Isn't it ever so satisfying when your hard-earned taxes are spent on something useful and constructive, such as the report of the National Mathematics Advisory Panel, which was charged with answering the question of why American students are falling behind the rest of the world in mathematical preparation. Of course, enormous piles of research has been done on mathematics education, and any math or science professor can tell you about the abysmal preparation of students who have gotten into college (and who are thus high school graduates).

The task of the advisory panel was basically to collate all the knowledge out there into a plan of action for fixing the disaster of math education in this country. Wouldn't it have been nice if math education researchers themselves brought all this knowledge together? Indeed, the mere existence of the panel shows that our problems go much, much deeper than any set of recommendations in a report can solve.

Fixing the problems with math education in this country will involve breaking a cycle that goes back decades at least. Until education professors face up to the elephant in the room, that poor students find an elementary education major an easy alternative, a million reports and recommendations like this one will be meaningless. All the experts in the world can make absolutely correct statements such as this:

Teachers and other educational leaders should consistently help students and parents to understand that an increased emphasis on the importance of effort is related to improved mathematics performance. This is a critical point because much of the public's self-evident resignation about mathematics education (together with the common tendencies to dismiss weak achievement and to give up early) seems rooted in the erroneous idea that success is largely a matter of inherent talent or ability, not effort.

But if elementary teachers consider themselves inherently "bad" at math, as I guarantee you many (if not most) of them do, they cannot possibly make their students understand that no one is just born bad at math.

There is a fundamental problem that has created this crisis to the point where any solution is at best decades away, if it is possible at all. Culturally, math illiteracy is not considered a deficiency. In fact, many people (including undergraduates taking required math courses) seem to wear it as a badge of honor. How did it get this way? We seem to take language illiteracy much more seriously (which is not to say that students are adequately prepared in reading and writing either). After all, while I have heard dozens of people proclaim their math illiteracy, I have never heard one person proudly announce that they cannot read or write. Parents don't think math is important, so their kids don't. Some of those kids go on to be elementary teachers. How do they get the degree when they continue to be illiterate in math? They do it within a system that gets many of the worst students, because instead of making them meet high standards, it accepts them adjusts courses accordingly. Minimum grade levels to graduate are meaningless; they just result in grade inflation. Programs focus more time on "methods" teaching than on content; even while all the skills in teaching methods in the world are useless if you do not have a firm grasp of what you are teaching.

Why are we passing poor math students through unchallenging courses that result in teaching degrees? One, because of the cultural stigma tied to proficiency in math. Two, because the profession of teaching today provides little rewards, even for the saints among us, which results in a high demand due to high turnover. Most teachers will tell you that it is not even about the pay and benefits - although the pay discrepancy between the teachers and administrators is a travesty. It is about a lack of autonomy in the classroom - due to government's and administration's love of relentless standardized materials and testing, which prevents even the smartest and most motivated teachers from using their abilities to teach creatively. It is also about parents who think schools are daycares for their snotty, insolent, bored-without-TV brats, instead of a controlled environment for children who have been taught respect to participate in the excitement of learning.

Thus, we have classrooms of teachers who know little content in certain areas and are even taught about teaching math in a way that reinforces its stigma:

Teachers and developers of instructional materials [added emphasis] sometimes assume that students need to be a certain age to learn certain mathematical ideas. However, a major research finding is that what is developmentally
appropriate is largely contingent on prior opportunities to learn. Claims based on theories that children of particular ages cannot learn certain content because they are "too young," "not in the appropriate stage," or "not ready" have consistently been shown to be wrong.

How is it that the people creating instructional materials have no idea what all the research has been telling us for years? This is a fundamental disconnect in most areas of education. The educational research that has been done decades ago and today is emphatically ignored by the people charged with actual education, such as school boards and administrators. For example, everyone knows that the best time for language acquisition (single or multiple) is early childhood. Yet, when is foreign language instruction begun in the U.S.? High school.

Another fairly futile recommendation:

...teachers must know in detail and from a more advanced perspective the mathematical content they are responsible for teaching and the connections of that content to other important mathematics, both prior to and beyond the level they are assigned to teach.

This seems to be a bizarre concept to students in math education. Even those at my campus planning to teach high school math - and thus who are essentially getting a content degree in mathematics with a few education courses - are known to complain, "why should I have to take this high-level math course when I will never teach this material?" The utter lack of interest in their major subject is astounding. Why are they math majors then? Because with such a shortage of math teachers, they are certain to get a job. And with teachers relatively uninterested in the subject they are teaching, the cycle of poor preparation continues.

