Friday, 11 July 2014

Math Lesson Networking

This is a post about a set of lesson studies I participated in from March to June. The actual name on the interoffice memos is "Cross School Math Networks", but I'm not sure that works for this post title, in that it neglects to point out that I'll reference actual lessons here that you can adapt. Though it does get across the fact that this allowed for networking between schools, so post titles are a bit of a dice roll.

Group work will also be looked at.

The initiative was launched in our board by Robin McAteer (@robintg - follow her) and Anne Holness, pitched as a chance for up to 2 people per school to put their names forward to collaborate with math teachers in other schools. I believe about 25 people expressed interest.


I missed the first meeting, in late March. Rehearsals for the school musical were starting to get serious, plus I was giving tests that day and wanted to be able to clarify any questions (makes marking much easier). As I understand it, they looked at the article "MindSets and Equitable Education" (Carol S. Dweck), and created lesson study groups, with a goal towards focussing on "at risk" learners. It did get summarized in the postlude.

The scheduling turned out to be rather interesting for the group I ended up in. We didn't plan this, but we had the following differences as we progressed through creating a lesson each over the term:
1) Grade 9. Lesson done mid-unit (analytic geometry). Groups had existed for a while.
2) Grade 10. Lesson to start unit (trigonometry). New groups were created for the strand.
3) Grade 11. Lesson to end unit (sequences/series). Groups previously non-existant.

We only did 3 studies, not 5, as one teacher was from a school with students from multiple courses together in a class, which we didn't visit.  We also had his student teacher for the first two lesson plans. One thing in particular that I sensed from those interactions was that the student teacher had more of an interest in targeting aspects of the activity towards specific types of "at risk" individuals. Meanwhile the rest of us were looking more at making the activity broad enough ("low floor") to appeal to everyone, even those who were "at risk" (with any tweaks to be done on the day). Though I may have misread that impression completely.

Mrs. Obagi, Mr. Gibson & Mr. McIntosh

We also had a peculiar advantage that each of us were teaching two sections of the course involved (9D, 10D, 11U), thus could attempt the same lesson twice, either before or after the observation, and make refinements as needed.


Plan: Students draw 5 lines, then attempt to describe their picture to other groups. The hope is this motivates relative slope, leading to parallel and perpendicular. Sample graphs of those were prepared in advance for group analysis in the second half.

Result: This was a good lesson... for motivating the Cartesian Plane. Many students described lines relative to a grid, not each other. There was some loss of engagement at the describing phase, regained with the sample graphs. Good comments/questions came from teacher prompt "explain to me what we just did". Everyone seemed motivated towards applying the concepts to a text-style question in the last 10 minutes.

Post-Refinement: Students draw 5 points, join 2, then look at using one of the other points to generate parallel or perpendicular. Removes the focus from a grid.

*"We didn't have to go all the way across the page." (In fact one group actually boxed in a small area)
*"How do I describe the first line in?" "19 squares down."
*"What's this direction?" "Vertical."
*"You'll have two points to find the slope ... there's no points at all!" (a line did not meet where chart paper grids crossed)
*"It's detailed ... but kind of confusing because it's so detailed." (about a line description)
*"Thank you, you beautiful soul, that's so much easier." (asked to draw a horizontal line)
*Teacher: What would have made this easier? Student: "Having axes."
*"I was going to ask what a reciprocal means."
*"Well it kind of makes no sense."


Plan: Give a tiny triangle, ask for the angles. Give proportionally smaller triangles, ask for the side measures of the tiniest. Give triangle cutouts and ask for them to be grouped. Motivation towards defining similar triangles.

Pre-Refinement: Don't show proportionally smaller triangles in order (focus jumps to patterning). Don't have an obtuse set with triangle cutouts (little thought involved). Supplemental: Complete the rest of this triangle to be similar if one side has length 100 cm.

Result: Tiny triangle took over 30 min, various methods used. Pacing very different in groups around the room. Some correct and incorrect use of "same" vs "similar". For some triangles were grouped into two sets, not three, but students did seem to get the main idea. Last 10 min class were told 'Create a study note of what the purpose was and what you learned'; exit card done next day.

*"We're starting trig? Yes! It's going to be easy from here on."
*"We can't measure it, so we must prove this is the same as that." "They're not the SAME..."
*"Ratios of corresponding sides are equal... do you understand this?" (reading from binder)
*"I don't think they're all equilateral, that seems way too easy."
*"Dividing 25.5 by 2, I divide the number first, then the decimal, then I add." (12.5+.25)
*"Is that 5 cm?" "I might have used the wrong side [of the ruler]." "That's 5 inches."


