In two articles, we describe how we are meeting the ambitious Key Stage 1 objectives of the 2014 mathematics curriculum through mathematical writing. The 2013 curriculum requires learners ‘to move fluently between representations of mathematical ideas.’ Children now have to study all four arithmetic operations and fractions as operators for small numbers from Year 1.
Schools have experienced three major waves of computer technology over the last 30 years: micro-computers, networked electronic whiteboards and now an emerging wave of individual one-to-one iPads and Android tablets. In the past, the UK government has found it difficult to demonstrate value from technology in schools. Despite 80 per cent of schools being rated good or outstanding by Ofsted, the country’s performance in international league tables has declined. Is it possible to do better this time? Do business-led school networks have a role in promoting one-to-one learning? Will the Coalition’s less directive approach to technology procurement allow schools to experiment with different models for learning and teaching? And, if they do, how will Ofsted assess schools as they begin to embrace a new pedagogy of flipped classrooms, enquiry and project-based learning?
Sociality Mathematics CIC has published the early results of its work developing an approach to formative assessment in Year 7 based on Gattegno's early algebra. A short video can be viewed here. Although there are many reports of the Cuisenaire-Gattegno approach being trialled in primary school, Fortran is the first reported study to reproduce Gattegno's findings at secondary level. The study was part funded by the Sutton Trust. The result is a successful proof of concept and a step change in the understanding of teachers and learners.
Recent results in a paper by John Jerrim and Alvaro Choi suggest that, although average math test scores are higher in East Asian countries, this achievement gap does not increase between ages 10 and 16. They conclude that reforming the secondary school system may not be the most effective way for England to ‘catch up’ with the East Asian nations in the PISA math rankings. Rather earlier intervention, during pre-school and primary school, may be needed instead.
What can self-organising biological systems tell us about learning and teaching? A good deal, according to Enrico Coen, a plant geneticist. In ``Cells to Civilizations'' he presents a unified account of the emergence of living organisms, and highlights common principles of development across levels -- from evolution and cell development, to learning and human culture.
Ian Benson and Anne Haworth are offering a 3 hour workshop for primary teachers and educationists at the ATM conference in Sheffield, UK on Wednesday 3rd April
Part 1: Early Algebra
Gattegno maintained that we can exhaustively identify the awarenesses needed in any domain and redefine teaching as the activity which leads students to cover this ground for themselves without missing any essential steps and without wasting time. To this end he developed the Cui curriculum and related textbooks. Like the proposed UK primary curriculum, Gattegno covers all four arithmetic operations, fractions and product tables at KS1. He did this by introducing algebra as a formal language first, before number. What does algebra look like to infants? We will cover Cuisenaire code, trains, staircases, patterns, decimal fractions and percentages in practical exercises.
Part 2 Metamathematics and Formative Assessment
Few primary teachers are familiar with mathematics as a language. What do teachers look for when they observe students working with rods and algebraic writing? What does Gattegno mean by equivalence and how does he harness the idea to create rich opportunities for students to learn? Reasoning with equivalence: colour, length, difference, parity, fractions as magnitudes, products. Reasoning about equivalence: sets, functions, domains, objects, arrows, permutations and combinations. Exercises with Complete Patterns. Formative assessment of student work in Years 1-6. Sessions based on eight years experience of re-introducing Cui in the Tizard network of primary schools.
''We had asked specifically about algebra.... Many respondents said that the Programme of Study (PoS) does not provide a coherent progression towards formal algebra, one claiming that it provided a weaker foundation than the current curriculum because the procedural approach discouraged thinking before acting.
``Many respondents suggested ways in which preparation for algebra could start in Year 1, and that expectations of algebraic thinking could be even more challenging if they were based on reasoning about relations between quantities, such as patterns, structure, equivalence, commutativity, distributivity, and associativity, and models and representations of these.
``There are opportunities to be explicit about this throughout the current draft in relation to mental calculation methods and understanding direct and inverse relations.....
``Algebra in primary connects what is known about number relations in arithmetic to general expression of those relations, including unknown quantities and variables. Operations need to be thoroughly understood in order to make this connection. The only generalising currently explicit in the PoS relates to pattern sequences and not other forms of relations.
``A coherent developmental strand for algebra should be made explicit, making clear the connections between knowledge of number, mental methods, generalizing, and representing relations between quantities and unknowns.'' (part 2, page 3)