As a teacher of mathematics since the mid-seventies, I have seen a lot of “waves” of reform come and go. All promise increased student performance with the implementation of the latest bells and whistles. Being a Floridian, I know that both the size of the “wave” and the impact it has depends on the distance between the surface and the bottom of the ocean. If I apply the reality to the analogy, I can conclude that the distance between the reform effort and students must be minimized if there is to be any measurable impact.
The latest “wave” of reform is the Common Core State Standards. These Standards define what students should understand and be able to do in their study of mathematics. There are two kinds of standards in the Common Core, Standards for Mathematical Content and Standards for Mathematical Practice.
Most of the Standards for Mathematical Content are familiar to teachers of mathematics. For example, Algebra Standard A.REI.4 says, “Understand that the graph of an equation in two variables is the set of its solutions plotted in the coordinate plane, often forming a curve or a line.” What is less clear is what does it mean to “understand” mathematics?
The Common Core State Standards deal directly with what it means to understand mathematics through the Standards for Mathematical Practice, standards that describe ways in which students should engage with the mathematics.
“One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.” CCSS
For example, many students have come to rely on FOIL as a “method” for multiplying binomials.
- Do students who apply the FOIL mnemonic really understand why (a + b) (a – b) = a2 – b2?
- Would they be able to multiply the binomials without using FOIL because they understand that it is an application of the distributive property of multiplication over addition?
- Do students recognize a4 – b4 as a difference of two squares?
- Can they explain how they could use the mnemonic to find the product of a binomial and a trinomial? Or do they just know how to FOIL?
To make this latest “wave” have an impact, we as teachers must get students beyond memorizing and applying rules. We must go beyond the Standards for Mathematical Content and incorporate the Standards for Mathematical Practice in our classrooms or there will be no impact on students. Connecting the mathematical practice to the mathematical content will help students demonstrate the true understanding that we are striving to achieve. As teachers of mathematics it’s up to us whether we catch the wave this time or continue to drown fighting the wave.