## Sunday, July 22, 2012

### Art photography in the maths classroom - thanks to flickr

So much of the mathematics content we teach appears to many students to be fairly dry on the surface - we hope we bring them on the journey to see its wonder, but there is nothing like a great visual to create interest and start a quality discussion. Recently I tapped into an amazing resource: flickr photos provided under Creative Commons licenses.  There are thousands of high quality inspirational photographs available just waiting to go into a mathematics lesson.

For the topic "The Quadratic Function", water was my visual theme. I use just one of these images for each lesson - we start the lesson with a full screen, high definition version and then I use smaller versions to create a visual link for transitions between lesson activities.

All images under Creative Commons. Click on the photograph for details.

For the topic "Locus and the Parabola" I blended an astronomy theme (think: parabolic reflectors) and more abstract designs:

All images under Creative Commons. Click on the photograph for details.

These images look spectacular in high definition projected onto a screen in class - and give an opportunity to engage in some broader discussion about the application of mathematics, and the relationships between art, science and mathematics. Interesting to see students also noticing the use of Creative Commons - a chance to model good practice and spread the CC message.

Because the creators of these images generously put their work 'into the commons',  I can use them in my own works and then in turn, share those works with other teachers without any copyright constraints - hopefully helping students in many other classes.

Finding Creative Commons licensed content on flickr

scroll down and select these boxes:

I also select "Interesting" which tends to return richer images. When using the image, make a CC attribution and provide a link back to flickr page. I also like to leave a thank you comment to author. And thank you flickr for coming to the CC party. Now if only Google would make CC searching available on its main image search page! I believe it is there - just hidden - and life is too short to look for hidden options.

## Thursday, July 5, 2012

### The monkey and the mathematician learn calculus

"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:

 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:
 Click on the image for a larger view
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:

 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....