Thinking Classroom in MHF4U

My advanced functions classes (grade 12 - similar to pre-calculus) are doing really well so far. I am mixing up VNPS (vertical non-permanent surfaces - i.e. whiteboards) with some direct instruction and a lot of explorations with Desmos. Here is the link to the Desmos activity I created to introduce increasing/decreasing intervals. Using the pause button was great to help focus everyone's attention as common misconceptions came to light.

Yesterday, we started investigating the remainder theorem. In their random groups, they divided f(x) = x² + 5x + 6 by x + 2, then by x + 3, then by x - 1. They also looked at f(-2), f(-3) and f(1). Then I asked what they noticed. Some really didn't notice much so I asked them to divide f(x) by (x+4) and find f(-4). They saw that the remainder from the division was the same as the value they calculated and that the value they were calculating was zero only if the divisor was a factor of f(x). (Note to future self: this was too scaffolded; fix for next year.)

So today we started with this: Find a factor of x³ + 5x² - 22x - 56. They were in new random groups of 3 and clearly did not make the connection to what they had started yesterday. Here is some of what I saw - lots to talk about!

We discussed our objective here - to factor this polynomial, which would allow us to sketch it. I asked something like "If something is a factor, what do we know?" The light bulb went on and they ran with that, finding at first one factor, then the remaining two using a variety of methods. Seeing that some were trying all integer values of x in f(x), I created a new question for them: Factor x³ + 6x² - 8x - 7. Those who had found all the factors to the previous question by systematically trying all integers starting at 0 soon got tired and asked if there was a better way (that was the point of the question). I suggested they look at the constant terms in their factors and the original polynomial. They were remarkably quick at putting those pieces together.

So, soon groups knew that only needed to try ±1 and ±7. They made more mistakes along the way (see below!), but there was progress. They found that they could not determine the other factors using the factor theorem, but had to divide by the first factor they had found. They understood the process and had ownership of it, having tried many paths that didn't take them where they wanted to go before figure out what would work consistently.

We didn't get through very many examples, but I firmly believe that it's better to work through one or two examples in depth, allowing students to find the pitfalls along the way and find their way out of them, than to spoon feed students a multitude of examples.

It's been ridiculously hot here the past few days so I feel somewhat incoherent and am not sure what point I was trying to make with this post anymore... I guess the takeaway I see is that using VNPS and VRG to let students explore and make mistakes is really powerful. I am really trying to get my students to do the thinking, not fall back on memorizing an algorithm, despite the fact that many think that is the best way to learn (ack!). I am trying to convince them that understanding the mathematics will take them so much further and be far more beneficial to them in the long run. That making attempts, some of which will fail, will prepare them for other times when they will struggle and not know where to start when solving a problem.

I'll end this with part of an e-mail I received from a student who graduated last year - I can't tell you how much it meant to me: "I have to say, I think your calculus class has been the most useful so far. The problem solving skills  I learned in that class have taught me to set up equations and approach problems from a different point of view (in multiple classes... especially chemistry)! The self learning technique also helped because that's pretty much all I do now before I go to a lecture. "

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.

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.

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.

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"