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| Devlin's Angle |
Evidently, then, the formal methods we teach in school are not the most efficient way to achieve mastery of everyday math. Moreover, on the face of it, they are not the most natural way to teach it. For an obvious feature of everyday math is that it is not inherently symbolic (in the familiar sense of being carried out my manipulating symbols on a paper or a blackboard). Rather, everyday math is a collection of mental abilities we humans have developed in order to reason about our world. Once we have mastered those mental skills, we generally use them in our everyday lives in an automatic, unreflective way, much as we use our ability to read. So why do we teach it in symbolic fashion?
Before I give the answer, notice that I am not asking why we teach symbolic math? There are at least two very good reasons for doing that. Rather, I am asking why we use a symbolic approach to teach the not-inherently-symbolic everyday math? And the reason I am making a big deal of this is that, while there is plenty of evidence that practically everyone can master everyday math when they need it in their lives, the majority of people fail to truly master symbolic math. Which means that the symbolic gateway to everyday math is actually not a gateway but a bottleneck - a bottleneck through which many people fail to pass on the road to intended mastery of something that could be mastered if learned a different way.
The reason for placing a bottleneck where there should be a wide-open gateway is that for practically the entire history of mathematics, we have had only one means of storing and widely distributing mathematical knowledge: written symbols. At first those symbols were written on clay tablets, making mathematics the one discipline that almost was handed down to Mankind on stone tablets. Then the clay gave way to parchment, then later paper. On a parallel track, teachers of mathematics and their students poured those symbols onto the less permanent medium of slate, then more recently blackboards and whiteboards. With several thousand years of people learning mathematics through the written symbol, it is hardly surprising that many think that mathematics has to be presented that way. Indeed, as far as I can tell, the majority of people think that mathematics is a set of rules for manipulating symbols. That is certainly how many teachers present it to this day.
But while much advanced mathematics is defined by means of symbolically presented linguistic structures (calculus being an obvious example), that is very definitely not the case for everyday math, which is a way of thinking about, and understanding aspects of, the world we live in.
Compare everyday math with music. No one, I am sure, confuses music with its symbolic presentation on paper using musical notation. Music is something you perform or listen to (both actions); the musical score written in the familiar abstract symbolic notation is just a way to store music and distribute it widely. Indeed, for many years, it was the only way to do so, but various generations of musical recording technologies, the most recent being the iPod, have made musical notation redundant as a storage and distribution mechanism, leaving the symbolic representation to the music professionals, though in many cases even professionals no longer bother to master the notation.
The moment we have the equivalent of the iPod for everyday math, we can ditch the symbolic bottleneck to everyday math mastery, and achieve the hugely important goal for modern society that every citizen has a mastery of everyday math.
Again, I should stress that my focus in this essay is the goal of a "logically capable, numerate" citizenry. Symbolic mathematics is so important in today's world that everyone should have some familiarity with it. One benefit of learning symbolic math is that it makes it easier to use everyday math in novel circumstances. It is also the portal to science and engineering, and every child should be given the opportunity to pass through that particular portal. But whereas lack of mastery of everyday math is a clear impediment to being a fully functioning citizen in twenty-first century society, many people get on just fine without mastery of symbolic math. It's the reliance on mastery of a symbolic representation as a means to teach everyday math that we need to ditch.
Actually, the iPod is not the right metaphor for learning everyday math, though it does point the way to the appropriate technology. In terms of learning, a better comparison is with the piano. As every parent knows, the best way for a child to learn music is to introduce a piano into the home, and let the child play it. Sure, some instruction can help, but that instruction accompanies the act of playing - or trying to play - the piano. The child listens to the music on her or his iPod, and then sits down and tries to reproduce the tune on the piano.
Sure, it takes a lot of practice, and many children give up - I was one of them - but no one goes around saying "I never understood what it was all about - I just did not get it." I suspect they would say just that if they were forced to master musical notation for ten years without being given the opportunity to play an instrument! But no music instructor would do such a thing. The reason is, they know, as do you and I, that people learn things best when they learn by doing, not by reading or performing paper-and-pencil exercises.
If you want to learn to drive a car, to ski, to play tennis, to play golf, to play chess, etc., you do so by doing it. It speeds up the process if you get help from a relative or friend who can already do it, and as you progress you are likely to benefit from seeking out a professional coach or instructor. But fundamentally, you learn by doing what it is you want to master.
If only we had an everyday-math equivalent to the piano, mastery of that crucial logical-quantitative skill set would become routine for everyone.
Well, now we do have that "everyday-math piano." Or rather, now we can have it. The reason I am writing this, is that after thousands of years of math instruction, we finally do know how to build one. I know because for several years I worked on a project to do just that.
More accurately, we know how to build them. For one thing we learned is that (as far as we could tell), unlike the piano, there is no single, universal "instrument" on which you can "play" (i.e., practice and learn) everyday math. Rather, it requires a fairly large orchestra of instruments. An instrument on which you can "play" one curriculum topic generally won't work for another topic. In fact, some topics (seem to) require several instruments to achieve understanding and mastery. Over a four-year period we designed and built several prototypes, and tested them on children.
That exploratory project ended some time ago. Unfortunately, because it was carried out with a commercial software company, I can't show you any of the prototypes we built, but I and some of the others who worked on that project are currently seeking funding to turn those prototypes into releasable products, so stay tuned. I can, however, tell you one of the key, basic design principles we used, and I can point to two products already available that do for a younger age group what we were trying to do for the middle school range.
