Number Sense

Children have the ability to "perceive" small numbers holistically without any counting. This is the starting stage of what is known as Number Sense.

The term was popularized by Stanislas Dehaene in his 1997 book "The Number Sense."

Multiple Meanings of the word Number

We need to remember that the word number has several meanings – the numeral, number representation and the quantity. In many cases, the meaning of the term "number" has to be figured out from the context in which it has been used. Hence it is better to clarify to children the exact meaning in which it is being used. It is a good idea to use the word numerosity to indicate the idea of quantity.

Neuroscience research is showing that the part of the brain which is related to math, actually has two parts; one which judges how much and the other which judges how many. Dr B Butterworth, author of “The Mathematical Brain” suggests that there seems to be a “numerosity module” or the "how many" area in our brain specialised for dealing with numerical representations.

Number Sense starts from the "how many" area. In evolutionary terms, the sensing “how much” has been more important than sensing “how many”. So the sense of “how many” has grown very slowly.

In the last few decades, anthropologists, have come across remote tribes, (like Piraha in the Amazon basin & Walpiri in Australia) who do not have a sense of numerical quantity beyond two. Their social & economic life seems to find the ideas one, two & many sufficient for their needs.

At early ages, number sense is the ability to mentally connect a small numerical quantity (a collection of pencils) with a number (say Three). This is not an easy idea for children. In their daily experiences, they are more likely to hear words like big, large & long more frequently than words like four or five. Hence the idea of a "measurable" quantity (how much sugar?) is easier for them to grasp than "countable" quantity (how many mangoes?). They can easily perceive the difference in quantity between 2 bags of rice. But the difference between 2 collections, particularly if they are not sufficiently different (basket of 5 apples vs basket of 6 apples), is difficult.

There are no words other than number words, as well as many & few (four, ten etc.) in a language to describe the idea of numerical quantity. Words like "less" and "more" are used in both counting and measuring situations. We use the word "number" to indicate a number by itself as well as the related quantity. There is no other way of mathematically describing a collection of mangoes except by using the word like "five". In contrast a quantity of rice can be described using various phrases as "a cup full" or "two handfuls" etc.

There are other difficulties when viewing a collection of say fruits. Piaget has shown that the same number of fruits when spread over a larger area is mistaken by children as "more". Also a collection of five mangoes appears more than a collection of berries. Hence numerical quantity is not a concrete quality like colour of shape which can be perceived or pointed out directly. On the other hand, colour (show me a green fruit) or shape (show me round fruits) is easy to directly point out and see.

From "measuring" context to "counting" context

Early in their life, children are more exposed to the idea of quantity in a "measuring" context. They need to be slowly introduced to the "counting" context. Number Sense is the development of this idea.

Due to reasons detailed above, the development of number sense in children takes a longer time. They acquire it over a period of time when they hear number words being used in contexts (Bring two plates, show 2 fingers) repeatedly. They also develop visual pictures or patterns of numbers. In homes where such interactions could be less, the development of number sense could take a longer time or may be less sophisticated. Hence children come to school with varying degrees of understanding of number sense.

But the discipline of math has been developed by a handful of highly intelligent mathematicians, very rapidly to very abstract levels, starting from the idea of “how many”. It has gone far ahead of the “how many” ability of humans.

Scientists & educators confirm that a sound foundation of number sense is very essential for mastering complex mathematical concepts. It enables us to master abstract concepts like numbers & their relationships to be stored & manipulated in the brain as visual images (both concrete and abstract). It provides for flexibility and fluency while thinking and working with numbers.

In pre-school, teachers generally do not realise the difficulty children have in understanding the idea of numbers. This could be because of 2 reasons.

1. Their syllabus focuses more on rote-learning and memorising number shapes & words and not much on understanding what numbers mean. If we think about it, the numeral "3" does not look anything like the idea of "three" for a child who is just getting exposed to the idea.

2. Most children have already acquired a rudimentary number sense at home and play before coming to school. Hence they manage to make the connection between numerals and numbers over a period of time.

But the reality is that children come to school with varying degrees of number sense due to the unevenness of their number experiences at home. There are quite a few students without a robust sense of numbers. These children find math very difficult and start lagging behind in understanding math. Hence incomprehension of math starts from the early classes.

Hence teachers in pre-school need to find out the level of number sense among their students. They need to build on these rudimentary ideas and build a robust number sense by providing children many situations involving concrete objects & life experiences. We will deal with such activities in the next chapter.

In one sense, if a child is able to visually point out that a basket of 5 oranges is "less" than a basket of 7 peanuts, we can say that the child has grasped the rudiments of number sense.

As children proceed to higher classes, the variety of numbers they encounter & the processes associated with them also become more complex. Hence in higher classes also teachers need to ensure that the number sense of the students is continuously deepened and strengthened. Since numbers are an abstract idea, number sense is also the ability to represent numbers in a variety of ways. We will deal with activities for strengthening number sense in Section 13.

In his article "Number Sense: the most important mathematical concept in 21st Century K-12 education" mathematician Keith Devlin underlines the importance of number sense for mastering concepts of math and as a corollary for 21st century K-12 education.

The article can be accessed here.

A recent groundbreaking study (quoted by Jo Boler) by Eddie Gray and David Tall, two British researchers found an important difference between low and high achieving mathematics students that they observed. The high achieving students consistently approached mathematics problems using what is known as number sense – they interacted with the numbers flexibly and conceptually. By contrast, the low achieving students used no number sense and seemed to believe that their role was to recall and use a standard method even when this was difficult to do.

Dyscalculia is a recently discovered learning disability which makes it difficult for children to understand numbers and numerical processes. Research is indicating that one of the primary reasons for this is the inability to develop a strong number sense at an early age.