Freefalling into Extension 2 Mechanics

This post refers to the NSW (Australia) Mathematics Extension 2 course - the highest level mathematics taught in our high school system, but should hopefully have relevance for anyone teaching introductory mechanics.

My class is just beginning to explore the harder mechanics content in our senior mathematics course, exploring the equations of resisted motion, so it was with great delight I brought the exciting work of Felix Baumgartner into my classroom:

Here are some reflections on introducing this topic:

Did you lay a strong foundation right at the start of learning calculus? Coming from a physics/engineering background, I taught all the calculus concepts right from the start as a study of rates of change, using this as a motivation for the mathematics. My classes started with motion using a motion detector even before we looked at the concept of the derivative. When we studied the properties of the derivative, the meaning of increasing, decreasing and stationary points, it was all done in the context of motion (lots of roller coasters!). When we studied the second derivative, we asked the question "Why?" - and looked at how much of the physical world works on the second derivative rather than the first. So by the time we moved on to specific topics of applying calculus to the physical world, my students already had wide exposure to the link between the mathematics and the physics.

Did you explain why we have equations of motion that connect acceleration to displacement? One of the hardest parts of the earlier work in our course on motion is understanding and working with these equations:

$$a = v \frac{dv}{dx},  a = \frac{1}{2} \frac{ d(v^2)}{dx}$$

By showing that acceleration is caused by forces, and that in turn forces are often dependent on position rather than time gives the motivation for these more complex equations. It's an excellent opportunity to tap into students' current science knowledge on gravitational and electrical fields - and again reinforces Newton's Laws of Motion (this time the second law). For students with a little more physics knowledge, it's very interesting to link the second form of this equation to the equation for kinetic energy $$KE = \frac{1}{2}mv^2$$ using some integration.

You can't emphasise Newton's First Law often enough.  It's amazing how many studies show that while university physics students can describe and use Newton's First Law of Motion, deep down they remain firmly wedded to the Aristotelian world view. I make it a point to emphasis the First Law of Motion each and every time I start a problem - and I keep an eye out for anyone hesitating or wavering. If necessary, I repeat my stories about ice-skaters and about the Voyager spacecraft continuing on their journey even though they have run out of fuel.

More diagrams, more diagrams! It is sad to see how few diagrams are presented in most mathematics textbooks when presenting the theory of dynamics. While there are diagrams in most worked examples, they aren't explicit in the construction of the diagram, leaving it for students (and teachers) to try to interpret why the diagram was done that way. I spent most of the introductory lessons on this topic just drawing pictures.

"That's why it's called RESISTANCE sir!" My favourite quote from one of my students. We were drawing diagrams and I was trying to find a clear way to show that the resisting force always opposes the current velocity direction. I was saying the word "opposes" a few times when one of my student yelled this out - we won't forget that in a hurry!

"Bait-and-switch" constants.  Your students are probably used to the little bait-and-switch games we play with constants of integration. The same games are played with the constants used for resistance forces:

$$R = kv,  R = mkv$$

Sadly many of our standard text books just switch on the fly between the two forms without explanation, adding in the mass whenever it's needed. It's very confusing for students (and teachers) when this is done so arbitrarily. And then it hit me: this is a totally legitimate game - we're just using a different constant to make our life easier. It sure would be nice to write:

$$R = k_1v,  R = mk_2v$$

I shouldn't complain since I happily went along with the game when integrating!  This is yet another small but cumulative thing that makes this topic challenging. I think it's important to be explicit about this little game.
Update: As explained in Robin's comment below, this trick is a bit too clever - verging on not being legitimate - because it gives the false impression that the resistive force depends on the mass - it doesn't. The trick only works for a specific case of the mass in this problem.

Students have difficulty seeing that the physics has nothing to do with the coordinate system. And it's not their fault - our teaching and our text books rarely show how arbitrary coordinate systems are - we just happily keep changing them to our convenience, potentially confusing our students. I think it's critical to draw lots of vector diagrams without any coordinate systems, and then make a clear and obvious choice with the class that we can choose any coordinate system that works for us.  We had a deep-learning moment in my class last week when I unwittingly applied a different orientation of the axes than was in our text book - a great opportunity to highlight this issue.

Terminal velocity is a really fun concept. Students are absolutely fascinated with it - the physical understanding is interesting, and the mathematical development is revelatory. We had some great discussion on different terminal velocities for different situations and these led directly into a more rigorous discussion of Felix's jump.

Don't think that girls aren't interested in watching extreme sport events. My class of fifteen girls was were absolutely riveted watching Felix make his jump. They insisted on watching the full length 10 minute video, totally transfixed for the duration. I recommend reading the Wikipedia page on Felix's "Red Bull Stratos" jump with your students prior to watching the video - it provides an excellent opportunity to discuss the language of motion, examine the different stages of the jump and provides meaningful context for this thrilling event.

Hold your breath and enjoy the whole jump:

The next post in this series looks at the wonderful Veritasium You Tube resources available for teaching mechanics.

