Developing Confidence in Mathematical Reasoning Bridget Culver Steve Orchard

Why? Why? Why? Why? Why? Why? Why? What is “developing pupils’ reasoning”? Developing pupils’ reasoning is NOT … Why? Why? Why? Why? Why? Why? Why?

of of of of of of of through through through through through through Learning … of through of through of through of through of through of through of … reasoning through

Purpose Of This Session Summarise What we have researched this year Let you think … What reasoning might be What reasoning might look like in the classroom Whether you might be interested in collaborating on this next year?!

Since this is all about reasoning … 52 + 53 = 105 34 + 35 + 36 = 105 Another? And another? Some more? All? Sure? Can you reason why? Could your students?

Developing Reasoning We worked on the principle that ALL students can reason. It is not based on ability – some will simply reason more complex work than others We know a student’s attainment, but we don’t know their potential… Mathematics is about…

conceptual understanding … confident, secure, flexible and connected factual knowledge procedural fluency conceptual understanding … confident, secure, flexible and connected

And resources and support to do so through giving… Time to think Time to be observed Time to prepare And resources and support to do so Time to discuss Time to develop Time to observe For me this… TIME… is the key. Some schools set curriculum time aside Some set training/discussion time aside Time that is always at a premium (in short supply!!). Time to reflect

Mathematical Reasoning ‘Reasoning’ - the action of thinking about something in a logical, sensible way… What do you think mathematical reasoning is? Why is mathematical reasoning important? What does mathematical reasoning look like for students? How do we develop mathematical reasoning in students? Reasoning…to be able to discover new facts based on those already known? Research by Nunes (2009) identified the ability to reason mathematically as the most important factor in a pupil’s success in mathematics..

National Curriculum Aims The national curriculum for mathematics aims to ensure that all pupils: become fluent …… reason mathematically ……. can solve problems ……. We are seeing these themes being explored in Key Stage 2 and Key Stage 4 Assessments. Aims The national curriculum for mathematics aims to ensure that all pupils: become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. The aims of the National Curriculum are to develop fluency and the ability to reason mathematically and solve problems. Reasoning is not only important in its own right but impacts on the other two aims. Reasoning about what is already known in order to work out what is unknown will improve fluency; for example if I know what 12 × 12 is, I can apply reasoning to work out 12 × 13. The ability to reason also supports the application of mathematics and an ability to solve problems set in unfamiliar contexts.

National Curriculum Look at the collection of problems. So how are these themes being explored in assessments? Look at the collection of problems. Can you dissect them and try to answer the following questions. What knowledge do students require to be able to solve them? How would being fluent and flexible with basic concepts assist students? What reasoning is involved? On your table are a selection of problems – the questions on the slide are on the footers of the pages. Discuss on your table – what do you think? Big Idea… Need to know where the students have come from in their learning Where they are. Where they are going so that we can best prepare them… Not focus on current ‘assessment’, but what skill would best suit for where they are likely to go… This will allow for better transition through Key Stages

Developing Mathematical Reasoning We looked at three ways that reasoning might be developed in the classroom: Planning for student responses Modelling ‘proof’ Reasoning structures

Planning for Student Responses We plan for “difficult points” We plan for likely misconceptions…. We may plan for the questions…. But how often do we plan for student responses? We designed lessons where the entire focus was on what responses the students might offer and how we could use that to engage and support ALL learners in the classroom. We designed lessons where the entire focus was on what responses the students might offer and how we could use that to engage and support ALL learners in the classroom.

Feedback Many who were reticent to contribute at first became more confident because they realised that they could answer without it being ‘right’ or ‘wrong’.

Modelling We modelled different ways of explaining or justifying results and then let students try to apply the skills to other problems Sometimes pictorially and sometimes algebraically

Feedback By seeing how it could be done more students felt confident to try other problems themselves (especially with the lower ability groups). What we don’t know is how much has been retained for next time.

Developing Mathematical Reasoning So what techniques can we use to develop mathematical reasoning? Explain why Barry is wrong. So have so many resources, which ones are most useful? Are they all?

Developing Mathematical Reasoning Which structures are most suited to this ‘context/stimulus’? Which are most suited to developing mathematical reasoning? Would something like this be useful in your department or school?

Summary Mathematical Reasoning is a key feature of the National Curriculum Mathematical Reasoning aids deep understanding of concepts There are variety of techniques which can be used in different circumstances Regular use of these appears to show improvement in a student’s confidence and ability to reason If you are interested in more work on this then please express your interest via Jurassic Maths Hub.