Meaningful Math Discussions: It’s about the Ideas, Not Where You Discuss Them

Our staff has been thinking hard about how teachers are using Investigations 3 to teach math in all of the different scenarios they are faced with this year. We’ve been visiting the remote classrooms of teachers we’ve collaborated with previously, to learn from teachers and students who are teaching and learning math online, and to see how the rigor and coherence of the curriculum is supporting them in that work. This series of blogs will share some of what we are learning. (Read an introduction to this series of blogs, as well as the first, second, and third in the series.)

Classroom Situation: This first grade class of 21 students has been fully remote since the start of the school year. The teacher chose to divide the students into two cohorts for math. She Zooms with Cohort A on Mondays and Wednesdays, with Cohort B on Tuesdays and Thursdays, and with the whole group on Fridays. She feels this allows her to do more formative assessment, hold more productive discussions, and for students to have more of a voice. Unlike previous years, she has also experimented with pairing students so that they work with the same partner over a period of time. When I visited, the class was working on Session 1.8 of Unit 4, Fish Lengths and Fraction Rugs. All names have been pseudonymed. 

We have been interested in learning about how teachers are using the curriculum to support them in providing meaningful math instruction remotely. So, I was excited to learn that a first grade teacher I’ve been visiting was going to use one of the Math Words and Ideas (MWI) animations to introduce the day’s session. The MWI is a digital resource designed to provide an overview of the year’s math at each grade. It illustrates ideas and problems, shares examples of student strategies, and poses questions that children (and families) can solve and discuss. We say that the MWI is designed to be used flexibly; here was an opportunity to see how! 

The students in this class had been thinking about longer and shorter, and measuring fish with tiles or cubes, for a week or so. On this day, students would engage with problems about how much longer (or shorter) one fish is compared to another; a challenging idea for first graders.

The teacher began by showing the Math Words and Ideas animation titled Comparing Lengths. This video tells the story of two friends who each caught a fish. It models measuring those fish with tiles and thinking about which fish is longer. Then, a question is posed, about how much longer one fish is. An automatic pause is built into the animation after the question is asked, to give students time to think about and solve the problem.

Interestingly, the teacher paused the video before it posed the question, with the following frame showing, and asked students to model the situation described in the story with their cubes.

So right now what you can see on my screen and with your own cubes is one fish that is 7 inches long, that’s Vic’s fish. And one fish is 5 inches. That belongs to the person telling the story.

She continued the video. It asked, “How much longer is Vic’s fish than my fish?” and then automatically paused. She gave students time to solve the problem and then asked someone to share how they had figured it out.

Eva: Well when I used my cubes it looked like it was like 2, Vic’s fish was 2 more cubes longer.

How did you arrive at that conclusion? What were you looking at? 

Eva: My cubes. I set them up and put them together and saw that it was 2 longer. 

I’m going to call on someone who can repeat what Eva said. How did she solve the problem? 

Caroline: She first set it up um with her cubes and then she thought that it it it had that, she, it had 2 more. She lined them up and then she saw 2 more cubes left on the one with 7 inches long. 

Nice job listening to another mathematician and explaining what they did. Thumbs up if you solved it similarly. Another strategy? 

Ailey: Well first of all I knew that 5 plus 2 is 7 so if I added 5 cubes, and 2, and connect them 1, 2, 3, 4, 5, 6, 7. 

So it’s almost like you took this fish and thought how many more cubes would I need to add to make it as long as Vic’s fish? Thumbs up if you thought about it like this. One more? 

Isai: My thinking was 5 plus 1 equals 6 and 5 plus 2 equals 7 so this will have to go 2 more to go to 7. 

The teacher summarized their ideas, using addition equations to represent their thinking so far. Then she played the rest of the video, which talked about the same situation in terms of how much shorter one fish was compared to the other. This allowed her to introduce how a subtraction equation could also be used to represent the problem.

After this 10-minute introduction, the teacher described the Math Workshop activities that students would work on for the remainder of the session, which included story problems about comparing the lengths of fish. She reminded them of the tools available and assigned math partners to breakout rooms. (Note: Read a Twitter thread about one student’s thinking about these kinds of problems, that took place in one of these breakout rooms.)

Reflections This class has me thinking, again, about how hard and thoughtfully teachers are working to engage students in meaningful math experiences, and how impressively students are responding.

• The MWI is designed to be used flexibly. The curriculum suggests that it can be used as a resource for students during class work or homework, and/or as a reference for families to better understand the work their children are doing in class. This teacher found yet another way to use the MWI; to introduce a new problem type.

Showing this video set this new kind of problem in the context of work they’d been doing—measuring fish and deciding which of two things was longer—and provided a familiar image that would help students visualize what the story was asking.

Building in an extra “pause-point” was also an important instructional decision. It gave students the opportunity to represent the situation for themselves, to model the story with their cubes. This meant that students had a representation of the story in front of them (and on the screen) that they could then use to reason about the question the video posed. Several students used those cubes towers to figure out how much longer Vic’s fish was.

• We’ve heard from many that engaging students in math discussions has been one of the more challenging aspects of remote teaching and learning. In this lesson, the MWI supported this class in discussing important mathematical ideas. In ten minutes, four different girls volunteered to speak. Three shared their strategy for solving the problem; one restated someone else’s thinking. Many other students participated by thinking about whether their strategy was similar to the person who just shared, and indicating this with a thumbs up. It was a lively conversation that got a range of strategies for solving “how much longer” problems “out there,” strategies that students then used as they solved similar problems in breakout rooms. I’m left thinking about how in-person interactions are not what’s required for meaningful math discussions; what’s necessary are teachers and students who are interested in and engaged by mathematical ideas and are excited to talk about them.

(Note: For more on how teachers are using the MWI to support math discussions in remote learning situations, watch Session 2 of our first Burst Series, Promoting Student Talk Using the Students Might Say.)