Here are two situations.

**Visual Pattern**

A visual patterns warmup routine, focused on writing expressions for the number of squares in the nth step of this pattern:

Groups come up with these two expressions and explain how they relate to the pattern:

Another way to write an expression that no student found looks like this:

The final expression is a different way of conceptualizing the pattern and offers a potentially useful perspective for future problems. Do I present it to students?

**Number Talk**

A number talk warmup routine, trying to find a strategy to mentally multiply or approximate 0.48*650. Students offer several different ways to break it down, either by breaking up the 0.48 or by breaking up the 650. Another approach that I’ve found useful for problems like this is to solve it in terms of percents — finding 10%, then 1%, then 50%, 2%, and finally 48%. Do I present it to students?

**When to Interject?**

There are multiple possible next steps.

- I could explain the additional perspective and why it’s useful
- I could present the expression or the outline of the number talk strategy and have students discuss in small groups to figure out how to assign meaning to it
- I could send students back to groups with an open-ended question of trying to find another method
- I could move on and keep my ideas to myself

I’m not sure what the best answer is. My fundamental struggle here is that these routines work with a great deal of student ownership. I’m doing plenty of work selecting and sequencing students to share their ideas and building norms for students to be attentive to each other and make connections. But I speak very little, and I typically let students present all of the math. Which is more important: norms of student ownership, or a pedagogically useful piece of more explicit instruction?

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Dr NicInteresting question. Maybe you could ask the students? Do they see you as a fellow learner and discussant, or as the sage handing down a “better” way?

dkane47Post authorInteresting! I hadn’t thought about that. Would be interesting for a number of reasons.

howardat58Try (n + 1)^2 – 2 but keep it quiet.

When they have thought for a bit just suggest the empty squares.

Regarding 0.48 x 650 there are lots of things to do here.

4.8 x 65 is a start.

Or multiply o.48 by 100, do the sums, then divide by 100.

Or write 0.48 as 0.5 – 0.02

Or a mixture ! I would look for alternative partial ways, not a “do it myself” thing.

dkane47Post authorInteresting w/r/t the number talk. I like that percent strategy but there were plenty more I didn’t think about.

Elizabeth RaskinI ask myself this question every day. I’m pretty sure there are times when I interject that I should and times I should but don’t. What I have noticed of myself is that I am a completely different teacher in each of my classes. For example, in my team taught inclusion class, I interject a lot more than in my advanced math class. My role in my inclusion class is much more guiding while my advanced students really just want me to stay out of their math processes.

Like you, I like to let the kids present their own work, but I like to think of myself as part of the class and will share my thinking with them as well…and since I don’t interject often I don’t think they see my ideas as the “end all, be all” of math. In fact, often the student response to when I show an alternative method is “I really like my way better.”

Oh, and I’m not above sneaking in my method when looking at “student work”. 🙂

dkane47Post authorYea one thing I’m struggling with here is the tension between using my judgment and interjecting when it seems appropriate, and the value of a routine that stays the same day after day to promote a safe space to take risks.

Aaron BieniekDylan, an important part of the pattern work, for me, is for students to be able to describe within the visual where the expression for the area comes from. So to explain 2n^2 – (n-1)^2 I might show two overlapping squares by drawing a dotted line around the 2×2 square inside of n=3. So I have 2 squares that are 3×3 (2n^2) but then I have to subtract the overlap which is 2×2 (n-1)^2.

My point is that to introduce the expression that no one thought of, you could reverse the reasoning. “Someone in first hour came up with this expression (n+1)^2 – 2. Explain using the visual how they arrived at that expression.” I feel like going both ways and explicitly tying to the visual raises the ceiling a little bit on tasks like this. (btw, your expression has a typo: (n+2)^2) Perhaps this is what you meant by “figure out how to assign meaning to it.”

For your percentage problem, I wonder if structuring the talk differently would be helpful. The idea would be how can you use your previous solutions to help with the next ones? Problems would be presented one after the other with discussion in between.

0.50 * 650

0.10 * 650

0.05 * 650

0.01 * 650

0.40 * 650

0.08 * 650

0.48 * 650

Then have the students meet with a partner to discuss how to apply this string to 0.63 * 475. I guess this isn’t an interjection though – it’s kind of after the fact thinking. So maybe you don’t interject and do something like this the next day, because now you know this type of thinking isn’t in their wheel house at the moment.

dkane47Post authorI really like your thinking here. One sentiment on Twitter last night was thinking about the purpose of the alternate strategies and trying to make clear to students what they are useful for by putting them in a situation where there is some need.

Brenda PayneShould the last expression read (n+1)^2 – 2?

dkane47Post authorWhoops. Nice catch, updated.