Category: Uncategorized

Teaching the Chain Rule inductively

I taught Calculus I every spring between 1996 and 2008. Perhaps the hardest topic to teach — at least for me — in the entire course was the Chain Rule. In the early years, I would show students the technique, but it seemed like my students accepted it on faith that their professor knew what he was talking about it. Also, it took them quite a while to become proficient with the Chain Rule… as opposed to the Product and Quotient Rules, which they typically mastered quite quickly (except for algebraic simplifications).

It took me several years before I found a way of teaching the Chain Rule so that the method really sunk into my students by the end of the class period. Here’s the way that I now teach the Chain Rule.

On the day that I introduce the Chain Rule, I teach inductively (as opposed to deductively). At this point, my students are familiar with how to differentiate y = x^n for positive and negative integers n, the trigonometric function, and y = \sqrt{x}. They also know the Product and Quotient Rules.

I begin class by listing a whole bunch of functions that can be found by the Chain Rule if they knew the Chain Rule. However, since my students don’t know the Chain Rule yet, they have to find the derivatives some other way. For example:

Let y = (3x - 5)^2. Then

y = (3x - 5) \cdot (3x -5)

y' = 3 \cdot (3x -5) + (3x -5) \cdot 3

y' = 6(3x-5).

Let y = (x^3 + 4)^2. Then

y = (x^3 + 4) \cdot (x^3 + 4)

y' = 3x^2 \cdot (x^3 + 4) + (x^3 + 4) \cdot 3x^2

y' = 6x^2 (x^3 + 4)

Let y = (\sqrt{x} + 5)^2. Then

y = x + 10 \sqrt{x} + 25

y' = 1 + \displaystyle \frac{5}{\sqrt{x}}

Let y = \sin^2 x. Then

y = \sin x \cdot \sin x

y' = \cos x \cdot \sin x + \sin x \cdot \cos x

y' = 2 \sin x \cos x

Let $y = \sin 2x$. Then

y = 2 \sin x \cos x

y' = 2 \cos x \cos x - 2 \sin x \sin x

y' = 2 (\cos^2 x - \sin^2 x)

y' = 2 \cos 2x

The important thing is to list example after example after example, and have students compute the derivatives. All along, I keep muttering something like, “Boy, it would sure be nice if there was a short-cut that would save us from doing all this work.” Of course, there is a short-cut (the Chain Rule), but I don’t tell the students what it is. Instead, I make the students try to figure out the pattern for themselves. This is absolutely critical: I don’t spill the beans. I just wait and wait and wait until the students figure out the pattern for themselves… though I might give suggestive hints, like rewriting the 6 in the first example as $\latex 3 \times 2$.

This can take 20-30 minutes, and perhaps over a dozen examples (like those above), as students are completely engaged and frustrated trying to figure out the short-cut. But my experience is that when it clicks, it really clicks. So this pedagogical technique requires a lot of patience on the part of the instructor to not “save time” by giving the answer but to allow the students the thrill of discovering the pattern for themselves.

Once the Chain Rule is discovered, then my experience is that students have been prepared for differentiating more complicated functions, like y = \sqrt{4 + \sin 2x} and y = \cos ( \sqrt{x} ). In other words, there’s a significant front-end investment of time as students discover the Chain Rule, but applying the Chain Rule generally moves along quite quickly once it’s been discovered.

How our 1,000-year-old math curriculum cheats America’s kids

A colleague recently pointed out an op-ed piece written by Prof. Edward Frenkel, a mathematics professor at the University of California. From his concluding paragraphs:

Of course, we still need to teach students multiplication tables, fractions and Euclidean geometry. But what if we spent just 20% of class time opening students’ eyes to the power and exquisite harmony of modern math? What if we showed them how these fascinating concepts apply to the real world, how the abstract meets the concrete? This would feed their natural curiosity, motivate them to study more and inspire them to engage math beyond the basic requirements — surely a more efficient way to spend class time than mindless memorization in preparation for standardized tests.

In my experience, kids are ready for this. It’s the adults that are hesitant. It’s not their fault — our math education is broken. But we have to take charge and finally break this vicious circle. With help from professional mathematicians, all of us should make an effort to learn something about the true masterpieces of mathematics, to be able to see big-picture math, the way we see art, literature and other sciences. We owe this to the next generations.

Here’s the whole editorial: http://www.latimes.com/opinion/op-ed/la-oe-adv-frenkel-why-study-math-20140302-story.html

I also should point out the thoughtful critiques of this article from mathematics educators that were published by the Los Angeles Times: http://www.latimes.com/opinion/op-ed/la-le-0308-saturday-math-teaching-20140308-story.html

Encouraging Students to Tinker

A recent blog post from Math Ed Matters had the following pedagogical insight:

How do we encourage students to tinker with mathematics? As a culture, it seems we are afraid of making mistakes. This seems especially bad when it comes to how most students approach mathematics. But making and then reflecting on mistakes is a huge part of learning. Just think about learning to walk or riding a bike. Babies are brave enough to take a first step even though they have no idea what will happen. My kids fell down a lot while learning to walk. But they kept trying.

I want my students to approach mathematics in the same way. Try stuff, see what happens, and if necessary, try again. But many of them resist tinkering. Too many students have been programmed to think that all problems are solvable, that there is exactly one way to approach each problem, and that if they can’t solve a problem in five minutes or less, they must be doing something wrong. But these are myths, and we must find ways to remove the misconceptions. The first step is to encourage risk taking.

A few months ago, Stan Yoshinobu addressed this topic over on The IBL Blog in a post titled “Destigmatizing Mistakes.” I encourage you to read his whole post, but here is a highlight:

Productive mistakes and experimentation are necessary ingredients of curiosity and creativity. A person cannot develop dispositions to seek new ideas and create new ways of thinking without being willing to make mistakes and experiment. Instructors can provide frequent, engaging in-class activities that dispel negative connotations of mistakes, and simultaneously elevate them to their rightful place as a necessary component in the process of learning.

Here are a few related questions I have:

  • How do we encourage students to tinker with mathematics?
  • How do we destigmatize mistakes in the mathematics classroom?
  • How do we encourage and/or reward risk taking?
  • What are the obstacles to addressing the items above and how do we remove these obstacles?

Source: http://maamathedmatters.blogspot.com/2014/04/encouraging-students-to-tinker.html