What exactly is multiplication?
After my three articles on the nature of multiplication in 2008 (June, July-August, and September) and then a fourth in January 2010, I promised myself to leave the topic alone for some years. In fact, I never set out to write that series at all. The first one was in response to the unexpectedly widespread reaction I received to what I thought was a fairly innocuous, throw-away comment I had made at the end of an earlier column.
In that initial posting, I had noted that it was not a good idea to tell students that “multiplication is repeated addition.” It turned out that large numbers of teachers thought it was just that, so I was driven to post what ended up as three columns, each one longer and more in-depth than its predecessor, explaining why the “multiplication is repeated addition” meme (MIRA) is not only wrong but an educationally dangerous one to propagate. The danger was demonstrated in dramatic form by the large volume of correspondence I received, and the plethora of public blog-threads that ensued, which showed that a great many citizens fervently believe that MIRA.
Why do I say danger? “After all,” some correspondents said, “even if MIRA is wrong, does it really matter if people believe it? As long as they can use a calculator properly and get the right answer, a false belief does no harm.” My answer to that is twofold. First, as an educator I have an ethical position. Those of us who teach mathematics, at any level, simply should not be in the business of spreading falsehoods.
Sure, we sometimes tell “less than the whole truth” in order to provide our students with a manageable path toward mastery of what can be difficult concepts. But when we do so we have an obligation (1) not to tell a blatant falsehood, and (2) leave open the doorway to later refinement, extension, or other modification when the student progresses further. The trouble with the MIRA story is that, as research shows, many students are left with the firm, but false belief that multiplication actually is repeated addition. Knowing that, we should stop telling that particular story.
The second part of my answer is that in today’s world we are faced with a great many decisions that depend upon an understanding of quantity. Some of them are inherently additive, some multiplicative, and some exponential. The behavior of those three different kinds of arithmetical operation differs dramatically, and thinking, say, additively when the problem is multiplicative (or even worse exponential) is likely to lead to some poor decisions that in many cases really are dangerous.
To give just two examples, poor numerical thinking about risk can lead to an unnecessary expenditure on airline security, as I pointed out in last month’s column, and a lack of appreciation for exponential growth can blind otherwise smart people to the catastrophic dangers of global warming. (I am making a mathematical point here; there are many other factors involved.)
It is a fact of life that many people will go through life without being able to do mathematics. When they need the services of a mathematically able person, they can surely find someone. But in a world built on numbers and arithmetic as much as on words and language, for our students to reach adulthood without understanding what the four basic operations of arithmetic are, for heavens’ sake, seems to me to be a situation that those of us in mathematics education should not accept.
Though the initial furor my original postings generated seems to have died down, I still receive emails from teachers and parents along the lines of “Okay, I see what you are saying, but can you then tell me exactly what multiplication is?” and from teachers who ask me to suggest how they should teach multiplication to young children. This column is in response to those requests.
By and large, my replies to those teachers who have written to me has been from the same perspective I adopted when I wrote my original three columns. I am a professional mathematician, but not a trained K-12 teacher. My expertise (and my credentials) are in mathematics, not in teaching. As a professional mathematician, I can (and I believe should) provide advice on the mathematics that is taught in schools, and constructive critiques of the way it is taught, but I do not think it is appropriate for me to suggest how to teach. Others are far more expert at that than I am. (I think it is no accident that the majority of professors of mathematics education are former teachers. It is hard to teach others how to do something you have not done yourself.)
Usually, I would refer my correspondents to two books that I myself found very useful in informing myself of issues of mathematics education. One is Adding It Up: Helping Children Learn Mathematics, authored by the Mathematics Learning Study Committee of the National Research Council, and published by the National Academies Press in 2001. The other is Terezina Nunes and Peter Bryant’s excellent 1996 book Children Doing Mathematics.
There are other books, and a lot of published research articles on the subject. For instance, in 1994, Guershon Harel and Jere Confrey edited a mammoth volume (414 pages) titled The Development of Multiplicative Reasoning in the Learning of Mathematics, that contains a wealth of research findings, insights, and useful advice.
(All three books are available on Google books.)
The problem with my response is that teachers rarely have the time to wade through a volume of several hundred pages, written primarily for the professoriate. I knew it, but they were the best sources I was familiar with.
What I can (and will) do here is try to explain my own understanding of multiplication. In some cases, I suspect that is what my correspondents were asking for. But in my replies to them, I never did more than make a few vague remarks about “It’s based much more on the real-world concept of scaling than on addition.” I was reluctant to go further because none of us really knows how we ourselves learn or understand mathematics, let alone how others do so or might find it profitable to do so. As a professional mathematician of many years standing, my understanding of multiplication may be different from that of others, and moreover aspects of it may be hard or impossible to convey to a child learning math for the first time. Having never taught at the K-12 levels, I would not know. The fact is, multiplication and multiplicative reasoning are complex and multifaceted. (That why the Harel-Confrey book I cited above fills 414 pages.)
Interestingly, the complexities of multiplication almost never arise in the daily activities of the professional mathematician. Indeed, a mathematician who has not reflected on how the subject can—or should—be taught at school level may not be aware of those complexities, though will be able to appreciate them at once the moment they are pointed out. (That was certainly my situation for much of my career.) For the mathematician, multiplication is an abstract, binary function on numbers (and other abstract objects) whose behavior is specified by axioms. We never ask the question “What is it?” nor do we seek to define it in terms of “more basic” functions. (I note in passing that at the level of formal, axiomatic mathematics, multiplication simply cannot be defined as repeated addition, since the latter is not a well-defined function, rather it is a meta-schema outside the axiomatic framework.) We just use the function in the way specified by the axioms.
