My daughter was stuck on a particularly challenge mathematical proof. She felt like she was close, but couldn’t quite push it further to resolution. The problem was to show that:
(1+cot^2y)(cos(2y) + 1) = 2cot^2y
She had a sheet of identities she could use as reference. Not really knowing how to get there from here, she played around with identities that seemed like steps in the right direction only to hit a roadblock. Stooping over her shoulder, I tried to play around with it and struggled.
I finally had to get my own sheet of paper so I could play around with it on my terms – my mind just works better with pen on paper. I love problems like this.
Not that there’s any immediate application for it in my life currently. At one point I worked on developing software for modems in military radios and there is, believe it or not, application for this in the real world. Cosines and sines are ways to model waves mathematically. Waves are all over the natural world – oscillating impulses of energy propagating through space. It all begins with a unit circle – a circle with a radius equal to one. To represent a position on that circle in terms of the angle, cosine represents the value of the x position, sine gives the y-position. Rotating around the circle at a certain speed, the x and y position oscillates between one and negative one in a periodic fashion that looks a wave. The frequency of the wave is the speed of the rotation. The mathematics is deeply embedded in my brain from years of calculus starting in high school extending deep into an Electrical Engineering undergraduate degree and as I said, practical application when I coded modem software for military radios not too long after graduation.
Anyway, it turns out the solution to the proof is fairly straightforward but requires a bit of pattern recognition, searching for steps that get you ever closer to the goal, the steps are as follows:
Step 1: (1+cot^2)(2-2sin^y) => using the idenity cos2y = 1-2sin^y, and then adding the 1’s together.
Step 2: 2-2sin^2y + 2cot^2y – 2sin^2ycot^2 => multiplying everything out.
Step 3: 2-2sin^2y -2sin^2(cos^2y/sin^2y) + 2cot^2y => convert cot^2y to cos^2y/sin^2y
Step 4: 2-2(sin^2y + cos^2y) + 2cot^2y => cancel out the sin^2y and factor out the 2.
Step 5: 2-2 + 2cot^2y => sin^2y + cos^2y = 1, pythagorean theorem,
Proof: 2cot^2y = 2cot^2
There’s a certain amount of endorphin kick solving this, but it’s also frustrating to get stuck on it. Frustrating and demoralizing. My daughter gets stuck on these problems, more often than not. And I know having your parent help you sucks for teenagers. Everyone’s brains are wired differently. Training has something to do with it as well – nature vs. nurture.
During the pandemic, with schools basically closed, I’m trying to get my son to take a music theory course with me on coursera. The video we watched today, we had to recognize chords and identify whether they were tonic or dominant. I got the basic idea, but I had trouble recognizing them by ear. My son could hear it far better than me – he’s better trained musically. Or maybe he has musical genes (from my wife) I just don’t have.
I don’t know.
Showing my daughter the proof she screamed, “why do I have to know this stuff anyway”. I don’t have good answers for it. It’s a basic existential question on the utility of school generally. Everyone understands the basic utility of math through algebra, the basic utility of reading comprehension and the importance of learning to write well. But school, especially a college prep school, pushes students far past this – reading books not especially enjoyable to read, especially for a young person without life experience, then being forced to write something enlightening about this book they could barely get through. Why do we teach writing in this precise way?
The problem with these question though is I have no idea. She’s 17. Nobody really knows what knowledge will end up being useful for her down the line. Most of it will largely not be. But I think there’s something essential about learning as much of the world as we possibly can, so we can make sense of it, recognize our place in it, and then perhaps have a shot to make some small contribution within it.
Steve Jobs took calligraphy in school for the fun of it. He then later revolutionized fonts on apple computers, largely because of this training he happened upon. Likely nothing that extraordinary will come out of any of this. Sometimes it’s just fun to use our brains – or not.
I don’t know.