Number sense is a skill that needs to be developed from a young age. An example of this would be displaying a concrete representation to a prep or year 1 student (see image below):
A simple problem for a person with a developed number sense. We could use subitising (sight recognition of quantity - explained later on) to recognise the quantity. However, a prep or year 1 student may count on their fingers to derive the total, or may already have the ability to instantaneously recognise the total. This is a process of early number sense. Gelman and Gallistel (1992) state that children as young as two years can learn to instantaneously recognise groups of two and three objects, and that most children by the age of four are capable of instantaneously recognising groups of four objects.
Recognising, identifying and writing numerals is a skill students must develop at an early age. This will form the basis of early number sense. Students will only develop recognition and an understanding of numerals with practice. In discussions with peers through tutorial activities, we looked at appropriate learning activities to help prep to year 2 students develop this skill. A process we could all agree would benefit students greatly was: recognise - teacher selects a named numeral from a randomly arranged group of displayed numbers; identify - students are to state the name of the displayed number; writing - students practice writing the numeral, focusing on numeral shape. This process will assist students in matching the symbol, to the name, to the shape. Teachers can use number charts, like the one below, to assist students in visualising the concrete amount for each numeral. Like our own classroom, a number line similar to this presented in clear view of all students can operate as a point of reference for all students.
There are four different types of numbers that must be recognised and developed through the early years. They are: cardinal numbers, ordinal numbers, whole numbers and integers. Numbers can be categorised according to use (Origo Education, 2007). Teachers may use the following activities to demonstrate to students the differences between each. Cardinal numbers are primarily used to count the number of objects in a set. Teacher could present the below image of a dog to the students, and ask them to count the number of legs. Using cardinal numbers, the student can use the descriptive statement "the dog has four legs". Zero (0) is given as the value of a set with no objects.
Ordinal numbers are used to indicate the position of an object in a numerical sequence. Teacher and students could look at Olympic swimming races to determine positions of each swimmer. Students would give the position order 'first, second, third ... eighth' to each swimmer depending on how they finished in the race. In discussions with peers, we decided that ordinal numbers could best be practiced after inter school sports carnivals. Students can use real life experiences to relate back to the content.
Whole numbers are zero and all counting numbers that don't include fractions or decimals. Integers are the whole unit distance between numbers on a number line. Teachers could demonstrate this by using whole number number lines. The below image (see example 1) illustrates a number line using whole numbers, increasing by an integer of 1. Activities incorporating number lines will develop a student's understanding of whole numbers, integers and multiples (see example 2).
When an understanding of numerals has been acquired by the students, teachers will begin to teach the process of reciting numbers in a particular order, or counting. Gelman and Gallistel (1992) states that counting involves a number of different principles. The most common: one-to-one principle, stable-order principle, cardinal principle, abstraction principle and order-irrelevance principle. Teachers may use the following learning experiences to develop the students ability to count.
One-to-one principle: When counting, only one number word is assigned to each object (Designed Instruction, 2010). Teacher may set an activity that sees students counting equal items for eating dinner. Students are split into small groups, and receive even numbers of plastic plates, cups, napkins and bowls. Counting will be required to ensure each student receives the same number of each item.
Stable-order principle: When counting, number words are always assigned in the same order (Designed Instruction, 2010). Students can use this principle when learning multiples. When counting in 2's, they can use the sequence 'two, four six, eight, ten ...' everytime they count.
Cardinal principle: Having knowledge that the last number name said describes the total quantity. In discussions in tutorials, we were advised that students at a young age may not understand this concept. The image below cleary illustrates 5 banana's. Asking a student without a developed concept of the cardinal principle "How many banana's altogether?", they would count each individual banana 'one, two, three, four, five'. Ask them again, you would get the answer 'one, two, three, four, five'. By developing this principle, students will begin to recognise that the number of banana's is in fact 5, which is the same as the last number stated in the counting process.
Order-irrelevance principle: When counting the number of objects in a set, the order they are counted is irrelevant, as long as each object is counted. Students could be assigned a task to count the number of buttons in a pile. As they are given no instruction to place the buttons in a particular order (rows, smaller groups etc) students may use any button to count, so long as all buttons are counted, and a total can be given. In discussion with peers, this method of counting may appear difficult for students beginning the counting process as no instruction needs to be given as to how the objects need to be counted.
Abstraction principle: Students realise that numbers can be used to count anything. The previous 4 principles apply to the abstraction principle. Similarly to the activity used in the order-irrelevant principle, the teacher can give students the same pile of buttons each, and ask them to count the total number using any counting method they desire. This will give students the opportunity to either: count each individual button, count in multiples (2's, 3's etc if they have the ability) or placing the buttons into rows or smaller groups of even numbers and counting the groups as the end (principle of multiplication that may be too advanced for younger years).