My concept of multiplication
The concept I have of multiplication when I am working within mathematics is operational rather than ontological, and is built on the axiomatically defined, abstract binary function that constitutes one of the two fundamental operations in a field (or more generally a ring). I know the properties of multiplication, both obligatory (such as associativity) and conditional (such as commutativity), how it relates to other functions (e.g. distributivity), and whether or not a particular instance has an inverse. I am comfortable dealing with it. I don’t ask what it is; all that matters are its properties.
I realize that my professional’s concept of multiplication is abstracted from the everyday notion of multiplication that I learned as a child and use in my everyday life. But my abstract notion of multiplication misses many of the complexities that are part of my far more complex mental concept of multiplication as a cognitive process. That is the whole point of abstraction. Though many non-mathematicians retreat from the mathematicians’ level of abstraction, it actually makes things very simple. Mathematics is the ultimate simplifier.
For instance, the mathematician’s concept of integer or real number multiplication is commutative: M x N = N x M. (That is one of the axioms.) The order of the numbers does not matter. Nor are there any units involved: the M and the N are pure numbers.
But the non-abstract, real-world operation of multiplication is very definitely not commutative and units are a major issue. Three bags of four apples is not the same as four bags of three apples. And taking an elastic band of length 7.5 inches and stretching it by a factor of 3.8 is not the same as taking a band of length 3.8 inches and stretching it by a factor of 7.5.
In fact, the nature of the units is a major distinction between addition and multiplication, and one of several reasons why it is not a good idea to suggest that multiplication is repeated addition, even in the one case where repeated addition makes sense, namely when you are dealing with cardinalities of collections. With addition, the two collections being added have to have the same units. You can add 3 apples to 5 apples to give 8 apples, but you cannot add 3 apples to 5 oranges. In order to add, you need to change the units to make them the same, say by classifying both as fruits, so that 3 fruits plus 5 fruits is 8 fruits. But for multiplication, the two collections are of a very different nature and necessarily have different units. With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first). The unit for the multiplier has to be sets of the unit for the multiplicand. For example, if you have 3 bags each containing 5 apples, then you can multiply to give
[3 BAGS] x [5 APPLES PER BAG] = 15 APPLES
Note how the units cancel: BAGS X APPLES/BAG = APPLES
In this example, there is a possibility of performing a repeated addition: you peer into each bag in turn and add. Alternatively, you empty out the 3 bags and count up the number of apples. Either way you will determine that there are 15 apples. Of course you get the same answer if you multiply. It is a fact about integer multiplication that it gives the same answer as repeated addition. But giving the same answer does not make the operations the same.
The MIRA fallacy becomes very apparent when you consider my second example, where I take an elastic band of length 7.5 inches and stretch it by a factor of 3.8. The final length of the band is 28.5 inches. But what are the units? What goes after the number 3.8 in the calculation
[3.8 – – -] x [7.5 INCHES] = 28.5 INCHES ?
The answer is nothing. It has no units. In this case, the 3.8 is a dimensionless scaling factor.
Incidentally, even when the initial length of the band and the scaling factor are positive integers, it makes no sense to view this example as one of repeated addition. If I take an elastic band of length 7 inches and stretch it by a factor of 3, its final length is
[3] x [7 INCHES] = 21 INCHES
But I did not take 3 copies of a 7 inch band and join them together (addition), rather I scaled (stretched) the 7 inch band by a factor of 3.
What about when you use multiplication to compute the area of a rectangle? If the rectangle is pi inches by e inches (where e is the base for natural logarithms), then its area is
[ pi in] x [e in] = pi.e sq.in.
or, in approximate numerical terms, 3.14in x 2.72in = 8.54sq.in. Again, notice the units. (Note too that repeated addition is not going to get you very far with this example.)
There are other applications of multiplication where the multiplier and multiplicand have different domain interpretations, but since I do not want to write a 414 page column, I’ll leave it at that with the hope that you get my point.
So what is my mental conception of multiplication? It’s a holistic amalgam of all the above and several variants I have not listed. That’s why I say multiplication is complex and multi-faceted. The dominant mental image I have is very definitely the continuous one of scaling, and I see all the others in terms of that. This means that my conception of scaling within this context is a very general one, that encompasses examples like my bags of apples. I can view the computation “3 bags each containing 5 apples gives 15 apples altogether” as “scaling” a bag of 5 apples by a factor of 3. In my experience, acquiring the concept of multiplication amounted to creating this mental amalgam—the amalgam that is my concept of multiplication.
I do not know how or when I acquired this scaling-centric concept of multiplication, and cannot really do it justice in words (unless I simply listed all its many facets), but I’ve had it as long as I can remember and it definitely is a single, holistic concept. To me, that concept is what multiplication is. As such, it is a numerical operation that corresponds to a very general form of scaling. Was I taught it, or did I just develop it over time? I do not know. Do other mathematicians have the same concept? Probably, though as I noted earlier, the ontological nature of multiplication rarely arises in professional mathematical activity, so likely very few have bothered to reflect on the matter.
