I have seen many of my teaching colleagues say “Wow! That’s beautiful!” This was not in response to seeing something beautiful in nature or a beautiful piece of art. This was in response to seeing a piece of mathematical work. I know that many students think of the math or science teacher as crazy when they hear the teacher say that there is beauty in math, but that inability of students to appreciate art in math points to something lacking in the way math is taught.
The general perception about math is that it is a difficult subject. Once this a student absorbs this perception form what he/she hears from the adults around him/her, it becomes difficult to accept math as a subject that can be enjoyed. The minimum that parents of a young child can do to help him/her excel in math is not to say that math is a difficult subject in front of the child. Parents, please do not poison the well of math before your child has had a chance to taste its waters. In today’s world, the general principle used for rearing a child is to let him/her explore and find out. But this principle is not applied to math. If you did not like math as a child, then please do not let your child know of your experience. After all, you did not like vegetables either.
The next problem lies with math teachers. There are specific things teachers of other subjects do, that math teachers need to do as well. English teachers teach students how to appreciate a piece of prose or poetry. They teach them about rhyme and rhythm as used in poetry, and about alliterations, allegories, climaxes and anticlimaxes in prose. Likewise, history teachers show students how to analyze a historical event in view of then current world situations. Very few, if any, math teachers will help students see the beauty in math. It would be very beneficial for students if math teachers actively showed students how or why a certain piece of math is beautiful. In the following paragraph, I will highlight three areas that teachers can focus their students’ attention to.
First, there is beauty in the process of math. Most math problems, besides the very elementary ones, can be solved in different ways. Each method will approach the solution from a slightly different angle. When a teacher presents different methods of solving a problem, then students can choose a method of their choice, but it is here that the teacher can help student appreciate that one method is more elegant, or reaches the solution in less time, or with less effort. I have jokingly told my students, more than once, that mathematicians are lazy people and always look for easier solutions. It would also be beneficial if students could be shown that certain solutions can more easily be coded for a computer program and would thereby reduce computational time and number of calculations needed to reach answer.
Second, there is beauty in the product of math. There are some math equations that are amazingly beautiful in themselves. Pythagorean theorem, quadratic formula, Euler’s formula, Heron’s formula are some examples. Euler’s equation is without doubt one of the most, if not the most, beautiful equations. It would be great if math teachers explained to students why these equations are considered beautiful. Talking about the history and background of these equations is a terrific way of introducing the humanity of mathematicians to students. Giving assignments that require students to read about the life stories of mathematicians involved in the development of these equations helps them empathize with mathematicians.
Third, there is beauty in the utility of math. Students will be more willing to learn a concept if they can see opportunities to apply it outside the classroom. I am a great advocate of learning for learning’s sake, but at the same time, I also want my students to be able to see where they will use what they are learning. I acknowledge that this may not always be possible for all topics in middle and high school, but I would encourage teachers to investigate the utility of topics, so they can at the very least tell students that there are people who use these concepts as part of their jobs. Imaginary numbers are one example of such topics. Electrical engineers use imaginary numbers to model electric machines. Even though not every student will become an electrical engineer, but if imaginary numbers are introduced with a mention of their use in engineering, then they become more palatable for students.
In this article, I have tried to convince math teachers to include the beauty aspect of math in instructions and given three areas where they can do this. This is not an exhaustive list of areas where teachers can talk about the beauty of math, but rather a starting point.