Devlin's Angle

The difference between teaching and instruction

2012-03-01T09:00:00.001-05:00

A brief discussion with a reader of my personal blog (profkeithdevlin) reminded me once again of the common confusion between instruction/training and teaching/learning.

As a child, I never experienced what I would now call mathematics teaching. (I revised my understanding of what teaching is, when, as an adult, I saw it in action on several occasions. Although I have no memory of having been taught that way, I guess it is possible that as a very young child I was.)

What I was presented with at school was instruction. The quality varied a lot, but looking back it was definitely instruction, not teaching. The teacher would explain some new concept or demonstrate to the class a method to solve a particular kind of problem, and then we would all work through several problems of the same type. And that was the procedure followed in all the math classes I can remember.

I quickly figured out how to play that game successfully – success in that case being measured by my being able to solve under exam conditions, problems like the ones the teacher had shown us and we had practiced in class and done for homework. Many of my fellow students did not master that game, and fell by the (well-populated) mathematical wayside.

The technique I mastered early for succeeding in that regime lasted me all the way through to calculus and on into university. Then things changed dramatically. Most of my professors provided little more than rapid summaries of new concepts and gave minimal instructions as to how to solve problems. I had to figure it out for myself afterwards, in collaboration with my fellow students, occasionally supplemented by going to the professor for help. It was at university then, when I discovered (collaborative) learning (the long sessions with my fellow students) and the power of real teaching (the activity that took place when I sat down with my professor to get help), and both were powerful and transformative.

As far as I can tell, most people in the US (and the UK) who last took a math class at high school have never experienced good mathematics teaching. Nor have many students who went on to take math classes at college level, but were not able to sit down one-on-one or in a small-group setting with the professor, as I did. All they have ever had is instruction. They often refer to it as teaching, since that is their only model. But it isn’t teaching; to call it that is to unintentionally insult the many thousands of good teachers out there.

Instruction is primarily one-directional, from an instructor (we should not use the word teacher here) to the student. Education in the instruction mode proceeds along the lines: first provide information, then give an opportunity to practice, then test.

Many students do learn to do well in this system. Some of the ones who do well actually learn what the course is supposed to be about, though others (and I suspect most) simply learn how to pass the course tests. Case in point: I got straight A’s on all my high school calculus courses (“freshman calculus” in US terms), but only when I was a doctoral student in mathematics faced with running problem sessions for math undergraduates did I actually start to understand calculus. At school I had merely learned how to pass the tests. At graduate school, five years later, I finally learned calculus, by way of trying to teach it.

The point is, unlike instruction, which is essentially unidirectional and provides no guarantee of learning that which is ostensibly being “taught,” teaching (the real kind) is bi-directional. In fact, you can’t separate real teaching from learning. They are simply two perspectives of the same human interactive process. From the teacher’s perspective it is teaching, from the student’s perspective it is learning.

For anyone who has experienced real teaching, what I am saying is obvious. Unfortunately, someone who has not experienced it likely has no idea what I am talking about. So let me give some examples that most people are familiar with.

Compare your school math classes with learning to drive, taking tennis lessons, being taught how to ride a bicycle, being taught to play a musical instrument, or being taught how to ski or improve your golf. Unless you were being seriously ripped off or shortchanged, each of those was highly interactive, with your teacher watching your performance and guiding you toward improvement.

Along the way, your teacher almost certainly gave you some instruction. Indeed, you might have stopped the learning activity and gone into a classroom where the teacher explained something at a whiteboard, or showed a video. Teaching and learning usually involve instruction. But giving and receiving instruction no more is teaching/learning than bricklaying is architecture. One is just a part of the other. An essential part, to be sure, but still just a part.

Long after I left school, I found myself visiting math classrooms where real teaching takes place, and nothing could be more different from what I experienced. Though I have observed many different styles of good teaching, two things they all have in common is that they are highly interactive and there is learning going on at the same time, as part of the process.

The distinction between instruction and teaching/learning becomes significant when cash-strapped education districts look to technology for assistance. For whereas technology can provide instruction and can provide teachers and students with resources to assist them, what is cannot do on its own is teach them. (Whether you think that is an inherent limitation of today’s technology or a fact of nature likely depends on your view as to how far artificial intelligence can go. I stated my position in my book Goodbye Descartes way back in 1998 and it has not changed since. But let’s leave that to one side, since my focus here is on the educational world today.)

The magnitude of the problem facing any teacher is made clear when you carry out a simple test that is all too infrequently done. (Actually, although simple in concept, it is time-consuming and difficult to carry out well, which is probably why it is done so rarely.) You sit down and talk with the student to find out what she or he has learned.

You might think that this is what the end-of-the-course test does, but nothing could be further from the truth. The well known educational consultant Marilyn Burns is an expert in carrying out such tests. Take a look at the following example of the kind of thing she discovers.

Watch this video on YouTube

If you are like me, when you heard Cena’s answer in the class, you will have concluded that the young girl understood place value representation. She certainly gave the right answer, and to those of us who do understand place-value, her verbally articulated reasoning screamed out conceptual understanding. But as the subsequent interview made clear, at best she has a rudimentary understanding in a particular context, and if truth be told we really have no idea what she knows.

The fact is, the human brain is a remarkable pattern-recognizing device. It will discern a pattern – usually many patterns – in a random display of dots on a screen. But is it the “right” pattern? Cena clearly recognized some pattern. But it is not clear what it was.

