Counting on counting

Learning quantities by instant recognition (subitising) is supported by literature as an ideal starting point. Research shows that three-year-old children can immediately recognise the greater of two sets in the range 1-4 if it matters to them. Some psychologists believe babies note changes to quantities in the number range 1–3. All number-based lessons for young students should include grouping activities to build up knowledge of number facts and later place value structures. A little practice every day, in an environment of risk-taking, can greatly enhance students’ factual knowledge and make the transition away from one-by-one counting easier when the time is right.

This session utilises a range of materials when looking at five-based grouping: a slavonic abacus, five-based tens frames, unifix (or similar) cubes, and students’ hands. Connecting two of these representations at a time is a powerful way to develop grouping knowledge. The examples below illustrate grouping tasks in the number range 5–10, but the tasks are transferable to smaller and larger quantities.

1.

Finger pattern pairs

An important connection is between the parts that make ten. If a student knows that in 7 + ? = 10 the missing number is three, then they may transfer that fact to the answer to the problem 3 + ? = 10.

For example:

• Show me seven fingers.
• How many more fingers make ten? How many fingers are you holding down?
• Write 7 + 3 =10 and say, “Seven plus three equals ten.”
• Show me three fingers.
• How many more fingers make ten? How many fingers are you holding down?

Variations

Students work in pairs. One student makes a number up to ten with their fingers. The other says the number and writes the numeral for it in the air.

Find different ways to make a number equal ten; for example, seven can be 5 + 2, 3 + 4, 1 + 6, and 0 + 7.

Finding ways to make numbers between 10 and 20 is possible in pairs or threes.

2.

Slavonic abacus and finger patterns

The Slavonic abacus is five-based. The purpose of the colouring is to enable instant recognition of a quantity without counting. Try not to use counting to confirm a quantity, as that is counterproductive to the intention of either knowing the quantity or working it out from known facts.

For example:

• Make a number in the range of five to 10 on the top row. Can you move all of the counters at once instead of one counter at a time?
• Show me that many fingers. Note that this gives all students time to work out an answer, and it also provides a way for you to see what each student is thinking.
• How many beads are there?
• How did you know there were eight?

Encourage grouping-based strategies, such as:

• I can see five and three.

and

• There are two missing from ten, so I held two fingers down.

Variations

Ask the students to convince a partner about how many beads have moved.

Ask the students to write the number for the beads on a mini whiteboard, then show you.

Move to “ten and” groupings such as ten and four to develop teen number knowledge. Students work in pairs to show that many fingers or write the number on a mini whiteboard.

3.

Five-based tens frames

Hold up a single tens frame, such as nine, for no longer than one or two seconds. The aim is for students to image the five-based patterns rather than count the dots one by one.

• How many dots did you see?
• Show me that number on your fingers.
• Write that number big in the air for me.

Discuss the structure that students saw. For example:

• I saw five and four.
• I saw one missing from ten.
• I saw three threes.

Variations

Play Tens frame Flash in pairs or threes. Players take turns to be the "flash" and show the tens frames, with the other students stating the number of dots on each tens frame as quickly as possible.

Instead of writing the number, talk to a partner about what you saw.

Write what is found with symbols like 8 + 2 = 10, 10 - 2 = 8.

Progress to showing two tens frames. Start with numbers less than five, for example, four and three. Move to ten and another tens frame for teen numbers, for example, ten and six. Try "close to ten" frames, like nine and eight, with another tens frame, for example, nine and five.

4.

Cube stack

Begin with a stack of ten cubes made from five of two colours. Like the slavonic abacus, the colours are used to support non-counting methods to establish quantity. Like the tens frame and abacus activities, students can match a quantity you hold up using their fingers or by writing the number. However, finding a missing part encourages part-part-whole knowledge.

Show the students a stack of cubes with some missing (put into your pocket). Show the stack for a second or two, then hide it.

Ask,

• How many cubes did you see?
• How do you know?
• How many did I put in my pocket?

Praise risk-taking even if the answers are incorrect, and try to offer knowledge that might be helpful. For example:

• I think there are eight because I saw five and two.
• Good work. This would be eight (showing five and three). Can you fix it?