Plan: Have a bin of variables from the sequences unit (a, n, d, r, tn, tn-1, Sn). Randomly draw 3 (repetition possible), and in groups, create 3 situational style questions with solutions. Switch problems with another group. In the last 20 minutes, compare answers. Motivation towards word problems, creating context.

Pre-Refinement: Up to 3 questions, at least one. Allow more time for creation, less time seems to be needed for solving. Add variable option "Your Choice" to promote engagement.

Pre-Run Anecdotes:
*"It can't be adding, it has to be multiplying."
*"We should give a harder number than 1,000 or they'll get it right off the bat."
*"Maybe we shouldn't give them the 57th term. We should think this through."
*"Why don't we build off the first problem." (for 2nd variable)
*"I don't think we're solving for d."
*"Was it 999?" "Close. You missed the last step." (I love that one.)

Result: Learning happened? I'll be honest, with this being my class, I felt more need to interact, and I wrote less down. Related, I felt like things were more chaotic than with my class the previous day - even though half a dozen students were absent. Though to be fair, I was also feeling a bit sub-par.

I know one group struggled with refining a single question for almost the whole time. I know I was able to motivate a group to try a third variable when I gave them "Your Choice". I was aware of a couple groups coming together to rewrite a question that had been presented in a confusing way, but was myself mentally tapped out by then. (Ended up using that one for the class test, but in changing the numbers to be smaller, messed one up. Derp.)

At any rate, my group members felt like this lesson had gone the best of the three - possibly because it was so completely open. One of the other teachers suggested doing an exit card before the period ended, which I did. I don't remember what I said, something about "what are you taking away from this experience". I particularly like what one student wrote, as (for me) it highlights my issues, as being both a pro and con of groups: "It was hard getting everyone to work together as we're used to tackling problems in different ways."


The entire set of teachers met again in June for a debrief. The idea of a Growth Mindset was covered again, and an "Effective Effort Rubric" was provided which identified approaches to processes as Fixed/Mixed/Growth. Groups also came up with some key points about what was learned from doing the study. "Less is more" was a key item for us, along with "have students set their own bar, choose own terms for entry". Now, what about the "at risk" component?

Group Summaries (click to enlarge)

Here's the thing with me. I don't feel I can focus on "at risk" learners when I'm still focussed on group dynamics. I'm not yet comfortable with the idea of groups themselves. More, it feels like a bit of a paradox to help an individual... when they're in a group.

Yet EVERY SINGLE LESSON that was developed was done with group work in mind. It does make sense. I see that. However it's not only something I continually have to wrap my head around, it's something I feel bad about for not being able to do effectively.

Something else that came up: OBSERVING IS A SKILL. You get a lot out of observing other classes - and on the flip side, it's so easy to miss things when busy with yours. Notably when helping individuals. Me, I'd say I much prefer to observe than to interact. Or to instruct than to interact. In fact there's a lot of things I rank higher than interaction. Yet interacting is back to being the cornerstone for group work. I'd say it's not my thing, but ironically, I had to interact with colleagues to come up with the lessons themselves. That was group work.

It was also pointed out to me on that day that I have a habit of "intentionally interrupting myself" as I reflect. I can see it, and I mention it here because I think I've veered from the point.


I came away from this with new experiences, having been able to see what other teachers are implementing in other schools. I always like that sort of thing, which is why I signed on. It was even said (at least in our group) that every teacher should experience this, the main barriers being (as always) time out of the class, and funding. More concretely, I got some new thoughts and strategies surrounding group work, which is a key weakness.

But is having those strategies the same as having strategies for "at risk" learners? I'm not so sure. Yet perversely, it's probably a stepping stone towards that - perhaps filling in a gap in MY OWN understanding. Am I being an optimist there?

The followup: I tried the same lesson technique to review for the Grade 11 exam. I cut up the various course expectations, put students into groups, had them randomly draw one, and then attempt to come up with a good "summary" question that they might see on the exam. As before, I feel like one class was more receptive, though one student (in the other class) said she would go along with it even though she felt more comfortable studying in a more traditional way. Which I let them do for the last couple days of the semester.

There's something to this. But it's not simple, at least not for me... and I can't figure out where it's going. Except maybe into more posts in the next school year. Do you have any insights?

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