Flatterland, a book review

Over the holidays, I read Flatterland by Ian Stewart. Stewart is an English mathematician and contributor to a branch of geometry called "catastrophe theory." Sounds fun, eh?

The subtitle of Flatterland is "Like Flatland, only more so." Stewart wrote it as a sequel to and extension of Edwin Abbott's 1884 geometrical satire, Flatland. One hundred years after the events of the first book, Victoria Line, great-great-granddaughter of the original A. Square, reads her ancestor's book (against her father's express prohibition) and finds in it a code for summoning a visitor from Spaceland.

Instead of a Sphere, Victoria receives a visit from a Space Hopper, a creature who helps her visit not just many dimensions higher than the third, but a number of other strange geometric worlds. In one she sees fractals everywhere. In another, parallel lines converge. In another she can see herself in the distance. Her investigation into curved space leads into a discussion of quantum physics and relativity, a theory the Space Hopper insists is poorly named, since it's based on the one thing that isn't relative - the speed of light.

So how's the book to read? My 8th grade son read it before me, and he found it mind-twisting and thought provoking - hard, but worth it. For me, with a year of college physics and no math beyond calculus, it was surprisingly accessible. Yes, it's difficult to visualize the worlds Stewart describes, but he assures us it's difficult for mathematicians, too. And its's fun to toss these ideas around lightly like oddly-shaped objects to juggle. The format consists mostly of dialogue between the Space Hopper, who is a little bit self-satisfied, and Victoria, who is by turns a stubborn teenager, a dense student, an independent reasoner, and an inspired explorer of new intellectual landscapes.

I wish the discussion of fractional dimensions were more thorough: I felt I almost had it when the story moved on. I enjoyed Moo-bius the Cow and the Klein bottles. Topology was fun to dip into, as was the discussion of infinity and parallel lines. The section on quantum physics and relativity took me back to the weirdest, most fun parts of physics. As for the story itself, while the plot is not enthralling, it's sufficient to carry the exploration of mathematics.

Who would like this book? Anyone who would like his or her mind stretched in weird ways. I'd recommend it especially for kids in grades 8-10 who like math even if they don't like calculation; people who like visual puzzles; and people like me who want their ideas of math and science tickled by new ways of thinking about and trying to visualize shape and space.

More than anything, Flatterland is valuable because it shares with us the freaky, imaginative side of mathematics and demonstrates that math is not just a matter of plodding through more and more difficult algebra and calculus. Instead, mathematics can become a great playground for the imagination.

Does anyone have another playful or challenging melding of math and literature to recommend?

High School Calculus Leads to . . . What?

High school calculus opens the door to careers in engineering, physical science, statistics, and all sorts of other math-heavy careers. That's what we've been told. And to be able to take calculus by grade 12, we need to make sure our kids are taking Algebra I in 8th grade. Yep, that'll do it. Hurry along, and we'll get ourselves a lot more scientists and engineers.

Except it's not working that way. Professor David Bressoud of Macalester College, just past president of the Mathematical Association of America, has been talking about this problem for several years. He points out that even as more students study and pass calculus in high school, college enrollment in higher level calculus and other advanced mathematics classes is staying flat or dropping. The percentage of US bachelor's degrees awarded each year in math-intensive majors has been falling since 1984. High school calculus may actually be discouraging students from pursuing science, mathematics, and engineering careers.

First, some facts. About 20% of high school seniors, or a full third of those going on to college right away, now take their first calculus class in high school. That's actually MORE than the number taking Calculus I each year in college. But only about 7% of bachelor's degrees, or maybe 110,000 BA's a year, are awarded in the math-intensive majors of physical science, mathematics, statistics, or engineering.

Of those taking calculus in high school (around 600,000 students a year), about half take the Advanced Placement calculus (AP) exam. The others either take AP calculus courses but skip the exam (perhaps because they don't expect to pass), or they take a non-AP class with "calculus" in the name.

Twenty-five percent of the 300,000 taking the exam, or about 75,000 students a year, earn high scores and receive college credit for calculus. These students--those who receive a score of 4 or 5 on the Calculus AB exam or a score of 3 or above on the BC exam-- do well. They are likely to succeed in more advanced math courses.

But what about the other 525,000 students a year who took high school calculus? Well, a third of them never enroll in any college math. They made it through that high school course, and now they're done. No more math, ever.

Another third place into pre-calculus or College Algebra. They move back two courses from where they thought they were as they finished high school.

Another 1/6 of these students fare even worse. They're kicked all the way back to remedial math. Their math skills are judged so shaky they end up paying for and taking a math course that doesn't even count for credit.

A lot of these students really thought they were doing okay in math. They worked hard. They did the homework. They made it through. They even took calculus! A friend has told me that the average high school math grade for students who end up placed in remedial math at Cal State University is 3.1, a good solid B.

What's to be done? Professor Bressoud suggests that we still have a lot more to investigate. What are the criteria for placing students in high school calculus? What are they actually studying when they get there? How are college math departments adjusting to students who have some exposure to calculus but aren't ready yet to go on? What happens to the motivation of those who don't receive college credit for it?

I would add some questions of my own. Are other countries, including those with centralized curricula, experiencing similar problems? Are programs like Agile Mind*, which uses technology to support the classroom math teacher, making a difference? And finally, what can we say about the experience of students who choose a non-calculus path, taking a course in Statistics or Mathematical Modeling their senior year of high school? Is it possible that another year consolidating skills in algebra or problem-solving might lead to greater success later on? Might experience exploring newer areas of mathematics like discrete math, probability, game theory, graph theory, or bioinformatics actually do more to prepare and motivate students than the competitive rush to calculus? We need to find out.

*Disclosure: The Noyce Foundation helps fund Agile Mind.