Friday I found I was answering a lot of "stop thinking" questions, in particular "I am right?" So today I was determined to leverage collaboration among groups to stop answering those questions.

**First Class**

Since there was no agreement on an answer to the triangle problem from Friday, I started the class again with this problem. A number of groups got incorrect answers quickly, so I paired up groups and said, "there's some disagreement here. Your job is to convince the other group you're correct." This worked well and most of the groups quickly had a correct answer.

At this point I didn't say answers were correct. I just said, "if you're convinced you have the right answer, here's another problem for you." This seemed to work well--eventually all groups had correct answers and moved on to the next problem.

The second problem was just to extend the triangle to 5 rows. Students had a lot more difficulty coming to agreement on an answer, and some groups came up with a different number each time they counted. So I suggested to the entire class that they could break it into the following smaller problems: count the number of upright triangles that are:

- 1 row high
- 2 rows high
- 3 rows high
- 4 rows high

There was much less agreement on this problem, and engagement was starting to drop rapidly, so I again said that if they were convinced they were correct they could move on to the next problem.

The third problem was as follows:

At a new development, a company is building roads and installing streetlights. The city bylaw states that there must be a streetlight at each intersection.

What is the maximum number of streetlights that would be required for:

- 1 road
- 2 roads
- 3 roads
- 4 roads
- 5 roads
(The original problem asked for n roads, but I decided that would be beyond my students.)

At this point there was a lot of shutting down. I thought perhaps it was due to students not knowing where to start, so I talked a bit about the benefits of writing something--anything--down, rather than just looking at the problem and thinking about it.

**Second Class**

I decided to start with the streetlight problem above. Initially students were drawing rectangular grids of streets, so I walked up to one group and drew the following, then walked way.

This was sufficient to get students thinking about non-rectangular grids.

At this point the frustration in the room was manifesting with students saying things like:

- What's the point in doing these problems if you don't tell us if we're correct?
- This isn't part of the course, so why should we do it?
- I want to work out of a workbook, not this way.
- Why should I explain my work to another student. I'm not the teacher; that's your job.

Thinking that this problem was too abstract, I decided to present another problem that would hopefully be more interesting and easier to grasp:

How many times in a 12-hour period does the sum of the digits on a digital clock equal five?

I was quite surprised that 4 out of 5 groups (yes I have a small class!) didn't even understand what the question was asking. Even after giving examples, there was still confusion. I have no idea what to make of that.

The disengagement was mounting, so I tried another problem:

Two people are playing a game of chance using two 6-sided dice. The dice are rolled and their product is recorded. If the product is odd, player 1 wins. If the product is even, player 2 wins. Decide whether this game is fair and convince another group you are correct.

Two of the five groups were successful in creating sample spaces and coming to the correct conclusion, without any prompting on my part. Unfortunately the other 3 groups were making minimal efforts.

Mercifully that class ended 5 minutes later :-).

**Conclusions**

- In Peter's presentations, he said a number of times that the initial few days can be rough and students will try to out-wait you, thinking that you'll eventually return to "traditional" teaching. I'm now seeing just how difficult this can be! For some reason the term "hitting the wall" came into my head when I was thinking about my afternoon class.
- I still believe that not outright telling students if they're correct will pay off in the future, but I'm struggling with how to make this work when there's so much push-back.
- I was having quite good engagement last week. I think engagement will return when the class problems are clearly related to Math 30-3 and students feel like the work is going to contribute to them passing the course.

Tomorrow I'm planning to give my classes some Sudoku problems.