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Support Productive Struggle in Learning Mathematics

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Support Productive Struggle in Learning Mathematics

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Support Productive Struggle in Learning Math.pdf

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Use and Connect Math Representations.pdf

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Section from NCTM Principles to Actions: Support Productive Struggle in Learning Mathematics

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Section from NCTM Principles to Actions: Support Productive Struggle in Learning Mathematics

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Support Productive Struggle in Learning Math.pdf

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Look For Sheet: Elicit and Use Evidence of Student Thinking

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Look For Sheet: Elicit and Use Evidence of Student Thinking

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Create a new document and edit with others at the same time -- from your computer, phone or tablet. Get stuff done with or without an internet connection. Use Docs to edit Word files. Free from Google.

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Heidi Fessenden's Blog: Who Talks? "Numberless" Graphs in Grade 2

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Heidi Fessenden's Blog: Who Talks? "Numberless" Graphs in Grade 2

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A class of second graders is having their first Hands Down Conversation (HDC). They are discussing what happens when you add 10 to a number, and what will happen if we count by 10 but start at 8. I'm sitting outside their circle, taking notes on everything they say. This is how the conversation ends.

David: It would be 0, 4,5, 8, then it would be 18, 8 at the end, 28, 8 at the end, 38, 8 at the end, 48, 8 at the end, 58, 8 at the end, and it would keep going.

Eddie: I get what you're saying, that there would be an 8 at the end of everything, but we kind of know that already.

Theo: Not everybody knows that. Babies don't know that. (He giggles).

There is a low hiss of voices around the circle. Some students don't like this way of talking.

I call a timeout to reflect on how our first hands down conversation has gone.

This class is not an easy one to facilitate. There are 22 students. 14 are boys. 8 are girls. More than half have an abundance of energy which makes it hard for them to sit still. Of those 14 boys, a majority (about 8) are quick-thinking, verbal, and impulsive, which means they often call out answers or comments before everyone has had time to think. Without careful management, this means that other students are completely excluded from the conversation.

After the second HDC, I list the names of the students who spoke on the board. We count how many spoke (7 of the 22 in the class). What do you notice? I ask. No one notices gender, so I ask how many boys spoke (5) and how many girls (2). We talk about some of the reasons so few children spoke. The same people talked over and over again. Some people tried to speak but their voices were quiet and they were drowned out by louder voices.

Before our next hands-down conversation, we work on a list of ideas for hearing more voices. Their ideas are spectacular.

Gender comes up a number of times. Girls suggest that boys should listen more and wait longer before speaking more than once. Each time the girls or I mention "boys," there is a murmur across the rug. The boys don't like it. I make a split-second decision not to write "boys" on our chart. The truth is that there is one girl who also spoke often and assertively, drowning out other voices, so this is not just a boy problem.

But I do tell them that men often talk more than women. I tell them that when grown-ups have meetings, the men talk more than the women, even when there are fewer men in the group. I say, "boys and men have to practice listening more and speaking less."

The next day, we have another hands down conversation. First we re-read our chart. We talk about what to do if two people speak at the same time (one person can say "go ahead" to the other). I ask the most talkative and impulsive boy to sit outside the circle with the teacher and a class list, and keep track of who speaks and how many times. That way he has an important job and won't overpower others so much. And I ask a quiet girl who rarely speaks to launch the conversation.

They do better. More students speak. More girls speak. And more boys speak than the week before. Some boys still talk a lot. But I can see others working really hard to make space for more voices. Some speak only once or twice. One stops himself a number of times and offers the floor to other kids. One girl tries asking another girl, "Would you like to speak?" At the end of the conversation we tally up the data: how many students spoke: 11. 7 boys and 4 girls.

Before our next hands down conversation, I wanted to show them the data of how their conversations had gone over the past two weeks. I wanted to present the data to them visually so they could make sense of it, see the positive trend of more participation, then hopefully increase that participation. I decided to do so with a variation of numberless graphs, a routine Kassia and Brian Bushart (among others) have been working on. My graphs aren't numberless, but they have few numbers, few words, and I shared them from simplest to most complex.

