What is mathematics?
Let me begin by considering what mathematics is all about. Mathematics is a subject that touches all of us in one way or another. Some of us use it for amusement, some of us use it to earn a living, and some of us find it difficult to accept that we cannot get along without it. Be that as it may, the ideas and activities bundled under name “mathematics” make it likely that we can all learn to love some of them by using them to amuse us.
Here are some examples of mathematical activities, achievements, and results. Using a ruler and compass to bisect a line segment, using long division to show that 12/4 equals 3, using the law of the excluded middle to show that Ö2 is irrational, using the diagonal method to show that the set of natural numbers can be put in one-one correspondence with the set of rational numbers, using an elaborate computer program to show that four colors suffice to color any map so that no two bordering countries have same color, and using limits to show that the cosine function can be used to calculate the slopes of the tangents of the graph of the sine function, they are all part of mathematics. So are constructing a magic square of numbers in which all rows, columns, and diagonals have the same sum, and calculating the shortest distance between two points in space. The list goes on.
Some of these aspects of mathematics are recreational because we use them to amuse ourselves. Playing monopoly, rearranging the faces of a Rubik's cube, constructing a magic square, and perhaps even playing chess or bridge, can be considered to be recreational forms of mathematics.
Studying the mathematics based on the theorem of Pythagoras and exploring the logical principles that justify the theorem are aspects of “pure” mathematics because we can study them out of intellectual curiosity, for they own sake. However, fractals, based on this theorem, are wonderful examples of some of the “recreational” aspects of the theorem.
Other aspects of mathematics are known as applied, because we use them to function in our daily lives. Electronic banking, buying a tank of gas for our car, sending a space shuttle into orbit, calculating the premium of a life insurance policy are all part of applied mathematics. Many mathematical games are given a sense of reality by casting them in an applied context.
A disturbing view
As we well know, there are many people out there who greet us with "Oh no, not mathematics. I was never good at that," when we introduce ourselves as mathematicians. I am proposing that we try to remedy this situation by teaching the community at large about mathematics using a recreational approach.
In his celebrated book, "Successful Intelligence," Robert Sternberg shows that many intelligent students fail to realize their academic potential because early in their lives, only one aspect of their intelligence was used to assess their intellectual capacity. Sternberg explains that what makes people successful in life is a combination of practical, creative, and analytic intelligence. Only the latter is usually taken into account when measuring intelligence using IQ tests.
I feel the same way about success and failure in mathematics. Many students end up believing that they can't do math or even dislike mathematics because they were initially introduced to only one aspect of our multifaceted subject. I believe that our teaching and learning of mathematics therefore stands to benefit from considering all three personalities of mathematics: pure, applied, and recreational. Here is an example of a student whose views of mathematics were profoundly changed by a recreational activity: A biochemistry student discovered that she can do mathematics by studying the mathematics of radio. She always wanted to know how radios work. With a good deal of probing, reading, and reflection, and some help from me, she succeeded. The following letter summarizes her achievement.
"Hi Dr. Szabo: I just wanted to thank you so much again for all of your patience and understanding during my project on how radio works. This really was something I had always wanted to do and it is great to have accomplished it, even though I needed a bit of a push to get me to finish the project. I am very happy that I did it. I noticed in class a few weeks ago, as we were going through some basic math formula, that they were more meaningful to me than they used to be: just like words---with a simple meaning. I feel like math is a bit less daunting and is more accessible than I it found before, and this has also made me glad that I took your course. Deborah"
A less successful story involves one of my tennis partners, Françoise, who told me over dinner one evening that at school she could always do algebra and geometry, but that she could never do “math.” What was she talking about? She wasn't trying to be funny, she wasn't lying, and she probably wasn't even confused. She had a point. There are all kinds of math. Most, if not all people, love some of it, dislike some of it, and wish they could grasp the rest of it. Let me give you some examples of what I mean. Let's begin by considering the role of mathematics in our daily lives. Imagine a day without math:
- No money, no stocks and bonds;
- No debts, but of course also no assets and no payday.
- No calendar, we'd never be late. No one hundred shopping days till Christmas.
- No computers, no radio, no television. A temporary blessing for some.
- No hockey. After all, we can't count goals, penalties, and scores. So what's the point? No winners or losers.
Perhaps a gentler world.
