. And it’s about time for mathematicians to come down and start kicking some ass over this, or we’ll be left with nobody capable of replacing us.
[...] I just have to add this quote from The Unapologetic Mathematician discussing this issue because it encapsulates what I have been trying to say for years though I [...]
Firstly- I recently watched the Karate Kid (again) and Mr Miyagi is awesome!
I must say that I slightly agree about learning the ‘algorithm’ first. This semester I might have spent too much time in actually trying to understand sequences and series (proofs etc) rather than doing the questions. I now realise that I shouldn’t have spent too much time in attempting to understand all the proofs since once I can do the questions and know what’s happening, the theory will start to make sense.
It’s a shame but I’ve used my calculator since year8/9 (since I’ve been 12/13) and my mental maths is pretty poor due to this. Thankfully since I’ve started university, I don’t use it as much as I used to. They shouldn’t be allowed in schools! (one of my college teachers could do calculations like 67 x 42 in his head!).
But it’s sad about what you say, that there might be students out there who could potentially go to Mars but are being ‘prevented’.
IMHO the teaching should incorporate the basics (by rote if that is what it takes) but at the same time the students should be gradually introduced and motivated to investigate pattern, structure and relationships.
Thereby reducing the subsequent dislocation that occurs on an introduction (or at every introduction) to higher levels of abstraction.
Logic should also be taught earlier rather than later.
Gasp! I watched the first video through the discussion of long division, and turned it off. I had heard that grade school math books have become large, verbose, and largely devoid of equations. Now I see that the kids are not even learning efficient algorithms of the simplest kind. Much of the power of math derives from efficient algorithms and the efficiency of equations and their manipulation. This is a tragic. Does anyone know if this trend also exists outside of the English speaking world?
[...] April 16th, 2007 in math, random Lately there have been a number of posts discussing the state of contemporary elementary maths teaching. While I certainly agree [...]
“Singapore Math” teaches concepts before algorithms.
I won’t bore you with examples of specific lessons, though.
I think the complaint with reform math is that it does neither. What they call “conceptual understanding” is probably not what a mathematician thinks of when he hears that phrase.
In terms of your students not understanding the specific application of long division to polynomial division, it’s not clear that their inability to extend this algorithm was because they weren’t taught the algorithm or because they were only taught the algorithm. “Conceptual math” advocates would argue that the whole reason they couldn’t extend it is because they didn’t understand it conceptually.
I have not met anyone who understands why the place-value algorithm for multiplication works without having learned how to do it first
To this I can only reply with anecdotes and we may have very different ideas of what “understands why” means in terms of children learning arithmetic. My ten-year old first learned multiplcation of “mixed” fractions by distributing over addition rather than the usual algorithm. He simply extended what he had learned from an earlier lesson on multiplication. I did eventually have to teach him something more efficient. When he was taught that decimals were fractions in “tenths” and “hundredths” he began converting all decimal problems to fractions (his idea), then performed the required operations, and finally would convert the answer back to decimal form. He appreciated being shown an easier way.
I have heard of several cases in which parents with degrees in math teach arithmetic to their children in this manner. It’s very slow-going while teaching the algorithm first brings instant gratification. Personally, I want both of them taught.
[...] The Unapologetic Mathematician. - The author’s of Everyday Mathematics do not believe it is worth students’ time and effort to fully develop highly efficient paper and pencil algorithms for all possible whole number, fraction, and decimal division problems. [...]
I am so glad I found your blog! You are fighting a very good fight, and you seem to be exceptionally well-armed. Hotcha!
There is a point that I’d like to make about “An Inconvenient Truth,” though, and at the risk of flaunting etiquette, I’d like to refer you to my blog post at The MathMojo Chronicles. It would be too long to post as a comment here.
Basically, the point of the post is that the “standard” algorithm is not necessarily the best in most cases, even though it obviously beats the poop out of the constructivist nonsense. There are other, more streamlined, even elegant (in the mundane sense, if it can phrased that way) and extensible algorithms.
Your “Karate Kid” reference is right on. I frequently do after-school programs in elementary schools around my state. Sad to say that many (most) of the children up to grade 6 still add on their fingers to certain extent. But in one hour, I get the little darlings doing rapid mental addition of nineteen one-digit numbers, simply by having them add playing cards (from ace, being 1, to 10) to find a missing card out of twenty cards.
They’ll do that over and over, doing nineteen or thirty-nine additions at a clip in order to find the value of the missing card.
If I’d have given them worksheets with twenty simple addition problems after school, they’d try to lynch me. But they’ll do hundreds of additions to “learn a trick.”
After they’ve done it a few times, and I write a long column of digits on the board, they laugh at how easy it is.
And they say you can’t learn math by magic!
Anyway, I really appreciate the quality of your blog.
All the best,
Brian, a.k.a. “Professor Homunculus” at mathmojo.com
[...] comes towards the end of a forceful but, I think, well-reasoned post which I found via Mathematics Weblog. As I understand it, the gist of the post is that in [...]
“Besides, there’s simply too much depth to any of these manipulations for any student to understand why before how”.
I think there can never be too much depth… It is easier to remember things when you know them really well. And you can not gain indepth knowledge if you avoid the depth
[...] many cases of basic techniques, intuition follows skill, rather than preceding it. It’s the Mr. Miyagi approach that I’ve written about before. Possibly related posts: (automatically generated)How much was [...]