Maria was precocious, and published an essay advocating higher education for women when she was nine years old. The essay was actually written by one of her tutors, but she translated it into Latin and delivered it from memory to an academic gathering in the garden of the family home. Her father also arranged for her to debate philosophy in the presence of prominent scholars and public figures. She disliked making a public spectacle of herself and asked her father for permission to become a nun. When he refused, she extracted an agreement that she could attend church whenever she wished, wear simple clothing, and be spared from all public events and entertainments.
This is on thing I say to all students. That to be a really good mathematician, you have to be lazy. Ok, so you look at that, and you sort of say right, you could go straight in, integrate it, put the values in and get zero. Or you could step back for a minute and say, is there a trick I could use that gets the answer without having to do a lot of work? And the answer is yes, there is. So before you dive into something, sit back, look at the problem, and see, is there a clever way I could do this that’ll save me time?
One of the strengths of Zena Hitz’s book Lost in Thought is in the variety of intellectual examples she offers. Hitz doesn’t limit examples of the intellectual life to literature and philosophy. Instead she reminds us mathematics and science are vital intellectual disciplines.
Her two major examples include Albert Einstein and Andre Weil the French mathematician. Hitz uses their two career paths to explore how environments of failure – Einstein’s stint at a patent office, Weil’s time in prison, provided the ingredients for mathematical breakthroughs to flourish.
The ingredients being the absence of needing to comply with their profession’s expectations. Their environments of failure allowed them to pursue their ideas for the sake of curiosity.
On Einstein’s patent office stint:
But it is a cloister for Einstein, since in the office there were no hotshot professors to impress, no university administrators to placate, no students to whom he had to justify his existence. It is, then, chiefly a place where the love of learning is put to the test, where ambition is frustrated, where his work has to run on its own power without the grease of seeking out carrots and avoiding sticks. In the quiet of the patent office the beauty of the structures of nature can take hold of him and display itself with clarity.
One might think that the reasons Weil produced better mathematical work in prison are straightforward: more free time, fewer distractions of ordinary life. But Weil jokes about prison’s advantage for “pure and disinterested research” and, echoing Einstein, praises the beauty of his theorems. So he too suggests that his work was nurtured by separation from social or political agendas, competition, social hierarchy, objects of ambition, the expectations of others. The pursuit of beautiful theorems might elsewhere be crowded out by things that seemed more pressing but that ultimately mattered less.
I was that stereotypical third grader who scoffed at his times tables and said, “When will we ever use this when we grow up?”
My theory on why many kids have a poor attitude towards mathematics is that we’re subconsciously taught to avoid problems. Whereas Mathematics is all about embracing problems.
Math wants you to make friends with problems. Spend time with problems, not run from them.
Problems are there to be solved!
True enough, the Altitude-on-Hypotenuse Theorum has yet to be an agenda item on any of my zoom calls. But the skill of problem solving still punches the clock everyday.
And the strategies for solving a math problem, can also be applied to any real world problem.
Grant Sanderson’s 7 tips for solving hard problems are below.
My real-world application take is in italics.
Hopefully at the end, you’ll hate math less.
Use the defining features of the setup
What are the rules of the game your playing? What are the inherent limitations?
Give things (meaningful) names
Naming things helps your mind organize ideas and outline solutions.
Leverage Symmetry
Identify what is similar. Are there any patterns? Have we seen this before in a previous problem?
Try describing one object two different ways
This reminds me of a practice the economist Tyler Cowen has. To improve his understanding of an argument, he’ll write out the point of view of the argument he opposes. Try the opposite of whatever strategy your using.
Draw a picture
Drawing, like writing is a form of thinking. As maker Adam Savage has stated: “Drawing is your brain transferring your idea, your knowledge, your intentions, from the electrical storm cloud at its center, through the synapses and nerve endings, through the pencil in your hand, through your fingers, until it is captured in the permanence of the page, in physical space. It is, I have come to appreciate, a fundamental act of creation.” Doodle. Stickfigure. Sketch. Create a visual form of the problem.
Ask a simpler version of the problem
What’s the smallest part of the problem we can solve first?
Floodlights and Goalposts 