Exploring inequality : an entry point to calculus

"Have you ever noticed .... ", I said to my senior maths class, as I walked in bearing a huge and very obvious glass bowl containing about 40 packets of Smarties, ".. how some people seem to have so much more than other people?"

"Taking it Back - Occupy Oakland" by Glenn Halog
http://www.flickr.com/photos/ghalog/6271929376/in/photostream/ CC-BY-NC-2.0

I then proceeded to "share" out the Smarties: first I gave 20 of the 40 packets to one student - making a huge pile on her desk. Her eyes popped out - while the other students looked with disbelief and some concern for their own anticipated share. I gave a wicked grin and 10 packets to the student next to her. To the rest of the class I handed out 2 or 1 packets - apart from a few students at the end of line who received nothing. Oh the looks they gave me!

And so we started a lesson exploring the question of how we could measure income distribution - a hook (although the class didn't know it yet) - to introduce our next calculus topic: integration.  Here are some notes on my first attempts at a lesson design using an idea from economics as a motivation why we might want to find the area between two curves.  But first a big thank-you to mathematics teacher Alastair Lupton who showed me how to bring the Gini Coefficient into the classroom and encouraged me to try it out in my classroom.

So here's the sequence I tried this year.

Step 1: Build interest in the problem. With strict instructions not to eat or worse yet - share - their Smarties, we looked at a short OECD video about the rising inequality in income distribution:

Depending on the time available, you might want to explore some other video material, perhaps some recent news footage of the Occupy movement protests, or look at some studies of global income distribution.

Step 2:  Thinking how to organise the data: I lined up the students, holding their very unequal distribution of Smarties. We ordered the line by 'income' and partitioned into 5 groups - helping the students see the organisation of the data into quintiles.  We returned to our desks and looked at some local and international data on income distribution, also organised into quintiles. Here is some recent Australian data:

Click on the image for a larger view.
Source: Australia Bureau of Statistics 6503.0Household Expenditure Survey and
Survey of Income and Housing User Guide 2009-10

Step 3: Ask the question: "How could we measure inequality?" This isn't easy or obvious. Give the class some time to explore ideas. Then it's time to look at how economists do it...

Step 4: Develop the idea of  graphing cummulative quintiles.  After trying some different ways to plot our quintiles, I showed the students how the economists do it: reorganising the data into cummulative quintiles. This allows us to make normalised curves which work for all situations, regardless of the size of the total income pool. We drew our first Lorenz Curves:

The Lorenz Curve is used to calculate the Gini Coefficient. The area A is the difference from total equality.
The larger the area A as a proportion of the total area A+B, the greater the inequality.
Source: Wikipedia Lorenz Curve Image by Reidpath,

To help explore the idea, we discussed what the Lorenz Curve would look like if one person had all the Smarties, and if all the Smarties were shared equally.  We also considered if the curve would ever go above the "Line of Equality" (it won't!).  We selected different data sets (see references below) and plotted them.  Here is the 1993 World Bank data for Nigeria plotted in GeoGebra, with a polynomial fitted to the curve:

By modelling the curve with a polynomial, we can use integration
to calculate the area under the curve and hence the area between the curves.
Data is entered into the GeoGebra Spreadsheet window, then plotted and
a function calculated to fit the data using FitPoly[].

Step 5: Ask the question again: how could we measure the inequality?  After looking at a few different data sets, students will quickly come to the conclusion that measuring the area between the line of equality and the Lorenz Curve will give us a nice single number. And now you have them hooked: here's a very interesting and practical reason we might want to be able to calculate the area between two curves.

Step 6: Declare a communist revolution.  I then ordered a redistribution of the Smarties so everyone was equal.  This was actually quite funny because several of my diet conscious students insisted they did not want any Smarties. Tongue-in-cheek I told them this was not an option - it was a revolution and everyone had to be equal whether they wanted it or not!  A nice opportunity to open up the discussion to different views about income distribution.  I gave my students a selection of recent articles from The Economist which seemed to provide a good balanced discussion on the topic.

Step 7: Begin the mathematical discussion on ways to calculate the area between the two curves. Your students will have many useful ideas! Try them out with the tools available. And now you're ready to start a calculus based exploration: What is the area under a curve? 

Where could you go with this lesson idea?

• Get students to make up a small poster using their data and stick them up on the wall. Then as you move through the Integration topic, you can refer to them in the context of each new idea.
• Once students know how to integrate, get them to model their curves as a polynomial - I like to use the GeoGebra FitPoly[]function - and then do calculate the integral, comparing their result to given Gini Coefficient for the data set.
• The student data makes for a great application of the Trapezoidal Rule : they can calculate the area without knowing the equation of the curve.  A good example of why you might want to use the numerical approaches to calculating integrals.
• Challenge activity: calculate the area under the curve using Simpson's Rule. If you only have the standard Simpson's Rule, you can't do it because there are an even number of data points! But there is more than one Simpson's Rule - challenge your students use the internet to find one that will work for this data. [Hint: Simpson's 3/8 rule will work].
• Apply the concept of the Lorenz Curve to another field of study. An interesting application is to social networks - some people contribute significantly more than others, while others 'lurk' in silence. I use edmodo with my class and there is a high degree of inequality in the number of postings per student - counting postings per students could make for an interesting Lorenz Curve.