After the students have found the missing part, reveal it from your pocket to check.

Variations

Students play in pairs, with one being the hider and the other the estimator.

Start with a number other than ten, for example, eight stacks made of five and three.

Students match stacks to finger patterns to help them find the number of missing cubes.

Students write equations for the stack problems, for example, 7 + ? = 10.

Progress to taking some cubes from each end. Progress to using two stacks of ten to start, depending on the number knowledge of the students.

Students need to simultaneously develop proficiency with number sequences going forwards and backwards by one and their capacity to apply those sequences to counting tasks. Ideally, students’ ability to say word sequences develops either ahead of or alongside their need to apply it. Therefore, students who can count collections up to ten should be learning number sequences beyond ten.

1.

Hundreds board

After practicing forwards and backwards counting by ones on a hundreds board, ask questions like:

• Can you find the number 8 without counting up or down?
• What number comes after eight?
• What number comes before eight?

Note that this activity practices numeral-to-word connection. Build collections alongside each other to get a three-way connection that includes materials.

2.

Slavonic abacus

A Slavonic abacus can help develop quantity-to-word connections and quantity-to-number connections if used alongside a hundreds board.

Say the numbers out loud with the students as you move the beads.

Note that when practicing the backward number sequence, it is the amount that remains, not the bead removed, that is counted. Zero is an important number to say at the end as an expression of the absence of quantity (no beads).

3.

Frogs in a bucket

Link the numbers before and after a given number to adding one and subtracting one from a given collection. Use toy animals or other objects and a plastic container so there is a loud "plunk" as objects go into the container. The Counting on counting frog animation gives an example of this with frogs in a bucket.

• We had 15 frogs, and one jumped out.
• How many frogs are in the bucket now?

Increase the number of objects in the container beyond ten so that students attend to the one more/one less principle rather than image actions inside the container. Avoid putting in the objects one at a time after ten. Throw them in as imaginary groups. For example, a nice sequence is

• 8 frogs and one more; 17 frogs and one more; 29 frogs and one more; 99 frogs and one more
• 7 frogs and one less; 15 frogs and one less; 27 frogs and one less...

Extend the problems to two more/less, three more/less, and beyond as students’ understanding and control of sequences grows. Where needed, link explicitly to the number sequence by referring to a hundreds board.

Jenny Young-Loveridge established long ago that students find counting a collection given to them easier than forming a collection of their own for the same number. “Counting before forming” does not mean a strict order for instruction; it is just an indicator of relative difficulty. Through experience, we want young students to consider

• What can be changed without altering the count?

4.

Changing the size, shape, and colour (attributes) of the objects

Begin with a set of objects in a readily subitised arrangement. Sets of plastic animals, cars, fruit, etc. are cheap and easily purchased. For example, begin with a set of six cars.

Tell the students to take a "photograph" (i.e., mentally) of the collection and stash it in their minds. Ask the students to close their eyes and say,

• I am going to change something. Try to hold the picture in your mind.

Change the objects for a different kind; vary the shape, colour, and size to add more challenge.

For example:

• Change the cars for teddies.

Ask the students to open their eyes and say,

• How many things can you see now?

Look for students to recognise that the number of items is unchanged, though the items vary from the original ones.

Aim for acceptance that the number is unchanged by means other than one-by-one counting, though use counting as a "fall back" strategy if needed.

Variations

Students may begin to say the number is invariant (never changing) because it has been the last few times, without necessarily accepting that the number is conserved.

Introduce the one more/one less principle by varying the number of new objects by one while still retaining the original layout. See if the students detect the change and trust invariance enough to build on it.

Make the variations more significant, such as adding or subtracting more than one and changing some parts of the arrangement.

Play "Odd one out" (see Counting on counting odd one out CM). Students must identify a reason for a card being different from the other two. One of the differences can be the number of items in the collection. Students build up their use of logic, particularly classification. Answers to Counting on counting odd one out CM: In the left-hand set on the first page, possible answers are: top card: teddies are sitting down; middle card: different amounts of teddies in each row; or bottom card: nine teddies (others have eight).