This problem bedevils all of us who seek to develop educational software or technologically-delivered courseware, from the free-to-all Khan Academy and the online classes given by MIT and my own Stanford, to for-profit spinoffs like Sebastian Thrun’s Udacity. They can be good. (I am planning to give my own online Stanford class this fall, and I surely would not attempt that if I did not see value in it.) But what they can be good at is providing instruction. They don’t teach and they do not guarantee learning of the intended material. For that, you still need a teacher, and the instructional material should be a tool that the teacher actively uses.

The kinds of problem exhibited by Cena are surely less worrying for older students, particularly students who have already learned how to learn – arguably the most important goal of schooling. But still there are dangers.

For example, my fall course will include lots of video, and there are known, significant problems with video instruction, as this video (sic) shows:

Khan Academy and the Effectiveness of Science Videos

(Khan Academy, the example in this video, is in the limelight at the moment, so tends to be the example of choice, but the problem with instructional video is a general one, and will be just as pertinent to my upcoming course. I’ve asked the editor of this column to reject comments focusing on KA as being off-topic.)

It’s tempting to try to overcome the Cena-type problem by introducing an interactive component. But that too does not work, as was discovered in 1973. In what rapidly became one of the most famous and heavily studied papers in the mathematics education research literature, Stanley Erlwanger exposed the crippling limitations of what at the time was thought to be a major step forward in mathematics education: Individually Prescribed Instruction (IPI).

But already this column has gone on long enough. If you want to find out about Erlwanger’s findings, skip over to my personal blog, profkeithdevlin, where I discuss "Benny's Rules" in the context of my work on mathematics education video games.

If You Don’t Have a Web Presence, Are You Doing Your Job?

2012-02-01T11:00:00.000-05:00

“The times they are a-changing.” - Bob Dylan

“Professor, when did you last post a blog or publish a webcast aimed at teachers?” - Me

Last month I promised I would say a bit more about my collaboration with the choral group Zambra to provide musical interpretations of mathematical equations. I will definitely come back to that in a future column, but right now I want to pick up a theme that emerged in a Finnish-American education summit at Stanford last month, for which I led the home-team organization.

In particular, take a look at this brief video clip of math-teacher/blogger Dan Meyer making a point in a panel discussion at the conference. (Sorry about the low sound level. The recording was from Dan’s flip-video camera.)

For a broad overview of the summit, see the report in The Huffington Post. To see Dan’s complete blog post on the panel, which includes the full, hour-long video recording, see http://blog.mrmeyer.com/?p=12592. I’ll come back to that panel in just a moment. (And in a future column, I’ll tell you about the other person on the panel, Karim Ani, the founder of that wonderful, innovative, math-educational resource Mathalicious.) First, the summit itself.

As you might imagine, given all the attention paid to the Finnish K-12 education system in recent years, in particular their multi-year success in the OECD’s international education outcomes comparison PISA tests, there was considerable outside interest in our event, both beforehand and afterwards. Finnish education is big news, and everyone wants to know how to replicate their success. The Finns themselves, meanwhile, continue to be bemused by all the attention. For success in PISA, or any international ranking, was never on their agenda.

In fact, they will tell anyone who is prepared to listen (few policy makers in the US are, it seems) that one of the worst things you can do in education is focus on “outcomes metrics” and test performance. The Finns themselves do not. Instead, they concentrate on education. (Though Americans typically understand the word “education” in a way that would be alien to any Finn under 40 years of age, who would have been educated after Finland abandoned the “fill up the brain with facts and techniques” model we still cling to, and embarked on a complete revamp of their school system that eventually put them at the top of the international league tables that were never in their sights.)

How do I know? Together with my colleagues at Stanford’s H-STAR institute, I have collaborated for many years with Finnish education researchers, I make frequent visits to Finland, I have spent time in a Finnish school, I regularly host Finnish education scholars when they visit Stanford, and I am on the Advisory Board of their nationwide interdisciplinary education research network, CICERO Learning. (The name stands for Cross-disciplinary Initiative for Collaborative Efforts of Research on Learning. Education researchers love these acronyms.)

But, as usual, I digress. Though there was a lot to learn from the summit, I want to focus here on one particular theme. What can education researchers do to ensure that the important observations, conclusions, and discoveries that result from their research find their way to the teachers who are in the classroom every day?

The degree to which what should be a major knowledge transfer pipeline is totally fractured (or maybe it never got built) was made clear to me by the huge blog-debate that ensued a few years ago when I alluded (in passing, I should add) to scholarship stretching back over fifty years about multiplication not being repeated addition and it not being educationally wise to perpetuate such a falsity. Just Google “Devlin repeated addition” to get to the story, and then Google “repeated addition” to appreciate the enormity of the knowledge-transfer failure. (I’m never sure whether to capitalize proper names that become verbs.)

For me, the fact that there are millions of people who, rather than examine the evidence and change their position, prefer to cling to what they were taught as children, is simply a fact of life (at least in the US). I can’t do anything about it. Evolution by natural selection and global warming suffer the same fate, and we Americans seem particularly prone to this head-in-the-sands behavior.

What I will try to do, however, is encourage my fellow academics to become much more proactive in disseminating what we have learned in mathematics education over the past, oh, fifty years. There is today no shortage of ways to do this. Particularly easy are blogs and webcasts for substantive content, Twitter and Facebook for shorter alerts to the former. Sure, it takes time to build those networks. But there is an audience out there of committed teachers who are eager for all the help they can get.

