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What skills and knowledge are required by students to make meaning of number and to develop computation skills?
What skills and knowledge are required by students to make meaning of number and to develop computation skills?
There are some underlying principles of counting and number that students need to understand. As we work through this we will practise and explore each.
As you do so ask, "When is this used in maths? When is this used in numeracy?"
As you do so ask, "When is this used in maths? When is this used in numeracy?"
1. Underlying principles of counting
1. Underlying principles of counting
1a. Underlying principles of counting - Stable order
1a. Underlying principles of counting - Stable order
The first principle of counting involves the student using a list of words to count in a repeatable order. This ordered or “stable” list of counting words must be at least as long as the number of items to be counted.
For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20.
Thinking deeper about stable order, we might consider rote counting from 0, counting on from a number (i.e.: “start at 6 and count to 18”) and counting backwards (i.e.: “count backwards from 15”) skills that are related to stable order.
1b. Underlying principles of counting - 1:1 correspondence
1b. Underlying principles of counting - 1:1 correspondence
One to one correspondence is the ability to match an object to the corresponding number and recognise that numbers are symbols to represent a quantity. Young children often learn to count without having an understanding of one to one correspondence (Kearns, 2010).
Understanding that each object in a group can be counted once and only once. Each object must be given one and only one number name. It is useful in the early stages for children to actually tag or touch each item being counted and to move it out of the way as it is counted.
Open the document below to read about 1:1 correspondence.
Supporting_Education_Ch_16.3a_One-to-one_correspondence.pdf
1c. Underlying principles of counting - Cardinality
1c. Underlying principles of counting - Cardinality
Understanding that the last number used to count a group of objects represents how many are in the group.
A child who recounts when asked how many items are in the set that they just counted, may not have an understanding of the cardinality principle.
1d. Underlying principles of counting - Conservation
1d. Underlying principles of counting - Conservation
Understanding that the count for a set group of objects stays the same no matter whether they are spread out or close together.
If a student counts a group of items that are close together and then needs to recount after you spread them out, they may not have developed an understanding of the principle of conservation.
1e. Underlying principles of counting - Order Irrelevance
1e. Underlying principles of counting - Order Irrelevance
The order in which items are counted is irrelevant.
Students have an understanding of order irrelevance when they are able to count a group of items starting from different places. For example, counting from the left-most item to the right-most and visa versa.
1f. Underlying principles of counting -Abstraction Counting Principle
1f. Underlying principles of counting -Abstraction Counting Principle
When we are counting a set of objects they do not have to be the same. The counting is consistent. We can count both tangible and intangible items - cars, dogs, cats, days, months, shadows, sounds etc. There are some things we can count and some things we cannot.
2. Underlying principles and processes - subitising (subitise) - recognise (how many in) small collections of objects and then increasing in size
2. Underlying principles and processes - subitising (subitise) - recognise (how many in) small collections of objects and then increasing in size
Read the article about teaching subitising
3. Underlying principles and processes - counting on, counting back, checking
3. Underlying principles and processes - counting on, counting back, checking
Watch this video and discuss the skills and knowledge being demonstrated by 5 year old Charlotte. Which key principles of understanding does she demonstrate?
Consider what activities might have been done with her to develop this understanding.
4. Underlying principles and processes - Counting in groups
4. Underlying principles and processes - Counting in groups
Counting in groups is an important skill. Children need to learn how to count in groups and understand that it is a technique we use to be more efficient when counting. Eventually it will lead us to an understanding of multiplication.
Children need to understand how to count in groups with objects, and also how to skip count which is strongly related to counting in groups (see next).
5. Underlying principles - skip counting
5. Underlying principles - skip counting
Students need to learn to count fluently forwards and backwards by twos, fours, fives, tens and hundreds starting at any number. They should begin by counting actual objects in groups so that they understand the concept (see above, Counting in groups).
Before achieving this, they will be able to skip count forwards fluently, but may experience difficulty counting backwards.
