One of the three main aims of the revised maths curriculum is to ensure that all pupils can reason mathematically in maths but it’s not always clear what we mean by this and how to incorporate it into lessons. So what is reasoning?
The NCTM tells us that: The ability to reason systematically and carefully develops when students are encouraged to make conjectures, are given time to search for evidence to prove or disprove them, and are expected to explain and justify their ideas.
It must become a part of the way we teach daily in order for our students to become proficient in their ability to reason.
Students therefore need opportunities to discuss their work in order to explain and justify their ideas. This discussion will help them clarify their ideas so they can then communicate them in writing. When writing it's helpful to provide some examples of the language used within reasoning.
It can’t be/it must be, because…
I already know…., so…..
This is the same/different because…
This is true/false because…
Changing a closed question to be open and have several answers can encourage reasoning since it requires divergent rather than convergent thinking.
For example, rather than asking a student to count objects, ask how many ways can they count the objects by grouping. Instead of asking what is 5 x 5, ask them to create calculations where the product is 25. You could then ask them to do the same for other square numbers and ask them what they notice.
When identifying shapes, rather than having the labels triangle, square and rectangle that match three images of those shapes, provide images of several types of triangles and rectangles (including squares) and ask them to say which are triangles and squares. Then ask them to explain what is the same and what is different about the shapes between and within the groups.
When working with fractions, rather than presenting a question such as 2/7 + 5/7, ask them how many calculations they can create where the two fractions add to one.
Examples of Activities and Questions
Spot the Mistake in this calculation
Missing Symbols and missing number calculations
In pairs, students create their own problems for others within a topic
Soduku and Futoshiki allow pupils to use logic and reason why a number belongs in a particular place. Students can play this in pairs where each of the partners takes it in turns to place a number since they will then need to explain their thinking about why a particular digit belongs in a square.
Is this statement always, sometimes or never true? Explain why…
Why must this calculation be wrong without working out the answer?
If we know…, what else do we know?
The answer is - what could the questions be?
Which two numbers must the answer be between, using estimates?
What am I? Guess who? Pupils are given clues or ask questions.
Can you spot a pattern? Explain what it is…
What comes next? How do you know? Is there a general rule?
What’s the same/what’s different? Which is the odd one out? Why?
Is this true or false? Explain why…
Games that include decisions (rather than those that just rely on the roll of a die) also encourage pupils to use reasoning. The following two examples can be used for a range of topics.
Let Me In!
The guards decide on the rule that will allow numbers inside such as even multiples of three. They can choose just one condition to make it easier. They do not disclose the rule.
Other players create numbers which may fit the rules.
When players show a number, the guards check if it fits the rule.
If the number fits, the guards allow it inside.
After they have tried 5 numbers, players write the rule on a folded piece of paper. The guards read out the suggested rules.
If no players suggest the correct rule, they have another 5 attempts.
All players who work out the rule correctly score 2 points.
This can also be played with shapes where pupils have to ask if their shape can be let in according to the hidden criteria.
This game involves rolling dice but the player can put the number rolled in any box, including those belonging to other people. This is a game that can be adapted for a variety of topics within number and played in pairs to encourage discussion.
One example for fractions is:
Each player draws a fraction where the numerator and denominator are replaced by boxes.
Roll a 0 to 9 die to create your starting number.
You can place the digit in any of the numerator or denominator positions (in your fraction OR your opponent’s fraction!)
On your turn, roll the die and agree with your partner which box to put the digit into.
Explain why you have chosen that box.
Each pair or player has their turn until the fractions are complete.
The player with the smaller fraction is the winner.
There are some excellent reasoning activities available for free.
NRICH provides a huge range of problems across the whole age range and when first published, students can submit solutions. They then select some of these student solutions as answers that can be shared with pupils.
NCETM provide online workshops that delve into reasoning within a range of topics.
White Rose Maths provides schemes of learning that include reasoning activities for each aspect of the primary maths curriculum.
For reasoning activities with a Christmas theme, visit the Christmas Giveaway at Mathsmoves.
While discussion helps reasoning, pupils need to be able to record their ideas.
Pupils will need support in explaining their reasoning; they need lots of modelling for effective recording. To encourage discussion, you can use one piece of A3 paper between two pupils and take it in turns to write the explanations. Students also need to practise explaining through diagrams, images, calculations and some words rather than writing long paragraphs.
Let students spend a few minutes swapping tables and viewing another explanation for the same piece of work. They can vote on which is the clearest and most concise explanation and they can display this on a learning wall which pupils can then reference as needed. The more confident students become with recording their reasoning, the more time you can spend working with groups of pupils on developing a greater depth of understanding.