Dan Meyer’s blog dy/dan, for instance, has 10,000 followers, mostly practicing teachers (I assume). Okay, that number is tiny compared to Kim Kardashian’s 12 million Twitter followers, but Dan’s 10,000 blog subscribers are all individuals who want to learn good math teaching techniques. Kim may entertain the world, but Dan and his followers want to change it. Those of us in the academy should help them. (Help Dan and the teachers, that is, not Kim, who seems to be doing just fine on her own.)

The fact is, thousands of teachers are already using social media as an extended staff common room, to learn and to exchange ideas about teaching. (In so doing, they are making up for the fact that the US does not structure teachers’ workloads to include lots of face-to-face discussion time the way the Finns do.) The result of all that internet traffic is that current best practices quickly proliferate across the Social Web.

The problem is, for the most part, what is circulating are the knowledge and the ideas already in the system, a lot of it dating from your grandparents’ day, some going back to medieval times. Much of what has been discovered in university schools of education over the past half century has not yet found its way to the people who really need it.

We academics have at our disposal the tools to change all that. And we can do it in the comfort of our homes, by going online for half an hour last thing at night, before we go to bed. It’s not just a matter of getting the information across. Credentialed academics bring to the discussion something that is sorely missing in most of the traffic on the edu blogs. Scientific evidence and validation. As the Kim Kardashian example demonstrates, number of followers is not a reliable measure of anything other than popularity.

Dan Meyer is a case in point. He clearly has a lot of influence in current US K-12 mathematics education. He reaches in excess of 10,000 teachers on a regular basis, and they in turn will interact with their own network. Much of his success comes from teachers recognizing that he has things of value to say, and that is why Google appointed him as a Teaching Fellow two years ago, why Stanford accepted him as a doctoral student in education, and why he was on that summit panel. In short, dy/dan is a GOOD THING. But on its own, 10,000 subscribers is not validation. There are a lot of other math-edu bogs out there that are not as good, and perpetuate education myths that the academic education community put to rest decades ago.

For all the ease of use, the polished look, and the reach of blogs, tweets, and webcasts, without scientific validation their content is just opinion. It may be good opinion, but it’s still opinion. What we need in education is a lot less opinion and a whole heap more empirically grounded facts. Those facts are there, but they are closeted away in university schools of education, and put on highly limited display only in articles published in scholarly journals, in books written for other academics, and discussed at academic research conferences.

Typical teachers do not have the time to even begin to sift through all of that valuable stuff. But they do have a half hour or so most evenings to skim through a few blogs and to occasionally tune in to a webcast. We academics need to be in that space, and we need to be there with a loud (but respectful and collegial)
e-voice.

To my mind, it comes down to what our obligations as academics are to the society that pays our salaries and provides us with the resources to do our work. Research is clearly one of those obligations. Teaching is another. The latter is where we need to change. In today’s world, when we can reach 10,000 as easily as 25, directing that teaching purely to the small number of students who are in our university lecture halls is no longer defensible.

For sure, there are mathematics researchers and mathematics-education researchers out there whose work is so good, it would be insane to ask them to do anything that takes them away from that work. But they are a minority in our profession. The rest of us should be evaluated differently.

Lest I leave you with the impression that I am trying to diminish the integrity or the prestige of the Academy, I’ll finish by noting that Stanford’s famed School of Engineering requires more than a strong record of publications and conference presentations in order to hire, tenure, and promote its faculty. They demand what they call impact. What that means is that every professor has to demonstrate an impact on society, say by their engineering research leading to a patent or finding its way into a product, or by their serving on a government panel and influencing policy.

Those of us in the academic world of education research should be held to a similar standard. Writing papers for one another and giving conference talks to our peers is an important part of what we do, but on its own it is not enough. If no teacher has learned of, or been influenced by, our work, why should we expect society to continue to support us? To my fellow academics, I ask: When did you last post a blog or publish a webcast aimed at teachers?

Patterns? What patterns?

2012-01-01T03:00:00.000-05:00

In the early 1990s I deliberately set out to help create a meme: mathematics is the science of patterns. My inspiration was an article written by Lynn Arthur Steen, published in SCIENCE Vol. 240 No. 4852, 29 April 1988, pp. 611-616. Steen’s piece was a high-level overview of mathematics as practiced by late twentieth century pure mathematicians, titled simply “The Science of Patterns.” In 1994, I published a full color book in W. H. Freeman’s prestigious Scientific American Library series with the title Mathematics: The Science of Patterns.

Both Steen and I were motivated by a desire to see major improvements in mathematics education, and (a closely related issue) an improved image and understanding of mathematics in American society. (This was long before the stakes were raised by the GOP declaring war on science.) But neither of us claimed originality to the view of mathematics captured by that catchy phrase. Back in 1940, the accomplished English mathematician G. H. Hardy wrote, in his book A Mathematicians Apology:

The mathematician's patterns, like the painter's or the poet's, must be beautiful,the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics. … It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind – we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.

The beauty to which Hardy was referring is, for the most part, a highly abstract, inner beauty, a beauty of abstract form and logical structure, a beauty that can be observed, and appreciated, only by those sufficiently well trained in the discipline. It is a beauty “cold and austere,” according to Bertrand Russell, the famous English mathematician and philosopher, who wrote, in his 1918 book Mysticism and Logic:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty – abeauty cold and austere, like that of sculpture, without appeal to any part ofour weaker nature, without the gorgeous trappings of painting or music, yetsublimely pure, and capable of a stern perfection such as only the greatest artcan show.