Skip counting is a gradually developing skill as students continue to expand the range of numbers with which they can skip count. They will also become able to skip count from any number, not just starting at zero, which results in the most familiar sequences (e.g. 0, 2, 4, 6 … instead of 1, 3, 5, 7 …).
Skip counting is important in the development of fluency in calculation, number sense and as the basis of multiplication and division. It is also important to help students move from calculating by counting by ones to using number facts. For example, instead of working out 12 + 4 by counting 12, 13, 14, 15, 16, students can immediately add 4, or possibly add 2 twice. This transition to using fluent number facts is a key to success throughout school.
HINT: Coloured translucent counters work best as you can still read the numbers underneath, which is important for younger children.
On a 1-100 (or 1-20, 1-30) chart, highlight every second number with a coloured counter. Discuss meaning of second. Look at pattern. Count by 2s. Maybe discuss even numbers.
On a 1-100 (or 1-20, 1-30) chart, highlight every third number with a coloured counter. Discuss meaning of third. Look at pattern. Count by 3s. Discuss in context of even numbers if the first lot of counters are still there.
On a 1-100 (or 1-20, 1-30) chart highlight every fifth number. Discuss and practise counting in 5s. What does fifth mean? Can we see any other patterns?
On a 1-100 (or 1-20, 1-30) chart highlight every tenth number. Discuss and practise counting in 10s. What does tenth mean? Can we see any other patterns? Does this pattern relate to the 5s pattern? The 2s pattern? The 3s pattern.
Keep going ….up to stage 3
https://denisegaskins.com/2008/09/22/things-to-do-hundred-chart/
6. Underlying principles - Use five and ten as reference points, including subitising with 10, 20 etc - related to place value
6. Underlying principles - Use five and ten as reference points, including subitising with 10, 20 etc - related to place value
We use a base ten number system so it is crucial that children build early reference points related to five and ten, which will lead on to 10s, 100s etc.
Activities to build this understanding:
Referencing 5
1. On a tens frame place five counters.
2. Class discussion about placement.
3. Do activities with 1 more, 1 less, 2 more, 2 less etc
Referencing 10
1. On 2 tens frames place ten counters.
2. Class discussion about placement.
3. Do activities with 1 more, 1 less, 2 more, 2 less etc
Friends of ten
1. Place counters of one colour to make less than 10.
2. Discussion of the friend of that number.
3. Open activity - what other friends can you make - keep record in group.
4. Discussion of subitising friends of ten and in other arrangements.
Games with friends of ten
Fill the ten frame - https://drive.google.com/file/d/0B7YM6iJAeDKVNTkwS3N5azBILXM/view
Two colours - https://drive.google.com/file/d/0B75cP9HMyXqtWGtEcThjSU8zbXM/view
Odd or even - https://drive.google.com/file/d/0B7YM6iJAeDKVMVZKNFBKdld3aG8/view (can do on ten frame or grid).
Ten frames to print - http://www.mathwire.com/templates/tenframemat.pdf
7. Underlying principles - place value
7. Underlying principles - place value
Click on the link below to read more about strategies to build an understanding of place value in young students
More advanced skills and knowledge related to place value using the ten frame
More advanced skills and knowledge related to place value using the ten frame
Using these strategies to learn basic understand will eventually assist student to be able to do more complex addition and subtraction problems. The tens frame can also be used to add and subtract groups of ten.
Click on the link to the left to play Wishball tens - there are six games:
Click on the link to the left to play Wishball tens - there are six games:
Wishball 10s and Wishball 10s Challenge
Wishball 10s and Wishball 10s Challenge
Wishball 100s and Wishball 100s Challenge
Wishball 100s and Wishball 100s Challenge
Wishball Whole Number and Wishball Whole Numbers Challenge
Wishball Whole Number and Wishball Whole Numbers Challenge
Try any two and think about how the game is supporting you to understand place value.
Try any two and think about how the game is supporting you to understand place value.
When you click on the link you will need to type in or paste in the following access code:
When you click on the link you will need to type in or paste in the following access code:
DWRVQG
DWRVQG
We could go on forever listing key understandings...
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