Transcript No Slide Title

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
Curriculum 2000 Key Stage 3
b) understand that transform ation of algebraic expressions obeys general
rules of arithmetic; transform algebraic expressions by collecting like
terms; by multiplying a term over a bracket; by taking outcommon
factors; by expanding the product of two linear expressions
Distinguish between words ‘equation’, ‘formul a’, ‘identity’,
‘expression’
Hong Kong
Singapore
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.
FORM 1: to understand the difference between a D1: open sentences; equations.
mathematical sentence and an equation
FORM II: m eaning of an identity and the
making of simple identities.
Japan
FORM I: terms in an algebraic expression
(collecting like terms, simplifying, use of
brackets).
LS: addition & subtraction of linear expressions.
LS-Grade 9: expanding and factorising simple
algebraic expressions; multiplication of a
simple linear expression.
Index notation in algebraic expressions
c) use index notation for simple integer powers
FORM I: simple idea of exponents.
Equations
d) set up simple equations; solve simple equations by using inverse
operations or by transform ing both sides
D1: equations; equivalent equations; solving
equations.
LS : to help pupils understand the meaning of an
equation and be able to solve a linear
equation.; the meaning of letters and solution
in an equation.
Linear equations
e) solve linear equations with integer coefficients, unknown oneither or
both sides; solve linear equations that require prior simplifi cation,
including those with –ve coefficients and –ve colutions
FORM I: simple linear equations in one
GCE A&D: solve simple linear equations in 1
unknown, their construction and solution in
unknown.
practical problems.
LS: solve a linear equ. in 1 variable by usingthe
attribute of equality ; to help pupils understand
the characteristics of linear fun ctions and
develop their abilities to use them.
Formulae
Direct proportion
g) set up and use equations to solve word and other problems involving
direct prop.; relate algebraic soln. To graphical representation
D1: solving algebraic problems; solving problems ES (Grade 6): to help children develop their
involvingfinancial transactions.
abilities to consider relations as two quantities
GCE-D: construct equations from given situations.
which vary in direct or inverse proportion; to
know the meaning of direct and inverse
proportion. To know their features by using
algebraic expressions and graphs in simple
cases.
Simultaneous linear equations
h) link a graphical rep. to algebraic solution; find an approx. soln. of a pair FORM II: simultaneous equations in two
of linear equations by graph. methods; exact soln. using algebraic
unknowns graphical and algebraic solns).
methods; graphs of equs. with no or infinite no. of solns.
Inequalities
i) solve ineq. In one variable and represent soln. on a no.line
Numerical methods
j) systematic trial and improv ement with ICT tools to find soln. of equ.
when no analytic method
FORM II: simple inequality and its solution on the
number line.
GCE –D: solve simultaneous linear equations in
two unknowns.
Curriculu m 2000 Key Stage 3
Hu n gary
Italy
France
Th e Ne the rl ands
MA2 Number and algebra
1. Using and appl ying numb er and algebra
Problem solving
a) explore connections in mathematics to develop flexible approaches
to increasingly demanding problems; select appropriate strategies
to use for numerical or algebraic problems
End of Grade 8: Interpreting
simpl e maths problems,
preparing a solution plan,
solving and checking the
problem on the basis of the
text.
BV: read, compare & in terpret
relationships between two
variables when solving concrete
problems using representations
of tables, graphs, words and
formulae.
MAVO: translate a chain of
elementary calculations into a
formula and vice versa.
b) break down a complex calculation into simpl er steps before
attempting to solve it
d) select efficient techniques for numerical calculation and algebraic
manipulation
Communi cating
f) represent problems and solutions in algebraic or graphical fo rms;
move from one form of representation to another; to get different
perspectives on the problem; pr esent and interpret solutions in the
context of the original problem. …
g) develop correct and consistent use of notation, symbols and
diagrams when solving problems
h) examine criti cally, improve, then justify their choice of
mathematical p resentation; present a concise, reasoned argument
End of Grade 6: Recognising
MS: to solicit and express hims elf
relationships, recording them
and communicate in a language
and developing the method of
that, though maintaining
learning mathematics.
complete spontaneity. becomes
LAGrade 7: to know the different
more clear and precise, also
representations of functions.
making use of symbols, graphical
Transforming a function into
representations, and so on,that
another representation.
facilit ate organisation of thought.
Grades7-10: Applying diagrams
and models in algebra and in
combinatori cs: appropriate
application of mathematical
language and a widening range
of mathematical signs.
End of Grade 6: Interpretation,
MS: understanding current scientific
analysis and translation of text
termino logy and expressing ideas
into the language of
in a clear, rigorous and concise
mathematics.
manner.
MS: to lead to the ability to be
precise, favouring a progressive
clarifi cation of concepts and
enabling the recognition of
analogies between different
situations, in order to find a
unified view of some central idea
(variable, function,
transformation, structure…)
Grades 7-10: Developing
MS: using and elaborating specific
discussion ability.
mathematical and experimental
scientific languages, which will
contribute to linguistic formation.
6 ième: learn to link ‘real
observations’ to diagrams, t ables
and figures.
4 ième: th e introduction of symbolic
algebra should be carried out
progressively looking for
situations which allow pupils to
give sense to algebra
BV: tr anslate simpl e relationships
between 2 variables from reality
into the representations tables,
graphs, words and formulae and
vice versa; translate between
representations; replace the
description of a relationship in
one representation by another
representation.
3 ième: in College letters are
introduced: as a means of
expressing measures (for
example the formul a for the area
of a rectangle), to represent
unknown values, to represent
variables,.
3 ième: use of figures, tables, graphs,
literal and symbolic expressions
Reasoning
Curriculu m 2000 Key Stage 3
Hu n gary
Italy
France
Th e Ne the rl ands
I) explore, identify, and use pattern and symmetry in algebraic
contexts, investigating whether particular cases can be generalised
further and understanding the impo rtance of a counter-example;
make conjectures.
End of Grade 6: Grouping
particular things through their
given aspects, systematisation.
Grades 7-10: Wording of
conjectures, patterns;
distinguish definitions from
theorems.
MS: recognise variable and invariant
properties, analogies and
differences; posing problems and
generating solutions.
3 ième: develop the ability to reason,
observe, analyse and think
deductively.
BV: recognise & int erpret the
characteristic properties of
simpl e relationships (such as
maximum & m inimal values)
which are useful in a given
situation.; determin e regularity in
number patterns.
MAVO: determine relevant variables
in a situation & construct an
appropriate table; determin e
regularity from t able of nos. &
describe this with words, graphs
& fo rmulae (including formulae
in words); d escribe the overall
development of a relationship
from a table.
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 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 — 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 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 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