Mathematics Developmental Continuum
Counting up to 20 Objects: 1.0
Supporting materials
Indicator of Progress
Students count the number of objects in a set of up to 20 objects. Success depends on students knowing the conventional verbal sequence up to at least 20 and being able to use the principles of counting which they have learned for counting up to 10 objects.
A major difficulty is caused by the irregularity of English number names in the teens. Some students confuse the 'teen' numbers with the 'ty' words when speaking and reverse digits when writing or recording the numbers. Larger numbers of objects in a group present special difficulties for some students as they have difficulty maintaining one-to-one correspondence.
Illustration 1: Teen numbers
When counting aloud by ones from 12 students might say: twelve, thirty, forty, fifty, sixty, seventy, eighty, ninety. There is often a slight hesitation before the student says twenty or one hundred.
Students will frequently write the teen numbers backwards to match how they are said (e.g. writing the 4 first in fourteen to produce 41). Teachers will often see students write the 4 first and then hesitate while thinking about where to place the 1. Even when the two numbers are placed side by side (for example 13, 31) students think they are the same number as they both use the same digits, a 1 and a 3.
Illustration 2: Counting in the teens
Students keep track whilst counting objects in a variety of ways. Some students touch the object; some keep track of the number of objects with their eyes while simultaneously nodding, tapping on their face, leg or the table. This means they are trying to coordinate eyes and head or fingers simultaneously while reciting the number sequence.
Larger numbers of objects in a group present difficulties for some students as they have difficulty maintaining the coordination of the verbal sequence with tracking the objects they are counting (that is, using one-to-one correspondence). This is evidenced when students start waving their hands randomly over the objects while reciting number names without any apparent coordination. Coordination depends on having the number name sequence secure and understanding the need to carefully coordinate.
Illustration 3
Examples of the types of tasks that would be illustrative of counting concepts 0 - 20, aligned from the Mathematics Online Interview:
- Question 1 - Estimate and count teddies scooped into a cup (holding at least 20)
- Question I – ‘First year of schooling detour’ - Selecting and counting 4 yellow, 3 green and 5 blue teddies
- Question III (a) (b) - ‘First year of schooling detour’ - Recognise number of dots on cards and match to the correct numeral
Teaching Strategies
The teaching challenge is to extend students' verbal counting skills and to strengthen the use of one-to-one correspondence for larger numbers of objects in a set. The number names in the teens in English do not follow a logical pattern, so they are hard to learn. In other languages it is much easier.
Activity 1: Verbal sequence involves assisting students to extend their verbal counting sequence especially the knowledge of the teen numbers.
Activity 2: Efficient counting strategies focuses on more efficient counting strategies, helping students use one-to-one correspondence for larger groups of objects.
Activity 3: Hidden collections assists students to use one-to-one correspondence with hidden groups of objects.
Activity 1: Verbal sequence
Reading and naming numbers
Provide students with a row of numbers on a strip of paper or a set of cards marked 0 - 20 placed in a line (Number Cards can be downloaded below). Initially encourage students to place their fingers on the number as they are saying the number sequence. Start and stop at various points. Particular emphasis needs to be focused on the teen numbers.
- Number Cards (PDF - 11Kb) - PDF document with numbers 1-20 to be presented as a strip or cut into separate numbers to make number cards.
What's missing?
'What's missing?' is an activity that can be used to reinforce the number sequence using the cards 0 - 20. Initially have the whole sequence in order. Ask the student to look away while you turn one card face down. Ask the child to say what number they think is missing, then let them check. This can be changed so that only part of the sequence is used with one, two, or three cards turned face down. Again emphasise the teen numbers.
Choose your card
'Choose your card' is an activity that means the students can choose their starting number from a range and then continue counting orally by ones until they reach 20 or the teacher stops them. Pay particular attention to the pronunciation of the teen numbers. Numbers like forty and fourteen will often be confused.
Counting together
Take every opportunity to count together and individually, for example
- How many students in our class today?
- How many pencils in the tin?
- How many days until the school concert?
Be sure to listen carefully to individual students as needed.
Activity 2: Efficient counting strategies
While students are successful counting small groups of objects, many need assistance to develop more efficient strategies in keeping track of the objects they are counting.
Ensure that students discuss good strategies for counting larger sets of:
- objects in circles
- objects in piles
- objects they cannot touch
- movements, sounds etc.
Activity 3: Hidden collections
Some students have difficulty with one-to-one correspondence with larger numbers of counters and need to keep recounting the counters they have already counted. Hidden collections is an activity that prevents students continually rechecking the counters they have just counted. Use a money-box type container with a slit in the top. Ask students to count the counters as they are being dropped in the container. After all the counters have been dropped into the tin, ask students to say how many there are and then encourage them to check whether they were correct.
Other languages
The number names in English are irregular in the teens. Other languages name numbers logically and with an emphasis on base ten. For example, in Asian languages, the translations of the names of the teen numbers are as follows:
ten, ten and one, ten and two, ten and three, ten and four, ten and five, ten and six, ten and seven, ten and eight, ten and nine, two tens, two tens and one, etc.
This system stresses the base ten system and is easy to learn.