In the 1990s, discussions about mathematics instruction in schools were plagued with a “war” in which two polarized groups disagreed on what mathematics instruction should include and how mathematics should be taught. On one end, some argued we should continue to teach with attention to students’ fluency with facts and algorithms in order to use procedures with efficiency. On the opposite end, others argued that students needed to develop their understanding of mathematical ideas so they could use concepts with meaning.
Unfortunately, there is now an attempt to resuscitate this “war” in North Carolina, with the Academic Standards Review Committee criticizing the Common Core State Standards for Mathematics. Like many polarized issues, this war represents only two small (yet vocal) groups, who position themselves at the extremes of the divide between fluency and understanding. Most mathematicians and educators are not debating: We agree that mathematics requires both procedural fluency and conceptual understanding. Neither is optional.
Consider the following problem, from an early National Assessment of Educational Progress: “An army bus holds 36 soldiers. If 1,128 soldiers are being bused to their training site, how many buses are needed?” About half of the 13-year-olds who answered this question said either 31 remainder 12 or just 31 buses.
These answers demonstrate students’ lack of understanding of the remainder and exemplify how problematic it is when students develop computational fluency without understanding. Students with understanding and fluency recognize that, because there is a remainder, 32 buses are needed to transport all soldiers.
What if a student with understanding but no fluency physically represented the 1,128 soldiers and then counted groups of 36 to find out how many buses are required? This response is as problematic as the first example! Students with understanding and fluency, who want to use a similar (yet efficient!) counting approach, can explain that we fit 360 soldiers in 10 buses, 720 in 20, and 1080 in 30. We still need to transport 48 more soldiers so we need two more buses for a total of 32.
There are, of course, other ways to solve this problem. But the important idea is that both fluency and understanding are important in mathematics. Again, neither is optional.
In the math wars, those who focused on fluency argued for teaching procedural efficiency in schools, with the idea that students would develop meaning on their own, perhaps later. Those who focused on understanding argued that students would develop efficient procedures on their own, perhaps later.
For over 20 years, however, research on the teaching and learning of mathematics has shown that teachers need to support students in developing both, by building procedural fluency from conceptual understanding. Students with initial understanding of mathematical ideas deepen that understanding by developing fluency with related procedures; at the same time, they begin to build new understandings of the next idea that will continue to deepen as they build fluency.
But here is the problem: Teaching in a way that develops understanding together with fluency is complex and difficult. It requires continuous search for a balance between these two components of mathematical proficiency and moment-to-moment decisions about when to push students forward and when to let them struggle a little longer with a novel mathematical idea.
So back to the Common Core. The guiding principles in these standards represent a serious, balanced attempt by mathematicians and educators to summarize, in one document, what we know about learning mathematics. Perfect? No. As we put these standards to work in classrooms and research on mathematics teaching and learning continues to unfold, adjustments will be needed to further improve the document. But the development of the Common Core involved experts from across the nation and took several years and rounds of public feedback to bring best current knowledge into action. To now use this document to restart a “Math War” in our state – and to replace it with new guidelines that once again prevent us from coming closer to an approach to mathematics that builds on both understanding and fluency – is just unfair to our children.
School mathematics that sees fluency and understanding as complements, not opposites, is common sense. We need to say no to attempts to reignite a war in our state and yes to supporting our teachers as they learn to implement mathematics instruction based on best current knowledge.
Paola Sztajn is a professor of elementary education at N.C. State University.