School algebra around the world Results of a

School algebra around the world Results of a study funded by the QCA Rosamund Sutherland Graduate School of Education University of Bristol with support from Hans-Joachim Vollrath, Federico Olivero, Antoine Bodin, Pieter Mans, Kam Yan Lai, Cher Ping Lim, Norifumi Mashiko, Lesley Ford, Carolyn Kieran, Rina Zaskis, Marj Horne, Tommy Dreyfus, Derek Foxman, Sarah Landau ICME 2001

Europe France, Germany, Hungary, Italy, The Netherlands The Pacific Rim Hong Kong, Singapore, Japan Canada Quebec, British Colombia Australia Victoria Israel ICME 2001

Some caveats Many countries are in the process of changing their mathematics curriculum. Study based on analysis of curricula, text books and examination papers. Relationship between teachers’ practice and these structuring factors is complex. Different ways of expressing curricula makes it difficult to make comparisons. ICME 2001

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Rosamund Sutherland: Singapore Special course for top 10% — repeating of years — exam at 15/16 Emphasis on ‘using and applying’ ‘formulate problems into mathematical terms, apply and communicate appropriate techniques of solution in terms of the problems’ Algebra introduced through generalising and looking for patterns. Special course incorporates more formal ideas of function than English curriculum & more emphasis on transforming and manipulating algebraic expressions. ICME 2001

Hong Kong Top stream separated — repeating of years — exam at 15/16 Little explicit mention of ‘using and applying’ & mostly with respect to word problems or science subjects. ‘Practice in translating word phrases into mathematical phrases’. Algebra not introduced through ‘generalising and looking for patterns’. More emphasis on transforming and manipulating algebraic expressions than English curriculum. ICME 2001

Japan No streaming in compulsory education — high stakes exam for senior high school (15 -18) No explicit mention of ideas related to ‘using and applying’. Emphasis on relationships between variables. ‘to help children develop their abilities to represent concisely mathematical relations in algebraic expressions and to read these expressions’ (6 - 12) Use of multiple representations in text books. Emphasis on transforming and manipulating. ICME 2001

Hungary Setting/streaming — repeating of years — no exam at 15/16 Algebra not introduced through generalising and looking for patterns. Explicit reference to ‘word problems’ ‘interpretation, analysis and translation of text into the language of mathematics’ Considerable emphasis on functions & transformation of functions logical connectives & proof Application of mathematics mainly in Physics & Chemistry. ICME 2001

France no official streaming — repeating of years - low stakes exam at 14/15 Algebra not introduced through generalising and looking for patterns. Not much algebra until Year 9 (age 12 -13) equivalent. But in Year 11 equivalent pupils are expected to start working with complex systems of equations & functions. Emphasis on resolving problems. ‘in all domains the resolution of problems is an essential objective’ ICME 2001

Italy No setting — repeating of years — exam at end of Year 9 equivalent Rather a gentle introduction to algebra in middle school, which is accelerated in Liceo where emphasis is on formal systems of equations, functions & transformations of functions (similar to France) Emphasis on communication ‘to solicit and express himself and communicate in a language that, through maintaining complete spontaneity becomes more clear and precise, also making use of symbols, graphical representations and so on, that facilitate organisation of thought’ ICME 2001

The Netherlands 4 types of secondary education with 1 st 3 years common to all (12 -15) Emphasis on relationships, connections between different representations and connections with reality. ‘The relationship between variables using multiple representations (tables, graphs, words and formulae) and recognising characteristics and properties of simple relationships’. ‘compare two relationships with help from corresponding table and estimate when they are equal’ ‘from specific points and the form of a graph make conclusions about a related situation’. ‘tables and graphs have their own advantages and disadvantages as representations’ ICME 2001

Israel Algebra not introduced as a means of generalising from patterns. Curriculum very much influenced by the modern mathematics movement. Year 7 (12 -13) Introduction to functions The concept of a function as a special relation between sets (domain and co-domain/range). Emphasis on multiple representation of function. Emphasis on both word problems and mathematical modelling. ICME 2001