What makes it so natural to learn to play the piano by actually playing a piano? (If that sounds like a silly "How else would you do it, stupid?" question, let me remind you that our children do not learn everyday math - a form of thinking - by doing it, rather by way of textbook and classroom instruction using symbolic manipulation.) Well, for one thing, the learner sitting at the piano is actually doing what she or he is learning. At first, they don't do well, of course. But they gain mastery by repeatedly trying to do the actual activity. Moreover, while the learner may benefit from some guidance, she or he does not need a teacher to tell them whether they are right or not. When you sit down at a piano and start pressing the keys, the feedback is instant. The piano itself - the sound that emerges - tells you if you have pressed the right keys. What is more, if you get it wrong, the instant, natural feedback from the piano tells you in what way it is wrong, and thus how to correct it next time: if the note was too high, next time press a key a little to the left, if it is too low, move a bit to the right.
More complex feedback and correction comes later, of course. But only when the learner has advanced considerably will it be necessary for an instructor (now in the role of a guide-on-the-side rather than a sage-on-the-stage) to provide feedback, and even then it is immediate, and the learner can at once try again. But at root, the piano itself tells you how you are doing, and provides constant, instant feedback of your progress. This is very different from handing in your math homework to the teacher and getting it back the next day - or several days later - with red ink all over it. Not only does that feedback come long after the learner's thought processes that produced the work, but it puts the teacher in an authority position as the arbitrator of what is right or wrong. And not even an arbitrator of the learner's thought process, which is what everyday math is all about, but of what she or he wrote down, encoded in symbols.
For sure, if a child has a personal tutor, then some of the problem with traditional math instruction can be overcome. The feedback can be instant, and a good instructor can talk with the learner to find out exactly what is causing them to make a particular mistake. To succeed, the teacher has to have a significant skillset, and establish the right rapport with the learner, but even in the best of circumstances it is the instructor who is telling the learner what is right and what is wrong, not the mathematics itself. The child learning the piano comes to know what is right and what is wrong by developing (through doing) an understanding of the instrument and the music, and they are the arbitrators of right and wrong, not the instructor.
In the case of everyday mathematics, it is all about thinking about and acting the world, so all we need to do to provide comparable learning experiences is build "instruments" that encapsulate key features of the world and let children learn by "playing" those instruments.
Well, it turns out that that phrase "all we need to do" hides a huge challenge. There were a number of really smart people working on the project I was part of, including some leading mathematics educators, and we found it very time consuming and difficult to find the "natural" way to encapsulate an everyday math skill in an "instrument" that, in addition to providing exercise in that particular aspect of everyday math thinking, was also sufficiently engaging and challenging to generate and maintain interest. Though we ended up with arguably thirty to thirty-five instruments that seemed to pass muster (not all were kid tested to any great extent, so it may be that some of them won't really work), I am by no means confident that every topic in everyday math can be covered in this way. But some parts can, and we won't know the inherent limitations in the approach until we try. Even if we can cover only some everyday math skills, that will be significant progress. Not just because that will enable all future citizens to acquire those particular life skills, but if they make progress in some parts of everyday math, that could generate sufficient momentum and confidence to master other parts through different means.
You will get some idea of what I am talking about if you go to the Apple App Store and download Motion Math. Produced by three students in Stanford's graduate program in Learning Design and Technology (LDT), it sets out to provide an instrument that a learner can, by "playing" it, acquire a deep understanding of the ordering of fractions. (Incidentally, I was not involved in the project, and saw the result only when the students first demoed it, just before they turned it into a commercial product.) It does not do "much" in terms of curriculum coverage. On its own, it doesn't even ensure full mastery of ordering of fractions. But it does have the important, natural control and feedback features I have been talking about, that facilitate learning by doing. That makes it one instrument that can go into the necessary orchestra.
Motion Math also highlights the two meanings of the word "play" that are relevant here. I have been talking about "playing" an instrument, like the piano. Motion Math is a casual (video-) game, "played" on an iPad or a smartphone. In this case, both meanings of the word "play" apply, and they probably will for all the instruments we will eventually put into our everyday-mathematics orchestra. For in designing and building that orchestra, we will be pulling from all the expertise in the video game industry.
The other example I want to give you is the Jiji video game produced by the MIND Research Institute, a nonprofit educational technology provider based in Southern California. This is a single game that contains many individual mathematical learning experiences. If you look at the individual activities, you will see they are all built according to the piano-like learning principle I have been talking about. The designers at MIND put symbolic representations to one side and develop new, native representations of basic math skills so their young learners can "play" with the mathematical concepts themselves. Only after a child has demonstrated mastery of working with the concept is she or he presented with the symbolic representation for the same operation. (And let me stress that the aim is not to "abolish symbolic notation," not even for everyday math. It is to replace a bottleneck with a wide gateway, so that everyone can pass through. Then, when everyone is through, we can turn to the symbolic stuff.)
There you have it. The future of everyday math education. The only question is, how many years before that future becomes the present, and modern societies can have the logically-able, numerate citizenry they need to function well?
For more details, see my recent book on mathematics education using video games, details in the endnotes below.
Talking of recent books, July 12 sees the US publication of my new book The Man of Numbers: Fibonacci's Arithmetic Revolution, the first book-length account of Leonardo of Pisa's writing of Liber abbaci and the world-shaking sequence of events that followed. If you think the first personal computing revolution took place in Silicon Valley in the 1980s, think again. What happened in Palo Alto and Cupertino was but history repeating itself. The first personal computing revolution occurred in thirteenth century Italy, with its origins in Pisa. Thanks to a recently discovered medieval manuscript in a library in Florence, we now know how the man popularly referred to as Fibonacci initiated that revolution, and thereby came to launch the modern commercial world. My new book tells his story.