Abstract versus concrete in mathematics education

In the New York Times, April 25 2008 Kenneth Chang wrote an article reporting on research done by Jennifer Kaminski, Vladimir Sloutsky, and Andrew Heckler at Ohio State University and reported in Science. I've quoted the core of that article below.

“The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.”
…
Though the experiment tested college students, the researchers suggested that their findings might also be true for math education in elementary through high school, the subject of decades of debates about the best teaching methods.
In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.
Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. “We told students you can use the knowledge you just acquired to figure out these rules of the game,” Dr. Kaminski said.
The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.
The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

I find the results of this study partly obvious and partly misleading. It is very obvious that when trying to teach a concept, giving examples with too many extraneous details will confuse the student. Imagine trying to devise checkers strategy when each of your checkers was a different shape. It would be very hard to avoid coming up with strategies that had nothing to do with the rules of the game, like: "a square checker should always jump over a triangular checker". Some degree of abstraction is necessary in order to allow the transfer of knowledge from one domain to another.

The study was done with undergraduate college students. I find it hard to believe that the researchers will replicate their results with children in grades K – 12, given what is known about the differences in learning with age. I also question the usefulness to practice of studies that report the statistical distribution of learning results from two teaching methods (say an "abstract" method and a "concrete" method). It is well established that different students have different learning styles, and that these styles may be determined, either by testing or informally by trained teachers. A teaching method that works well for a student with a strong verbal-procedural learning style may work poorly for one who has a visual-kinesthetic learning style. If one style is predominant in a population and the abstract style works better for that style, the results will be better on average, but it might be a great mistake to teach all children using that method, in effect writing off the students with the minority style.

The system the researchers used is isomorphic to the cyclic group with 3 elements, or addition modulo 3. There was a concrete model where the elements were measuring cups filled 1/3 full, 2/3 full, and 3/3 full, which I found the easiest to understand. The abstract symbolic model where the elements were circles, diamonds, and an irregular figure seemed much more mysterious and indeed Richard Weiss pointed out to me that it is possible to construct two non-isomorphic addition tables that fit the abstract model. See the New York Times article for more detail.

One reason given by proponents of multiple concrete representations is motivation, related to the idea that students must see mathematics as relevant to their lives before they will invest effort in learning it. I've always felt this is dubious, because in most textbooks or classrooms the supposedly real-life scenarios seem cooked up. I feel there are two ways to enhance motivation through curriculum design: (1) Go all out. Get students interested in some activity that excites them and requires real mathematics, such as the design of computer games, building a robot or a racecar, etc. The problem with this approach is that it fails to address the testing mania which grips our country. Students who learn mathematics this way are going to acquire skills in a non-standard order, and will not know some topics that they need to pass high-stakes tests. (2) Teach mathematics through problem solving. Eliminate, as much as possible, phony word problems. Instead tap children's innate curiosity and competitiveness with questions like: "Can 2467432 be a perfect square (no calculators allowed)?"

The King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker & Co, 2006

This biography of one of the greatest geometers of the twentieth century was a mixed bag for me. Donald Coxeter was a fascinating and brilliant man, and a list of his correspondents and admirers is a veritable who's who of twentieth century mathematicians, artists, and scientists. Donald was long-lived, even for a mathematician. He lived from 1907 – 2003, and was mathematically creative until nearly the end. A man of this rich creativity presents a formidable challenge to a biographer; how to sketch the trajectory of his life while neither getting bogged down in details on the one hand, or being superficial on the other.
A search on Amazon reveals that this book is the first published by Ms. Roberts, and the book definitely has some rough edges. I feel she needed more editorial guidance than she received. Ms. Roberts, who is not a mathematician, has apparently interviewed a number of Coxeter's colleagues at great length, and she has let them do the talking. As a result, I feel the absence of a strong author's voice, and a resulting fragmented picture of Coxeter. I found the book hard to plow through in places, which really surprised me given the strong interest that I have in Coxeter's work.
The flip side of this is that it is wonderful to read accounts from such luminaries as John Conway, Walter Whitely, Freeman Dyson, M. C. Escher (through his son, George), Buckminster Fuller and Douglas Hofstadter (who wrote the forward).
I found the glimpses of Coxeter's life outside of mathematics to be quite fascinating. He was a pacifist and a vegetarian, and seems to have been highly regarded as a person by most who knew him. His early life included problems with a withdrawn mother and an overbearing father, and a brief psychoanalysis by Freud's student, Stekel. I would have liked to have been told more about Coxeter the man.
Perhaps the best compliment I can pay the book is that I came away from it wishing that I had known Coxeter, and determined to read more of his work.