Recreational aspects
What I'd like to propose in this paper is that there are many mathematical topics that can first be introduced in recreational form. For the moment, let us think of recreational math simply as math that's fun and popular. This obviously includes a vast array of topics. Here are some examples:
- Mathematical riddles for developing problem solving skills.
- Logic games for developing analytical skills.
- The mathematics of gambling for developing statistical skills.
- Famous sets of natural numbers such as the Fibonacci numbers for seeing mathematical patterns in natures.
- The construction of calendars using greatest common divisors.
- Prime number, large primes, and the distribution of primes.
- Tiling problems and the connection between music, art, and mathematics.
- Boolean algebra and truth.
- The role of the law of the excluded middle in our daily lives.
What Françoise thought of as math probably involved complicated arithmetic and the manipulation of complex objects such as showing that the following equation holds:
Sometimes word problems are also rigged up as brain teasers, cast in an irrelevant mathematical contexts. Here is one of them. An express train traveling at 100 km/h leaves Montreal for Toronto at the same time as a slow train traveling 50 km/h leaves Toronto for Montreal. Which is farther from Montreal when they meet? The frequent incorrect answer: The slower train. Why would anyone be trapped by this type of a nonsensical question? It surely says something about the power of deception inherent in school mathematics.
The list of sixty
If we examine the list of topics identified by the American Mathematical Society as comprising the field of mathematics, we notice that mathematics is grouped into approximately sixty general categories. Most of their names are not reminiscent of any mathematical topics taught in school or college. Many familiar topics, such a Galois theory and projective geometry, are not even listed. They are subsumed under more general topics in the list. However, buried in this formidable list are also some new starting points for an effective and enjoyable study of mathematics. By this I mean mathematical activities that can be pursued for fun and leisure.
Do we have the time? Do we have the option to include some of our favorite recreational topics in our teaching? Certainly. Why don't we remove some of the irrelevant material we now teach and replace it with recreational topics. I suggest that we tune up our curriculum just a frequently as we tune up our cars. I find that student course evaluations are a great source of inspiration in this regard. If we want to meet the needs of a numerate society by keeping our students interested, it is a good idea to take such reviews seriously.
What is recreational mathematics?
Let us return to the question of identifying recreational aspects of mathematics. Here are two attempts to define them. Recreational math is math that we do for fun. Since most mathematicians probably think that their work is fun, this definition may be too narrow. So let us try again. Recreational math is math that many do for fun.
Example. Monopoly
Consider the fact that 90 million copies of the Monopoly game have been sold over a period of fifty years. Surely this means that a large number of people find the mathematics of business and optimization to be recreational.
Example. The Rubik Cube
Another example is the Rubik Cube. Over 200 million cubes were sold in the first three years of its appearance on the market. Thus aspects of combinatorics and geometry certainly have mass appeal and are considered fun by millions of people.
Example. Sports Statistics
The fascination with the statistics of professional sports points to another aspect of quantitative recreational reasoning.
In his books on numeracy and the mathematics of everyday life, John Paulos gives us many other examples of recreational aspects of mathematics. So does Robert Northrop in his book on mathematical riddles. It would actually be a fascinating challenge to design school- and college-level mathematics courses around games such as Monopoly, Rubik Cube, or some of the activities discussed in the books by Paulos and Northrop. We could use the Polya method which consists of the idea of using “leading problems” to teach and learn of mathematics. I have used this idea in my linear algebra book and have built the entire course around the problem of solving linear systems. I am happy to report that I have managed to attract an enthusiastic audience for the course.
The AMS classification of mathematics has no category called “recreational.” Perhaps most mathematics has some recreational aspects. For this reason, it may not really matter what mathematics we teach or learn. What may matter is how we teach it. The famous Cambridge mathematician G. H. Hardy once observed that Euclid's reductio ad absurdum is one of mathematics' finest weapons and provides us with interesting recreational possibilities. The fact that we can show in mathematics that certain things are impossible is one of the great virtues of our subject. Students are fascinated by the logic involved.
