Strengthening Number Sense 1

We have seen that Number Sense is a complex & very critical skill. Having a sound number sense means acquiring a variety of skills associated with numbers, their properties, relations & operations.

Hilde Howden in an article in 1989 in Arithmetic Teacher gives a very comprehensive definition of number sense. “Number sense can be described as good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of con- texts, and relating them in ways that are not limited by traditional algorithms. Since textbooks are limited to paper-and-pencil orientation, they can only suggest ideas to be investigated, they cannot replace the "doing of mathematics" that is essential for the development of number sense. No substitute exists for a skillful teacher and an environment that fosters curiosity and exploration at all grade levels”

We give below some other activities, which can be done with students at various class levels, to deepen number sense

1. Seeing/ Recognising Numbers (without the need to count)

   1. Show students a pattern of dots for a few seconds and ask them to identify the number

   2. The dots can be either random or in a pattern to aid recognition. Some examples are ten grapes arranged as a bunch, dot patterns on dice etc.

   3. The idea can also be done with coloured patterns to represent fractions or decimals. (3/4 as 3 squares coloured red out of a 2 by 2 square pattern

2. Math Chats & Discussions

   1. The idea is to encourage students to use mathematics in their casual conversations. For example, a student can say “I went to a shop and purchased a shirt worth Rs 430”. Other should continue with the idea put forward by the first student. It could be a comment or a math problem related to the first comment. This can be continued with teacher helping out where necessary. An example could be “How did you pay for it”. The discussion could be about modes of payment or about shirt sizes.

   2. Through chats, they realise that math plays an important role in understanding and interpreting a wide range of their life experience. It also promotes a familiarity with math ideas and flexibility while thinking about them.

   3. Every time we ask a student to articulate his/ her thinking we enable them make new connections or crystallize their ideas just by trying to put them into words or trying to put them into writing.

   4. Math discussions can also explore various strategies to solve the same problem and probe the reasoning behind a strategy,

   5. All the other activities suggested here can also become the basis for math chats.

   6. Ideally math chats should be done as an “ice breaker” activity at the beginning of the class.

3. Identifying number patterns in a structured grid

   1. 1 to 100 in a 10x10 grid. Horizontal rows give +1/-1 relations. Vertical rows give +10/-10 relations. Diagonal cells can be interpreted as other patterns

   2. Multiplication tables from 0 to 10 arranged in a 10X10 grid. Diagonal shows Squares. All products are arranged symmetrically about the diagonal.

4. Writing visual patterns with numbers

   1. A “5” in the dice as 2 + 1 + 2

   2. A bunch of 10 grapes as 4 + 3 + 2 + 1

5. Skip Counting –

   1. Starting at various numbers (not only 0, or tens or hundreds) including mixed fractions (4 ¾) and decimals (78.3)

   2. In increasing or decreasing steps of - Whole numbers, fractions (1/2 , ¼ etc) & decimals ( by tenths (0.1) by hundredths (0.01)). Can be a whole class activity passing from student to student

   3. As a whole class or group activity passing from student to student. If the skip counted numbers are written on the board in certain tabular forms, the patterns in the sequences can easily be brought out.

   4. If there are a fixed number of students, then the ending number can be estimated.

6. Partitioning a number/fraction in as many ways as possible

   1. 6 as 1 + 5 or 2 + 4 or 3 + 2 + 1 etc

7. Open-Ended Problems – Pose problems that could have multiple answers, rather than just one solution. For example – give me two numbers whose total is 11. Give me 2 numbers whose product is 36.

Estimations – Have assessment papers which only require estimations. Hence the time given could be less than for a regular test.

Making a number with other numbers & operations– without and with restrictions

1. Paying a certain amount using currency and coins. Pay Rs 100 using both Rs 20 notes and Rs 5 coins

2. Make 100 in any way you want (40 + 50 + 2 + 8)

3. Make 125 using just 3 numbers (40 + 50 + 35)

4. Make 100 using 3 numbers, one addition and one multiplication ((5 + 15)*5)

5. Make 0.03 in any way you want (0.01 + 0.02)

6. Make 0.3 using 2 numbers and one division (3 / 10)

7. Make 0.3 using 2 numbers and one multiplication (3 X 0.1)

In the higher classes, as children start encountering more sophisticated numbers & representations. Multi-digit numbers (using place value), fractions, decimals, numbers with non-decimal bases and integers are some examples. For each of these representations, the nature of the number sense varies (52, 5 + 2, 5*2, 5/2, 0.52 all have to be interpreted in unique ways). The idea of number sense has to be revisited periodically to anchor their understandings in their prior knowledge as well as prepare them for concepts that they still need to learn. The next chapter we would deal with some of these activities.