The Poincaré Conjecture: In Search of the Shape of the Universe, Donal O’Shea, Walker & Company, 2007

I recommend this book highly. It explains the history, context, and importance of the Poincaré conjecture, as well as many of the attempts to solve the problem, culminating in the successful solution by Grigory Perelman. The writing style is lively, and the explanations seem to be about as clear as possible, given the complexity of the mathematics, the amount of material covered, and the relatively short length of the book (200 pages + 75 pages of footnotes and appendices). The book is semi-popular; while the target audience is probably people with an undergraduate degree in mathematics or the equivalent, there is enough historical and personal narrative to appeal, I would think, to the non-mathematical reader. The book ends with a page-long appreciation of the civilizations and people who erected an edifice of thought that culminated in the statement and proof of this conjecture, which makes a major contribution to the understanding of the universe in which we live.

O’Shea shows how the history Poincaré conjecture is the history of much great mathematics, extending from ancient to modern times, and including such intellectual giants as Euclid, Gauss, Riemann, and Poincaré, and such stellar contemporary mathematicians as Milnor, Smale, Freedman, Donaldson, Thurston, Yau, and Hamilton.

For me, a major side benefit of reading this book was its recommendation of Jeffery Weeks’ book, The Shape of Space, which explains many of the big ideas of modern topology and geometry while demanding little mathematical background, only a willingness to think hard and work at visualization.

My Philosophy of Mathematics

Inspired by Where Mathematics Comes From by Lakoff and Nunez, reviewed below, I am setting forth my own philosophy of mathematics. It is probably not original; in fact it seems like common sense. It also seems a bit like Kant’s philosophy of ontology, if I understand Kant. Before presenting my own ideas, I’ll summarize the major philosophies of mathematics prevalent today.

Lakoff and Nunez discuss the three major philosophies of mathematics prevalent today and put forth their own, called the philosophy of embodied mathematics. To me, these philosophies are rather like the proverbial blind men trying to describe an elephant. The man touching the trunk says an elephant is like a fire hose. The man touching the tail says the elephant is like a rope. The one touching a leg says an elephant is like a tree. Each blind man correctly describes a part of the whole.

The philosophy of formalism states that mathematics is the production of valid formulas that follow from a small number of axioms. A proponent, Bertrand Russell, defined mathematics as “the subject where we never know what we are talking about, nor whether what we are saying is true.” For formalists, mathematics exists independent of meaning. This seems preposterous on the face of it, and I doubt that many working mathematicians hold it to be true. However, it contains a grain of truth in that I (or any mathematician that I can think of) would be disinclined to accept as true something that could not, in principle, be given a formal proof. That Godel has shown that there are propositions about arithmetic that cannot be proven true or false does not invalidate this.

The Platonic philosophy of mathematics, or Platonism, states that mathematics is concerned with the discovery of truths about a realm of abstract mathematical ideas, and that this realm has an objective reality outside of any human minds. This is the philosophy that most working mathematicians intuitively have. The deeper one enters into a mathematical subject matter, the more it seems to have an objective reality. However, this view has been pretty well torn apart by the philosophers. Many of the arguments against Platonism are well summarized in Where Mathematics Comes From and elsewhere, and the gist of them seems to be the impossibility of a physical being knowing a non-physical reality. My philosophy of mathematics, given below, contains a modified form of Platonism, which I think meets the objections.

The post-modernist explanation of mathematics states that mathematics, like all systems of ideas, is purely a product of the culture in which it arises. In its extreme form, this philosophy is not only wrong, it is pernicious. If there is no objective reality and truth is what society wishes it to be, then we are in the world of Orwell’s 1984, where Winston Smith’s reeducation by the state is complete when he is willing to believe that two plus two is five. Extreme postmodernism is the philosophy of totalitarianism, and it seems to me that a professor who believes this theory is in the wrong line of business. That said, a moderate form of postmodernism does illuminate mathematical thought. Mathematicians, like everyone else, are products of their culture. The areas of mathematics that are deemed important and the methods of proof that are accepted are determined, in large part, by culture.

