1. Reasoning With Primary Aged Children
Thinking Out Loud
Reasoning With Primary Aged Children
Debbie Streatfield and Faye Heath

2. Objectives for the workshop:
Understand the importance of developing children’s reasoning skills in primary schools;
Know a range of ways to develop children’s reasoning skills within the primary mathematics curriculum.
Thinking Out Loud

3. The importance of reasoning
‘Mathematical reasoning, even more so than children’s knowledge of arithmetic, is important for children’s later achievement in mathematics’
‘Mathematical reasoning is learning to reason about the underlying relations in mathematical problems they have to solve.’
‘In the context of time pressures, more time should be allocated to developing children’s reasoning, than to practising calculation skills.’
From: Development of Maths Capabilities and Confidence in Primary School Terezinha Nunes Peter Bryant Cathy Sylva and Rossana Barros 2009
Thinking Out Loud

4. Talk Vocabulary Correct use of vocabulary
Speaking in complete sentences
Opportunities to talk
Modelling and scaffolding
Thinking Out Loud

5. Thinking Out Loud It is … because … It could be … because …
Promoting the vocabulary of reasoning:
It is … because …
It could be … because …
It can’t be … because …
It won’t work because …
If … then …
It would only work if …
If … is … then … is …
I know … because …
I notice that …
We could try …
Thinking Out Loud

6. Putting reasoning at the heart of maths learning
Pupils need to understand that they are expected to:
think before doing
notice things
make decisions based on what they notice, know and understand.
From: NCETM Primary and Early Years Magazine No. 93
Thinking Out Loud

7. Thinking before doing Put the boxes in order
How spread out are the marbles?
Put the newspapers in order
What other approaches do you use that support your children to think before they do?
Thinking Out Loud

8. Noticing things
NCETM Video
Thinking Out Loud

9. Noticing
the student [need to] asks themself what is similar about the questions and what different, what it is about the context which enables the technique to work, what sorts of difficulties might the technique encounter in different situations, etc.
From: John Mason Asking Mathematical Questions Mathematically 1998
Thinking Out Loud

10. 8 + 5
Grid pairs such as those shown below, or other (correct)
variations on them, were produced by different pupils:
Either or or

11. Making decisions based on what they notice and understand
True or False? = =
Explain your reasoning.
Write three more pairs of equivalent calculations using =
as a starting point.
Year 5 example from NCETM – Teaching for Mastery Questions, tasks and activities to support assessment
Thinking Out Loud

12. The importance of planning effective questions
Questions can be invitations to pupils to do something, say something or suggest something
What is important is the thinking that the questions generate e.g. reasoning about what the child is being asked to do; on choices to be made; on representations that support understanding (theirs or others); in order to justify the answer
Thinking Out Loud

13. Cuisenaire/number rods
If the dark green is a half, which rods could you use to make one and a quarter?
If the yellow is a half, which rods could you use to make one and a half?
Which rods would you choose to represent a half, and why?
Can you create your own effective questions?
Thinking Out Loud

14. Supporting Fluency
Reasoning about what is already known in order to work out what is unknown will improve fluency
From: NCETM Primary and Early Years Magazine No. 93
Thinking Out Loud

15. Progression in reasoning
Step One: Describing: simply tells what they did
Step Two: Explaining: offers some reasons for what they did. These may or may not be correct. The argument may yet not hang together coherently.
Step Three: Convincing: confident that their chain of reasoning is right and may use words such as, ‘I reckon’ or ‘without doubt’. The underlying mathematical argument may or may not be accurate yet is likely to have more coherence and completeness than the explaining stage.
Step Four: Justifying: a correct logical argument that has a complete chain of reasoning to it and uses words such as, ‘because’, ‘therefore’, ‘and so’, ‘that leads to…’
Step Five: Proving: a watertight argument that is mathematically sound, often based on generalisations and underlying structure.
From: Reasoning: The Journey from Novice to Expert (Article) Nrich website
Thinking Out Loud

16. NCETM developing opportunities and ensuring progression in the development of reasoning skills
+COUNTING
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number
count backwards through zero to include negative numbers
interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers, including through zero
use negative numbers in context, and calculate intervals across zero
count, read and write numbers to 100 in numerals; count in multiples of twos, fives and tens
count in steps of 2, 3, and 5 from 0, and in tens from any number, forward or backward
count from 0 in multiples of 4, 8, 50 and 100;
count in multiples of 6, 7, 9, 25 and 1 000
count forwards or backwards in steps of powers of 10 for any given number up to
given a number, identify one more and one less
find 10 or 100 more or less than a given number
find more or less than a given number
Spot the mistake:
5,6,8,9
What is wrong with this sequence of numbers?
True or False?
I start at 2 and count in twos. I will say 9
What comes next?
10+1 = 11
11+1= 12
12+1 = 13
……..
45,40,35,25
I start at 3 and count in threes. I will say 13?
41+5=46
46+5=51
51+5=56 ……
50,100,115,200
38 is a multiple of 8?
936-10= 926
= 916
= 906
…….
950, 975,1000,1250
324 is a multiple of 9?
= 7706
= 8706
= 9706
177000,187000,197000,217000
When I count in 10’s I will say the number 10100? =
–10000 = =
-80,-40,10,50
When I count backwards in 50s from 10 I will say
-200
The temperature is -3. It gets 2 degrees warmer. The new temperature is -5?
Thinking Out Loud

17. Diamond 9
Thinking Out Loud

18. Reflection Thinking Out Loud
What elements of developing effective reasoning do you see used most frequently and effectively in your school?
What would you like to develop?
How might you go about developing this?
Thinking Out Loud