5.

Changing the spatial layout of the objects

Piaget’s original experiments with spatial layout involved two collections of objects. First, he laid out two collections, one above the other, like this:

Children were asked the equivalent of,

• Are there more blue teddies or green teddies, or are there the same number?

Most young children tend to equate the length of a collection with its number.

Next, one collection was spread out like this, and the question was repeated:

Young children often believed that the number had changed as the objects were rearranged and opted for blue as the "biggest" collection. Piaget called trust in the count under change to spatial arrangement “conservation of number.”

Variations

Similar activities to those given above for changes to colour, size, shape, and so on. can be carried out with spatial arrangement. 

Begin with a set of objects with some similar characteristics so that students can easily recall the objects that are present. Ask your students to take a photograph of the collection in their minds. Ask them to close their eyes and tell you what they can see. This promotes attention to structure—the organisation of a pattern. Encourage structural responses such as:

• There were two of each animal.
• The animals were in a hexagon.
• Three on top, three on the bottom.

Vary the spatial layout of the objects by expanding or contracting the length or area of the collection. Ask the students to open their eyes, then tell you if there are still the same number of objects and how they know.

Make minor variations to the total number as well as the spatial layout so that students are required to trust the invariance of the count and build with it or take from it.

Domino count. Students will need 13 plastic containers, a whiteboard pen, and at least one set of dominoes.

The task is to create "bins" for all dominoes that have the same number of dots in total. Students use the whiteboard pen to label the bin with the numeral and word that match the total. For example, the 8 (eight) bin will hold these dominoes:

Once the bins are created, students can arrange the dominoes in a pattern like this (for the six bins). They can also write the number facts for each domino.

Ask questions like,

• What comes next in the pattern? [4|2]
• Have you already got that domino?

Look at which bins have the most dominoes in them and which have the least. Ask why that happens. Discuss what zero means in this context:

• Nothing of something—no dots!

6.

Developing the abstraction principle

The abstraction principle involves the idea that non-tangible items such as sounds, touches, and ideas can be counted. Developing students’ capacity to count items they cannot see and feel is important because we want them to realise that numbers are ideas. Students ready for extension can carry out thought experiments with imaginary objects and anticipate what happens with real objects. One way to encourage abstraction is to connect visible and tangible objects with sounds and touches. 

Play the game, Messages, with the students in pairs. The students sit back to back, with one student facing the teacher. The teacher shows, then withdraws a pattern card (Counting on counting pattern cards CM) to the first student in the pair. That student turns around and gently “pokes”  or taps the number of dots they saw on their partner’s back. The partner shows how many dots they felt by holding up that many fingers. Both players turn around to check to see if the number of fingers matches the dot pattern.

Play the game Pattern Match. Put the students in groups of three or four. Each group needs a set of pattern cards and a set of digit cards. Spread the pattern cards out so they are separated individually and lying face down on the mat. Shuffle the digit cards and put them in a pack, face down, in the centre. Players take turns to turn over the top-digit card, then use their memory to turn over one of the pattern cards. If the cards match, the player keeps both cards and has another turn. Once all of the cards are matched, the player with the most pairs wins. Discuss with the class how they recognise some patterns instantly. Look for students to use combinations of smaller groupings, e.g.

• I know it’s seven because four and three are seven.

The initial technical aspects of counting involve a one-to-one match between objects in a collection and the set of whole number names and symbols. Once the whole number counting scheme is learned, students need to develop a more sophisticated idea of what counting does. The cardinality principle involves understanding that the last number counted tells about the whole collection, and the last word is not simply naming that last counter. Counting strips (Counting on counting number strips CM) used with transparent counters are very useful in helping students appreciate cardinality as well as the significance of the one-to-one and stable order principles.

1.

Ask the students to count a set of seven counters, making sure that they use at least three different colours. Place your own set of five counters on the strip, one colour at a time, beginning at one. Ask the students to copy you using their collection, starting with a given colour.

2.

After placing the third counter, ask the students,

• What number will the last counter go on?