The philosophy of embodied mathematics has its own problems, and also supplies valid insights. This philosophy states that mathematics does not have an objective reality but is not totally culturally determined either. Instead, “Mathematics is the product of human beings. It uses the very limited and constrained resources of human biology and is shaped by the nature of our brains, our bodies, or conceptual systems, and the concerns of human societies and cultures.” Lakoff and Nunez take special aim at Platonism, which is what they see at the center of an elitist “romance of mathematics”.

I think that mathematics is more than a product of human beings. It can only be understood as a blend of internal and external reality. Let me offer an analogy. Imagine we are in a forest, and I (pointing) say “this is a tree” and “that is a tree”. Most people would agree that the trees are actually there, but I am saying much more. I am saying that both objects I am pointing to are instances of the same kind of entity, namely “tree”. So the concept of tree is rooted both in the external entities (which are apparent to any sentient beings, not just humans) and internal concepts and abilities, including language and the idea of a category of “tree”. In the same way, mathematics is implicit in the regularities of the universe but must be made explicit by human thought. It is both a feature of the universe and a product of our minds.

There is no way to prove this, but it would be hard to imagine that an alien that was intelligent in a way that would enable it to talk with us would not believe that 2 + 2 = 4. On the other hand, imagine that this alien, like Saint-Exupery’s Little Prince, lived on a very small spherical planet. Such an alien might have no idea of Euclidean geometry, because that would be irrelevant to its environment. The ratio of the circumference to the diameter of a circle (C/d) would not be pi, but rather it would be variable. A circle drawn around the equator would have C/d = 2, and progressively smaller concentric circles in the northern hemisphere would have ratios closer and closer to pi. Whether the alien would regard pi as an important number or not is unclear. Given the ubiquity of pi in our mathematics, it is hard to believe that it would not appear in alien mathematics as well. Our ideas about mathematics are not just about us, and not just about the external world, they are about the complex and interpenetrating interaction between the two.

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, George Lakoff and Rafael E. Nunez, Basic Books, 2000.

Recently a student of mine – a middle school teacher – asserted that he did not believe that 0.999… = 1. I tried several methods of convincing him that he was wrong; for example calculating 1 = 3(1/3) = 3(0.333…) = 0.999…, and showing that 0.999 … < 1 led to a contradiction. I browbeat him into submission, but I don’t think I convinced him. After reading the book under review, I see I could have handled things better. I learned that in non-standard analysis, 0.999 … < 1 is true, and more importantly I learned that understanding even the simplest infinite processes involves mastering some tricky metaphors.

This is an important and flawed book. It has generated much commentary, pro and con, and if your curiosity is whetted by my review I suggest you check the reviews of the book in Amazon.com. Like The Number Sense by Stanislaus Dehaene (reviewed below), this book applies cognitive science to an analysis of mathematical thought. But unlike that book, Where Mathematics Comes From goes far beyond an analysis of arithmetic skill, and analyzes some very sophisticated concepts of higher mathematics. Also, compared with Dehaene’s book, this book does not depend very much on laboratory science, but depends almost entirely on theoretical cognitive science.

This is a long book, and a somewhat difficult read, unless you happen to be well versed in the jargon of both mathematics and cognitive science. Even though the authors cover a lot of territory in 450 pages, and the quality of the writing is generally good, I had the feeling that I was reading the same thing over and over. If you want an “executive summary” to get the gist of what the authors believe they have accomplished, I suggest looking at the section “A Portrait of Mathematics” on pages 377 – 379.

In the first four chapters the authors describe the brain’s innate arithmetic, which is quite rudimentary and similar to the innate arithmetic of many other species, and then details how humans have learned to extend these basic concepts to an arithmetic which enables efficient calculation and obeys certain “laws”, such as the commutative property for addition. This section sets the tone for the rest of the book, by introducing basic cognitive mechanisms that the authors believe explain how mathematics is invented and understood. These include grounding metaphors, which yield basic, directly grounded ideas, and linking metaphors, which yield abstract ideas.

