Friday, 24 March 2017

Curve Sketching in a Thinking Classroom

I ended my first full unit teaching in a Thinking Classroom (you can read about the beginning of this journey for me here) in my calculus classes just before the March break. I surveyed my students and had mixed feelings about the results. Many suggested 20 minutes of notes at the beginning of class, yet many of these same students also said that they never looked at the notes I post on Google classroom each day. I e-mailed Alex Overwijk to ask for advice, saying that I felt like I was doing something wrong, or at least not entirely right. His response went along the lines of "they don't like being uncomfortable, they don't like having to struggle and you aren't doing anything wrong". I think that what led to me feeling as I did had a lot to do with the very skill-based nature of the unit. They were taking derivatives and taking more derivatives and then they took some more derivatives. This led to groups taking turns doing questions. The questions were not challenging enough to require them to work together as a group to solve them. I'm not sure how to change that for the let's-learn-how-to-take-derivatives unit, but I will ponder that some more before we get there again next year.

This past week we have been working on the elements of curve sketching. Using the first and second derivatives to help determine intervals of increase and decrease, local maximum and minimum points, intervals of concavity, points of inflection, etc. I have continued to use visible random groups and they have continued to work on the VNPS (whiteboard/chalkboard) for almost the entire class each day. I have tried to be more intentional about what I do work through with them - mostly at the beginning of the class. The questions have been more interesting and more challenging for them. I am really pleased with their efforts. I am finding that they are putting all the pieces together more easily and that I am also thinking more deeply about the material. And it's fun! At least twice this week the bell rang at the end of my afternoon class without anyone in the room being aware that it was the end of class. They didn't want to stop. It's incredible how much fantastic work they are producing and how well they can explain it all to me. I am not quite sure how much gushing is appropriate, but my students are awesome. I snapped this picture of some of them this morning and it makes me happy and proud to look at it.



Here is the progression I used for the week (apologies - I got lazy and didn't include all the answers). They did not all get through every question yesterday and today, but I believe that they all have a solid grasp of the material. I continue to post filled-in notes at the end of each day should they wish to review the work or try any of the questions on their own.

Monday, 6 March 2017

Full Unit in a Thinking Classroom

I was going to blog about the last few days in my calculus class, including "Leibniz Day", but I just can't. I found out at the end of the day today that a student I taught all of last year passed away over the weekend. Sometimes teaching is really hard.

Here is the docx version of what I did and here is the PDF version.

Wednesday, 1 March 2017

Quotient Rule in a Thinking Classroom

I learned a lesson yesterday when my students far exceeded my expectations. One group had completed the entire sequence I came up with to "discover" and apply the quotient rule within 30 minutes of the start of class. This is when I am really glad that I have taught the course many times before and know the material inside out. I let that group start on their homework, something that never really happened during class time even before I switched to a the thinking classroom model, while I found another question worthy of their time. I could have just thrown an uglier question up on the board for them, but I wanted one that would make them think and, hopefully, challenge them. I found a good one and they got back up and worked at it for most of the remainder of the class.

The challenge for me is to set up my sequence of questions in a very intentional way, making sure the progression is neither too little nor too much at a time. But I also have to make sure that I have planned enough challenges to keep them going and keep them thinking. Getting that right will take a little more practice.

After that class I fixed things up for my afternoon class, adding the new question into the sequence along with another harder question that would make any algebraic misconceptions come to light. It turned out to be a little too tricky for most groups and they all got stuck. I adjusted by giving them the answer so they knew what they were trying to get, but many groups eventually abandoned that question and moved on. After that class I rearranged my sequence again, putting that question last.

As I learn to adjust and plan better hopefully I will get better at finding that sweet spot of just-right difficulty progression and quantity.

Monday, 27 February 2017

Product Rule in a Thinking Classroom

Today was day 3 of running my classroom as a thinking classroom - visible random groups (of 3) working on vertical non-permanent surfaces. We started with a quick demonstration to show that the derivative of a product is not the product of the derivatives. Students then worked through a product rule "discovery" activity that I have been using for years. You can find it here. I wish I could give proper credit for it, but I do not remember who shared it with me. 

Here is the setup:

Students worked out expressions for length, width and area and for their rates of change before completing the following table.


Their job was to work with the numbers they came up with in the table to figure out a pattern that worked for each row - that would be the product rule.

It is always interesting to see who comes up with it quickly and who takes a little longer (often because they are trying really complicated things!).

Once I felt like the majority of students had found the pattern, I sent them off to their VNPS to work through today's sequence of questions. 