On the other hand, scaling is a natural physical concept, and abstracting from physical scaling to the numerical operation of multiplication is no more difficult than abstracting from the physical activity of combining two lengths together to give the numerical operation of addition. The advantage of approaching multiplication from scaling is that the resulting numerical operation works in all cases, whereas the MIRA approach works only for positive integers. (Sure, you can tell stories to extend the resulting RA notion to rationals, but it is contrived, and the final jump to real numbers is problematic. The scaling approach gets you to real numbers in one go, where real numbers are identified with lengths of lines.)
So, for all those who have asked me, that is what I understand by multiplication: a somewhat generalized notion of scaling built directly on a physical intuition. And though, as I keep stressing, I have no experience teaching elementary mathematics, I cannot for the life of me see why multiplication is not taught that way.
It may be that one reason some of us are more successful in mathematics than others is that we manage to form good mental concepts early on in the learning process. (I hesitate to use the phrase “the right concepts”, since I do not know that there is a unique correct concept of multiplication, and am very aware of my own difficulty of articulating mine in words.)
Devlin’s Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR’s Weekend Edition. His most recent book for a general reader is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books. Follow him on Twitter at @profkeithdevlin.
FEBRUARY 2011Bad math, bad thinking: the BMI and DNA identification revisited
THE SERMON
This month’s column is about the importance, in today’s scientific, technological, and statistics-driven society, of clear thinking. I present two simple mathematical arguments, one to show that a seemingly scientific emperor actually has no clothes, the other that another emperor has so many garments available, we need to be careful which ones we choose to display.
My first argument shows that the much-vaunted body mass index (BMI), used by doctors as a basis for the advice they give to patients and by medical researchers worldwide as a measure in countless studies, is little more than a disguised measure of waist-size, which even your granny knew was a perfectly sensible way to tell if you carried too much fat.
My second argument shows that in the FBI DNA database, the chances are actually better than 50% that two unrelated entries share the same 9-locus DNA profile. The significance of this is that a 9-locus match is generally assumed as enough to guarantee a reliable identification that can form the basis of a guilty verdict. Indeed, for many years, DNA databases stored only 9-locus profiles.
Neither conclusion shows that current practice is inherently dangerous. The BMI is really little more than a fancy way to refer to your girth, and as I indicate, the DNA calculation I give addresses just one aspect of profile databases. In both cases there is more to be said – in the case of DNA identification a lot more. In fact, my point is that DNA identification requires considerable statistical understanding. The two calculations I present are both elementary, requiring just a few minutes scribbling on a piece of scrap paper. But they are, I think, illuminating. The surprising, and worrying, thing is that no one seems to have bothered to pause and make them before.
SERMON OVER.
In my May 2009 column I discussed the absurdity of the body mass index and in October 2006 I looked at the danger of false conviction in a DNA Cold Hit identification. In both cases, I presented what I thought was a fairly sound logical argument to support my claims, but I still receive a slow but steady stream of emails from people who read those columns and try to convince me I am wrong. By and large, their objection to my BMI claim is that the wretched number (and the formula behind it) must make sense, otherwise the CDC, the bulk of the medical profession, and the medical insurance companies would not use it. For my DNA identification rant, the objection is not so much that the FBI uses it, rather the conclusion I reached runs against our intuitions.
Now, I have no problem with people disagreeing with me. Heavens, I was a mathematics department chair for four years and a dean for eight, and before that a father of two daughters growing up through adolescence to adulthood, so I have had my fair share of disagreements. What bothers me is that some folks seem (1) all too willing to accept something simply because its looks scientific and some august body advocates it, and (2) unable to adjust their intuitions when faced with evidence that they mislead us, as they do on occasion.
It bothers me because I am in the education business, and the primary goal of education is to help people develop the ability to think for themselves and to reach conclusions based on evidence and rational thought.
It bothers me particularly because, as the ancient Greeks recognized, when taught well, mathematics is one of the best mental disciplines to develop analytic thinking skills. At which point I cannot miss a chance to point you to this excellent video.
But I digress. Going back to the BMI, some simple algebraic reasoning should be enough to show that the emperor has no clothes.
BMI rant revisited
Recall that, ignoring a constant, the BMI formula is
BMI = M/H2
where M is your mass (weight) and H your height.
To a reasonable approximation, the human body can be thought of as an ellipsoid with circular cross section. (I am looking at the logic behind the BMI here. What matters is how the various parameters affect each other; the exact numbers are not significant. You could model the body as a cylinder if you preferred.) The length of the ellipsoid is the person’s height, H, and the diameter of the cross section at its mid-point (the waist) is W/pi, where W is the waist size (girth). The volume of the body is then
V = (pi/6).H.(W/pi)2 = (H.W2)/6.pi
Hence, if D is the average density of human tissue
M = D.V = (D.H.W2)/6.pi
Thus:
BMI = M/H2 = (D.H.W2)/6.pi.H2 = (D.W)2/6.pi.H
In the case of an adult, H is, of course, a constant, as is D (more or less). Thus the formula reduces to
BMI = K.W2
where K = D/6.pi.H is a constant. So what is the BMI really measuring? Your waist! Now just fancy that, the size of your waist indicates whether you are overweight or not! Who would have guessed?
Which begs the question, why doesn’t the medical profession just use that far simpler metric? Beats me.
