What is Number Sense?

Early in my teaching, I was able to identify when students struggled with numbers. I didn’t have the sophistication or necessary tools to pinpoint exactly where the problem originated, or the gaps that needed filling, but it was glaringly obvious as children dragged their way through math, seeming to fall further behind daily, that something was not right. Because I didn’t have the understanding of mathematics and how children think, my only support for these children was grasping at straws. More practice and more games didn’t improve their number sense, but made the parents, children, and me feel like were trying, and perhaps making some marginal improvement.

Over the years, I have had conversations with many teachers about number sense and place value. What does it mean? What does it look like? What can children do and say when they understand numbers? What do we do when children struggle with numbers? As it turns out, this is a complex topic, that many feel ill-equipped to answer, and researchers struggle to find a common definition. Many teachers rely on the place value skills enumerated in textbooks and older standards. But, does that really mean a child understands numbers? If you teach a child how to compare, order, and round numbers, does this mean they have mastered number sense?

There are a variety of definitions of number sense in the field of mathematics. NCTM (2012) defines number sense as follows:

“Number sense refers to a person’s general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems (Burton, 1993; Reys, 1991). Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms (Howden, 1989).”

McInstosh, Reys, and Reys (1992) defined number sense as “a propensity for and ability to use numbers and quantitative methods as a means of communicating, processing and interpreting information. It results in an expectation that numbers are useful and that mathematics has a certain regularity (makes sense).” They further delineated their definition with the clear framework detailing number sense below, with the understanding that overlap occurs.

Number Sense

Key Components	Student Understandings
Knowledge and facility of numbers	A sense of orderliness of numbers (place value,and relationships and ordering between and among number types)Multiple representations for numbers (symbolic, equivalencies, decomposing, and comparisons) Sense of relative and absolute magnitude of numbers (comparing to physical and mathematical referent) System of benchmarks (mathematical and personal)
Knowledge and facility with operations	The effect of operations (whole numbers, fractions, decimals)Mathematical properties Relationship between operations
Applying knowledge of and facility with numbers and operations to computational settings	Relationship between problem context and computation (exact vs. approximate)Awareness that multiple strategies exist (invent, apply, and select strategies, determining efficiency) Inclination to utilize an efficient representation and/or method Inclination to review data and result for sensibility (reasonableness)

Framework for Number Sense (McIntosh et al. 1992)

McIntosh et al. make it clear that number sense is not just about knowing what to do, but rather, must be entrenched in what makes sense. Thinking must be involved at all levels of working with numbers, and therefore, numbers and procedures cannot be taught in isolation. This framework very much supports the goals of Common Core.

The goals within Common Core, NCTM, and McInstosh et al. are further supported by the goals of mathematical proficiency as defined by The National Research Council. The Council determined five interwoven and interdependent strands of mathematical proficiency. In their report, Adding It Up Helping Children Learn Mathematics (2001), they clarify the importance of depth, clarity, precision, flexibility, and reflection in student thinking as delineated by the strands below:

• conceptual understanding—comprehension of mathematical concepts, operations, and relations
• procedural fluency—skill in carrying out procedures
 flexibly, accurately, efficiently, and appropriately
• strategic competence—ability to formulate, represent, and solve mathematical problems
• adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
• productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

If we reflect upon how number sense is defined, then what does number sense look like? Students with strong number sense can subitize easily, understand magnitude (relative size of numbers), have established cardinality, can strategically decompose (break down numbers) to make computation easier, understand the importance of ten within our number system, understand number relationships, and have proportional reasoning. They determine the efficiency of their strategies, the reasonableness of their answers, and understand the application and context within which operations are used. These concepts cannot be taught in one lesson or unit. They are ongoing experiences students need as part of their math education throughout the grade levels. When students emerge from classrooms focused on these concepts, they come to understand that math is about relationships, not memorization, and flexibility rather than rigid rules.

Therefore, number sense entails far more than the traditional chapter on place value. Proficiency cannot be measured by a skill set, or regurgitation of memorized procedures and rules. Students need dynamic experiences with numbers. They need to be able to dive into the depth of numbers, explore their uses, their flexibility, their application, their differences, their nuances, and their reason. And, while numbers are both abstract and concrete, they need to be seen as something that makes sense. Ultimately, learning mathematics must be with understanding. When we present math as a series of rules and explain to children how to follow a procedure step-by-step, we have actually robbed them of the opportunity to develop both number sense and mathematical proficiency. As this is how our system is designed, many children receive passing grades throughout school, only to falter later on, finding there is no solid foundation to support more advanced mathematics.  Common Core is calling us to change our practices so that children emerge from the classroom as mathematical thinkers that demonstrate the ability to adapt to the ever-changing world awaiting them, rather than as mini-calculators with an isolated set of memorized skills.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up : helping children learn mathematics / Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education, National Research Council ; Jeremy Kilpatrick, Jane Swafford, and Bradford Findell, editors. Washington, DC : National Academy Press, c2001.

