Guessing In Math

Q: Which way to Millinocket?
A: You can’t get there from here.

I’ve been a fan of the “Sidney Awards” that David Brooks gives out around the start of the New Year. Originally dubbed the Hookie Awards, the name was changed the next year to signify a more adult designation. The awards are a tribute to the philosopher Sidney Hook. The winners are essays that Brooks deems enlightening and good reads. I appreciate getting reading suggestions from all sources and during the holiday period am apt to read things I wouldn’t otherwise get to during the school year. This time around, unfortunately, I had already read two of the essays – Michael Lewis’ piece The End in Portfolio Magazine, which Tom Friedman had recommended earlier in his column (Brooks shouldn’t be allowed to choose Sidney winners from among essays his fellow columnists at the New York Times recommend so readers like me have more of a selection) and an essay by “Professor X” in the June 2008 issue of the Atlantic, In the Basement of the Ivory Tower, very depressing but definitely a good read.

Professor X is an adjunct instructor of English, teaching introductory classes at night at two different colleges, one a small private college, the other a community college, doing this to pay the bills till he can find a better gig, i.e., a traditional tenure track position. The students are adults going to school part time, almost certainly to get some certification for work. The crux of the piece is that many of these students are not prepared for college, are uninterested in what is taught in the English class largely because they don’t read, and while as a business proposition their paying tuition is a money maker for the hosting college (adjuncts don’t make very much and by holding the classes at night they are utilizing space that would otherwise be idle) as an ethical proposition attending college for many of the students is a sham because these students aren’t ready. The implication is that things are better for the students who attend full time during the day. Perhaps.

At a New Years Eve dinner party hosted by my good friend Larry DeBrock, I got to talking with another friend, Denise, now retired but who had been my wife’s boss and I believe her job title was Associate Provost for Human Resources, a very important position on Campus. I’ve known Denise for a long time. Her husband Wally, also now retired, was a faculty member in the Econ Department and an expert bridge player. He taught several of us assistant professors a stripped down “Standard American” bidding system in bridge which we used when we played over the lunch hour and occasionally in the evenings. It was good fun, an entertaining diversion from work.

Somehow during the conversation Denise brought up to another friend, Diane, who is from out of town but for many years has come to the DeBrock’s house for New Years, that I would on occasion tutor Denise’s daughter in math, both in high school and in college. Really what would happen is that Wally would give me a phone call and we’d talk through some homework problem she was assigned, a quid pro quo for the bridge lessons. Wally thought taking Math was important and had made an agreement with his daughter that she’d go beyond the required courses at Illinois. So she dutifully took Differential Equations, kind of an odd course for an Accounting Major to take. Barbi, the daughter, has done spectacularly well in her Accounting career. But she really had no clue about Differential Equations. She got through by memorizing the solutions to the problems in the textbook, an awesome feat of will but not a very satisfactory way to learn the math.

Denise and Wally both grew up in the San Francisco Bay area and both attended Berkeley as undergrads. Wally also did his graduate work in Economics there. Denise went to graduate school later, after they had moved to Illinois. During the dinner party Denise confessed that she struggled with math in high school, algebra and geometry were opaque to her, let alone trig. Somehow she got by. She was a very good student at Berkeley and reported she thrived in the large intro classes, doing better than most of her peers.

I’ve taught a lot of Intermediate Microeconomics, which requires a fair amount of analytic geometry, the type you’d learn in high school math. Many of my students would struggle with the math. Somehow, they too got by on the math requirement, but cognitively math was a foreign language to them, one they didn’t speak. Even though Illinois is a selective and highly regarded university, and I’m talking about full time students, the traditional 18 – 22 year olds, the situation with math for the non science or engineering students is pretty much the same as what Professor X describes for his English students.

Something is terribly wrong here.

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I used to believe in the math gene, that I have it while my wife doesn’t, and that with our kids I wasn’t sure but it seemed like the older one has it and the younger one doesn’t. Until this year (trig) the older one was getting A’s, seemingly easily, and if he’s doing fine I’d just as soon keep my nose out of it, let him learn on his own, try to tone down the pressure that inevitably comes from being the child of academic parents. The younger one had some issues with algebra last year and geometry this year so we’ve gotten into the routine where I check his math homework.

Last year in addition to the checking I spent substantial time coaching him on the math to talk through the thinking and help him conceptualize about what he is doing. Once in a while we seemed to make progress. Much of the time, though, it was tough sledding. In other contexts, such as the history he’s learned from playing computer games, he is extremely fluid and gets to the result very quickly, seemingly without effort. I believe he has an expectation that math should be the same way, instead of a sequence of steps, some a labor to take. This expectation impedes his learning.

