This week I had the chance to work with a third grade teacher, Andrea! Her class is just about to begin their unit on multiplication and division so she wanted to do Talking Points to see what they knew, and were thinking about, in relation to these operations. During our planning we discussed the ways in which this Investigations unit engages students in these ideas, misconceptions students typically have, bounced around ideas, and played with the wording of the points. Being my first talking points activity with third graders, I was so excited to see how students engaged in the activity. I have found that even during Number Talks, the younger students are very eager to share their own ideas, but listening to others is difficult.
In looking for how students “saw” multiplication and thought about operation relationships, we designed these Talking Points..
Andrea introduced the activity and we did the first talking point as a practice round in which we stopped the groups after each of the rounds to point out the important aspects. We pointed out things like,”I liked how Bobby was unsure and explained why,” and “I liked how everyone was listening to Becky when she was talking,” and “I thought it was great proof when Lily drew something really quick to support her thoughts.” Then we let them go and walked around to listen as well!
During the Talking Point round, some things we found really interesting were:
- How difficult is for them to sit and listen to others without commenting. Not like it is not hard for use as adults though, right? 😉
- How much students struggled to say why they were unsure. Sometimes it was not knowing what the word division meant, yet they struggled to articulate what it was within that talking point that was confusing them. What a great thing for them to be able to think about!?
- How they related the dot images they had been doing in class to multiplication and division.
- How they thought about inverse operations. They said things like, “I don’t know what division is but if I can use subtraction with addition, I probably can use division with multiplication.
We had pulled two of the points that we wanted to discuss, whole group, afterwards, “I can show multiplication as a picture.” and “We can use multiplication problems to solve division problems.” We put them up and just asked them what their table had talking about. The conversation was amazing. Hearing how they thought about multiplication as groups of but 50 x 2 means “fifty two times” while 2 x 50 means “2 fifty times.” We also heard how someone at their table had changed the way they thought about something. And the division conversation was so great and for the students who were unsure because they did not know what division meant, it felt really organic to come out that way… from them, not us.
Of course, we followed with a journal write:) We gave them three choices to write about…
I was so impressed by the way they wrote about their thinking, by 5th grade, they will be amazing!!
The student above, during the Talking Points, said that he could show multiplication as a picture because “an equation IS a picture.” It was lovely to see him make the connection to a visual for an equation in his journal.
I wish the quality of this picture was so much better but her pencil was so light it was hard to see! She does a beautiful writing about how exactly someone at her table changed her mind with such an articulate way of talking about multiplication and division!
This student above explains perfectly why teaching is so difficult…”…sometimes we have facts about math, we all have a different schema. We were taught differently than other kids.” I am curious to hear more about her feeling about the end piece, “some kids know more then other kids.” Is that ok with her and she understands we all will get there just at different times?
These last two were two different ways in which students reflected on the dot images they have been doing in relation to multiplication!
What a great class period! I cannot wait to be back in this class to see how students are working with and talking about multiplication and division!
~Kristin
MathMinds 
From the future, is there a way to make sure that *somewhere* along the line, “multiplication and division” includes division (as more than “the other number in the times table)?
The adults I work with almost *never* have a real understanding of division. the people who think “what’s half of 100” is an easy obvious question tell me they do not know what 100 divided by two is.
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This is the first I’ve heard of “talking points.” What does it look like in a classroom? On the sheet you pictured, you have a box for tally marks. How do they work? What are you tallying?
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Hi Laura! I described them a bunch in this post and there are links in there to Elizabeth’s blog who did a TON of work with them: https://mathmindsblog.wordpress.com/2014/11/13/talking-points-2d-geometry/
Hope this helps!
Kristin
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You might want to make that last point a bit more specific and say when I multiply by an integer. That lays a little groundwork for later and makes this observation into something that doesn’t need to be unlearned when you hit fractions.
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You might want to make that last point a bit more specific and say when I multiply by a counting or natural number. That lays a little groundwork for later and makes this observation into something that doesn’t need to be unlearned when you hit fractions or that pesky 0 for that matter.
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I agree it is definitely a misconception later. Teaching 5th grade previously I was curious to hear the convo. Since we were using this as a formative, discussion point we didnt feel as if students solidified an “answer” during this time. It was more for us to hear how they think around it.
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If I’ve understood these talking points correctly, some of them are meant to be wrong or debatable.
My experience with unlearning is that it can be quite exciting. Like when, kids who have never seen an irrational number press 3 and then the square root button. “Hey! What is that really big number?” We explore square numbers, squaring integers, and then look at the inverse, square roots. “So the square root of 100 is ten, what is the square root of 99? Try it out.” So, the old simplicity is broken, but in a memorable way.
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Yes, you understand correctly! There is no right or wrong answer that is solidified at the end because, like you said, some are wrong or are debateable based on interpretation.
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