Canada — BC All pupils follow same course until age 14/15 then can choose between Applications of Maths and Principles of Maths — no exam at 15/16 Algebra introduced as a means of generalising from patterns. Grade 8 ‘it is expected that students will analyse a problem by identifying a pattern and generalise a pattern using mathematical expressions and equations; test mathematical expressions or equations by substitution and comparison of patterns; display in graphic form the table of values created from an algebraic equation and draw conclusions from the pattern created; translate between a verbal or written expression and an equivalent algebraic equation. Relatively formal work with polynomials and functions by Key Stage 4 (age 13 -15) equivalent. ICME 2001

Canada — Quebec All pupils follow same course until aged 14/15. Examination at age 15/16 (multiple choice and problem solving) An emphasis on presenting problems in which “algebra is a more efficient means of solution than arithmetic: ”. A strong emphasis on links between representations. A relatively delayed introduction to symbolic algebra. Considerable emphasis on functions and properties of functions for those studying at higher levels at age 15/16. ICME 2001

Australia —Victoria No official streaming — no official repeating of years — no exam at 15/16 Algebra introduced as a means of generalising from patterns. Curriculum organised into: expressing generality, equations and inequalities, function, reasoning and strategies. Expressing generality: use a method of algebraic manipulation such as factorisation, the distributive laws and exponent laws, and elementary operations and their inverses to re-arrange and simplify mathematical expressions into equivalent alternative forms. ICME 2001

Algebra as a study of systems of equations Some countries place more of an emphasis on algebra as a study of systems of equations (for example France, Hungary Israel and Italy) than other countries. This then tends to develop into a more formal approach to functions and transformations of functions. ICME 2001

Word problems and modelling The idea of introducing algebra within the context of problem situations is evident within most of the countries studied, although these ‘problem situations’ are sometimes more traditional word problems (for example in Italy, Hungary, France, Hong Kong) and are sometimes more ‘realistic modelling situations’ (Canada, Australia, England). In general where there is more emphasis on solving ‘realistic problems’ there tends to be less emphasis on symbolic manipulation (for example in Canada (Quebec) and in Australia (Victoria)). ICME 2001

Functions and graphs The curricula differ in their approach to the introduction of graphs, with some countries (for example Hungary, France, Israel, Japan) predominantly introducing graphs within the context of the treatment of functions and the transformation of functions and other countries (for example England, Australia) introducing graphs within the context of modelling realistic situations. Both British Colombia and Quebec in Canada place more emphasis on transformations of functions than is the case in England Australia. ICME 2001

Computers and algebra The majority of the countries make an explicit reference to the use of computers in the curriculum for example in the Netherlands: use a simple computer program when solving problems in which the relationship between two variables plays a part. ICME 2001

Similarities across anglosaxon countries There are similarities between the algebra curricula of the countries, England, Australia and Canada (BC) and in particular this relates to an emphasis on algebra as a means of expressing generality and patterns. The Singapore curriculum also reflects this aspect of algebra and indeed the Singapore curriculum is very influenced by the English examination system. ICME 2001

The Pacific Rim countries The algebra curricula of the three Pacific Rim countries studied (Hong Kong, Singapore and Japan) are not particularly similar. ICME 2001

Symbolic algebra There appears to be an earlier emphasis on the use of symbolic algebra in Japan than in any of the other countries studied. Japanese is not an alphabetical language and so meanings which Japanese pupils construct for these literal symbols are likely to be different from those constructed by pupils in countries which do use alphabetical languages. ICME 2001

Differentiation In general the nature of the schooling system (comprehensive or not) seems to influence the way in which algebra is introduced. With the exception of Japan in countries in which there is a comprehensive education system (England, France and Italy up to age 14 -15, Canada, Australia) there is less of an emphasis on the symbolic aspects of algebra during this comprehensive phase. ICME 2001

Differentiation In France and Italy the expectations on students with respect to algebra when they enter the Lycée/Liceo increases substantially in comparison with what is expected in the ‘middle schools’ within these countries. ICME 2001

Questions How are differences in curricula related to cultural and historical differences? How is algebra research influenced by curriculum in a particular country? How is curriculum in a particular country influenced by research? ICME 2001