Wednesday, June 25, 2014

One Resource a (Week)Day #4: Calculus Applications and Sustainability

I picked up several Calculus books from school to take home to skim through over the summer. I am actually pretty happy with my Calculus curriculum from this year for its exploration-based approach, but I am (of course) looking for supplemental resources to further enrich the curriculum next year.

One book that I picked up is called Calculus and Its Applications, authored by Marvin L. Bittinger. The Post-It that I had slapped on it says "student-friendly", which was just my initial impression after flipping through it for 30 seconds. As a teacher, I try to be creative in my own presentation of material in the classroom, but I always appreciate a textbook that is straight-forward and easy for students to reference in hindsight, with a "classic" organization of ideas and skills. For example, our previous textbook had a weird content layout where they tried to present everything at once. Although some of it is innovative, that's not really a student-friendly text, in my opinion, because without a teacher's guidance, that textbook is daunting and likely to remain on the shelf untouched.

Anyway, this textbook by Bittinger is heavy on applications, which I like. I tried to interleave my class with economic, physics, and basic max/min problems as much as possible this year, but this book goes quite a bit further to research real mathematical models generated from real sets of data. The examples are (dense and therefore) more suitable for college students, perhaps, but I appreciate how they regularly cite their sources to corroborate their use of real data, and there are also little side panels that offer more information on, for example, the process and limitations of radioactive dating. The book also includes some applications with which I am not familiar, including the elasticity of demand and some environmental sustainability problems. (I really liked the practical tie-ins, although realistically I think that some of the econ bits are just a tad bit too abstract for my Calculus students, who found even marginal cost, marginal revenue, and marginal profit to be elusive notions this year without concurrently taking an economics course.) Certainly, this would be a nice companion text for someone who's looking to apply Calculus in their other subject studies, and probably a really nice text for a university-level math prof, who is looking to expand the appreciation for Calculus as a useful tool of analysis.

The student-friendliness that struck me initially is evident in the structure of the text. The front inside cover has a "Summary of Important Formulas" for easy reference, and the index of applications is also unusually placed in the very front of the book. The book starts off with a quite comprehensive review (Ch. 1) of prerequisite algebra skills, which I imagine most Calculus teachers can assign as independent review. Ch. 2 is when they start to move into limits and other Calculus-related concepts.

As a secondary school teacher, what I like about this book (besides the no-fuss organization) is the applications and some attempt at bringing technology into the curriculum.

I liked this investigation of limits using the Table feature of the calculator, and will no doubt adapt this for my classes next year. (I liked the start of the activity, but I think they could have pushed it a bit further to include numerical analysis of limits with rational functions.)

The book had a really gentle way of transitioning from the tabular / arithmetic limit approach to the graphical approach. They said that you should draw an "X" on the endpoint that the function is approaching from either side (+ and -), and if the two X's overlap, then the limit exists. It's nothing fancy, but the language is really student-friendly and I liked the tactile approach.

Further into the book, they also used movie stats and technology to model a logistics curve, and then used that to motivate the finding and analysis of derivatives. I think this is pretty cool, because you can modify the structure slightly to have students pick their own movie / album sale / etc. and to do their own analysis based on their own personal interest, and it introduces them to a useful functional form that isn't always studied in earlier courses.

Also really cool: There are problems in this book that have to do with sustainability, which I am always looking to bring into my classroom. (I am convinced that we all need to be teaching sustainability, the same way that we all teach literacy and we all teach technology. It's an essential 21st century value that they must have when they leave our classes.) The problems pictured below have to do with an application of integral Calculus in predicting how much longer certain ores or mines will last before they are depleted.

Here's another problem/discussion having to do with how to hunt animals sustainably and to maximize our return without damaging the ideal animal population. The problem is quite involved, but mostly because the ecological concepts within it are not ones that we are accustomed to thinking about, from a mathematical standpoint.

I hope you have enjoyed this installment of One Resource a (Week)Day! I feel pretty great so far about the few resources that I have looked at. I know that there are lots of invaluable digital resources out there as well, but I plan to delve into them when I am down in New Orleans. This way, I won't be carrying any textbooks with me to and fro.

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