1. T HE NEW NATIONAL CURRICULUM : WHAT IS THE ROLE OF THE TEACHER EDUCATOR ? Anne Watson AMET 2013

2. P URPOSE OF STUDY Mathematics is a creative and highly inter- connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high- quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.

3. P URPOSE OF STUDY creative inter-connected discipline history’s most intriguing problems science, technology and engineering financial literacy ability to reason beauty and power enjoyment and curiosity

4. A IMS 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 appropriate knowledge rapidly and accurately. reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing and communicating an argument, justification or proof using mathematical language can solve problems with increasing sophistication including non-routine problems expressed mathematically or requiring mathematical modelling, by breaking them down into a series of steps and persevering in seeking solutions.

5. A IMS 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 appropriate knowledge rapidly and accurately. reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing and communicating an argument, justification or proof using mathematical language can solve problems with increasing sophistication including non-routine problems expressed mathematically or requiring mathematical modelling, by breaking them down into a series of steps and persevering in seeking solutions.

6. A IMS practice increasingly complex problems over time conceptual understanding recall apply following a line of enquiry conjecturing relationships and generalisations developing and communicating an argument justification or proof non-routine problems expressed mathematically requiring mathematical modelling persevering

7. A LSO move fluently between representations of mathematical ideas make rich connections majority of pupils will move through the programmes of study at broadly the same pace. progress based on the security of understanding pupils who grasp concepts rapidly should be challenged those who are not sufficiently fluent with earlier material should consolidate their understanding

8. O FSTED SUBJECT SURVEY VISITS Teaching is rooted in the development of all pupils’ conceptual understanding of important concepts and progression within the lesson and over time. It enables pupils to make connections between topics and see the ‘big picture’. Problem solving, discussion and investigation are seen as integral to learning mathematics.

9. Constant assessment of each pupil’s understanding through questioning, listening and observing enables fine tuning of teaching. Barriers to learning and potential misconceptions are anticipated and overcome, with errors providing fruitful points for discussion.

10. T HE ROLE OF THE TEACHER EDUCATOR To embed problem-solving throughout mathematics teaching and learning

11. P ROBLEM SOLVING – THREE KINDS Procedural : Having been subtracting numbers for three lessons, children are then asked: ‘If I have 13 sweets and eat 8 of them, how many do I have left over?’ Application : A question has arisen in a discussion about journeys to and from school: ‘Mel and Molly walk home together but Molly has an extra bit to walk after they get to Mel’s house; it takes Molly 13 minutes to walk home and Mel 8 minutes. For how many minutes is Molly walking on her own?’ Conceptual : If two numbers add to make 13, and one of them is 8, how can we find the other?

12. OTHER ROLES OF THE TEACHER EDUCATOR Looking at the challenges for teachers in the new curriculum bearing in mind: Their likely school experience – varied but including procedural text-focused work Their recent work towards the 2007 curriculum – more problem-solving, functional mathematics Current pressures in school – test-focused, acceleration, grade-trade New intentions – same curriculum for all, increased conceptual challenge Habits in school – levels, three-part lesson etc.

13. creative inter-connections history STEM finance reasoning beauty & power enjoyment & curiosity practice increasingly complex problems over time conceptual understanding recall & apply follow a line of enquiry conjecture relationships develop & communicate justification & proof non-routine mathematical problems mathematical models perseverence shift between representations make connections enrichment consolidation What will teachers find new/difficult? same pace progress based on understanding

14. T EACHER EDUCATOR FOCUS inter-connections (TE) reasoning (TE) beauty & power (TE) enjoyment & curiosity (TE) creative (web) history (web) STEM (web) finance (web)

15. T EACHER EDUCATOR FOCUS practice increasingly complex problems over time conceptual understanding recall & apply follow a line of enquiry conjecture relationships develop & communicate justification & proof mathematical models perseverance (SBTE?) non-routine mathematical problems (web?)

16. T EACHER EDUCATOR FOCUS shift between representations make connections enrichment consolidation same pace (SBTE) progress based on understanding (SBTE)

17. T HE ROLE OF THE TEACHER EDUCATOR To develop the teacher’s capability to ask important questions such as: How does this idea connect to the rest of the curriculum? What kinds of reasoning are required/made possible by this mathematical idea? How does this idea make learners more powerful? What sources of curiosity are lurking in this idea?

18. M ORE QUESTIONS How can questions become more complex over time? What timescale – hours, days, weeks, years? What does it mean to understand this idea? What has to be recalled and how? What has to be noticed to know when to apply this? What can be enquired about/conjectured and how? What needs justifying/proving and how? What does this idea contribute to a modelling perspective? What needs perseverance and over what time period?

19. M ORE QUESTIONS What representations are useful and how do they relate? How can this idea be presented so that it connects? What deeper understandings/applications are possible? What aspects of this idea might need consolidation? What is a reasonable learning goal for everyone, and how will enrichment and consolidation be managed?

20. I MPLICATIONS FOR MATHEMATICS TEACHER EDUCATION Subject-specific focus Working on mathematics Range of experiences to reflect on Questioning habits to plan and evaluate teaching Distinguishing between what is learnt by being told; what is learnt by watching others do mathematics or teach mathematics; what is learnt by doing mathematics; what is learnt by teaching mathematics