Chapters 5 – 7 study the linking metaphors that determine algebra, logic, and sets, and chapters 8 – 11 deal with “The Embodiment of Infinity”. The concept of infinity underlies most of modern mathematics including various number systems (integers, rationals, real numbers, and complex numbers). It appears in many different guises from points at infinity in projective geometry to cardinal and ordinal infinities in Cantor’s theory. The authors introduce a “basic metaphor of infinity” (BMI) that is supposed to account for our understanding of all these concepts. Chapter 11, “Infinitesimals”, is perhaps the most mathematically interesting part of the book. The authors present the hyperreal numbers of Robinson and Keisler, which include infinitesimal and huge quantities and provide be an intuitive and direct way of dealing with calculus. They also introduce a system of their own invention, the granular numbers, which is a subset of the hyperreal numbers that seems to be easier to use.

Chapters 12 – 14 continue the discussion of the infinite by critiquing the program of modern analysis that was pioneered by Dedekind and Weierstrass and continues to this day. While the authors profess the highest admiration for the intellectual achievements of these men and their followers, the subtext here is that their (Dedekind’s and Weierstrass’) purpose in separating analysis from its roots based in an intuitive geometric understanding was to obfuscate the subject and make mathematics the preserve of a specially trained elite. I disagree. My understanding is that the development of technology was beginning to make the traditional conceptions of mathematics inadequate. For example, on page 307 the authors approvingly cite James Pierpont’s (1899) list of “prototypical properties of a curve” including that it is continuous and has a tangent. With this definition, it would be difficult to have a consistent theory of Fourier series necessary for the analysis of radio waves or to solve differential equation with a driving function given by a step function in electrical engineering. The Mandelbrot set and related constructions are rooted in the real world (Mandelbrot’s prototypical example is the coastline of England.) and these “monstrous” sets seem to describe nature better than classical curves and regions.

Chapters 15 and 16 contrast the author’s implications for a philosophy of mathematics that is grounded on the human mind –“embodied mathematics”– with other philosophies that either posit mathematics existing outside the real world (Platonism and “the romance of mathematics”) or see mathematics the manipulation of essentially meaningless strings according to given rules (formalism) or as a cultural construct. I will describe my own view in a separate posting.

The last section of the book is a “case study” in four parts shows that the methods developed in the book can be used successfully to teach mathematics in a way that focuses on meaning. The subject is Euler’s famous equation e^(i*pi) = -1. This is quite good, though I might do a few things differently

E5. Duck, Duck, Goose

This problem was sent to me (as is) by Walter Carter of Seattle.

Some children have made up a simple version of the game “Duck, Duck, Goose”. In this game a group of people stands in a circle, and the person who is “it” taps the first person on the shoulder and says “duck”. The next person is tapped and called “duck” The next person is tapped and called “goose”, and the process is repeated. Every person who is called “goose” must sit down when they are tapped.

If there are a million people in a circle, and they are labeled sequentially from 1 to 1,000,000, and the tapper starts at person 1 going around and around until only one person is left standing, then what is that last person’s number?