Thinking beyond the mathematics:

• Talk to the economics teachers at your school. I discovered mine do teach the Gini Coefficient, but they don't go into how it is calculated.  I think it could be a very powerful lesson to develop a  sequence of combined economics/calculus lessons with an economics teacher at your school. The more I explored the subject, the more interesting I found it. Options to consider include: the effects of taxation policy on the Lorenz Curve; the differences in the Gini Coefficient between different types of economies; differences within one country over a time series; challenges to the validity of the measure; economic and social arguments on the topic of income distribution.  All highly suitable for deeper mathematical and social science exploration.
• Take some time out to look at the Gap Minder website which options to view the data through the Gini Coefficient.

Resources

• The Wipedia pages on the Lorenz Curve and the Gini Coefficient are a good starting place, with good entry points to more nuanced discussion on the use of the Gini Coefficient.
• Some excellent Australian data and good explanatory notes in the ABS publication 6503.0 Household Expenditure Survey and  Survey of Income and Housing User Guide 2009-10. See Table A1 in Appendix 1, and a high quality discussion on the Gini Coefficient in Appendix 3.
• International data from World Bank: World Development Report  2000/2001: Attacking Poverty - Table 5 Distribution of Income or Consumption 
• The October 23, 2012 issue of The Economist contains some excellent articles on the challenges of income inequality seen from a pro-Free Market view. I found these particularly interesting given one could hardly call The Economist left wing!

Some teaching reflections:

• The students really loved the lesson - they were engaged and it was interesting.
• I planned carefully for my 'inequitable Smarties distribution'. Our class was well established and we knew each other well enough that my students would know I was up to something and trust me when I played this game. I also made sure the students who didn't receive Smarties were the most resilient, confident students.
• I did however make the mistake of trying to do this opening lesson in a single 50 minute period - it wasn't enough time and I rushed it, making it less student centred than I had hoped. This lesson needs a double period to do it justice. 
• Is it worth taking the time out from a busy course to do this activity? I think so. Once I realised I could leverage this work into my teaching of the Trapezoidal Rule, Simpson's Rule, the area between two curves and also do some polynomial modelling, I saw it was a lesson that  just "keeps on giving".
• Coming from a physics background, it was wonderful to find an interesting and practical application of calculus to a completely different field. Many of my students are planning a career in business and are interesting in economics - here was something to show them the calculus applied to money as much as to speeding particles!

This is part 2 of a sequence of posts on teaching integration. 
Part 1: Slicing and Dicing.  Part 3: Integration in the world around us

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....

Getting personal with rates of change

The key to mastering calculus seems to be gaining a good understanding of rates of change, how this relates to the idea of a function and then seeing how we can use the tools of algebra and geometry to develop the gradient function. Now as exciting as it is for some of us to play with a quadratic or a cubic function, I recently discovered, quite by chance, a very personal and highly engaging way to explore rates of change of a function.

This data is all about ME!

I was exploring the introductory concepts with a Paul*, a teenager just starting on the calculus road, when he made the connection that he was experiencing a very dynamic change process: his height had started shooting up in the last few years and very soon he expected to be nearly 2m tall.  He knew he was experiencing a "growth spurt", growing at a faster rate than when he was younger. Just as I was wondering how we could use this connection in our exploration, Paul told me his parents had been marking his height on the kitchen wall for the last 10 years. Wow! This was exciting - some real data we could plot and explore.  Paul measured the markings off the kitchen wall and made a table showing his height at different ages. And here's what we were able to do with that data in GeoGebra:

In the process of this exploration we uncovered many ideas about slope, functions and the use of modelling, each time applying them in a context Paul had a profound personal interest in - we even named our  polynomial the "Paul function" in his honour. I'm certain he will never forget the idea of rates of change, the gradient function or the power and fun of modelling a function based on data points.

Do your students have a wall somewhere with their heights measured over the last 15 years? If so, I highly recommend working this into your calculus activities.

I'm a deeply indebted to the work of Mary Barnes, in particular her 1999 series "Investigating Change", on teaching and learning calculus, still available at Curriculum Press. Google Books previews are available. Thanks also to a new Google+ friend and teacher Steve Phelps for showing me how to use the GeoGebra FitPoly function. Using GeoGebra to build a gradient trace function comes from the original GeoGebra documentation by Marcus Hohernwater, however I have found in practice it can take students some time to understand what is going on. Priscilla Allan's YouTube demo shows a good way to use colour to make it clearer and I have extended her idea to actually show the trace point moving along the x-axis, prior to adding in the gradient value. 


And special thanks to Paul (*not his real name of course) and his parents for allowing me to share this story of our ongoing exploration of mathematics.