Students who understand cardinality know that the last counter will go on the digit seven.

3.

Get your students to remove the seven counters from their strips.

4.

Ask:

• We started with the blue counters first. What will happen if we start with a different colour, like yellow? What number will the last counter go on?

5.

Invite students to share their ideas. Get them to check their predictions by replacing the counters, starting with the other coloured counters. This activity is about the order irrelevance principle.

6.

Repeat the activity with larger collections of counters and different mixes of colours. Thirteen and 24 are good numbers, especially the latter because students need to image the extension of the strip. After three different collections, students are likely to say that the order of counting the colours makes no difference, i.e.,

• The last count is always the same.

Be aware that they may be saying this because it has occurred three times in sequence, not because they recognise the cardinal (measuring) nature of counting.

Variations

Integrating several principles of counting will show whether students trust the count (cardinality) and can build on that trust.

1.

Demonstrate the activity above, using other examples of collections on the number strip. Include examples where the counters are not correctly placed (put more than one on the same number or leave gaps). Students should recognise that the objects must be in one-to-one correspondence in invariant sequence. Look for students to recognise that some last counts do not always tell the correct whole amount.

Begin by asking the students to put a given number of counters on their number strip. Using one colour is useful. For example, place eight red counters on the numbers one through eight.

Tell the students that you are going to add three green counters to the collection.

Ask:

• How many counters will there be then?
• How do you know?

Students may tell you that the next numbers covered will be nine, 10, and 11. Note that it is important that students realise that the result of adding one or removing one from a collection is given by the next number after or before in the number sequence.

2.

Progress to students imaging the addition of counters. Imaging can be developed by asking problems that are beyond the numbers of the strip, e.g., 19 counters, add four. Imaging can also be developed by turning the strip over so the digits are not visible and still placing counters on the strip.

Removing counters from a collection is more complex than adding on, for two reasons. First, the backward whole number counting sequence tends to be harder to master, especially in the teens and back from the low thirties. Second, the answer to subtraction is the number of counters that remain rather than the number of the last counter that is removed. Both reasons can be responsible for errors in early subtraction.

Ask the students to make a collection, say nine, and place the counters onto their number strip.

Tell them you are going to take away two counters from the strip.

• How many counters will be left then?

Expect the students to anticipate the correct action (removing the counters from 9 and 8).

Look for students to apply cardinality by telling you that seven counters will be left.

3.

Progress to examples that develop imaging through turning over the strip but still putting on counters and going beyond the scope of the numbers strip, for example, 21 – 3 = ? and 100 – 5 = ­­­?.

Put the students in pairs to pose similar problems, with one student making a collection of counters with mixed colours and the other student getting them to predict the result of adding or removing up to five counters of one colour. The students’ predictions can be checked, if necessary, by placing counters on a number strip.

The comparison of two collections is a common task in real life. Along with joining collections and partitioning collections, comparison forms the suite of problem types to which addition and subtraction are applied. Comparison offers students a chance to trust their counting and apply their understanding of the one more and one less principle.

1.

Pose the following problem to the students using the pictures from Counting on counting storyboard pictures CM to make a storyboard. The pictures can be snipped and copied to an Interactive Whiteboard or made into cardboard icons and fastened onto a storyboard using Velcro dots. Alternatively, use equipment available in your class. Here is an example:

• Here are 5 puppies and 6 bones. Is there a bone for each puppy?

2.

Discuss how the problem might be solved. Students may suggest matching the puppies and bones in one-to-one correspondence to find out if any bones are left over or if any dogs end up with no bones. Counting both collections and using the whole number sequence to anticipate the difference is a more sophisticated strategy because it applies easily to collections of any size. Using one-to-one matching to anticipate the result of 99 bones with 101 dogs would be a lengthy process.

3.

You might illustrate the difference by using two number strips with different coloured counters. Build five and six like this:

Visually, students can see that one more addition to five results in six. So, in the puppy situation, there will be one extra bone.

4.

Vary the puppies and bone problems by increasing the numbers and differences up to three. This will encourage anticipation by trusting the count and making one-to-one matching more cumbersome. Keep the differences to no more than three and use spatial grouping to encourage grouping strategies to count the number of puppies and bones.

