This past week, I attended a less-than-inspiring AP conference for AP Calc, as I am teaching the course for the first time come September. Though some parts were helpful, the presenter spent almost all of the 8 hours every day just lecturing about Calculus and going through mediocre worksheets with us. He was a perfectly warm and friendly guy, but he was also sloppy, disorganized and often slightly incorrect, not to mention not creative at all. I was pretty disappointed. [Disclaimer: People have given me far better reviews about AP conferences in the past… I think it depends on the presenter organizing].
But, while watching the Calculus curriculum being presented methodically on the board (without any distractions because my wireless wasn’t working), I was struck by how confusing it must be to stare up at a mess of disorganized mathematical notation. I decided to brainstorm ways to improve the taking-notes-from-the-board aspect of my own course – to make my notes more “sticky” in my students mind and to make them more useful for the problem solving. We can all inspire some day to have a completely student centered, inquiry based, problem solving classroom, but even in those there is certainly room for (and a need for) teacher directed instruction… and that can always get better too.
Inspired by Square Root of Negative One Teach Math’s loop to convert logs to exponents to logs and Sam’s Riemann Sum setup, I tried to think of ways to use visual ways to connect conceptual math with notation (which is probably the biggest hangup with my students), to basically create a sort of intermediate form to help make the abstraction make more sense. Here are a few ideas I had… keep in mind I haven’t tried any of these with my students.
1. A Beefier Number Line for Graph Sketching
Problem: One of the things I noticed this past year is that my students would dutifully make number lines to test the derivatives but would sometimes totally forget what they were doing in the process. Also, many would mix up the first and second derivative.
Solution: Have the students immediately interpret their results with visual indications of increasing/decreasing and concave up/concave down. Make the separations on the second derivative number line be double lines instead of one to reflect the double prime part of the second derivative notation.
2. A Point-Slope Picture for Point-Slope Form
Problem: Anytime there are multi-step problems, many students either try to memorize algorithms or get completely overwhelmed calculating one thing that they lose other parts in their work.
Solution: Draw a picture of a tangent line and let the point be the O in POINT and the line be part of the L in slope. Then, finding these two items gets you everything you need to find the tangent line. Maybe arranging them vertically and carrying the final part of each step out to the side might keep students more organized. The bonus is that this is a picture that fits with the math and not just a forced acronym.
3. Enhancing Volume Integrals With Pictures of Cross Sections
Problem: The hardest part of figuring out the volume of solids is setting up the integral. Students have trouble figuring out what area equation to integrate and then which variable to use when integrating (i.e. which way to go).
Solution: Draw the cross-section near the solid and an arrow in the direction in which you are accumulating cross sections (or on the problem words if you skip the picture). Then draw the same shape next to the integral sign and an arrow. Inside the shape of the integral write the area equation as you would see it in geometry, and above the arrow write a d-whichever-way-the-other-arrow-goes. Then replace the area equation with something else that is in terms of the whatever in d-whatever. Works for the disk and washer methods in volumes of revolution too.
Okay, so maybe those aren’t all THAT helpful, but I personally prefer thinking about small changes when I have so much on my mind about the school year. Though these are obviously not replacements for deeper understanding, maybe they could be crutches to help students go from something that might make sense to them to the abstraction of notation. Main point: I’m going to pledge to sit down and try to think about how to make notes more “sticky” before every unit.
from @bowmanimal
Continuous Everywhere but Differentiable Nowhere 
I think these would certainly be helpful. They also remind me of “The Back of the Napkin” by Dan Roam – a book I keep meaning to reread now that I’m about to start my first year of teaching. The book is targeted for business folks, but I think there is a good deal we could use from it.
wow, thanks for the book suggestion! i’m going to add that to my summer list.
This is one of the places where technology can be really useful, especially for the cross-sectional diagrams. Check out this TED talk on teaching math without words for an expansion on your thoughts here: http://www.youtube.com/watch?v=7odhYT8yzUM
that was awesome – i teach almost 100% English Language Learners, so that was very informative. i’m curious what you mean about the cross-sectional diagrams though – i have used a few programs to draw these with the computer and plan on doing that a bit with my students next year, but would love if you have a more specific idea. it’s tough because, at least for ap, part of my teaching goals are to get them to pass a test that is very language heavy.
Having taught AP Calculus for a few years myself, I can tell you that I like these ideas. I know they would work with my students because they tend to be under-prepared for calculus and visual learners. I like your idea of making ideas and notes “sticky.” I am transitioning to teaching for a virtual school this year, so I will definitely need me virtual whiteboard time with my new students to be “sticky!” Thank you for the ideas!
Great ideas! It’s interesting that students get so caught up in the *math* of calculus when calculus is so visual. It’s true that they lose sight of what they are looking for and tend to memorize algorithms rather than use pictures to derive the equations they need. The _only_ problem I can see is the point-slope idea when the slope does not fit so perfectly with the “L”. But that’s just my OCD talking ;)
Sorry the conference was, for the most part, unhelpful. At least you got a chance to remember what it is like to be a student bored by the disorganization and inaccuracy of a bad lesson. There’s something to be learned from that…
haha, i was thinking about that too with the L on the slope. this is the type of thing that i’ll try with my students and they’ll make it so much better (like the logs loop at the top of the page).
Nice post Sam. This is very helpful.
Hi Guillermo,
Actually this was a post by my friend and guest blogger Bowman. As you noted, he’s doing a bang up job.
Sam
Beefier number lines is standard. I doubt I ever did a graphing problem without them. Second example is clear too, though for emphasis I would include the generic point slope formula to be clear what was going where. For the third one, I agree, for those whose drawing skills are less than perfect, it makes sense to do a 2-d picture. But with a little bit of practice one can draw the proper discs and washers to emphasize the intuitive form of the integral, dx is a hint as to how the integral is set-up in the first place as a limit of sums where disc or washer has a real thickness.