Help? Multivariable Calculus!

I need help! GACK!

So I’ve decided that my one week of lazing about and reading (finished Middlesex and the teen novel It’s Kind of A Funny Story, and started Heat) is officially over [1]. It’s time to get to work!

A mere two hours ago, I hoisted the Multivariable Calc book that I’ll be teaching from next year from the pile on the floor to my desk and gave it my first run through. It looks okay [2].

I’m designing this course from scratch, and I wanted to ask for some advice for anyone who has taught it, or has ever designed a course from scratch…

  1. Anything — print or web resources, jokes, songs, videos, pictures? Also, it doesn’t have to be about multivariable calculus; any advice on how to design a really awesome course from scratch would be much appreciated! How long did it take you, what resources did you draw upon, did you make a general outline of topics or a specific day-by-day schedule, did you write your assessments beforehand or during the school year once you’ve gauged the students’ abilities, and the other million questions that I’m thinking of.
  2. Does you know of any good software for graphing in 3-D that is open source (read: free) and works on a PC? I know of SAGE, and OCTAVE, and the like, but I’m wondering if those programs are a bit overkill for this course. Is there something less bulky out there? Maybe even a really powerful 3-D graphing calculator that people like? 

    UPDATE: I just remembered that SAGE came out with the online SAGE notebook, which is what I think I’ll probably implement! It’s like MAPLE in terms of the command line, and it seems extraordinarily powerful. 
     

  3. How do you teach your students to graph in 3-D by hand? How do you do it on the board? SmartBoard?
     
  4. Have you ever taught a class with 3-5 students before? Do you treat it like a regular class — with lecturing but with more individualized attention? Or did you teach it seminar style? What would a seminar style math class look like?
     
  5. Do you have any good investigative activities or projects for multivariable calc? Or that you do in calc that can be extended?
  6. Have you ever just thrown out teaching from a textbook and used an online textbook? Or mixed and matched textbooks? Or taught without any book?
My big idea at the moment is to make a course with no exams. These are kids who are accelerated enough to have taken AP Calculus before their senior year. I want to expose them to the idea that math can be a big series of puzzles. That math can be investigative instead of regurgitative. That math can be collaborative. That math can be hard and challenging and rewarding if they persevere. And since we have a tiny class (maybe 3-5 students), this is definitely possible. So as I said, no exams. Instead, we’ll have nightly homework assignments with the more fundamental and basic questions, and then “problem sets” with investigative problems due at the end of each chapter. The problem sets can be worked on alone or with others, but the write-ups need to be done alone, and they will be graded on correctness and clarity. Who knows, maybe I’ll even teach them to use LaTeX (well, MiKTeX) to write up their solutions.

Hopefully I’ll use this blog to post about the evolution of the course design as the summer progresses… so don’t change that RSS reader!

[1] Other books that are lined up to be read this summer are Fight Club, The Kite Runner, and The Adventures of Huckleberry Finn (all for school); if I have time, I also want to read Lazarus’ Closed Chambers, Tartt’s The Little Friend, Dewey’s The School and Society and The Child and the Curriculum, Pais’ The Science and the Life of Albert Einstein, and finish up the second half of Gogol’s Dead Souls

[2] I’m using Anton’s Calculus, Early Transcendentals, 8th Edition. My school uses the first half of the book in calc classes and I don’t want to make my students buy a second book. My initial opinion: the book is okay but seems to be unnecessarily dense in places, and could have left a number of sections out. The exercises at the end of each section are quite good.

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5 comments

  1. I am going to teach a similar class — AP calculus BC, to kids who already had a year of calculus. I don’t have any suggestions though because this is my first year teaching.

    I want to do what you said, show the students “That math can be investigative instead of regurgitative. That math can be collaborative.” I’m thinking of making it a sort of exploratory course where in addition to multivariable calculus we look at other areas of math they might encounter in college: discrete math, statistics, linear algebra, mathematical modeling, complexity theory, network science, etc.

  2. Anne, I’d love to get any ideas, investigations, activities that you come up with — any branch!

    Discrete math is the most obvious place to get investigation accessible for high schoolers, I think.

    But it might be fun to teach them a unit on plain old “problem solving” where they get competition problems (from AMC, AIME, etc.) and talk about solving them — and methods to solve problems (e.g. trial and error, induction, etc.). It’s about intuition, dead ends, and trying various approaches. (If you are ever in need of these sorts of problems, I have a bunch of places I can recommend.)

    If you do anything with chaos/complexity theory, can I suggest looking at Steven Strogatz’s “Nonlinear Dynamics and Chaos.” It’s in paperback, and is totally investigative (using MATLAB for some great investigations), and many parts are accessible for students who have learned some very basic multivariable calc. In fact, now that I think about it, our mathclub investigated the logistic map and bifurcation diagrams (from the book), and that didn’t need anything past Algebra II and learning how to plot sequences on their TI-83s.

    I pushed for my school to allow me to teach a course based around this book instead of the multivariable calc course, but unfortunately it was a no-go.

  3. I taught a course as a graduate student called “calculus with technology” at Michigan Tech. We used Mathematica to do real world problems in calculus, and the course was mainly directed toward the mostly engineering students at the university. I’m not so sure that it was successful.

    I actually wouldn’t bother with worrying too much about teaching 3d software or LaTeX, as I think those tools are useful, but might ultimately distract from the task at hand – teaching calculus.

    As for the answer to your nowhere differentiable continuous function – the most common example is to take a bounded sum of progressively more dense and less tall sawtooth functions.

    By the way, I arrived at your site via presurfer.com which linked to NxE’s fifty most influential female bloggers (congratulations) – and I’m from Denver, so I clicked you.

    So sorry to hear about your dog. Please feel free to email me with any calculus questions.

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