November 2011

Making Sense of Math in the Standards Era

P. Mark Taylor

Most states have adopted the Common Core Standards (CCS) or will be adopting them in the next few years. One key feature of the CCS is the Standards for Mathematical Practice, which describe the expertise K–12 mathematics teachers should develop in their students:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

Standard 1 seems relatively basic, but it can be the most challenging aspect of teaching math.

A couple of years ago, I was a middle grades mathematics consultant for two school districts in which teachers were struggling through their first year of implementing new state standards. I spent a lot of time in classrooms with students and teachers who seemed to think Standard 1 was a new concept in mathematics. After all, many teachers and students had made it through math by memorizing facts and procedures without truly understanding them.

Although they wanted all students to understand the how and why of math, teachers believed there was too much other content to cover in the time available. Some teachers believed not all students were intellectually capable of making sense of mathematics.

Ensuring every student makes sense of mathematics is not just a good idea, it is the job teachers were hired to do. Here are some hands-on activities math teachers can use to help their students make sense of mathematics as suggested in Standard 1:

• Make conjectures. Rather than jumping in to solve a problem using memorized equations, students use inductive reasoning to estimate or guess answers and then use their understanding of math to test their conjectures.
• Use concrete objects or pictures to conceptualize and solve problems. This promotes visual thinking and helps students "see" the sense of the math concept. Yes, manipulatives have a place outside elementary school.
• Verbalize concepts. Verbalizing math concepts—orally or in writing—helps clarify and solidify their meaning as students explain patterns, concepts, tables, graphs, drawings, and diagrams.
• Use technology. Online rulers, protractors, calculators, spreadsheets, statistical packages, and dynamic geometry software help students dig deeper into math concepts.

Is there an age or grade level at which it is no longer appropriate to use concrete models for hands-on learning experiments? Consider this: calculus, the most abstract mathematics in the K–12 curriculum, emerged from the study of concrete objects. The most natural development of mathematical knowledge at all grade levels includes hands-on activities.