1. Graham Griffiths, David Kaye and Ruth Moulton Pythagoras, Phi, Probability and Pi; taking the p, and putting it into mathematics. Presentations by trainee teachers and their views of mathematics

2. New training course New qualifications for numeracy teachers Greater emphasis on pedagogy and mathematical thinking Opportunity to explore teachers views and beliefs about Mathematics

3. Current interest in effect of views of mathematics on teaching include Paul Ernest: “Teaching reforms cannot take place unless teachers’ deeply held beliefs about mathematics and its teaching and learning change” Sue Sanders: difficulties in identifying teachers views

4. Our Study Questionnaire about beliefs of mathematics, teaching and learning – to position our teachers Assignment - to find out about views of mathematics

5. Questionnaire Based on one by Malcolm Swan -used with 63 teachers Half our teachers (15) completed the questionnaire Teachers own perception of views of mathematics Useful to make a comparison with Swan’s cohort Gave us confidence our teachers were not untypical

6. Categorisation using the Questionnaire Give percentage value to each of three statements about views on Maths Maths teaching Maths learning Categorisation from study by researchers at Kings College, London Connectionist Discovery Transmission

7. Categorisation of Questions Mathematics is.. a given body of knowledge and standard procedures. A set of universal truths and rules which need to be conveyed to students. Transmission a creative subject in which the teacher should take a facilitating role, allowing students to create their own concepts and methods. Discovery an interconnected body of ideas which the teacher and the student create together through discussion. Connectionist

8. Categorisation of Questions Learning is.. individual activity based on watching, listening and imitating until fluency is attained. T ransmission individual activity based on practical exploration and reflection. Discovery interpersonal activity in which students are challenged and arrive at understanding through discussion. Connectionist

9. Categorisation of Questions Teaching is.. structuring a linear curriculum for the students; giving verbal explanations and checking that these have been understood through practice questions; correcting misunderstandings when students fail to 'grasp' what is taught. Transmission assessing when a student is ready to learn; providing a stimulating environment to facilitate exploration; avoiding misunderstandings by the careful sequencing of experiences. Discovery a non-linear dialogue between teacher and students in which meanings and connections are explored verbally. Misunderstandings are made explicit and worked on. Connectionist

10. Categorisation of teachers based on questionnaire. C (2) D (1) TD (1) TC (2) CD (1) TDC (4) T (4) We would expect most people to hold a mixture of beliefs Using mean scores our teachers were broadly similar to Swan’s – if not a little more connectionist

11. Assignment Deliver and oral presentation of own understanding of mathematics and its position in society “ We suggest that you start with a mathematical problem and solution …. Choose a problem that is at the edge of your own mathematics knowledge (just out of your ‘comfort zone’) ” Criteria included 1. Analyse role of mathematics in the world at large 2. Discuss popular perceptions of mathematics & impact on learners 3. Discuss historic and cultural development of mathematics 4. Analyse links and connections that can be made between areas in mathematics 5. Analyse the processes within a mathematical problem

12. Presentations Our anticipation teachers would put a problem at the centre of their presentation In the event teachers had different focii topic focus: e.g. history of numbers conceptual focus: e.g how do we multiply decimal numbers application focus: e.g use of π problem focus: e.g predicting number of victims in outbreak of Ebola fever

13. Presentations Variety of views of mathematics Might have been influenced by criteria Where not problem focused difficulty satisfying criterion 5 (process of problem solving)

14. Future Work We hope to repeat the study during the next course Hope to also compare approach to assignment with teaching practice

15. Problem History and social uses of probability calculations, links with other areas of maths. Process of solving the problem and the solution itself. Likely hood of winning the lottery and how to maximize jackpot winnings. K Topic Connections to other areas of maths. Application in every day life. What fractions are and their history. What fractions of weekly earnings are spent and saved. G Conceptual History. Problems of calculations with time. Links to everyday maths. How is time measured. How to measure time D Application Simple problem in context of construction. Stages of problem solving. Links to other areas of maths. History and learner perceptions of Pythagoras’ theorem. Length of wood needed for an A frame that is part of the roof of a house C Categoris- ation Other areas addressedMain EmphasisProblem Posed Trainee Categorisation of Presentations

16. 8649 Totals 5421 No response 13 TDC 11 C 1 CD 1 D 11 TC 1 TD 112 T TopicApplicationConceptualProblem Presentation Beliefs Summary categorisations of all 27 teachers