Friday, May 27, 2005

Math Teaching

One curious phenomenon is the way that some liberals oppose good math teaching for fear of being associated with conservatives. Take this article by Barry Garelick, identified as a mathematics analyst for a government agency:
I work for the federal government, which has a program that gives employees a chance to work on Capitol Hill to gain experience and knowledge of legislative and congressional procedures, which is valuable information when working in government. I applied for and received a six-month detail to work in a Democratic senator’s office. Senator X (so called, in keeping with mathematical convention to describe a class of variables, because, as I was also to learn, both the good intentions and the shortcomings of Congress are institutional) was interested in establishing a science project to nurture a “homegrown” breed of scientists and engineers who would then support that state’s burgeoning technology industry. Since I thought a likely place to start would be math education, the staffers working the education issue asked me to see what I could come up with.

I compiled a list of questions that I sent by e-mail to various mathematicians involved with the math education issue. The questions focused on the quality of textbooks and teaching, with emphasis on algebra and geometry. I also wanted to know whether K–6 texts taught arithmetic well enough to prepare students to learn algebra.

The nice thing about working on the Hill is that you almost always get responses to e-mails and phone calls. Fifteen minutes after I sent an e-mail to Harvard mathematics professor Wilfried Schmid, he called. I found out that his initiation into the world of K–12 math education was similar to mine—through his daughter. He explained how she was not being taught her multiplication tables. He was shocked at the math instruction she was receiving in the 3rd grade. Its substance was shallow, memorization was discouraged, students were kept dependent on mental crutches (her teacher made her work with blocks or count on her fingers), and the intellectual level was well below the capability of most of the kids in his daughter’s class.

Schmid’s reaction to the problems of math texts and teaching was similar to that of other mathematicians I talked to in the course of my Capitol Hill assignment, particularly those with children. Those dialogues led me to develop an ad hoc theory that I will postulate (this means I don’t have to prove it): Hell hath no fury like a mathematician whose child has been scorned by an education system that refuses to know better. In Schmid’s case, he talked to parents, school boards, and ultimately with the Massachusetts commissioner of education. Along with others, he succeeded in revamping Massachusetts’s math standards, much to the dislike of the education establishment and textbook publishers.

The controversy over K–12 math education has come to be known as the “math wars.” Like Schmid, mathematicians have been active in this debate, as has the “mathematics community” at large, including not only mathematicians at the university level, but teachers and others involved in the education establishment. They believe that students must master basic skills (the number facts, standard algorithms for adding, subtracting, multiplying, and dividing) in tandem with larger concepts about mathematics.

On the other side of the debate are the followers of an education theory that promotes discovery learning, minimization of both teacher instruction and repetitive drills, and a disdain for standard procedures (algorithms). The math being protested—by the mathematics community—is called a variety of things: “reform math,” “standards-based math,” “new new math,” and, most commonly, “fuzzy math.”


* * *

Concept still trumps memorization. Textbooks often make sure students understand what multiplication means rather than offering exercises for learning multiplication facts. Some texts ask students to write down the addition that a problem like 4 x 3 represents. Most students do not have a difficult time understanding what multiplication means. But the necessity of memorizing the facts is still there. Rather than drill the facts, the texts have the students drill the concepts, and the student misses out on the basics of what she must ultimately know in order to do the problems. I’ve seen 4th and 5th graders, when stumped by a multiplication fact such as 8 x 7, actually sum up 8, 7 times. Constructivists would likely point to a student’s going back to first principles as an indication that the student truly understood the concept. Mathematicians tend to see that as a waste of time.

