School algebra around the world

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• 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

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

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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.

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SingaporeSpecial course for top 10 repeating
of years exam at 15/16
Rosamund Sutherland

• 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.

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Hong KongTop 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.

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JapanNo 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.

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HungarySetting/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.

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Franceno 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

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ItalyNo 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

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The Netherlands4 types of secondary education
with 1st 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

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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.

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Canada BCAll 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.

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Canada QuebecAll 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.

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Australia VictoriaNo 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.

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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.

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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)).

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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 and Australia.

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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.

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Similarities across anglo-saxon 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.

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The Pacific Rim countries

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

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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.

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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.

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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.

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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?