Mcintosh, A., Reys, B. J., & Reys, R. E. (1992). A Proposed Framework for Examining Basic Number Sense. For The Learning Of Mathematics, (3), 2.

NCTM (2012). Illuminations. Cited on December 30, 2012 from http://illuminations.nctm.org/Reflections_preK-2.html.

The Pliability of Numbers

My sister told me in recent years that she doesn’t believe in subtraction – that there is no such thing. I thought she was crazy, and clearly, she had a different interpretation of math than what I was taught.  When I was growing up, numbers were viewed as rigid, set in stone. They were black stamps that etched the pages of math books in their perfectly organized rows. Hardened rules were taught, and if executed precisely, led to exacting calculations. Pages and pages of work droned on until it was declared that you had reached math proficiency and passed, or math despair and had failed. We became perfect little calculators (or not). There wasn’t much room for free thinking. It wasn’t really necessary; you just had to follow the rules. I am a rule follower, so I marched to the beat of the classes, claimed my A’s, and moved on, never reflecting upon mathematical principles until recent years. My sister on the other hand, well, she doesn’t believe in rules imposed by others, she would much rather create her own. Therefore, her supposition about subtraction should not have been a surprise.  In her world, there is no subtraction, only positive and negative numbers – you just have to keep the sign with the number. At first this may sound ludicrous, but in reality, she demonstrates a superior understanding of numbers and operations. It does not match what is taught in traditional mathematics, however, it rings true, even in higher level math. If you have a conversation with mathematicians, they see numbers and operations differently than often what is taught. The traditional algorithms used in the classroom today came from hundreds of years of mathematicians figuring out the shortcuts. In the classroom, we often teach the shortcut that uses the least amount of paper. No wonder students get so confused, they never get a chance to develop the understanding of how that shortcut developed.

I have a theory that there are four categories that most people fit in regarding their experiences with mathematics. The first have an innate understanding of numbers and principles. They see beyond what is taught in the traditional classroom. It just always made sense. The second is like myself, the rule follower/memorizer. They usually get good grades and are seen to be good in math, although they may not really have much understanding beyond following the prescribed steps. The third group understands how to work with numbers and operations, but it may not match how it was taught, which led to frustration and either a distaste for math courses or struggling grades. The last group just hate math. It never made sense and they couldn’t remember all the rules. I realized early on when I became a classroom teacher that the rules seemed to be conflicting, and at times, counterintuitive. For example, If you want to compare numbers, then start with the largest digit. If you want to add numbers, start in the ones place, which, as it turns out, is not how most students would naturally solve an addition sentence. We often feel tied to the way we were taught because it is what is familiar, regardless of what category of mathematician we associate ourselves with.

I am now trying to create a new mathematical path for my students. One where they can see connections and relationships, where numbers make sense. I now realize that numbers and operations are more like clay than rock. They can be molded and shaped into different forms. Their flexibility allows you to reorganize them and shape them into friendly equations that make mental math and everyday calculations simpler. Understanding of operations allows you to estimate and compute in diverse ways. I contend that the mantra in traditional mathematics echoed those of Nike, “Just Do It!” However, I have developed a new mantra, one that can only be used when we are taught to think about the pliability of numbers and operations – “Don’t work for the numbers, make the numbers work for you!”  This understanding took me time to develop. I had to retrain my brain and develop a deeper understanding of numbers and operations. I had to change my perception of math, and that it was more about understanding relationships than following the rules. When working with students now, I focus on how numbers can be represented, what the operations truly mean, and how numbers, strategies, and operations are related. Students are demanded to think, analyze, and make connections within mathematics. Flexibility is the norm rather than the outlier.

If you are anything like I was in my earlier perceptions of math, you may be wondering what this flexibility looks like. I know when I was first introduced to this idea, I could not see outside my box and had someone show me just a few ways of solving other than how I was taught. The rest of the ways, I learned later on either through my students teaching me or through research articles. You can download samples of strategies used by my students over the years. The names I give them are not the official names, but rather, what we have called them as a class. It can be uncomfortable to stretch beyond our own experiences, but once we do, we have an understanding that could never have been reached before.

In America, we live in a culture deeply entrenched in traditional algorithms, and it can be challenging to see beyond, or to help others understand why it is important to teach math in a different way. I spend a lot of time educating the community on how and why I am teaching this way. I often show parents articles on the importance of problem solving for today’s students, such as chapter two from Making Sense: Teaching and Learning Mathematics with Understanding by James Hiebert et al. I encourage you to read the article, The Harmful Effects of “Carrying” and “Borrowing” in Grades 1-4 by Constance Kamii, which I have also shared with parents during conferences. Reading these articles may feel uncomfortable, but change often is.