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.
Sometimes I get asked how my son came to be so accelerated in math. I’m sure some assume we must be doing a shallow just-the-basics curriculum or that I stand over him Tiger-mom style as he whips through pages of worksheets. Neither is the case, and the math study we do is both very deep and the most joyful part of our school day. There are a few secrets though:
- Math is his thing. It happens to be my thing too, so this creates a synergy between teacher and student that is hard to force. Simply put, it makes magic. This is the most important ingredient to math acceleration. I would not be able to coerce the amount of learning we do: it comes from the intrinsic interest of the student.
- We homeschool, and do so year-round. Nothing beats a one-on-one learning environment. With homeschooling we can take time to work through concepts that require more attention and skip the busywork, all with my student getting immediate feedback on mistakes. The lack of a long summer break means we can keep moving forward without a loss of learning or time wasted on extensive review, and although we do take plenty of vacation time, we also do math on most weekends and while traveling. I estimate the combination of summer and weekend work probably doubles the amount of math we get done in a year compared to a student in public school.
- I make it fun. Hands on activities, games, and a unique approach to skill-building along with humor and a light hearted attitude by me, keeps us moving forward even when work gets tough.
- Variety. The first three points might be obvious but the issue of “what to study” in the first place is not quite as clear and what I’m here to talk about because it applies to all learners, accelerated or not, math-loving or less so.
The biggest lesson I’ve learned in working with gifted children is that no matter how brilliant the student is, stamina and attention span are still strongly tied to chronological age. Without a sustained attention span the amount of material one can cover is going to be limited. What does this mean in practical terms? It means even if you have a rare Kindergartner who can do long division, they probably can’t do a whole page of practice problems the same way a fourth grader could. In fact, they can probably only do one problem before they are bored or mentally exhausted. Even students who are not extremely advanced will get bored with too much repetition, and this can hinder math progress as well as leave students thinking math is uninteresting and uncreative when nothing could be further from the truth.
My radical solution is to give up on the idea of mastery, albeit temporarily. Some educators (though fortunately not all) are obsessed with the idea that students must memorize their multiplication tables, perform spectacularly on timed tests, and get 100% on assigned work before they can move onto more interesting stuff, but this is entirely backwards. The more advanced the student is, the more backwards it is. To let you in on a secret: the narrow track we’ve created for math education is almost entirely artificial. Math is such a broad and rich field that there’s simply no reason you have to stay on path. You can mix arithmetic practice with probability with some beginning algebra with number theory and cover more than one topic at once. I have especially high regard for discrete math which is all but ignored by traditional K-12 schooling. Of course all students should master multiplication tables and become automatic at traditional algorithms, but not to the exclusion of other learning – by exploring multiple threads of math at the same time its much easier to keep interest high, tedium low, work longer, and gradually build stamina for more difficult work.
Just the other day I came across this profound statement in Art of Problem Solving: The Basics, Volume I:
We strongly feel that a student should learn all subjects simultaneously. There are two reasons for this. First, it helps to convey the interconnectedness of it all; how geometry naturally leads to coordinates and how those coordinates make it easy to define conic sections and the complex plane; how counting leads to probability, the binomial theorem, and number theoretical ideas. Second, it all sinks in better. Overloading on a single subject can cause students to acquire a surface understanding which doesn’t connect to any deeper comprehension, and is thus rapidly lost.
In my next post I’ll give some specific examples of how to include more variety in your students math education.