In my mind, I picture two contrasting ways of structuring a Calculus course (and a hybrid way of doing something in between):

1. The traditional way: You introduce limits/infinity, followed by derivatives and their applications, followed by integrals and their applications, followed by mixed Calculus applications practice. Maybe you'd have a research component in the start of the year, in which kids do some research on all the modern applications of Calculus. From the various areas of applications, each student would choose one area to complete a capstone project on. This gives you chance to give them written feedback on their capstone project throughout the year, and their final paper to be submitted at the end of the year would have to contain an explanation of the relevant Calculus concepts, show example calculations that are applied within their area of research, incorporate technology as a way of enhancing/verifying their results, and they would present their projects to their peers as a way of solidifying the entire class's understanding of the applicability of Calculus.

2. Working backwards: You first spend some weeks taking the kids through the many ways of analyzing real-world problems using basic differential and integral concepts. They do everything on their calculator only, and they don't learn any of the algebra skills until after they have mastered the interpretation/application of results in the setup and analysis of a problem. Then, you peel away one layer of the magic and you work on all the core algebra skills of differentiation and integration without calculator, while consistently referring back to the applications that they have already seen, so that they don't forget the point of these algebra skills. Then, after they master doing all the same contextual applications manually and verifying their differentiated/integrated results via graphing calculators, you go on to more advanced concepts such as related rates problems, which cannot be explored solely on a calculator. Eventually, once the students have learned all the ins and outs of the core Calculus skills, you finally peel away the last layer of the magic and tie those "algebra shortcut" skills to the ideas of limits and infinity. The advantage of leaving limits to last is that the kids would already know what the derivative and integral formulas should look like by this point, and in taking the limits, they can self-monitor correctness. This "backwards" structure provides them with the full picture by the end of the course, of where everything comes from, but hopefully you have provided so much reinforcement of the applications that no kid will walk away from your class with only vague ideas of where Calculus is useful or how all the pieces are interconnected.

I talked briefly to one of my department chairs, and he is very supportive of the latter approach as an experiment. That makes me pretty excited! I will wait until I have chatted with the other Calculus teachers to start specific planning, since I have never taught Calculus as a full course before, but I think that our school culture is very supportive of teachers trying new things. So, I am going to try to wiggle my way towards a somewhat backwards organization of the course, because that makes the most sense to me.

I did some quick research, and these are the Calculus applications that I would like to introduce, either at the beginning of the year or at the start of each unit prior to developing those specific skills. Are there other ones that I am leaving out, but that are very common/useful for students to know?

Calculus applications

1. Optimization

* Maximum enclosed area

* Build a box (hands-on activity)

* Max/min rotating distances from a fixed point (incorporate technology)

2. Instantaneous rates / steepness

* Falling ladder analysis (exploratory starting with diagrams/tables)

* Learning curve, length of list vs. memorization time (lead-in with experiment)

* Marginal cost / marginal revenue / marginal profit

* Incremental effect of education/experience on income

* Rates of change within different savings account setups after fixed time

3. Summation of variable quantity

* Physics applications (distance, velocity, acceleration)

* Total work required/done in stretching a spring

* Total interests paid on a loan over time

* Supply and demand analysis – total opportunity cost for producer

* Supply and demand analysis – total consumer surplus, total producer surplus

* Architectural integration for accurate surface area

* Center of mass (difficult)

4. Average of a function

* Average velocity

* Average day length over a season

* Average power of an AC circuit

* Moving average of stocks (mini-project?)

* Maximum enclosed area

* Build a box (hands-on activity)

* Max/min rotating distances from a fixed point (incorporate technology)

2. Instantaneous rates / steepness

* Falling ladder analysis (exploratory starting with diagrams/tables)

* Learning curve, length of list vs. memorization time (lead-in with experiment)

* Marginal cost / marginal revenue / marginal profit

* Incremental effect of education/experience on income

* Rates of change within different savings account setups after fixed time

3. Summation of variable quantity

* Physics applications (distance, velocity, acceleration)

* Total work required/done in stretching a spring

* Total interests paid on a loan over time

* Supply and demand analysis – total opportunity cost for producer

* Supply and demand analysis – total consumer surplus, total producer surplus

* Architectural integration for accurate surface area

* Center of mass (difficult)

4. Average of a function

* Average velocity

* Average day length over a season

* Average power of an AC circuit

* Moving average of stocks (mini-project?)

5. Related rates (we won't work on these until later in the year)

I strongly endorse the 2nd approach. No limits until you need them. You might want to look at Matt Boelkin's open source calc activities: http://opencalculus.wordpress.com/about/download-active-calculus/

ReplyDeleteIf you decide on 2, I will be eagerly following your blog. Brave approach.

ReplyDeleteIf you decide on something like 1, I want to suggest a 1b. I don't think limits ought to be first. I taught calculus from the book for many years, then took a few years away from it, and then taught it twice last year. I loved how it went. I did a bit of hand-waving for limits, and spent the first three weeks (community college, Calc I in about 17 weeks) on understanding the idea of a derivative. Then I did derivatives of polynomials and jumped to graphing (which is way later in our textbooks). Then trig functions and product and quotient, and a bit about sound waves (Shawn Cornally inspired). Next unit, all the rest. Then limits before integration. I have lots of materials that I hope to get posted on my blog. Now much of it is at this google doc.

I recommend the second approach. Perhaps build it in a project-based manner as well :)

ReplyDelete@Sue, I like your google doc. It looks like what my brainstorms typically look like, just some rough notes and a bunch of links. I'll be sitting down next week to pull up something similar, after I work out some more concrete ideas.

ReplyDeleteI found the second backwards approach to be very interesting! I hope all went well during that year. How did you come up with this approach and do you think that Calculus in all classes should be approached that way?

ReplyDeleteI found the second backwards approach to be very interesting! I hope all went well during that year. How did you come up with this approach and do you think that Calculus in all classes should be approached that way?

ReplyDelete