People have very different attitudes to mathematics. While some love it, some find it very difficult and some even hate it. Even though it is true that mathematics is built on an axiomatic foundation, a strong case can be made for the ultimate foundation of mathematics being its beauty. Richard Feynman, an American physicist known for expanding the theory of quantum electrodynamics, says, “To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”
Educators who see the beauty at the center of mathematics and can make their students see it that way, are more likely to be able to get their students’ attention and teach them more effectively. Also, as well as facilitating the study of sophisticated mathematics, this puts mathematics in its proper place so as better to understand the value of what it has to say to human beings. To see the beauty and the pleasure in mathematics can change the negative attitudes of some students and help educators in teaching mathematics.
Often it seems that we pursue mathematics education from either a structural or an applications point of view. From a structural point of view, we insist on building up all of the tools one may need in a sequential, logical order, because an educator will need the students to know all of the smaller pieces before they can build any of the larger ideas. An analogy for this would be if we forced somebody to study all of the nails, screws, bolts, and tools to build a house before we let them even see the plans for the house. This is one of the main reasons that most people who study mathematics in their school years think that it is a pointless exercise in playing with formulas and has no significance in real life. For these people, mathematics might be helpful only in keeping track of their checkbooks after graduation. Some students think that they can calculate whatever they need using computers, but sometimes this is not very effective because students may not understand the logic behind the problems and the results do not mean anything to them or they are unable to detect errors.
The applications point of view leads to making up “word problems” that appear to be about the real world, but everyone knows that they are highly artificial. It also leads to focusing at higher levels on only the applications. Hence, for instance, in calculus we spend a lot of time plodding through various physical applications, without letting students see the bigger picture. Or we spend time in liberal arts mathematics talking about things such as modeling and linear programming, which yield great applications, but are generally tedious and do not give most students much appreciation for mathematics. If we see the beauty at the center of mathematics, as well as introducing ideas that may have application or may build some tools, we can bring students to have a much bigger picture of mathematics at a much earlier stage in their mathematical development.
Educators can include some fun topics in courses that they teach. For instance, they can encourage students to discover the amazing number patterns in nature, such as the Fibonacci sequence in pine cone spirals, pineapples, and cauliflowers, in which the number of pieces increases in the following manner: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ... (add the last two numbers to get the next). Another example is the “golden ratio.” In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is a mathematical constant, approximately equal to 1.6180339887 and recent research shows that people think that the shapes and figures in this ratio are more interesting and aesthetically pleasing to the human eye.
Sometimes, even a small algebra trick can miraculously bring students to love mathematics and be more focused, and then more interested in the deeper aspects later on. Take a look at this symmetry:
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321.
Here are a few more examples showing the beauty of mathematics visually with numbers:
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10 = 1111111111
9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888.
Students’ minds can be broadened by seeing the surprising differences that arise when we move to non-Euclidean geometry. Fractal shape examples in nature, such as snow crystals, and things like the Mandelbrot set, which is a set of points in the complex plane the boundary of which forms a fractal, can be introduced with a background and give rise to amazingly beautiful images and ideas. Even such deep and thought-provoking ideas as these can be understood by students when they have curiosity, creativity, and an open mind.
All these examples and others like them can inspire people to see the beauty of mathematics and give them a better understanding and a sense of the expanse of mathematics. With a little more discovery of and exposure to the more beautiful aspects of mathematics, students are much less likely to feel any hatred for mathematics and may develop a much greater appreciation for the creation of the universe.
Ali Kemal Unver is a postdoctoral scholar at the University of California, Los Angeles.
* The golden ratio can be derived by the quadratic formula, by starting with the first number as 1, then solving for the 2nd number x, where the ratio [x+1]/x = x/1 or (multiplying by x) yields: x+1 = x2, or a quadratic equation: x2-x-1=0. Then, by the quadratic formula, for positive x = [-b + sqrt(b2-4ac)]/2a with a=1, b=-1, c=-1, the solution for x is: [-(-1) + sqrt([-1]2 -4*1*-1)]/2*1 or [1 + sqrt(5) ]/2. See the second reference for details.