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1a. What skills and knowledge are required by students to make meaning of numbers and basic computations?
1a. What skills and knowledge are required by students to make meaning of numbers and basic computations?
Some underlying principles of numeracy understanding are described below - (not in any particular order) - as we work through these understandings we will practise each activity/understanding
Some underlying principles of numeracy understanding are described below - (not in any particular order) - as we work through these understandings we will practise each activity/understanding
When teaching we need to use concrete objects and representations and correct maths language - a great teaching model for developing skills and understanding
When teaching we need to use concrete objects and representations and correct maths language - a great teaching model for developing skills and understanding
Watch this video about the importance of using concrete materials (objects) and pictures in early numeracy learning. Click on this link or click on the video to the right and use the Pin code SMPDWL (copy the pin before you click on the link).
Children need to count collections of all different types of objects!
After watching the video make sure you understand that when you remove concrete materials for counting the child is dealing with abstract ideas and understandings.
Discuss: what happens if you move to written computation methods too quickly?
Discussion - Why do we use stuff (concrete materials) when teaching new concepts in Maths, particularly in the Early Years? There are some articles below which we will come back to after our initial activities....
Discussion - Why do we use stuff (concrete materials) when teaching new concepts in Maths, particularly in the Early Years? There are some articles below which we will come back to after our initial activities....
How and when do we move students from using concrete materials to abstract ideas
How and when do we move students from using concrete materials to abstract ideas
1. Good concrete activity is good mental activity - pp 3-7
2. Why Teach Mathematics with Manipulatives? 1 longer page but based on research
3. Manipulatives in the Primary Classroom (or print version https://nrich.maths.org/content/id/10461/Manipulatives%20in%20the%20Primary%20Classroom.pdf ) 7 pages
4. Concrete-to-Pictorial-to-Abstract (C-R-A) Instruction or Concrete - Representational- Abstract (C-R-A)
5. Using manipulatives in mathematics learning 1 longer page
6. Simulations in maths in high school 1 page with some links to some simulations and also https://illuminations.nctm.org/SuccessStories.aspx?id=4269 (Middle school using Illuminations resources)
Extra reading From Objects and Images to Mathematical Ideas
Refer to the video of Rathmell's Triangle (above)
1. Underlying Principles - Stable order
1. Underlying Principles - 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.
2. Underlying Principles - 1:1 correspondence
2. Underlying Principles - 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
3. Underlying Principles - Cardinality
3. Underlying Principles - 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.
4. Underlying Principles - Conservation
4. Underlying Principles - 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.
5. Underlying Principles - Order Irrelevance
5. Underlying Principles - 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.
6. Underlying Principles -Abstraction Counting Principle
6. Underlying Principles -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.
7. Underlying Principles - Rote counting and Recognising numerals
7. Underlying Principles - Rote counting and Recognising numerals
Supporting_Education_Ch_16.3b_Number_sense.pdf8. Underlying principles - subitising (subitise) - recognise (how many in) small collections of objects and then increasing in size
8. Underlying principles - subitising (subitise) - recognise (how many in) small collections of objects and then increasing in size
9. Underlying Principles - counting on to add and counting back to subtract; using to check answers
9. Underlying Principles - counting on to add and counting back to subtract; using to check answers
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.
10. Underlying principles - Use five and ten as reference points, including subitising with 10, 20 etc - related to place value
10. 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
click on the image above and then on the 'open in a new tab' icon.
11. Underlying principles - Counting in groups and skip counting
11. Underlying principles - Counting in groups and skip counting
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. So they might have 7 objects and they might count 3, and 2 more is 5 and 2 more is 7.
Children also need to understand how to skip count which is strongly related to counting in groups . With skip counting they count a group of objects using pairs, or threes, or fives but they use the same number throughout.
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.
Use a 100 chart to count in 5s, 10s, 20s etc.
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/
12. Underlying principles - place value - problems with written algorithms and underlying understandings
12. Underlying principles - place value - problems with written algorithms and underlying understandings
Click on the link below to read more about strategies to build an understanding of place value in young students. This is one of the areas in which children have the most challenges and moving to written methods too quickly or without building the requisite underlying understandings has been found to be a major cause of problems.
Research shows that may children have inaccurate or faulty number concepts and use rote-learned procedures with little regard for the quantities represented by mathematical symbols (they add the digits in the tens column without understanding that they represent tens)
Base-ten blocks are widely used to teach place-value concepts but children do not perceive the links between numbers, symbols and models (they use the wooden cubes to add but do not have a firm understanding of how each block relates to the digits in the written algorithm).
Click on the link to the left to play Wishball tens - there are a range of games that suit students at different levels of understanding:
Click on the link to the left to play Wishball tens - there are a range of games that suit students at different levels of understanding:
Try one or two and think about how the game is supporting you to understand place value.
Try one or 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
More advanced skills and knowledge using the ten frame
More advanced skills and knowledge 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.
We could go on forever listing key understandings... here are a few more to think about!
Children need a huge range of mental computation skills and strategies such as:
Children need a huge range of mental computation skills and strategies such as:
doubling, adding multiples of ten, adding numbers that end in 5
number pairs (inside the tens frame and above eg 8+5)
and they need two digit strategies such as:
• Aggregation (eg: 28+35: 28+5=33,33+30=63),
• Wholistic (eg: 28+35: 30+35=65, 65-2=63)
• Separation right to left (eg: 28+35: 8+5=13, 20+30=50, 13+50=63)
• Separation left to right (eg: 28+35: 20+30=50, 8+5=13, 50+13=63)
and they need to know their times tables and be able to use an empty number line.
Mental computation skills are not just knowing your times tables. Times tables are one very small part of mental computation. If children are using mental computation skills they are working out a problem without using written records.
For example, they might look at two prices and work out which is the best value for money. Or consider the difference between traveling on an Express train and an all stops train. we will revisit Mental Computation skills in Links between processes and strategies (page 1c)
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