Sunday, January 16, 2011

Why we can't trust math professors...

At least on k12 math education...

Consider the following math professors: Keith Devlin (Stanford), who wants grade schools to de-emphasize calculation skills; Dennis DeTurck (Penn), who wants grades schools to stop teaching fractions; and Jordan Ellenberg (University of Wisconsin) and Andrew Hodges (Oxford), who criticize grade school math as overly rote and abstract. And consider all the math professors featured in these three recent Youtube videos (an extended infomercial for the so-called "Moore Method," yet another re-branding of guide-on-the-side teaching and student-centered discovery learning--thanks to Barry Garelick for pointing me to these!).

Every last one of these math professors sounds like yet another apologist for Constructivist Reform Math. Each one of them can be--and in some cases is--readily cited by Reform Math acolytes and by the Reform Math-crazed media as such. And, yet, had these professors actually been subjected to Reform Math when they were students, it's hard to imagine that any of them would have enjoyed the subject enough to pursue it beyond grade school. 

Indeed, unless the mathematician in question has actually sat down and looked at the Reform Math curricula in detail, and imagined him or herself subjected to it, year after year, in all its slow-moving, mixed-ability-groupwork, explain-your-answer-to-easy problems glory, we should not trust what he or she has to say about it. Indeed mathematicians in general, unless (like Howe and Klein and Ma and Milgram and Wu) they take the time to examine what's going on with actual k12 math students in actual k12 classrooms, are especially unreliable judges of current trends in k12 math. Here's why:

1. Grownup mathematicians remember arithmetic as boring and excessively rote, and tend therefore to downplay the importance of arithmetic in general, and arithmetic calculations in particular, in most students' mathematical development.

2. Unable to put themselves in the shoes of those students to whom math doesn't come as naturally as it does to them, they tend also to downplay the importance of explicit teaching and rote practice. Some mathematicians take this a step further, imagining that if one simply gave grade schoolers more time to "play around" with numbers, they'd make great mathematical leaps on their own.

3. As more and more college students show worsening conceptual skills in math, mathematicians tend to fault k12 schools for failing to teach concepts and "higher-level thinking," not for failing to teach the more basic skills that underpin these things.

4. In upper-level college and graduate math classes, attended disproportionately by those who are approaching expert-level math skills, student-centered learning is much more effective than it is in grade school classrooms, where students are mathematical novices. Not enough mathematicians read Dan Willingham and appreciate the different needs of novices and experts.

Now there are two specific ways in which mathematicians can provide valuable insights for k12 math instruction:

1. They are perhaps the best source on what students need to know in order to handle freshman math classes. 

2. To the extent that they remember what it was like to be a math buff in grade school, and to the extent that they take a detailed look at what's going on right now in k12 math classrooms, they can offer insights into how well suited today's curricula and today's classrooms are to the needs and interests of today's budding mathematicians.

But the fact that mathematicians are really, really good at math does not, in itself, make them reliable sources on what works in k12 math classes. Quite the contrary.


Barry Garelick said...

Thanks for this post. I would add to the list of mathematicians to not trust: William McCallum, Uri Treisman, Hyman Bass.

They make the same mistake in judgment that some teachers, administers and school board members do when talking about the NSF-funded atrocities like Investigations in Number, Data and Space, and Everyday Math. They remark that they never understood math this well when they were learning it in grade school, but what they are seeing is their adult insight and experience. They owe their understanding and proficiency to what they now decry as ineffective, believing in the illusion that the math in these programs is deep. What they fail to believe (and refuse to believe when it is pointed out to them at the countless school board meetings where parents protest the programs) is that many if not most children cannot make the connections the adults are making who already have the experience and knowledge of mathematics.

As Katharine pointed out, in upper level math classes in college, the students have larger domain knowledge, so that inquiry-based learning can be more effective. I say "can be" because even the most knowledgable mathematicians have to be told things once in a while.

Even R. L. Moore, who is being deified in the series of informercials that Katharine posted, used his method in grad level course in math, stating that his method would not be appropriate for, say, freshman calc.

Back in the era of the 60's New Math, discovery learning was a staple of those programs, but given slightly less emphasis. Conceptual understanding played a big role, and procedural fluency was downplayed. One of the mathematicians who worked on the geometry textbook that came out of the New Math era (published by SMSG) was Edwin Moise. The geometry book was excellent--perhaps because one of the people working on the project--Floyd Downs--was a high school math teacher who knew what worked and didn't work with students.

Moise went on later to write a textbook in calculus. I used that textbook in my freshman calculus class. I still have it, and a glance through it shows how much he relied on a "spiral process" and presented problems with only a hint of the prior knowledge needed to make the inferences necessary to solve it. Instructors got wise to the book fast and skipped the various chapters that were too theoretical. In looking through it recently, I could see it's a book that's useful once you know calculus, or perhaps used for honors students who have already had a sufficient background in calculus in high school. But for the first-time learner, the high end theories were too much. That said, it's still a lot better than the reform calculus series taught on many campuses.

