Create Like a Mathematician?!

written by: Dr. Tony McCaffrey
Most students do not think of mathematicians as being creative. Math was created long ago, so their story goes. Geometry was created 2,300 years ago; algebra 1,200 years ago; and calculus 350 years ago. No math has been created since long ago, they speculate.

Nothing could be further from the truth! Mathematicians are creating new math every day and the creative act is as exciting as creating new art, stories, science, etc.

For example, Network Theory is one hot math area where many new things are being discovered. It explores how things are connected together, such as our social relationships. Most people have had the experience of being far away from home and running into someone who knows one of their family members or neighbors. They might exclaim, “It’s a small world!”

Mathematicians now study how these networks are structured, how they form, and how resilient they are to change. It turns out that many things have this small-world structure: the Internet, food chains, electric power grids, etc. This information about small-world structure can be used to help protect power grids from terrorist attacks, for example.

Game Theory is another hot math area, which explores how different sides can make moves in a contest. It has implications for business negotiations, economics, war, sports, investing, treaty negotiations, etc. The movie, A Beautiful Mind, shows mathematician John Nash winning a Nobel Prize in economics for his work on noncooperative games. Ten other game theorists have also won Nobel Prizes.

Like a detective, a mathematician looks for clues and patterns where no one has found them before. Then, the mathematician forms hypotheses about those patterns and tries to prove that they are correct. Once something is proven true, it is true forever! It is exciting to try to prove something that will be true for the rest of time!

This detective-like activity is exactly what my students do in a new Eagle Hill math class called Mathematical Reasoning.

For example, for a 100-piece jigsaw puzzle, how many joins does it take to complete the puzzle? One join occurs when two clusters of pieces are connected together or two individual pieces are connected together. When examining the many different orders of putting the pieces together (5,050 different ways to order the joins for a 100-piece puzzle), the students looked for patterns. However, it would take them most of the school year to examine all the orders.

So, the students learned a valuable lesson. When given a complex problem to solve, solve a simpler one first. A 100-piece puzzle is too complex to start with, so they should start with simpler puzzles (e.g., 4-piece) and see if the pattern they detect for the small puzzles also applies to larger puzzles. The students formed hypotheses about the patterns they noticed and tried to represent and demonstrate them mathematically. Their discovery: no matter how you assemble a 100-piece puzzle, it will always take you 99 joins! Counterintuitive, but true! In fact, for any puzzle with pieces, it will always take n-1 joins to put it all together.

At times, I even give my students problems that no one knows the answers to. Some of the problems are famous unsolved problems and I tell them if they can find a pattern and show it is correct, they would be world famous by the end of the week.

Certainly, this class is about how mathematicians work and create. More generally, however, it is also about how to approach any problem in which there is no clear path forward. Too many problems are assigned in school that exercise only the equation or technique that was just learned. There is little or no mystery to what to do. But real life problems and intriguing problems are the ones where you do not know how to start. You have to explore. You have to wander around until you see an entry point or a pattern that calls you forward. It is an art form and authors such as George Polya in his classic book, How to Solve It, show you how to roam around in a purposeful manner and not be afraid of dead ends. It teaches you “patient problem solving,” as mathematician Dan Meyer might say.

Being a mathematician is like being a detective where the whole world is your “crime scene.” Clues, patterns, deductions, and discoveries abound for the brave searcher who has the curiosity and perseverance to look for patterns where no one has found them before.

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Learning Diversity is a blog hosted by Eagle Hill School where educators, students, and other members of the LD community regularly contribute posts and critical essays about learning and living in spaces that privilege the inevitability of human diversity.

The contributors of Learning Diversity come together to engage our readers from a variety of disciplines, including the humanities, social sciences, biological sciences and mathematics, athletics, and residential life. Embracing learning diversity means understanding and respecting our students as whole persons.

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