President, Colorado Council of Teachers of Mathematics
The spring issue of the Colorado Mathematics Teacher continues the focus on the Standards for Mathematical Practice. In my last message, I highlighted Mathematical Practice 1: Make sense of problems and persevere in solving them. In this message, I’d like to turn our attention to Mathematical Practice 3: Construct Viable Arguments and Critique the Reasoning of Others.
Mathematical Practice 3 is at the heart of learning mathematics and engaging in rich mathematics tasks. This Practice requires students to construct arguments based on conjectures they make, to justify and communicate their conclusions and to respond to others’ arguments. Extending beyond the study of mathematics, this Practice is embedded in the core standards of language arts (e.g., cite specific textual evidence to support conclusions, write arguments to support claims using valid reasoning evidence), science (e.g., construct and argue for explanations and defend interpretations of the corresponding data) and social studies (e.g., evaluate both primary and secondary sources and develop interpretations defended by evidence from a variety of these sources). Thus a focus on supporting students’ expertise with Mathematical Practice 3 will reap benefits in mathematics, language arts, science, and social studies.
Keeping in mind that the Standards for Mathematical Practice are the varieties of expertise that students should develop, teachers are accountable for establishing a supportive classroom environment and selecting tasks that connect the Practice Standards with the content standards. As was true in Mathematical Practice 1, it is also essential that students have rich tasks if they are to develop the expertise with Mathematical Practice 3 that includes the opportunity to explain and justify their thinking and critique others’ strategies. Also essential are established classroom norms to support and encourage students to engage in this type of discourse. These norms will ensure a classroom environment where students feel safe to share their thinking and challenge others’ ideas without fear of ridicule. Such norms might include the ideas that student sharing and discussing of reasoning and justification are vital for learning mathematics, the understanding that mistakes are opportunities for learning, and the expectation that students listen to others.
After a supportive environment has been established and students have been given a rich task to solve, the teacher’s role is to incorporate strategies that promote student discourse. Examples of such strategies include private think time, think-pair-share, turn and talk. Teacher’s use of questioning and wait time will also support students in explaining and justifying their thinking. Such questions might include: How did you decide what the problem was asking you to find? What mathematical evidence supports your solution? How can you convince someone else that your solution makes sense? Will it still work if...? How did you decide to try that strategy? How did you test whether your approach worked?
Did you try a method that did not work? Why didn’t it work? Would it ever work? How do you know?
As you collaborate with your colleagues in grade-level or department teams, I encourage you to include a focus on Mathematical Practice 3 in your daily instructional planning. It is a vital practice that will reap benefits for students across multiple content areas and grade levels.