I remember the moment well – it was almost bedtime, and my son, already in pajamas, asked me about the derivation of the quadratic formula. Yeah, yeah, that’s kind of thing just happens in our family. Since it had been a while (like 25 years) since I had studied this myself I looked on the internet for the best way to present the idea of completing the square.
I loathe Khan Academy, but decided it might be useful for this kind of refresher. Boy was I wrong! The opening of the video started off ok as he discussed the fact that quadratic equations that are perfect squares are super easy to solve. Check! If we can make any quadratic look like its a perfect square we can solve that easily too. Great! I won’t bore you with the details, but needless to say it went downhill from there as the video immediately dived into a bunch of manipulations and never once explained things conceptually. You see, completing the square is actually really easy to understand if you actually draw a square. Hey look, we’re taking a rectangle and cutting off some bits to make it a square and adding another little bit to finish it off! You can draw it out and show the student visually exactly what’s happening. Here’s an example of a site that gets it right.
This is the crux of my problem with Khan Academy – a lot of procedural mathematics with little focus on understanding. It might be ok for test prep review when concepts have already been mastered, but I absolutely cringe when I hear of homeschoolers using it as their primary math program or parents of gifted kids using it to accelerate. These kids need to be challenged, inspired, and taught problem solving skills, not memorization of mindless algorithms. For a child that is behind or already turned off from math, seeing numbers fly around the screen with no idea of the underlying concepts could lead to even more confusion. Real math is not about manipulation, its about knowing how to think and set up a problem.
I do think there are some good videos out there for learning math, for instance AOPS has some that follow along with their books, Education Unboxed has outstanding teacher training videos for using Cuisenaire rods, and The Great Courses has some cool lectures that we’ve really enjoyed. Additionally, online courses can be a great way for motivated older students to learn. However, when it comes to elementary and middle school math in particular, I really believe that human interaction, discussion, hands-on manipulatives, and instant feedback are essential components of learning how to think about numbers. For a more detailed and articulate critique of Khan Academy I highly recommend this article from the Washington Post: Khan Academy: the revolution that isn’t. The title says it all.