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Students count the number of objects in a small set. Success depends on several understandings, one of the most important of which is using one-to-one correspondence.

While some students can recite the number sequence accurately (i.e. say 1, 2, 3, etc.) they have difficulty maintaining one-to-one correspondence when counting a set of objects.

For more information, see: Learning to count

To ascertain whether students can use one-to-one correspondence, ask students to count a set of counters you have placed in a pile in front of them. Children not using one-to-one correspondence will not co-ordinate saying the number names with taking the counters one by one. Some will say more than one number per object; others will take more than one object per number.

Show the student two sets of 6 counters arranged as shown below, spread out or in a group. Ask the student which group has more. Successful students will count each of the sets of counters and find that they contain the same number of counters. Some students will look at the two sets and say that the longer row of counters contains more. Even when they have counted both sets, some students will say that both sets have six counters but the top line of six has more. The appearance dominates the students' judgement.

Examples of the types of tasks that would be illustrative of one to one correspondence concepts, aligned from the Mathematics Online Interview:

- Question I - ‘First year of schooling detour’ - Selecting and counting 4 yellow, 3 green, 5 blue teddies
- Question 8 (b) (c) - Read cards from 0-10, show 7 and get that number of teddies

The following activities involve students using all their senses to develop the co-ordination that is essential to one-to-one correspondence. That is, the teaching strategies use movement of the body, hearing sounds, using eyes and feeling with hands.

Activity 1: Counting and body movements provides several ideas to help to use one-to-one correspondence while counting movements using their bodies.

Activity 2: Counting sets of objects provides several ideas to help students to use one-to-one correspondence while counting concrete materials.

Activity 3: Matching the written number to the quantity provides several ideas to help students to use one-to-one correspondence while counting representations of objects.

Activity 4: Variations on counting suggests some very important counting variations.

These activities simultaneously extend students' verbal counting skills and improve the use of one-to-one correspondence.

Note on Subitising

Subitising is the ability to identify the number of objects in a set without counting and when there is no special arrangement (e.g. not in a dice pattern).

Research shows that children as young as two years of age can subitise 2 or 3 objects*. There is quite a lot of evidence that babies and some animals* can do it too.

Adults can subitise about 5 or 6 objects. Test yourself; the objects can be randomly arranged, not in a standard pattern e.g. as on a die.

* of course they may not know the name of the number - but they know how many there are as a quantity!

Keeping track of counting

Students need to track the counting sequence as they are saying and pointing to numerals. The numbers 1 to 10 (and later beyond) can be written on a strip of paper or individual cards (marked 1 - 10) can be placed in order (download below). Students must put their fingers on the written symbols as they are saying the number words. Gradually increase the number sequence.

- 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.

How many movements?

Students count the number of claps the teacher makes. This can be the number of beats on a drum, taps on a triangle. Students count aloud and aim for rhythm.

Can you do this?

Can students make the number of movements given by the teacher e.g. clap three times, hop three times, skip five times, nod six times? Students count aloud as the actions are done.

How many stairs?

A good task for young students is counting the number of stairs in a staircase. They find it extremely difficult to coordinate walking up or down stairs while reciting the verbal sequence in co-ordination. Some students think the number of stairs appears to change depending on whether the students count faster or slower than they step.

Place a group of objects (e.g.: shells, leaves, counters, teddies, boats, cars) on the table. Ask the student to count how many objects there are. Watch carefully and see if you can determine how the student decides how many objects there are.

- Does the student give an instantaneous response? Was this response correct or incorrect? Can they explain how they worked out how many objects there were?
- Does the student touch each object as they count?
- Does the student move each object as they count?
- Does the student track the objects with their eyes while touching their face, tapping their leg or under the table as they count?

These behaviours demonstrate that students have an understanding of one-to-one correspondence.

Shuffle the Number cards (PDF - 11Kb) and ask the students to choose a card.

- Can they place that number of objects next to the card?
- Can they draw that many objects?

Can students predict how many objects there are if

- one more is added to the group?
- two more are added to the group?
- if one object was taken away?
- if two objects were taken away?

Introduce the counting of pictures of objects on a printed page. How do students keep track of the objects they have counted?

- Does the student look at the objects and give an instant answer (i.e.: can they subitise)?
- Does the student touch each picture?
- How does the student check that they have counted every object only once?

Counting objects in a row is not all there is to counting. Students need to develop good strategies for keeping track of what objects they have counted in many other situations. This is an important skill that can be discussed in class.

Counting objects in a circle.

How do you remember where you started? (e.g.: "I keep a finger of my left hand on the starting object and point at the other objects in one-to-one correspondence with my right hand").

Counting a pile of objects.

How do you remember which ones you have counted? (e.g.: "I move them across to the other side of the table one by one").

Counting moving objects.

How do you know which ones you have counted? Very hard, and it is important to recognise that this will be a difficulty. (When a student recognises that this task will be difficult for them, they are demonstrating metacognition).

Kalmus, H. (1964) Animals as Mathematicians, Nature 202, 1156 - 1160

Gelman, R. & Gallistel, C.R., (1978) The Counting Model. In *The Child's Understanding of Number. (pp 73-82)* Cambridge Ma: Harvard

University Press.

Reys, R., Suydam, M. & Lindquist, M. (1984) *Helping children learn mathematics*. New Jersey. Prentice-Hall