And let's not forget that there is seemingly always a role played by big business when policy makes no sense:

Mathematics textbooks are much smaller in many nations with higher mathematics achievement than the U.S., thus demonstrating that the great length of our textbooks is not necessary for high achievement. Representatives of several publishing companies who testified before the Panel indicated that one substantial contributor to the length of the books
was the demand of meeting varying state standards for what should be taught in each grade. Other major causes of the extreme length of U.S. mathematics textbooks include the many photographs, motivational stories, and other nonmathematical content that the books include.

Why all the extra garbage in these math books? (For an extreme example of textbooks with gratuitous material that schools should definitely avoid, see this YouTube video.) Interestingly, the panel's report does not spell out the obvious reason: publishers' profits. The textbook industry has gotten completely out of control, filling books with junk to make them longer to justify the cost, and coming out with new editions every couple of years, which forces school districts to spend the money to replace their entire inventory, rather than order a few replacement books. In science, this can be justifiable given the rapid increase in knowledge and thinking that occurs - textbooks that define only two biological kingdoms, for example, would not be useful in preparing students in biology. But school-level math has been the same for decades, if not centuries. Discoveries on the frontiers of mathematics do not change how you do long division.

There are so many problems that the panel's report goes on and on. One suggestion the panel makes that has merit is the idea of having math specialists teaching at the elementary level. Like art and music teachers, they would travel from class to class an alleviate the burden of teaching math from all the teachers who hate it. The only way to break our current cycle of math phobics creating more math phobics is for kids to realize at an early age that math is interesting, and fun for everyone, not just for geeks. Math is part of what makes us human.

Another more radical suggestion is that there be no mathematics teaching at all until middle school. This may seem counterintuitive to the recommendations of the panel, but the main reason most students are terrible in math when they get to middle school is that their aversions have been so reinforced they are already lost causes - poor teaching at the elementary level has already convinced them they are no good at math so they don't even try. What if we waited to teach math, so that students haven't already closed their minds before they have a teacher who is actually interested in math? Why not use the time in elementary school to teach a foreign language? Since many current college students today cannot do middle-school-level math, no one can claim that it would be impossible for students to catch up at that point. As the report points out, the same simple concept is often taught year after year after year in elementary school, which adds to the boredom factor.

Whatever we do, it must somehow involve changing the greater American culture that looks down on the enjoyment of math as geeky, and the hatred of math as cool. Maybe someone could make some Einstein and Von Neumann action figures for Happy Meals. If the culture does not change, the performance of American students in math will not either.

Labels: education, mathematics

Is mathematics an emergent property of sociality? I posed this intriguing question to a mathematician colleague, who is also an evolutionary biologist, and he said yes. The question came up because I have argued that rules are actually a social construct; a solitary species needs few or no rules governing its interactions with other individuals of its species, because other than mating or occasional territorial conflict, it has almost none. Individuals in social species, by contrast, are completely dependent on rules to survive and reproduce, because interactions with other members of the species are constant, and determine standing within a social group, and thus generally reproductive success.

Most evolutionary arguments applied to humans are tenuous, because of cultural complexities that overly our basic biology. Complicating the picture further, aberrant behavior (that which does not comply to a given social norm) is also probably more common among humans than among other social animals, because 1) we have chemical treatments that suppress some symptoms of such conditions, 2) we have easy access to addictive products which our brains did not evolve to cope with, such as drugs, junk food, slot machines, etc., and use of these can lead to self-destructive behavior, and 3) many aberrant people are smart enough to overcome or disguise their problems enough to fit in somewhat. So, there are many ways in which humans seem to get away with behaving in socially maladaptive ways, without suffering reproductive consequences, as other social primates probably would.

However, we did evolve as a social species, and much of our behavior is indeed a legacy of that evolutionary history. The playing of games is an example. Games are all about rules. Kids love learning new games, because their brains are wired to learn rules -- particularly rules for navigating in real society, but an artificial society with artificial rules will do. Whether it is sports or war games or pin-the-tail-on-the-donkey, humans love games. Games with complex rules are more fun to learn for many of us, but those with fairly simple rules but complex strategy, such as Go or hearts or chess, usually capture the most active minds. It is our love of rules that make us despise the referee who makes a bad call. In our minds, if a rule is broken, the entire game should be void.

