Inspiration and Mathematics

In my multivariable calculus class this year, we’ve been holding a regular “book club” during our long blocks. (Don’t ask… we have a rotating schedule and every seven school days we have a 90 minute class.) Right now we’re reading Edward Frenkel’s Love and Math: The Heart of Hidden Reality.

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In the introduction, Frenkel criticizes the teaching of math:

What if at school you had to take an ‘art class’ in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo da Vinci and Picasso? Would that make you appreciate art? Would you want to learn more about it? I doubt it. [1] … There is a common fallacy that one has to study mathematics for years to appreciate it… I disagree: most of us have heard of and have at least some rudimentary understanding of such concepts as the solar system, atoms and elementary particles, the double helix of DNA, and much more without taking courses in physics and biology. And nobody is surprised that these sophisticated ideas are part of our culture, our collective consciousness.

So many whirling thoughts came up while I was reading these passages. One thought led to another to another to another. Writing this post is an attempt to start recording them and to get them a little more codified in my mind! It is still going to be a hot discombobulated stream-of-consciousness mess. #sorrynotsorry

I wonder if I asked my kids “what is mathematics?” right now, what they would say. I am doubtful that their answers will include the adjectives and verbs that I personally would say.

I wonder if I asked my kids “what is is going on in the field of mathematics?” right now, what they would say. I’m guessing a lot of blank stares.

I wonder what my kids would say if I asked ’em “what courses exist in college for mathematics?”

I wonder what my kids would say if I asked them to name a mathematician who is alive?

I wonder if the word “mathematics” was changed to “astronomy” or “physics” or “biology” if their answers would be different.

There are ideas that my kids learn about modern physics (in popular culture, in classes) which spark their imagination, blow their minds, make them curious and full of wonderment at the weirdness and strangeness of the world. Special relativity. Quantum mechanics. Quarks and the structure of atoms. They are exposed to these ideas, even if they don’t have the mathematical capabilities or abstraction to attack them rigorously. And these ideas have a powerful effect on some kids. (I know I wanted to be a physicist when I first learned about these ideas!)

But what do my kids learn about modern mathematics — from school or popular culture? Are there any weirdnesses or strangenesses that can capture their imagination? Yes! Godel’s incompleteness theorem. Space filling curves. Chaos theory. The fact that quintic and higher degree polynomials don’t have a general “simple” formula always works like the quadratic formula. Fractals. Higher dimensions. Non-euclidean space. Fermat’s Last Theorem. Levels of infinity. Heck, infinity itself! Mobius strips. The four color theorem. The Banach-Tarski paradox. Collatz conjecture (or any simply stated but unproven thing). Anything to do with number theory! Anything to do with the distribution of primes! But do they capture students’ imaginations? No… because they aren’t exposed to these things.

Where in our curriculum do kids get inspired? Where does awe and beauty fit into things? When do we ever explicitly talk about beauty in mathematics? When a kid has a rush of insight and makes a visible gasp, what do we do in that moment? What has to already be in place for a kid to make that gasp?

We need to expand how we frame mathematics in high school so it isn’t seen as “Algebra I, Geometry, Algebra II, Precalculus, and Calculus.” These course names aren’t mathematics.

We need to consciously and regularly introduce a bigger and more modern world of mathematics to our kids. How? Having kids read when the New York Times publishes an article about a mathematician or mathematical result! Using resources like Numberphile and Math Munch and Vi Hart videos. And… I don’t know.

We need to provide space and time for kids to explore an expanded vision of what math is, and have choice in having fun and playing with this expanded vision of math. (My explore math project is an attempt to do that — website here, and posts one, two, and three here.)

We need to have mathematical lore, stories we can tell students. Galois duel! Ramanujan’s inexplicable genius! What are mathematical stories that can be passed down from generation to generation? (Does a good resource exist for this? Tell me!) [Update: The internet went down when I was going to edit this post by mentioning we need stories and people who aren’t just white men!]

Do we have Feynman or degrasse Tyson-esque figures we can point to? Dynamic popularizers of the subject that have entered the public consciousness?

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Maybe what I’m trying to say, if I had to distill everything down to the core, is:

(1) Can we find a way — in our existing schools with our set curricula and limited time — to expand kids notions of what mathematics is by exposing them to notions external to the Alg-Geometry-Alg II-Calc sequence. And if we can do this well, will it help inspire more kids to be interested in mathematics? 

