Construct Viable Arguments in Math

math skills parenting for success Nov 02, 2022

One morning a tour group passed through my second grade classroom.  The students were chattering in small groups about different math tasks they were doing.  Manipulatives were spread around the room and colored pencils lay askew on the tables.  The room had a productive hum to it as students worked together to find solutions and explained their thinking using blocks.  As the tour group left the room I overheard one parent whisper to another, “Oh, this is not a good math classroom at all – the students are TALKING.”

When I was in school my middle school math classes were silent. If we spoke, the entire class would suffer consequences.  Perhaps the parent above had a similar experience when they were in school.  I have to laugh at my own experiences now because we know so much more about how children learn and we know that those conversations are powerful ways of building understanding.  I have seen the impact of students talking to each other while solving problems.  Often I will hear, “I didn’t think of that!” as a student discovers a solution they like better than the one they used.  

Construct Viable Arguments and Critique the Reasoning of Others

In fact, the current Math Practice standards support conversation.  Math Practice #3 encourages students to “Construct viable arguments and critique the reasoning of others.”  Using this math practice, students consider the models that others have used and engage in conversation about the best ways to solve problems.  

Kindergarten students might be sharing a model they built of a problem using unifix cubes or pattern blocks.  Fifth graders might sketch their solution using open arrays and then explain the trajectory of their thinking using precise mathematical language.  

To be skilled at Math Practice #3 students also need to be good listeners.  They need to have an open mind and carefully consider the details of what the other speaker is saying.  They need to be fully present in the moment.  If the listeners are open-minded, the speaker will feel comfortable sharing more freely.  

The kind of deep listening described above takes practice.  Beyond being useful as a math skill, it is so needed as a social skill if people are to come together and understand each other. 

Connecting with Others' Ideas

The skill of listening and learning about one another is supported in Jo Boaler’s book, Limitless Mind, where she discusses the power of collaboration.  “New ideas and directions come from people reasoning with each other, setting out ideas, and considering the ways they are connected to each other."[I]

Historically, seeking help to engage in discussion around math problem has been seen as a weakness.  However, Jo Boaler has turned this idea around, “Connecting with another person’s idea both requires and develops a higher level of understanding.  When students work together… they have opportunities to make connections between ideas, which is inherently valuable for them.”[ii]  REQUIRES and DEVELOPS a higher level of understanding!  That is so different than previously held ideas about collaboration.

Boaler comments, “Parents of high-achieving students sometimes complain that their student is being used to educate others, their student could just as easily work alone and zoom through material…Despite the fears of some parents of high-achieving students, the students in this approach (one of working in groups and valuing the contributions of others) who improved the most in math achievement were the highest achievers.”[iii]  We can change our thinking and see collaboration as a strength, not a weakness.  

Math Challenge Through Conversation

When we look to challenge students in math, rather than looking for worksheets and online games, let us engage children in conversation about their thinking.  Ask them to explain their thinking about a problem and in turn, share how we were thinking about the problem.  Talk about similarities and differences between the ideas.  Have them construct a viable argument about why they think their answer is correct. Sometimes even a third approach is needed to sort out one of the differences.  Considering the problem from these different perspectives will deepen the child’s understanding of the problem and make them a stronger mathematician.

Engaging children in collaboration and teaching them to really listen to each other's ideas will not only support their math learning but will support important social and emotional skills of valuing diversity (of people and ideas) and living in community with others. 

[i] Boaler, Jo. Limitless Mind. HarperOne, 2019, p. 200.

[ii] Boaler, Jo. Limitless Mind. HarperOne, 2019, p. 168. 

[iii]Boaler, Jo. Limitless Mind. HarperOne, 2019, pp 187-188.