"Even a monkey can differentiate" - that's how I described the rules based approach that seems to dominate so many students' (and teachers') interaction with calculus. Coming from the "teaching for understanding" camp, I made a very deliberate and careful attempt in my first teaching of calculus to emphasise understanding as opposed to a formulaic, mechanical approach to the subject. And yet - a few weeks later, I've come to embrace my inner monkey. There is a place for mechanical, automated rule based thinking in mathematics - and I'm now leaning to the view we need to make room for both the monkey and the mathematician.
Here's the monkey at work:
No disrespect - WolframAlpha is an amazingly powerful tool, but it reminds us differentiation can be done without understanding. |
As I worked through the basic rules of differentiation with my class, I found myself continually looking at the rules from the 'monkey' viewpoint as well as the 'understanding' viewpoint.
Differentiation from first principles
Monkey: "Substitute in the values correctly, expand, pray you can factorise out the bottom, then shrink the delta-x to zero."
Mathematician: Understanding the central principle. The meaning behind every element of the fundamental equation is pivotal - it's like a little prayer in our holy canon. If you have to memorise the formula, you haven't understood it. Visualise the image of the secant becoming a tangent and just write down the description of the process: $f'(x) = \lim_{\Delta x \to 0}\frac{f(x + \Delta x)-f(x)}{\Delta x}$. OK - now release your inner monkey and finish the work.
The Chain Rule
An exploration using Marc Renault's amazing Chain Rule analogy interactive gives our mathematician side a boost here. For our monkey side, we developed the language of 'inside' and 'outside' to describe composite functions - modelled on Russian dolls. Here's how I summarised the two approaches:
Differentiation from first principles
Monkey: "Substitute in the values correctly, expand, pray you can factorise out the bottom, then shrink the delta-x to zero."
Mathematician: Understanding the central principle. The meaning behind every element of the fundamental equation is pivotal - it's like a little prayer in our holy canon. If you have to memorise the formula, you haven't understood it. Visualise the image of the secant becoming a tangent and just write down the description of the process: $f'(x) = \lim_{\Delta x \to 0}\frac{f(x + \Delta x)-f(x)}{\Delta x}$. OK - now release your inner monkey and finish the work.
The Chain Rule
An exploration using Marc Renault's amazing Chain Rule analogy interactive gives our mathematician side a boost here. For our monkey side, we developed the language of 'inside' and 'outside' to describe composite functions - modelled on Russian dolls. Here's how I summarised the two approaches:
Click on the image for a larger view |
The Product Rule
I'm a big believer in showing the geometric justification - and it's more credible than the limits sleight-of-hand pulled by high school text books. That's for the mathematician. For the monkey, we learn the rule - and I like a cross-product type visualisation:
The Quotient Rule
Last but not least, the quotient rule. I think it's important for the mathematician to see the connection to the Chain Rule and the Product Rule ("so that's why there is squared in the denominator!") For the monkey - well it's another pattern to get into the habit of using:
Who's more important: the monkey or the mathematician? As much as I initially laughed at my inner monkey, I've come to value him. I don't think we need to choose between the modes of working - there is value in both. I suspect it's about 'reducing cognitive load' - with a reliably functioning monkey, we can perform low-level functions without too much thought, saving our awareness to concentrate on the more complex ideas at hand. The only danger with that monkey is too many bananas and we can forget the meaning behind the operations....
I'm a big believer in showing the geometric justification - and it's more credible than the limits sleight-of-hand pulled by high school text books. That's for the mathematician. For the monkey, we learn the rule - and I like a cross-product type visualisation:
Click on the image for a larger view |
Last but not least, the quotient rule. I think it's important for the mathematician to see the connection to the Chain Rule and the Product Rule ("so that's why there is squared in the denominator!") For the monkey - well it's another pattern to get into the habit of using:
Click on the image for a larger view |
Who's more important: the monkey or the mathematician? As much as I initially laughed at my inner monkey, I've come to value him. I don't think we need to choose between the modes of working - there is value in both. I suspect it's about 'reducing cognitive load' - with a reliably functioning monkey, we can perform low-level functions without too much thought, saving our awareness to concentrate on the more complex ideas at hand. The only danger with that monkey is too many bananas and we can forget the meaning behind the operations....
Hi,
ReplyDeleteI'll be teaching my first calculus class starting in August. I appreciate your thoughts. I'm wondering as I look at the mathematician thought bubble for the product rule: What is the sense behind that visualization? It looks like the amount that an area is increasing. (It's 9am after only 4 hours of sleep; so forgive me for not catching some obvious connection between the relationship between the two fundamental theorems of calculus.) If increasing area is an appropriate representation of differentiation, why don't we also want to know about that corner du*dv?
The idea is to show that the tiny corner in the end delta u * delta v is indeed extremely small compared to the delta u * v + delta v * u - so it is reasonable to throw the little corner away. I personally think of it as a second order differential so it's always going to 'an order of magnitude' smaller than the other delta pieces (not sure if this legitimate thinking).
ReplyDeleteYou've inspired me to write another post soon about some of the key things I tried to do in my first calculus class...
now I get it! cool graphics :)
ReplyDeletebtw, I also have a blog and a web directory, let me know if u want to exchange articles.
emily.kovacs14@gmail.com
Emily :)