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.

Discourse and Debate

As a teacher, I have spent many years developing my understanding of children in the 7-9 year old range. Now that I have my own child, I find it infinitely fascinating to watch him grow from the ground up. I suppose that was the great interest of Piaget, and in watching my son, I realize he is a unique individual with unique experiences like all children. However, watching him make mathematical sense of our world is both delightful and entertaining at times. He was a late bloomer when it comes to speaking. It wasn’t until shortly after age two that he began really expressing himself verbally, and that we could gain some insight into how he processed his world. We had spent much time counting, especially the 13 stairsteps in our house. Counting to 13 seem to come easily to him, learning three numbers at a time. What was especially interesting was when he first began speaking in 2-3 word phrases and he noticed that both of his grandmas were in the kitchen with him. He looked back and forth out them, and said with delight, “Two gras!” Since then, he has shown how he explores our world mathematically by choosing to count different objects and people. Although he may not really understand the concepts, it is fascinating to hear him use mathematical terms in his daily conversation. He has expressed numbers such as 20 1/2, 100, and 1000, and looked at objects and said, “I’m trying to find how many inches,” as he demonstrates his own form of measuring. His learning is certainly not linear, and he is absorbing far more than what I intentionally work on with him.

So how does this relate to my topic? It is through conversation that I gain insights into my son’t thinking. As a toddler, these discussions are rather short, however, questioning still plays a role. How many do you see? Which is more? How much do you want? The importance of dialogue does not change as children get older. Unfortunately, we don’t have the leisure of much one-on-one time with a classroom full of students, and it is easy to fall into the pattern of teacher talking and students “listening.” However, learning is not a passive role, and without discourse, we don’t really know what students are thinking. Conversations play a central role to eliciting student misconceptions, conjectures, and big ideas. Conversations can allow for us to probe student thinking, scratch away the surface, and develop enduring understanding. Conversations will tell us more than any test alone.

So how do we give students more time to talk? Put them in the driver’s seat. Instead of leading the lesson, pose problem situations that students can grapple with and debate. Be the facilitator, rather than the leader of the classroom conversation. And when kids really don’t agree, then organize a debate. Students love this opportunity to defend their thinking, and it forces them to analyze the nuances of the problem to develop a greater understanding. It is also memorable, which has a lasting effect.

Other tools for giving students more talk time include pair-share thinking frequently throughout a lesson, conferring with students one-on-one, in pairs, or small groups during independent work time, and strategic student share outs. When students are ready to support each other in their learning, I put them in collaborative groups to discuss their strategies and efficiency with concepts. I call these small group  activities “Math Chats.” They are differentiated cards that groups of 3-4 work with to discuss their thinking behind a concept. Students are encouraged to offer different perspectives and analyze how their strategies make the problem friendlier.

Subitizing: More Than Meets the Eye

Subitizing is a relatively new concept for me. Sadly, it is not an integrated part of the mathematics curriculum yet. Therefore, children often have little experience with subitizing. When I first learned about this concept, I thought it pertained only to kindergarten, first grade, and struggling students beyond those years. However, I have now seen that children of all ages benefit from subitizing.

What is subitizing, and why is it important? Clements and Sarama (2009) define two types of subitizing. The first, perceptual subitizing, pertains to the ability to both perceive intuitively and simultaneously the amount in small number sets. No counting is neccessary, you just know the amount when you see it. Children develop the prerequisite skills for perceptual subitization at a young age. According to Clements and Sarama, children begin naming collections of 1, 2, and 3 from ages 1-2. By age three, children can also create collections made of 1-3 objects, sometimes 4. Perceptual subitization up to a collection of 4 occurs at age four, and the recognition of sets of 5 develops at age five. To encourage the development of perceptual subitizing, parents and teachers should play snapshot games where children see an organized picture of dots, squares, and other simple geometrical shapes organized in a linear fashion for couple of seconds, then determine the number.

The second type, conceptual subitizing, relates to the ability to instantly see the parts, and join them together to make a whole. For example, given the picture of five arranged with three and two dots, a conceptual subitizer would see 3 and 2, and know that makes 5. Again, no mathematical operations may be consciously enacted, but rather, an instant recognition that the parts make that whole. As children progress, subitizing helps with the visualization of operations and mental math. Conceptual subitizing to five and ten begins at age five. By age six, children are able to conceptually subitize to 20. At this age, five and ten-frames are helpful organizers for promoting subitizing at these higher levels. Skip and counting and place value with subitization begin at age seven, and by age eight, children see multiplicative relationships, such as 5 groups of 10, and 4 groups of 3, which makes 50 and 12, so 62 dots. As you can see, children benefit from subitizing activities well into third grade and beyond. Below is a table to organize these milestones in subitizing.

