His opening is spot on: “I teach high school math.  I sell a product to a market that doesn’t want it, but is forced by law to buy it.”  He goes on to argue that the problem with math education, a problem exacerbated by most textbooks, is that it discourages what he terms patient problem solving.  Problems in textbooks rarely reflect the types of problems one encounters in real life: textbook problems usually supply you with just the right amount of information, and the question is frequently just a matter of plugging values into an appropriate formula.  More complicated questions are frequently presented in multiple parts, so that students rarely have to think long and hard about any one thing.  Instead, they are offered bite-sized pieces of a larger problem, any one of which is relatively simple.  Taken together, the net result may be the solution to an interesting problem, but by holding students’ hands like this along the way, they never develop a taste for solving difficult problems.

This hand holding, Meyer argues, trains students to become impatient when they encounter an actual problem.  If a question isn’t broken down into easily digestible pieces, any one of which can be solved by plugging and chugging into one of a short list of formulas, students are trained to throw up their hands.  In a world fraught with problems, however, this does a disservice not just to students who study mathematics, but all students.

The video is a little over 10 minutes, but if you have the time I’d encourage you to watch it.  Meyer talks about potential solutions to this failure of our educational system, and compares the current state of affairs to watching episodes of Two and a Half Men.  And despite the presence of a fraction in the title, the comparison is not favorable.

Enjoy!

(Hat tip to Patrick for sending me the link.)

Late last year, a study was published in Proceedings of the National Academy of Sciences which tried to pin down origins for the gender gap in mathematics education.  As I’ve discussed before, the gender gap in math education is shrinking, and has been shown to be less about biology and more about culture – in cultures where gender equality is weaker, the gender gap is stronger.  Nevertheless, even in American culture, the gender gap still persists, and this study by Sian Beilock and others has tried to figure out how, if the gender gap is culturally based, it comes about in young students.  The original study can be found here, while a discussion of the study that was featured in the news can be found here.

Professor Beilock and her colleagues tried to correlate young students’ math anxiety with the math anxiety of their teachers.  In particular, they looked at 1st and 2nd grade students, of whom a vast majority (over 90%) have teachers who are female.  The study assessed the math anxiety of the teachers and measured the math achievement of the students at the beginning and end of the year.  Here are the results, taken from the introduction to the paper:

There was no relation between a teacher’s math anxiety and her students’ math achievement at the beginning of the school year. By the school year’s end, however, the more anxious teachers were about math, the more likely girls (but not boys) were to endorse the commonly held stereotype that “boys are good at math, and girls are good at reading” and the lower these girls’ math achievement. Indeed, by the end of the school year, girls who endorsed this stereotype had significantly worse math achievement than girls who did not and than boys overall.

These findings make intuitive sense, and lend further support for the need to better our mathematics education at all levels, or at the very least require primary educators to study mathematics more seriously.  Teaching mathematics with confidence is not something that comes automatically, even for those who may have been good at math in their early years.

It’s interesting that boys weren’t more likely to endorse the view that boys are good at math and girls are good at reading if their teacher had math anxiety.  I’d be curious to see what the case is in a classroom led by a male teacher, both with and without math anxiety.  Given the dearth of male primary educators, however, this type of data may be harder to acquire.  In any event, the lesson here is clear: if you want your daughter to not fear math, it wouldn’t hurt to demand that her teachers not fear it either.  Or at the very least, demand that any math fear be exhibited only by male teachers.  That may be a cheaper solution.

I’d also be interested in knowing whether this trend can be reversed by a suitably competent teacher.  If a group of 2nd grade girls is taught math by a woman who is unqualified, but in 6th grade is taught by a woman who is exceptional, can this help undo the damage that the 2nd grade teacher has done?  I would hope so.

It's too bad Prezbo doesn't have lady parts.

