Patterns from ideas
“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”
– G.H. Hardy
It is a tragedy that mathematics is seen as anything but empowering and anything but beautiful. Painters might try to express an idea through the visual, and poets through language, but mathematics take the opposite approach. In mathematics, the ideas and questions derive the patterns and tools. And so, while aesthetic appeals and written norms shift over time or run into barriers of culture and language, mathematical ideas persist above the fray. Mathematics is incredible because it is adaptable, and uniquely human.
For instance, we can suggest the idea that things with seemingly different attributes can hold a unifying relationship (or pose the question “is there any way to compare circumstances that hold different values?”). Well, one man may give 2 of his 20 pieces of silver, and another man 5 of his 50 pieces of silver, but we say that – despite their different circumstances – both men have tithed. Why? It is because 2 and 20 can be found to have a unifying relationship with 5 and 50 – the same ratio of 1:10. See how this idea persists and this question is addressed: the Greeks encoded the idea of an ideal, “golden ratio” into the Parthenon (and as such the golden ratio now goes by φ, or phi, in honor of the sculptor who supposedly built the statues there), Leonardo da Vinci illustrated de divina proportione, and today we seem to find this ratio occurring naturally in the depths of the seas and the vastness of the cosmos above.
We look to ourselves and decry injustice since we do not find a unifying relationship when comparing the rates of incarcerated black and Latino men with that of white men. We debate the equity of taxes: should tax rates hold a constant ratio regardless of total income, or should the ratio of cost-of-living to total income be elevated as a more important consideration? Values may evolve, language may change, but mathematical ideas and their compelling questions remain as focal points surrounded by such ever-changing patterns and conclusions. How incredible it is that humans can discern and communicate these patterns that exist all around us.
Mathematics is not compelling and it is our fault
Yet much of modern mathematics education subverts the natural flow from mathematical idea to mathematical pattern, tool, or conclusion. In fact, teachers often forget that there is even a compelling math idea or question that spins the wheel of mathematical intrigue. We simply hold up the derived pattern or tool and expect our students to marvel at its usefulness.
Think of it like this – if you were to explain Superman to me by showing him turning back time by spinning the Earth backwards (or traveling faster than the speed of light, whatever you believe), what would be the result?
Besides taking some issue with the physics involved, I’d probably end up with more questions than when we started (wait, so this guy can fly? In space? He hallucinates voices from clouds? What’s with the spandex?). If I like you I might take your word for why Superman is great – “alright, so he has super powers that he uses to save people,” but I’m certainly not compelled by Superman or see what he has to do with, well, anything.
This is one reason why superhero movies always have the origin story – so that we can understand the implications of the hero’s actions and can relate to them as characters. It ups the stakes of the climax in act 3 of the story, and ensures we’re really rooting for our hero to win. If I actually get to watch the entirety of Superman then I’ll find the ideas of loneliness and duty and love. If I understand the context, the significance of what Superman does here (going against the ethereal voice of his biological father Jor-El and instead following the advice of his “Earth father” Jonathan Kent to turn back time and save Lois Lane) is incredibly more compelling. Superman turning back time simply becomes a tool to answer the actually meaningful question of “when should we follow rules? Who should we listen to when two rules conflict?”
Math lessons are often akin to showing students Superman flying around the Earth and expecting them to simply appreciate the significance of this moment. It is no wonder that students feel intimidated, disengaged, and disinvested in the beauty of mathematics. From my observations in math classrooms this quarter here is a sample of quotes I have collected from students in response to the question “What is your favorite subject? Why?” and if it wasn’t math, “What keeps math from being your favorite?”:
“I don’t like measurement because there is too much to do.”
“I don’t like equations because they’re confusing.”
“I like science more than math because I get to make stuff.”
“I like math because I’m good at addition.”
“I like math because I like counting.”
“I liked algebra because it was the basics. This (geometry) is harder.”
“I never liked math because all of those numbers.”
“I like science because I get to explore things.”
“I like science because we get to create things.”
“I like social studies because I can learn about different people and my ancestors.”
I get some of the following themes from these and other quotes…
So how can mathematics compel students?
Just like good storytelling, there isn’t one answer to the question of “how do we make this compelling?” Luckily there have been some incredible folks who have put a ton of work already into answering this (I’ll bring in the work of Stanford/Jo Boaler, Dan Meyer, Grace Chen, Rico Gutstein, and others), and I have leaned on some common threads in their incredible work to start constructing a framework. Repeatedly we see that conceptual, idea-driven planning empowers and compels students in learning mathematics. In my next post I will discuss how to construct lessons from a mathematical idea in greater detail, but as a preview the general framework I have sketched out is as follows:
- Gather Kindling
- Define a compelling question/problem that the mathematical idea addresses/overcomes.
- Define how students will demonstrate application of the mathematical idea to address/overcome the compelling question/problem.
- Sketch the intermediate knowledge, skills, language, and reasoning that bridge the question/problem and the answer/solution.
- Set a Spark
- Access cultural, academic, and/or critical student knowledge to spark students’ connection to the compelling question/problem.
- Ignite the Fire
- Present opportunities for students to construct new knowledge about the mathematical idea(s) that addresses the compelling question/problem.
- Fan the Flames
- Present opportunities for students to apply new knowledge to address the compelling question/problem.
- Present opportunities for students to extend or connect new knowledge towards other compelling questions/problems.
- Present assessment opportunity for students to demonstrate application of the mathematical idea in answering the compelling question/problem.
- Provide some form of resolution to the compelling question/problem and set up a future problem.
I look forward to sharing some of the what, how, and why of the above framework with you all in the next post.