Reasoning Reveals and Propels Understanding

Wayne Snyder

How would your students solve this problem? 8 + 4 = ? + 5

What answers would you expect to see and what percentage of your students would get each of the varied responses, both correct and incorrect? And could you tell why they got those answers?

This problem, taken from Carpenter, Franke, and Levi (2003), epitomizes the issues and challenges facing us as we try to embrace the new paradigms of the Common Core State Standards (CCSS) in mathematics. Though there have always been math educators who have championed mathematical thinking, a quick review of curricula, text books, tests, and teaching practices finds widespread emphasis on processes and computational skills and little mention of teaching, soliciting, and assessing the mathematical thinking necessary for in-depth understanding and appreciation of mathematics.

In the Making Algebra Accessible Project (MAAP), described in Brown, Coddington, Snyder, Baldridge, and Orona (2013), our team of math and teaching professionals set out to research how to help move K–6 teachers into an understanding of Early Algebraic Reasoning (EAR), and how to teach and assess EAR among their students. In professional development workshops and monthly seminars, the teachers wrestled with various EAR-type problems. They created concrete and pictorial representations of the problems, constructed function tables of the situations, and looked for patterns. They wrote and shared their explanations and listened to and discussed those of others.

These teachers then took some of the same problems back to their classrooms and asked their students to investigate them. The students used manipulatives and drawings to represent their solutions and shared their thinking via poster paper and on camera. The teachers learned what the students were thinking, where strengths and misunderstandings were, and how to adapt instruction accordingly.

One of our first areas of focus, illustrated in the problem above, was the equal sign. This sample problem revealed rampant student misconceptions of how the equal sign functions in this context. Our student observations matched results similar to what Carpenter et al. (2003) found, where only 5 percent of students in grades 1 and 2, 9 percent in grades 3–4, and 2 percent in grades 5 and 6 got the problem correct. As a mathematics problem, these results are abysmal, especially since the equal sign is the foundation of algebra and quantitative aspects of the sciences. Misconceptions about the equal sign have been found to continue even among undergraduate students (Weinberg, 2010).

Without eliciting the mathematical thinking that lies behind the answer, there is no way to tell where understanding breaks down. Only by providing the structure, environment, and opportunity for students to share the reasoning behind their answers can we dislodge misconceptions.

When we asked students to explain the reasoning to their solutions for 8 + 4 = ? + 5, either written or verbally, students made it clear that most believed that the equal sign means "output." Our most common answers matched those found by Carpenter et al. (2003)—12, 17, or either 12 or 17—and the explanations consistently focused on adding the total numbers together without concern for the placement of the equal sign in the problem, and how that might change its function. Many students added only the numbers on the left, others added all the numbers, and still others believed that adding either set of numbers was acceptable. Students have apparently learned the meaning of the plus sign, but not the meaning of the equal sign.

Eliciting students' thinking is not just about determining their misconceptions; it is also about understanding and encouraging their correct mathematical reasoning. Those students who did get the correct answer had a variety of explanations. Most described some way of balancing the sums on the two sides. But some saw other patterns, such as, "5 is one more than 4, so 7 has to be one less than 8." This type of mathematical reasoning surprised the teachers and further helped them to appreciate the power of requiring students to explain their reasoning, rather than simply giving an answer to be considered right or wrong. Teachers celebrated and made students comfortable sharing individual reasoning, and this allowed teachers to formatively check understanding and misconceptions within student reasoning.

Mathematical reasoning is a required part of the new paradigm of mathematics education, underscored and made explicit in the Common Core Standards for Mathematical Practice. Reasoning does not take the place of mathematical processes; rather, it strengthens the understanding, retention, and application of processes. Teaching for mathematical reasoning involves allowing and requiring students to express and to share their reasoning; listening to their explanations; and responding, guiding, and celebrating their mathematical thinking. All students benefit from both greater mathematical reasoning skills and the ability to describe and share their reasoning. Our results found that, with appropriate training and support, teachers' paradigms of mathematics education can be shifted to include eliciting and sharing mathematical thinking, and that this shift toward reasoning as the cornerstone for teaching and learning pays off in deeper understanding for both teachers and students.

References

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heineman.

Brown, S. A., Coddington, L., Snyder, W., Baldridge, K., & Orona, B. (April 28, 2013). Quantitative models as tools for unpacking student thinking. Paper presented at the annual meeting of the American Educational Research Association, San Francisco.

Weinberg, A. (2010). Undergraduate students' interpretation of the equals sign. Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education, Raleigh, NC.

Wayne Snyder is an assistant professor of practice in teacher education at Claremont Graduate University, Claremont, Calif.

ASCD Express, Vol. 8, No. 24. Copyright 2013 by ASCD. All rights reserved. Visit www.ascd.org/ascdexpress.