But I digress – as I often do when I reflect on the natural beauty of mathematics, a beauty more fundamental than that of the best poetry or art, the latter capturing the beauty in life and the human cognitive system’s response to its existence and environment, mathematics providing a glimpse of the even deeper beauty of the universe itself, the very substrate on which we and everything we know exist.

The one obvious drawback with the “science of patterns” meme is, and always was, that it does not stand up to reflection. Not because it misses the target. Rather, it is, like many memes, altogether too general. It only captures mathematics for someone who already knows what mathematics is; it does not serve as an explanatory definition. “What kinds of pattern?” is the obvious initial response from someone not well versed in mathematics who meets our meme for the first time. Practically any science (including the life, human, and social sciences) can be described as “the science of patterns of type X” for a suitable X.

Of course, both Steen and I had the answer at the ready. His entire SCIENCE article was devoted to providing examples of the kinds of pattern whose scientific study constitutes mathematics, as was my considerably longer book. Our hope was that that catchy phrase would provoke curiosity to read our explanations and learn what it was we were trying to convey.

If I were asked to provide a “dictionary definition” of mathematics as it is practiced today, I would come up with something along the following lines.
Mathematics is:

The systematic study of number, shape, form, structure, relations, motion and change, and other concepts, represented by precisely defined abstractions; and
the development and application of procedures for reasoning with those concepts; and
the use of rigorous logical argument to determine truth on the basis of accepted assumptions (axioms); and
the application of that methodology to the real world.

More accurate than our meme to be sure; but nothing like as catchy. And decidedly not destined to become a meme.

Still, accuracy aside, thinking of mathematics in terms of patterns is far more reflective of the bulk of contemporary mathematics than is the computational-centric view of the subject that still seems the dominant one in society at large. It also leads naturally to comparisons with other “pattern-delineated” human activities. Music for example.

With its high degree of abstraction and its formal, symbolic language, music has long been one of my favorite examples when trying to explain mathematics – and its attraction – to non-mathematicians. Yet like all good comparisons, music and mathematics serve to illuminate each other by their differences as much as their similarities.

The similarities and the differences between mathematics and music motivated me to collaborate with a choral group (Zambra) a few years ago. The idea was that we should work together to find musical interpretations of some of my favorite mathematical equations. The result was a stage show called Harmonius Equations. We started with – no prizes for guessing the right answer here – Euler’s Identity. Here is the result:

Next month I’ll take a look at the rest of that collaboration.

Christmas Trees from the Land of Santa Claus

2011-12-01T03:00:00.000-05:00

I’m writing this month’s column from the Arctic Circle, Rovaniemi in Finland to be precise, home of the University of Lapland, where I am visiting to give a lecture, and home too for the real Santa Claus, or so they say around these parts. Surrounded by trees as far as the eye can see - which is not very far at this time of the year when the sun never rises above the horizon - my thoughts turned to trees even taller than those in the Finnish forests. Infinite trees that can be found only in the conceptual forest where mathematicians tread.

This brief foray into the fascinating world of the infinite extends my discussion last month of the Recursion Principle, and shows not only that infinity has a habit of surprising us, but even seemingly “obvious” constructions and proofs involving infinity can require powerful axioms, in the case of this month’s column not only the Recursion Principle again, but a principle far better known to non-professional mathematics aficionados called the Axiom of Choice.

So fasten your seat belt and prepare for a wild ride into the world of infinite trees.

To the mathematician, a tree is a well-founded, partially ordered set such that the set of all predecessors of any element is totally ordered, having a unique minimal element (called the root of the tree). Well-founded means that every nonempty subset has a minimal element, so there can be no infinite chains going downwards.

At first encounter that formal definition looks pretty complicated, but the notion is a pretty simple one that a diagram quickly makes clearer. A tree typically looks like this.

In terms of its basic structure, this is not unlike the trees that surround me here in Lapland.

To keep the diagram simple, I have drawn only a few nodes (the points). The partial order is represented by the lines that connect nodes. Each node can have any number of immediate successors in the tree (i.e., nodes immediately above it in the partial ordering), possibly an infinite number. Some nodes may have no immediate successors (and hence no successors at all).

A totally ordered path through the tree that follows the connecting lines is called a branch. Because nodes can have more than one immediate successor, a branch going up through the tree can at some node split into two or more separate branches. But going in the opposite direction, downwards, from a node there is exactly one branch, leading all the way down to the root of the tree. Thus, when you climb a tree, you have opportunities to “branch out”, like Robert Frost in his famous walk through the wood, but there is only one way to get back down to the bottom.)

Mathematical trees arise in studies of family descendancy in history or genetics, such as the tree that represents all women and their female offspring, though in those studies the trees are conventionally drawn growing downwards, with their roots at the top.

Since the unique branch that leads up to a given node is a well-ordered set (i.e., totally ordered and well-founded), each node can be assigned a unique “height” in the tree. The root has height 0, all immediate successors of the root have height 1, all their immediate successors height 2, etc. All the nodes having a specific height N in the tree are said to constitute the N’th “level” of the tree.

Trees that arise in history or genetics are finite, and hence there will be one or more nodes that have maximum height in the tree and constitute a “top level”, but mathematicians consider infinite trees. When they started to do that, they encountered a number of surprises, as is often the case with the infinite.

One initial curiosity is that whereas infinite trees can have infinite paths going upwards, heading in different directions, all paths downwards are finite and end at the root. (A path, whether it goes upward or downward, has to be specified in terms of step-by-step instructions: start here, then go there, then go there, etc., possibly into the transfinite. More specifically, you need to specify a function from an ordinal number into the tree.)

