Susan Jo Russell: Noticing Regularity to Constructing Arguments @ NCTM Boston 2015

My school found me a sub for the day on Friday, so that I could attend NCTM's National Conference in Boston. Much of the PD I have been sent to recently has had more general education foci -- technology, managing faculty-student relationships, and creativity to name a few. So, I was really excited to get some mathematics specific professional development.

Since I was only there for 1 day, planning which sessions to attend was tough -- as usual. Doesn't it seem like the best presentations are always scheduled during the same block? On Friday, that block was in the morning when I had at least 4 presentations highlighted. So, when my first choice presentation started slowly and then experienced tech problems, I stepped out to head for Susan Jo Russell's talk on Noticing Regularity and Constructing Arguments just a few rooms down. I am sure glad I did.[4]Leaving a presentation before it is over ALWAYS makes me feel terrible. It is something I have forced myself to start doing. There is no use sitting through an hour of something unless it will be useful.

SJR was talking about short lessons her company (TERC) has been developing and testing  The lessons focus on looking for patterns and then helping students learn to generalize them. She starts the kids off with examples from addition, but I thought we would have more fun looking at one of the multiplication lessons.[5]Also, I missed pretty much all of the addition part. For example:

What do you notice as the factors increase by 1?
6 x 4 = 24 6 x 4 = 24
6 x 5 = 30 7 x 4 = 28


Many students notice that the product increases as the factors increase, but they can be more specific about how much the products will increase; the product increases by an amount equal to the unchanged factor.

This style of questioning is very similar to something I do with my fifth grade students each fall. I work on developing their understanding of equality (too many think that 2 + 4 = 7 - 1 and similar equations are "not allowed"). As part of that work, they solve equations by comparing the left side to the right side. For example:

z + 153 = 275 + 152

I lifted this idea from a great Thomas Carpenter book and have been really pleased with how it has worked. You can see here that students would be making a comparison that is similar to the multiplication example above.

One area I have been struggling with is how to help my students develop the ability to generalize and explain the patterns they see in these equations. While they can often tell me the value of the variables, they often cannot generalize or explain the pattern. Doing so really hits the CCSSM Standards of Mathematical Practice, so it is definitely something I want to be better at teaching.

SJR's presentation provided a three-step framework for helping students develop their ability to generalize and justify. Here are the three steps paraphrased:

  1. Look at examples (see above)
  2. Articulate the pattern (the product increases by an amount equal to the unchanged factor)
  3. Modeling and/or represent the pattern (SJR used an equal groupings model for the multiplication)

I would like to add/suggest an additional step:

  1. Extend the pattern. (In my above example, that might mean talking about what happens when a factor increases by 2)

SJR also showed videos of 4th graders grappling with these ideas.[6]And doing quite well too with the support of some great discussion. Hearing her speak and watching the videos made it clear that I need to step up my game in 2 places: 1) giving students more time to articulate the pattern and 2) helping students model the patterns as a means for justifying and extending. I am excited to try that next fall!

Follow Up:

  • In case you are interested, I found a copy of the slides from a similar presentation by SJR. Check it out.
  • I found an interesting and related problem on Steve Wyborney's blog. Do you see how it is connected to SJR's multiplication example?