Number Concept, Number Sense & Numeracy

In my last blog post, I posted a video of Dave Willard explaining that taking the time to focus on mastery of number sense made grade level concepts more accessible. However, it occurs to me that it might not be clear what "number sense" is. We have various terms that we may throw around: number concept, number sense, numeracy. But what do these terms really mean? And how do they relate to mastering grade level concepts? Let's unpack these terms a bit...

Number Concept

According to Mahesh Sharma, number concept is the "backbone of math" which I suppose means that without it we would be amorphous blobs lacking the structure to engage in thinking and talking about patterns and quantity and space. Number concept involves understanding 
number,
 its 
representations and 
applications. It should be mastered in kindergarten. However, if older students are found to be lacking number concept, this is the first place to intervene. Mastery of number concept includes:

1. Lexical entries for number (all varieties of number names—reciting number names in order—and knowing the difference between number words and non-number words), 
2. Meaningful counting (one-to-one correspondence with sequencing and conservation of number), 
3. Recognition and assignment of a number orally to a collection of discrete objects without counting (e.g., standard visual clusters of up to ten objects instantly), 
4. Representation of a collection—visual cluster of objects up to ten in its written form (a visual cluster of seven objects graphical representation, e.g., 7), 
5. Assignment of a graphical representation when the name of number is heard (hears s-e-v-e-n and writes 7), and 
6. Decomposition and re-composition of a collection (cluster into sub-clusters, 45 addition sight facts).

These aspects of number concept must be integrated to help students develop flexibility.

Number Sense

Number sense builds on number concept and is implied by multiple CCSS Math Practices (as noted below). It involves the integration of:

1. Number concept.
2. Mastery of number facts (appropriate to the grade level).
3. Place value (appropriate to the grade level).

Number sense allows students to move fluidly between equivalent representations of quantity (MP 2), understand the structure of and relationships between quantities (MP 7), use estimation and visualization to make sense of problems and solutions (MP 1) and apply their understanding of quantity to any context or situation (MP 2).  Number sense helps students attain numeracy, but it also serves as the foundation for algebraic reasoning and mathematical problem solving.

Numeracy

Numeracy is the ability to execute all 4 arithmetic operations (addition, subtraction, multiplication and division) on whole numbers:

1. Correctly
2. Consistently
3. Fluently
4. Efficiently (in standard form)
5. With Understanding

We must use number sense to attain numeracy. Without number sense, it would be impossible to execute operations "with understanding," and this is perhaps the most important of the 5 criteria mentioned above. This should be mastered by the end of 4th grade, but obviously, it's not the end of the line. Everything else builds on a student's numeracy. Algebra is generalized arithmetic, after all. If our students can execute all operations with number sense and deep understanding, we have set them up for success in learning about proportionality and algebraic reasoning.

And it all begins with number concept. Many difficulties in learning mathematics can be traced back to some deficiency in number concept. So, even if you're not a kindergarten teacher, it makes sense to know what to look for and how to intervene.

What to look for and what to do?

If a student lacks number concept, he or she will demonstrate:

• Difficulty estimating size and magnitude of numbers, 
• Inefficient problem solving methods (like sequential counting), 
• Lack of fluency with additive facts, 
• Correct methods but without understanding why, 
• Inability to de-compose and recompose a number.

While I was attending Lab School last summer with Mahesh, I had the opportunity to follow him around as he worked one on one with students ranging in age from kindergarten to fifth grade. I observed that whatever the age of the student, Mahesh always began by checking the student's number concept before moving on. Here's an example of what that looked like with a fifth grade student:

Mahesh started by holding up a visual cluster card with 7 diamonds on it.

Mahesh: What is this number? 

The student paused. I could see his lips moving as though he was silently counting, but Mahesh interrupted him. 

Mahesh: Do you see a four? 

Student: Yes. 

Mahesh: Show me. 

The student touched the four items he saw. 

Mahesh: Write it.  

The student wrote a "4." He used an inefficient method to write it, and Mahesh corrected him. 

Mahesh: Do you see another four? 

Student: Yes. 

The student touched the four items he saw. Mahesh and the student sped up. 

Mahesh: Do you see a two? A three? A five? A six? 

The student touched the cluster at Mahesh's prompting. 

Mahesh: Can you give me all the facts that make seven? Can you give them to me systematically? 7 plus... 6 plus... 5 plus... and so on down to zero? 

The student was able to do this smoothly for: 7 + 0, 6 + 1, 5 + 2, and 4 + 3. Then he got hung up on "3 + 4" So, Mahesh moved to the Cuisenaire rods. 

Mahesh: Can you take out one of each color to make the staircase? Do you have all of them?   

The student made the staircase.

Mahesh: Touch the number when I say it: 2, 7, 5, 2, 9, 2, 9, 7, 5, 3, 1, 8, 6, 5. 

The student touched the rod on the staircase that corresponded to the number. 

Mahesh: Show me a length of 15. Can you place it horizontally? Put it in order the way you would write it.

Mahesh: Show me 17... 12... 19... 14... 16. 

The student did this easily. 

Mahesh: Take any two rods that make 8.   

The student took out two fours.

  

Mahesh: How do you know it's equal to 8? How can you prove it? Prove it. 

The student arranged the fours end to end parallel to the eight rod.

Mahesh: Any other combinations? Can you arrange them systematically for me? 

The student arranged them this way.

Mahesh: Read them from top to bottom. Touch them as you read them. 

Student: Zero plus eight is eight. One plus seven is eight. Two plus six is eight... etc. 

Mahesh covered up the rods with a sheet of paper. 

Mahesh: Read them now. 

After the first two combinations, the student had difficulty with this. Mahesh revealed the rods. 

Mahesh: What is happening to this number, this first addend? 

Student: It's going down by one. 

Mahesh: What's happening to this number, the second addend?  

Student: It's going up by one. 

Mahesh covers the rods again.

Mahesh: Good. Read them again now.

After the student was able to do this fluently (quickly, orally, without looking at the rods), Mahesh moved on to making 9 and repeated the whole process.

The intervention above took place in about 15 minutes, and we can see how quickly Mahesh was able to diagnose that the student needed to build number concept.  With fifth grade students, we might be tempted to have our interventions target grade level gaps without taking into account their shoddy foundation.  When we fix the foundation first, the rest will be more stable.  How do we fix the foundation?  The best tools for building of number concept are dominoes, visual cluster cards, playing cards and Cuisenaire rods.  The best methods emphasize mastery of clustering rather than counting.

In my next blog post, I will be describing how number sense integrates number concept, fact mastery and place value to teach numeracy skills.  I was thinking I'd do a concept from 3rd grade, but if you have any special requests (concept or grade level) please let me know...