The ability to reason has become statutory and many would argue essential to success in mathematics. Reasoning is the act of thinking about something in a sensible or logical way and allows children to make sense and make links between the mathematics they are using. The primary national curriculum lays outs its aims:
[Children should] reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language.
We will explore what reasoning might look like, how we might explicitly teach children to reason and consider the role of vocabulary and language in reasoning.
According to Drury (2014), a fifth of children fall below national expectations by the end of primary school yet teachers who teach for mastery with reasoning expect every single child in their classrooms to succeed regardless of their socio-economic background, prior attainment, gender or race. When teachers, and other adults, stop acting as if mathematics is for ‘clever people’ we can transform attitudes and achievement in mathematics: this starts with having high expectations of all learners and giving them the tools and skills to reason mathematically. Differentiation may look quite different and may be considered by depth of understanding of a given topic, whereby children may have encountered similar questions in a variety of different contexts.
What might reasoning look like?
Many schools and organisations are planning lessons which include fluency, reasoning and problem solving. Teaching for mastery requires high expectations for all learners and may be achieved using mastery principles such as the use of manipulatives and many representations, effective and precise use of mathematical vocabulary and by allowing enough freedom to reason and make links with the mathematics they are encountering. Reasoning should be preceded by essential skills and known facts to allow children to apply them to a problem. In the KS2 problem below, children will need an understanding of area (multiplying), how to divide numbers mentally and perhaps using formal written methods. This needs to be explicitly taught. A suggestion of how to do this follows.
- Explore the area of the square and consider what we would need to know about it to have an area of 64cm².
- Each side of a square is equal, so each side must equal 8cm because 8 x 8 = 64.
- As one side of the square equals 8cm, then all of the sides on the shape must equal 8cm as the square and the hexagon have sides of the same length. The equilateral triangle also has sides of the same length so must be equal to 8cm.
- Children need to know what a perimeter is and that there are 8 sides to the whole shapes outline.
- Therefore, 8cm x 9 =72cm and the perimeter of the shape is 72cm.
How can we reason?
The example above demonstrates the complexity of enquiry, conjecturing, relationships, making links and justification. This level of reasoning may not come naturally to children so we may need to give the children worked examples.
The role of vocabulary.
Effective reasoning comes from the precise use of mathematical vocabulary. Without precise vocabulary, reasoning can become wholly and vague. Some strategies to support with this may include:
Here the teacher can explicitly teach and give definitions of the words the children will need to use in order to reason accurately. This strategy does rely on children making the links and knowing where and how to start with a problem.
STEM sentences allow a teacher to guide the children through the reasoning process, often omitting the variable with a sentence for the children to change or discover. These may guide the children through the process and allow them to consider the key vocabulary in which to reason.
I say, you say, we say.
As some children may be reluctant or not as confident with reasoning, modelling and explicitly teaching vocabulary is essential. Using the “I say, you say, we say” mantra, the teacher may say a sentence, then choose targeted children to repeat the sentence back including the variable or focus vocabulary, then the whole class can be encouraged to say it. In doing so, all learners are exposed to the language and vocabulary of mathematics and hearing it in the correct context.
Over time, this strategy may be scaled back to allow the children more open freedom for reasoning. Children should be encouraged to read, speak and write the reasoning STEM sentences to use and apply the vocabulary. An example of STEM sentences can be found below.
A language rich mathematical philosophy.
As the importance of precise vocabulary is clear to be seen for reasoning mathematically, time and effort should be given to explicitly teaching it. Many schools are now using the White Rose Maths scheme which includes many excellent reasoning tasks. However, it is placed together in blocks or units such as place value, shape and measure or statistics. Great mathematicians do not use vocabulary is units, they apply it to all areas of mathematics. One way to achieve this can be through starter activities to constantly revisit vocabulary such as: yesterday, last week, last month and last term:
This approach may embed vocabulary and put it at the core of the mathematics curriculum, liberating the ability to reason. Using and applying the vocabulary precisely is key here and should be developed carefully within your school’s curriculum. When done effective, the results can be transformative.
Since the introduction of the renewed primary curriculum in 2014, SAT questions are increasingly requiring links to be made between areas of mathematics. The example below shows the complexity of some questions expected by the end of KS2.
In 2018, some of the KS2 SAT papers required written reasoning to gain marks:
The mark schemes highlight the need for precise language in order to gain marks:
Resources to support this approach:
The shop at http://mrbeeteach.mrbee.org/