How students come into school

• No number exposure
• Rote counting
• Some idea

Three dimensions of mathematical understanding

• Procedural understanding - procedural fluency and numeracy strategies
• Conceptual understanding - visual conceptual understanding
• Problem solving  - applying procedural and conceptual

True Counting Vs Rote counting

• Knowing the why eg. why is 5 bigger than 4
• Why does 3 look different to 6
• That what you’re counting doesn’t have to be the same
• Using random items to count rather than always using ‘maths’ equipment

Most kids learn through rote - memorisation

• Need to include sets
• Counting 1, 2, 3, 4 around the set
• Need to know the purpose of counting
• Features don’t matter when I’m counting I just want to know the quantification not the features
• How much altogether
• Generalisations don’t apply to numbers
• Needed for addition and subtraction foundation helps to elminate the ‘I can’t do 5-7 so I’m going to swap it to 7-5 need understanding of what it actually means to take something away
• Saying ‘I’m trying to find out how many when i’m counting’
• Need to tag/touch items as counting to ensure they’re making connections

= Principals about how to count

• Cardinal principal - Counting 1, 2, 3 ,4 knowing that once we go past 1 it no longer exists, 2 it no longer exists the whole group is 4

= principals about what can be counted

• Abstraction - I can have 1 chair and I can have 1 pen
• Irrelevance principle - counting in a line … saying you can go back to something to count don’t always have to be in a line.
• Counting on - being able to hold 1, 2, 3 in head and counting up from three without adding on

• Tracking / tagging should be taught…
• Touching
• Hovering
• Eye tagging
• Making different physical group sizes so spreading out, not in a line, stacking etc.

Try counting in a rectangle - do they know when to stop or do they keep counting around the shape

Change from always being in a line

Vary the way you arrange counters

• Numicon
• Tens frames
• Piles
• Tally marks
• Lines
• Containers
• Counters with two colours - turning them over to count

Helps with learning when to stop - I’ve counted the set now I need to stop

How many XX are in this picture

Number lines that don’t start at zero - zero being last concept to be taught

Subitising

1-4 instantly see no matter what arrangement it has

Above or seeing the pattern quazi subitising is above only see it if it’s in pattern otherwise can’t see it

Digits and numbers

Ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Numbers can be made up of more than one digit

Hundreds board can cause issues with realising 10 is two digits

Number charts

Knowing that once they’ve counted they don’t have to count again

Tagging with mats

Looking at where it moves within the sets is it part of the set are the sets the same?

Using set lines for different counting

Ensuring that students know that 5 means the whole set not just the tag at the end

Different sets of things to count

• Events - sleeps
• Imaginary objects

Order of counting is irrelevant being able to count from any counter not just in a line

Last digit to be taught 0

• I have nothing
• There is nothing in the tens, nothing in the ones
• Absence of quantity

Compensation

• Increasing and decrease as compared to addition and subtraction
• Part whole thinking how can I break my numbers apart
• There are different ways to make a number 3+3, 4+2, 5+1

Being able to see the equals sign

Knowing that part whole can be done with any number

Knowing what to do with broken numbers when adding by circling the whole set and left to add is the rest

Knowing when subtracting that I subtract 3 to get 10 and then I need to subtract 4

13-7

Place value

Ensuring children understand ten more ten less and by colour coding students can see the digit follow through

Do they have visual spatial awareness

• Lego models
• Making patterns
• Seeing patterns in pictures not tracing but drawing patterns from looking at the picture

Counting assessment tool

• Email her for a copy