# School Arithmetic is to Mathematics, as Making Sounds is to Music: Some pedagogically supported transitions from arithmetic to mathematics

Music arises from the coordination of sounds to form relationships, whether in sequence as in a tune, or simultaneously, as in chords and polyphony. Arithmetic in school tends to focus on the performance of calculations (as this is readily testable), whereas mathematics is about the recognition, articulation, and study of structural relationships between, in the case of arithmetic, numbers.

Participants will be invited to engage in some tasks which may spark discussion about pedagogical actions which may promote and support a transition from arithmetic as calculation, to mathematics as awareness and exploitation of structural relationships between numbers.

BIO: John Mason has been teaching mathematics ever since he was asked to tutor a fellow student when he was fifteen. In college he was at first unofficial tutor, then later an official tutor for mathematics students in the years behind him, while tutoring school students as well. After a BSc at Trinity College, Toronto in Mathematics, and an MSc at Massey College, Toronto, he went to Madison Wisconsin where he encountered Pólya's film 'Let Us Teach Guessing', and completed a PhD in Combinatorial Geometry. The film released a style of teaching he had experienced at high school from his mathematics teacher Geoff Steel, and his teaching changed overnight.

His first appointment was at the Open University, which involved among other things the design and implementation of the first mathematics summer school (5000 students over 11 weeks on three sites in parallel). He called upon his experience of being taught, to institute active-problem-solving sessions, which later became investigations. He also developed project-work for students in their second year of pure mathematics. In 1984 he wrote Thinking Mathematically with Leone Burton and Kaye Stacey, which has turned into a classic (translated into five languages), and is still in use in many countries around the world with advanced high school students, with graduates becoming school teachers, and with undergraduates in courses in which students are invited to think about the nature of doing and learning mathematics. Learning and Doing Mathematics was originally written for Open University students, then modified for students entering university generally.

At the Open University he led the Centre for Mathematics Education in various capacities for fifteen years, which produced the influential Routes-to Roots-of Algebra, and numerous collections of materials for teachers at every level. His principal focus is thinking about mathematical problems, and supporting others who wish to foster and sustain their own thinking and the thinking of others. Other interests include the study of how authors have expressed to students their awareness of generality, especially in textbooks on the boundary between arithmetic and algebra, and ways of working on and with mental imagery in teaching mathematics. The book Researching Your own Practice: The Discipline of Noticing is one manifestation of a lifelong collection of tactics and frameworks for informing the teaching of mathematics. Along the way he has articulated a way of working developed at the Centre which provides methods and an epistemologically well founded basis for practitioners to develop their own practice, and to turn that into research.

With his wife Professor Anne Watson at the University of Oxford he has tried to support the teaching of mathematics and the development of mathematical thinking through publications such as Questions and Prompts for Mathematical Thinking; Thinkers; and Mathematics as a Constructive Activity: learners generating examples. His book with Sue Johnston-Wilder, Fundamental Constructs in Mathematics Education, provides research students with access to hundreds of extracts from the mathematics education literature, and together with Alan Graham, their books Developing Thinking in Algebra; in Geometry; and in Statistics provide teachers with a lifetime of tasks through which to inform their interactions with students.

In 2006 he became a Senior Research fellow in Mathematics Education at the University of Oxford where he takes on various tutoring and supervisory roles. Since being retired from the Open University as Professor Emeritus he continues to respond well to invitations to work with students, teachers and teacher educators in many countries around the world, creating applets to support his workshops and his mathematical thinking.

John retired from the Open University in 2009 after 40 years, but continues to work on mathematical problems, most of which arise from considering pedagogical issues in school and university mathematics. He develops applets to assist him in his thinking, which others might find useful to use with learners to foster and promote mathematical thinking.