Mathematics could well be defined as the study of number and shape. And measurement connects shape and number. So, to learn, appreciate, and marvel at the beauty, power and reach of mathematics, one should first understand number and shape.
Developing the Concepts and Skills of Numbersense
Improving numbersense in children involves mastering the component skills and at the same time developing the ability to integrate them. Integration takes place when they apply skills in meaningful situations in efficient ways. Mastering component skills means developing
(a) Number concept,
(b) Arithmetic facts, and
(c) Place value.
Mastery of any mathematics concept, skill, or procedure means that the child has (a) understanding, (b) efficient strategies for arriving at it, (c) fluency (i.e., an arithmetic fact should be answered in 2 seconds or less orally and 3 seconds or less in writing), and, (d) can apply it in problem solving contextually.
To achieve fluency in reading with comprehension, a child needs to acquire a set of concepts and skills:
- Mastering the alphabet,
- Acquiring a large collection of sight words,
- Understanding and acquiring phonemic awareness,
- Relying on sight words and phonemic awareness (learning to decode an unfamiliar word by chunking it into familiar, manageable sounds, and, then blending these sounds them into reading that word).
This process is successfully achieved when a knowledgeable and sympathetic adult helps the child to practice the component skills.
In the early stages of the reading process the mastery of two key component concepts—a robust sight vocabulary and understanding of phonemic awareness, practiced with adults’ guidance help children to become independent readers.
Similarly, in achieving early fluency in numbersense, the key concepts/skills relate to number concept are:
Numberness: This is the process of integrating by
a. Identifying a collection of objects (e.g., a cluster of objects) by visually scanning it,
b. Associating the collection to an orthographic image, and,
c. Calling the name of the orthographic image and the collection by the name of the number. Essentially, it means assigning a symbol to the quantity represented by the cluster of objects.
Decomposition/recomposition: This means seeing a number as made up of component smaller numbers (e.g., visually recognizing a cluster of objects as a union of sub-clusters and vice-versa—the cluster of five objects contains in it a cluster of three objects and two objects), and
Sight Facts: Using the visual decomposition/recomposition process one sees a number is made up of two smaller numbers (i.e., 5 is 2 and 3). A sight fact is like a sight word. These are called sight facts as the fluency of these facts is arrived by constant visual exposures just as in sight words. Thus, one achieves the mastery of 45 sight facts (See Figure 3).
By the help of numberness, sight facts, making ten, teens’ numbers, and decomposition/ recomposition, one can develop any addition fact efficiently and effectively and then with usage one can master it.
For example, to derive the addition fact: 8 + 6
8 + 6 = 8 + 2 + 4 (using sight facts of 8 and then of 10)
= 10 + 4 = 14 (knowing the teens numbers); or,
8 + 6 = 4 + 4 + 6 (using the sight facts of 8 and then of 10)
= 4 + 10 = 14 (knowing the teens numbers), or,
8 + 6 = 2 + 6 + 6 (knowing sight facts of 8)
= 2 + 12 (knowing double of 6)
= 14 (knowing sight facts of 4, and teens numbers); or
8 + 6 = 8 + 8 –2 (knowing sight facts of 8)
= 16 –2 (knowing doubles of 8)
= 14 (knowing teens numbers); or
8 + 6 = 7 + 1 + 6 (knowing sight facts of 8)
= 7 + 7 (knowing doubles of 7)
= 14 (knowing double of 7).
It is important that children derive these facts in several ways to develop flexibility of thought, fluency, applicability, and deeper understanding.
Similarly, one can derive a subtraction fact: 17 – 9 = 10 + 7 – 9 = 1 + 7 = 8; etc.
Developing the Concept of “Numberness”
When we look at the following visual cluster card representing 5, we can see the four sight facts of 5.
By visual observation of the above VC CardTM for number 5, the child forms the image of the number 5 as five one’s (1 + 1 + 1 + 1 + 1), a figure (orthographic representation, 5), as a visual cluster (as in above figure), and its relationship with other numbers (by decomposition/re-composition of the visual cluster) and understands the number 5 as 4 + 1; 3 + 2; 2 + 2 + 1. The integration of these component skills indicates that the child has acquired the concept of numberness (in this particular case the “fiveness”)
Decomposition/Recomposition of Number
Decomposition/recomposition is to numbersense as phonemic awareness and phonological sensitivity is to the reading process. Individuals with an understanding of decomposition/recomposition are able to relate and connect numbers with each other. In the absence of decomposition/recomposition, children use inefficient and laborious strategies like counting one number after the other or both the numbers. With decomposition/recomposition, they can relate numbers better and arrive at novel and efficient strategies.
