ICME 2001

School algebra around the worldResults 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

EuropeFrance, Germany, Hungary, Italy, The Netherlands

The Pacific RimHong Kong, Singapore, Japan

CanadaQuebec, 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.

Curriculum 2000 Key Stage 3 Hong Kong Singapore Japanb) understand that transformation of algebraic expressions obeys general

rules of arithmetic; transform algebraic expressions by collecting liketerms; by multiplying a term over a bracket; by taking out commonfactors; by expanding the product of two linear expressions

FORM I: terms in an algebraic expression(collecting like terms, simplifying, use ofbrackets).

D1: collecting like terms.D2: algebraic manipulations; expanding products;

factorisation.D2: transformation of formulae.GCE-D: transform simple & more complicated

formulae.GCE A&D: manipulate directed nos; expand

products of algebraic expressions.

LS: addition & subtraction of linear expressions.LS-Grade 9: expanding and factorising simple

algebraic expressions; multiplication of asimple linear expression.

Distinguish between words ‘equation’, ‘formula’, ‘identity’,‘expression’

FORM 1: to understand the difference between amathematical sentence and an equation

FORM II: meaning of an identity and themaking of simple identities.

D1: open sentences; equations.

Index notation in algebraic expressions

c) use index notation for simple integer powers FORM I: simple idea of exponents.Equationsd) set up simple equations; solve simple equations by using inverse

operations or by transforming both sidesD1: equations; equivalent equations; solving

equations.LS : to help pupils understand the meaning of an

equation and be able to solve a linearequation.; the meaning of letters and solutionin an equation.

Linear equationse) solve linear equations with integer coefficients, unknown on either or

both sides; solve linear equations that require prior simplification,including those with –ve coefficients and –ve colutions

FORM I: simple linear equations in oneunknown, their construction and solution inpractical problems.

GCE A&D: solve simple linear equations in 1unknown.

LS: solve a linear equ. in 1 variable by using theattribute of equality; to help pupils understandthe characteristics of linear functions anddevelop their abilities to use them.

FormulaeDirect proportiong) set up and use equations to solve word and other problems involving

direct prop.; relate algebraic soln. To graphical representationD1: solving algebraic problems; solving problems

involving financial transactions.GCE-D: construct equations from given situations.

ES (Grade 6): to help children develop theirabilities to consider relations as two quantitieswhich vary in direct or inverse proportion; toknow the meaning of direct and inverseproportion. To know their features by usingalgebraic expressions and graphs in simplecases.

Simultaneous linear equationsh) link a graphical rep. to algebraic solution; find an approx. soln. of a pair

of linear equations by graph. methods; exact soln. using algebraicmethods; graphs of equs. with no or infinite no. of solns.

FORM II: simultaneous equations in twounknowns graphical and algebraic solns).

GCE –D: solve simultaneous linear equations intwo unknowns.

Inequalities

i) solve ineq. In one variable and represent soln. on a no. line FORM II: simple inequality and its solution on thenumber line.

Numerical methodsj) systematic trial and improvement with ICT tools to find soln. of equ.

when no analytic method

Curriculum 2000 Key Stage 3 Hungary Italy France The NetherlandsMA2 Number and algebra1. Using and applying number and algebraProblem solvinga) explore connections in mathematics to develop flexible approaches

to increasingly demanding problems; select appropriate strategiesto use for numerical or algebraic problems

End of Grade 8: Interpretingsimple maths problems,preparing a solution plan,solving and checking theproblem on the basis of thetext.

BV: read, compare & interpretrelationships between twovariables when solving concreteproblems using representationsof tables, graphs, words andformulae.

b) break down a complex calculation into simpler steps beforeattempting to solve it

MAVO: translate a chain ofelementary calculations into aformula and vice versa.

d) select efficient techniques for numerical calculation and algebraicmanipulation

Communicatingf) represent problems and solutions in algebraic or graphical forms;

move from one form of representation to another; to get differentperspectives on the problem; present and interpret solutions in thecontext of the original problem. …

End of Grade 6: Recognisingrelationships, recording themand developing the method oflearning mathematics.

LAGrade 7: to know the differentrepresentations of functions.Transforming a function intoanother representation.

Grades7-10: Applying diagramsand models in algebra and incombinatorics: appropriateapplication of mathematicallanguage and a widening rangeof mathematical signs.

MS: to solicit and express himselfand communicate in a languagethat, though maintainingcomplete spontaneity. becomesmore clear and precise, alsomaking use of symbols, graphicalrepresentations, and so on, thatfacilitate organisation of thought.

6 ième: learn to link ‘realobservations’ to diagrams, tablesand figures.

4 ième: the introduction of symbolicalgebra should be carried outprogressively looking forsituations which allow pupils togive sense to algebra

BV: translate simple relationshipsbetween 2 variables from realityinto the representations tables,graphs, words and formulae andvice versa; translate betweenrepresentations; replace thedescription of a relationship inone representation by anotherrepresentation.

g) develop correct and consistent use of notation, symbols anddiagrams when solving problems

End of Grade 6: Interpretation,analysis and translation of textinto the language ofmathematics.

MS: understanding current scientificterminology and expressing ideasin a clear, rigorous and concisemanner.

MS: to lead to the ability to beprecise, favouring a progressiveclarification of concepts andenabling the recognition ofanalogies between differentsituations, in order to find aunified view of some central idea(variable, function,transformation, structure…)

3 ième: in College letters areintroduced: as a means ofexpressing measures (forexample the formula for the areaof a rectangle), to representunknown values, to representvariables,.

3 ième: use of figures, tables, graphs,literal and symbolic expressions

h) examine critically, improve, then justify their choice ofmathematical presentation; present a concise, reasoned argument

Grades 7-10: Developingdiscussion ability.

MS: using and elaborating specificmathematical and experimentalscientific languages, which willcontribute to linguistic formation.

Reasoning

Curriculum 2000 Key Stage 3 Hungary Italy France The NetherlandsI) explore, identify, and use pattern and symmetry in algebraic

contexts, investigating whether particular cases can be generalisedfurther and understanding the importance of a counter-example;make conjectures.

End of Grade 6: Groupingparticular things through theirgiven aspects, systematisation.

Grades 7-10: Wording ofconjectures, patterns;distinguish definitions fromtheorems.

MS: recognise variable and invariantproperties, analogies anddifferences; posing problems andgenerating solutions.

3 ième: develop the ability to reason,observe, analyse and thinkdeductively.

BV: recognise & interpret thecharacteristic properties ofsimple relationships (such asmaximum & minimal values)which are useful in a givensituation.; determine regularity innumber patterns.

MAVO: determine relevant variablesin a situation & construct anappropriate table; determineregularity from table of nos. &describe this with words, graphs& formulae (including formulaein words); describe the overalldevelopment of a relationshipfrom a table.

ICME 2001

SingaporeSpecial 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.

Rosamund Sutherland:

Rosamund Sutherland:

ICME 2001

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.

ICME 2001

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.

ICME 2001

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.

ICME 2001

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’

ICME 2001

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’

ICME 2001

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’

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 — 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.

ICME 2001

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.

ICME 2001

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.

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 and 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 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.

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?