The Standard

The third standard of mathematical practice, construct viable arguments and critique the reasoning of others, is all about mathematical discourse. Students need to be able to engage in productive and active conversations about math both with their peers and with adults. There also needs to be an opportunity for students to share ideas with their peers, listening to one another  and asking clarifying questions to build understanding and then suggesting ideas to improve or revise. Allowing students time to discuss their problem solving builds ownership and gets them invested in talking about math.

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The first half of the practice asks students to construct arguments that fit different situations such as explaining their reasoning, proving why something is right or wrong and then being able to communicate why. In order to first construct the argument, students need to be able to explain their thinking with confidence. They then need to be able to justify their reasoning with accurate language and vocabulary. Constructing arguments does not only have to be done for their own math problems, but also when looking at how someone else solved a problem.

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The second half of the practice, critique the reasoning of others, wants students to ask questions, get clarification and point out areas they agree or disagree with when discussing math. Students need to be able to listen respectfully to how other students have solved problems while also explaining the reasoning behind a different strategy that could have been used. When errors occur, they need to be able to identify the flaw in the argument and explain what it is.

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The Classroom

In the classroom, constructing viable arguments and critiquing the reasoning of others is actually pretty simple to implement. Encourage students to talk about their problem solving processes both with the teacher and their peers. Time may need to be spent establishing expectations about how talking about math should look like and sound like. Students will also need to be given time to practice using these expectations. In the younger grades, this might mean spending time sitting quietly and looking at who is talking. Anchor charts with sentence frames may also be extremely useful as more mathematical discourse is implemented into the classroom routines. However, anchor charts can become a crutch if used for too long so once students have started discussing without the sentence frames, take the anchor charts down.

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When it comes to encouraging students to construct viable arguments, have them talk with each other about how they solve problems. Observe the students while creating a plan for the ideas and thoughts that should be highlighted for the entire class. Call on students that have unique arguments and give them the floor to share their thinking. This allows students a safe space to critique the reasoning of others because they can ask questions and offer suggestions to their group before joining the whole group.  It is also important to be continuously asking students questions about how and why they solved a problem. That way they are always thinking about their intent when choosing a strategy so that they can justify it. Lower elementary students may need to use concrete, hands-on materials to build and justify their argument because they are still developing the mathematical vocabulary needed to verbalize an argument. 

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Finally, the use of open ended tasks are valuable because whenever a student has the ability to solve a problem in their own way, there is an opportunity for constructing arguments and critiquing the reasoning.

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Question Stems

• Why did you use _____ to help you with this problem?
• How do you know this was the best approach?
• How do you know?
• Why does that make sense to you (or not make sense)?
• Does that make sense?

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Sentence Frames

• I disagree with ____ because I think _____
• I like how ______ said _____, but I think _____
• My thinking is different from _______ because ________

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Picture Books

Persuasive

• Don’t Feed the Bear by Kathleen Doherty
• Duck! Rabbit! by Amy Krouse Rosenthal
• Hey, Little Ant – By Phillip and Hannah Hoose
• The Day the Crayons Quit by Drew Daywalt

Conflict Resolution

• The Sandwich Swap by Queen Rania of Jordan Al Abdullah
• The Squirrels Who Squabbled by Rachel Bright
• Talk and Work It Out by Cheri Meiners
• Henry and the Kite Dragon by Bruce Edward Hall

Conversation Skills

• My Mouth is a Volcano by Julia Cook
• Millie Fierce by Jane Manning
• We Disagree by Bethanie Deeney Murguia‍
• Where Are You From? By Yamile Saied Méndez

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Next Steps

Now that you’ve got a basic understanding of the third Standard of Mathematical Practice, there are many places that you can dive deeper to learn even more. If you are just beginning to teach the act of agreeing and disagreeing in your classroom, no matter what age level you teach, this article about teaching how to handle disagreements will be helpful. It is also helpful to remember what soft skills are needed to have conversations with each other. The United States Department of Labor has developed a curriculum to promote workforce readiness skills and this unit on communication skills is full of great resources. If you are more interested in promoting mathematical discourse in your classroom, there is a book called “Mathematical Discourse: Let the Kids Talk! –Helps teachers to get students talking about math and explain their problem-solving methods and reasoning (Grades K-12)” by Barbara Blanke as well as an article by Houghton Mifflin Harcourt all about mathematical discourse.

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Sara VanDerWerf, MDE, will also be hosting a webinar for the third mathematical practice on November 13th, 2024 at 7:00 AM which you can find by registering here.

If you’d like more information, support, or guidance on developing a better understanding of Mathematical Practice #1, please reach out to our Math Team here at Resource Training and Solutions.

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Mindy Strom

Math Lead

Email: mstrom@resourcecoop-mn.gov

Phone: (612) 505-7997

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Megan Klaphake

Math Coach

Email: mklaphake@resourcecoop-mn.gov

Phone: (218) 770-0026

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References:

^{SanGiovanni, J. (2019). Using the mathematical practices effectively in the classroom. https://www.mheducation.com/unitas/school/explore/research/reveal-math-using-mathematical-practices-effectively-classroom.pdf} 

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^{Illustrative Mathematics. (2014, February 12). Standards for mathematical practice: Commentary and elaborations for K–5. Tucson, AZ. Retrieved December 29, 2018 from http:// commoncoretools.me/wp-content/uploads/2014/02/Elaborations.pdf}

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^{Flynn, M. (2017). Beyond answers: Exploring mathematical practices with young children. Stenhouse.}