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## Reasoning and Making Sense in Mathematics: It’s a K–12 Focus

By NCTM President Linda M. Gojak
NCTM Summing Up, September 6, 2012

Since developing its bold Agenda for Action in 1980, NCTM has provided mathematics educators with several groundbreaking publications that have defined a vision for change in mathematics education. Although mathematical content is important in all of these publications, they also focus on the significance of changing traditional approaches to teaching mathematics—change that is critical to offering every child the opportunity to achieve his or her maximum potential in mathematics. Central to these pedagogical shifts is instruction that supports student reasoning and sense making about the mathematics they are learning.

In 2009 NCTM released Focus in High School Mathematics: Reasoning and Sense Making, a framework for high school mathematics—the first in the Focus in High School Mathematics series. Although the mathematics examples are taken from high school mathematics content, the descriptions of reasoning and sense making are as applicable to the primary grades as they are to high school. Now is the time to extend the focus on reasoning and making sense to the elementary and middle grades.

For students to understand mathematics deeply, they must have the opportunity to make sense of what they are doing. In a traditional approach, the teacher shows the students a procedure by demonstrating each step, and students complete several similar examples following the teacher’s explanation. I call this the “follow the rules” approach. The problem with this method is that students have a litany of rules to follow and often become confused about when to follow which rule. When taking a test, students ask themselves, “Which way do I move the decimal point?” Or, “Am I supposed to flip a fraction here?” Traditional approaches do not promote deep understanding or adaptive reasoning. And we wonder why our students struggle on the high-stakes tests!

In a classroom that promotes reasoning and sense making, the teacher begins the class with a rich task designed to give students a chance first to explore mathematical concepts by making connections to previous knowledge and then to use various strategies to complete the task. Students are likely to begin their work with some cognitive dissonance. By thinking through what they already know and can use, trying an approach, considering whether an answer is reasonable, and sharing their thinking with classmates, students not only make sense of what they are doing, but also develop their own understanding of the mathematics—not the teacher’s understanding, not a classmate’s understanding, but their own understanding.

Consider an example: A sixth-grade class is beginning the study of division of decimals. The students have a good conceptual and procedural understanding of whole number division. The teacher begins the lesson by posting the following on an interactive white board.

Without doing any calculating, work with your group to make a list of everything you know about the quotient for each of the following.

82.5 ÷ 1.2

39.5 ÷ 0.95

436.2 ÷ 0.63

16 ÷ 0.05

Students work together in small groups and spend the next 10 minutes talking about each example. They have an entry point into the task because they understand whole number division and they know what questions to ask themselves. No rules here! No moving the decimal points around! The richness of this task lies in the opportunity for students to extend their previous knowledge of division to a new set of numbers, decimals. While the students are working, the teacher circulates among the groups, listening to their discussions, asking carefully constructed questions, taking note of students who may need more help, and planning the class discussion that will follow. In the class summary, students explain their reasoning. This is the first step in the development of this topic, and the groundwork has been laid for students to continue to build understanding based on reasoning and sense making.

Whether you teach preschool, elementary school, middle school, high school, or college, consider steps that you can take to transform your mathematics classroom into an environment that promotes reasoning and sense making for all students.  It is not an easy transition, but there are NCTM reasoning and sense making resources to help with this transformation. It will be different for you, it will be different for parents, and it will be different for your students. However, the benefits will make the added effort worthwhile.  In an era of common standards and assessments, unless we all, from the primary to the college level, work to shift our instruction to include reasoning and sense making for every student, our efforts to improve mathematics achievement for all students will not succeed. It’s time to make the shift happen!