Counting, Conjectures, and Claims

2015-05-01 12.16.12

Last week I got to visit three wonderful schools in Michigan to think and learn with teachers about math talk, reasoning and number sense. We got to plan, teach and debrief together, which is one of my favorite ways of learning with a group of teachers!

With one group of 3rd grade teachers, we decided to think more about choral counting routines and how we could use them to get kids talking, reasoning, and looking for patterns and structure.

The team chose to count by 3s starting at 21. Before teaching, we thought about what mathematical ideas and patterns this count might bring out and how we could record the count to support those ideas. This group of teachers also wanted to think more about students journaling around math, so we thought about taking time at a certain point in the count to have kids write about their thinking.

In the above picture are the numbers we said and recorded in our count. Students immediately noticed how the digit in the ones place repeated at a certain point in the count, as well as patterns related to how many times numbers in a particular decade would appear. Students noticed that while we were skip counting by 3s, there was a jump of +30 between the numbers in the same column. When we asked students to predict the number that would go in the orange box below 60, some counted by 3s from 78 (the last number we said together), but many used the +30 noticing.

But things got really interesting when one student spoke up and said “Hey, and 30 + 60 is 90! You can add the first two numbers in the column together to get the one below.”

“That does work here, doesn’t it?” I said, writing it down. “I wonder if that works everywhere or just here? I wonder how we could figure that out.”

In many ways I wish we had had more conversation here, especially since the students were new to this routine and to the idea of pursuing and investigating claims and conjectures, but for the sake of time and getting to the journaling we decided to move ahead with our plan to have them write.

We asked students to write the number they thought would go in the second orange box (the blank box in the picture) and write about the patterns they noticed that helped them figure this out.

As a group of teachers we read through the students’ journal entries. Some kids wrote about skip counting from 90. Some kids wrote about how they knew the ones place would be a 2 and the number would be 100 and that 2. But a good number of kids wrote that the number would be 114 because 42+72=114. This was fascinating because they ignored the previous patterns they had noticed in favor of following this claim set forth by a student. This claim had power! (I think the idea that you could notice something that would lead to a claim was powerful for this group.) The students wanted to make a generalization, find a pattern, find a rule so much that they ignored evidence against their claim. It was fascinating and a beautiful error, in an inaccurate sort of way. 🙂

This count and journaling also provided amazing material for teacher discussion. The thoughtful teachers I worked with talked about how they might push students towards ways of investigating conjectures and claims, how you might prove or disprove this claim, and the importance of this kind of reasoning for wider mathematical understanding.

All this great thinking, teaching, and debriefing in a brief little time with this team of teachers. I was only sad that I didn’t get to stay for next class meeting to see where students would take this next!

But it has pushed my thinking as well. First, it made me reach for Russell, Schifter and Bastable’s Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades. It also made me reach out to Twitter-thinkers Kristin Gray (@mathminds), Mike Flynn (@MikeFlynn55), Simon Gregg (@Simon_Gregg) and Elham Kazemi (@ekazemi) to clarify my thinking on claims and conjectures (which I’m still working on!). I can’t wait to think and teach more.


3 thoughts on “Counting, Conjectures, and Claims

  1. Kassia, you know how interesting this is to me! As someone who’s been trying out counting circles this year it’s fascinating (it helps that this is a Grade 3 lesson, which is the same as Year 4 which I’m teaching this year). Also the claims. (How many other mistakes are due to free-floating unchallenged claims in students’ heads??)

    I love it that you get to work with teachers in teams like that, planning, teaching and debriefing with the journals.

    I’m also thinking of using journals, instead of just loose leaves as I do now. My colleague in secondary, Jim Noble, uses journals very artistically, but I like the idea of them being a place for thoughts, mistakes, stuff that isn’t visually beautiful too, a mixture maybe. Something for September, the new year perhaps…

    1. Simon,
      I was wondering if they all latched on to the claim because I didn’t immediately shut it down and point out that it wasn’t always true. Maybe, just as students have to learn that being asked, “How do you know?” doesn’t mean you’re wrong, they also have to learn that “That’s an interesting observation. I wonder if that’s always true,” doesn’t mean that the claim is true. Hard to say, because I didn’t know this particular group of kids beyond this one lesson.
      I’ve also been doing a lot of thinking about journaling and have some ideas I want to try next year for sure! We should compare notes next September. 🙂

  2. So me being me I tried this out today with my third graders. We had some similar conversations and I was surprised to see that although we had discussed the +30 from one row to the next that some kids still said they had skip counted to get the missing numbers. A few even ended up writing a number smaller than the previous row- until it came time to explain their thinking. I also wany to be better about having the kids journal!

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