We got through a full week of Calculus AB without a droll lecture. The first two days were mostly for introductions and just some review, and to sort of “warm-up” my engagement/facilitation skills. They need to practice their engagement, as well. Students are trained to believe that teaching means showing examples and giving problems like the examples. We’ve trained them well at this and then wonder why they can’t solve a problem.

Day 1:

After quick introductions around the table we moved to graphing. I used a Dan Mulligan technique of engaging a class while having them talk the math. Here’s how it works. You work in partners and they sit back-to-back, one facing the board (board partner), and one facing away at a desk with pencil and paper (drawing partner). They can sit shoulder-to-shoulder if that works better for the classroom.

The board partner must clasp his hands in his lap (this was amazingly fun to watch and police). An image is projected on the board and the board and started a timer. The board partner described the image to the drawing partner who drew what he heard.

The images began with a smiley-face, then to a graph of a line, and then to more intense graphs with holes, asymptotes, cusps, max/mins, jump and point discontinuities, etc. All that “stuff” they learned in Precalculus. The trick is to have two, or ideally, three of each “type” of graph. One to let them completely fumble and get wrong. Then they can talk about what they could have said to get it right. This is the real power of the math-talk. This is where they discuss and agree on proper vocabulary. They try again a second time and get it right, and a third time to switch roles and do it again.

This is a powerful conversation. A student seeing an image, putting it into words, another student hearing words and putting it into an image. Then talking about what words they should use. I’d sometimes stop and ask which teams used terms like increasing, decreasing, concavity, turning points, just to give them some pointers. I really didn’t need to, they figured it out fast. I heard them say domain and range to set up the axes, piece-wise, closed point, open point.

Exactly zero people checked out during class. Zero heads on desk. Zero whining.

I’m looking forward to doing this with derivatives and second derivatives later in the year. Here’s the document with the images.

Day 2:

Still in introduction phase, with no specific learning targets identified. Although this really did work out to teach average rate of change and just the idea of getting a rate of change from something that changes. We used the two videos I spoke of on this post . We used this document to instruct them how to do it. We watched the video of the speedometer. We talked about what we saw, how fast it went, how long it took, etc. Then we watched them only the odometer and told them to make the speedometer out of it. I gave them this rubric and instructions of how it should be completed.

I have their first drafts, and the data looks good. This is a work in progress that I hope to have some nice projects completed. Jury still out on this one, but so much potential to come back to when I do integral calculus and block out the odometer and show the speedometer and ask them how far the Lamborghini went.

Day 3:

Whiteboarding! I have seen teachers online talk about their big whiteboards and I wanted to try it. I bought an 8×4 ft board at Home Depot for $11.73 and they cut it for free giving me 6 big whiteboards.

The activity consisted of writing equations of tangent and normal lines. I walked them through the 10 steps and they passed the whiteboard around their group of 4 taking one step each. The steps:

- Draw a x- and y- axis
- Label axes with tick marks
- Draw a curvy graph (not necessarily a function)
- Draw a point on the graph and label its coordinates
- Draw a tangent line to the graph through the point
- Measure the slope of the line (need a 2nd point on the line)
- Write the equation of the tangent line
- Draw the normal line to the curve through the point
- Find the slope of the line using the tangent line
- Write the equation of the normal line

They did this at least 4 times until everyone had at least done each part once.

I felt the first three days went well. The engagement aspect of this sort of teaching is astounding. They are actively engaged for the entire block with no effort from me to get them “back on task”. Some things I want to remind myself about active engagement:

- Nothing is fun forever. These activities must be short and concise and have a learning target visible for them to know why they are doing it and when they can stop
- Get used to chaos and learn to “hear through it” or stop it. There were times when I could stand at a group and watch them finish a round of 10 steps of drawing in 3-4 minutes. While they were drawing they may have been talking about something non-math related. I allowed this. We’ll see if that has repercussions as the year goes on. Be alert, watch for questions all the time, and walk the class constantly. If nothing else is going on, ask them deeper questions about what they are doing.
- They must feel success, even if I need to intervene. It is not ok to hope they got something out of it. If I need to jump in and give them guidance, that is better than letting them flounder hopelessly. The first time they walk out of that class confused they’ll say “she doesn’t teach, she just walks around asking questions”. This is not acceptable, not in the beginning.
- Always get an exit card. See above. Still working towards that.

Your post gave me lots to think about. I’m going to try your day 1 activity soon. EXACTLY the type of activity my kids will eat up. Thanks!

This is a great activity to get them talking about math. Good luck and let me know how it goes!

This sounds great. Scribd now charges for downloads. Would you be willing to put this in dropbox, or to email it? (suevanhattum on hotmail)