Stanford mathematician Keith Devlin takes a deeper look at the notion of number sense described in his previous essay All the mathematical methods I learned in my university math degree became obsolete in my lifetime.
Why Is number sense important? Number sense is important because it encourages students to think flexibly and promotes confidence with numbers. Authors Ann Carlyle and Brenda Mercado anthropomorphize this delightfully in their 2012 book Teaching Preschool and Kindergarten Math as children “making friends with numbers”. Or, as math educator Marilyn Burns observed in her 2007 book About Teaching Mathematics, “students come to understand that numbers are meaningful and outcomes are sensible and expected.”
Again, coming from a focus on mathematics education for children with learning disabilities, Russell Gersten & David Chard wrote in 1999, “Just as our understanding of phonemic awareness has revolutionized the teaching of beginning reading, the influence of number sense on early math development and more complex mathematical thinking carries implications for instruction.” (Gersten & Chard, 1999)
The fact is, students who lack a strong number sense have trouble developing the foundation needed for even simple arithmetic, let alone more complex mathematics. In one study of 180 seventh-graders conducted by the University of Missouri in 2013, researchers found that, “those who lagged behind their peers in a test of core math skills needed to function as adults were the same kids who had the least number sense or fluency way back when they started first grade.” (Neergaard, 2013) Now connect the dots to the sobering fact that 1 in 5 U.S. adults lacks the math competency of a middle school student—leaving them woefully unqualified for most jobs.
How do you teach number sense? A large body of research has shown that number sense develops gradually, over time, as a result of exploration of numbers, visualizing numbers in a variety of contexts, and relating to numbers in different ways.
Burns suggests the following key, research-based teaching strategies to build numbers sense:
- Model different methods for computing: When a teacher publicly records a number of different approaches to solving a problem–solicited from the class or by introducing her own—it exposes students to strategies that they may not have considered. As Burns explains, “When children think that there is one right way to compute, they focus on learning and applying it, rather than thinking about what makes sense for the numbers at hand.”
- Ask students regularly to calculate mentally: Mental math encourages students to build on their knowledge about numbers and numerical relationships. When they cannot rely on memorized procedures or hold large quantities in their heads, students are forced to think more flexibly and efficiently, and to consider alternate problem solving strategies. (Parrish, 2010)
- Have class discussions about strategies for computing: Classroom discussions about strategies help students to crystalize their own thinking while providing them the opportunity to critically evaluate their classmates’ approaches. In guiding the discussion, be sure to track ideas on the board to help students make connections between mathematical thinking and symbolic representation (Conklin & Sheffield, 2012). As noted in Classroom Discussions: Using Math Talk to Help Students Learn, the goal is “not to increase the amount of talk but the amount of high quality talk.”
- Make estimation an integral part of computing: Most of the math that we do every day—deciding when to leave for school, how much paint to buy, what type of tip to leave in a restaurant, which line to get in at the grocery store relies not only on mental math but estimations. However traditional textbook rounding exercises don’t provide the necessary context for students to understand estimating or build number sense. To do that, estimation must be embedded in problem situations.
- Question students about how they reason numerically. Asking students about their reasoning—both when they make mistakes AND when they arrive at the correct answer—communicates to them that you value their ideas, that math is about reasoning, and, most importantly, that math should make sense to them. Exploring reasoning is also extremely important for the teacher as a formative assessment tool. It helps her understand each student’s strengths and weaknesses, content knowledge, reasoning strategies and misconceptions.
- Pose numerical problems that have more than one possible answer: Problems with multiple answers provide plenty of opportunities for students to reason numerically. It’s a chance to explore numbers and reasoning perhaps more creatively than if there was “one right answer.”
Burns, Marilyn. About Teaching Mathematics: A K-8 Resource. 3rd ed. Sausalito, CA : Math Solutions, 2007.
Carlyle, Ann, and Brenda Mercado. Teaching Preschool and Kindergarten Math: More than 175 Ideas, Lessons, and Videos for Building Foundations in Math, a Multimedia Professional Learning Resource. Sausalito, CA: Math Solutions, 2012.
Conklin, Melissa, and Stephanie Sheffield. It Makes Sense!: Using the Hundreds Chart to Build Number Sense. Sausalito, CA: Math Solutions, 2012.
Gersten, Russell and D. Chard, David. “Number Sense: Rethinking Arithmetic Instruction for Students with Mathematical Disabilities.” The Journal of Special Education 33.1 (1999): 18-28.
Neergaard, Lauran. “Early Number Sense Plays Role in Later Math Skills.” ABC News, 2013 http://www.abc2news.com/news/health/early-number-sense-plays-role-in-later-math-skills
Parrish, Sherry. Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5. Sausalito, CA: Math Solutions, 2010.