What is great teaching and how to get it ‘Developing primary teachers’ maths skills… educating not training - a sample of the Primary Maths Programmes funded by the London Schools Excellence Fund Ruth Williams, Lampton School’ Ruth Williams SLE

What is number sense?

What is number sense? The term "number sense" is a relatively new one in mathematics education. It is difficult to define precisely, but broadly speaking, it refers to "a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms" (Bobis, 1996).

1. Dot arrangement Consider each of the following arrangements of dots. What mental strategies are likely to be prompted by each card? What order would you place them in according to level of difficulty?

2. My Numbers 1.5 10 24670 2 327 6 Number of miles on my odometer Number of sisters I have How old my car is Number of cats I’d like to have My door number Number of years I lived in my house

Arithmetic Proficiency Arithmetic Proficiency: achieving fluency in calculating with understanding An appreciation of number and number operations, which enables mental calculations and written procedures to be performed efficiently, fluently and accurately.

Arithmetic Proficiency Arithmetic Proficiency: achieving fluency in calculating with understanding Public perceptions of arithmetic often relate to the ability to calculate quickly and accurately – to add, subtract, multiply and divide, both mentally and using traditional written methods. But arithmetic taught well gives children so much more than this. Understanding about number, its structures and relationships, underpins progression from counting in nursery rhymes to calculating with and reasoning about numbers of all sizes, to working with measures, and establishing the foundations for algebraic thinking. Ofsted report – Good practice in primary mathematics

Developing mathematical skills MISCONCEPTIONS INCREASED TEACHER CONFIDENCE STRATEGIES RESOURCES INCREASED MOTIVATION AND ENGAGEMENT PROBLEM SOLVING IMPROVED PRACTICE RESILIENCE & PERSEVERANCE IMPROVED PUPIL OUTCOMES

Mathematics in action How would you do 672 – 364? How would you expect to see it being taught? Would you expect to see the same strategy every time? © columinate 2013

Conceptual Understanding Mathematics in action How would you do 672 – 364? How would you expect to see it being taught? Would you expect to see the same strategy every time? BALANCE Procedural Fluency Conceptual Understanding INTEGRATION © columinate 2013

Developing mathematical skills NUMBER SENSE AND SKILLS FLUENCY STRATEGIES TYPICALLY SUCCESSFUL MATHEMATICIAN CONCEPTUAL UNDERSTANDING

Mathematics in action 672 – 364 what next? © columinate 2013

Mathematics in action 672 – 364 what next? How would you extend the more able? © columinate 2013

Mathematics in action 672 – 364 what next? How would you extend the more able? How can you deepen understanding rather than just increasing procedural fluency? © columinate 2013

Mathematics in action 672 – 364 what next? How would you extend the more able? How can you deepen understanding rather than just increasing procedural fluency? What about estimation and justification? © columinate 2013

Mathematics in action What about estimation and justification? Improving teaching and learning by deepening understanding © columinate 2013

What is great teaching and how to get it Situations seen: Theme of lesson: Calculate squares, cubes and roots Extending the most able: Use the 6 laws of indices Ruth Williams SLE

What is great teaching and how to get it Ruth Williams SLE

Final Thought: “Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.”