Dr. James Tanton has produced a top-quality series of instructional videos on Quadratic Functions. I watched all of them in an afternoon — it was the visual equivalent of a book that I couldn’t put down! Check out his fantastic material right over here.
Following is a summary of what I found so compelling about Dr. Tanton’s work on quadratics.
Coherence. James begins by treating quadratic sequences of numbers, then proceeds to discuss the algebra of quadratics, followed by material on graphing quadratics, then fitting quadratics to data, then applications of quadratics. The whole course feels like one continuous story — as it should, but often doesn’t. We know that the general study of quadratics — ellipses, hyperbolas, circles, and parabolas — hinges on the technique of “completing the square.” James rightly places his emphasis on this important technique, which he employs throughout the whole instructional sequence. Need to solve a quadratic equation? Just complete the square. Want to find some special features of the graph of a quadratic function? Just complete the square. Want to know how to derive that classic Quadratic Formula? Just complete the square.
Rigor. All of the mathematics James presents is logically sound and inter-connected. Nothing “comes from left field.” And his insights were incredibly instructive for me. For instance, I learned how to take a quadratic sequence and develop a formula for it *without using the regression feature on my calculator!* James teaches us to view sequences as combinations of simpler sequences — elegant, powerful, beautiful! Learn his technique in Lesson 1.2. Another instance: James points out how to easily graph functions like y = 3x^2 + 21x + 100 by exploiting symmetry — be sure and check out Lesson 4.4! And finally, James makes the Quadratic Formula completely transparent in Lesson 3.1. I knew how to do this already, but in a much uglier way! His method is crisp, clean and consistent with his earlier lessons.
Focus. James shows students how to write formulas for quadratic sequences, how to solve quadratic equations, how to graph quadratic equations, and how to use modeling with quadratics to solve problems. What more could you want? Well, according to the standard curriculum, there are several more things one could want, but James lays out his case against these extraneous items! In a nutshell, he argues that the theory of quadratic functions is essentially a story about *the symmetry of squares.* This observation applies equally well to equations and to graphs (not to mention sequences). His fundamental criterion for including/excluding topics is this: Does it contribute to, or take away from, the story of symmetry? Here are some of the things James purposefully omits from his course on quadratics:
- The general Quadratic Formula (he considers this optional material)
- The Discriminant of a Quadratic Expression
- Factoring (!!!)
You can read James’s reasons for excluding these topics in the material in Lessons 7.1-7.4. I personally found his arguments very natural and compelling, but I suspect that many will find his view extremely controversial! Add your view to the comments section below.
Entertainment. James begins with a short sketch of the history associated with the study of Quadratics. Galileo and the Leaning Tower of Pisa! I was hooked, and my students were hooked. He employs a light and humorous style — achieve victory over Queen Quadrika! But first we have to fend off attacks from the Quadrilites!
Attention to ideas, not jargon. James scrupulously avoids the use of jargon — he does not use the standard terms “parabola,” “vertex,” or “axis of symmetry” until the very end, after students have learned to make graphs! On the other hand, he *does* bother to explain the term “quadratic.” Doesn’t “quad” mean 4? So why do polynomials with x^2 get called “quadratics?” Listen to James’s answer in Lesson 2.1. Note that this is exactly the opposite of the standard approach — students will have no idea why the very object of study is called “quadratic,” and yet they will be bombarded with jargon such as “vertex” and “axis of symmetry” from day 1.
Human. About 0.01% of the time, James makes a calculation error! I’m not being a snotty critic — quite the opposite. I feel that it’s important for students to see that even the “best and brightest” mathematical minds are apt to make mistakes occasionally. Too often I think students are made to feel that there are “those that get math,” and “those that don’t.” I’m therefore grateful that James’s presentation isn’t *too* perfect.
Effective. It’s always legitimate to ask the question: “That may work well on a whiteboard. But does it work with students?” I have been using James’s materials in my Honors Algebra 2 classes for the last week. Here is a sample of student work on a “Level 3” problem:
Go ahead and check out this one-minute introduction to the study of Quadratics! And James — if you’re reading this, I’m eager to see more material under the “Courses” page!