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We talked about Number Sense in chapter 4.2 as the capability which helped humans to invent numbers and create the discipline of mathematics.
We also saw that as new kinds of numbers & operations are introduced at various stages of school, flexibility of thinking about them also has to practiced. Number Sense is also about becoming fluent and flexible in handling & using new kinds of numbers and their operations.
Some of the aspects of a “strong number sense” which have been identified by mathematicians and educators are listed below. Many of them may overlap with others as various areas of math itself are interconnected. So they are just to be seen as guidelines. Each person should adopt processes which they are comfortable with.
Visualising Numbers
Using the Place Value System - Using the bundles/sticks or the sheet/strip/piece representation
Geometrically - Prime Numbers as lines, square numbers as squares, composite numbers as rectangles
Understanding numbers in various representations
Meaning & value of 328, ½ , √3, , 4.05 etc
Relationship between numbers
If 32 is half of 64 then 64 is double of 32
If 8 X ½ =4, then 8÷1/2 will be 16
Understanding & comparing magnitude of numbers
¼ is more than 0.2. (25>20)
Counting in 2s takes a longer time than counting in 3s to reach from a given number to another
Marking a number line evenly with numbers, where only the starting and ending numbers have been given
Using numbers & operations flexibly
Calculate 46 + 25 as 46 + 4 + 21 = 50 + 21 = 71
18 X 9 as 20X9 – 2X9 or 18 X10 – 1X18
Represent numbers flexibly depending on the context
48 as 50-2 or 40+8 or 45+3
50 + 48 can also be seen as 50 + 50 -2 = 100-2 =98
Extend operations
If 7 X 8 = 56, what would be 7 X9?, 8 X 9?
Use relations between the end numbers when multiplied by certain numbers. This will help in figuring out factors of a number.
Use relations between end numbers when squared
Note that both the end numbers total to 10! (1 + 9, 2 + 8, 3 + 7, 4 + 6)
Make reasonable estimates & spot unreasonable answers
Have a feel for number and see that 567 + 328 must be something close to 900
Sense that the problem 23 + 38 = 511 is wrong since it has to be around 60
Use connections between operations
45 +58 ->45 +60-2 -> 105-2 ->103
36 X 5 -> 360 ÷ 2-> 180
Use numbers in thinking
Quantify thinking and guessing
Exercises in Mental Mathematics
The topic of mental mathematics is wrongly seen by most as an exercise in memorisation where the actual process is carried out in the mind instead of paper & pencil. It is actually meant for developing strategies as listed above for developing fluency.
Ability to do math mentally. Mental math is not performing the operations imagining a piece of paper and a pen. It is the ability to visualise both numbers & operations on them.
Estimation& Rounding Off
Estimation is a very important skill in daily life for which a good number sense is necessary.
It is a tool to detect possible errors in calculations and to check for reasonable answers. This is a vital set of skills for all students.
When we go shopping with cash, we need to have an understanding of what we need to buy, the approximate costs and the total amount of money involved.
When we plan a party, we need to estimate what & how much food & ingredients we need to buy. There should neither be a shortage of food or a waste of food.
When we go to a shop (which normally will not give a printed receipt) and buy a variety of items, we need to estimate the balance that we should be receiving.
A good number sense will help us make fairly good estimates.
Rounding a number to the nearest 5s or 10s is one of the steps in estimation. The numbers become easy to operate on and arrive at approximate answers. For example, 48 X 53 can be approximated to 50X50 which is easy to estimate as around 2500.
Chapter 13.3 gives the various patterns that can be identified in a collection of 36 tokens.
Chapter 13.4 gives various ways in which a multiplication sum 18 X 5 can be visualised and computed.
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