Logic is full of puzzles and paradoxes. Many of them are routinely used to construct games and recreational mathematics. Even some of the proofs in logic can be intriguingly entertaining. For example, Russell's paradox is an amazing example of a mathematical construction that can be used for entertainment and recreation. Let S be the set of sets that are not members of themselves. Then S belongs to S if and only if S does not belong to S. How do we get out of this jam? We can do so by making a definition. There are sets and classes. The object S in Russell's paradox is a class, but not a set. By their very nature paradoxes of course defy resolution. Students love this: Even mathematicians get stuck!
Russell's paradox and a whole lot of mathematics are based on the assumption that given any statement A, the statement “A or not-A” is true. In logic, this assumption is known as the law of the excluded middle. If we don't accept the law of the excluded middle, we don't have to accept a lot of math. Happy? The first half of the 20th century was preoccupied with the consistency of mathematics. Classical mathematicians, led by Hilbert, battled intuitionist mathematicians, led by Brouwer. Their bone of contention? The law of the excluded middle.
A perhaps more accessible and hence more enjoyable use of the law of the excluded middle may be the proof that “there are more real numbers than there are integers.” The proof is based on a remarkably plausible construction which lends itself to mathematical recreation. Here is the basic idea.
Example. A diagonal construction
Count the decimal expansions of the positive real numbers between 0 and 1 using the infinite set of natural numbers 1,2,3,… Answer: it can't be done. Challenge: prove it. Here is a two-step solution. List all expansions schematically. Then construct an expansion that cannot be in the list. Consider the following list:
Let
As we can see, the sequence b is different from any of the sequences matched with the natural numbers 1,2,3,4,5,… Instead of playing with magic squares, we could play with diagonal sequences using actual numbers, not double-subscripted terms. For example, we could try to build a sequence that is not in the list
0.12345
0.30098
0.67800
Solution: Any sequence of the form 0.216xx, for example, will fail to be in the list. It may not take more than this little game to make the key idea of the uncountability proof of the real numbers plausible.
Example. Quaternions and rotations
In the 19th century, Hamilton discovered the quaternions in his attempt to understand how far we can extend the complex number system by abandoning the requirement that multiplication is commutative. Today, computer game designers consider quaternions the greatest invention since sliced bread. They use them to speed up three-dimension rotations.
Example. Triangles and squares
Geometric figures have always been a source of enjoyments. Paper folding, tiling, and other forms of geometric amusement have been around for a long time. One of the most well-known and beautiful geometric facts is the theorem of Pythagoras mentioned earlier.
Theorem. The sum of the areas of the squares on the two shorter sides of a right-angled triangle is equal to the area of the square on the longest side of the triangle.
A beautiful illustration of this theorem are the Pythagorean trees consisting of iterations of triangles and squares. Here is a simple tree built by iterating the construction.
By iterating the construction further, we get a bushier tree:
This example shows that one of the exciting new recreational activities of mathematicians is the creation of beautiful pictures using fractal geometry.
Over 2,500 years ago, Pythagoras discovered, as pointed out in the theorem, that if x and y are the two shorter sides of a right-angled triangle, and if z is the hypotenuse, then
More than 2,000 years later, Fermat contemplated this equation and wondered whether there was something unique about it. He thought and claimed to have proved that for all n>2, then the equations
have no positive integer solutions. The proof that Fermat's conjecture was correct is one of the great contributions made to mathematics towards the end of the 20th century. The proof constitutes a major triumph of the human spirit. Not only are the techniques used to prove this theorem clever and elegant and settle an age-old problem, they also lead to scores of applications, including applications to the practical and important field of data encryption.
Example. The Möbius strip
Geometry certainly has its surprises. It is possible to construct one-sided three-dimensional surfaces known as the Möbius strips. Their construction can keeps school children amused for hours. Today the idea is used to put a twist in printer ribbons to allow printers to use both sides of a ribbon. Here is Escher’s beautiful rendition of such strips.
Example. The trisection of angles
Some of the most spectacular results in mathematics are negative. They prove that certain things cannot be done. A theorem of algebra states that it is impossible to trisect an arbitrary angle with ruler and compass. What is the bizarre recreational content of this theorem? Many people take up the challenge and trisect angles. Sometimes it is quite difficult to find the errors in the constructions. In his paper on a proposed angle trisection, mentioned in the Reference below, Roger Cooke discusses one such example.