For example,

• How many dogs are there?
• How many bones are there?
• Is there a bone for each dog?

On the number strip, the problem solution looks more obvious:

Note that the difference between the two can be found by counting up from 14 to 16, or by counting down from 16 to 14.

5.

Progress to imaging by turning the number strips over and building each collection. Without the numerals present, students will need to image the counting on or back process. 

Variations

Increase the size of both collections so students can generalise the process of finding differences by addition or subtraction.

Independent activity about differences might take two forms.

1.

Provide the students with copies of the pictures from Counting on counting storyboard pictures CM. Using these pictures, students can make up their own comparison problems for another student to solve.  

For example:

• “How many more cars are needed to fill all the garages?”

After students have completed their problems, put them into groups of four to share. Bring the class together to discuss strategies for solving the problems. The problems can be made into a book of difference problems for independent work or class discussions.

2.

Alternatively, use or make up two dice with appropriate numbers. For example, one wooden cube might be labelled 4, 5, 6, 7, 8, 9, and the other 8, 9, 10, 11, 12, 13. The two cubes are rolled, and sets are made of the numbers that come up, e.g., 12 cars and 9 garages. Students work out the difference between the numbers, e.g.,

• Three more garages are needed.

The most significant understanding that develops from counting is that collections can be partitioned and recombined without the conservation of numbers being disturbed. Part-whole understanding, as it is commonly referred to, is an extension of Piaget’s principle of conservation of number. Two components of part-whole understanding are significant:

• Trusting the count as collections are partitioned and recombined
• Appreciation of the strategic use of partitioning to make calculations easier and for transfer to similar calculations

Animals on the farm

This activity is initially aimed at the first component, though it can be extended to include the second component.

1.

Draw a farm with paddocks joined by fences on a sheet of paper or the whiteboard. Counting on counting farmyard CM has a template if you want to create laminated sheets for student use. Plastic farm animals are commonly available at toy retailers and dollar shops. Use counters to represent animals if need be. Ask a student to put ten animals on their farm or use a single farm as an instructional focus.

2.

Tell your students to watch as you move one or two animals across the “bridges” to a different paddock. Ask,

• How many animals are on the farm now?

Some students are likely to recount, while others may accept that the total number is invariant. Repeat moving animals until the students accept that the total number of animals has not changed. Note that your students may indicate acceptance of the previous number based solely on repetition rather than conservation. So the repetition needs to be disrupted so students can apply trust to the count.

3.

Move two or three animals to different paddocks, then put another animal on the farm in an obvious way. Ask,

• How many animals are on the farm now?

Students who trust their previous counting are likely to realise that, in effect, one more animal has been added, so the next number, eleven, gives the new number of animals. Similarly, move some animals around, then remove two animals.

4.

Have students work in pairs with a Farmyard card (Counting on counting farmyard CM) and some counters or plastic farm animals. They begin by putting ten animals on the farm. Then they can take turns to rearrange the existing animals, then add or remove up to three animals, all while their partner is watching. The partner must then work out how many animals are now on the farm. Encourage your students to avoid counting the animals unless it is to confirm the number.

5.

An extension of the activity is to introduce a “barn” in the form of a plastic cup. Begin with a given number of animals on the farm, say ten. Tell the students to close their eyes. Move some animals around, and then place a plastic cup (barn) over the animals in one paddock. Tell the students that the same animals are on the farm, but some are in the barn. Ask them how many animals they think are in the barn. Discuss their strategies for working the missing number out. Encourage risk-taking with a focus on grouping strategies; close approximations lead to known and trusted facts.

6.

Students can play the barn game in pairs. They can record their answers as equations, e.g., 4 + 2 + 4 = 10. The number of animals they work with can be varied to create more challenges.

7.

The farm scenario can be used to develop powerful strategies for working out 9 + and 8 + facts. Be aware that students will need to know the "teen" code as prior knowledge, e.g., 10 + 4 = 14 (fourteen means four and ten). Start with nine animals on one side of the farm and five on the other.