Trajectory of Subitizing

Age	Type of Subitizing	Example
1-2	Precursor to subitizing: Name small collections up to 3	I see two grandmas.
3	Precursor to subitizing: Create small collections up to 3 or 4	I can count three crackers.
4	Perceptual up to 4	**** I see four stars.
5	Perceptual to 5	***   I see five stars.    **
5	Conceptual to 5	***  I see three and two **   stars. There are five stars
5	Conceptual to 10	* * * * * * * I see 3, 3, and 1, which makes 7.
6	Conceptual to 20	* * * * * * * * * * * * * I see 5, 5, and 3, so that makes 13.
7	Conceptual with place value and skip counting	I saw tens and twos, so 10, 20, 30, 32, 34.
8	Conceptual with place value and multiplication	I saw groups of tens and fours, so 5 tens makes 50, 4 fours makes 16, so 66.

All information adapted from Clements and Sarama (2009)

I encourage you to try out subitizing with your students or children. Where are they in their development? I have seen eight year olds that are conceptual with place value and multiplication, and 14-year olds who are at perceptual to 4. Experiences with numbers, both counting and visualizing, are crucial for number development, regardless of the age. Once foundations are set, subitizing can be used to teach more complex topics. I have used both subitizing and arrays to develop understanding of multiplication, algebraic properties, and division. If you are new to teaching Common Core, then my new unit on teaching multiplication using subitizing and array cards may be just for you! These cards are intended to be part of a program that also includes problem solving. You can check out my 17 page free download to see for yourself!

Clements, D.H., Sarama, J. Learning and Teaching Early Math The Learning Trajectories Approach (2009). Routledge: New York, NY.

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.

Solving the Problem

Do you remember that feeling you had as a child or teen when you got to the problem solving part of a lesson or unit in your textbook? Perhaps you felt joy in finally getting to do something somewhat more challenging than completing the 50 number sentences you just solved. Or maybe you felt fear that someone might actually notice that you really have no idea what you are doing. Worse yet, indifference may have become your bland taste for math and none of it mattered to you anyway. For me, the word problems always seemed like an enigma. Often it was just for extra credit – a nonessential part of mathematics. Other times it was assigned and I prayed for the magical key words to guide my way to a solution. I always felt uncertain. Having just practiced a set of rules for a page of tedious equations, the word problems didn’t fall into the same category as the rote memorization of the lessons practiced. It was disjointed and out of sync. Even those memorized key words didn’t help most of the time. They didn’t match multi-step problems or ones not involving algorithms at all. We all know the joke about the train –  If a speeding train is heading towards Los Angeles at 84 mph….Who cares? I’m not on that train.

I often find students in the same quandry. Upon entering 2nd or 3rd grade, students mistakenly believe that to solve a word problem you take the numbers, throw in an addition or subtraction sign, and then find the answer. The answer doesn’t need to make sense since the problem may not have to begin with.

My feelings about problem solving changed drastically when I went through training and implemented Cognitively Guided Instruction (CGI). Problem solving became the basis for my lessons rather than the afterthought. It became both my tool for teaching and the students’ access for understanding. I could finally really see what my students understood, and determine precise next steps for instruction. It took time to cultivate a classroom (and a teacher) that values problem solving. Through conversation and strategic share outs, students had real access to the underlying complexities within math that had evaded them. I came to understand mathematics with much greater clarity than before. And you know what? My students don’t need 50 problems to develop that understanding too. As it turns out, problems solving is the key to understanding, not the afterthought.

This is a sample format for differentiating word problems for your students. The sample on the left is a subtraction problem (SRU: Separate Result Unknown). There are multiple number choices for students to choose from depending on their comfort with the problem. The box is for the plan. I expect students to try at least two plans before trying a new number set to solve. I don’t always use a formal paper, as students have math journals to work in. However, it is nice to have a sample to send home or collect to collaborate and examine with colleagues. I have a rubric attached to my Snowmen Subtraction Word Problems that can be used to grade word problems. Again, I don’t grade all of the student’s word problems, but I do keep anecdotal records that tell me how they thought about a problem and how I need to push them to the next level.

What does understanding look like?

In education, determining what understanding looks like is one of our biggest quandaries.  We examine learning goals from multiple perspectives, and with our given resources, often fall short. Learning is complex, and therefore, understanding is too. It is much easier to determine proficiency in skills and rote tasks. Often we accept mastery of skills to be synonymous with understanding, and this is where I would contest is one of our greatest shortfalls in education. It is far easier to see if a child gets the answers right or wrong, than if the child understands the how and why. When I first began my journey into the teaching profession, I was asked to take a proficiency test in math to determine if I was current enough in my skills to bypass a more current math methods course. It was the first time anyone ever asked me to explain why and how an algorithm for regrouping worked. I thought it was really strange at the time. My whole life, I never questioned the “whys” of math. Of course, I am a rule follower, so I did very well. I did not need to understand, I just had to keep to the rules. My husband is my opposite. The rules never made sense and he had his own way of thinking about numbers that did not fit into the rules. Therefore, he did not do well in his mathematics courses. Guess who is the better mathematician? It wasn’t until I stopped “teaching” my students and started listening to how they thought that I truly began to understand mathematics. And, learning to teach in a way where I was the real learner has proven to be a far more complex and rewarding methodology than any textbook I have seen.