Although new to WyzAnt, I have tutored mathematics for the past four years at college. One thing I notice among many students is a great deal of annoyance towards mathematics. They feel mathematics to be too abstract, rigorous, relentless, and just plain boring. Most students either prefer a subject they will end up using in "real life", or a subject that gives them a sense of wonder.

However, the amazing thing about mathematics is how truly wonderful it is for me. Most people who see my attitude towards mathematics (including others who are reasonably adept at math) find this odd or misplaced, and I fully understand their lack of sympathy. Perhaps you are one of them. But I can assure everyone reading this there is something truly mysterious about mathematics that breaches the very foundations of astonishment and awe.

Take prime numbers for example. It's pretty clear that you can take any whole number and decompose it down into a product of prime numbers (i.e. 180=2⋅2⋅3⋅3⋅5). This might make you think "Which numbers are prime?" You start to list them out and find 2, 3, 5, 7, 11, 13, 17, 19,... wait, how many are there? ...23, 29, 31, 37, 41,... this doesn't seem to be ending or even repeating! Why should it never end? I mean, not that this really helps me build better bridges or spaceships or anything, but I just kind of want to know. Well, if it ever ended, you'd be able to multiply all primes together. Why should that matter? Because if you could multiply all primes together, you could also add 1 to their product, and this new number would need to be divisible by a new prime.

Or take Zeno's Paradox: Let's say I want to get from 0 to 2. I first walk to 1. Then I walk 1/2 more. Then I walk 1/4 more. Then I walk 1/8 more. Then I walk 1/16 more... etc. Will I ever reach 2? Well if time wasn't a constraint, then the answer is yes! Suppose S=1+1/2+1/4+1/8+1/16+..., whatever it is. Well then (1/2)S=1/2+1/4+1/8+1/16+.... Wait a minute, that's just the same thing as S without the first 1, or S-1. So S-1=(1/2)S. Solving for S gives S=2.

You might think "Well ok, infinite sums must evaluate to something if you keep adding smaller and smaller terms". However, this isn't true. The infamous harmonic series 1+1/2+1/3+1/4+1/5+1/6+... grows to infinity.

Did you know the square root of 2 is irrational?

What about the square root of 3?

What about Pi?

I think theorems as mysterious and subtle as the Fundamental Theorem of Calculus are just astonishing. In symbols, it says ∫ƒ(x) dx =F(b)-F(a). In words, this says the area under ƒ(x) between the x values a and b is equal to the length F(b)-F(a) (for physics people, don't worry, the units work out).

Who was Pythagoras? Why is the Pythagorean Theorem true? Why is it important?

It's facts like this that make a mathematician pursue the depths of logic and reasoning. It isn't about programming better software, designing better satellites, or building better structures. It's just about curiosity.

Unfortunately, in our education system curiosity is not a practical motivator for learning mathematics. Mathematics isn't seen as a fine art that has been refined over the millennia. It's nothing more than a tool that can be used to better our lives.

For me, I agree that mathematics is a tool. However, saying mathematics is nothing more than a tool is like saying the internet is nothing more than Facebook; perhaps it's partly true, but thinking of it this way completely narrows your view of what you can actually do with it. And when I say “…do with it…” I don’t mean the practical applications in the “real world”, whatever that even means. I mean the fact that I can use trigonometry and calculus to prove 1-1/3+1/5-1/7+1/9-1/11+…=Pi/4. I mean the fact that I can use algebra and calculus to prove Pi is irrational. I mean that I can then use these two facts to prove that there are infinitely many primes in a different way from the one I mentioned earlier! These kinds of facts are exciting in a different way than when we get excited about some job opportunity or an A on an exam, but that doesn’t mean it isn’t “real” excitement.

We need to pursue mathematics in a creative way. When I tried to inspire my students by showing them a cool little math fact here and there, I found that they tended to like me more and were more enthusiastic about learning math. It isn’t about persevering through it, like it’s some kind of torture. It’s about relishing in the spectacular sea of mysteries, journeying through the labyrinth of learning, and embracing the only study that just does puzzles for a living. We need to bring about the tales of truth, the poetry of proofs, and the weirdest form of wonder you will never fully appreciate.

