I am increasingly convinced that “construct viable arguments and critique the reasoning of others” (Standard for Mathematical Practice 3) may be the nine most important words in all mathematical practice.

First, the practices of constructing arguments and critiquing reasoning represent a powerful cross-disciplinary approach and are as relevant in mathematics as they are in English language arts, science, and social studies. They help break down the subject-specific silos and support the understanding that communication, justification, and critique are essential components of learning.

Second, this practice is a joy to embrace because it represents a long-overdue shift away from the multiple-choice, skill-based, rule-based focus that emerged, to the detriment of all students and teachers, with No Child Left Behind. Test-prep memorization and regurgitation have little to do with thinking and reasoning and cheat students out of the opportunities consistently found in highly effective schools and gifted classrooms.

But the primary reason this practice is so important is its implications for teaching. In order for students to construct viable arguments, they must be asked why or be expected to explain their thinking or be directed to convince the class. In a classroom where the shared expectation is that all answers need to be justified, students consistently construct arguments. They communicate their understanding and demonstrate their thinking process. No longer is it acceptable to stop at “83” or “9 square centimeters.” Instead, they individually and publicly justify why the answer is 83 or 9, thereby giving the teacher and the entire class an opportunity to discuss alternative explanations.

Similarly, in order for students to critique the reasoning of others, they must be in a classroom where reasoning is made public and open to review and comment. A classroom culture that values critique rather than the one right way to get the one right answer is a culture in which students are far more actively engaged in their learning.

Again, in place of a single approach or justification, many approaches and justifications surface, thereby strengthening everyone’s learning. In addition, given the learning opportunities of mistakes and misconceptions, critiquing others’ reasoning helps address common mistakes and misconceptions, also to everyone’s benefit. It is a joy to observe a class focus on the mathematical reasoning of three different groups of students and then discuss approaches: what was the same, what was different, what was correct, what was flawed. What better preparation for life and citizenship could we hope for?

Imagine a world in which teacher preparation, coaching, support, and evaluation all focused on the degree to which any lesson included opportunities both to construct and effectively implement viable arguments and to critique the reasoning of others!

Leinwand, S. (2009). Accessible Mathematics: Ten Instructional Shifts That Raise Student Achievement. Heinemann.