Number: Taking handfuls
View Sequence overviewOrganising a collection helps us to count accurately.
Some ways of arranging collections make it easy for us to see how many items there are in the collection, without needing to count every item.
Each group
A large quantity of small items which students can take a handful of (for example: counters, pasta, dried beans). A student handful should be around 10 to 20 items.
Each student
How big is a handful? Student sheet
Task
Introduce the idea of a ‘handful’ to the students. For example:
I had a handful of blueberries for morning tea today. I wondered afterwards: how many blueberries might have been in my handful? How big is a handful?
Present students with a large quantity of small items (e.g. counters, pasta, dried beans). Ask students to take a handful of items each.
Pose the question: How many items do you think might be in your handful? Encourage reasonable estimates, not the ‘right answer’.
Pose the task: Carefully count the items in your handful. Organise your collection in a way that makes it easy to see how many items you have.
Allow students time to count and organise their collection.
Once students have organised their collection, ask them to show their handful to someone else and compare the ways they have arranged their collections. Prompt them to consider: Which arrangement makes it easier to work out the total number of items in your collection? Why?
After comparing, students may rearrange their handful if they would like to.
Ask students to use How big is a handful? Student sheet to record the final way they arranged their handful and how many items were in their handful altogether.
‘Seeing’, not counting
Students are asked to arrange their handful in a way that makes it easy to see the total number of items in the collection. Did you notice that this goal says, ‘see the total number’ and not ‘count the total number’? What does it mean to see a total in a collection?
Subitising is the immediate recognition of numbers without counting. For example, when 5 is rolled on a die, most people recognise immediately that it is 5 without needing to count. They have constructed a mental object for 5 that makes counting unnecessary.
Clements (1999) distinguishes between two forms of subitising: perceptual and conceptual subitising. Perceptual subitising is the instant recognition of a number without counting. The example above is a form of perceptual subitising. Generally, children subitise in this way for collections of 5 or less objects.
Conceptual subitising is used for numbers greater than 5. Although it may seem that a total can be seen instantaneously, a little more thinking is involved. With conceptual subitising, numbers are thought of in terms of their subitisable parts. For example, the total 7 can be quickly recognised by seeing 5 and 2, or 3 and 4.
This task is designed to build students’ conceptual subitising skills. Students are asked to arrange their ‘handful’, a collection of more than 5 items, in a way that makes it easy to see the total number of items in the collection. Students need to think about how they can arrange their larger collection using smaller subitisable sets. By doing this, students are exploring and building their knowledge of part-part-whole relationships of numbers.
Students are asked to arrange their handful in a way that makes it easy to see the total number of items in the collection. Did you notice that this goal says, ‘see the total number’ and not ‘count the total number’? What does it mean to see a total in a collection?
Subitising is the immediate recognition of numbers without counting. For example, when 5 is rolled on a die, most people recognise immediately that it is 5 without needing to count. They have constructed a mental object for 5 that makes counting unnecessary.
Clements (1999) distinguishes between two forms of subitising: perceptual and conceptual subitising. Perceptual subitising is the instant recognition of a number without counting. The example above is a form of perceptual subitising. Generally, children subitise in this way for collections of 5 or less objects.
Conceptual subitising is used for numbers greater than 5. Although it may seem that a total can be seen instantaneously, a little more thinking is involved. With conceptual subitising, numbers are thought of in terms of their subitisable parts. For example, the total 7 can be quickly recognised by seeing 5 and 2, or 3 and 4.
This task is designed to build students’ conceptual subitising skills. Students are asked to arrange their ‘handful’, a collection of more than 5 items, in a way that makes it easy to see the total number of items in the collection. Students need to think about how they can arrange their larger collection using smaller subitisable sets. By doing this, students are exploring and building their knowledge of part-part-whole relationships of numbers.
Ask students to display their student sheet next to their handfuls in preparation for a gallery walk.
Review the original task (Organise your handful in a way that makes it easy to see how many items you have) and ask students to think about what they expect to see as they complete the gallery walk.
Ask students to consider the following questions as they look at others’ work: Which arrangements make it easy to see how many are in a handful? Why?
Conduct the class gallery walk.
Focus this Connect discussion on which arrangements made it easy to see how many there were. You might make comparisons to familiar objects that are designed for easy counting or recognisable collections, for example, playing cards, dominoes, or dice. Also look at those arrangements that are not easy to quantify quickly. As a class, consider how these handfuls might be rearranged so that it is easy to see how many there are without needing to count.
Discuss: Which arrangements made it easy to see how many are in a handful? Why?
- Students might think about patterns that made numbers easy to count. For example, subitising patterns such as those on dice.
- Some students may use groups of 10 when there are more than 10 objects.
Ask the students to return to their own collections and rearrange their items a final time, using the strategies they have seen to make their collections easier to count.
Again, ask students to show their arrangements to someone else and compare the different ways they have arranged their collections.
Discuss with the students what they noticed about how others had organised their collections.
Explain: Organising a collection helps us to count accurately. Some ways of arranging collections make it easy for us to see how many items there are in the collection, without needing to count every item.