One of the central aims of the National Curriculum for mathematics is Reasoning.
“The national curriculum for mathematics aims to ensure that all pupils...
reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language”
The National Curriculum (DfE, 2013)
Although it is a central tenet, the National Curriculum is very light on both the detail and progression in this key skill. There are occasional references through words synonymous with reasoning. For example, pupils are required to ‘make connections’, ‘make deductions’ and there is one reference to ‘conjecture’ in the year 5 non-statutory guidance.
Reasoning is rather nebulous and is inextricably linked with problem solving in mathematics. Some mathematical commentators believe that the skills of reasoning cannot be explicitly taught. My opinion though is that this is, at best unproductive and, at worst, unhelpful. The skills involved in reasoning might be tricky to identify but they underpin much of the entire curriculum and are crucial aspects of pupils’ learning behaviours. The skills are obvious in reading. Both inferential and deductive reasoning are requisites in refining understanding to ‘read between the lines’ - to attain a deeper sense of a text. The skills are also crucial in science, history and geography to mention just a few more areas of the curriculum.
To help teachers develop these skills in classrooms, it is essential that they are made explicit in terms of both the actual foci and their progression. In this way, teachers can develop rich and effective opportunities for reasoning in mathematics. The concepts in mathematics will be the context; reasoning will be the way in which pupils ‘go about’ their maths. The ‘tentacles’ of reasoning permeate and connect concepts. In this way, it holds mathematics together and is the vehicle to develop deeper thinking.
Reasoning is the pathway to success. The National Curriculum agrees – it just doesn’t help much. Teachers need support to make this a reality in the classroom. Just as learning doesn’t happen by osmosis, neither does teacher development.
Three aspects need development:
- explicit progression and teaching of specific skills
- tweaked activities that provide rich opportunities to practice
- refinement of language (teacher-pupil and pupil-pupil dialogue) and representation
All three provide an effective starting point.
Ways forward: think big; start small
Explore the HfL progression in ‘working mathematically’ document. It identifies a range of explicit reasoning skills. Learning prompts are outlined in phases and the development of each sub-focus can be tracked.
- Make connections
- Draw conclusions
Key reasoning skills, ‘Progression in Working Mathematically’ (HfL, 2015)
Use these to review your curriculum offer and design both opportunities to teach the skill but also high quality opportunities to practice. Think about how teachers model what effective reasoning sounds and looks like. Make sure you provide opportunities for pupils to review, evaluate and importantly refine their conclusions and explanations. As a school, build in opportunities to share pupils’ representations and establish what effective reasoning should look like. This helps to not only establish expectations but can also be used as models for pupils.
Develop teachers’ questioning and the language of reasoning. Think about the kinds of question stems that support pupils to think deeply about what they are doing, how this is effective and how it can be improved. Teachers can select a range of stems to frequently use and develop good habits such as those below. A few years ago, I worked with a group of teachers with this focus. They selected some key questions in lessons, pinned up copies by their board and explained the purpose to pupils. Teacher modelling is key. I spoke to some of the teachers a short while later. They were delighted to reveal that not only were they now more habitually thinking about good questions but that pupils now asked these of each other!
Another practical tip is to use language response stems such as, “It could / couldn’t be because... It will / won’t work because... If I started again, I would...”
My top tip for this is to handwrite them on large strips of card and again stick them by the board. Don’t staple them! That way they can be taken down and used. When you want a pupil to refine their sentence, provide them with the strip and time to rehearse.
The National Curriculum in England Key stages 1 and 2 framework document
(September 2013). Department for Education (DFE-00178-2013).
Progression in Working Mathematically. HfL Maths team (2015).