This month’s curricular competency focus is explain and justify mathematical ideas and decisions. This curricular competency is the same across grades K-12 and is included in the Grades 10-12 courses with the addition of “in many ways“.
This competency falls under the organizer of “Communicating and Representing” is also connected to the Core Competency of Communication, particularly the aspect of explaining and reflecting on experiences.
Elaborations are suggestions for educators to consider as they plan for developing this curricular competency:
- mathematical arguments
What is a mathematical argument?
A mathematical argument is the debate and discussion of a mathematical problem or task. This involves the explanation and justification of the reasoning, problem-solving process and the solution. As stated by Small (2017), the ability to create a sound mathematical argument is developed over time.
A common instructional routine in our district is Number Talks. During this routine, students are asked to share their mental math strategies for solving questions involving number operations. Part of this routine is defending or “proving” their solution through their strategy explanation. Other students may agree with, build on or argue with the strategies used. A focus of this routine is both building mathematical discourse structures as well as building the listeners, connectors and reflectors needed in a mathematical community. During Number Talks, students listen to each others’ explanations and justifications and then also use mathematical language to communicate their own mathematical arguments. Before orally sharing their explanations to the whole group, students are often given the opportunity to turn and talk, or think in their head to formulate and rehearse their explanations.
In the book Teaching Mathematical Thinking, author Marian Small (2017) suggests the language that develops during mathematical argumentation and discourse may sound like this:
“I agree with ______ because _______.”
“I didn’t understand why you __________.”
“I disagree with ___________ because ____________.”
“I wonder why you _____________.”
“What if you had _____________.”
Small (2017) provides some examples of open question that nurture mathematical argumentation. For example, for grades 3-5 students:
Liz says that when you multiply two numbers, the answer is more likely to be even than odd.
Do you agree or not? Why?
And for grades 6-8:
A store employee noticed that an item’s price had been reduced by 30% and realized it was a mistake. So she added 30% back to the reduced price. Avery said the price is the same as it used to be but Zahra disagreed.
With whom do you agree? Why?
What tasks like these are we presenting to students to intentionally nurture and practice the development of explaining and justifying mathematical ideas and decision-making?
Mathematician Dan Finkel shares the importance of conjectures and counterexamples in his playful instructional approach. More information can be found on his website mathforlove.com
In the following example from Dan, a student made a conjecture that if you multiply both factors by two, the product will stay the same. Can you think of a counterexample that disproves this?
In their book But Why Does It Work? Mathematical Argumentation in the Elementary Classroom (2017), authors Susan Jo Russell et al share an efficient teaching model focused on mathematical argument for developing the ability of students to justify their thinking and engage with the reasoning of others. Their model supports students in:
- noticing relationships across sets of problems, expressions or equations
- articulating a claim about what they notice
- investigating their claim through representations such as manipulatives, diagrams, or story contexts
- using their representation to demonstrate and explain why their claim must be true or not
- extending their thinking from one operation to another
In their book Teaching with Mathematical Argument (2018), authors Stylianou and Blanton suggest that a focus on justification and explanation of thinking can celebrate the diversity of thinking within our classrooms. From their book:
“How can argumentation be a goal and an expectation for all students? One strategy is to embrace students’ use of diverse strategies. This diversity can then be used to plan cognitively demanding instruction that includes argumentation and that allows all learners to build from their own thinking and access their peers’ thinking to develop their understanding of new concepts. Rich, open tasks that invite argumentation are challenging because of their open nature. However, their openness also allows access to students who struggle in mathematics. Being open implies having more than one entry point, which makes such tasks accessible to students who often struggle to follow one particular procedure.”
By honouring the diverse thinking of the learners in our classrooms, we are also nurturing the important idea that there isn’t “one right way” to do or think about mathematics. Creating entry points for all students to explain and justify mathematical ideas is part of creating a safe mathematical community for all.
Some questions to consider as you plan for learning opportunities to develop the competency of explaining and justifying mathematical ideas and decisions:
How do we support students and families in understanding that explaining and justifying your answers and processes is an important part of mathematics?
What problems and tasks are we presenting to students to intentionally nurture and practice the development of explaining and justifying mathematical ideas and decision-making?
What visual and language supports might support students as they engage in mathematical discourse and argumentation?
What opportunities do students have to notice patterns and relationships, make conjectures and generalizations across mathematical concepts? What ways could they share and explain their mathematical ideas by using materials, pictures or diagrams, stories or contexts or numbers and symbols?
How might technology provide access for students or transform the way they are able to explain and justify their mathematical ideas and decisions?
But Why Does It Work? Mathematical Argument in the Elementary Classroom
by Susan Jo Russell, Deborah Schifter, Virginia Bastable, Traci Higgins, Reva Kasman
Heinemann Publishers, 2017
Teaching with Mathematical Argument: Strategies for Supporting Everyday Instruction
by Despina Stylianou and Maria Blanton
Heinemann Publishers, 2018
by Marian Small
Teachers College Press/Nelson, 2017
Promoting Mathematical Argumentation by C. Ramsey and W. Langrall (2016). Teaching Children Mathematics (volume 22), number 7, pages 412-419.