Another case in point was illustrated in an article that appeared last fall in the New York Times. It described a 4th-grade class in Ossining, New York, that used a constructivist approach to teaching math and spent one entire class period circling the even numbers on a sheet containing the numbers 1 to 100. When a boy who had transferred from a Catholic school told the teacher that he knew his multiplication tables, she quizzed him by asking him what 23 x 16 equaled. Using the old-fashioned method—one that is held in disdain because it uses rote memorization and is not discovered by the student—the boy delivered the correct answer. He knew how to multiply while the rest of the class was still discovering what multiples of 2 were.
So that depicts the situation. How do some liberals respond?
Though academic debate about mathematics curricula will no doubt continue, the field of argument is increasingly muddied by politics. It was in this context that I began my investigation into math education in 2002. I recall meeting with Senator X’s deputy chief of staff and two other staffers not long after completing my research on math curricula and the battles that had shaped—often, misshaped—them. “So what are your ideas on how math and science education can be enhanced?” they asked. My answer was something like, “You can enhance a car by painting it, but if the car has no engine, it’s not going to do much good.” This was not what they were expecting to hear. Nor were they expecting to hear that Lynne Cheney had also taken up the cause of anti-fuzzy math. At that point, the discussion took a decidedly troubling turn. These staffers—Democrats—now worried that they could not support policies that were also advocated by the wife of a powerful Republican.

I told them about the open letter from the two hundred mathematicians and urged them not to confuse the message with the messenger. “This is a real issue,” I said. “Kids aren’t learning the math they need to learn.”

I had discussions and sent e-mails in the hopes that I would at least get a chance to brief Senator X on the issue and, perhaps, persuade him to ask some tough questions of NSF when it came time to fund their programs. But I felt that at any moment everything was going to be whisked away.

And one day it was. The staffers in my office talked with other Democratic staffers on the Hill, who told them that it would be wise to stay away from the “fuzzy math/Lynne Cheney/Bush agenda” issue. Ultimately the staffers I was working with told me they couldn’t take a chance on having Senator X “come off like Lynne Cheney.”

This development was not surprising to any of the mathematicians with whom I had been working—most of them Democrats, like me. The senator was never briefed, and no investigation into NSF was launched. I was thanked for my hard work. I went back to my regular job and started tutoring middle school students in math at a school in D.C. while continuing to work with high school students in my neighborhood. That year a 9th-grade girl was having problems in geometry and came to me for help. “What seems to be the problem?” I asked. “I don’t know how to do proofs,” she said.

“I know,” I replied. “Don’t worry. It isn’t you.”

All politics is local, I decided.
Cheating kids out of an education in basic math is inestimably more harmful to their actual lives than anything that they might be taught about evolution.

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19 Comments:

Blogger P. G. said...

I suspect this is unfair. I don't know anything about how children best learn and retain math. Neither, however, do mathematicians -- although they no doubt know more about the subject than I do, the ability to prove the riemann hypothesis or whatever is unlikely to be of much utility in figuring out how to teach long division to children.

The people who do know how to perform that teaching are teachers and child psychologists. If they say that this so-called "fuzzy math" helps children learn and retain the same concepts that they were being taught in the un-fuzzy math, I say go to it.

I simply don't see what the hell some mathematician, working on things like cryptography and number theory and linear algebra and all the rest would possibly have to say about how to learn 7th grade math.

Also, the supposedly anti-being-associated-with-republican-math comments made by the democratic staffers -- while absurd and appalling if true -- weren't quoted, except for one quote about Lynne Cheney that looks suspiciously out of context since it doesn't say in what respect the coming-off would occur. All the rest of the supposedly incriminating statements are characterizations by this source of what he understood the democrats to mean.

5:29 PM  
Blogger Stuart Buck said...

As far as I can tell, this isn't a debate about the best technique for getting kids to learn the multiplication tables. If that were the case, teachers might have a good idea about what techniques are successful.

This debate is on a meta-level: Should kids be taught the multiplication tables at all? Some people apparently say no, that kids shouldn't be "drilled" with boring facts, that they should focus on higher concepts. Others -- whether mathematicians or lay people such as myself -- are entitled to believe the obvious truth that kids don't learn the multiplication tables unless they are taught to do so. It's not (usually) a natural ability. As well, if kids aren't taught the simplest mathematical tasks, it's no wonder that when an item costs $4.76, and I give the cashier $5.01, the cashier has no idea why I handed him the penny until he punches the numbers into his machine.

