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Sheffield, William J.. 2021. “Algebra Teaching in School”. Loughborough University.https://doi.org/10.26174/thesis.lboro.14216810.v1.
Algebra Teaching In School
by
William John Sheffield B.Sc.
A Master's Dissertation submitted in partial fulfilment of the
requirements for
Education of the
1988.
the award of the degree
Loughborough University
of M.Sc. in Mathematical
of Technology, January
Supervisor: P.K. Armstrong M.Sc.
(S)by William John Sheffield 1988.
Abstract
Abstract
Algebra is seen by many school leavers as the source of
mystification, frustration and failure in their mathematics course.
To give algebra more meaning and purpose across the whole ability
range is an immediate challenge to the mathematics teacher.
From a brief description of what can be said to constitute
algebra in the context of the secondary school, there follows a
discussion of its relevance and justification for inclusion in the
mathematics syllabus for all school pupils.
A variety of algebraic structures are considered (vectors,
transformations, matrices, sets, groups), even though their
inclusion in school mathematics is likely to be reduced with the
introduction of GCSE assessment. Of more immediate significance
for the practising teacher are the suggestions for making
generalised arithmetic more meaningful and motivated by realistic
situations and investigations.
Previews of current research into childrens errors in
algebra, give insight into childrens perception of the use of
letter and the difficulties they encounter with notation.
Increasing use of micro-computers in mathematics lessons
has implications for the teacher of algebra. They will influence
the content of the algebra syllabus as well as the teaching method.
Acknowledgements
The valuable advise and assistance given to me by my
dissertation supervisor Mr. Peter Armstrong is gratefully
acknowledged. Without his encouragement this dissertation would
never have been finished. Without his constructive criticism it
would not have been worth writing.
The use of the facilities and
Sibthorp Library, Bishop Grosseteste
appreciated.
the help of the Staff of
College is also greatly
The help of MEDU. (Micro-electronics Education Development
Unit) at Bishop Grosseteste College, Lincoln, whose expertise at
word-processing, created order out of chaos, is also acknowledged.
The encouragement, interest and invaluable help given by
Eric and Mary Middleton is warmly appreciated.
Lastly my long-suffering wife Sheila, who had to be without
a husband whilst he locked himself away reading and writing. In
addition, she revived her rusty art of typing to prepare this
manuscript. The original handwritten draft was so illegible and
disorganised that any errors must remain my responsibility.
Contents
Contents
Abstract
Chapter1 The Nature of Algebra 1
Chapter2 Why Teach Algebra? 6
Chapter3 Generalised Arithmetic 11
Chapter4 Manipulation 22
ChapterS Children's Errors and Mis-Conceptions 41
Chapter6 Other Algebras in School Mathematics 54
Chapter7 The Role of Proof in School Algebra 80
Chapter8 Algebra and GCSE. 86
Chapter9 Micro-Computers in the Teaching of Algebra 98
Bibliograph 103
Chapter 1
The Nature of Algebra.
Of all mathematical topics taught in schools today
perhaps the one in which many pupils appear to achieve the least
is algebra. In the past it was a subject restricted to the more
able students who would be required to use algebraic techniques
in post 16 courses. But increasingly more pupils are now
expected to learn some algebra. And yet pupils responses' to
algebra teaching are often disappointing. The Bath Study
"Mathematics in Employment 16-18 "registered" [33] a very strong
impression that algebra is a source of considerable confusion and
negative attitudes among pupils".
skills on numbers and symbols
They carry
without any
out manipulative
interest or
understanding. Topics such as sets and matrices seem pointless.
They engage in time filling activities to learn the conventions
of a language they will never use. The mathematical needs of
employment recorded in the Cockcroft Report (1981) [17] found
little use being made of algebra "Formulae, sometimes using
single letters for variables but more often expressed in words or
abbreviations, are widely used by technicians, craftsmen,
clerical workers and some operatives but all that is usually
required is the substitution of numbers in these formulae and
perhaps the use of a calculator ••••• It is not normally necessary
to transform a formula; any form which is likely to be required
will be available or can be looked up. Nor is it necessary to
remove brackets, simplify expressions or solve simultaneous or
quadratic equations". For many pupils then, the ability to answer
the algebra questions in a public examination appears to be the
ultimate goal of algebra teaching.