To be sure, I have a pretty shrewd idea why Quetelet introduced the BMI back in the early nineteenth century. (See my earlier column for the history.) He was looking at public records to determine society trends and doubtless he had lists that gave people’s heights and weights, since these are measured and recorded all the time. But the tables almost certainly did not give waist sizes, since those are rarely measured, except when we buy trousers. So he played around with the data he had until he found a correlation. The correlation he found was with what he called the body mass index. It required taking the square of a person’s height. It certainly had an air of scientific authority and precision—after all, it came from a mathematical formula! But in reality, all he had done was find a way to estimate people’s waist sizes.
And the medical profession has been chanting the BMI tune ever since.
The above alternative formula for the BMI does provide a bit of insight that I find lacking in the original. For the purposes of a population study (such as Quetelet’s), H will be a variable. My new formula then looks like this:
BMI = C.(W/H).W
where C is the constant D/6.pi. The ratio W/H, your girth-to-height-ratio, is a factor whose relevance we can all surely appreciate. But notice that there remains an additional, multiplicative W factor. This shows that girth is a particularly significant factor; the BMI increases linearly with girth even after you have taken account of the girth-to-height-ratio.
Of course, the BMI still comes down to waist-size—doubly so as it turns out! And nothing I say here takes away from the issues I raised last time, including the fact that using a formula obtained by a statistical correlation across a population to make individual health and lifestyle recommendations and decisions has no scientific justification.
This isn’t rocket science, folks. Take another look at what Ed Burger says in the introductory remarks in that video. Please. Then get your physician to watch it.
I suggest we henceforth rename the BMI the “Bogus Mathematics Index”.
Cold Hit DNA identification rant revisited
Moving on to DNA, you’d better read my original rant to get the relevant background, since this is a complex issue, and I’m going to assume everything I said there.
For the sake of argument, suppose the government has a database of 13-loci DNA profiles from 1 million individuals. (The FBI’s CODIS database has roughly 4.2 million such profiles. As a size of the database increases, the kind of problem I am talking about becomes more significant. So the calculation I am about to do is more conservative than in reality.)
The probability of a random match on a single locus is about 1/10. That means, the match occurs purely by chance; it does not provide identification. (This is an empirical fact I learned from the experts, as you can too.)
In seeking to identify a crime perpetrator based on a DNA profile (and nothing else), suppose the FBI lab finds a match with a crime-scene sample on 9 loci. (This is actually pretty good. Crime scene samples are often damaged or degraded, making a full 13-locus match impossible. Again, I’m just passing on what the experts say here.)
Naively, if the RMP on a single locus is 1/10, you might think that the probability of a match on 9 loci being a result of pure chance is (1/10)9, that is, 1 in a billion. Hence, you conclude, the FBI have surely got the guy. Guilty.
Well, he might be guilty. But in the interests of justice you should at least check the math. And there is a lot of math you can do to get an overall sense of how likely things are. One calculation you might make—and statisticians have looked at this issue empirically for a real DNA database (Arizona’s) and obtained results consistent with my calculation—is to find out how likely it is that the 1 million entry database contains two different individuals whose profiles are identical on 9 loci. When you perform that calculation, you find that the chances of that happening are better than 50%. That’s right, the odds are actually in favor of there being a random match on 9 loci. Surprising, no?
You can find my calculation of this result here. Lay readers may find it a bit intricate, since I use some techniques to handle large numbers efficiently. But I can explain the argument using a simpler example where the numbers are not so big. The same fact that seemingly unlikely events can actually occur quite often (like two people having the same DNA profile) is often demonstrated with what is called the Birthday Paradox.
The question, is how many people do you need to have at a party so that there is a better-than-even chance that two of them will share the same birthday?
Most people think the answer is 183, the smallest whole number larger than 365/2. But that is not the case. In fact, you need just 23 people in the room. (The analog to the DNA case is how big does a database have to be for the chances to be better than half that two entries share the same profile on 9 loci. In that case, the answer is not 23, but a million entries will do it, and CODIS already exceeds that number.)
Here is how to get that answer. To figure out the probability of finding two people with the same birthday in a given group, it turns out to be easier to ask the opposite question: what is the probability that NO two will share a birthday, i.e., that they will all have different birthdays?
With just two people, the probability that they have different birthdays is 364/365, or about .997.
If a third person joins them, the probability that this new person has a different birthday from those two (i.e., the probability that all three will have different birthdays) is (364/365) x (363/365), about .992.
With a fourth person, the probability that all four have different birthdays is (364/365) x (363/365) x (362/365), which comes out at around .983. And so on.
The answers to these multiplications get steadily smaller. When a twenty-third person enters the room, the final fraction that you multiply by is 343/365, and the answer you get drops below .5 for the first time, being approximately .493.
This is the probability that all 23 people have a different birthday. So, the probability that at least two people share a birthday is 1 – .493 = .507, just greater than 1/2.
With that simpler example in hand, you might now want to take another look at my DNA calculation. It’s essentially the same.
Now, I am most definitely not trying to denigrate the use of DNA profiling. It is the most reliable and powerful means at our disposal to catch criminals (and to free wrongly convicted innocent victims). What I am trying to do is ensure that citizens understand the math sufficiently to make a proper evaluation of DNA evidence. That original figure of 1 in a billion is terribly and dangerously misleading, and could lead to innocent people being convicted. Equally, the fact that on mathematical grounds alone the CODIS database is more likely than not to include two individuals whose DNA matches on 9 loci is potentially misleading, and used by a skillful defense attorney might result in a guilty person getting away with the crime. That elusive, crucial concept the truth lies somewhere between those two extremes.