5:44 PM  
Blogger John Thacker said...

The people who do know how to perform that teaching are teachers and child psychologists. If they say that this so-called "fuzzy math" helps children learn and retain the same concepts that they were being taught in the un-fuzzy math, I say go to it.

Except, no. There have been repeated studies about what sorts of teaching methods work best. They all show that Direct Instruction and other methods score better not just on mastery of facts and techniques, but even on things like self-esteem. (Turns out that knowing things helps self-esteem.) The educationists don't care. Teachers hate teaching by rote and by script, for one thing. And all the educational theory disagrees with the methods that scientific studies have shown to work better. The education theory people, seeing that the studies and the theory conflict, choose to ignore the studies.

5:48 PM  
Blogger John Thacker said...

Here's a link to a listing of some of the wide-ranging studies examining various instructional methods. Direct Instruction (a particular direct instruction method and curriculum) scored extremely high on all the tests and does very well. There are many websites about it. It is, however, hated by the educational establishment for reasons given in the article.

Their opposition is based on theory which they dogmatically hold to in the face of all data and evidence.

6:01 PM  
Anonymous Anonymous said...

Really, Paul Gowder, if you have any intellectual integrity, it would behoove you to offer a current-day example of a situation where educational theory actually reflects empirical quantitative research before you begin exalting education theorists. As the other commentators note, that just isn't the way it works.

7:23 PM  
Blogger P. G. said...

Anonymous: there's no call to attack my "intellectual integrity." I openly said that I don't know the right answer, but that I'd trust child psychologists and teachers over mathematicians to get to it, in the absence of any more direct evidence in my posession, simply because of their specialized study of the area. What's dishonest about that?

John then provided new information. If, as he said, education theorists have just ignored the scientific evidence, then obviously their theories lose credibility. I've no problem with that. (I do note that the pdf John linked to trumpets a meta-analysis. Meta-analysis, as ordinarily practiced, is often questionable because its practitioners have a habit of attempting to compare wholly different studies from wholly different populations, etc. But without data on the specific study, it's hard to tell.)

Stuart: what's the difference between this meta-level and an ordinary level debate? Can we agree that both sides of whatever debate is going on want kids, when they get out of the education level, to be able to take two numbers, multiply them together, and come up with the correct answer? If so, then I think it's immaterial whether they do so via memorizing tables or learning techniques, as long as they actually do it.

If not, then of course I agree with you, the actual learning shouldn't be stripped out of the education.

But if it's just rote versus non-rote learning of the same stuff -- well, that's pure technique.

2:02 PM  
Blogger Sam Chevre said...

Can we agree that both sides of whatever debate is going on want kids, when they get out of the education level, to be able to take two numbers, multiply them together, and come up with the correct answer?

No; amazing as it sounds, skills like these are exactly what is in dispute. (I was once a math teacher, my girlfriend is a math teacher). A lot of educators consider that learning how to multiply is unnecessary--just learn how to use a calculator to get the correct answer. This variety of "math knowledge" is absolutely crippling; it is almost impossible to become good at algebra, or even at fractions, without a good command of how to do multiplication and division without a calculator.

3:17 PM  
Blogger P. G. said...

Sam: to the extent your observations reflect general trends, I retract everything I said in the previous comments -- of course that's bloody ridiculous, and if my fellow left-wingers are pushing it, they need a good slapping upside the head.

1:02 PM  
Blogger John Thacker said...

The National Council of Teachers of Mathematics is filled with people who are amazingly pro-calculator, to the point of not wanting to teach memorization of any basic skills. It's insane. I don't think that they reflect all left-wingers, certainly, but the education establishment generally favors Democrats (since Democrats vote for higher educational spending of all sorts) and puts tremendous sums into backing them. It takes a lot of effort and courage for Democrats to oppose the education establishment. However, if the opposition even looks like it's being led by Republicans, then the educational establishment is easily able to rally its troops and get Democrats to oppose it.