To give algebra more meaning and purpose across the whole
ability range is clearly an immediate challenge. It is a task
without a simple solution. Childrens' misunderstandings, lack of
motivation and algebraic errors are not easily remedied. Perhaps
Page 1
Chapter 1
more than any other branch of mathematics, algebra teaching needs
a re-think. The way in which it is taught, the materials and
techniques used, the resources available (textbooks, calculators,
computers) will require the skills and imagination of the
teachers to motivate youngsters. Algebra must not be seen solely
as a manipulative skill with symbols. Opportunities for pupils to
take an active part in the learning of its techniques and
conventions should arise from investigations and practical work.
There is no clear definition of what is meant by algebra.
The layman's view is usually narrow, restricted to "using letters
instead of numbers" and solving equations. In fact the division
of mathematics into various branches: arithmetic, geometry,
algebra, statistics, trigonometry etc., although at times
convenient, is rather arbitary, there being so much connection
between them. Concepts and techniques involved in one are applied
in another. But there are features of school mathematics which
are essentially algebraic in nature.
The classical algebra most commonly met in schools is
the algebra of real numbers. This is sometimes called cosa
algebra from the Italian word "COS A" for "the unknown quantity".
It is a form of generalised arithmetic characterised by a
symbolic language. Proficiency at this type of algebra involves
many skills. The ability to express generality in words or
symbols. Learning the language of algebra, i.e. understanding the
meaning ·of symbols and manipulating the symbols. Substituting
into formulas and simplifying expressions. Creating and solving
equations. Algebra is also used for codifying the laws of
operations: (commutative law, distributive law etc.). 2 + 3 = 3 + 2 is making a statement about those specific numbers whereas a +
b = b + a is a concise way of making a statement about all real
numbers. In this context algebra is not just restricted to number
Page 2
Chapter 1
theory. Laws and relations between other structures and in other
branches of mathematics are most concisely expressed in algebra.
An B = B~ A is an algebraic statement of a commutative law
applied to sets. At a more advanced level the language of algebra
is also essential for proofs in number theory as well as in
geometry and in other branches of mathematics.
Classical algebra is clearly an important part of the
school maths. curriculum, but the nature of algebra is much wider
than that. A modern algebraist would view algebra more as the
study of structure and relationships. Symbolisation, in the form
of algebraic symbols is not strictly necessary for us to
recognise an activity as being essentially algebraic. A
comparison and classification of abstract structures
characterises this view of algebra. The use of symbols or a
codified language is merely a convenience for the sake of brevity
and to enable an abstraction to be applied to a number of
situations. Let me give an example. A child may make the
following observation:
Mummy is bigger than me and Daddy is bigger than Mummy so
Daddy must also be bigger than me.
This is an algebraic statement which could be symbolised
as:
A(B and B(C =~ A<C
The symbols allow us to recognise the algebraic
structure (namely' the transitivity of the relation "is bigger
than ") in a form which will enable us to use the abstraction in
other situations. In set theory:
Page 3
Chapter 1
ACBI\BCC=rA(C
In sport: Team A beat team B and team B beat team C, so
if team A play team C, then team A should win.
The recognition and identification of the essential
features of an algebraic structure are basic skills involved when
algebra is viewed in this way. At school level this involves the
study of sets, groups, vectors, matrices, probability, logic,
patterns, symmetries of shapes and motion geometry. Symbolism and
manipulation are still important skills, but the concepts
involved are not necessarily numbers. Relationships are studied
in the abstract.
It is the abstract nature of mathematics that make it
such a difficult (almost incomprehensible) subject for many
children. Applying mathematics to "concrete" or "real life"
situations (essentially mathematical modelling) provides us with
yet another view of the nature of algebra. The modeller would
view algebra as a symbolic language. Letters being used to
represent real variables. The manipulation of these variables
being according to rules which model the real situation to which
these variables apply. Clearly with this as our definition of
algebra we are again not restricted to the algebra of real
numbers. Skills of manipulation, symbolisation and recognition of
structure are still essential, but a modeller would stress the
usefulness and applicability of the language of algebra to
interpret the real world.
Summary. I have singled out four views of the nature of algebra,
namely:
(1) Generalised Arithmetic (the algebra of real numbers).
Page 4
Chapter 1
(2) Mathematics as an axiomatic system where the
statement of laws and the processes of deduction and proof are
algebraic processes.
