**Designing Powerful Calculus Lessons**

Workshops at Santa Clara University (August 17, 2017) and Cal State Maritime (April 20th, 2018)

- What is your experience teaching with group work or non-lecture methods?
- What is a question you hope to address in the workshop?

- Question. How do you figure out hard mathematics?
- Goals for student group tasks

- Daily team problem solving on large surfaces
- Online HW, computer graded, symbolic aerobics.
- Supplemental videos by Arek Goetz, but by halfway, no one was watching
- The groups work at large surfaces. Blackboards, and when wall space is available, I use static paper which turns walls into whiteboard space. Lacking wall space, you can use easels or small whiteboards.

**Norms.**I tend to model these through enforcement. I tell students if they solve my task right away, I gave them the wrong task. They are supposed to build up their brain muscles. I'm their personal trainer.**The Worksheets.**http://betterfilecabinet.com/f08/226ws.pdf. The videos use Worksheet 1. Sometimes I introduce the problem before giving out the worksheet. https://www.youtube.com/watch?v=-2Ex4rWhSc4**Divide; them into groups, they do the first task, then hand them the worksheet for part 2. (10)****Grouping Students****Some other theories:**- Set roles, like in Complex Instruction. Organizer, reporter, questioner, resource monitor
- Heterogenous or homogeneous “ability” grouping

**My approach**. I go back and forth between openly random groups (no more than 4) and letting them pick. I don't do any "ability" based algorithms, on purpose. Students are fast and slow, but speed depends on the task.

https://www.youtube.com/watch?v=Gbdw7rWNSu8- Randomizing; algorithms: count off modularly to N, count off and divide by N and find your remainder, hand out cards when they arrive, find at least one person you haven't worked with.

**Managing the Group Work**- Circulate quickly. Don't stay with a group for more than a minute or two.
- Scan large surface work in the room to see people's progress. Go prod groups that are either "done" or stuck/far behind.
- Have in your pocket some scaffolding questions and also extensions.
- If they think they're done with task, ask "Is this everyone's answer? So everyone can explain this?"

https://www.youtube.com/watch?v=ApLed2m4-eo- Ask; a random person (not always the most certain or uncertain looking one) to start the explanation
- I spin the Marker/Pen Of Fate and ask whomever it points to.
- Then ask another person to continue.
- If time, ask them a follow-up to check their understanding.
- If anyone seems disconnected from group answer, be sure to ask them to continue or confirm the answer. If they are openly confused/disagreeing, tell the group you'll come back in two minutes and they should convince everybody. Then loop back.
- If they are really done, tell them to pat themselves on the back, then give them another task. Either continue to the next class-wide task, or give the a bonus problem/extension.

- If stuck, DON'T GIVE THEM A HINT YET. Ask them what they've tried.
- If they abandoned a good strategy, tell them that was a good idea and they should go back to that.
- Praise any interesting strategies, fruitful or not.
- If they're stuck, try to ask them a key simpler question rather than suggesting a strategy. "Is this a graph of
*constant*hoisting?" - Leave the group with a productive next question as soon as possible.

**Checkpoints**- Many tasks have key checkpoints students need to pass to make progress.
- For me, that's usually one or two key steps/ideas per task. If there are more mandatory chokepoints in a row, you may have to re-think your task.
- If a lot of people are stuck, you can have them stop and look around the room to get ideas. Or have people share a partial result, like "what did people get for the derivative at x=0?", particularly if people disagree. Then just send them back to groups without resolving the disagreement. "Who's right?"
- There might be 2-3 checkpoints per class session that I want to discuss as a whole class. For those, push along the slower groups until they get at least a partial result. Then give a mini-lecture.

**Whole Class Discussions****De-briefs**. Good format is to take a couple of different answers. Especially if you know there are multiple ways. I usually warn people that I'm going to call on them.- Don't just reward the bold, quick and loud. You can ask a question and then wait for four hands before letting people talk.

https://www.youtube.com/watch?v=yNfJNxag6a4 - I'm; not organized enough to use clickers, but I use a thumb poll for easy interactivity. Up, down and in-between.

https://www.youtube.com/watch?v=PT5LgbNr0CE - When; someone gives a "crazy" creative answer, you need to give it status. https://www.youtube.com/watch?v=761cw1yitaA
- Enjoy; it publicly and also make people follow up mathematically. Spend class attention on it. In fact, on the first day, you should incite/invite crazy creative answers. People need to know they can use their whole creative minds and that they can do math with their own personality.

**Mini-lectures.******These are good to wrap up a task**using proper academic language and technical vocabulary**.- Now students have been wrestling with the idea in their head, so you can do a model solution of a related problem, or a more general solution. Be sure to connect back to the task.