Ask,

• How many animals are on this (left) side? (nine) How do you know?
• How many animals are on this side? (five) How do you know?
• How many animals are on the farm altogether?

8.

Encourage students to use grouping strategies to count the number of animals. One-by-one counting is helpful to illustrate the power of "up over ten" strategies.

9.

Once the answer is accepted (14), move one animal from right to left to form ten and four. "How many animals are on the farm now?" provides the same example of developing 9 + facts.

10.

Record equations to help the students see patterns that can lead them to know facts, e.g.,

• For 9 + something, just take one off the something and make it a teen number.

Similarly, for 8 + facts, two animals need to be moved.

11.

Extend the problem up through ten to more complex examples, e.g.,

• There are 29/79/99 animals on this side and five animals on this side. How many animals are there altogether?

Skip counting is the process of counting in multiples of two or more. Some researchers, such as Anghilerri, believe that learning skip counting sequences assists students in developing multiplicative thinking. In the early years of primary school, counting in twos, fives, and tens is sensible, as these sets of multiples have patterns that make them easy to remember. The patterns can also lead to divisibility rules, such as “a whole number can be divided into equal sets of five if the number ends in 0 or 5"; e.g., 35 divides equally by five. Sometimes the sequences can be extended to include threes and fours, although these sequences are harder to remember.

As with other counting, it is vital to connect number patterns with quantities and words and to use different senses. Prediction of further numbers in a counting sequence helps to develop generalisations about which numbers belong and do not belong.

1.

Begin with body percussion and say a one-by-one sequence. Have students touch their chests on odd numbers and their knees on even numbers (or similar). Move to "being goldfish" on the odd counts by mouthing but not saying those words. So only the even numbers are spoken as knees are touched.

2.

Use a hundreds board to highlight the spoken numbers. Interactive hundreds boards are readily available online.

3.

Ask predictive questions such as

• Will we touch our knees when we say twenty-six? How do you know? What about 39?
• How can we know if a number is even?

4.

As students become proficient at reciting the skip counting sequence by twos, tens, or fives, apply these sequences to quantity. For example, you might get 14 students to stand up and ask questions like:

• If they line up in pairs, will each student have a partner? Will there be a person without a partner?”

If appropriate, use this activity to develop a definition of numbers as odd or even.

5.

Put bundles of ten ice block sticks on the mat in sequence, or have students hold up hands of five to develop skip counting of tens and fives.

6.

Use the slavonic abacus and the hundreds board alongside one another. Move some counters on the abacus explicitly in sets of two.  Ask the students how many beads you have moved. For example:

Note how trusting the count is needed to accept that the skip count of “2, 4, 6, 8” gives the total number of beads. Extend the difficulty of the problems by:

• Moving more sets of two counters, e.g., over ten, over twenty, etc.
• Masking what is moved so all students can hear the click of movement requires double tracking of the skip counting sequence and the number of counts
• Reversing the process by asking how many sets of two you have moved, e.g.,
  • I moved 14 beads. How many twos?
• Introduce an odd bead but still privilege counting by twos, for example:

7.

At some point, look at the skip-counting sequences purely in symbolic form. Use a hundreds board display with highlighting to show the multiples.

Ask pattern-based questions:

• Will 68 be in the counting by two sequence? How do you know? Will 100? Will 91?
• What is the same about all the numbers in the counting by five sequence? The counting by ten sequence
• What numbers are not in the counting by two sequence? How are these numbers the same? How are they different?

8.

Link the charts to the problem with quantity. For example:

• Here are eight pairs of shoes. Where is the total number of shoes on our charts? How do you know?

Note that students need to identify that the twos sequence applies and that the answer is the eighth count. You might record this fact as 8 x 2 = 16 and address what the symbols mean. Multiplication is expected later, starting at Level 2, but the ideas can be started much earlier.

9.

Link skip counting to divisibility with problems like:

• Here are nine children. Can they all find a partner? How do you know?"

• Here are 18 children. Can they all find a partner? How do you know? 

(Look for students to make links to the answer for nine children.)