In any case, when it comes to talking about effective teaching methods, the recently established Institute of Education Sciences, a part of the Department of Education, is a good place to go. It was established in 2002 as part of all the NCLB stuff, over the opposition of parts of the educational establishment. It's the DOE's official institute to study data about what educational methods work better and propose various experiments to test different methods. The education establishment fought its creation; that should tell you something about how well they can be trusted.

6:52 PM  
Anonymous Anonymous said...

The notion that people who actually work in mathematics don't know anything about how mathematics is learned is really quite hilarious. Naively one might think that their, uh, personal experience with learning math, as well as their knowledge and appreciation of what math actually is, from the ground up, might give them some, y'know, insights. At least in aggregate, and when compared with the population at large. But no - obviously "child psychologists" know better. *sigh*

In general, it is my impression that the real problem lying at the root of the 'fuzzy math' push is that the people we currently task with teaching math at this level, for the most part do not seem to actually like math. They find multiplication tables "boring", and memorization of course is just "rote", and so on. These descriptions we always hear of "standard" math aren't coming from 8 year olds, they're coming from teachers! In other words, our math teachers don't like the subject of math. That's fine (it's their right and all), but it's also obviously a bad policy result, and has two effects:

-It makes them *bad* teachers. They hate what they have to teach so don't do it well; as well, they can't help but convey their disdain for the subject to the students, if only inadvertently.

-It causes them (understandably, actually) to try to alter what math is. Since the subject of math is boring for them, why not just change it? It would be more fun to teach if math were different than it is. So, they set about making math different. That's what this is about.

Teachers do not want to teach math, so they clamor to teach something else. They have invented that "something else" to teach in lieu of math, something that they find more interesting and exciting to be teaching in a classroom. Contra gowder above, 'fuzzy math' policies are all about making teachers happy and allowing them to have more fun in the classroom, not about successfully teaching students math - which they do not actually seem to want to do.

It would, obviously, be better to shift the focus to the students and teaching them math. But of course, teachers wield political power and 8 year olds do not.

Their parents do however....

12:11 AM  
Blogger P. G. said...

Anonymous: that's ridiculous in several respects.

1. The idea that mathematicians would know how to teach elementary arithmetic more than anyone else is absurd. Mathematicians have no special training or transmissible insight into multiplication tables. They have special training and transmissible insight into complex systems and linear algebra and primes and all that stuff. How does this make them any better at teaching the basic math that is learned in school by children?

By analogy: I'm a lawyer. Does this mean that I'll be better at teaching high school civics than a high school civics teacher? Does this mean I have any idea what needs to be done to instruct children in whatever it is that high school civics teaches? Certainly not! Now if they wanted to know about the First Amendment, or substantive due process, or Title VII, or 42 U.S.C. 1983 and qualified immunity, or why law & econoimics is stupid, I could teach that better than the high school civics teacher, but those aren't what gets taught in high school (as far as I know). And I strongly suspect that the high school civics teacher knows just as much about "how a bill becomes a law" and that sort of high-school twaddle as I do.

The same applies to mathematicians versus teachers and child psychologists. Any schoolteacher (or child psychologist) worthy of the title knows how to multiply two integers and get the correct answer just as well as anyone on the faculty of the math department at MIT. What other knowledge have the latter to add? What insight do matrices and number theory have to offer a child in the mechanical task of multiplication? On the other hand, the former have years of experience in actually teaching children, plus the benefit of psychological research, to add. Advantage: teachers and child psychologists.

2. I think everyone finds multiplication tables and memorization boring and rote, children included. Didn't you? I know I did. In fact, contra your allegation that the descriptions of rote math never come from 8 year olds, I don't think I've ever met an intelligent kid who didn't think memorizing stuff like multiplication tables (which they may not have described as rote because they don't know the word) is about as boring it's possible to get without actually being strapped down to a chair with your eyelids propped open. The same goes for memorizing battle dates in history class instead of being actually permitted to develop an understanding for why the battles happened.

In fact, although I recall being taught the multiplication tables, I don't think I ever bothered to learn them. I don't automatically remember what 8x7 is, for example, right now: I have to stop for a second and think about it. Fortunately, all my elementary school math tests provided ample time to do that stopping-and-thinking, and I don't see what's wrong with that.

