Developing Mathematical Thinking and Reasoning

1
Developing Mathematical Thinking and Reasoning
NQT Meeting February 2008
2
Objectives

• To discuss the importance of mathematical
  thinking and reasoning
• To consider effective ways of helping pupils to
  investigate, explain, predict and generalise
• To explore progression in activities that promote
  reasoning in mathematics

3
Children need .

• questions / problems that require them to reason
  (and make decisions).
• scaffolded questions to help develop reasoning
  skills.
• to think about what they know and use what they
  know.
• to be given opportunities to ask questions,
  especially What if ? questions.
• to explain, predict and generalise.

4
Key features and elements of reasoning

• Understand the question
• Use logical thinking
• Make connections
• Apply what you know
• Recognise and explain patterns and relationships
• Make sensible estimates
• Make informed predictions
• Look for patterns in results
• Investigate general statements (own ones in Y4)
• Ask What if ..? questions
• Explain and then generalise about what has been
  found out

5
Purposes of questioning

• to motivate
• to assess
• to promote reflection, analysis or enquiry

Questioning can . . . . . . . .

• focus attention
• arouse interest
• stimulate thinking
• find out what children know
• invite everyone to participate
• probe pupils understanding

• diagnose misunderstandings
• invite questions from pupils
• get pupils to explain, predict or give reasons
• help pupils to express what they think, believe
  or know
• help pupils apply their learning

6
Questions can be used to assess .

• knowledge
• use of mathematical language
• use of models
• methods and strategies
• reasoning skills
• understanding

7
3
4
2
1
This is an addition wall.
NB could
also be subtraction / difference wall The
value of each brick can be found by adding the
pair of numbers on the row below. What is the
number that needs to be put into the brick at the
top? Repeat for numbers 4, 11, 12 and 23
8
Effective use of resources

• Flik-flaks
• Wall
• Counting machine
• Counting stick
• Number / money fans
• Target boards

9
Problem solving target board
10
Problem solving target boards
  20
  16
  6
  17
18
  12
  10
  15
  3
11
Reasoning across mathematics strands

• Counting and understanding number
• Dice / card games, logic statements
• Calculating
• Bens addition, Whats my rule? , Number line
  jumps
• Shape and space
• Sorting shapes, Coordinates, patterns
• Measures
• Money amounts, Most likely to .
• Handling data
• A Country Walk, Kingscourt BBs

12
Comparing money amounts
0
1
2
Stay standing if the amount you have on your card
is
greater than 40p
less than 160p
greater than 0.75
greater than 50p but less than 1.25
13
How old is Granny?from Apex Maths 2 (Cambridge
Univ. Press)

• Jan and Chris visited their granny. It was her
  birthday.
• She gave them five clues to help them work out
  her age.

1. I am younger than 60 but older than 50.
2. If you add the digits of my age, you get an
even number.
3. Both of the digits in my age are odd.
4. The two digits are different.
5. If you add the digits in my age, you get a
two-digit number that has next-door digits.
14
ICT has the potential to .

• Enhance demonstration and mathematical
• modelling
• Improve the quality of interactions and
• discussion
• Help to develop childrens ability to explain
• and reason

15
Odd one out?

• Tell yourself
• Tell a friend
• Tell a pen friend

16
11
5
16
Guardian of the Rule
15
19
28
34
1
11
4
7
16
3
61
100
9
12
14
10
8
17
3p
2p
4p
3 x 2 6
5p
100 is 320
18
True
False
or
I can make four TU numbers with two different
digits. I can add numbers in any order and the
answer is always the same. All triangles have
three sides. I can make 10p / 20p / 1 using
four coins.
19
Balance the number sentence

• Your job is to balance different number
  sentences
• You can use only these numbers
• 6 , 3 , 5 , 4 , 1 , 2
• You dont have to use all of the numbers in each
  number sentence
• You cannot use the same number more than once in
  the same sentence
• You can use , - , x ,

6 3 4 - 2
20
Number Sequence

• Look at these sequences
• 5, 7, 9, 11, _ , _ , _ ,
• 1 , 4 , 7 , _ , _ , _ ,
• What are the next three numbers in each sequence?

Will the number 100 be in the sequences? Explain
your answer.
Pause it!
20 Cards
Monty
21
Value added

18

16
14

Notebook
Nelson Thornes 3
22
Total squares
Use 8 different dominoes. The sides of each
square must have the same number of dots as the
middle number.