The fact is, in the era of DNA identification, judges and juries simply cannot avoid getting to grips with the relevant math. Identification hinges on those calculations. There may be no way of avoiding bringing mathematicians into court to explain how the calculations are done. But for that to be effective, those judges and juries need first to learn (and accept) that human intuitions about probabilities are hopelessly unreliable. That can prepare the way, not for mathematical laypersons to learn how to do the calculations themselves—the experts can do that part—rather how to follow the calculations and evaluate the answers. For it is on those answers that justice will ultimately depend.
Time to watch that video again. And this time, watch the two others in the series. (You will see them if you watch the video embedded above on Youtube.) Though on the surface they are about learning math, the real message is—as several of the participants say in different ways—learning to think.
Devlin’s Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR’s Weekend Edition. His most recent book for a general reader is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books. Follow him on Twitter at @profkeithdevlin.
MARCH 2011Learning Math with a Video Game
According to 2008 figures from the Pew Research Center, 97% of today’s K-12 students spend many hours each week playing video games. By the time they graduate from high school they will have spent some 10,000 hours doing so. During the course of that game play, they will acquire a vast amount of knowledge about the imaginary worlds portrayed in the games, they will often practice a skill many times until they are fluent in it, and they will attempt to solve a particular challenge many times in order to advance in a game. Would that they devoted some of that time and effort to their schoolwork!
Since video games first began to appear, many educators have expressed the opinion that they offer huge potential for education. The most obvious feature of video games driving this conclusion is the degree to which games engage their players. Any parent who has watched a child spend hours deeply engrossed in a video game, often repeating a particular action many times, will at some time have thought, “Gee, I wish my child would put just one tenth of the same time and effort into their math homework.” That sentiment was certainly what first got me thinking about educational uses of video games 25 years ago. “Why not make the challenges the player faces in the game mathematical ones?” I wondered at the time.
In fact, I did more than wonder; I did something about it, although on a small scale. I wrote a simple mathematics education game for my children’s elementary school, in which the player had to use the basic ideas of coordinate geometry in order to discover buried treasure on an island, viewed in two dimensions from directly above. While it did not offer the excitement of the commercial entertainment games my own children were playing, the children at the school seemed to enjoy playing my more modest effort. Perhaps offering a prelude of things to come, the game even made a financial profit. I wrote it for free, and the school PTA sold copies to other local schools. I think we sold five or six copies in total, each one on an audiocassette tape. I did not give up my day job.
And that was about the extent of my foray into the world of video games until around 2003, when, as a result of my location in the heart of Silicon Valley, I got to know some professional video game designers. Talking with them, and with video game scholars, over the years not only aroused in me once again the huge educational potential of the medium, but also gave me insights into the organizational and engineering complexities involved in commercial video game production.
Yet whenever I have tried some of the math ed video games being produced, I have been significantly underwhelmed. For the most part, they do not teach mathematics at all, rather they test what has already been learned elsewhere. Moreover, that testing is largely restricted to automatic recall of basic number facts and rapid use of arithmetic skills. Nothing wrong in that. I wish I’d had access to an enjoyable video game to help me practice my multiplication tables when I was in elementary school. What has left me unsatisfied is that those games (and we are really still talking about first-generation mathematics education video games here), while succeeding in achieving their (modest) educational design goals, fall way short of the potential I, and many other, are sure the medium offers.
On problem with the majority of math ed video games on the market today that will quickly strike anyone who takes a look, is that they are little more than a forced marriage of video game technology and traditional mathematics pedagogy. In particular, the player of such a game generally encounters the math in symbolic form, often by way of a transparent screen overlay on top of the game world.
But video-game worlds are not paper-and-pencil symbolic representations; they are imaginary worlds. They are meant to be lived in and experienced. Putting symbolic expressions in a math ed game environment is to confuse mathematical thinking with its static, symbolic representation on a sheet of paper. It’s like the early would-be aviators who tried to fly by building ornithopters—machines that added flapping wings to four-wheeled cycles. Those pioneers confused flying with the only instances of flying which they had observed—birds and insects. Humans achieved flying only when they went back to basics and analyzed the notion of flying separately from the one particular implementation they were familiar with. Similarly, to build truly successful math ed video games, we have to separate the activity of doing mathematics, which is a form of thinking, from its familiar representation in terms of symbolic expressions.
Mathematical symbols were introduced to represent mathematics first on parchment and slate, and still later on paper and blackboards. Video games provide an entirely different representational medium. As an interactive, dynamic medium, video games are far better suited in many ways to representing and doing middle-school mathematics than are symbolic expressions on a page. We need to get beyond thinking of video games as an environment that delivers traditional pedagogy—a new canvas on which to pour symbols—and see them as an entirely new medium to represent mathematics.
This will not be easy. I know that from personal experience, having spent a large part of the past five years working on a well-funded project trying to do it. But I am convinced it can be done—though to what extent and how well remains to be discovered. Sadly, I cannot describe the particular project I worked on, which had proprietary elements, but my new book Mathematics Education for a New Era: Video Games as a Medium for Learning, just published, summarizes my current thinking on the subject. I’ll come back to the topic of mathematics education video games in future columns.