Children learn decomposition/recomposition by seeing patterns of arrangements of objects as in dominoes, dice, playing cards, Rek-n-Rek, Ten-Frame, etc. However, it is best achieved through the use of Visual Cluster Cards and Cuisenaire rods. For example, breaking (decomposing) the cluster into two sub-clusters of 2 and 3; 1 and 4 and derive the relationships: the four addition facts 5 = 1 + 4 = 2 + 3 = 3 + 2 = 4 + 1 are called sight facts of 5 (See Figure 2).
Just as sight words are learned by visual exposure and repetition, these number relationships are sight facts, which are also taught through visual exposure and oral repetition. The repetition should take place first systematically and then these sight facts should be practiced and recalled at random by asking a range of questions.
1 + 4 = ? 2 + 3 =? 3 + 2 = ? 4 + 1 = ?
5 = 1 + ? 5 = 2 + ? 5 = 3 + ? 5 = 4 + ?
1 + what number = 5. 4 + 1 = ? What two numbers make 5? What number + 3 is 5? Etc.
Once children have mastered the sight facts of a number orally, the teacher should ask them to write them systematically.
1 + 4 = 5, 4 + 1 = 5; 5 = 1 + 4, 5 = 4 + 1
2 + 3 = 5, 3 + 2 = 5; 5 = 2 + 3, 5 = 3 + 2
This process should be repeated for the first ten counting numbers.
Thus knowing a number means an ability to write the number, use it as a count, recognize the visual cluster and its component clusters as smaller numbers that make the number. This is true for all ten numbers. They should be able to see and 45 sight facts for the first ten counting numbers. The 45 addition sight facts are:
Without the idealized visual image of these numbers as clusters and the decomposition/ recomposition process, children have difficulty in developing fluency in number relationships. Most dyscalculics and many underachievers in mathematics have not learned number concept in this proper form.
An effective method of developing the number concept and the sight facts is using Visual Cluster Cards. This process can begin with dominoes, dice, and other such materials that aid in forming these cluster patterns for numbers in the mind’s eye. However, the prolonged use of counting objects delays this automatization process.
Cuisenaire rods and Visual Cluster Cards are efficient tools for developing, extending, and reinforcing the decomposition/ recomposition of numbers achieved through the color and length of the C-rods and visual cluster patterns respectively. Using Cuisenaire rods, for example, the number 10 can be shown as the combinations of two numbers as follows.
The above arrangement can be summarized into the 9 sight facts of number 10 using decomposition/recomposition.
10 = 9 + 1 = 1 + 9
10 = 8 + 2 = 2 + 8
10 = 7 + 3 = 3 + 7
10 = 4 + 6 = 6 + 4
10 = 5 + 5
The same process is used for finding the sight facts for other numbers: 2, 3, 4, 5, 6, 7, 8, and 9; sight facts relate to only these numbers.
Once children have formed these combinations, the teacher helps them to make these combinations fluent. Both Visual Cluster Cards and Cuisenaire rods help children to create visual images of these decompositions and help in acquiring fluency. Number concept is the beginning of the development of numbersense.
How to Begin Teaching Subtraction Sight Facts
Once children have mastered the addition sight facts, they can easily learn the related subtraction sight facts. For example, beginning with the visual image of a number as in Figure 1 (here the number is 5) and then using the process presented in Figure 2, one can derive the sight facts related to number 5. The teacher shows the card representing number 5 (Figure 1).
Teacher: Look at the number of objects on this card?
Then, she covers a sub-cluster of the cluster of five objects. (See Figure 2).
Teacher: I hid some objects on the cards. How many pips on the card are hiding?
Teacher: Great! Look at the card, now. How many objects are showing?
Teacher: We will read this as: 5 take away 2 is 3. We write this fact as: 5—2 = 3.
She repeats this process for other sub-clusters of 5 and derives the subtraction facts of 5. These are:
5 – 1 = 4, 5—4 = 1; 5 – 2 = 3, 5—3 = 4.
The process is repeated for other numbers and all the subtraction sight facts are mastered with practice.
 See an earlier post on Sight Facts and Sight Words on this blog and also Sharma (2015) Numbersense a Window to Dyscalculia in The International Handbook on Dyscalculia (Steve Chinn-Editor)
 Visual Cluster CardsTM is a deck of sixty cards representing the ten counting numbers in multiple forms of clusters in four suites (club, spade, diamond, and heart). VCCs are used for teaching and learning number facts and later to learn the concept and operations on integers. They are used for playing Number Games for (Number, Addition, Subtraction, Multiplication, Division, Integer, and Algebra Wars). Through these number games, students learn and reinforce arithmetic facts and achieve fluency.