Lockhart’s article lambasts the current state of mathematics education in this country. Some of his main points are the following:

• Mathematics is an art form, but unlike other art forms like music or painting, is not understood as such by the general population. As a result, students are not exposed to the beauty of mathematics, and are instead taught through drill and memorization, which effectively kills any natural curiosity the student may have.
• The most important part of mathematics lies not in the facts or theorems that students memorize, but in the arguments that show why these facts must be true. By stripping away the beauty and elegance that lies behind many of these arguments, students don’t develop an appreciation for (or a real ability to do) mathematics.
• The only class that does emphasize proof (high school geometry) sterilizes the process so much that all the beauty is drained from the arguments.
• Math education spends too much time trying to force artificial connections to the real world, rather than exposing the natural beauty that lies within mathematics. Most word problems don’t actually reflect any type of problem that one would find in the real world.

• Lockhart has no love for the endless drilling that goes on in current math classes (the type of drilling that continues all the way up through calculus). But to what extent are drills a necessary evil? If you want to become a concert pianist, you’d better practice your scales. Nobody will argue that drills are particularly taxing, but they do have their purpose in other arts – shouldn’t they in mathematics as well?
• Many are quick to point out the one major problem with comparing mathematics to other art forms: mathematics has wide applicability to other fields, whereas other art forms do not. Lockhart argues that even though this is the case, the essence of mathematics isn’t its practical consequences. This may reflect his own personal bias (after all, he was a researcher in analytic number theory), and while it’s a bias I share to a certain extent, I doubt that this is a universal belief among mathematicians in general.
• I often find that students feed into the current system of teaching the facts rather than the ideas, because the facts are easier to check on standardized tests. Most students want to know a technique for solving a problem, and couldn’t care less about why the technique works, where it came from, or most importantly, its limitations. In essence, I see a tremendous lack of curiosity. Much of this seems to stem from a desire to get a good grade (which may lead to a good job), rather than wanting to learn for learning’s sake. However, this is a problem that goes beyond mathematics – to what extent are the problems Lockhart address indicative of broader problems in education?

Give it a read – if nothing else, it will give you something to think about.

With an international average of 500, American 4th and 8th grade students scored a respectable 529, on par with the Netherlands, Lithuania, Germany, and Denmark. As might be expected, however, we’ve still got a significant way to go when it comes to competing with other countries. Hong Kong made the top of the list, with a score of 607.

Of course, this data by itself doesn’t do much to explain what factors may be driving our improvement. One suggestion is that we are simply aiming higher. The article linked above ends with the following:

The congressionally appointed National Math Panel recently called for sweeping changes in how schools teach math, pushing for a greater emphasis on algebra and higher-order problem solving. [Brookings Institution researcher Tom Loveless], a member of the panel, says the changes would go a long way toward improving our international ranking. “We’re making progress, but we’re several decades from being first in the world,” he says.

Maybe, then, the problem isn’t that we’re not making progress – it’s that we’re just not making progress fast enough. The question then becomes: how can we identify what’s working, and crank it up so that we can get our kids to a competitive level on an international stage as quickly as possible?

It’s a heavy question, indeed. I’d encourage you to ruminate on it. Perhaps some School House Rock will help to inspire you.

Of course, I doubt this is all the teachers’ fault – elementary and middle school teachers are a rare commodity in this country, and kids need someone to teach them math. An understaffed school will do what it takes to make sure there’s somebody at the front of the classroom. And I certainly don’t envy those teachers out there who may not feel so confident in their math ability, but are nevertheless trying to impart all that mathematical know-how because nobody else will.

One thing that is a little confusing is that when this story made the news rounds, most headlines made a statement like: Teachers are only one chapter ahead of their students. The press release for the study, however, makes no such claims – instead, the main point seems to be simply that there are too many teachers without a major or a certificate in the subject they are teaching. This, of course, is an issue that needs to be addressed, but it doesn’t seem to be nearly as dramatic as the headlines would claim. If you’ve taught 6th grade math for five years, even without a certificate or a major, chances are you certainly know the material well enough that you aren’t only one chapter ahead, even if you are a lousy teacher. Similarly, some people have a natural talent for teaching, and can become good math teachers even without much of a mathematical background – just ask The Wire‘s Roland Pryzbylewski.

