As we expect every child to read fluently with comprehension by the end of third grade, we should expect every child to have mastery of numeracy with understanding by the end of fourth grade so that they can access and learn mathematics easily, effectively, and efficiently. Then, they can appreciate the reach of mathematics, its utility, power, and beauty. To do so is to understand number, acquire numbersense and build the brain for fluency in numeracy and beyond. Through learning, practicing, and applying knowledge of the number concept, numbersense, numeracy, and mathematical way of thinking children will have access to higher mathematics.

When I ask teachers in my workshops, from elementary through high school and college, what their major concern is in teaching students mathematics, comments about numbersense are the most frequent:

- Many of my students do not have “good” numbersense. How do I develop “good” numbersense in my students so that I can teach my curriculum at grade level?
- I wish my students knew their facts. No numbersense!
- How can I teach my grade level material, when they do not even have a sense of the place value of whole numbers? No numbersense!

The concept of numbersense is important for learning higher mathematics and also for day-to-day living. When I evaluate children from elementary to high school and even college students and adults, for learning problems in mathematics, most times it is the lack of mastery of numbersense that is at the base of many students’ difficulties in mathematics.

When I ask teachers: What do you mean by numbersense? Everyone gives his/her own definition. Numbersense is a key concept, but the meaning of this term is only vaguely understood. A term that is not well-defined cannot be effectively taught and assessed. Therefore to teach and to assess numbersense, it is important to know:

- What is the meaning of numbersense? How do we define it?
- How does it develop in children?
- How do we develop and teach it?
- What student behaviors must be evident when it is present?
- What are the component skills necessary to learn and master it?
- How do we know the child has acquired it? How do we assess it?
- What are the levels of its achievement? How should it manifest at each grade level?
- What is its role in learning other mathematics concepts?
- What are the implications if it is not acquired?

Till these questions are answered in the teacher’s mind, the concept cannot be developed in children and assessed. A teacher’s instructional methodology for this concept depends on how it is understood. In the next several blogs, I plan to answer the questions posed above.

**What is Numbersense?
**From Kindergarten to upper elementary school, three major concepts form the foundation of arithmetic. They are also essential elements and building blocks in learning higher mathematics concepts, skills, and procedures. These are

**number concept**,

**numbersense**, and

**numeracy**. Numbersense depends on the mastery of number concept, and its mastery is essential for the development of numeracy. The mastery of the concept and numbersense skills is the integration of three major components and skills.

- Number Concept
- Arithmetic Facts
- Place Value

When students have mastery of these individual skills, they develop competence in numbersense and numeracy by integrating these skills.

Numbersense is a developmental and hierarchical concept. The type and level of mastery of arithmetic facts and place value varies from grade to grade; therefore, the concept and skills related to numbersense change and become complex and more demanding from grade to grade. The following are the non-negotiable component skills for the development of numbersense at each grade level. This does not mean nothing else other than these needs to be learned at these grade levels. Non-negotiable skills[1] at any grade level mean that other concepts, procedures, and skills can be mastered easily if these non-negotiable skills are mastered.

**1. Numbersense at the end of Kindergarten**

- Mastery of number concept
- Mastery of 45 sight facts
- Place value (two digits)

**2. Numbersense at the end of First Grade**

- Mastery of number concept
- Mastery of 100 Addition facts
- Place value (three digits)

**3. Numbersense at the end of Second Grade**

- Mastery of number concept
- Mastery of 100 subtraction facts (assuming the 100 addition facts have been mastered)
- Place value (four digits)

The major goal of the first three years of mathematics curriculum (K through second grade) is to master **additive reasoning: **understanding the concepts of addition and subtraction, fluency of addition and subtraction facts, mastery of addition and subtraction procedures, applying these skills into solving problems, and knowing that they are inverse operations of each other. It means that: (a) given an addition problem, one can transform it into a subtraction problem and vice-versa, 23 + 12 = 35, 12 + 23 = 35 and 35 – 12 = 23, and 35 – 23 = 12, (b) when two numbers (10 and 9 are subtracted) from a given number (27), then the (27 –10 = ?, 27 – 9 = ?), then are being subtracted from a given number, then the remainder is larger from the given number in the case of the smaller number being subtracted from it (27 –10 17, 27 –9 = 18), (c) the difference of two number (51—29 = ?) will remain the same when the problem is translated by a number ((both numbers are translated by 2 units: 52 – 30 = 22 and 51 –29 = 22), etc.

**4. Numbersense at the end of Third Grade**

- Mastery of number concept
- Mastery of 100 multiplication facts (multiplication tables from 1 through 10)
- Place value (at least 5 to 7 digits and ultimately any digit whole number)

**5. Numbersense at the end of Fourth Grade**

- Mastery of number concept
- Mastery of 100 division facts
- Place value (any number of digit whole number) and to hundredth place

The major goal of the third and fourth grades mathematics curriculum is to master **multiplicative reasoning: **understanding the concepts of multiplication and division, fluency in multiplication and division facts, mastery of multiplication and division procedures, applying these skills into solving problems, and knowing that they are inverse operations of each other (given a multiplication problem, one can transform it into division problem and vice-versa).

**Numeracy
**Numeracy is both dependent on numbersense and aids in the development of numbersense; in this sense it is the culmination of numbersense. Numeracy is the ability and facility of a student to execute four whole number arithmetic operations correctly, consistently, and fluently in the standard form with understanding. By the end of fourth grade, every child should have mastered numeracy.

[1] See an earlier post on this blog on non-negotiable skills at elementary school level.