(2) Are there ways for us to keep an focus on beauty, the unexpected, awe, and wonderment in our classes? And find ways to record, highlight, and amplify those moments for kids when they happen? Why I love mathematics is because of all of these moments! Maybe focusing on them would help kids love mathematics?

UPDATE: Annie Perkins has a great blogpost which captures some of the exact same ideas and feelings here. But she’s more eloquent about it. So read it. Updating it here so it is archived for my own thinking on this.

 

[1] This notion has so many resonances with Paul Lockhart’s A Mathematician’s Lament. Which I highly recommend.

 

 

17 comments

  1. For stories about mathematicians try to get hold of “Men of Mathematics” by E T Bell. Yeah, well, there weren’t many women involved in the old days, but you should find something elsewhere on Ada Lovelace, who worked with Charles Babbage.

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  3. I generally use technical publications (e.g., IEEE Spectrum, ACM, SWE, etc) to connect kids to some of the practitioners of mathematics. There is also the Association of Women in Mathematics.

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  4. I wholeheartedly agree with everything you have written. The real challenge is not for you, or me, or other commenters to bring the passion and beauty back into mathematics, because we probably already endeavour to do this. The real challenge for us is to bring this message to the maths teachers who will not read this blog or any of the fantastic resources it mentions. And I reckon that’s probably at least 70% of math teachers out there. Our mission begins with bringing the passion, beauty and love of mathematics back to teachers first- through conversation, professional development, membership in some of the many amazing mathematics focus groups. If we can reinspire more teachers, it may inspire more students. Oh, and thank you for your blog, which inspires me.

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  5. I’ve been a subscriber for about a year now, but this is the first time I felt compelled to respond to one of your posting. What follows is in response to your request for ” stories and people who aren’t just white men!”
    I’ve been teaching at a private all-girls high school in Brooklyn Heights/Metro Tech area for the past 3 years. Since my calculus class is not tied to a year-end regents, I have some flexibility to make little detours from a rigid curriculum. So, for the past 3 years, I’ve assigned a written/oral project on Women in Mathematics. The assignment is typically to be done during March, Women’s History Month. This year, because of the Easter calendar, the oral presentations were done at the beginning of April, and lasted for 3 days. Also, for the first time, there was a strong representation of 20th century mathematicians and women of color.
    I have my students report on the culture and societal norms of their mathematicians in addition to their mathematical achievements; then they give an oral presentation and share their particular mathematician with the entire class. The oral presentations follow in chronological sequence. The idea is for them to see how conditions towards women mathematicians changed over the course of time and to become aware of the fact that several women were mentored by other women.
    A google search will turn up lots of resources for the reports. The main ones I’ve given the students are: https://www.agnesscott.edu/lriddle/women/women.htm. This is the main data bank for the topic. Another one is: http://www.math.buffalo.edu/mad/wohist.html, which is a site for Black Women in Mathematics. Finally, Evelyn Lamb writes a blog for Scientific American, called Roots of Unity, where she has posts on women mathematicians.

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  6. Thank you for articulating this so well. It is hard to say why math is treated so differently than other content areas – both by students and by their teachers. In many English classrooms, it is safe to write a bad poem, or in science classrooms, students are willing to make conjectures or try things before they have learned the “right” way. But that willingness to take risks and try things is nearly absent from most math classrooms. Most students wait to be shown the way to get to a specific answer that is “correct.” The strongest students are often the most uncomfortable when asked to do exploratory work.

    I am deliberate in including bits and pieces of interesting math in my algebra classes, even if they fall outside of my regular curriculum. For example. fractals are of particular interest to me, and the big ideas are accessible to even young students. But I admit that it is sometimes uncomfortable being fully transparent about this with colleagues. The class time spent on this means time taken away from the regular curriculum – for which we already don’t have enough time. This is true, but for me, helping students form a positive relationship with math and cultivating their curiosity clearly outweighs coverage. We need to prioritize this curiosity, and build in ways to make our classrooms safe places for students to explore and be creative.

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  10. Three “coffee table” math books for the classroom that spark kids imaginations (and conversations!) The Ultimate Book of Optical Illusions, Al Seckel, Sterling Publishing, 2006; The Mathematics Devotional, Clifford Pickover (I know you recognize that name!) AND The Math Book, also by Clifford Pickover, both published by Sterling Press, 2014 and 2009 respectively. Hmmm. Maybe I need to check out this Sterling Publications book list- they may have more!

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