In the meantime, if you want to get a good background into the educational potential of video games, I highly recommend the book What Video Games Have to Teach Us About Learning and Literacy, by Prof James Paul Gee of Arizona State University. And if you want to experience a video game that will really challenge your (logical, not mathematical) problem solving ability, try the free demo version of Portal. If you are like me, you will willingly pay the $14.99 price to convert your free demo into the full version. (BTW, the little animation you see on the demo website is not the game; rather it is an “instructional video within the game”. The game itself has high production values.)
Devlin’s Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR’s Weekend Edition. His most recent book for a general reader is Mathematics Education for a New Era: Video Games as a Medium for Learning, published by AK Peters/CRC Press. Follow him on Twitter at @profkeithdevlin.
APRIL 2011Finger counting
For more superb videos from the hugely talented Vi Hart, go to https://www.youtube.com/@Vihart.
Devlin’s Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR’s Weekend Edition. His most recent book for a general reader is Mathematics Education for a New Era: Video Games as a Medium for Learning, published by AK Peters/CRC Press. Follow him on Twitter at @profkeithdevlin.
MAY 2011NOTE: In assembling this archive, where possible I have updated URLs that I discovered no longer worked. In the case of my new Twitter account, with @keithdevlin already taken, I first created an account as @nprmathguy, but soon realized that was too narrow an identification, and switched to @profkeithdevlin. The @nprmathguy tweets in this post were indeed mine.
The Keyring Problem: NCTM fumbles the ball
Last month, this tweet from NCTM caught my eye:
“What a nice little problem to form the basis of a class discussion,” I thought. Okay, it is perhaps a bit contrived. How often has anyone needed to know how many different arrangements there are? But that aside, it has a number of features that give it great pedagogic potential:
I should add that one solution seemed to me to be the best, but I suspect that others would argue for an alternative. (My answer is 3,840. I’ll come back to it in just a moment. Before you read on, you might want to see what answer you get.)
From what I have said so far, it is clear that I follow the NCTM’s Twitter feed, and on that basis you may infer, correctly, that I have an interest in K-12 mathematics teaching. You may even suspect, also correctly as it turns out, that I am a member of NCTM. Indeed, the week before that tweet was published I attended their annual conference in Indianapolis. Not as a math teacher, for I am not qualified to teach mathematics at K-12 level and have no experience doing so; rather as a mathematician interested in K-12 teaching and believing that my knowledge of mathematics may be of assistance to those who do teach in our nation’s schools, to those who train them, and to those who provide them with educational materials.
ASIDE: Why do I make this disclaimer? Because our society appears to be riddled with individuals who, on the basis of having been taught K-12 math as children, seem to think they know how math should be taught. (In some cases, those individuals freely admit that they were not taught well and never really got it, yet that does not prevent them advocating one method or another, even the one that did not work on them.) In stark contrast, I do not find many people who, on the basis of having flown in an airplane, think they could fly one, or indeed have anything useful to tell a pilot. Nor do I find people who think their experience being treated by a physician makes them qualified to operate on a patient or pontificate on medical practice.
Teaching, like many things in life, might look simple from the outside, but it assuredly is not. Now, it is true that the likely outcome of a non-pilot taking control of a jet airliner or a non-physician treating a sick patient would be dramatic and tragic, in that people could die. In contrast, if someone not adequately educated in mathematics and untrained in mathematics pedagogy teaches a math class, the children are not likely to die. The worst that could happen are one or more of the following:
Yup, the outcome is not dramatic, and no one gets killed. But nonetheless tragic, don’t you agree?
What makes it particularly hard to appreciate the importance of good teaching is that, unlike flying a plane or performing heart surgery, it is very difficult—though not impossible—to measure the outcomes of mathematics teaching. (Standardized tests are about as useful in determining whether an individual can think mathematically as measuring a tennis player’s bicep to see if she or he has a good serve.) Thus, the lack of benefit—or worse, negative outcomes—of inadequate or bad math teaching tend to remain hidden. This may go a long way to explain why so many people think that teaching math is not a skillful profession the same way that being an airline pilot or a doctor are. (Though the large number of adults who claim they cannot do math, and even hate the subject, should give us a BIG CLUE that something, somewhere is not right in many of our nation’s classrooms, and has not been for at least three generations, and probably longer.)
But I digress. Back to that NCTM keyring problem. I had just one nagging worry when I read that tweet. Was the person who posed it falling into the familiar trap of defaulting to one particular interpretation, and posing the problem as one that (wait for it) has a “right answer.” Given the wealth of materials that NCTM publishes and promotes on good mathematics teaching, the odds were against this, but I had sufficient worry to tweet a reply:
Sadly, my fear proved to be correct. Later that day, the NCTM tweeter (or is the appropriate actor a “Twitterer”?) published the answer that was expected:
I responded
Of all organizations to make such a blunder, the NCTM is surely one of the worst. No, I am not going to cancel my membership of NCTM, which does a lot of good work. (Besides, some of my best friends are card-carrying NCTM members.) The fact is, the organization is made up of people, and people make mistakes. All the time. Moreover, instant media like Twitter (and an MAA column like mine) increase the risk of something going out that a bit of reflection would have prevented. There is absolutely nothing wrong with getting something wrong. That is how we learn.