But a far bigger surprise came from trying to generalize a fairly simple result about infinite trees called Koenig’s Lemma: an infinite tree, all of whose levels are finite, must have an infinite branch (i.e., a path that goes all the way to the top of the tree - though even there you have to be careful, since an infinite tree does not necessarily have a top level).

Proving Koenig’s Lemma is fairly straightforward. You apply the Recursion Principle, which I discussed in last month’s column. Using that principle, you define an infinite branch by recursion. Call the branch you will construct B. To determine B, you specify for each natural number N the node in B on level N, and you do that in such a way that B(N) always has infinitely many nodes above it. B(0) is the root of the tree. With B(N) defined, for B(N+1) you take any immediate successor of B(N) that itself has infinitely many successors.

A simple induction proof shows that for every N, with B(N) suitably defined there always will be at least one node above it on level N+1 that has infinitely many nodes above it, so the above recursive definition works. Here is that proof.

Since the tree is infinite, there are infinitely many nodes above the root. Level 1 is finite, so at least one node on level 1 must have infinitely many nodes above it. (Because a finite union of finite sets is finite.) Likewise, if the node B(N) has infinitely many nodes above it, then because level N+1 is finite, at least one node above B(N) on that level must also have infinitely many nodes above it, and can be taken for B(N+1). So the recursive definition is sound.

But note that this process of defining an infinite branch assumes that it is possible to make infinitely many choices. (At each stage N we choose for B(N+1) one node above B(N) on level N+1 that has infinitely many nodes above it.) Unless the tree has some additional structure that provides a uniform means of making this selection, this requires a basic assumption about infinite sets called the Axiom of Choice. Though the Axiom of Choice in its full generality has been the subject of much discussion in the history of mathematics regarding its assumption as an axiom for mathematics, the simple version used for proving Koenig’s Lemma, called the Axiom of Dependent Choices, has generally been spared most of the attention.

Most people find Koenig’s Lemma intuitively obvious and accept the proof I just sketched. The surprise comes when you try to generalize it. The most obvious generalization is to uncountable trees all of whose levels are countable. (An infinite set is called uncountable is it cannot be put into a one-to-one correspondence with the natural numbers. In the 19th century, Georg Cantor famously proved that the real numbers constitute an uncountable set.)

An uncountable tree all of whose levels are countable has to have an uncountable number of levels, so the natural numbers are not sufficient to number them; you need the transfinite ordinal numbers, which extend counting beyond the natural numbers. But apart from that tweak, at first encounter most people feel sure they can generalize the proof of Koenig’s Lemma to show that an uncountable tree all of whose levels are countable must have an uncountable branch. (Again, this will require the Axiom of Choice, this time to make an uncountably-infinite number of selections.)

That was definitely my reaction when I came across this question as a first-year graduate student in set theory. I was therefore surprised when I sat down and tried to sketch the proof and ran into an unexpected obstacle at limit ordinals. Those are transfinite ordinals that do not have an immediate predecessor, a phenomenon that does not arise for the finite ordinals (aka the natural numbers together with 0).

My difficulty turned out to be more than my inability to make the proof work. The generalization is false. In 1934, a Polish-American mathematician called Nachman Aronszajn constructed a specific uncountable tree that has only countable levels yet has no uncountable branch.

For anyone who feels comfortable with transfinite recursion (i.e., recursion that allows you to define functions whose domain extends into the transfinite ordinals), it is easy to work out Aronszajn’s construction if I give you a couple of clues. A cryptic clue, that will help get you started but won’t spoil your enjoyment when you get the construction yourself, is that it is important to think rationally. With that, you should give it a try.

STOP READING NOW IF YOU WANT TO TRY IT ENTIRELY ON YOUR OWN.

Here are some more extensive clues to Aronszajn’s construction. The tree will have an N’th level for every countable ordinal N. The nodes on level N will be strictly increasing, bounded N-sequences of positive rational numbers. You define the tree by transfinite recursion (using the Axiom of Choice as well as the Principle of Transfinite Recursion). To ensure that the recursion works, you give every node countable-infinitely many immediate successors, and do it so as to preserve the condition that for every level N, for every N-sequence S of rationals on level N, for every countable ordinal M above N, and for every positive rational Q greater than the supremum of S, there is an M-sequence on level M that extends S, whose supremum is less than Q. From here on it’s all your’s. Enjoy!

How multiplication is really defined in Peano arithmetic

2011-11-01T15:00:00.000-04:00

A familiar joke in mathematical logic (the field I worked in for the first twenty years of my career) is the following hypothetical entry in a mathematical dictionary:

Recursion
See “Recursion”.

Though recursion is ubiquitous in modern mathematics, even at the most basic level of the analysis of the arithmetic of natural numbers, it is a subtle concept, easily misunderstood. At its heart is the always problematic step from the finite to the infinite. Having taught set theory and the foundations of mathematics for many years at various universities, I long ago learned that many students never do fully grasp it.

Many sources gloss over it. For example, the Wikipedia entry for the Peano Axioms for the natural numbers, which on the whole is pretty good, refers to “recursion” to justify its definitions of addition and multiplication on the natural numbers, but the Wikipedia page it links to punts when it comes to properly explaining the all important Recursion Principle, let alone proving it (the set-theoretic Recursion Theorem), giving a sketch proof of uniqueness (easy) but not the crucial existence part (which many people find tricky). [My comments refer to the pages when I accessed them on October 31, 2011.]

