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Published 2014 Revised 2021
This article forms part of our Reasoning Feature, and complements the article Reasoning: Identifying Opportunities.
Developing reasoning skills with young learners is a complex business. They need to learn to become systematic thinkers and also acquire the ability to articulate such thinking in a clear, succinct and logical manner. In many classrooms more progress is being made with developing the systematic thinking than with the elegant communication. There needs to be equal emphasis on both these aspects of reasoning and in both we need to consider progression. What would we expect from a novice reasoner as opposed to an expert reasoner? How can we help young learners to progress to expert level?
Progression in reasoning
At NRICH we see a five-step progression in reasoning: a spectrum that shows us whether children are moving on in their reasoning from novice to expert. Children are unlikely to move fluidly from one step to the other, rather flow up and down the spectrum settling on a particular step that best describes their reasoning skills at any one time.
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. This is the beginning of inductive reasoning.
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. This is called inductive reasoning.
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. This is also called deductive reasoning.
Let's look and see what these can look like in some children's solutions to an NRICH problem: Sealed Solution.
We need to bear in mind that when looking at these solutions we are assuming that children are able to communicate their thinking in a written form that is understandable by another without any verbal interpretation from the child themselves. In the classroom we are more likely to find out about children's reasoning through listening to their talk as they explore a mathematical challenge or talking with them ourselves.
Here is the task Sealed Solution:
Here's a solution that was sent to us from Ieuan in Prague:
At first I was randomly picking numbers, and on my first attempt doing it I found a solution:
0 + 7=7 5 + 3=8 9 + 4 =13 6 + 8=14 2 + 1=3.
And then I tried using a system from then on, of adding a number to the smaller number then subtracting one from the smaller number, but it did not go very well because when I converted 5 + 3 to 4 + 4 I realised that you cannot do that.
Then I found out something quite clever that from one onward: each 2 numbers have the same amount of possibilities (I think by this Ieuan means that pairs of consecutive numbers have the same number of possibilities). For example 2 and 3 have two possibilities, 4 and 5 have have three, 6 and 7 have four, 8 and 9 have five and it goes on forever! So I wrote down all the possibilities for 7, 8, 13, 14, 3.
Then I shortened it, so if I use 13 as an example: 13+0, 12+1, 10+3, 9+4, 8+5, 7+6. Then I would take off 13+0, 12+1 and 10+3 and do that for all the rest! So when I had all the possibilities I did two attempts without succeding then I got one and I started explaining it on here, but I realised I had found the same as my first attempt.
Then I did one attempt and I found another: 3+0, 8+6, 9+4, 7+1 and 2+5.
We would say that this child is showing some reasoning ability as they describe how they move from a random approach to a more systematic approach. He uses phrases such as, 'I found the same as my first attempt' and shows how he moved to spotting a pattern: 'each two numbers have the same amount of possibilities'. However, he is not yet at the stage of being able to explain his reasoning clearly - he says he thought it would be best to make the biggest totals first yet the reason is not given. His reasoning ability would seem to be ahead of his communicating ability, which is very common in young learners.
How does that first solution compare with this next solution to the same problem?
Year 6 pupils from St John Fisher Harrogate Magic Maths Club:
We started off by thinking of all the possible ways of making the totals. This took a long time.
We thought that it would be best to make the biggest totals first, using the bigger numbers to make them:
14 = 9 + 5, 13 = 6 + 7, 1 + 2 = 3, 4 + 3 = 7 and 8 + 0 = 8.
Some of us did it the other way round, making the smallest totals first, with the smallest numbers:
1 + 2 = 3, 4 + 3 = 7, 8 + 0 = 8, 7 + 6 = 13 and 9 + 5 = 14.
We could also come up with pairs randomly but it's quicker to use a strategy.
7 + 0 = 7, 5 + 3 = 8, 9 + 4 = 13, 6 + 8 = 14 and 1 + 2 = 3.
Like Ieuan the way that these children describe their problem-solving process suggests that they are developing some reasoning fluency: 'we thought that it would be best to make the biggest totals first, using the bigger numbers to make them'. However, they have yet to offer an explanation as to why they chose to do that.
Let's look at Rebekah's explanation for Sealed Solution and see how well she is beginning to justify her choices:
36 (10 numbers): 5 + 5 + 5 + 5 + 5 + 3 + 3 + 3 + 1 + 1
38 (10 numbers): 1 + 1 + 1 + 3 + 3 + 5 + 5 + 5 + 7 + 7
37 (9 numbers): 5 + 5 + 5 + 5 + 5 + 5 + 5 + 1 + 1
Here the child is moving into proof by showing that you cannot make an odd number with an even number of odd numbers. They are using mathematical generalisation. However, the chain of argument could be developed to show exactly why this can't work by showing why you can't make an odd number with an even number of odd numbers. Specific examples can be helpful for illustrative purposes yet they
do not constitute a watertight proof. This is a key development in reasoning with children and we need to model for them how to move beyond specific numerical examples to a general proof that holds good for all examples. So, this chain of reasoning needs to show that two odd numbers always make an even and that ten odd numbers is made up of five pairs of odd numbers. Therefore, there will be a
sum of five even numbers, which must be even.
What do you think of the chain of reasoning in this other solution to the same problem, Make 37? Has the child shown a complete chain of reasoning? Sometimes we can be glad that a child has given a reason at all. Sometimes it may be appropriate to challenge them further to provide a complete chain of reasoning. How true is this of your classroom? Is this a useful development for you?
Let's look at some examples of NRICH solutions to see how well they fit these criteria. Let's look at Ieuan's solution to Sealed Solution again and this time, look at the communication of the reasoning. How do you think Ieuan is doing on the journey to becoming an expert communicator of reasoning using the criteria we offered above?
First, a brief reminder of the problem:
In addition to 'organisational' tools the development of visual methods of proof, such as diagrams, may be helpful to explain reasoning. Particularly at primary level, visual proofs can be very useful as way to express a reasoned argument.
Beyond that we will want to offer children the opportunity to understand algebraic concepts and notation to help them express their reasoning in a succinct and elegant manner. Take a look at these two solutions to Eggs in Baskets and see whether you think the algebra has helped achieve this:
The Brown basket has one more egg in it than the Red basket.
The Red basket has three eggs less than the Pink basket.
How many eggs are in each basket?
In the 'words' version the child is showing evidence of moving towards becoming an expert communicator of reasoning. Their argument has a system to it - they try one egg in the brown basket and see what happens, then two eggs and then three. They are using the problem-solving skill of trial and improvement and systematically trying out different numbers of eggs. They are able to express their thinking in a clear, succinct and logical way. Each sentence is compact and uses logical argument words such as 'if there were ”¦ there would be' and 'so finally'.
Here's the algebraic version. How does it help develop the communication? To what extent has it 'translated' the words in the first solution into mathematical symbols?
Supporting children to move forwards with reasoning
Let's take a look at two children's solutions to Shape Times Shape: