As I sit in the airport, waiting to catch a flight to Sacramento, CA, where I will speak at a conference on equitable teaching practices, I reflect upon what I see daily in classrooms across demographics and districts. I regularly see classroom instruction that pulls all students to the middle, challenging a few, and ignoring many. Many teachers facilitate mathematical lessons by posing surface-level questions in class. Unfortunately, these questions do not require rigorous thinking or reasoning but reinforce memorization.
Students with great mathematical potential often fail to develop effective reasoning and argumentation skills when the primary focus of their math classes is calculation skills. They navigate their school lives successfully, reciting math facts and obtaining a superficial understanding of mathematics. The students can quickly shout out responses because they came to class with the answers memorized. Unfortunately, they miss opportunities to analyze new situations, develop creative solutions, and persevere as they work through complex problems.
Students who come to school to develop their conceptual and procedural knowledge of mathematics spend their time in math class observing their peers shout out answers while attempting to grasp bits and pieces of information they gather from conversations between their teachers and a few select students. They do not have enough time to recall the information or process potential solutions to the question, and as a result, they miss out on learning opportunities. They rarely discover their innate ability to make sense of mathematics and solve problems with confidence.
Students performing far below grade level are often lost in the rapid-fire dialogue and miss the opportunity to engage in thoughtful conversations that foster learning and build connections between simple and complex mathematical representations.
Compare the following two questions and consider the responses students might give.
What is 7 x 6? versus What strategy can we use to find the product of 7 x 6?
The first question is a gathering question. Gathering questions require immediate answers, the rehearsal of known facts and procedures, and lead students through a procedure. With the gathering question, “What is 7 x 6?” There is only one correct, short-answer response, 42.
The second question is a generating question. “What strategy can you use to find the product of 7 x 6?” There are multiple correct responses. Students can share their skills and knowledge while utilizing a variety of strategies such as skip counting, derived facts (7 x 5) + 7, and the distributive property (7 x 5) + (7 x 1). Students can also utilize academic vocabulary while explaining their strategies, e.g.,” I decomposed, rounded to a friendly number, or counted my multiples of 7 or 6.” Generating questions generate ideas and discussions by soliciting information from the group. Generating questions can help students make connections between representations and ideas (Boaler & Brodie, 2004). Richer language and reasoning flourish in students’ responses.
It can be tempting for teachers to follow the path of least resistance. Asking above-grade-level students to answer low-risk gathering questions quickly, setting the pace for instruction, while developing a superficial understanding of mathematics. In contrast, when generative questions are missing from classroom conversations, students new to a grade level or its content miss the opportunity to develop critical thinking and reasoning skills.
“Teacher questioning is the main instructional tool educators have to surface student reasoning and understanding so they can build from what students know to develop deeper knowledge and the ability to make important mathematical connections (NCTM 2014, p. 35).”
We can cultivate mathematical communities when teachers pose generative questions, and students explain and justify their ideas to their peers. Students acquire new ideas, compare the effectiveness of their strategies with those of others, and disagree with and critique the ideas of their peers. None of this can happen when we overly rely on gathering questions for our instruction.
If questioning is the primary instructional tool for developing mathematical reasoning, are we pulling students to the middle by using our questions to encourage the recitation of known facts, or are we elevating all students and their thinking by generating new ideas and connections to their highest heights? Why? How do you know?
References
Boaler, J. & Brodie, K., (2004). Proceedings of the twenty-sixth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Toronto: OISE/UT.
Huinker, D., & Bill, V. (2017). Taking Action: Implementing Effective Mathematics Teaching Practices in K-Grade 5. National Council of Teachers of Mathematics.
