The strange looks combined with the gasps of horror are starting to get to me. I thought that after all these years I was getting used to it, but it happened again at a recent conference.  At Stanford, no less.  I had made the mistake in public company of mentioning how much I loved math and getting kids excited about it through WISE.  “Really?  Uhmm, that’s nice,” they said just before wandering off to refresh their drinks.

But in defense of the common man, I am here to tell you that math is simply misunderstood.  Worse yet, even the broader institution of K-12 Math Education (note the capital letters to indicate the authorities that comprise “Big Math”) as a whole fails rather drastically to understand what math is.  There are even a number of Universities that don’t really get it (see below). So it’s not really all that surprising that our schools and even our culture at large fail to grasp the significance, the beauty, the elegance, or the joy of practicing mathematics.

When you say “math” at a party, most people remember those painful moments in high school of rote formula memorization and the mechanical repetition of plugging in numbers, the otherwise meaningless manipulation of abstract symbols to achieve higher test scores.

And while yes, the manipulation of abstract symbols can be a useful tool in math, the mechanics and minutia of symbolic manipulation are all too often mistaken for what math really is.  Exercises in notation are the obvious and visible attendants when people are really doing math.  But to anyone unversed in the practice, it is not obvious at all that anything else is going on.  Is it any surprise that the medium is mistaken for the message? Math must be what most people can see, just as written paragraphs are what literature is all about.  So what is the message, you ask?  What is math?  What are all the assiduous formula manipulators missing?

There is logical structure to our universe.  How things are, and can be arranged, how they move, what colors and shapes they comprise, how things sound, how they feel, and what they will do in just a few moments; all of these things can be organized and understood by realizing underlying patterns and regularities.  Then we can use these patterns and concepts to understand and predict other things.  We have developed written languages and notation to illuminate these patterns, the symbols and operations of written mathematics, and better yet, we have developed the symbolic language in such a way that their very structure and regularity don’t just describe like verbal languages but also mimic the very structure and pattern of the reality that they describe.

So ultimately, math is about patterns, logical and inter-related processes, puzzles, models, consistencies, inconsistencies, causality, correlations, and all sorts of profoundly interesting ideas.  Practicing real math is nothing less than a creative exercise in discovery, realization, insight, and deduction.  But sadly, most school curricula mistake facility with the tools of notation to be the sole component of math, leaving out the heart and the joy of it all.  Justifiably bored and disinterested students flee as a result.  I take heart from the fact that what they are fleeing isn’t really math, but an empty simulacrum thereof. But someone needs to tell people that what they fear is not math itself, the wonders of which they might actually enjoy.  I hereby offer two points in particular that will help illuminate the difference between math itself, and its written expression most often practiced in schools.

The Real World and Mathematical Abstraction
The first issue is the prevalent confusion between the overall subject of math; the ideas, the patterns, the observation of the real world and physical objects, and the ‘mathematical’ abstractions thereof, as written in equations.  Both are important and fundamentally intertwined, but rarely clearly related in practice.  Most schools tend to focus on the mechanical practice of the latter, i.e. “…do every odd problem (for which the answers are in the appendix) in chapter 7.”  The operations on the symbols seem arbitrary things to memorize and repeat without any apparent connection to reality or future life other than through the example problem template at the start of the chapter.  Many curricula try to offer “real-world” type problems, but often fail to clearly link the real-world components and steps in solving life’s actual problems with the power of written abstraction that we can use to reason about the real world without having to actually touch it.

Other schools, such as those using Montessori programs, focus on the real world with physical manipulatives, beads, blocks and so on, but then fail to clearly relate what the kids can touch and arrange with their abstractions that can be manipulated mentally and with paper absent the blocks.  Students in these types of programs learn to manipulate and reason physically, but not how to reason abstractly without the manipulatives. Nor can either set in isolation appreciate the intimate relationship between the real and the abstract or the power of understanding various and complementary approaches and representations of fundamental truths.

Neither type of program successfully relates the abstract to the real in such a way that both are clear expressions of the other, and that math is at the same time both grounded in reality and still open to intellectual creativity and insight in the abstract.  This fundamental beauty of mathematics is that the language we have invented to talk about math has as its very essence, a logical structure and process which mimics the real world, but in its abstraction, distills a purified essence of the real world that can be approached intellectually without requiring a descent into the messy and noisy real world.  We can use the abstraction to think about and learn about the real.  We can use the real to refine the abstraction, which in turn further illuminates the real in a continuing cycle of learning.  The real world and its abstraction are simply two sides of the same coin, but it is a very rare program indeed that clearly articulates this realization, and an even rarer one that specifically trains students to use and relate both sides of the coin.

Interestingly enough, even university researchers often miss this point.  Not long ago, a friend pointed me to a New York Times article a few months ago about an Ohio State study claiming that manipulatives and real-life examples did not seem to help in learning math.  You can read the full research article here with a subscription to Science.  Let’s just say that the conclusion of the study surprised me enough to pony up a few dollars to read the full article on the Science site.  Much to my relief, I found the study deeply flawed in its very structure of asking whether students learned math better by studying the abstraction or observing the real world.  Just by asking the divisive question, they missed the fundamental truth of mathematics as an abstraction of the real world with a deep understanding of both and their interrelationship as necessary for true enlightenment.