After I projected each graph, I simply asked "What do you understand?" and "What do you wonder?" They discussed the graphs for more than 30 minutes. The turn and talks were loud and animated. Best of all, nearly everyone spoke in the large group, even students who rarely participate or who, when they do speak, are often confused.

And here are their observations:

Here is the second graph.

Finally, the third graph.

Some of my favorite observations:

The girls and boys combined to make everyone.

The blue bars show who spoke on Week 1

The reds are taller than the blues (because Week 1, only a few boys and girls talked)

Girls still talked less than boys Week 2

The reds are the same on slide 2 and slide 1 (because the total number of students stays the same!)

The green is bigger than the blue (because more people talked Week 2 than the first week)

How many kids spoke anyway?

We didn't talk much about the actual numbers of students who spoke, but we did spend some time discussing whether there are 7 or 8 girls in the class. Besides the kids who just looked around the rug and counted, there was some discussion of how you could use a graph with only a few numbers on it to figure out how many each bar represented. One student described how he imagined these numbers in between the 6 and the 12.

I know most adults who look at these graphs will be interested in talking about the dynamics of our hands-down conversations. What do these data mean? How can we change them? How did the class talk about these questions and decide to move forward?

But what I loved about today's conversation what how easily the second graders stuck to low-inference observations. Only once did someone make a suggestion to the class, and it was after the comment that "the red has to be bigger than the blue." We talked about when that might not be true: when would the red and blue (or green) be equal? Only if every single student participated in the hands-down conversation.

Looking at data this way, and starting with what we understand, opened the door for everyone to participate. It was the perfect low-floor, high-ceiling task. Jayvaughn, a student who raises his hand to participate all the time but who is often confused and hard to follow, observed that "the reds are taller than the blues." And in fact (as I told the class), this is how grown-ups make sense of graphs like these. We look for trends overall. Figuring out exactly how many girls spoke or how many boys are in the class is less important than the trends, than observations like the fact that each time the reds were taller than the blues. After Jayvaughn made that observation, I asked if someone else could explain what his observation meant in terms of real life. Tyrell explained that this was the case because "week 1, only a few boys and girls talked."

I could go on and on about each of their observations and why I like them all so much, but I'll stop myself here. What I most was reminded of was how much depth and understanding can come out of just talking about what we see. This is why we spend so much time noticing and wondering (in math and in all subjects). When we slow down and just observe, we make the most sense for ourselves.

A Note on Gender Identity (added 2/24/18)

One thing I thought about but didn't include in this post originally was the gender binary implied in this whole data analysis. This is a class where everyone identifies, so far, as either being a girl or a boy -- but this analysis I encouraged them to do did not question that gender binary in any way. I am not sure how to hone my students' ability to pay attention to the gender dynamics of who speaks in class without reinforcing the gender binary, but I'm sure there are good ways. I'd love suggestions.

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Who Talks? "Numberless" Graphs in Grade 2

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A class of second graders is having their first Hands Down Conversation (HDC). They are discussing what happens when you add 10 to a number, and what will happen if we count by 10 but start at 8. I'm sitting outside their circle, taking notes on everything they say.

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Section from NCTM Principles to Actions: Pose Purposeful Questions

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Section from NCTM Principles to Actions: Pose Purposeful Questions

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Look For's Implementing Tasks that Promote Reasoning and Problem Solving

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"Look Fors" Implementing Tasks that Promote Reasoning and Problem Solving

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Section from NCTM Principles to Actions: Facilitate Meaningful Math Discourse

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Section from NCTM Principles to Actions: Facilitate Meaningful Math Discourse

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Robert Kaplinsky's Ignite Talk on Productive Struggle

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Robert Kaplinsky's Ignite Talk on Productive Struggle

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Math Class Needs a Make

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Today's math curriculum is teaching students to expect -- and excel at -- paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. Dan Meyer shows classroom-tested math exercises that prompt students to stop and think.