Only two things are indefensible about getting something wrong: not learning from the mistake and not trying again. We Americans and British (I’m both) typically find it hard to admit an error. (Though a crucial factor in the continued entrepreneurial success of Silicon Valley, where I live, is that not only is failure accepted, if someone does not repeatedly fail they are viewed as not trying hard enough.)
We need to overcome that fear of failure. (My friends in Japan tell me it is much worse there.) The fact is, with the keyring problem going out into the Twittersphere under the logo of the NCTM, we have a wonderful news hook on which to hang a valuable reminder of the power of what the blogging math teacher (and now Stanford doctoral student) Dan Meyer refers to as “WCYDWT teaching. (“What Can You Do With This?”)
The keyring problem is, as I indicated at the start, a great problem for investigation. There is no single right answer. It all depends on how you interpret the question. As occurs so often with applications of mathematics, you have to start by turning an underspecified question into something precise. With answers ranging from 12 to 3,840 (Any advance on those limits, anyone?), it is clear that how you introduce the precision can make an enormous difference.
The NCTM tweeter presumably modeled the situation as: “How many ways are there to order five points on a non-oriented circle having no distinguished point?” I’m not going to give any of the calculations, by the way. I may not have K-12 classroom experience, but I’ve done enough math teaching in my life to know that good teaching is a matter of helping students figure it out for themselves.
This, incidentally, is why Khan Academy (which I have endorsed elsewhere and will continue to do so) does not teach anyone how to do math (i.e., how to think mathematically), any more than a dictionary, thesaurus, and grammar book can teach someone how to write a novel. What Salman Khan does is provide an excellent instructional resource for some of the tools you need to do math. A particularly gifted and motivated individual could likely use it to learn mathematics on their own, particularly if they had access to a mathematician who could assist them when they are stuck.
Thank you for doing that, Sal. Your site is valuable, and you deserve all the attention it has garnered. The danger I see in all the media coverage of late is if people think that the provision of a tool that can play a part in improving the nation’s mathematical abilities is actually the entire solution. (As far as I am aware—and I have met Sal a few times and we have talked about his material—he himself has not claimed that, though others seem to have come close.)
The NCTM tweeter’s interpretation of the keyring scenario is an easy one for experienced math teachers to slip into without realizing it, because it is the one that leads to the easiest computation. But as I pointed out (at length, as is my wont) in my May 2010 column, interpreting in simplistic ways problems formulated in real-world terms is tricky and not without significant dangers.
Unfortunately, formulating abstract mathematical and logical puzzles in seemingly real-world terms (word problems) is a tradition that goes back to the very beginnings of mathematics several thousand years ago, and has been used ever since. As I noted in that earlier essay, this is fine for those of us who enjoy puzzles and learn to interpret the problems in the intended way—to “read the code”—but is problematic for everyone else. And in this case the everyone elses constitute the majority of people!
Here is how I arrived at my answer. In a drawer at home, I have several key rings I have acquired over the years. So many that I suspect that, from that one source alone, a future historian might be able to retrace many places I have visited and institutions I have interacted with. Those keyrings all have one feature in common, namely a topologically circular ring, and the majority share a second feature, namely to the ring is attached some kind of object, often a badge or shield, sometimes a ball-shaped object, an occasional pen or flashlight, etc. I’ll add that some of those badge-like attachments have two distinct faces and are rigidly attached to the ring, some have two distinct faces but the entire badge can be rotated on an axle, and some are symmetrical so that the entire keyring looks the same from both sides. (“What difference can these features make?” one or two of your students might ask. If they don’t, you should ask them.)
I also have an even larger collection of keys. (Why do we keep old keys, by the way? I have no idea what the majority of them lock, and for sure they no longer fit anything I still have access to.) A very small number of those keys are symmetrical: when I turn them over, they look identical. (Again, is this significant to how we model the problem?) But the vast majority do not have such symmetry.
When I modeled the keyring problem, I based my model on my nostalgia-based collection of real keyrings and my irrational collection of old keys. In other words, I did what I think any sane, sensible person would do outside the math class: I took the problem at face value, as asking about real keys on real keyrings. I based my solution on what I most commonly saw in my drawer.
I then had to decide what constitutes an “arrangement”. Again, I interpreted this word (I almost said “key word”, but decided that would be too tacky, though puns are their own reword) in the way I thought most natural. The problem asked about different arrangements of keys, not about permuting them around the ring. In particular, if I change the orientation of one of the (asymmetrical keys), I change the arrangement. Again, there is a story to be told, and a decision to make, when symmetrical keys are involved. And I wondered for a while what to do about having two or more identical keys on the same ring, when the mathematical model you choose could depend on why the question was asked in the first place.
In the end, I came out with 120 x 32, namely 3,840. A far cry from the NCTM’s 12. Which of us is right? Neither. There is no right answer for the problem as posed. It’s a modeling task—using mathematics to analyze something we encounter in the world. For sure, I think my answer is more in accord with real keys on real keyrings, and hence more realistic, than the NCTM’s. But the value of the problem is that it has so many different answers, and shows that building a mathematical model is a tricky thing.
Solving a math problem that arises from a situation in the world can be hard. But nothing like as hard as formulating the issue mathematically in the first place. And as I argued in another earlier column in
July 2010, in the U.S., our economic future and our security depend on our excelling at the latter kind of activity. Put frankly, a crucial economic reason why it is important for as many Americans as possible to know how to solve math problems is so they can think mathematically about real world problems.
