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## Two by Four

The first strategy I’d like to share for embedding formative assessment (Remember, formative assessment isn’t always a thing. It’s feedback for learning) is referred to by the consultanty-types as “two by four”. While generally working in the field of consulting, I should point out that I’m not much of a consultanty-type. I’m not a fan of fancy processes and the equally silly names we give them. So, for classification purposes only, I present to you the two by four.

One of Dylan Wiliam’s 5 key strategies is to activate students as instructional resources for one another. This strategy is my go-to one, mostly because I’m trying hard to embrace the philosophy of the revised program of studies, which suggests I should orchestrate experiences from which my students can extract their own meaning. When I active them as instructional resources for one another, they learn from each other, rather than from me writing notes on the board.

This strategy is really simple, and can be used to teach just about anything I can think of in our curriculum. Two students learn something through some kind of structured activity. They check in with two more students who learned the same thing and compare strategies and results. When they agree, they move on.

Let me tell you how this worked in a class I tried recently. I was invited to model formative assessment in a Math 20-1 class (Pre-Calculus 11 elsewhere in the WNCP). The topic of the day was multiplying radicals. In my classroom days, I would have written some rules on the board, done some examples, and then given an assignment for practice. The students would not have been activated much at all.

For this lesson, I built a structured activity that I hoped the students would work through and learn what I needed them to learn about multiplying radicals. That activity is posted below.

Some notes about how it went:

• First of all, I should point out that this does not need to be photocopied and handed out. It would work just fine as a set of questions that the teacher poses to the class. I ran it off, because I was going into a stranger’s classroom and I wanted to make it as simple as possible for the students.
• As the students worked, I circulated and listened in. I helped when needed and as little as I could. I regrouped them at the appropriate times and with the appropriate people. By the end of class, almost all of the students taught themselves how to multiply radicals. I gave exit slips at the end (another formative assessment strategy) and the exit slips showed me that most students had met or exceeded my expectations. Two students didn’t manage to learn the material. This next part is huge. Wait for it…
• It wasn’t my class. I didn’t know many of the student names. Despite that fact, when I saw the two exit slips that showed little or no understanding, I described to the classroom teacher exactly who I thought those two students were in terms of where they were sitting. I nailed it. Because I was circulating and working closely with the class, I knew exactly who was getting it and who wasn’t. That’s feedback, folks. That’s formative assessment, folks. The old me (remember, the one who stood at the board writing example after example) would have had no idea who was getting it and who wasn’t. Unit tests were frequently sources of great surprise for me.
• I was fascinated by the strategies they were coming up with. On the first page of the activity, all students concluded that $\sqrt{a}\times\sqrt{b}$ was equal to $\sqrt{ab}$. What they did when I introduced coefficients on the next page was really interesting. The room was split into two camps when they came across $3\sqrt{2}\times6\sqrt{3}$. Many of the pairs of students got right to it and multiplied the coefficients, then multiplied the radicals, and came up with $18 \sqrt{6}$. They confirmed their answers using their calculators, and moved on. The other camp wanted to rely on the only rule they knew at that point ($\sqrt{a}\times\sqrt{b}= \sqrt{ab}$). Their solutions looked like this: $3\sqrt{2}\times6\sqrt{3}= \sqrt{18}\times\sqrt{108}=\sqrt{1944}=18\sqrt{6}$. They were converting to entire radicals, multiplying, and then simplifying. It worked, but it was highly inefficient. So when it became time to let the pairs meet with other pairs to see what they had learned, I made sure that each group of four contained one pair with the efficient strategy and one pair with the inefficient strategy. Students were taught the more efficient strategy by other students, rather than by me. That’s formative assessment.

I did something similar in a Math 30-1 (Pre-calculus 12) class earlier this year. The description and resources are available here.

### 5 Responses

1. I really need to try this! Thank you, John.

2. I am getting ready to do addition, subtraction, and multiplying radicals this week. LOVE THIS! I am now starting to think about how I could adapt this to include the addition and subtraction of radicals as well. THANK YOU SO MUCH for sharing this!!!!
–Lisa

• You’re probably already there, but I think if I was going to build one for addition and subtraction, I’d build on the algebra of combining like terms, introduce radicals, and include some calculator checks so that they can discover the rules are really the same for like terms and like radicals.

3. […] these strategies, but any guidance you can give will help. John Scammell shared what he did with multiplying radicals earlier and I am using that here in the near future. How do you create these kinds of materials? […]

4. […] This is a lesson I built as a demo during my travels. In it, I attempt to have students learn from each other, rather than from me writing everything on the board. The technique I am using is activating students as instructional resources. Right after I used this lesson, I described it in great detail on my blog. […]