“Based on my experience in math education, I find that many pedagogical structures or moves sound great in theory, but are often easier said than done. Because of the complexity, asking students to construct mathematical arguments has been one of those things for me. Fortunately, over the years, I have had the joy and privilege to work with Jody and Chepina, whose thinking around math argumentation is grounded in theory and paired with practical and actionable advice and structures. I am so excited for others to share in their deep thinking and look forward to seeing the impact their work will have in classrooms.”
Mathematical argumentation as an opportunity for curiosity
Students bring curiosity and wonder to the classroom every day. When we鈥檙e attending to their ideas, we can find more opportunities for mathematical argumentation in our math lessons. Let鈥檚 look at how these opportunities arise in a first-grade classroom.
The lesson and card sort activity
The purpose of the lesson was for students to identify equivalent addition expressions by sorting cards, each with a different expression. Students quickly noticed that there is more than one way to get the same value. They could even begin to see the commutative property in action when shown cards with addends in opposite order: 4 + 3 = 3 + 4.
While sorting addition expressions, rather than organizing the cards in piles by their value, one student named Jenna organized the cards in columns. This student-led creative modification to the card sort structure allowed for different noticings and wonderings to emerge. She started by creating a row across the top with cards showing one addend of 0. At first, she wasn鈥檛 consistent with the top card being 0 + x or x + 0, but over time changed them so that they were all 0 + x. Then she filled in the last column with expressions equal to 10. As she added cards, she started to change the order so that the first addend on the cards increased going down the column and the second addend decreased.
As Jenna added each card to her organizational structure, the teacher asked where that card would go and how she knew. When asked about the 2 + 6 card, for example, Jenna said, 鈥淏ecause this is counting to 8, and this鈥濃�攕he pointed to a gap鈥斺�渉as to be 7, because [the 6 is] 2 less than 8. It fits here because these are all twos鈥� (in the row). Jenna was coordinating several characteristics of rows and columns within the structure of the chart.
As Jenna continued to fill in the chart, she noticed yet another pattern. Pointing to the step pattern, Jenna noticed that, “There is a stair step. The pattern keeps going. There鈥檚 one more way to make the total as it gets bigger.鈥�
We could state this conjecture more precisely as: “For any whole number n, there are n + 1 ways to add two whole numbers to get a sum of n.鈥� Jenna was making sense of big math ideas and noticing structure embedded in arithmetic.
Reflecting on the experience
As we step back and reflect on what we experienced with Jenna, we wonder what could happen next. How might Jenna justify her thinking? She noticed the stair-step pattern and multiple ways to arrive at the total. What might she say if we asked her, 鈥淲hy is that happening?鈥� Or if we gave her a tool such as linking cubes and asked if she could use the cubes to show why that works? Are there other questions that could have nudged her to extend her thinking, such as, 鈥淲ill that always work?鈥� or 鈥淲hat numbers does it work for?鈥� Is there a tool or representation that would help her continue her reasoning?
We can also think about what might happen if we shared Jenna鈥檚 idea with the other students. How might they respond to Jenna鈥檚 noticing? Would her ideas lead others to see and use structure in similar ways? How would they make sense of her ideas? Perhaps this is an opportunity to engage students in each other鈥檚 ideas.
Opportunities for curiosity
The opportunity for Jenna and her classmates to make sense and explore their natural curiosity emerges from a classroom environment that鈥檚 playful and filled with wonder鈥攚here children are given time to explore and interact with materials and each other. We noticed that as Jenna progressed through the cards, she refined and added on to her thinking. This is evident in the first row of cards. She grouped cards with 0 as an addend, then began to sequence them, and later considered the positioning of the addends to 0 + x. As we might infer from the interaction of the teacher, there isn鈥檛 one right way to think about the task, nor one way for the teacher to encourage students to think about it. We hear the teacher ask Jenna to share her reasoning: It鈥檚 not a question posed to evaluate Jenna鈥檚 thinking, but rather to gain insight into her thinking鈥攕omething the teacher is genuinely curious about.
In addition to the classroom environment, the card sort also presented an open-ended opportunity. Students made sense of the sort in many different ways, some finding related pairs and starting to identify (not yet naming) the commutative property, others grouping problems with a common addend in piles. All students had access to the task and time to make sense of it.
We share the story of Jenna as one of many instances where young children have shared their brilliance with us. Their wonder and curiosity inspired us to explore their ideas along with 国产三级片�2 teachers. We saw students notice, wonder, conjecture, justify, and extend ideas that led to a deep understanding of key mathematical concepts while integrating mathematical argumentation.
We share ideas like the brilliance from Jenna in our new book, , where we also make connections among instructional routines, center, and card sorts. Our book also discusses supporting students in curious exploration, building on what they already bring to the classroom as a way to bring opportunities for mathematical argumentation into our lessons.
Rumsey, C., & Guarino, J. (2024). . Solution Tree. Bloomington, IN. ISBN: 9781960574367
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