The skill of knowing INSTANTLY what 8x7 is has no social, scientific, artistic, or economic value as compared to the skill of knowing how to determine, in half a second, what 8x7 is ("I remember that 8x8 is 64, so subtract 8..."). What's the situation where you need the rote version? I suppose if you're in a war using some truly ancient (or broken) piece of field artillery maybe and you need to calculate a trajectory before the enemy gets you, but I think you've got bigger problems than the multiplication tables at that point.

So why should children be subjected to something that bores them when they can (if they can) learn it better another way? Do you learn better in boring ways or interesting ways?

Of course, I say again that this is all subject to the baseline position that the goal is to get the kids out of school with the knowlege of how -- by whatever means -- to use their brains to get the numbers multiplied together. I reject any theory that does not purport to reach this goal. However, within the set of theories that do purport to reach this goal, I trust the ones coming from the teachers and child psychologists.

9:39 AM  
Blogger karrde said...

Guys, I'm a latecomer to this discussion.

I myself am working as a graduate student at a research-oriented Tech University.

The various instructors at this TechU routinely complain about the kind of mathematical education that students have received in their elementary/high school experience. There seems to be a rising percentage of students who have all the mathematical understanding of a trained animal pushing buttons on a calculator.

I have run into this in comments by other college-level instructors, especially the ones who are extremely knowledgeable about mathematical instruction at the lower levels, and they all complain that there is far too much training in calculator use and far too little training in mechanics of mathematics.

This can be taken with a grain of salt--however, I think that this is indicative of a strange trend in professional education.

Since the book Why Johnny Can't Read was published, a large debate has ensued in educational circles about the difference between "sight-reading" and "phonics" in teaching 6-year-olds how to read. The "sight-reading" pros claim, rightly, that most adult readers don't use phonics very heavily. They claim that if we can teach little children this process, we'll save everyone the hassle of phonics.

The "phonics" side might admit that many adults have learned a large vocabulary by sight, but adamantly insist that reading is easier to learn if the students know how to decode strange words phonetically.

Long debates are held over this subject. The evidence at hand seems to indicate that children who are taught phonics can decode most of the English language at the end of their second/third year, while the sight-reading students may have 1,000-3,000 words memorized by that time.

But the debate seems to involve some principle or idea which is deemed sacred to both sides, so the facts are drowned out by the shouting.

The debate over "fuzzy math" versus "drill and memorization" is made out of the same stuff--lots of noise and heat, little light.

All I can ask is this--did the scientists who designed the Apollo 11 (or the SR-71 reconnaisance airplane) learn their mathematics by drill, or by these "fuzzy math" methods? What about the civil engineers who designed the Golden Gate bridge, Mackinac Bridge, Brooklyn Bridge? Or the Wright Brothers, before they built their Flyer?

If so many scientific and engineering geniuses of the past learned their mathematics by drill and memorization, then perhaps it has something going for it.

There's also something I heard an old curmedgeon mention about middle-school and high-school math. He mentioned that band members go to practice every day. Football players drill their passes, blocks, tackles, and formations every day. By analogy, he says good math students should do lots of addition, subtraction, multiplication, and division every day. (Or is that lots of algebraic solving, geometric proofs, use of trigonometric identities, and solutions involving limits...?)

1:27 PM  
Blogger P. G. said...

Really, the ideal would be to teach it all. Extend the school year and pack in both the memorization and the conceptualism.

To the extent the memorization is necessary (and assuming it is) it shouldn't be abandoned wholesale, but it's a strange kind of poverty to condition kids to spout information without any internal connection between that information and the world or the abstractions that lead to the information.

It really is like history teaching. What possible point is there in memorizing the fact that battle X happened on date Y and being able to spout it on an examination without having a conceptual understanding of why the two countries were at war?

Are we trying to produce thinkers or repeaters?

2:38 PM  
Anonymous Anonymous said...

