Teachers who take a careful look at students' work—and listen to the accompanying thinking—have a powerful tool for supporting students' learning. In our work with teachers, we've seen teachers use students' work and discussions about it to promote mathematical thinking, explore students' understandings, and inform future instruction. Even the work of the youngest students can help teachers understand their math reasoning. To illustrate, consider a lesson we recently observed.
The Hungry Caterpillar Through a Mathematical Lens
Ms. Stewart, a kindergarten teacher at Hilltop Elementary, an urban school in the Pacific Northwest, was reading Eric Carle's The Very Hungry Caterpillar aloud. As she read, she invited students to use a sheet of paper and pencil to keep track of how many things the caterpillar ate. After she finished reading, students talked with a partner about the mathematical strategies they used for keeping track. She prompted them to use their recording sheets to explain how they counted the number of items the caterpillar ate and to justify their answer.
While student pairs discussed their strategies, pointing to their recording sheets, Ms. Stewart listened in. She was curious to learn more about her students' mathematical thinking, specifically their understanding of number names, counting sequences, and connecting counting to cardinality. She also hoped to select relevant work to share in the upcoming whole-group discussion.
Ms. Stewart regularly uses children's literature to engage her kindergartners in discussing mathematics. She finds powerful learning opportunities for students when she approaches story contexts and illustrations with a mathematical lens. During this read-aloud, her goal was to promote lively discussion about counting and cardinality.
Promoting—and Documenting—Student Thinking
As the reading progressed, students derived their own strategies for figuring out how many things the caterpillar ate. Their drawings provided evidence to support their mathematical reasoning. This documentation of students' thinking helped Ms. Stewart assess their understanding. As she brought the group back together, she announced, "I'm going to invite a few students to share their work with the class. Use your recording sheet to show us how many things the caterpillar ate and how your recording helps prove your answer. Niko, would you get us started?"
As Niko shared his work (shown in Figure 1), he explained, "I think the caterpillar ate 25 foods. I made a tally mark every time he ate something. Then I counted the marks 1, 2, 3 …." [counting up to 25]
Figure 1. Students' Recordings of How Many Things the Caterpillar Ate
Student work published with parental permission.
Ms. Stewart repeated Niko's strategy: "We see that you made a tally mark every time the caterpillar ate something. We hear you saying each number and counting in order from 1 to 25." Turning to the class, she added, "Let's count out loud and say the numbers as Niko points to his tally marks." After they'd counted aloud together, Ms. Stewart displayed Niko's sheet on the front wall and invited Izzie to share her work. She chose Izzie because Izzie, although still counting by ones, had used a different strategy and got a different answer than Niko did.
Izzie, whose work also appears in Figure 1, said, "I think it was 26 foods. I wrote a letter for the day it was when he ate something."
To clarify Izzie's strategy, Ms. Stewart asked: "We see your letters. How did that drawing help you know how many things the caterpillar ate?"
Pointing to the numbers above the letters of the week, Izzie responded "I counted 1, 2, 3 … up to 26. Then, I knew it was 26 things that he ate."
"Both you and Niko used your drawings to help count and to know the total number. You also used your drawings to support your answer," Ms. Stewart said. "Let's hear from one more student."
Fiona next explained as she displayed her work, "I drew the foods. I counted them. Then I got 26, like Izzie, but different than Niko."
"We see your numbers above the foods. Can you explain those to us?" asked Ms. Stewart.
"That's 1 because there is one apple. Then that's 2 for two pears. Each number goes with how many things the caterpillar ate that day. Then it is 26 at the end after all of this stuff. He ate A LOT!" explained Fiona.
"Okay," Ms. Stewart summed up. "We have two different answers, 25 and 26. Let's see if we can use our recording sheets to reason together about these different answers and what might be the difference in our solutions."
As the discussion continued, students engaged in sense-making talk, with their recording sheets playing an essential role in supporting their explanations—and in understanding their classmates' thinking. The class discovered that the confusion about 25 or 26 foods was caused by differing perceptions of whether the last food item—a leaf—counted as food. Eventually students agreed a leaf should count as food, at least for a caterpillar, and agreed on an answer of 26.
Exploring Students' Learning
After school, Ms. Stewart looked through the recording sheets to gain insight into her students' mathematical understanding. She noted how their skills and understandings corresponded to the math standards for their grade level in the Common Core State Standards.
This teacher could see that most students had written numerals above any symbols they used to represent what the caterpillar ate, similar to the strategy Niko and Izzie used. This helped her know that students were learning how to write the numbers from one into the mid-twenties and to represent a number of objects accurately through a written numeral (which corresponds to Common Core Math Content standard K.CC.A.3).
Ms. Stewart gleaned important information through reflecting on how students discussed their work in pairs and in the whole group. She heard students making sense of the relationship between numbers and quantities, in this case to answer a "how many?" question; as students counted objects, they said number names in standard order, pairing each object with one—and only one—number name. In developmentally appropriate ways, then, these students demonstrated an understanding of important mathematical concepts and the ability to apply them.
Ms. Stewart also considered what students' work revealed about their progress in using good mathematical practices. The Common Core standards' seven practices of mathematically proficient students include making sense of problems and persevering in solving them, constructing viable arguments, and critiquing the reasoning of others. Ms. Stewart's students generated their own strategies for counting and recording how many items the caterpillar had eaten, thus showing they could make sense of problems. In the group discussion, they were able to explain the reasoning behind their strategies and consider different representations of the problem to help them analyze their own—and others'—reasoning.
Informing Instruction
As she studied their work, Ms. Stewart wondered how these learners were connecting their counting to cardinality and how they were understanding that the last number name a person says when they count a set of objects tells the total number of objects counted. She decided to plan a class activity and discussion that would help her learn more about how students were making sense of counting and cardinality.
Knowing that most of her students have demonstrated the ability to keep track of items in a set and to explain their strategies, Ms. Stewart might next decide to challenge students to keep track of something represented with more complexity, perhaps in a different children's story. She might incorporate counting, grouping, or subitizing into a future activity, asking students to record their thinking on paper and use this work to support their thinking in group discussion.
Responding Purposefully
Note that Ms. Stewart purposefully responded to students' work in ways that promoted deeper learning. Her in-the-moment verbal responses and prompts supported students in explaining their thinking, engaging with others' thinking, and making and justifying claims. For example, when she asked students to use their work to show how many things the caterpillar ate (and show how their drawing proved their answer), she pressed students to engage in explanation and justification, practices that encourage deeper levels of learning.
Ms. Stewart also motivated, rather than discouraged, students who struggled toward conceptual understanding. When the disagreement about 25 or 26 emerged, she honored both answers by saying, "Okay, we hear that we have two different answers, 25 and 26" and then invited students to investigate further. Students had the chance to share their thinking, puzzle through the debate, and revise their solutions.
This description of how one teacher used a popular children's book to encourage math reasoning reflects some important ideas about using student work to support learning. It shows that engaging students in open tasks (in which they derive their own strategies) and connecting those tasks to discussion is one of the best ways to gain information about students' learning. Teachers can review work samples to accomplish many instructional goals, such as giving students opportunities to examine their own and one another's work; engaging students in collaborative sense-making; grounding questions and discussion prompts within focal strategies; and providing feedback that supports learning.
There are many ways to set up open-ended math tasks that produce informative student work. Connecting the task to appealing children's literature is a great strategy. By viewing The Very Hungry Caterpillar through a mathematical lens, Ms. Stewart provided students with accessible math ideas through a shared, enjoyable experience. The work generated in such activities is an invaluable resource for both teacher and students.
Author's note: All names and school names are pseudonyms.
