ATM Annual Conference, Swansea University, Wales
``Progress in mathematics is not merely a progress from easy exercises to complicated exercises, from short theorems to long ones. It should be a progress from the concrete to the abstract, from the particular to the general, from diversity (many little patterns) to unity (fewer,larger, patterns), from the comparatively unstructured, to the comparatively structured.'' A. G. Sillito paper to the Scottish Education Committee, Mathematics Syllabus Committee, April 1963 cited in Mathematical Reflections: contributions to mathematical thought and teaching written in memory of A. G. Sillito, ed ATM, CUP, 1970
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PREMIERING: A VIDEO INTERVIEW WITH Gattegno’s collaborator Madelaine Goutard
PRE-READING (optional): AN ONLINE REPORT ON GATTEGNO/GOUTARD FROM A TIZARD SCHOOL
SESSION DESCRIPTION:
The tizard primary schools are exploring the use of multimedia technology to overcome the obstacles to wider adoption of Gattegno’s mathematics curriculum. We introduce students to the Cui approach to algebraic writing by naming particular patterns made with Cuisenaire's colored rods. These names are interpreted as instructions for a user (or computer program) to draw images using a software tool called an algebra checker. The manipulation of the contents of a box of Cuisenaire rods is represented in software as a process of structured drawing with colour- coded rectangles we call brix(TM). Attendees will learn how to re-organise Gattegno's “Mathematics with Numbers in Colour” as a common learning procedure for algebraic and number systems. And, they will learn about the experience of teachers as researchers, such as the tizard alpha site in Devon, who have introduced the Cui approach to mathematics to their schools.
Delegates are encouraged to bring laptops to take away the software tools and textbook to try the tizard approach for themselves.
REFERENCE:
From the UK Williams report on Primary Mathematics. ``Within the primary curriculum there is a clear and logical pattern, which builds on the EYFS, through number and counting to more complex and abstract concepts in mathematics. This approach has much to offer and, where it is implemented well, builds children’s confidence so that they feel ‘at home with number’. However, some schools have developed schemes and use programmes that first stress the concrete, abstract and algebraic aspects of mathematics, and then apply them to understanding number and calculation. All the programmes of this nature that were observed to be successful invariably gave the children a considerable amount of practical experience with structured materials. For example, ‘Cuisenaire’ resources were used very effectively in one school visited by the panel, where the defining criteria for success were undoubtedly the enthusiasm and expertise of the head teacher and the staff for this approach.'' (para 220, page 61)

is published by Lambert Academic Publishing.
with the Developer menu in safari to inspect the local database and create brix diagrams programmatically. Users can toggle between a database view (shown with the yellow sticky notes below and in the listing of the content of the selected database) and the presentation view (shown using the colored brix, here black (k), brown and red (nr) and orange (o).
Just a few moments ago, Apple introduced Safari 5 here at Apple WWDC in San Francisco. This is the next major version of our favorite web browser. And with Safari 5 Safari Extensions, there is a new way for our young mathematicians to add
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functionality to Safari. Follow this thread to monitor the building of SoMa - sociality's open media architecture.


Session Code A16.15-17.45``for our purposes, the commutative squares in the pullback and pushout diagrams pertain to statements about cognitive (sub)systems'' what do transitive inference etc, Philips et al, 12/2009, plos (pdf below)

``I believe that the biggest flaw in education is to assume that young people cannot grapple with abstraction at an early age. The whole world is an category of abstractions to children. The problem with teaching children the neat mathematical stuff that is there in the world is that there are far too few educated adults to re-enforce the ideas.''Scott Carter, 12/19/09 n-category cafe, page 6

Dr Ian Benson is a Cambridge mathematician and Stanford computer scientist, presently Principal Researcher for the trans-atlantic Tizard programme, and Visiting Professor at Kingston University (UK).
Tom Kilsby is Year 6 teacher and Mathematics Coordinator at Tizard primary school, West London

Tizard is a seven year old transatlantic research and development partnership for math reform: joining parents, schools, teachers and pupils in primary schools in UK, US and France. I am the principal investigator. Copies of my peer-reviewed books and research papers can be read via academia.org. Accounts on our work have been published in Aspect, the magazine of the UK organization for school improvement professionals, the Cambridge Computer Laboratory Ring Magazine, the magazine of Churchill College Cambridge and Prospect Magazine:



The Tizard Schools are re-evaluating a curriculum developed by Dr Caleb Gattegno, and known to the US and UK Departments of Education in the 1950s and 1960s as the Cui programme. William Brownell, Dean of the Education School at UC Berkeley and math educator evaluated the Scottish and English Cui programmes for the US government in 1966:

Brownell was described in Shulman (1974) as a leader in the study of the psychology of school subjects. Shulman observed that Brownell's ``brilliant experimental work on the teaching of arithmetic reads as well today as it did forty years ago.'' (Lee Shulman, the psychology of school subjects: a premature obituary, Journal of Research in Science Teaching, 1974, vol 11, pp319-339). Brownell challenged the whole concept of mental age as a criterion of readiness in learning mathematics. (W. A. Brownell. Readiness and the arithmetic curriculum, Elementary School Journal, v38, 1938, pp 344-354)
Gattegno wrote in 1955 ``the historic development of culture, if it has something to bring to our understanding of the (then) present moment, can be entirely foreign to what a mind stimulated in a new way can or could do, unforeseen in the former experience of the group. Too rigid a determinism, coupled with a slightly sentimental historicism, risk making us ignore whole continents potentially present in the mental universe.''
Brownell and Gattegno's voices remain in a minority. A substantial majority of mathematicians and educators continue to follow Maxwell's Genetic Principle: that ``the best way to guide to the mental development of the individual is to let him retrace the mental development of the race.''

Standardisation based on 19th Century curriculum unfoldings has its critics, but as Christopher H. Tienken, Editor AASA Journal of Scholarship and Practice argues it remains the main determinant of state policy in US (Common Core CCSSI) and UK (National Curriculum).


We spelt out the opportunity that is being missed in this Prospect magazine blog. You can read about our part in the campaign to revise the UK National Curriculum in Mathematics to remove extraneous content from Year 6, to make room for algebraic thinking and more time to work on fractions in the attached article from Prospect Magazine:

Tizard is part funded by the UK Department for Education (DfES), Department for Business, Innovation and Skills (DTI), Shuttleworth Foundation, Apple Inc, Devon, Leicestershire and London schools and the Rita and Rowland Benson Trust. It is hosted at Stanford Computer Science.
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