Paul,

First of all, there's almost no such thing as "special training or transmissible insight into multiplication tables" unless by this you refer to the higher mathematics built upon them. And yes, that is what mathematicians are trained in, higher mathematics. Perhaps even more importantly, mathematicians are highly likely to be people who like mathematics and I would argue that this is probably even more important when it comes to being a good teacher. As you say, the multiplication tables are boring, but at least mathematicians know how they are structured and abstracted, and that there is light at the end of the tunnel.

You seem to have this idea that for any given subject there's some secret, psychological-biological method or sequence of incantations - this "what needs to be done" thing you keep referring to - for inserting it into childrens' brains (it's different for each subject, I presume), and only "child psychologists" (and, of course, junior high school teachers who have gone to the correct conferences) know them. That is BS and I wonder where you got this idea.

Yes, the multiplication tables are boring, but they are building blocks. "Conceptual understanding" comes later. To refrain from teaching someone the building blocks in the hopes that he will stumble on the "conceptual understanding" is downright cruel. A small fraction will do fine, the larger majority will be left behind. All because teachers disdain the idea of "just" teaching basic math facts (like you said, 8 year olds don't know the word "rote". "Are we trying to produce thinkers or repeaters?" is an adult's question and sentiment).

And your lawyer analogy is a bit off. I am not aware that "civics" per se is a building block of lawyerin'. But pick something that is - How a bill becomes a law? Public speaking? How to construct an argument? Elementary "if..then" logic? I don't know, you tell me. But yes, I *do* reckon that you, as a lawyer, would have a nice insight into [insert subject that is a building block of lawyerin' here] that the general public - even "child psychologists" - would not tend to have.

You asked why one needs to know the memorized answer to 8x7. One doesn't. The real priority is to *get past* the multiplication tables so you can stop talking about them and do real stuff! But (1) that doesn't mean you can just skip them. And (2) this "fuzzy math" stuff sounds like they spend years and years pondering them. It's a *waste of time*. These math facts are less equivalent to Dates in history than they are to Words in English. You need to know a minimum of Words to be able to construct sentences. You don't want to waste years and years thinking about "Words" before starting to write.

Teaching math is *not* like teaching history, at least the version of history you allude to that consists of 'dates'. Math is hierarchical; you build upon what you learned before. Stalling this process at age 8 to assuage teachers' concerns that they're being "interesting" enough, is really quite criminal.

Just get past the dang tables and move on, for crying out loud. It really shouldn't even take that long.

10:54 AM  
Anonymous Barry Garelick said...

Mr. Gowder:

I am the author of the article that spurred the discussion on math education. You stated:

"Also, the supposedly anti-being-associated-with-republican-math comments made by the democratic staffers -- while absurd and appalling if true -- weren't quoted, except for one quote about Lynne Cheney that looks suspiciously out of context since it doesn't say in what respect the coming-off would occur. All the rest of the supposedly incriminating statements are characterizations by this source of what he understood the democrats to mean."

I can assure you the quote was not out of context and that the paraphrasing of staffers' comments was quite accurate. The staffers with whom I was working conferred with other Democratic staffers who were familiar with the issue and were informed that this issue had political "baggage". The "fuzzy math" issue was one associated with Lynne Cheney and was considered by Democratic staffers to be a Bush agenda item. Taking on this issue would also have necessarily involved addressing the teacher's unions which is something the staffers of Sen. X probably did not want to do. People who work on the Hill and know how it operates did not find this aspect of my article surprising. This "guilt by association" aspect of Hill politics goes on, unfortunately.

Barry Garelick

11:19 AM  
Blogger P. G. said...

In that case, let me say that I'm embarassed to be associated, if only by joint party membership, with such people. This is regardless of the merits of the issue (which I still dispute). It's to my mind totally unethical to avoid something that one genuinely believes is good policy simply because it's associated with one's opponents.

Note however that this kind of moral compromise is shared by both parties. Witness the unholy alliance between the Christians and the Capitalists, one that hasn't been seen since Jesus kicked the moneylenders out of the temple...

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