Online programs such as Five Little Ducks or Ten Fat Sausages can assist in promoting the development of number concepts. They use song to explore counting back.
Five Little Ducks Ten Fat Sausages
Another aspect of early number sense is subitising. As mentioned previously, subitising refers to the rapid, accurate, and confident sight recognition of quantity (Kaufman, 1949). As experienced adults, we easily recognise the number of dots on a dice face, or total marks in a tally. To assist in the development of this skill, we can play games or run activities for students in prep to year 2 that would be appropriate learning experiences to gain familiarity with this sense. Games that include dice, dominos, tally markings, or playing cards can assist students with sight recognition of quantity. In my own learning experiences from tutorial tasks, I have witnessed the benefit this skill can have on the learning process. As a mature adult with a developed number sense, sight recognition comes naturally.
Once students have some understanding of number sense, as teachers, we can develop number concepts with the use of the beginning processes.
Recognising, identifying and writing numerals is a skill students must develop at an early age. This will form the basis of early number sense. Students will only develop recognition and an understanding of numerals with practice. In discussions with peers through tutorial activities, we looked at appropriate learning activities to help prep to year 2 students develop this skill. A process we could all agree would benefit students greatly was: recognise - teacher selects a named numeral from a randomly arranged group of displayed numbers; identify - students are to state the name of the displayed number; writing - students practice writing the numeral, focusing on numeral shape. This process will assist students in matching the symbol, to the name, to the shape. Teachers can use number charts, like the one below, to assist students in visualising the concrete amount for each numeral. Like our own classroom, a number line similar to this presented in clear view of all students can operate as a point of reference for all students.
Whole numbers are zero and all counting numbers that don't include fractions or decimals. Integers are the whole unit distance between numbers on a number line. Teachers could demonstrate this by using whole number number lines. The below image (see example 1) illustrates a number line using whole numbers, increasing by an integer of 1. Activities incorporating number lines will develop a student's understanding of whole numbers, integers and multiples (see example 2).
One-to-one principle: When counting, only one number word is assigned to each object (Designed Instruction, 2010). Teacher may set an activity that sees students counting equal items for eating dinner. Students are split into small groups, and receive even numbers of plastic plates, cups, napkins and bowls. Counting will be required to ensure each student receives the same number of each item.
Stable-order principle: When counting, number words are always assigned in the same order (Designed Instruction, 2010). Students can use this principle when learning multiples. When counting in 2's, they can use the sequence 'two, four six, eight, ten ...' everytime they count.
Cardinal principle: Having knowledge that the last number name said describes the total quantity. In discussions in tutorials, we were advised that students at a young age may not understand this concept. The image below cleary illustrates 5 banana's. Asking a student without a developed concept of the cardinal principle "How many banana's altogether?", they would count each individual banana 'one, two, three, four, five'. Ask them again, you would get the answer 'one, two, three, four, five'. By developing this principle, students will begin to recognise that the number of banana's is in fact 5, which is the same as the last number stated in the counting process.
Abstraction principle: Students realise that numbers can be used to count anything. The previous 4 principles apply to the abstraction principle. Similarly to the activity used in the order-irrelevant principle, the teacher can give students the same pile of buttons each, and ask them to count the total number using any counting method they desire. This will give students the opportunity to either: count each individual button, count in multiples (2's, 3's etc if they have the ability) or placing the buttons into rows or smaller groups of even numbers and counting the groups as the end (principle of multiplication that may be too advanced for younger years).
Online programs such as Five Little Ducks or Ten Fat Sausages can assist in promoting the development of number concepts. They use song to explore counting back.
Another aspect of early number sense is subitising. As mentioned previously, subitising refers to the rapid, accurate, and confident sight recognition of quantity (Kaufman, 1949). As experienced adults, we easily recognise the number of dots on a dice face, or total marks in a tally. To assist in the development of this skill, we can play games or run activities for students in prep to year 2 that would be appropriate learning experiences to gain familiarity with this sense. Games that include dice, dominos, tally markings, or playing cards can assist students with sight recognition of quantity. In my own learning experiences from tutorial tasks, I have witnessed the benefit this skill can have on the learning process. As a mature adult with a developed number sense, sight recognition comes naturally.
Once students have some understanding of number sense, as teachers, we can develop number concepts with the use of the beginning processes.
An example of this would be utilising the process of sorting. Previously, we discovered that sorting refers to grouping objects in accordance to one or more attributes. Students could use their developed number sense to sort a pile of buttons into small groups, based on the number of holes through the middle. A scenario may involve a teacher explaining to the students that there are 16 buttons in each pile, and that sorting them by hole number will create 4 even groups (see image below). Students will understand that the correct answer to the scenario will leave them with 4 groups of buttons, with even numbers in each. Students with a developed numbers sense may start asking questions that relate to basic multiplication or division (4x4=16, or 16/4=4).