It seems that mathematics is universal, a truth that existed before humans and that they discovered. But to humans at least, mathematics is also all about rules, and perhaps the way that we perceive mathematics is filtered through our obsession with rules. We all learned them at the beginning of every school year for a decade. "Addition and multiplication are commutative. The transitive property says that if a=b and b=c, then a=c. The distributive property says that a * (b + c) = a * b + a * c" and so on. If you take higher level math classes in college, you discover that there are other mathematical systems with different rules; for instance, matrix multiplication is not commutative. So math is indeed a world of many rules that apply one way in one context but another way in a different context, very much like the rules of social interactions -- for example, it is inappropriate to wear a bikini at the opera, but just fine at the beach.

Although many would protest the truth of the statement, humans are wired for math. (If you hate math, it is not that you are "no good" at it; it is because the way it was taught to you made it painful and boring. This is a persistent problem that will likely never be corrected on a large scale, because of the vicious cycle of elementary school teachers who dislike math and barely get through it in college, go on to teach it poorly, cause their students to dislike it, and so on.) The interesting question is, would, or could, an intelligent solitary species have developed math? Some would say the question is completely moot because only a social species would have evolved brains as large as ours, because sociality requires a larger brain to navigate the intricacies of social interactions, in addition to the basic needs of finding food and mates and defending oneself. It is perhaps a chicken-and-egg question. But what is no question is that complex rules govern sociality, human brains are therefore wired to learn and use rules, and mathematics is a system of rules. Mathematics, very much like religion, is likely a byproduct of our success as a social species.

Labels: brain, evolution, humans, mathematics, sociality

My recent dearth of posts has only to do with a temporary priorities shift. I have roughly two discretionary hours per day (and that assumes that housework counts as "discretionary") and I elected for the past week to spend it finishing up a quilt top I began almost a year ago, which I decided to display here to prove my excuse. (Although the quilt is neither about biology nor music, it is about math and art, which I decided are close enough.)

The quilt (above) was designed to showcase the eleven regular and semi-regular Archimedean tilings of the Euclidean plane known to those who have studied geometry. Tilings are patterns of polygons which fit together on a flat surface, with no gaps. (There are also multiple tilings of polyhedra in three-dimensional space, for example a soccer ball, which consists of hexagons and pentagons. These can be fit together to form a (nearly round) polyhedron, but cannot fit together in a plane - if a soccer ball were flattened out, some gaps would appear between the shapes. ) This website demonstrating each of the tilings in the Euclidean plane provided me with the color coding.

A regular tiling is one in which one type of regular polygon (a polygon with equal sides and angles) can be fit together repeatedly in a plane with no gaps. The three regular tilings are certainly familiar to most people, who see them used in floors all the time: squares (block 2 above), hexagons (8), and triangles (10). The eight semi-regular tilings use regular polygons of mixed shape to cover a flat surface. These are: octagons and squares (block 1 - also often seen on floors); hexagons and triangles (two ways, block 3 above and the entire quilt); squares and triangles (also two ways, blocks 4 and 7); hexagons, squares, and triangles (9); dodecagons and triangles (5); and dodecagons, hexagons and squares (6). These can be easily derived by figuring out which combinations of angles in the polygons add up exactly to 360°, which is necessary for the tiling to be flat. For example, the angles in a regular hexagon are 120°, so three hexagons can meet at a flat vertex. Squares have 90° angles, and you can either fit four of them together, or combine two squares (180°) with three triangles of 60° each to form a flat 360° vertex. And so on.

Of course, several of these are well known to quilters the world over. Blocks 2, 4, and 10 are most often seen in quilts because they involve squares and triangles, which are relatively easy shapes to cut out, and do not involve sewing into corners. The reason the construction of the blocks took me nearly a year is because the majority of them do involve sewing into corners, which means you cannot assemble the block by sewing straight lines only. Sewing into corners is especially hard on a machine, which I used, and when the angles are acute, which they get to be in the blocks with four or five shapes meeting at a vertex. Blocks 3 and 7 were the real killers for this, and each took me months because I didn't have the patience to focus on them for any length of time. Plus I'm not particularly experienced with this in the first place, so I did a lot of seam ripping.

Because I am a hand quilter, it will probably be another year (at best) before the quilt is finished - it is a very large quilt. So, when more gaps appear in the blog, well, that's just one of the other things I'm doing with my time.

Labels: art, mathematics