This year I’m coming to believe that the behavior is a consequence of how math is taught. Homework is a set of perhaps twenty problems from the textbook, most in the drill category, many little perturbations of what came previously. The emphasis is not on derivations, but rather on getting familiar with a class of problems. This encourages learning math as a bunch of formulas to be applied rather than figuring things out based on primitives and context. Partly because I get home from work late many evenings and partly because I’m not happy with these assignments, this year I’ve done much less coaching and instead have simply done the checking. I do have to figure them out, at least the first one in a category, and sometimes that takes time. I need to draw diagrams, go through some steps, some of which are not immediate, to get to an answer. My son, and this I’m sure he gets from me, tries to do this in his head. There often aren’t any diagrams or only scant drawing. Sometimes he gets stuck and doesn’t know how to do a problem at all. Then he’ll leave it blank and ask for help, but that’s help on getting the answer, not on how to work the problem.

Because we’ve been anxious about grades and getting into a good college, I asked my older son to show me his trig book so I might help him study for his final. He said he was having difficulty with formulas of the form cos( + ), the cosine of the sum of two angles. There’s a chapter on this and related formulas so I looked at his book to see what that chapter said. It suggested the student memorize the formulas! This is not the way to teach math.

There actually is a book called The Math Gene. This short review is worth the read. Keith Devlin, the author of the Math Gene, makes a compelling case that math ability is a property of our species and it is intimately tied to facility with language. Those who are good in math developed some affinity for it early on but everyone is capable of establishing such affinity. The question then is why so many don’t.

My guess is that at some point, possibly in Elementary School, maybe in Middle School or High School, most kids hit a blockade with math, their prior method of learning math no longer seems to work. They become frustrated and ashamed. They need a way to work their way through this, but they don’t have that capacity on their own. They need coaching. The best sort of coaching, in my view, would help the student develop his own capacity so he can make further progress on his own. Too much tutoring is aimed at getting students through the next test. Damn the test. The stakes are too high to worry about the test. We’re talking about developing a lifelong capacity for abstract thinking. If the kid is stigmatized, what else would you call it if he passes the test but doesn’t have a fundamental understanding of what is going on, that lifelong capacity will not fully develop.

If the math teacher has 20 students per class (most have more) and teaches 5 different classes (which I take is the standard load), that’s a lot of students who potentially need individual attention. It’s probably not fair to expect the teacher to be the coach, though perhaps in some cases that can happen. Most kids won’t have a parent who was a math major in college and who earned a doctorate in economics, a field akin to applied math. At some point in most kids’ schooling the parent won’t be able to serve as the coach. How does this problem get solved?

I’ve been seriously stuck in math a few times in my life. For at least a couple of those I was having emotional problems simultaneously, though I’m not sure about cause and effect. In the first semester of my sophomore year in college I was taking both Abstract Algebra and Analysis at MIT. Both were over my head. It was frightening. I was a math major and I was supposed to be a bright guy. I transferred to Cornell the following semester. As a junior there I took a Topology course from a really great teacher who forced us to talk through all the concepts, many of which were similar to those from Analysis (open and closed sets, compactness, etc.). I found his homework interesting and I put in a great deal of time working through the problems. What I couldn’t do as a sophomore in one setting I could do as a junior in a different place.

We need the great teachers who help the students work through their conceptual struggles and inspire the students to figure things out on their own. Many of those teachers should be working with Middle School students or High School students, but probably only a handful at a time, so their work would differ form what the regular teachers do.

* * * * *

There is a school of thought that students should spend their time on open ended real world problems as those are authentic. Certainly such problems offer good motivation for learning because students will have interest in the circumstances and thus in identifying remedies. But by the very nature of these real world problems, students will not be able to test whether a proposed remedy is effective unless it is actually implemented. Consequently, I believe there to be substantial value for students to work on closed ended problems but those that have some originality to them – they are unlike other problems students have encountered so the students must use some ingenuity in finding the solution. They will themselves know whether they’ve found a good solution, because math provides a way to logically test whether the solution holds. It is the ingenuity and the logical testing that we want students to learn.

Students who do learn these things will address the complex real world problems in a different way than their peers do. The math savvy students will construct an abstract representation of the real world problem, analyze and solve that, then use this solution as the basis of their plan for the real world problem. They may then critique their own abstraction by considering what was omitted in the process and how including that might affect their solution. They will develop a sense about how much of the complexity can be accounted for into the model before it gets too unwieldy to analyze. In other words, they’ll understand the limits of their own ingenuity.

Most math classes in Middle School and High School don’t have students work original problems, except perhaps for extra credit. (Some examples of such problems are the 12 coin weighing problem, the modified donkey theorem, this site shows why the donkey theorem fails as a general proposition but it holds if the angle is the largest in the triangle, and determining how many cubes of side 1 can fit into a sphere of radius 2.) When I attended Cardozo High School (1969-72) outside the Math Office there was posted a “Problem of the Week.” I can’t recall whether those who got a correct solution had their names posted or not. My sense is that you’d try these for the challenge rather than for the recognition. Puzzles engage people; witness the popularity of Sudoku today.