However, the amazing thing about mathematics is how truly wonderful it is for me. Most people who see my attitude towards mathematics (including others who are reasonably adept at math) find this odd or misplaced, and I fully understand their lack of sympathy. Perhaps you are one of them. But I can assure everyone reading this there is something truly mysterious about mathematics that breaches the very foundations of astonishment and awe.

Take prime numbers for example. It's pretty clear that you can take any whole number and decompose it down into a product of prime numbers (i.e. 180=2⋅2⋅3⋅3⋅5). This might make you think "Which numbers are prime?" You start to list them out and find 2, 3, 5, 7, 11, 13, 17, 19,... wait, how many are there? ...23, 29, 31, 37, 41,... this doesn't seem to be ending or even repeating! Why should it never end? I mean, not that this really helps me build better bridges or spaceships or anything, but I just kind of want to know. Well, if it ever ended, you'd be able to multiply all primes together. Why should that matter? Because if you could multiply all primes together, you could also add 1 to their product, and this new number would need to be divisible by a new prime.

Or take Zeno's Paradox: Let's say I want to get from 0 to 2. I first walk to 1. Then I walk 1/2 more. Then I walk 1/4 more. Then I walk 1/8 more. Then I walk 1/16 more... etc. Will I ever reach 2? Well if time wasn't a constraint, then the answer is yes! Suppose S=1+1/2+1/4+1/8+1/16+..., whatever it is. Well then (1/2)S=1/2+1/4+1/8+1/16+.... Wait a minute, that's just the same thing as S without the first 1, or S-1. So S-1=(1/2)S. Solving for S gives S=2.

You might think "Well ok, infinite sums must evaluate to something if you keep adding smaller and smaller terms". However, this isn't true. The infamous harmonic series 1+1/2+1/3+1/4+1/5+1/6+... grows to infinity.

Did you know the square root of 2 is irrational?

What about the square root of 3?

What about Pi?

I think theorems as mysterious and subtle as the Fundamental Theorem of Calculus are just astonishing. In symbols, it says ∫ƒ(x) dx =F(b)-F(a). In words, this says the area under ƒ(x) between the x values a and b is equal to the length F(b)-F(a) (for physics people, don't worry, the units work out).

Who was Pythagoras? Why is the Pythagorean Theorem true? Why is it important?

It's facts like this that make a mathematician pursue the depths of logic and reasoning. It isn't about programming better software, designing better satellites, or building better structures. It's just about curiosity.

Unfortunately, in our education system curiosity is not a practical motivator for learning mathematics. Mathematics isn't seen as a fine art that has been refined over the millennia. It's nothing more than a tool that can be used to better our lives.

For me, I agree that mathematics is a tool. However, saying mathematics is nothing more than a tool is like saying the internet is nothing more than Facebook; perhaps it's partly true, but thinking of it this way completely narrows your view of what you can actually do with it. And when I say “…do with it…” I don’t mean the practical applications in the “real world”, whatever that even means. I mean the fact that I can use trigonometry and calculus to prove 1-1/3+1/5-1/7+1/9-1/11+…=Pi/4. I mean the fact that I can use algebra and calculus to prove Pi is irrational. I mean that I can then use these two facts to prove that there are infinitely many primes in a different way from the one I mentioned earlier! These kinds of facts are exciting in a different way than when we get excited about some job opportunity or an A on an exam, but that doesn’t mean it isn’t “real” excitement.

We need to pursue mathematics in a creative way. When I tried to inspire my students by showing them a cool little math fact here and there, I found that they tended to like me more and were more enthusiastic about learning math. It isn’t about persevering through it, like it’s some kind of torture. It’s about relishing in the spectacular sea of mysteries, journeying through the labyrinth of learning, and embracing the only study that just does puzzles for a living. We need to bring about the tales of truth, the poetry of proofs, and the weirdest form of wonder you will never fully appreciate.