Conceptual coherence In mathematics, new ideas, skills and concepts build on earlier ones. If you want build higher, you need strong foundations. Every stage of learning has key conceptual pre-cursors which need to be understood deeply in order to progress successfully. When something has been deeply understood and mastered, it can and should be used in the next steps of learning. A key element of mastery is that ideas are deeply understood and don’t need to be re-taught. However, important that these ideas are used and built on as this provides the necessary consolidation.

The Secondary Teaching for Mastery Specialist Programme Session 1: What is Teaching for Mastery?

Secondary Teaching for Mastery Programme What it is Collaborative effort to develop teaching for mastery in local schools What it isn’t Prescriptive training by ‘expert’ teachers Discuss aims of programme. A collaborative effort to define TfM at Secondary, to consider 5 big ideas from primary and explore relevance Not prescriptive training resulting in expert teachers – all on a journey towards mastery. By the end of our careers we still won’t have mastered it, but will have a better understanding Aim to initially develop own practice and departments so they can demonstrate key elements of TfM and in turn develop TfM across other neighbouring schools.

What is ‘Mastery’? A mastery approach; a set of principles and beliefs. A mastery curriculum. Teaching for mastery: a set of pedagogic practices. Achieving mastery of particular topics and areas of mathematics. The essential idea behind mastery is that all children need a deep understanding of the mathematics they are learning. A mastery approach; a set of principles and beliefs. This includes a belief that all pupils are capable of understanding and doing mathematics, given sufficient time. Pupils are neither ‘born with the maths gene’ nor ‘just no good at maths.’ With good teaching, appropriate resources, effort and a ‘can do’ attitude all children can achieve in and enjoy mathematics. 2. A mastery curriculum One set of mathematical concepts and big ideas for all. All pupils need access to these concepts and ideas and to the rich connections between them. There is no such thing as ‘special needs mathematics’ or ‘gifted and talented mathematics’. Mathematics is mathematics and the key ideas and building blocks are important for everyone. 3. Teaching for mastery: a set of pedagogic practices that keep the class working together on the same topic, whilst at the same time addressing the need for all pupils to master the curriculum. Challenge is provided through depth rather than acceleration into new content. More time is spent on teaching topics to allow for the development of depth. Carefully crafted lesson design provides a scaffolded, conceptual journey through the mathematics, engaging pupils in reasoning and the development of mathematical thinking. 4. Achieving mastery of particular topics and areas of mathematics. Mastery is not just being able to memorise key facts and procedures and answer test questions accurately and quickly. It involves knowing ‘why’ as well as knowing ‘that’ and knowing ‘how.’ It means being able to use one’s knowledge appropriately, flexibly and creatively and to apply it in new and unfamiliar situations.

What is ‘Mastery’? Is Teaching for Mastery just good teaching? How does it differ from standard practice in UK Secondary Schools? What is their understanding of TfM? Opportunity to brainstorm ideas?

Conceptual Understanding What image is helpful? Procedural Fluency Conceptual Understanding INTEGRATION 6 6

Conceptual vs procedural knowledge Mathematical knowledge, in its fullest sense includes significant, fundamental relationships between conceptual and procedural knowledge. Students are not fully competent in mathematics if either kind of knowledge is deficient or if they both have been acquired but remain separate entities. When concepts and procedures are not connected, students may have a good intuitive feel for mathematics but not solve the problems, or they may generate answers but not understand what they are doing. Not just emphasising both but aiming for the integration of the two. Heibert and LeFevre - Conceptual and procedural knowledge in mathematics: an introductory analysis (Chap 1 of ‘Conceptual and procedural knowledge: the case of mathematics’ (1986).

conceptual understanding - comprehension of mathematical concepts, operations, and relations procedural fluency - skill in carrying out procedures flexibly, accurately, efficiently, and appropriately strategic competence - ability to formulate, represent, and solve mathematical problems adaptive reasoning - capacity for logical thought, reflection, explanation, and justification productive disposition - habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. The article can be obtained from: https://www.ru.ac.za/media/rhodesuniversity/content/sanc/documents/Kilpatrick,%20Swafford,%20Findell%20-%202001%20-%20Adding%20It%20Up%20Helping%20Children%20Learn%20Mathematics.pdf

5 big ideas 5 big ideas from Primary Mastery Specialist Programme Are these relevant to Secondary?

Coherence Messages Small steps are easier to take. Focussing on one key point each lesson allows for deep and sustainable learning. Certain images, techniques and concepts are important pre-cursors to later ideas. Getting the sequencing of these right is an important skill in planning and teaching for mastery. When something has been deeply understood and mastered, it can and should be used in the next steps of learning.

Representation & Structure Messages The representation needs to pull out the concept being taught, and in particular the key difficulty point. It exposes the structure. In the end, the children need to be able to do the maths without the representation A stem sentence describes the representation and helps the children move to working in the abstract (“ten tenths is equivalent to one whole”) and could be seen as a representation in itself There will be some key representations which the children will meet time and again Pattern and structure are related but different: Children may have seen a pattern without understanding the structure which causes that pattern

Fluency Messages Fluency demands more of learners than memorisation of a single procedure or collection of facts. It encompasses a mixture of efficiency, accuracy and flexibility. Quick and efficient recall of facts and procedures is important in order for learners’ to keep track of sub problems, think strategically and solve problems. Fluency also demands the flexibility to move between different contexts and representations of mathematics, to recognise relationships and make connections and to make appropriate choices from a whole toolkit of methods, strategies and approaches.

Mathematical Thinking Messages Mathematical thinking is central to deep and sustainable learning of mathematics. Taught ideas that are understood deeply are not just ‘received’ passively but worked on by the learner. They need to be thought about, reasoned with and discussed. Mathematical thinking involves: looking for pattern in order to discern structure; looking for relationships and connecting ideas; reasoning logically, explaining, conjecturing and proving.

Variation Messages The central idea of teaching with variation is to highlight the essential features of a concept or idea through varying the non-essential features. When giving examples of a mathematical concept, it is useful to add variation to emphasise: What it is (as varied as possible); What it is not. When constructing a set of activities / questions it is important to consider what connects the examples; what mathematical structures are being highlighted? Variation is not the same as variety – careful attention needs to be paid to what aspects are being varied (and what is not being varied) and for what purpose.

Comparing Lessons Look out for the 5 big ideas: Coherence Representation & Structure Variation Fluency Mathematical Thinking Lesson 1 in comparing fractions – only considering those with same denominator or same numerator Restrict learning to this key point