The widespread lack of appreciation for the Recursion Principle (it’s a principle you use when you are developing Peano arithmetic, a theorem you prove when you are doing set theory) lay behind many of the erroneous comments I received in response to my series of articles on multiplication not being repeated addition (not even on the natural numbers). Those articles appeared on MAA Online in June 2008, July-August 2008, September 2008, and January 2011.

If you really understand the Recursion Principle, which is what is required in order to construct addition from the successor function and to define multiplication from addition, then you will know that addition is not repeated successor (though oddly, no one ever claimed that to be the case) and multiplication is not repeated addition.

ASIDE: This is not an article about mathematics education. How teachers introduce addition and multiplication is another issue. I am not, repeat not, suggesting we teach recursion in the K-12 system. My complaint throughout that previous exchange was that, whatever and however we teach in our schools, we should not be stating things that are flat wrong. In this column I want to explain the recursion principle, its importance, its power, its need in mathematics, and how it is used, so you too will know why it is flat wrong to tell kids that multiplication is repeated addition. Remember, even if you don’t see the purpose of all the mathematical navel-inspection that led to the formulation of the principle, some of the students in your class may well find themselves wanting or needing to understand it in a few years time. While it is often pedagogically wise to say less than the whole truth, we should not lie to our students.

Okay, I’m back off my pulpit. The Recursion Principle is a rabbit-out-of-the-hat existence assertion. When used to construct addition from the successor function in Peano arithmetic, it tells you that there is a function from NxN to N that, lo-and-behold, can in any specific instance (i.e., for any pair of specific natural number arguments) be calculated by repeatedly applying the successor function a fixed number of times. Likewise, when the recursion principle is used to construct multiplication from addition in Peano arithmetic, it tells you that there is a function from NxN to N that, again wonder-of-wonders, can in any specific instance (i.e., for any pair of specific natural number arguments) be calculated by repeatedly adding a fixed number of times.

You need to invoke the Recursion Principle in order to obtain addition and multiplication because without it you just don’t get those functions. Period.

Let’s see how the Wikipedia entry I referred to earlier handles the Peano constructions of addition and multiplication.

(click image for full size)

If you read that Wikipedia account carefully, you will see that it does not actually tell you how to define addition or multiplication. Rather it tells you the properties those functions will have once they have been defined. The closest you get to an indication of how the two functions are defined is that word “recursively”, which as I noted earlier, links to another Wikipedia page that also does not tell you how the function is defined.

The definition is actually not difficult, provided you are used to working in abstract set theory. But since most people are not, authors typically leave it out, an instance where saying less than the whole truth is, in my view, justifiable. (Though I think they should explicitly note that what is being missed out is fundamental and important.)

Let’s re-write the Wikipedia definitions of addition and multiplication in standard functional notation. Thus, addition is a function P:NxN -> N such that for all numbers a, b,

1. P(a,0) = a

2. P(a,S(b)) = S(P(a,b))

and multiplication is a function M:NxN -> N such that

1. M(a,0) = 0

2. M(a,S(b)) = P(a,M(a,b))

The Recursion Principle guarantees that such functions exist. In general form, the principle says (and this is the version for binary functions over N):

Given a function H:NxN -> N and a number c, there is a function F:NxN -> N such that

1. F(a,0) = c

2. F(a,S(b)) = H(a,F(a,b))

In words, 2 says that for any given first argument a, the value of F where the second argument is the successor of b is defined in terms of the value of F where it is b. The function H tells you precisely how those two values are related.

Actually, the Recursion Principle also guarantees that the function F is unique. But that part is easy to prove from the two conditions, using induction. (Wikipedia sketches the proof.)

There are variants of the principle for unary functions and for n-ary functions for all other n, as well as for other domains besides N.

Here is how the function F is defined within set theory. Remember, in set theory, a function from NxN to N can be defined as a set of ordered triples (x,y,z) of numbers such that for each pair x, y of numbers there is exactly one number z such that the triple (x,y,z) is in the set.

Let W be the family of all sets of ordered triples T of numbers such that

1. (a, 0, c) is in T, for all numbers a

2. if (a, x, y) is in T, then (a, S(x), H(a,y)) is in T.

Then let F be the intersection of the family W. It is a routine exercise in set theory to prove that F is the required function. (That last sentence means that a mathematician having some familiarity with set theory will find it routine. It is a typical early exercise in an undergraduate course in set theory.)

So now you know.

This may all seem like a great deal of fuss about nothing. But what is going on here is really very deep. Much of modern mathematics involves finding ways to handle the infinite - calculus exclusively and spectacularly so. Mathematicians learned over many years of painful lessons that the step from the finite to the infinite is a tricky one that requires considerable finesse. In particular, you have to exercise great care to set it up correctly and do it right. The Recursion Theorem is one of those crucial bridges that allow us to go beyond the finite to the infinite, to extend human intellect from its finite physical limitations to the infinite world beyond that our minds can construct. By getting the mathematics right, we can make that step with total confidence. Confidence both in that abstract world itself and in the concrete conclusions it allows us to reach about our lives, our science, and our technologies. That is huge for Humankind. To say “multiplication is repeated addition” is like saying “differentiation is division”; for both boil down to saying that the infinite is no big deal, little different from the finite. That’s not just a lie, it is to deny two thousand years of human progress.

Mathematics: A Recyclable Tool for the Modern Era

2011-10-03T03:00:00.000-04:00

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.

