When you hear the word argument, you might think of a heated dispute or a clashing of opposing sides. In the mathematics classroom, however, the practice of argumentation involves making claims, supporting them with evidence, evaluating the reasoning of others, and making sense of mathematical ideas. This mathematical practice is identified by the Common Core Standards as central to the work of K-12 mathematics. It's the practice through which a mathematical community determines what will be accepted as true.
While students in the upper grades may engage in more formal argumentation and deductive proof, young children can and do engage in supporting their reasoning with evidence, making sense of the arguments of their peers, and making conjectures about mathematical relationships. For example, students might argue that 5+3 is the same thing as 3+5 by modeling both expressions using unifix cubes.
Or students might move toward making more general statements about how and under what conditions a mathematical relationship is true. For young students, this might sound like, “It doesn’t matter what order you add the numbers in, you will always get the same amount.” This is related to, and perhaps builds on the claim above, but implies this relationship will be true for any number.
We’ve been experimenting lately with activities that can support young learners to engage in argumentation. Each of these activities can be a 15 minute discussion in your classroom, so they can be woven into the mathematics work you already do with your students. The activities encourage argumentation by inviting students to (1) compare and evaluate, and (2) look for patterns and make conjectures about what they're noticing.
Which One Doesn’t Belong
The activity Which One Doesn’t Belong (WODB) was developed by Christopher Danielson, a mathematics teacher, blogger, and author. In WODB, the teacher shows students a collection of four mathematical objects (shapes, numbers, expressions, etc.) and asks them to decide which one doesn’t belong and why. The activity supports students to begin making specific claims about shared properties among specific objects by noticing and comparing those properties, and supporting their claims with reasoning. Because there are lots of possible answers, there are lots of opportunities to make and evaluate claims. In this video, you’ll see both a kindergarten and first grade class engaging in Which One Doesn’t Belong.
[video_embed src="https://www.teachingchannel.org/videos/evidence-supported-claims-nsf/emb..." width="480”]
True/False or Same/Not the Same
In a True/False activity, the teacher shows students an equation and asks them to decide whether it is true or false. Students have to compare and evaluate the two quantities to decide if they're equal or not, and use their reasoning to work toward consensus. In this video, you’ll see a kindergarten class engaging in a version of True/False, which we’ve referred to as Same or Not the Same? The teacher presents students with two images instead of numerical expressions, and the students begin making claims about whether the images are the same or not, and to support their ideas with reasoning.
[video_embed src="https://www.teachingchannel.org/videos/kindergarten-math-lesson-nsf/embe..." width="480”]
In Related Problems, students begin moving toward more general arguments by looking for patterns in a set of related problems. This activity can support students to begin to articulate properties of numbers and operations. In this video, you’ll see how the activity supports first grade students to construct and support general claims and conjectures about how addition works.
[video_embed src="https://www.teachingchannel.org/videos/reasoning-about-addition-nsf/embe..." width="480”]
Give these activities a try and let us know in the comments below what you and your students learn!
For More on Argumentation with Mathematics:
Which One Doesn’t Belong Resources
- Which One Doesn't Belong?: A Shapes Book, Teacher's Guide with Student Book, by Christopher Danielson. This set, with the accompanying Teacher’s Bundle, is a source of sets of four shapes to use with Which One Doesn’t Belong?, along with some guidelines for constructing sets and implementing the activity.
- Watch Kristin Gray try a Which One Doesn’t Belong? with third graders.
Blog Posts from Teachers
- Follow #wodb or @wodbmath to see teachers' posts and conversations on what they're learning about Which One Doesn’t Belong?
- Connecting Arithmetic to Algebra (Professional Book): Strategies for Building Algebraic Thinking in the Elementary Grades, by Deborah Schifter, Susan Jo Russell, and Virginia Bastable has many examples of young children noticing and making claims about numbers and operations, including strings of related problems.
- But Why Does It Work? Mathematical Argument in the Elementary Classroom, by Susan Jo Russell, Deborah Schifter, Virginia Bastable, Traci Higgins, and Reva Kasman will be published April 2017 by Heinemann. This book offers a model for teaching argumentation and lessons for elementary classrooms.
- Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School, by Tom Carpenter, Megan Loef Franke, and Linda Levi has many examples of young children noticing and making claims about numbers and operations, including the use of True/False equations.
- Beyond Answers: Exploring Mathematical Practices with Young Children, by Mike Flynn, provides stories of young children engaging in the mathematical practices from the Common Core through classroom dialogue, teacher reflections, and examples of student work.
- Developing Essential Understanding of Mathematical Reasoning for Teaching Mathematics in Grades Pre-K-8, published by NCTM, provides images of children’s work in argumentation in the context of the learning that happens in PreK-8th grade.
- Visual Patterns is a website curated by Fawn Nguyen and targeted for middle school students, but young children can also develop argumentation by analyzing the visual patterns, figuring out what goes next in the pattern, and why.
This work was made possible through support by the National Science Foundation.
Elham Kazemi is a professor of mathematics education at the University of Washington. She loves to design professional learning opportunities for teachers, coaches, and principals to learn together about children’s mathematical thinking.
Kendra Lomax is a teacher educator at the University of Washington. She collaborates with local teachers to learn about children’s mathematical thinking and designs job-embedded professional learning opportunities through projects like Teacher Education by Design. Curiosity about children’s mathematical ideas is at the heart of her work.
Alison Fox is a teacher educator and doctoral student at the University of Washington. She works in collaboration with local school districts to develop powerful ways of supporting teachers, coaches, and principals to learn together about teaching mathematics.