1. Number Sense
Cherri Moseley

2. Outcomes What is number sense?
How does children’s number sense develop?
What are the most effective teaching methods and resources?
How can research findings inform classroom practice? 
This PD has been predominantly influenced by Anghileri’s book

3. 2 x ? x 9 = 90

4. Overview of the session
Definitions
Activities
Strategies
Pedagogy

5. What is number sense? Mcintosh et al (1992)
“Number sense … reflects an inclination and an ability to use numbers and quantitative methods as a means of communicating, processing and interpreting information. It results in an expectation that mathematics has a certain regularity.”
“Number sense refers to a person’s general understanding of number and operations along with the ability and inclination to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for handling numbers and operations.”
Discuss what you think Number Sense is on your tables.
Do the two quotes above help?
Can you come up with a working definition?

6. Examples of numerical fluency or number sense?
= = = 65 Which is bigger? ½ or 3/7 If you know … 7 x 8 = 56 … x 0.8 – – 2 x 1.5
First equation is mathematically incorrect!
½ or 3/7 is one of the questions from the ICCAMS project (Increasing competence and confidence in multiplicative structures)

7. Possible components of number sense
Place value
Quantity
Relationships
Symbols
Vocabulary
Representations
Operations
Pattern

8. Place value Positional place value is
where the digit of a number is placed, e.g. in 345, the digit 3 is positioned in the hundreds.
Additive place value is
when all the individual values of the digits are added together to give the whole number e.g = 345
Multiplicative place value is
when we multiply the digit by its position to get its true value
Rising Stars Mathematics
Task 1
Gattegno chart activity – whole class activity – pointing and reading on next slide
Task 2
Individual Gattegno charts – handout - in pairs, cover a 2/3/4/5 digit number - guess what it is; now ask ‘what is 10 more, 10 less, 100 more, 1000 less, a tenth more/less
Task 3
Play game first; 2 teams; choose 3 digit cards and make the largest number
Play game in pairs
Task 4
Place value handout – in pairs, use digit cards to choose a 3 digit number. Multiply/divide by 10/100/1000 etc
What is happening? Talk about balance scales/numbers growing bigger

9. 1000
2000
3000
4000
5000
6000
7000
8000
9000
100
200
300
400
500
600
700
800
900 10 20 30 40 50 60 70 80 90 1 2 3 4 5 6 7 8 9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09

10. Counting Subitising Number system Magnitude
Quantity
Counting
Subitising
Number system
Magnitude
Counting – take a pile of counters; how many? Cardinality. Counting. Now organise them so we could quickly ‘see’ them without counting.
Subititing – a few slides to look at ……
Number system – you have just used it! But there are rules …. Smallest to largest, place value,
Magnitude – who has the biggest number (in counters) on the table? How do you know? How could you make it bigger/smaller?
Did you all assume each counter was worth 1??????

16. Relationships Fact families Derived facts Connections:
commutative associative distributive
inverse equivalence
Fact families – pick a number sentence/equation – write the fact family – did you write 8?
Choose an example from the delegates, model on flip chart using the bar model.
Derived facts – using your fact family derive some facts.
Discuss definitions and give examples; if you don’t know, how would you find out?

17. What is a symbol? Zero Equivalence and inequalities
Symbols
What is a symbol?
Zero
Equivalence and inequalities
Ask delegates what symbols they can think of – record on flip chart.
Is zero a number? Discuss. 4

19. Vocabulary The importance of using the correct vocabulary
augend + addend = sum
minuend – subtrahend = difference
multiplicand x multiplier = product
and factor x factor = multiple
dividend ÷ divisor = quotient

20. 13 Representations Pictorial and concrete
Are your classroom resources accessible e.g. trugs? 13
Trugs – remind yourself about the variety of manipulatives
How many different ways can you represent 13?
Reference Making Numbers book

21. Operations
18 x 5
A farm shop sells about 72 eggs each day. How many days will 300 eggs take to sell? What time will the last egg be sold?
“When arithmetic is taught as a logical structure of connected processes and results, rather than a sequence of standard procedures, children will learn that there is flexibility and choice in solving problems.”
(Anghileri, 2000, p.139)
18x5 – tell delegates to show this visually;
Youcubed activity – 18 x 5
Includes calculating – but also the fluency and adaptability and choice
Halving and doubling
Eggs problem comes from Anghileri’s book p.139

22. What’s in the box? What patterns exist?
Draw a box – delegates suggest a number; it either goes in the box or outside depending if it fits the pattern.

23. The pedagogy of number sense
Open
Flexible
Mistakes/misconceptions
Pair and group work
Making connections
Less is more
Playing
More than one answer/method/explanation
One question … five different ways
What changes do you need to make?
What changes does your school need to make?
Talk through then give time to discuss

24. Possible components of number sense
Place value
Quantity
Relationships
Symbols
Vocabulary
Representations
Operations
Pattern

25. Where is it going? Look at the sample GCSE questions
Which ‘number sense’ strategy or strategies would you use to solve the question?
Refer back to list at the beginning of the day

26. Outcomes What is number sense?
How does children’s number sense develop?
What are the most effective teaching methods and resources?
How can research findings inform classroom practice? 

27. “If teaching approaches change so that children learn connections then the outcome could be a new generation of mathematical thinkers who will be autonomous learners driven on by their fascination with numbers”
Anghileri, 2000, p.138

28. Cherri Moseley cherri.moseley@btopenworld.com