Review of the Number Sense

The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997
This book, written by a noted neuropsychologist, explores the new field of mathematical cognition. That is, it attempts to root our understanding of the development of mathematics in the biology of the brain. It is one of those rare books written by a pioneering researcher in a scientific field who is also an excellent writer – in English as well as presumably in his native French. I think it is particularly valuable for those of us in education, because in order to teach mathematics we must understand how children actually acquire mathematics. While there is much to learn here, I also found much to disagree with, and I will deal with these points below. Perhaps the major drawback to the book may be its date of publication, since Dehaene indicates that the ten years following the writing of the book promise to be a time of unparalleled scientific advance in the field.
The book is organized into nine chapters:
Chapter 1, “Talented and Gifted Animals”, discusses scientific research that shows that many animals have innate primitive arithmetic skills, which enable them to add, subtract, and compare small integers. Calculations and comparisons of numbers become less accurate as the numbers involved increase beyond three.
Chapter 2, “Babies Who Count”, sets forth the contention, supported by ingenious research, that shows that, similar to animals, human babies as young as a few days old also have innate arithmetic skills, enabling them to understand and manipulate small integers.
Chapter 3, “The Adult Number Line”, discusses the conception that human adults have of number. Much of this chapter has to do with discovering the extent to which we can manipulate numbers very quickly, that is, without visible thought.
Chapter 4, “The Language of Number”, discusses the ways different cultures name numbers, and the effect this has on calculating abilities.
Chapter 5, “Small Heads for Big Calculations”, applies the results covered in the previous chapters to the difficulties of teaching arithmetic to children.
Chapter 6, “Geniuses and Prodigies”, presents case studies of a number calculating prodigies and mathematical geniuses, and attempts to show that their abilities are not different in kind from that available to any intelligent adult.
Chapter 7, “Losing Number Sense”, discusses the relationship between brain function and number sense as revealed by studying people who have lost various parts of their number sense due to lesions in particular parts of their brains, or to other brain injury.
Chapter 8, “The Computing Brain”, shows how modern advances in brain research cast light on relationship between calculation and the brain. The tools of positron emission tomography (PET) and electro- and magnetoencephalograpy are described, and some results obtained by applying these tools to mathematical cognition are discussed.
Chapter 9, “What Is a Number?” moves into the philosophy of mathematics. Dehaene tackles questions such as the merits of the formalist, Platonist, and intuitionism theories of mathematics, and the relationship between mathematical truth and reality.
A book of this wide coverage is bound to be controversial. I recommend reading it yourself and making up your mind about some of the controversial issues, but I’d like to bring up a few places where I disagree with the author.
It seems to me that one of the dangers of neuropsychology is that of reductionism, and although Dehaene is a sophisticated thinker I don’t think he escapes this.
I take issue with his apparent assumption, which seems unsupported by data, that ability to perform arithmetic calculations is strongly correlated with the ability to do higher mathematics. Among mathematicians I have known, some excel at arithmetic, some are poor, and many are in between. The type of thinking that is involved in geometry, for example, seems to have little to do with arithmetic ability.
I find particularly problematic his discussion of mathematical geniuses, for several reasons. First, he lumps together the self-taught Indian mathematical genius Ramanujan with autistic super-calculators and idiot savants. To me, this is as if one compared Shakespeare with a pre-typewriter clerk who filled thousands of pages of commercial transactions. Both men may have had unusual ability to produce fast legible handwriting, but we would only call one a genius. Second, Dehaene makes clear that he believes that anyone could be a super-achiever in mathematics or arithmetic if they devoted enough time and effort to the enterprise; that there is nothing special about the brain (or mode of thinking) of the genius. This is speculation, and I prefer the opposite speculation of Oliver Sachs, whose prime-number generating autistic twins seem not to calculate but rather to see the integers “directly, as a vast natural scene” or Ramanujan, who described his own mathematical discoveries as being handed to him by a Hindu god while he slept. Non-believers can imagine that Ramanujan’s unconscious mind allowed him to make his discoveries operating in a way that might be totally different from his conscious mind.
Another oversimplification is Dehaene’s belief that young Oriental students do better than Western students at learning mathematics because the Eastern languages have shorter more user-friendly names for the digits. He seems to not consider the cultural differences that lead Oriental families to value hard academic work more than Occidental families do, which by itself is enough to explain differences in achievement.
In terms of pedagogic implications, Dehaene’s research has led him to the belief that the human brain is not well designed for calculation: “Ultimately, [innumeracy] reflects the human brain’s struggle for storing arithmetical knowledge”. He therefore feels that “by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them to concentrate on meaning.” This is a position I have long shared; however I am now teaching middle-school mathematics teachers, and they mostly report that their students, who have grown up using calculators, are grossly innumerate. Since many algorithms of elementary algebra have counterparts in arithmetic algorithms, these students are not able to progress in algebra. I now advocate getting children to a state of competence in calculation before letting them use the calculator freely. However, I agree with Dehaene on the usefulness of concrete computational representations (manipulatives) in the classroom.
Dehaene gives a good description of the basic theories of mathematical epistemology: Platonism (mathematical objects have a reality, and the mathematician discovers this reality rather than inventing it), formalism (mathematics is about the formal manipulation of strings of symbols following basic laws of logic), and intuitionism (mathematics is a construction of the human mind, so that alien intelligences would create different a mathematics different from the human one.) He comes down for intuitionism, but it seems to me that his dismissal of Platonism is entirely too glib. He asks, rhetorically, “If these [mathematical] objects are real but immaterial, in what extrasensory way does a mathematician perceive them?” I would argue that they are perceived in the same way that we perceive a coherent world from the streams of sense data that enter our brains. We create our mental worlds, and this seems to be true whether or not the basis of the world is “material” or whether is grounded in ideas. Both the material and mathematical mental worlds are subject to laws of internal consistency, and both are subject to judgment by members of a community.