By Matt Larson, NCTM President
February 20, 2017
In a recent President’s blog post on the need to make homework comprehensible, I referred to the Fordham Institute Report, Common Core Math in the K–8 Classroom: Results from a National Survey. The report offers another interesting finding: “The math wars aren’t over.”
The authors of the report observe, “The Common Core math standards seek to bring a peaceful end to the ‘math wars’ of recent years by requiring equal attention to conceptual understanding, procedural fluency, and application (applying math to real-world problems). Yet striking that balance has not been easy. We see in these results several examples of teachers over- or underemphasizing one component to the detriment of another” (p. 6).
I found this statement particularly striking. This over- or underemphasis may be less a function of independent teacher actions in the classroom than a result of teachers doing their best to interpret and implement what they find in their curricular materials, which, as the Fordham report indicates, are aligned (or not) with the Common Core State Standards to varying degrees. The over- or underemphasis of a particular component may also reflect teachers’ trying hard to comply with mandates at the district or school level.
It is critical to appreciate that the over- or underemphasis phenomenon is not a new one. Mathematics teachers in the United States today are just the latest generation of U.S. educators to be caught in a 200 plus–year pendulum swing between an overemphasis of rote practice of isolated skills and procedures and an overemphasis of conceptual understanding, with their respective overreliance on either teacher-directed or student-centered instruction.
It all began in 1788 (the same year that the U.S. Constitution was ratified), when Nicolas Pike published the first major U.S. mathematics textbook, entitled Arithmetic. The process that Pike recommended for teachers was to state a rule, provide an example, and then have students complete a series of practice exercises very similar to the example. If that teaching process sounds familiar, it is probably because that was the way you experienced math instruction as a student yourself. This process became the U.S. script for teaching mathematics and is deeply embedded in our culture—expected by the vast majority of students and parents alike.
In the 1820s the pendulum swung for the first time when Warren Colburn published a series of texts, including Colburn’s First Lessons: Intellectual Arithmetic, Upon the Inductive Method (1826). Colburn recommended that teachers use a series of carefully sequenced questions and concrete materials so that students could discover mathematical rules for themselves. By the 1830s the pendulum was swinging back with the publication of the Southern and Western Calculator (1831) and The Common School Arithmetic (1832), which once again emphasized direct instruction of rules and procedures, taught the “good old fashioned way.”
The late 1950s and 1960s were the era of “New Math.” Proponents of new math worked to make the pendulum swing from rote learning to discovery teaching approaches that emphasized developing students’ understanding of the structure of mathematics, how mathematical ideas fit together, and the reasoning (or habits of mind) of mathematicians. The 1970s and 1980s saw the pendulum swing in the opposite direction yet again as these two decades became known as the “back to the basics” era, with a focus again on procedural skills and direct instruction.
In 1989 NCTM gave birth to the standards-based education reform effort with the release of Curriculum and Evaluation Standards for School Mathematics, and subsequently NCTM followed up this transformative publication with a series of other standards publications, culminating in Principles and Standards for School Mathematics (2000). By the mid-1990s over 40 states had created state math standards or curriculum frameworks consistent with the NCTM standards. But by the late 1990s the pendulum began to swing back to the basics when the “math wars” erupted in California and then spread across the country as parents and others demanded a renewed emphasis on procedural skills and direct instruction.
And that brings us to the latest perceived pendulum swing: the Common Core State Standards and the associated myriad of misinformation and misinterpretations surrounding them, as well as the historic and seemingly inevitable pushback that now benefits from and is fueled by social media. So what should a mathematics teacher caught up in these historic and continual pendulum swings do? My advice: Seek balance.
In many ways it seems as though we live in a world that is out of balance—pushed to extremes—that has “lost the middle” in various ways. To move mathematics teaching and learning forward, we have to resist the urge to be pushed to extremes. We have to do our part to break the historic cycle of pendulum swings. As Hung-Hsi Wu, professor emeritus of mathematics at the University of California–Berkeley, wrote nearly two decades ago, “Let us teach our children mathematics the honest way by teaching both skills and understanding.” This is essentially what the Common Core authors argue when they state, “[M]athematical understanding and procedural skill are equally important” (National Governors Association and Council of Chief State School Officers 2010, p. 4).
Over a decade ago the National Research Council (NCR) published Adding It Up (2001), which promoted a multifaceted and interwoven definition of mathematical literacy:
Procedural fluency and conceptual understanding are often seen as competing for attention in school mathematics. But pitting skill against understanding creates a false dichotomy … Understanding makes learning skills easier, less susceptible to common errors, and less prone to forgetting. By the same token, a certain level of skill is required to learn many mathematical concepts with understanding, and using procedures can help strengthen and develop that understanding. (p. 122).
We need to return to, promote, and implement in our classrooms the NRC definition of mathematical literacy. The goal of mathematics education is not complicated. We want students to know how to solve problems (procedures), know why procedures work (conceptual understanding), and know when to use mathematics (problem solving and application) while building a positive mathematics identity and sense of agency. How? Why? And when? These questions are the very essence of rigor in the Common Core. We can ask ourselves simple reflective questions at the end of each lesson and over the course of a unit:
Are my students learning how to solve a problem (or problems)?
Are my students developing an understanding of why certain solution strategies work?
Are my students learning when and where they might apply particular solution strategies?
Did the experiences in my class help my students see themselves as capable doers and users of mathematics?
If the answer to all four questions is yes, that is more than likely a good sign. If we stay focused on all four learning goals, and resist (or dodge) the pendulum swings in any one direction while steadily building students’ positive mathematics identity, perhaps the constant criticism that moves the pendulum will stop. When we stay the course and let students engage, learn, and develop their understanding, skills, and abilities to use mathematics, our students will be the beneficiaries of our efforts to find and maintain that balance.
For more information, see Balancing the Equation: A Guide to School Mathematics for Educators and Parents.
NCTM President's Message: The Elusive Search for Balance
By Matt Larson, NCTM President