Math, Art, and Music:  Which of these is least like the others?
This is a trick question, because to many people’s surprise, they are all very similar.  My second point is that Math, as most commonly practiced in repetitive school calculation exercises, bears little, if any, relation to the creative and intellectual exploration of math outside of school.

Real math is not about mindlessly copying example problem steps on homework to nail the SAT test. It is about wonder, and exploration, and discovery, the untangling of interesting puzzles, a creative exercise at every step. I could go on in this vein for a while, but someone has already done a better job of it than I could myself.  If you are even vaguely curious about what real math is like, or fear its evil twin that currently dominates our schools, or just worry about the technical and economic strength of our nation, you MUST read “A Mathematician’s Lament,” by Paul Lockhart. There are some truly sublime sections on musical and artistic analogies with mathematics, and the most on-point critique of K-12 math that I have ever seen.  As a friend of mine recently mentioned, “At first I thought I would get tired of it, but ultimately, I found it to be perfectly on target.”  Amen brother.  If there were anything I would add to the piece it would be about math’s fundamental utility as a tool in science and physics and how each field and approach further illuminates the others.  Here are a few of my favorite excerpts, but seriously, follow the above link to read the whole thing:

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.  The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity– to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs– you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.
If your art teacher were to tell you that painting is all about filling in numbered regions, you would know that something was wrong. The culture informs you– there are museums and galleries, as well as the art in your own home. Painting is well understood by society as a medium of human expression. Likewise, if your science teacher tried to convince you that astronomy is about predicting a person’s future based on their date of birth, you would know she was crazy– science has seeped into the culture to such an extent that almost everyone knows about atoms and galaxies and laws of nature. But if your math teacher gives you the impression, either expressly or by default, that mathematics is about formulas and definitions and memorizing algorithms, who will set you straight?

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The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.

But a problem, a genuine honest-to-goodness natural human question– that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of mankind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them). A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions. A triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way?

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So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and level of experience. By giving them time
to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.

The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one. Of course it’s easier to post a set of rules than to guide aspiring young artists, and it’s easier to write a VCR manual than to write an actual book with a point of view.

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There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun.

About Paul Lockhart from the MMA web site:

Paul is a mathematics teacher at Saint Ann’s School in Brooklyn, New York. His article has been circulating through parts of the mathematics and math ed communities ever since, but he never published it. I came across it by accident a few months ago, and decided at once I wanted to give it wider exposure. I contacted Paul, and he agreed to have me publish his “lament” on MAA Online. It is, quite frankly, one of the best critiques of current K-12 mathematics education I have ever seen. Written by a first-class research mathematician who elected to devote his teaching career to K-12 education.

Paul became interested in mathematics when he was about 14 (outside of the school math class, he points out) and read voraciously, becoming especially interested in analytic number theory. He dropped out of college after one semester to devote himself to math, supporting himself by working as a computer programmer and as an elementary school teacher. Eventually he started working with Ernst Strauss at UCLA, and the two published a few papers together. Strauss introduced him to Paul Erdos, and they somehow arranged it so that he became a graduate student there. He ended up getting a Ph.D. from Columbia in 1990, and went on to be a fellow at MSRI and an assistant professor at Brown. He also taught at UC Santa Cruz. His main research interests were, and are, automorphic forms and Diophantine geometry.

After several years teaching university mathematics, Paul eventually tired of it and decided he wanted to get back to teaching children. He secured a position at Saint Ann’s School, where he says “I have happily been subversively teaching mathematics (the real thing) since 2000.”

He teaches all grade levels at Saint Ann’s (K-12), and says he is especially interested in bringing a mathematician’s point of view to very young children. “I want them to understand that there is a playground in their minds and that that is where mathematics happens. So far I have met with tremendous enthusiasm among the parents and kids, less so among the mid-level administrators.”

So Where Can You Find REAL Math Materials?

The single program I have ever seen that best captures and relates the importance and interrelationship of the real world and abstract expression in math is Henri Piccioto’s Algebra Lab materials. They offer a fantastic combination of cleverly-designed manipulatives together with sets of problems using the symbolic abstractions, all supplemented with open-ended creative problems that really get you thinking and understanding the BIG PICTURE and how everything is related.  The system explicitly helps students develop facility with approaching problems from either physical or abstract directions, and how one approach can inform and illuminate the other.  Students can directly observe how different approaches can be used to best advantage for different types of problems. The crowning component of the system is the great set of open creative questions that get students exploring patterns and symbolic reasoning on their own using the physical and abstract tools they have developed.

The Lab Gear that comes with the program is comprised of a series of carefully-sized blocks that help articulate what algebra is really about, arrangements and grouping, and what things like factorization really mean.  You fiddle with the blocks, and get an intuitive feel for how things really are, and then learn about the abstractions, the principles and equations.  You expose fundamental truths of the world and learn how to discuss them!

Here are some animated graphics that were created by George Collison of INTEC
(International Netcourse Teacher Enhancement Coalition) to demonstrate the Lab Gear materials in use to help illuminate polynomials and factoring.

Anyone else want to get together and write some materials on Trig, Calc, or Pre-Calc?  At the very least, leave your comments below as to where you have found strong materials that make the real-to-abstract connection!