# Beautiful Mathematics: What we miss, and what students think!

According to Merriam Webster’s dictionary, mathematics is defined as “The science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations”.  This definition certainly contains a lot of items, and concepts.  Mathematics seems so complex, but like a great tool man.  Mathematicians and students alike are always learning to use and adding more tools to their tool box in order to solve problems. Perhaps it is the oversimplification of the beauty of mathematics, and the removal of exploration within math that causes students to see math as boring.  In many ways, math explains the world. The more I learn about the ways that math is expressed, seen, and utilized in the universe, the more I am amazed at the beauty of it.

George Hart used mathematical patterns to form the amazing construction of overlapping pencils below.  When we observe mathematical patterns and fully understand them, then we can use them to create extraordinary structures.  The pencils connect in such a way that is appealing to the viewer.  I am no artist, but the mathematical patterns are beautiful in this mini sculpture.

George Hart’s Overlapping Pencils

Penrose Tiles

Literally, anything that is in the world you can apply mathematical operations to in order to solve for some unknown, or measure something.   I think as teachers, we should be encouraging our students to exercise their minds by exploring whatever phenomena they find interesting and looking at the mathematics involved in the process.

That could be in music finding the frequency of the vibrations of a guitar string, or the pitch the strings form.  The frequency could be calculated and a function could form for the frequency of the vibrations based on the note.

It could be in sports, in a football game considering the possible functions of the flight of a ball when it is thrown over 60 yards over the top of a 6’4″ defender.  The students could use a quadratic function to model the flight of the ball path, and try to find the quadratic function that most quickly gets the ball to the receiver.  The flight path of the ball is a beautiful piece of math.

The exploration could come in their favorite horror movie, measuring the pattern of the facial expressions of the terrifying figures, and looking for the similarities in shape in order to find the most terrifying shapes.  The student could use the similarities in shape to construct their own horrifying figures, and see the beauty/effect that the mathematical operations used to make the pattern have on the creation of the shapes.

Math is all around, intriguing problems are infinitely available, yet as teachers we often choose problems that do not magnify the beauty in the world, and the beauty of mathematics itself.  The patterns, operations, and functions that express those operations involved in the pattern are the building blocks of the beuaty of mathematics.

Some students may find beauty in numerical patterns or operations, but as teachers we must realize that for some this will take time or may never occur for students.  Instead they may experience mathematical beauty in the pattern of the seeds of a sunflower, or the shape of an octagon, or even in the graph of a function.  Mathematical beauty comes in shapes, sizes, patterns, numbers, and history.  If we oversimplify math to operations with numbers, and cliche story problems many students will miss out on the “spark” which will inspire them to explore mathematics they are interested in!  Of the picture below we may ask ourselves, “What is the function of the helix?” Or invite students to ask whatever questions they like.  Studying helix’s is a higher level of mathematics usually taught in Calculus, but students that get excited about the topic may work as far as there teacher is willing to enable them, in my opinion.  Obviously it would take time for a student to explore helix’s, but if a teacher combines the current standards with the students exploration the student may be able to reach higher and farther mathematically.

Hurricane Helix

If you are still not convinced of the variety of beautiful shapes to explore check out this video

Beautiful Math Patterns

I’m convinced that if we are able to change the way students look at math problems we need to break some stereotypes.

Stereotype #1

Math is boring!

In our teaching how we present mathematics plays a large role in adding to or removing this stereotype.  If we present material in cliche, and ways that only present numbers with little purpose in connecting to the world, then we make mathematics boring.  Instead teachers need to allow students to explore areas of the world they are interested in and see the mathematics in the process.  When teachers have students make their own examples, and solve their own problems then math comes to life!

Stereotype #2

Mathematical Stereotype Speed

The article above explains the stereotype and how the common core is beginning to change the stereotype by having students develop a deeper mathematical understanding that goes beyond the procedural level.  We need to teach students that speed is only one component of problem solving skills, and is no where near the most important part of being a mathematician.  We need to encourage students to dive deeply into problems rather than rushing through them because the longer they take the more discoveries they will make and the better mathematicians they will become.  Students need to be encouraged to fully explain their process and shown that math is so much more than finding answers.

Stereotype #3

Math fields are for Guys and Nerds.

Men, women, young and old should be enabled by their teachers to enjoy the beauty of mathematics in whatever form it may appeal to them.  We need to present mathematics in cool ways and relate it to a variety of areas of life.  Male and female teachers alike need to encourage girls and jocks that like math to explore more deeply into their particular areas of interest in the field.

As teachers, we need to become the people that inspire, encourage, and enable our students to see the beautiful mathematics in the world that is all around us!  Now I challenge you to find how many sunflower seeds are in the image at the top of this post and send me your answer plus an explanation.