If you are a professional educator (and if you are reading this blog the chances are high that you are), you likely noticed what seemed to be opposing philosophies behind the two books I published earlier this year.
In my recent book about the use of video games in mathematics education I seem to be arguing that, since middle school mathematics is grounded directly in the real world, it is more effective to let people learn it in a situated fashion in a video game, rather than by way of the more abstract symbolic written representation found in books, that we are all familiar with.
On the other hand, in my book The Man of Numbers about Leonardo of Pisa's thirteenth century blockbuster Liber abbaci and its more user-friendly version Book for Merchants, I describe how capturing generations of accumulated arithmetic know-how in the then novel symbolic form developed by the Indian mathematicians and distributing it in book form, led to the western financial revolution that began in medieval Italy.
Indeed, in my own "price-of-a-latte" Leonardo e-book companion Leonardo and Steve I make (and back up) the claim that the introduction of symbolic arithmetic was a user-interface breakthrough that was ever bit as pivotal in generating widespread acceptance of Hindu-Arabic arithmetic as Steve Jobs' Apple Macintosh computer was in turning us all into computer users starting in the 1980s.
So which is better when it comes to mathematics education, situated or abstract? The answer is that both have advantages. That's not the kind of simplistic, black-or-white, pick-one-or-the-other answer that Fox News likes to pretend is how humans operate, but the fact is, there is truth in both. Which has greater applicability for any one purpose depends both on that purpose and on the prevailing circumstances. For acquiring basic mathematical thinking skills, the situated learning that can be provided in a video game is demonstrably far more effective. The great advantage of abstract symbolic mathematics, on the other hand, is that, once mastered, a particular technique can be applied in many different situations.
This is why the recent New York Times op-ed by mathematical powerhouses Sol Garfunkel and David Mumford is a strong argument in favor of quantitative literacy education but essentially orthogonal to questions about mathematics education, a point made well by my fellow MAA columnist David Bressoud this month.
The way that the abstraction of mathematics can lead to applications in different domains is perhaps best illustrated when someone takes some mathematics developed to solve one particular problem and applies it successfully in a very different domain. One of my favorite examples is found in a 2003 paper by Lawrence Sirovich in the Proceedings of the National Academy of Sciences, titled A pattern analysis of the second Rehnquist U.S. Supreme Court. By tabulating several years worth of decisions by the Court as a rectangular matrix, Sirovich was able to analyze the data using the same mathematics used to compress images (represented in digital form by a rectangular matrix of pixel values), thereby uncovering (a more accurate description might be, confirming the strong suspicions many of us already had) persistent political bias in their split decisions. (I discussed that fascinating piece of research in my Math Guy slot on NPR's Weekend Edition at the time.)
Another equally impressive, and newsworthy, example of the same transferability of mathematics that becomes possible when it is represented in abstract form arose recently with the publication on arxiv.org of a fascinating analysis of basketball shooting strategies: The problem of shot selection in basketball: "The shooter's sequence", by Brian Skinner of the University of Minnesota.
Skinner applied to basketball the mathematics used to analyze traffic flow on the roads. Most drivers want to minimize the time it takes them to get to their destination, but to do so they have to negotiate their way through all the other drivers trying to do the same thing. In basketball, the team with the ball wants to get it across the court and into the basket, but they have to negotiate their way past all the opposing players. Those two scenarios don't seem particularly close, but it turns out that at the level of abstract mathematics, they are close enough. Surprisingly, the traffic math does work for basketball.
One of the interesting results to come out of mathematical analyses of traffic flow was that the overall commute period can sometimes be sped up by closing a heavily used road. This basically shakes things up by forcing the drivers who normally use that road to take another route. Some drivers may end up taking longer to get to their destination, but overall, the average commute time can go down.
In basketball the analogy is to not field a star player, and Skinner shows that the same conclusion follows on the court. That might sound crazy, but in fact it had already been observed to be true. It's called the Patrick Ewing theory, after the high-scoring New York Knicks player. Analysts had noticed that when Ewing or other high scoring players like him were absent, their team was more likely to win. The rest of the team had to adjust their play to make up for the star's absence, and that resulted in a win. Of course, it is only a one or two game phenomenon. It would not make sense for a team to trade away their stars. It's basically a shaking up effect, just as in traffic flow.
The main focus of Skinner's paper, however, is when it's best to take the first opportunity to shoot, or when it's better to wait for a high quality shot that is more likely to go in. One of his results is that the more seconds there are left on the clock, the choosier a team can be about which shot to attempt, but everyone already knew that. It's pretty obvious. What is new, however, are precise numbers about how often to attempt a shot.
For example, suppose team A and team B each have the same chance of scoring on a given shot but that A passes the ball twice as fast as B. Let's also assume that both teams have the same ball turnover rate and have plenty of seconds left on the shot clock. Conventional wisdom says team A's best prescription for success is to shoot twice as often as B. But the math shows that if team B shoots, on average, say every 20 seconds, then team A should shoot every 13 seconds rather than every 10. The extra 3 seconds allows team A to be more selective about which shots to take, and in a game where the difference between winning and losing often is a matter of seconds, that can be a winning strategy.
Whether or not Skinner's mathematics has a significant effect on how coaches direct their players is an open question. My point here is that his paper provides yet another wonderful example of how abstract mathematics (hopefully written on recyclable paper) can be taken from one domain and re-used in another. To go back to the Garfunkel-Mumford article and Bressoud's response, being a successful and productive citizen in twenty-first century America requires a basic level of Quantitative Literacy, but for this country to maintain its current status as world innovation leader (the only appealing future I can see for us) we need a substantial number of our citizens with sufficient genuine mathematical skills who can apply mathematics in novel situations, either by developing new mathematical techniques or by recycling old ones.
To this end, I would be interested in hearing from any CEOs or HR directors of major companies, particularly in the tech arena, as to what specific problem solving talents or skills they look for in new employees. True, I suspect few high tech CEOs or HR folk are regular readers of MAA Online, but some readers may know such individuals, in which case please let them know of this request. My question is deliberately under-specified, since I do not want to prejudge the kinds of answers I receive. To give just one illustration (for sure not typical of the responses I might get), I suspect few employers of the nation's mathematical talent are interested in a proven ability to solve a second order differential equation, but they might well be swayed by an ability to apply differential equations in a novel way to solve a critical and uniquely different problem the company faces in supply chain management. Different employers might look for different manifestations of that kind of ability. If the education world is aware of what the job market looks for (and many in the mathematical education world do read MAA Online), we can start to change our education method to meet that demand. And for readers who find such a suggestion an offensive "sell-out," I would suggest they read my Man of Numbers book to learn how modern arithmetic and algebra were developed precisely to meet the needs of trade and commerce.