The First Arithmetic Textbook in the Western World

2011-09-01T03:00:00.000-04:00

Image courtesy of the Riccardiana Library in Florence.

The image above shows the initial page of the first arithmetic textbook in the western world. Written (in vernacular Italian) around 1290 CE by an unknown author in Umbria, it begins with the declaration: “This is the book of abacus according to the opinion of master Leonardo of the house of sons of Bonacie from Pisa.”

Wait a minute? Doesn’t everyone know that Leonardo of Pisa’s Liber abbaci was the first arithmetic textbook? Isn’t that the story I tell in my recent book The Man of Numbers: Fibonacci’s Arithmetic Revolution?

Not quite. Liber abbaci, completed in 1202, was the first comprehensive description of modern arithmetic (using the Hindu-Arabic numerals, decimal place-value representations of numbers, and the basic arithmetic algorithms we all learn in school) in the western world. But it was not really a textbook. Its mammoth length and the fact that it was written in Latin made it accessible only to scholars.

Smart man that he was, Leonardo recognized that, and wrote a shorter, simpler book, aimed at merchants and businessmen. Unfortunately, though copies of most of Leonardo’s works survived (no original Leonardo manuscript did), there are no known copies of his smaller book for merchants. We know he wrote it because he refers to it in Liber abbaci, calling it his liber minoris guise (“book in a smaller manner”), and because other authors referred to it. Those other references suggest that it most likely comprised material from the first ten chapters of Liber abbaci together with parts of one of Leonardo’s other books, De Practica Geometriae, but exactly what its contents were, no one knew.

Until 2003, that is, when mathematics historian Rafaella Franci, who directs the Center for the Study of Medieval Mathematics at the University of Siena, published the results of a remarkable study of a manuscript she came across in the Biblioteca Riccardiana in Florence. The manuscript is anonymous, and occupies pages 1 to 178 of the library’s Codex 2404. The work itself is undated, but dates in some of the problems place its writing at around 1290, and the vernacular language used places it in the Umbria region.

The book (known as Livero de l’abbecho — “Book of calculation” — from the anonymous author’s introduction) is divided into thirty-one short chapters. It is not an original work. Roughly three-quarters of the problems are faithful translations into the vernacular of problems in Liber Abbaci. Of particular note to a modern reader, it includes Fibonacci’s famous rabbits problem, though recast in terms of pigeons. The book was clearly written for a wide audience, since the author began each problem type with simpler problems than those in Liber abbaci.

Could this be a copy of Leonardo’s lost “book for merchants”? Or did the unknown author get the material from Liber abbaci? Since the writer gives no indication of having any particular mathematical skill, it seemed unlikely he could have carried out the difficult task of simplifying the sophisticated descriptions found in Liber abbaci. Moreover, his text included material on geometry that is hardly mentioned in Liber abbaci. It could, however, have come from De Practica Geometriae.

Even more intriguing, the Umbrian’s book had three chapters on calculating interest and depreciation, a topic not covered in Liber abbaci at all. This material was obviously included to make the treatise more useful to merchant readers, but where did it come from?

So here we have a manuscript written around 1290 that provides a much simplified account of the novel, and for the times highly sophisticated mathematics in two of Leonardo’s long, dense scholarly texts, coupled with some novel material on financial matters. The author of the manuscript gives no indication of having any particular mathematical ability, so he must have simply copied it from a single source. It is, in short, a medieval equivalent of a photocopy. But a photocopy of what?

There was only one mathematician at that time who had the ability to write the photocopy’s source: Leonardo. Which means that the manuscript in the Riccardiana Library is not only the oldest known textbook on practical arithmetic in the western world, it must in fact be a copy of Leonardo’s lost “book for merchants. A conclusion that has been confirmed by other studies since Franci reported her findings.

For more details, see my recent book The Man of Numbers and the companion e-book Leonardo and Steve, (where I compare Leonardo with Steve Jobs, a man of our age very much in the news of late).

The First Personal Computing Revolution

2011-08-01T03:00:00.001-04:00

The young man could hardly contain his excitement. He was sure the invention he had just seen could change the world. It would usher in a new era of personal computing. No longer would a businessman or trader have to rely on a member of the select brotherhood of computing professionals to crunch the numbers. He could do it himself.

The people who showed him the invention were fascinated by how it worked, but they clearly did not see what the young man could: its huge commercial potential. As so often happens in history, the right person was in the right place at the right time. Not only had the young man shown mathematical talent at an early age, he had grown up in what was then the acknowledged world capital for innovation, particularly in the business world. He also had the savvy to know how to make the invention available to ordinary citizens. The trick was to package and market it to them directly, in a way that they could at once appreciate and understand.

Within a few years, the young man had succeeded beyond all expectations - save perhaps for his own far-sighted vision. The personal computing revolution was underway, new businesses were being created, new ways of carrying out international trade and commerce were developing, new financial institutions were being established, and new fortunes were being made. The world would never be the same again.

To readers familiar with the story behind the development of the Macintosh computer, the above sounds like a description of Steve Jobs' legendary visit to the Xerox Palo Alto Research Center (PARC) in December 1979. Working in secret for several years, PARC's hundred or so computer engineers and designers had been developing most of the components of modern personal computing, including the windows-mouse-pointer system for displaying files and navigating through the computer's memory, the Ethernet, the laser printer, and a concept called object-oriented programming that made such systems possible. Although Xerox, the parent company, was unable to see the commercial value of what their money had bought, Jobs certainly did. Four years later, Apple released the Macintosh computer, a consumer implementation of PARC's system, followed soon afterwards by the Apple LaserWriter, that made another PARC invention, the laser printer, into a successful consumer product. The modern era of easy-to-use, personal computing was born.

