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The ABCs of Number Sense

Number sense.  A phrase that we hear a lot in the math world. Some kids have it and some don’t. Many adults lack number sense! But what do we mean by number sense, and how do we help our students develop a strong sense of number?

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I think it is important to have a working definition of number sense, and I like this one:

“…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)
—from NCTM’s Illuminations website

Notice how it incorporates the ideas of understanding number and operations, flexibility, mathematical judgments, and problem-solving.  It also makes clear that we learn mathematics to use it.

Children come to school with a sense of quantity and number (Sousa, 2007), and just as the primary grades lay the foundation for strong literacy skills, they also set the stage for number sense and numeracy.

Now that we have a common definition for number sense, and we understand its importance, let’s turn to how we help our students develop it. Surprisingly enough, it’s as easy as ABC!

A is for Accountable Math Talk

What does your math classroom sound like?  Who does most of the talking during your math instruction–you or the students?  Most of us experienced quiet math classrooms growing up with little student collaboration, but all that is changing.  Wonderful books like Number Talks (Parrish), Classroom Discussions in Math (Chapin), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz) expose us to the importance of student discourse and provide guidance on developing a classroom that thrives on rich mathematical discussions.  In her book Math Sense: The Look, Sound, and Feel of Effective Instruction, Christine Moynihan says it this way:

“I want noise and plenty of it–productive, purposeful, and meaningful noise–from everyone, students and teachers alike.”

So how does this look in your classroom?  Simply put, it looks like you talking less and your students talking more!  From the first day of school, you want to create a culture in your classroom that supports respectful and productive discussions.  As you plan your lessons, look for opportunities for students to talk with each other. If you find yourself talking too much during a lesson, have the students do a quick turn and talk.  Develop rich questions that engage and challenge students and guide them toward the learning, rather than handing it to them on a silver platter.

Relinquishing the role of sage on the stage will not be easy.  But you will be amazed at the rich conversations your students will have and the deep learning that will result when you step back and let them do the talking.

B is for Bruner’s Concrete, Pictorial, Abstract Approach

Which is greater? One-eighth or one-fourth?

Ask a class of 5th graders this question, and you might be surprised by how many students answer one-eighth.  One reason for this misconception is that many students lack a mental image for these fractions. If our students are to have strong number sense, it is essential that our mathematics instruction provides students with the experiences necessary to develop deep, conceptual understanding.

The concrete, pictorial, abstract (CPA) approach to learning originated in the 1960’s as a result of the work of psychologist Jerome Bruner.  It is often referred to as CRA, with the word representational taking the place of pictorial.  Bruner (and countless researchers since) determined that mathematical content was best learned when it followed a progression from concrete, hands-on experiences, to pictorial representations linked to the concrete learning, before finally reaching the abstract, or symbolic, stage.  To illustrate the sequence using a fraction example, students would first have numerous hands-on experiences with concrete models of fractions, such as Cuisenaire rods, fraction strips or tiles, fraction circles, etc.  At some point, students would begin drawing representations of the concrete materials they are using.  Notice how the concrete and representational overlap. Finally, the abstract, or symbolic representation would be introduced, again overlapping with the previous stages.

Now think about how math is taught in many classrooms–with textbooks and worksheets.  Do you see the problem?  Not very concrete, right?  That’s not to say that we need to throw out all of our textbooks and worksheets.  We just need to make sure that we are consciously and consistently providing concrete experiences for our students, and that’s actually pretty easy to do.  When topics are presented in abstract form on a worksheet or in a textbook, provide manipulatives for students to use and then have them represent their solution pictorially.  So students model 2 + 3 (abstract) with counters (concrete) and then draw their solution in their math journal (pictorial).

C is for Composing and Decomposing Numbers

With the introduction of the Common Core State Standards for Mathematics, the phrase compose and decompose numbers positively exploded onto the mathematical scene. Don’t believe me?  A Google search on the phrase ‘compose and decompose numbers’ now returns over a quarter of a million hits!

In plain terms, composing and decomposing numbers is basically the understanding that numbers can be combined (composed) to make bigger numbers and bigger numbers can be broken (decomposed) into smaller numbers. In other words, 2 and 3 can be combined to make 5 (composing) and 5 can be broken into 2 and 3 (decomposing).

While Kathy Richardson notes that “If basic facts are to be foundational, they must be based on an understanding of the composition and decomposition of numbers”, the power of part/part/whole thinking goes way beyond fact fluency.  Consider the following examples showing how the ability to fluently decompose and compose numbers allows for flexible computation with not only basic facts, but also with multi-digit computation and even measurement concepts.

Kindergarten and 1st grade are critical for developing this deep understanding of composing and decomposing numbers.  Activities should be differentiated so that each child is working on his or her own “number” as determined by some variation of a “hiding assessment”. For example, show a student a train of 5 linking cubes, then put them behind your back and break some off.  Show the partial train to the student and ask “How many more to make 5?”  If the student knows all the combinations for 5 with automaticity, try the combinations for 6.  The number that a student stumbles on becomes their number.  Use a variety of activities such as number bracelets, dot cards, and Shake and Spill to differentiate composing and decomposing practice.

Starting a new school year is both exhilarating and stressful. There is so much great research about instructional best practices that we often feel overwhelmed about where to start.  As you head back to school this fall, just remember your ABCs!

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  1. Awesome! Love the reference to Jerome Bruner. I am a firm believer that we need to do all three together, instead of separately. So for your fraction example, have them use the Cuisenaire rods but during the same lesson have them also draw a picture of what they did with the rods plus attach the symbols/abstract to the picture. That way kids see how all three connect together. For instance I often see textbooks that have a lesson on using Base 10 Blocks for subtraction (or addition), then another lesson on drawing sticks and dots to represent the blocks and solve the problems, and then another separate lesson on the algorithm, thinking that the use of the blocks and the drawings helps build understanding of the abstract algorithm, but many kids don’t make the connections on their own. If you do the concrete of the blocks along with having them draw a representation and write the symbols all in the same lesson, then it becomes easier to see the connections. Plus then you get multiple days of doing all three together instead of three days of separate activities.

    1. Absolutely, Christina! I think that is a common misconception about CRA–that they are separate and distinct stages. They definitely must overlap.

    2. The rekenreks look amazing! Uh, I think I’d have the kids make them with plastic canvas. They’ll appreciate the finished products!

  2. This is great, love all the ideas. How do we translate this thinking to our high school students, is it too late for them?

    1. It’s never too late, Irma! It might be more difficult, because they have been taught in a certain way for a longer period of time, but I think the students would be very open to it. Certainly adding accountable talk to high school math classes would appeal to the social nature of that age group. CRA should not be an elementary only instructional practice. Learners need concrete and representational experiences for any new skills.

  3. Thank you for explaining exactly how to assess what “number” they are on. This is something I want to work really hard at this year and knowing how to assess it is super helpful.

  4. This is a little off subject, but would you give me your opinion of what the term “fluently” equates to in math terms? Teachers at my school see it as being able to write or say math facts quickly. I was told, by a math instructional coach, that it means the ability to know different combinations of numbers that make a number. For example: 2+3 and 4+1 equals to 5. Perhaps both are correct? I have a problem with timed fact tests, but teachers that I know use this to assess whether or not students are fluent with their math facts. I would appreciate your opinion on this also. Thank you so much and I always enjoy your articles!!

  5. Concrete and pictorial representation is so important. A lot of students use visual clues to help them process information so when they construct or draw a problem it helps them to become more fluent in math. Number sense is critical for math in the upper grades. By implementing the CRA approach will help close the gaps in the students number sense.

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