Yes, the keyring puzzle is a great problem, and definitely one that NCTM should be promoting.
Devlin’s Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR’s Weekend Edition. His most recent book for a general reader is Mathematics Education for a New Era: Video Games as a Medium for Learning, published by AK Peters/CRC Press. Follow him on Twitter at @profkeithdevlin.
JUNE 2011Wanted: a mathematical iPod
For some years now, I have been using the term “everyday mathematics” to refer to the following collection of topics: basic number concepts, elementary arithmetic, arithmetical relationships, multiplicative and proportional reasoning, numerical estimation, elementary plane geometry, basic coordinate geometry, elementary school algebra, quantitative reasoning, basic probability and statistical thinking, logical thinking, algorithm use, problem formation (modeling), problem solving, and sound calculator use.
These are the basic quantitative and logical abilities that are essential for anyone to function well in, and contribute to, modern society. The world we live in requires a degree of mastery of each of these mathematical topics.
“Mastery” in this context means not just being able to perform calculations with fluency. It is also important to have a good conceptual understanding of numbers, arithmetic, and reasoning, particularly in the context of real-world applications. A person can have computational skills without much conceptual understanding, though that invariably leads to problems if any further progress in mathematics is required—and many people find that it is.
For example, a child can learn the multiplication tables by rote to the point of rapid, automatic recall, without having any understanding of what multiplication is or how it relates to things in the world. Indeed, many successful adults never fully understand multiplication, though they can use it correctly in certain circumstances. This is demonstrated dramatically by the widespread belief in the adult population of the US that multiplication of positive whole numbers is repeated addition. (See my January 2010 column for and the earlier columns referred to there.)
When something as basic as multiplication of positive whole numbers turns out to be conceptually fairly complex, you know that everyday math involves a whole lot more than rote learning of a few facts. You can learn to calculate with numbers without any real understanding of the underlying concepts. But applying arithmetic to things in the world, to quantities, and understanding the relationships between those quantities, requires considerable understanding of those underlying concepts.
One obvious (and characteristic) feature of everyday math is that it can all be done in the head. You don’t need a paper and pencil. In large part this is because the objects it deals with are all in the physical environment we live in. Indeed, everyday math is a collection of mental skills for understanding and reasoning about our environment.
A less obvious feature of everyday math, well known to mathematics educators, is that practically everyone is able to achieve mastery of those skills if they find they need it in their everyday lives or jobs. (See my 2000 book The Math Gene for a summary of the main evidence.)
Though it is possible to express everyday math in formal symbolic terms, you don’t have to. When you do, however, a curious thing happens. Ordinary people who become proficient in everyday math in their daily lives to the point where their accuracy rate is around 98% when they do it mentally in a real setting, find that their performance level drops to around 37% when they try to do it symbolically. (I describe this in The Math Gene as well, where I also provide an explanation.)
Given the huge importance of everyday math in today’s society, it clearly makes sense that we should teach it in the most effective way. Quite clearly, teaching it symbolically does not work for a majority of people. So why do we do so?
The answer is technology. For two-and-a-half thousand years, symbolic representation was the only way we had to store and widely distribute mathematics—even basic, everyday mathematics. But what if there were an alternative method of storage and distribution for everyday mathematics, from which people could learn how to do it. One that would produce that near perfect 98% accuracy performance exhibited by ordinary citizens when they need it in their everyday lives.
Such a storage/distribution device would be equivalent to the iPod for storing and distributing music so anyone could listen to it and appreciate it, a storage device that circumvented the formal, symbolic musical notation that for many centuries was the only medium we had for storing music. Wouldn’t it be cool if there were such a device?
Well, it turns out the technology already exists to build such “mathematical iPods”—devices that store everyday math in a native fashion, without the need for symbolic representations written on paper. How do I know? One of the advantages of living in Silicon Valley, as I do, is having an opportunity to see, and try out, new technologies. Unfortunately, the limits on the length of my column mean there is not enough space for me to give any details here. Unlike Monsieur Fermat, however, I will be able to pick up the story next month.
To be continued …
Devlin’s Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR’s Weekend Edition. His most recent book for a general reader is Mathematics Education for a New Era: Video Games as a Medium for Learning, published by AK Peters/CRC Press. Follow him on Twitter at @profkeithdevlin.
JULY 2011Students should learn everyday math the way they learn to play a musical instrument
In my last column I defined what I mean by “everyday math” and pointed out that the term refers to a collection of quantitative and logical skills that are essential for everyone who wants to play a full role in twenty-first century life in an advanced nation. I also observed that it is best learned in an experiential, situated fashion. Indeed, studies have shown that almost all ordinary people can achieve a performance level of 98% accuracy when faced with using everyday math in their everyday lives. Yet, when presented with what a mathematician would regard as the very same problems in paper-and-pencil, symbolic form, their performance drops to around 37%. (So to the individuals concerned, they clearly are not the same problems.)
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 create new “instruments”, based on what we learned, and turn them 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—via 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 several 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.
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 then 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.
Devlin’s Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR’s Weekend Edition. His most recent book for a general reader is Mathematics Education for a New Era: Video Games as a Medium for Learning, published by AK Peters/CRC Press. Follow him on Twitter at @profkeithdevlin.
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