But the young man I was referring to lived eight hundred years earlier, in medieval Italy. The personal computing revolution that began in Silicon Valley in the 1970s and 80s was actually the second the world had seen. The man who started the world's first personal computing revolution lived in the Italian city of Pisa from around 1170 to maybe 1250. His name was Leonardo Pisano (Leonardo of Pisa), but he is better known today by a nickname given to him by a historian in the early nineteenth century: Fibonacci, a name derived from the Latin filius Bonacci, which translates literally as "son of Bonacci".

Most people associate his name with the "Fibonacci numbers," a sequence generated by starting with the pair 1, 1 and obtaining each successive number by adding together the previous two: 1, 1, 2 (=1+1), 3 (=1+2), 5 (=2+3), 8, 13, 21, 34, 53, 87, etc. These numbers arise frequently when you count things in plants (such as leaves, petals, spirals), for which there is a scientific explanation, and are believed by some to be of use predicting the behavior of the financial markets (which you have to take as a matter of faith). The sequence's connection with Leonardo is tenuous to say the least. It was known long before he was born, and there is no evidence Leonardo had any interest in it whatsoever. In his arithmetic book Liber abbaci, published in 1202, one of the many hundreds of problems he gave required the reader to generate the sequence to get the answer, and that one connection led a French mathematician to name them the Fibonacci numbers in the 1870s. That is all there is to it.

But Leonardo has a far more substantial claim to fame than the number sequence now named after him: he started the first personal computing revolution and thereby launched the modern financial and commercial world.

"Ah," I can almost hear you sigh, "Devlin is going to describe how Leonardo's Liber abbaci introduced Hindu-Arabic arithmetic into Western Europe."

Well, sort of. But the story is far more complex than you may have heard. We know from the introduction Leonardo wrote in Liber abbaci that, as a young man, he traveled to the North African trading port of Bugia, where he saw Arabic speaking traders using a strange new method to carry out the calculations to complete their trades. We also know that upon his return to Pisa he did write his book Liber abbaci (the double-b indicating that it was the "Book of Calculating", not the "book of the abacus"). Moreover, the book did describe the powerful and efficient Hindu-Arabic arithmetic. So far, this sounds just like the Steve Jobs - Xerox PARC episode, with Leonardo's Liber abbaci playing the role Apple's Macintosh computer subsequently would. But then the story becomes murky.

Though Liber abbaci was undoubtedly one of the first descriptions of Hindu-Arabic arithmetic in western Europe, and though the decades that followed saw the fairly rapid dissemination of the new methods, with hundreds of supposedly derivative arithmetic texts being written, the problem that always faced anyone tempted to read causality into this sequence of events was that there were no records showing how the material passed from the parchment leaves of Leonardo's manuscript onto the pages of those subsequent texts.

In fact, there is good reason to believe that hardly anyone ever read Liber abbaci. For, though writers of the subsequent texts copied passages (and often entire books) from one another, virtually none of those texts had any passages in common with Leonardo's much more scholarly tome - that in an era where all books were handwritten and where it was accepted practice for writers to copy freely from the works of others.

That tantalizing (and frustrating) puzzle was finally solved in 2003, when the close examination of a hitherto ignored thirteenth century manuscript in a library in Florence provided the final, crucial step of a two-hundred-year train of painstaking archival forensics work that stretched back to the end of the eighteenth century. As a result, we now know that Leonardo of Pisa played a role in the development of today's world every bit as great as Copernicus or Galileo. Indeed, it turns out that the parallels between the introduction of Hindu-Arabic arithmetic to western Europe in thirteenth century Italy and the introduction of modern personal computing (as an easy-to-use consumer product) in 1980s California are remarkable, down to fine details.

I'll give you one of those fine details. When Jobs left PARC, the first computer he developed having a windows-mouse-pointer interface was called the Lisa, but it was too large and too expensive to be a successful consumer product, and he abandoned it for the smaller, cheaper Macintosh he developed next. Likewise, Leonardo realized that Liber abbaci was too big and contained too much heavy-duty mathematics to be useful to the commercial men in Italy, and so he wrote a smaller, cheaper, simpler version. Unfortunately, though copies of Liber abbaci survived, there were no known extant copies of his smaller book. Until, that is, the start of the twenty-first century, when University of Siena mathematical historian Rafaella Franci opened the pages of Codex 2404 in the Riccardiana Library in Florence.

Next month, I'll tell you about (and show you some pages from) that recently discovered manuscript, and explain how it allowed historians to piece together what is probably the greatest episode in the development of modern society you never heard of.

In the meantime, I talked with NPR host Scott Simon about Leonardo and his book Liber abbaci, in my Math Guy slot on Weekend Edition (July 16), and you can learn more of his story there.

This month's column is adapted from the introduction to my recent e-book Leonardo and Steve: The Young Genius Who Beat Apple to Market by 800 Years. In that book (known in the publishing world as a "short" - it's under 15,000 words) I draw out the many parallels, some of them uncanny, between the two personal computing revolutions.

NOTE: The above collage (which I put together myself from publicly available sources) includes a page from a thirteenth century manuscript copy of Liber abbaci, courtesy of the Siena Public Library in Italy. The line drawing, from an etching of unknown but relatively recent origin, is purportedly of Leonardo, but widely believed to be a work of fiction.