(3) Algebra as the study of structure.
(4) Algebra as a symbolic language to model the real
world.
Page 5
Chapter 2
Why Teach Algebra In Schools
There comes a point in the learning of mathematics when
progress can be severely impared without the techniques and
language of algebra. In this sense mathematics is a hierarchical
subject. Decimal fractions will be difficult for children until
they have some understanding of place value. Algebra can not be
fully appreciated until real number arithmetic is mastered. The
study of calculus requires confidence with equations. Trigonometry
relies on some knowledge of geometry, and so on. Anyone who studies
mathematics at a higher level, or who uses advanced maths in their
work (Engineers, Scientists, Statisticians, Accountants etc.) must
learn algebra.
There are many post 16 courses offered by Schools and
Colleges which require an ability and knowledge of
mathematics. Sciences , Geography, Engineering, Computer Studies,
etc. all make mathematical demands on their students. In the case
of algebra these demands may take the form of specific techniques
required. For example the ability to rearrange formulas or to solve
simultaneous equations. But more generally an understanding of
algebraic concepts will be expected. The ideas of variable,
relation, function, symbolisation, manipulation etc may well be
essential. Any child with the ability to study mathematics at a
high level should be taught sufficient mathematics in school to
enable their studies to continue. In particular they should be
taught algebra.
Of course some youngsters when choOSing their courses at 16 + will
opt for subject which make much less demand on mathematical
ability. Their choice should be based on knowledge not ignorance.
Their options should not be limited because of a restricted
Page 6
Chapter 2
mathematical curriculum in school. A failure to teach algebra would
impose an unnecessary and avoidable restriction on a study choice.
The requirements of post 16 education is a strong
justification for the teaching of algebra to children
average ability. But the authors of the Cockcroft Report
of above
[17] point
out: " ••• we believe that efforts should be made to discuss some
algebraic ideas with all pupils". (para 461) although they do add
the rider " ••• formal algebra is not appropriate for lower
attaining pupils". So in what ways will algebra enhance the
mathematical experience of and prove useful to the average child?
Whenever a relationship is expressed in words or symbols we
are using algebra. Many examples can be seen in everyday life. The
Cockcroft report gives the following example (para 77) of a formula
used in nursing:
child's dose = Age x Adult dose.
Age + 12
Until a few years ago (the current method is much more
complicated) the qualifying income for rate rebate was calculated
with this formula:
Q = 10(R - S) + A + 278 C
where Q is the qualifying income, R is the annual rate bill, S is
the Sewage bill (if included in R), A is the needs allowance and C
is the number of children.
It is interesting to note here the use of letters to
represent quantities rather than numbers. This is contrary to the
accepted mathematical convention, but more of that in future
chapters.
Page 7
Chapter 2
Other examples of formulas can easily be found. But other
relationships which are not expressed in a formula are essentially
algebraic. How to work out the cost of hiring a car given the daily
rate and mileage charge? What is the time in New York if it is
midday in London? How does the builder work out how much concrete
he needs for house foundations? How many rolls of paper are needed
to decorate a room? How can the housewife adapt a recipe to feed
her family?
To a greater or lesser extent algebra can
used in a variety of everyday situations. Not just in
be seen to be
the form of
an explicit formula. But whenever a relationship is understood to
exist between variable quantities then algebra is being applied.
A further justification for the teaching of algebra the way
it enhances the whole mathematical experience of the child. It
supports such aspects as: problem solving, logical deduction,
abstraction and generalisation. Mathematics is the only school
subject to devote itself to working with abstractions rather than
the concrete reality of sense experience. In the early phases of
mathematical learning, youngsters are presented with abstractions
that are closely connected with concrete situations. A number of
different situations are presented with the same abstract notion
(be it size, volume, addition, space, time etc.) in the hope that
the concept is understood. The process of abstraction is to
recognize what is common to a number of different situations and to
discard what is irrelevant. The power of algebra is its ability to
make the rules of a conceptual structure explicit.
The ability to generalise is an important mathematical
process and the most efficient way in which a general statement can
be expressed is in the form of algebraic language. This is most
powerfully seen in other branches of mathematics.
Page 8
Chapter 2
In arithmetic: a (b + c) = ab + ac is making a statement
which is true for all values of a, band c.