The last of these asked them to come up with the product rule for three terms. I loved what some of the groups did. They extended the introductory activity to 3-D, added height and worked through the numbers again! (sorry that the picture quality is terrible)



Here's a group that came up with a conjecture for the product rule with three terms and tested it out. I would like to say they did this instead of asking me if they were right, but they did jump right to it when I said they should check it for themselves.


It was really exciting to see such fantastic work, at such a high level from all my students. I love how they trust that they will be able to tackle all the questions I give them and believe in themselves enough to try.

P.S. All of my planning is on one getting-bigger-by-the-day Word document. I'll post the whole thing at the end of the unit.

Friday, 24 February 2017

Why Should You Try a Thinking Classroom?

Overheard in calculus class this morning (day 2 of full thinking classroom):

  • "I don't think this is going to work"
  • "Why?"
  • "Maybe we can try..."
  • "Shouldn't that happen because..."
  • "What does the graph look like?"
  • "What if we..."
  • "Can you verbally explain what you did?"
  • "Can you tell us what we did wrong?" (one group to another group)
  • "We were real mathematicians today"

Thursday, 23 February 2017

Thinking Classroom - Day 1

After last week's workshop with Peter Liljedahl I decided to go full-on thinking classroom in both my calculus classes. I told them that they wouldn't be taking notes today and that they would be working in groups at the boards around the room. We talked a little about what the derivative function is and how we find it, along with the issues that would arise if they tried to find the derivative of y = x^729 from first principles. Next they each chose a card to determine their group. Off they went to their whiteboards/chalkboards and they started on the first question. Even though many had already been told the power rule, I made them "convince me" (and themselves) by finding each derivative from first principles. Here is the order of the questions they did:


They noticed patterns in parts (a) and (b) and were able to explain why the derivatives of (c) and (d) were the same as (a). Part (e) went better than expected and generally confirmed their conjectures. The results from parts (f) and (g) were confusing for many and I found that it was helpful to rewrite the question and get them to write the answer in the same form in order for them to see the pattern still held. They got stuck trying to do part (h) from first principles so needed to find the derivative another way.

At this point we stated the power rule as a group and turned to proving it. In the past, I have gone through the proof with my classes and many students' eyes have glazed over as they completely tuned me out. This time I gave them the expansion of x^n - a^n, we talked about how many terms there would be in part of it and let them try the proof. At least one group in each class finished the proof on their own! And all groups made good headway with it which helped them stay engaged when I showed them the full thing. I think they thought it was kind of cool!

I gave them two more questions after the proof:


The first was no problem and the second was done incorrectly by almost 100% of groups. We stopped there for today and I asked them to write down a summary of what they had learned. I didn't do anything else to close the lesson as I felt like it wasn't needed.

Here is the sequence for tomorrow:


Overall I thought today went well. I have done enough of this type of work with students that I was very comfortable and my students were great. There were a few times when I took a marker (there was only one marker/piece of chalk for each group) and handed it to a particular student, but in general they took turns doing the questions. Those not writing the solutions were watching what was going on, looking for errors. There were some good discussions going on today, but I anticipate more tomorrow due to the nature of the questions. There were some groups that would call me over to check their work, but they got a lot of "What do you think?" and "Convince me" and "Are you sure?" so I suspect that will diminish as we continue. I had to ask a few students to put their phones away, but it was not really an issue. They all did math and were all thinking and even those who came in knowing the power rule learned something new.

Saturday, 18 February 2017

Here Goes!

As I wrote in my last post, I am jumping into the full VNPS-VRG-thinking-classroom in my calculus classes. I have two classes - one in the morning, the other in the afternoon - so I will be able to tweak my plan in between and hopefully really make progress with both my planning and implementation. We have finished our first unit on limits and introducing the derivative function. On Wednesday we will start with the derivative rules (power, product, quotient, chain) so that's what I'm trying to plan out. Sheri Walker and I thought through this progression together yesterday so I've tried to tie things together and include some of what I expect to see. The intro will be to the entire class, the sequence are the questions I will give each group of 3 at their whiteboards/blackboard. Groups should progress at different rates so my job is to circulate, observe and help keep them in flow. The last question (part i) may be for the speed demons or for everyone - we shall see! I am also trying to keep in mind what comes next.


You may notice that I have a number of unanswered questions. If you can help me think through those, I would be most appreciative.

My plan is to edit my document after I have done both classes and then post the new file in case it might be useful to others.