We did more of these on the Math Team, with the difference that the contests were timed, three 10 minute slots with two problems per slot, so you had to get an insight quickly or you wouldn’t be able to do it at all. Since the problems had novelty to them, being good at doing these problems meant you had to be a good guesser. I believe that is a skill that can be learned as long as there is some recognition beforehand about what constitutes a good guess.

To get insight into these problems the trick is to come up with a picture or a framing question that lends itself to analysis and that gives a ready view of the problem. Consider the cubes inside the sphere problem. That was one of problems from the Math Team. I didn’t get it. I didn’t come up with the appropriate visual. But I thought about it again, not too long ago (more than 35 years after I missed it the first time around) this time having in mind Rubik’s Cube. If each of the little cubes within Rubik’s Cube has side equal to 1, can the full Rubik’s Cube be squeezed into a sphere of radius 2? The answer is no, it can’t. You can’t even get one face of the cube in because the diagonal of a face will be longer than 4 since the face itself is 3x3. But you can get in the cubes that form the x, y, and z axes comprised of the middle cube and each cube that shares a face with the middle cube (there are 7 1x1 cubes that make up these axes). It turns out you can squeeze in a few more 1x1 cubes in addition. So now there is a way to count what’s possible and come up with an answer.

I don’t remember ever being told to find a picture to solve a math problem although in common parlance in math class we talk about “seeing” the solution. Just what is it that we see and how do we find it? It’s as if we’re lost in the woods, looking for a way out. How do we find a path that works?

Practice matters for this. We get better at finding the path over time. My sense of why we see some kids with high math ability and others not is that some have early success and hence they continue practicing. Others have an early stumble and they stop. Over time the lack of practice translates into, “I’m no good at math.” Had the kid recovered quickly from the early stumble, he may have done very well in math thereafter.

If that’s right, then teaching such kids we need to do two things. First, we need to watch for those early stumbles and make sure the kid overcomes that soon before a permanent scar forms. Then we need to keep it interesting so the kid wants to practice. If practice is a drudge, most kids won’t do it. Instead, they’ll get by.

Beyond that there may still be differences across students in math ability. But what we observe mostly, say from how students do on standardized math tests, is performance of a bunch of kids who’ve been stigmatized about math along with the performance of a smaller set of kids who weren’t so stigmatized, so the variation in ability seems much greater than what it actually is.

* * * * *

Math may seem esoteric and not especially relevant to students, particularly if the field they choose to pursue is unlike physics, engineering, or economics, which clearly rely on math quite a bit. In my view, however, it is very useful in a host of other areas. Consider such down to earth activities as printing out a picture from your digital camera and resizing the image to best fill the page, or redesigning a kitchen, or doing household finance. All require some basic math. So certainly an argument can be made that some math competency is essential for adult functioning irrespective of field of endeavor.

I want to advance a different argument. Math is critical for us as thinkers. It helps in any domain where abstract thought is helpful, which for me is just about everywhere. The type of thinking one does in math is very useful in writing. It helps in making arguments, in seeing relationships between different ideas, and in developing a sense of elegance in thought. This is not merely about becoming proficient with logic. It is about being conscious of representations and becoming comfortable reasoning this way.

Perhaps more importantly, good math knowledge is crucial for decision making, particularly the type of decisions that executives make. It forces the decision maker to be cognizant of the assumptions being made and to develop a sense of humility – the assumptions may not be valid. It encourages the decision maker to avoid making conclusions that are leaps of faith, not based on the assumptions, and to encourage an empirical rather than a faith based way of assessing the goodness of any particular decision.

This idea doesn’t seem to have found its way into the press. When folks like Tom Friedman talk about math education, see this interview between Friedman and Daniel Pink for example, they argue that math is (part of) the gateway to the next generation of World products and therefore math knowledge fuels the economic engine. This piece by Michael Lewis and David Einhorn on the End of the Financial World as We Know It is getting closer on the role math plays, as a check on irresponsible decision making and as a general validation device, but it still gives the impression (I don’t doubt it is accurate I’m just questioning whether it's desirable) that the guy who knows the math is the nerd in the back room. The big wheels ignore it in their own thinking. In this particular case the argument is that there was information about the Madoff scandal dating as far back as 1999. But the information stayed within a relatively closed loop. It didn’t disseminate broadly. Might that be in part because people were incapable of explaining why it was a Ponzi scheme?

So I believe we’ve undersold the importance of math understanding and that, in large part, has enabled our culture where so many otherwise well educated people are math phobic, while many others think they know math when what they really have is formulaic mastery but little to no knowledge on how to use math to solve actual problems. This is a cultural issue as much as an educational one. We need to change this culture. Doing so will require a big time investment of people. And the aim should be clear – sophisticated math understanding, not graduate level math analysis but rather understanding mathematical argument, being able to make such argument, and being able to evaluate the argument of others. This needs to be a universal outcome from our education system, not just the domain of an elite few.