**1. Base need**- Shelter, health, food.
**To get a good grade**- To have a fair shot to get a good grade

**2. Safety**- To be respected in class
- To ask questions honestly
- To have personal boundaries
- To be comfortable*
- adjustment period to being pushed to be active

- Teacher to be in charge

**3. Love and belonging**- To be liked by the teacher
- At least for the teacher to want them to succeed

- To be liked by peers
- To feel belonging to the college

- To be liked by the teacher
**4. Esteem needs**To understand

To get the right answer

To show off their talents

- Students can show off: quickness, humor, creativity, worldly experience, academic language, drawing/visual skills, independence.
- want to build on their power and harness it for math power
- Full brain, not guessing at the answer we want to hear

**5. Personal growth**

**(look at Worksheets 2, 3, 4). Take general comments.**- Non-routineness. It's okay to start off with a warm-up/check-up routine task, but if all questions are routine, the fast students will be bored/over-confident and the slow students will feel depressed.
- Don't over-structure. Flag Hoist could have been a step-by-step worksheet: "now estimate the slope at three points", "describe what the hoister is doing at t=0, 2 and 4. You can always ADD hints and structure. Figuring out what to do is part of the adventure and creativity.
- Almost anything that requires interpretation, or deciding something due to a calculation.
- Anything they could argue about using math. "What the most realistic?"

- Converting between representations.
- Graphs and tables are good because whole groups can look at them together.
- Graphs, tables, verbal, symbolic, kinesthetic

- Routine tasks, prematurely!
- You can take something you were going to lecture on, and let them try it first. Then if no one gets it, do a mini-lecture. Even if they get it, it’s good to sum it up.
- I ask them to write a derivative definition from idea of limit of average rates of change. Then you can mini-lecture about it because they’ve been thinking about it.
- They can try a task from a later chapter.
- Or it can be a routine task without the scaffolding or context (so they don't know which approach to take).
- Almost any word problem they don't have a rote recipe for solving.

**Differentiation**(pedagogy, not calculus): content, process and product- Interesting partial results.
- Sub-cases solved. Solved with simplifying assumptions. Intriguing approaches that dead-ended for interesting reasons.
- You want people lifting their appropriate weight.

- Admit multiple solution paths.
Then in de-brief have multiple approaches presented. Ideally, make connections between them.

Also, good for non-boring aerobics. Compare different methods. E.g. 6.4. Compute this derivative using product & quotient, and then do this using only constant, sum, diff and power. & 8.5. ( (x^3)^5 )' using chain rule, using power rule. Practice for algebra, exponents, exercises derivative rules, has a built-in check.

- Interesting partial results.
- A lot more thoughts on picking and using rich tasks at Hsu, E., Kysh, J., and Resek, D. (2007). Using Rich Problems for Differentiated Instruction. New England Mathematics Journal, 39, 6--13. http://bfc.sfsu.edu/papers/HsuKyshResek-RichProblems.pdf (The; Complete Director's Cut)

- Students. About four weeks.
- About 25% are really into it right away.
- About 50% go along with it, seeing if you'll stick with it.
- About 25% are suspicious and may take a little longer. They want to see if they can get a good grade in this format. So have an early Big Fun Quiz. Give them praise, build on their connections to classmates...

- You. A semester or a lifetime.
- How do you know how hard a problem is, or how rich it is?
- You learn through time and honest observation. I always write a short sentence after the class in my day plan noting how hard the task was and improvements for next time.
- Don't change everything at once. I often have to back off of something too-ambitious after week 3.

- Calculus I With Group Work, Fall 2008 (Materials, worksheets, teaching outline)
- The Better File Cabinet, a database of Treisman workshop calculus and precalculus problems.
- Hauk,S., Speer, N. M., Kung, D., Tsay, J.J., & Hsu, E. (2016). Video cases for college mathematics instructor professional development. Retrieved from http://collegemathvideocases.org.
- Bressoud;, D., Mesa, V., Rasmussen, C. (Eds.), Insights and Recommendations from the MAA National Study of College Calculus. Washington, D.C.: Mathematical Association of America. http://www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf
- Hsu;, E., Murphy, T. J., & Treisman, U. (2008). Supporting High Achievement In Introductory Mathematics Courses: What We Have Learned From 30 Years of the Emerging Scholars Program. In M. Carlson, & C. Rasmussen (Eds.), Making the Connection: Research and Teaching in Undergraduate Mathematics. Washington, DC: Mathematical Association of America. Print ISBN: 978-0-88385-183-8. Available at http://bfc.sfsu.edu/papers/Hsu-Murphy-Treisman-final.pdf
- Hsu;, E., Kysh, J., and Resek, D. (2007). Using Rich Problems for Differentiated Instruction. New England Mathematics Journal, 39, 6--13. http://bfc.sfsu.edu/papers/HsuKyshResek-RichProblems.pdf (Full Director's Cut)

*First posted August 16th, 2017*