Reasoning: Where to start?

    Published: 31 October 2018

    In this guest blog, Lucy Wilson, maths subject leader at Wheatfields Junior School in St Albans, shares how she and her team successfully improved children's ability to reason mathematically. 

    A school example

    We believed that focusing upon developing reasoning will provide us with a core strategy to deepen pupils’ learning further and will help to increase the proportions of pupils reaching the expected and higher standards.

    Description of approach

    Across the school, we noticed that when pupils were provided with activities that required reasoning, they fell down. It is a clear feature of the end of key stage testing regime in mathematics. Many pupils struggle to ‘reason’ to an answer in some of the test questions, often not because they do not understand the mathematical concept being focused upon but because they struggle to articulate their thinking clearly. We wanted our pupils to develop a deep, long-term and adaptable understanding of Maths. Although the school focus upon reasoning was not solely for the purpose of performing better in tests, it would be a very positive outcome of deepening pupils’ understanding and the strategies they use to work on their mathematics.

    Scrutiny of work in books also identified that the opportunities pupils were given were not as reasoning-rich as we would like. Opportunities were missed. The quality of tasks and pupils’ reasoning responses in books needed both development and refinement.

    We have previously focused upon developing teachers’ questioning with good impact but we also now realised that developing reasoning means much more. It involves developing pupils’ representations and explanations in their books. In this way, it also provides greater evidence of the security of learning. Similarly, the misconceptions that arise provide exploration points for pupils to review, justify and prove further.

    Teachers were familiar with some of the language of reasoning. Many of the words were frequently used in classrooms. However, evaluation indicated that teachers were unsure about what this needed to look like during teacher inputs and outcomes in books.

    Teachers needed support to develop their understanding and the tools to seize as many reasoning rich opportunities as possible. This had two elements - explicit opportunities to develop reasoning and building in every day opportunities as part of effective teacher-pupil and pupil-pupil dialogue.

    Exploring the HfL progression in ‘working mathematically’ document provided a starting point. It helped us identify the range of explicit reasoning skills. The learning prompts in each phase supported us to understand each sub-focus. It also helped our assessments and tracking.

    We also accessed the suite of ‘Reasoning-Rich Classrooms’ courses provided by HfL. This was incredibly helpful in considering both source material and a range of effective classroom approaches to develop this focus.

    Following the training days, we decided to explore two approaches. The first focused upon developing both teachers’ modelling and pupils’ use of reasoning language. The second focused upon activity design and the structure within lessons to build this learning up.

    Staff meetings were used to work on these focuses. The subject leader attended planning meetings with other year groups to support staff further. Firstly, we considered our teaching inputs and how well we were using these to promote and encourage reasoning. We considered different ways to start lessons with a problem, practical activity or statement that would allow collaborative learning to take place to make use of the reasoning language. We created language sentence starter mats from the course material which are now always accessible on tables and displayed on our working walls. Teachers planned how to use this language during lesson inputs to model or refine and develop children’s comments and ideas further. Our marking began to acknowledge more where children had used reasoning language and became much more focused on this rather than correct answers or procedures.

    We also looked at challenges provided to the children for self-selection and considered to what extent these all promoted and required the skills of reasoning. We wanted to ensure that it was not just challenge 2 and 3 that involved elements of reasoning and that all children were accessing ‘reasoning rich’ activities. In planning sessions, teachers considered how carefully chosen questions (such as what can’t it be? How do you know?), reasoning style activities (such as find the fault and fix or odd one out) and tweaks and adaptations to these (by adding a new dimension or rule) could then encourage more reasoning to take place.

    It was also necessary to change our mind-sets on recording and expectations in books to ensure the evidence of their reasoning was prevalent. For example, we considered the value of a photo and to what extent this really showed the depth of understanding through that task. Speech bubbles are now used frequently alongside photos of tasks to enable students to demonstrate their understanding and use the language of reasoning.

    Reasoning language

    We developed a range of language prompts to use in the classroom. They appear on working walls. Instead of becoming wallpaper, they are frequently taken down and focused upon. Crucially, they are modelled by the teacher and used by pupils.

    Many of our pupils are very quick to respond to questions but did not always provide the mathematically detailed responses we were looking for. “I just knew it,” was a common response. These language prompts and the structure provided by teachers helped to develop pupils’ internal thinking and now lead to far more refined explanations. Pupils are provided with both prompts and time to improve the quality of their explanation on their own and with others.

    The modelling of what a ‘quality explanation’ sounds like by the teacher, by peers and time for pupils to move beyond a simple ‘because’ have been the catalysts for some vastly improved pupil reasoning.

    Working wall

    Activity design

    Teachers spend a lot of time looking for activities for mathematical reasoning. Though there are many sites to explore for ideas, we recognise that it is exactly that – ideas. The focus of development here, therefore, was about helping teachers refine an activity to ‘make it better’. We asked teachers to give themselves a little bit of time to reflect and explore the activities they have devised to think about how they could be improved to develop better pupil thinking. When considering an activity, we now ask ourselves if there is an opportunity for reasoning to take place.

    Reasoning example 1

    We have also focused upon ensuring that, as teachers, we model how to engage in rich activities. We know that it isn’t simply a matter of providing a great activity. That might provide the opportunity but won’t guarantee successful learning. Teacher modelling is the key and has improved pupils’ access and response. They are aware now of what it needs to look like. In some cases, we initially reduce the mathematical demands of a task so that pupils can focus their energies upon using their reasoning skills to work mathematically.

    Reasoning example 2

    Previously, we have focused upon developing our marking. In particular, providing pupils with improvement points to re-explore. But now we are more aware of the need to use these improvement prompts as part of the next lesson. If they are significant learning steps then they should be explored in the next lesson together.

    Impact and recommendations

    The impact has been very positive. Book looks show that pupils are all readily engaging in reasoning activities and teachers are able to assess responses with greater clarity of pupils understanding and help to consolidate or extend further. The language of reasoning can be heard in daily maths lessons across the school and is becoming habit. Teachers feel more confident planning and delivering lessons which are underpinned by the skills of reasoning. Our SATS results in 2017 also indicated a 10% improvement in standards. 89% of pupils achieved ARE in comparison to the previous year of 79%. 34% of pupils also achieved the higher standard. In 2018, these improvements have been maintained.

    This impact was also further validated in the recent Ofsted inspection:

    “[senior leaders] have ensured that teaching and learning have improved since the last inspection. For example, the work to improve the teaching and learning of mathematics has resulted in a curriculum that is a strength of the school. In all classes visited, pupils regularly carry out reasoning and problem-solving activities that stretch pupils of all abilities and develop resilience and fluency with numbers.

    Progress in mathematics last year was a bit slower than in reading at the end of Year 6. The changes made by the mathematics leader have been very effective in ensuring that there is consistently good-quality mathematics teaching and learning across the school.”

    Short inspection of Wheatfields Junior Mixed School, Ofsted, 6th June 2017

    We are pleased with the impact so far but plan to continue to develop this further. We will continue to ‘milk’ activities for as much reasoning as we can get and plan to develop children’s responses even further through pictorial representations and written and oral answers. Integrating reasoning as part of the pupils’ mathematical journey within extended tasks is another area of development. Supporting pupils to pause, review and consider refinements to work ever more effectively is an example of this.

    Wheatfields Junior School is a two-form entry Junior school in the St Albans district of Hertfordshire. The school was rated ‘good’ by Ofsted in June 2017.


    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).

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