In geometry: A =TT r:2 is true for all circles.
In trigonometry: a = b
sin A sin B
= c --sin C
is true for
all triangles.
It is this characteristic of succinctness which makes
algebra such a powerful tool. Through algebra one can neatly
summarise a wide range of mathematical laws and relationships.
Manipulating the laws and relationships can lead to logical
deductions and proofs. In this sense, algebra truely integrates and
is essential to all of mathematics.
Finally a competance at algebra can enhance a childs
ability at problem solving. Mathematics is embodied in the idea of
problem solving. There is little point in being able to calculate
numbers, solve equations, draw graphs or any of the skills of
mathematics unless these skills can be organised to solve problems.
Algebra not only provides a tool for problem solving, but by
developing logical thought, can enhance a child's ability to devise
his own strategies for problem solVing.
Summary: I have highlighted the following justification for the
teaching of algebra:
(i)To learn the techniques required in post 16 courses in
mathematics or other subjects.
(ii)Algebra is used in everyday life.
Page 9
Chapter 2
(iii)The process of abstraction and generalisation is an
algebraic process.
(iv)Algebra is a powerful tool
mathematics.
in all branches of
(v)Studying algebra can improve a child's problem solving
ability.
Page 10
Chapter 3
Generalised Arithmetic
Seeing a generality
The ability to generalise is an important aspect of
mathematical competence. To recognise a pattern to identify commom
features of several situations, to abstract rules or techniques
from a number of different applications are all central aspects of
mathematical activity. Algebra is the language in which
mathematicians express generality.
There are many ways in which a teacher can present
children with opportunities to generalise. Childrens first
experience of area is usually through counting squares, but they
quickly see a short way of counting squares in a rectangle. If a
rectangle consists of 8 rows of 9 squares then there are 8 x 9
squares altogether. The child has extracted a general rule for all
rectangles. The relationship between length, breadth and area has
been recognised and understood. The precise way in which this
relationship can be expressed using the language of algebra can be
made meaningful to the child. The starting point is not the
formula A = lb with the teacher struggling to explain the meaning
of the symbolism, but an understanding of the relationship between
variables followed by a desire to express the relationship
concisely.
Firstly the child must "see" or recognise a relationship
or pattern and then be able to "say" or explain the relationship in
words. The need for precision and conciseness lead naturally to
symbolism in the recording of the relationship. This sequence :
Seeing, Saying, Recording is an important way for children to
understand and appreciate algebra.
Page 11
Chapter 3
Let uS take some examples: Geometric patterns are
frequently suggested as suitable vehicles for investigation in
maths lessons. See
Mathematical Education.
L.
O.U.
Mottershead (1985) [30]. Centre for
(1985) [15]. B.Bolt (1985) [4].[5].
The following example is taken from "Routes to / Roots of Algebra"
O.U. (1985) [15].
BORDERS
It is required to put a border of squares around the
following shapes. How many squares are needed to make the border?
D EB The amount of guidance given by the teacher will vary with
the age. experience and ability of the class but most children will
draw the border and count the squares or maybe the teacher will
suggest such an approach.
- - -
Now the teacher must attempt to make the children "see"
the pattern by asking such questions as :
Page 12
Chapter 3
"How many squares will make a border for the next shape in
the series?"
"Do you need a picture?"
"How many will make a border for a square 10 long and 10
wide?"
"What about 100 squares long and 100 squares wide?"
It would be tempting at this stage to introduce x. A
border for a square x long and x wide. But such an approach may be
premature. The child should firstly be asked to explain, to "say"
the method he has used to solve the problem in words or even in
pictures. There are several possible ways of doing this. Perhaps
the most straightforward is as follows:
o o
o o The border consists of 4 times the length of the square
plus 4 for the corners as shown in the diagram. This would lead to
the formula
b=4x+4
Page 13
Chapter 3
Alternatively a child may see the pattern this way:
I I I I
The border consists of 2 times the length of the square
and 2 times the length plus 2. This would give the following
formula:
b=2x+2Cx+2)
A child with a good understanding of area may even see
that the border is the difference between the area of the outer and
the inner square.
He would get this formula:
2-b=Cx+2)(x+2)-x
Page 14
The investigation can be extended
Rectangles or to even more complicated shapes.
Chapter 3
to other shapes.