Amy P said...

"...who wants grades schools to stop teaching fractions..."

Good heavens. That's terrible. Fractions are the sine qua non for almost everything. Isn't it much truer that we wait far too long in elementary school before engaging fractions? My 3rd grader is just about done with the Kumon Grade 6 Fraction workbook and it has been really a good experience. Fractions (particularly multiplying and dividing them) is so much more dynamic and interesting than standard "big number" arithmetic.

My husband is a math PhD and I once read to him the math curriculum for the Archdiocese of Arlington (VA) when we were thinking of sending kids there several years ago. Anyway, each year had dozens of marginally connected math topics in elementary school, followed by a death-defying leap into traditional algebra. When I read it to my husband, he thought it sounded interesting. Then I explained that the kids got those same topics each year, just with the addition of even more topics. He was instantly much less of a fan. A lot of mathematicians who think they are fans of reform math may just need more information and more exposure to it to have a more informed opinion.

Barry Garelick said...

I would add to the list: William McCallum, Hyman Bass.

There are, however, some mathematicians who do not like reform math and they should be noted:

Jim Milgram (Stanford), Stanley Ocken (CUNY), Steve Wilson (Johns Hopkins), Hung-Hsi Wu (Berkeley).

Lsquared said...

Please don't lump the Moore method in with the rest of them. It's amazing: Moore method classes are what turned me into a mathematician. Yes, it's a guide-on-the side way of teaching--a lot of the talent and teaching goes into the preparation of really good notes (theorem lists) for the students to prove (that and knowing when to step in and redirect and when not to--it's not a teaching method I'd trust a non-mathematician to use). It is, so far as I can tell, the very best way to teach students how to prove theorems.

I'm told that Moore himself taught courses as low as college algebra and trigonometry that way--I wouldn't have any idea how to do that. Keep in mind, though, that the Moore method is good for training mathematicians because it's focus is on proving and learning to prove--Moore himself was looking for students who would make good mathematicians, even when teaching college algebra and trigonometry. I wouldn't argue that the Moore method is the best choice if who you are training is future engineers and physicists, but it is absolutely the best for training future mathematicians.

ChemProf said...

Yeah, but that means the Moore method isn't appropriate for K-12. If the price of gaining a couple of mathematicians is turning off a room full of future scientists and engineers, the price is too high. At that level, they haven't sorted out yet.

Deirdre Mundy said...

Wait... lists of theorems to prove? Isn't that essentially High School geometry?

Also, I've found most of my mathematician friends rarely deal with numbers greater than '2'. So of COURSE, from their perspective, calculation is useless. Perhaps it would be better if we recognized that applied math and pure math are two completely different disciplines that both happen to have a passing aquaintance with the sets of real and imaginary numbers....

ChemProf said...

Thinking more about this, I'm also reminded of something I see in advising. When I have a student who loves calculus and thinks they want to major in math, I recommend that they take Real Analysis (which in my college is the first major proofs class) as soon as possible. That's the class that really shows what mathematicians do, while Linear Algebra or DEs won't.

Barry Garelick said...

Dierdre: GOOD POINT!!

For all the complaining that the refomers do about how students should be able to apply prior knowledge to new situations rather than the alleged traditional approach of "Here's the method, now here's a bunch of problems to solve using the method", proof-based geometry offers the opportunity to do that all the time. Not necessarily to prove the major theorems, but to prove various propositions, using theorems, postulates and definitions. There is no set algorithmic approach in doing proofs; rather, they involve a strategy and an understanding of what is going on in the problem and what sequence of steps will get you from the given hypothesis to what must be proven. But geometric proofs are thought to be de passe.

Jordan Ellenberg said...

Just to set this straight. I think memorization of arithmetic facts, like multiplication tables, are vital for kids learning math, and I don't think I've ever said otherwise. Anyone who's ever taught first-year students math can see that a student lacking in computational fluency is at a severe disadvantage when trying to master college-level material. And computational fluency starts with knowing a lot of arithmetic facts by heart, without having to think about them.

I do tend to hope that this is not all the K-12 curriculum will do.

Also, the "Moore method" is not a re-branding of something newfangled; it's quite old. And most working mathematicians find it quite weird, albeit intriguing, and would never use it, or anything like it, in a classroom.

Finally -- and probably most importantly -- I certainly agree that math professors are trained at teaching college students; we are not, and I think seldom claim to be, authorities on K-12 education.

Katharine Beals said...

Thanks, Jordan Ellenberg, for your clarifications!

Regarding the Moore Method, what I should have written instead of 'an extended infomercial for the so-called "Moore Method," yet another re-branding of guide-on-the-side teaching and student-centered discovery learning' is: 'an extended but content-free infomercial for the so-called "Moore Method" that makes this method sound like another re-branding of guide-on-the-side teaching and student-centered discovery learning.'

In other words, it seems that the Youtube videos I link to don't do the Moore Method justice. They certainly don't capture anything weird about it that would explain why most working mathematicians would never use it. So now I'm intrigued, and would love to learn more.