“Both/And Thinking” and the Five Strands of Mathematical Proficiency

Too often in education examination of important questions can devolve into shouting matches, producing cacophony rather than providing clarity. Why does this happen? Frequently we fall into a trap, what the National Association for the Education of Young Children (NAEYC) essay entitled “Moving from Either/Or to Both/And Thinking in Early Childhood Practice” describes as

“a recurring tendency in the American discourse on education: the polarizing into either/or choices of many questions that are more fruitfully seen as both/and. For example, heated debates have broken out about whether children in the early grades should receive whole-language or phonics instruction, when, in fact, the two approaches are quite compatible and most effective in combination.” (Developmentally Appropriate Practice in Early Childhood Programs, p. 23)

Both/And thinking marks the work of the National Research Council (NRC) in their landmark monograph Helping Children Learn Mathematics, which outlines steps for cultivating mathematical proficiency in our students. Rather than joining in the endless (and fruitless) “math wars,” the National Research Council, under the aegis of the National Science Foundation and the U.S. Department of Education, conducted a study of what research reveals about “successful mathematics learning from the preschool years through eighth grade.” The NRC carried out a three-part charge:

1. Synthesize the rich and diverse research on pre-kindergarten through eighth grade mathematics learning.
2. Provide research-based recommendations for teaching, teacher education and curriculum for improving student learning and to identify areas where research is needed.
3. Give advice ad guidance to educators, researchers, publishers, policy makers, and parents. (p.7)

The National Research Council’s “five strands of mathematical proficiency” offer a rich and complex view of the teaching and learning of mathematics—one which flies in the face of the “either/or” thinking that has dominated the shed-more- heat-than-light arguments of the past ten years. The NRC’s meta-research identifies “five strands” of mathematical proficiency. Rather than advocating only computational fluency or only conceptual understanding, the NRC notes the following:

“When people advocate only one strand of proficiency, they lose sight of the overall goal. . . . [S]ome people who have emphasized the need for students to master computations have assumed that understanding would follow. Others, focusing on students’ understanding of concepts, have assumed that skill would develop naturally. By using these five strands, we have attempted to give a more rounded portrayal of successful mathematics learning” (pp. 12, 10).

In other words, we needn’t choose between computational fluency and conceptual understanding. Our students need both—and more. The NRC’s five strands of mathematical proficiency are as follows:

• Conceptual understanding: a student’s grasp of fundamental mathematical ideas. One not only knows isolated facts and procedures but one knows why a mathematical idea is important and the contexts in which it is useful.

• Procedural fluency (computing): skill in carrying out mathematical procedures flexibly, accurately, efficiently, and appropriately. This includes being fluent with procedures such as adding, subtracting, multiplying, and dividing whether using pencil and paper or mentally. It also can include procedures from algebra, geometry, and statistics. Fluency can be defined as having the skill to perform the procedure quickly, accurately, and flexibly.

• Strategic competence (applying): ability to formulate, represent, and solve mathematical problems. A student should learn not only to solve routine problems but also apply the same problem solving techniques to invent a solution to a non-routine problem.

• Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification. Reasoning connects all areas of math through logical connections between concepts and situations. Reasoning also interacts with the other strands of mathematical proficiency, especially when one is trying to solve a problem.

• Productive disposition (engaging): habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. Students who are engaged in mathematics believe they can solve problems as well as learn concepts and procedures even if it requires effort. They view mathematical proficiency as an important part of their future.

These five strands are interwoven and interdependent. It is difficult to develop one strand without impacting another. Learning all of the strands of mathematical proficiency solidifies not only one’s knowledge base but also one’s flexibility in using these skills in new or novel situations. And that’s the point: 21st century students must, above all else, be adept at learning how to learn. As educators, we must both model this kind of learning AND create the optimum context for it. Both/And Thinking can help us do precisely that.

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