Math teachers in action. You can tell Pryzbylewski means business.

This type of data is more powerful than looking at SAT scores, for instance, because exams administered in a mathematics competition are notoriously difficult. There are thousands of students who will score an 800 on the math section of the SAT, and so this test offers no way to distinguish between them. Looking at this other data, however, allows us to gain a much deeper insight into the abilities of students in the U.S. with an aptitude for mathematics.

The data suggests a couple of things. First, contrary to the Gender Gap theory I have discussed before, there are many women who perform extremely well on these exams. While the data can’t support or refute the Gender Gap theory conclusively, it does show that indeed there do exist women who are good at math.

The second conclusion, which is somewhat broader, is that a majority of students in the US who excel in these exams are either foreign (for example, the Putnam exam can be taken by any undergraduate in the US or Canada, not just citizens), or are children of immigrant parents.

The combination of these two points is highlighted in the breakdown by country of the women who compete in these types of competitions. The article informs us that, regarding the makeup of the teams sent to the International Mathematics Olympiads,

All members of the United States team were boys until 1998, when 16-year-old Melanie Wood, a cheerleader, student newspaper editor and math whiz from a private school in Indianapolis, made the team. She won a silver medal, missing the gold by a single point. Since then, two female high school students, Alison Miller, from upstate New York, and Sherry Gong, whose parents emigrated to the United States from China, have made the United States team (they both won gold).

By comparison, relatively small Bulgaria has sent 21 girls to the competition since 1959 (six since 1988), according to the study, and since 1974 the highly ranked Bulgarian, East German/German and Soviet Union/Russian IMO teams have included 9, 10 and 13 girls respectively.

The analogy between math and sport is certainly a rich one. Both require hard work and discipline in order to excel. Both should be included in any child’s education. And both attract people to their summer camps.

At the same time, there is quite a wide cultural divide that is perceived between these two groups. Athletes are put on pedestals (literal and metaphorical), and their toned physiques are heralded as the pinnacle of human achievement. They are also widely regarded for their dedication, their determination, and nobody questions their hygiene. Sadly, the same cannot be said for mathematicians.

To combat this inequality, why shouldn’t it be the case that top performers from both fields should be able to have their face on a Wheaties box? Certainly breakfast is the most important meal of the day for both athletes and mathletes – we should emphasize this point by highlighting the achievements of mathematicians on the orange box we all know and love.

2. Put a mathematician on The Simpsons.

Sure, The Simpsons doesn’t hold quite the cultural sway that it used to, but its longevity shows that it has carved out an enduring place for itself in our culture. It is still quite popular, and has its share of devotees, and for this reason many people still pay attention to what it has to say, even if it may have been eclipsed by other series (animated or otherwise) in recent years. In this respect, it is a bit like the Hillary Clinton of prime time television.

Therefore, it stands to reason that having a mathematician lend their voice to an episode of The Simpsons, if handled in the right way, certainly couldn’t hurt to flip the cultural perception of mathematics on its head. The Simpsons has featured hundreds of guest stars: celebrities, heads of state, authors, athletes … the list goes on. Moreover, a move to bring in a mathematician would not be entirely unprecedented – Stephen Hawking has made not one, but two guest appearances on the show, and although he is a physicist rather than a mathematician, it would not be such a huge leap to move from a guest star of the former occupation to a guest star of the latter.

Of course, we could give the show the benefit of the doubt, and assume that despite their best efforts, producers have been unable to find mathematicians who would be willing or able to participate. Should this be the case, I am willing to humbly submit myself for such duties. I believe I am able to shoulder the tremendous responsibility that such an opportunity would entail.

Stephen Hawking: Bringing theoretical physics
to the forefront of pop culture since 1999.