Example. Primitive recursion
Many simple algorithms can serve as the basis for mathematical group games. Take, for example, the definition f(0)=1 and f(n+1)=nf(n). The first person to calculates the sequence f(0),…,f(10) wins. A similar game can be played using the Fibonacci sequence or any other sequence defined by primitive recursion. Players can make up their own rules. Let f(0)=5, and f(n+1)=f(0)n². The first person to calculate the sequence f(0),…,f(10) wins.
Example. Linear independence
Take any 3 x 5 matrix and find the first column from the left that is linearly dependent on the columns to its left. For example, suppose that
is a given matrix consisting of five columns. Then the first three columns are linearly independent, and the fourth column is the first column linearly dependent of the first three. The challenge is to change the matrix A to a matrix B in which one of the first three columns is linearly dependent on the other two, and where the fourth column is linearly independent of the first three. Student can use all of their knowledge of linear algebra to refine this game.
Example. Orthogonal matrices
In geometry we are often interested in studying transformations between geometric objects that preserve distances. Such transformations are represented by orthogonal matrices. An orthogonal matrix is a matrix whose columns are of length 1 and whose columns are mutually orthogonal. Two columns are orthogonal if their dot product is zero. Orthogonal matrices are frequently used in computer animations. The challenge is to construct one. After spending a long time trying to find a numerical example, students are always in awe of the fact that the Householder construction
works. For any nonzero vector (x,y,z), the constructed matrix is orthogonal.
Example. Singular value decomposition
Computer games involve massive instances of matrix inversion. Since they require time and substantial memory to perform, it is often useful to replace matrix inversion by matrix transposition using the orthogonal matrix closest to a required matrix inverse. The following theorem makes this often possible.
Theorem. For every real matrix A there exist two orthogonal matrices U and V, and a diagonal matrix D for which A=UDVT.
If we replace the matrix D by an identity matrix, we get the orthogonal matrix UVT which is the closest to the inverse of A, measured in the Frobenius norm. The square of the Frobenius norm is the sum of the squares of the entries of A. The square of the norm is therefore the sum of the squares of the Euclidean lengths of the columns of A. So indirectly, we have another recreational application of the theorem of Pythagoras. Students have a lot of fun calculating the matrices UVT with the help of computer algebra systems and using them to construct simple computer animations.
The moral of the story
In many ways, recreational math is simply an enjoyable way of looking at traditional math. My own solution to making math fun consists of using mathematical software in my teaching and research. By removing some of the manipulative obstacles from the teaching and learning, students are able to understand and enjoy the conceptual aspects of the subject. Thus instead of spending hours trying to calculate the eigenvalues of a matrix, my students play a maze game. They find winning paths by identifying mathematical truths and avoiding mathematical falsities. The statements involved in the game are the mathematical theorems learned in the course, together with falsified variants that must be identified and shown to be false. I use Maple, Mathematica, and Scientific Notebook for this purpose.
Mathematics goes far beyond what we have chosen to teach. A closer look at the topics selected as our mathematics core shows that this core is not always relevant to our audience. But mathematics is alive and well. There is enough to choose from to satisfy the curiosity of most of those who claim they can't do math or don’t like math. My view is that most of us can do some math. It is our job to identify which math our students can do and build on it.
References
- Roger L. Cooke, A Proposed Angle Trisection. University of Vermont, 2001.
- Howard Eves, An Introduction to the History of Mathematics. Holt, Rinehart and Winston, New York, 1964.
- Solomon Garfunkel, For All Practical Purposes (5th edition). W. H. Freeman, San Francisco, 2000.
- Eugene P. Northrop. Riddles in Mathematics. Penguin Books, London, 1944.
- John Paulos, Numeracy. Knopf, New York, 1991.
- John Paulos, A Mathematician Reads the Newspaper. Basic Books, New York, 1995.
- George Polya, How to Solve It (2nd edition), Princeton University Press, 1957.
- David Singmaster, The Unreasonable Utility of Recreational Mathematics. First European Congress of Mathematics, 1992.
- Robert J. Sternberg, Successful Intelligence. Plume, New York, 1997.
- Robert R. Stoll, Set Theory and Logic. W. H. Freeman, San Francisco, 1963.
- M. E. Fred Szabo, Linear Algebra: An Introduction Using Maple. Academic Press, Boston, 2002.
- M. E. Fred Szabo, Graduate Student Success. Harcourt Brace, Toronto, 1995.
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