Po-Shen Loh’s Quadratic Method Isn’t Groundbreaking, but That’s Not the Point
When I was in high school, I watched Po-Shen Loh give a talk on this really amazing formula that relates 1/89 and the Fibonacci sequence.
The fact that the entire Fibonacci sequence could be wrapped up so elegantly with a single fraction blew my mind, and Po-Shen explained it so playfully and effortlessly that I still remember the result to this day.¹ I clicked on Po-Shen’s recent video about a new way to solve quadratic equations hoping to be similarly blown away by some grand insight into the nature of parabolas, but what I got instead was, at first glance, a little disappointing.
The 4-minute clip was underscored by a stereotypically intellectual soundtrack, complete with rhythmic strings and tinkly piano arpeggios. Visuals included extreme whiteboard close-ups, shots of Po-Shen flipping through a perfectly stacked collection of ancient mathematical tomes in a room way too dark to be good for reading, and at one point, a literal lightbulb turning on. The whole thing felt a bit over-the-top.
As for the video’s mathematical content, recall that to find the roots of x² + bx + c, you want to find two numbers that sum to b and multiply to c. Po-Shen’s idea was to observe that the two numbers must be the same distance from their average, so we may write them as b/2 + u and b/2 – u, then solve (b/2 + u)(b/2 – u) = c for u. To me this seemed not much better than using the quadratic formula or my favorite technique, guess-and-check. I felt like I was missing the point. Why was this video getting so much hype, to the point where Po-Shen has a new section on his website entirely dedicated to it?
The answer eventually dawned on me: I am not the video’s intended audience. As a math major, I’ve come across methods far more disgusting² and unmotivated than, say, completing the square. But if I put myself in the shoes of a young student learning about quadratic equations for the first time, I can see how Po-Shen’s approach has merit. Though his approach may be equivalent to traditional methods, it feels easier (perhaps because it draws upon the intuition that parabolas are symmetric), and a small change in how material is presented could make a huge difference in students’ understanding.
After my initial let-down and subsequent appreciation came a twinge of embarrassment: How come I hadn’t thought of Po-Shen’s method before?? The methods I learned in school seemed alright to me, so I stuck with them. Never did I consider trying something different (despite considering myself someone who believes in math as a creative art and all that), which I think this speaks to the prescriptive power of education. All too often, education ends up being about learning a specific set of facts, with no room or incentive for students to explore outside the confines of a curriculum. I think Po-Shen echoes my sentiments at the end of the paper he wrote expanding on his video. He writes,
May this story encourage the reader to think afresh about old things; seeing as how progress was made on this 4,000 year old topic, more surprises certainly await the light of discovery.
Regardless of your (loving? hateful?) relationship with the quadratic equations, let Po-Shen’s (b/2 + u)(b/2 – u) = c be a reminder to think afresh the old things in your life.³
1. The same cannot be said for the material on my upcoming algebra exam…!
2. Most proofs from this Fourier analysis class I took used a technique called “produce inequalities out of nowhere that coincidentally bound the term exactly how the theorem requires.”
3. I still think the Fibonacci thing is cooler though.