We could even find the number of squares needed to make a
border 2 (or more) squares thick. Our aim is always in this sort of
investigation to get students to recognise a pattern, to be able to
express the pattern in words, and finally to use algebraic symbols
and letters to represent variables in order to write a formula.
The variety of geometric investigations leading to
possibilities of generalisation and formula construction are
virtually limitless. Patterns of dots, squares, triangles to be
extended. Number of squares on a chequer board (such as a chess
board). Numbers of cubes required to make certain solid shapes.
Building up shapes using straws, matchsticks, pipe cleaners. The
inventive teacher need never run out of ideas. See C.S. Banwell, et
aI., (1972), [3], and A. Wigley, et aI., (1978), [39].
Page 15
Chapter 3
But generalisation is not restricted to geometric
patterns. It occurs in all branches of mathematics. A child who
has learned and understood that:
1/7 + 1/7 = 2/7
should be able to cope with 1/51 + 1/51 even though he has never
met fifty oneths before. He has made the generalisation that:
+ anything
1 1 2
anything the same thing
Making the generalisation explicit by
writing:
x x x
is the process of generalised arithemetic, or algebra.
In the field of statistics the child learns that the mean
of four numbers is found by adding the numbers and dividing by
four. For five numbers we divide by 5. In general we divide by
the number of numbers. Introducing the letter n to stand for the
"number of numbers" is another example of a natural and meaningful
way of introducing generalised arithmetic.
Introducing algebra into problems of everyday arithmetic
can too often be seen to be artificial or contrived. Here is an
example from S.M.P. Book C. (1969).
"My brother is two years older than me. For example, when
he started school I was 3 and he was 5. When I started school, I
was 5 and he was 7.
Page 16
Chapter 3
On a pair of number lines, draw a mapping diagram to show:
my age ~ my brother's age
On squared paper, plot some ordered pairs representing
this relation. What do you notice? If x represents my age and y my
brothers age, what is the connection between x and y?"
A simple relationship between two variables giving rise to
a straight line graph and the equation y=x+2. The connection
between x and y is easily seen, its symbolic representation by
graph and equation have the advantages of being understandable and
meaningful to the majority of children. If it was required to find
the brothers age when my age is given, the appropriate value could
be read off the graph or found using the formula. But of course we
wouldn't really need algebra to crack this sort of straightforward
arithmetical nut. The use of letters and algebra is an unnecessary
contrivance difficult to justify. We are using algebra simply for
the sake of it, children will not see the point.
Here is a better example of how algebra can be introduced
into a problem of arithmetic.
CAR HIRE
Hiring a car costs ls. 75 per day plus 9p per mile. Seeing
how to work out the cost of hiring the car over several days for a
given mileage is not quite as straight forward. The formula:
Cost =~S.75 x number of days + 0.09xmileage
clearly makes the relationships explicit. The algebra is not only
Page 17
Chapter 3
accessjhle to the child, but also relevant and convincing. We are
using a technique appropriate to the problem.
Extending the problem to compare the cost of hiring the
above car for a week with another car hire company which offers a
weekly rate of ~100 with unlimited mileage would involve the
solution to the following equation:
8.75 X 7 + 0.09x 100
In order to find the "break even" mileage for which the 2
costs are the same. Algebra is not being introduced into the
problem just for the sake of it, but as a valid and convenient
method of solution. There is also the additional pay-off in
gaining a better understanding of the general relationship between
mileage and cost. It is important for the teacher to choose
everyday problems so that the appropriate mathematical ideas really
do increase insight and understanding. The algebraic technique
must be seen as being useful and appropriate.
Recording generality.
When a child has recognised and understood a relationship
between variables, the next stage for the teacher is to get the
child to record on paper the generality. This is usually met with
some reluctance. Firstly because children who have explained
something in speech cannot see the point in writing it down. More
important it is a lot harder to explain something in writing. It
is a more indirect form of communication, less easily modified,
permanent and rigid. But there are important reasons why the
teacher should insist on written expressions of generality.
From a practical point of view it enables the teacher to
Page 18
Chapter 3
communicate with a large number of pupils at the same time, and
assists him in diagnosing those pupils with misunderstandings or
difficulties. But of more concern are the pay-offs for the pupil.
The requirement for written expression forces the students to
clarify their thoughts and to express themselves precisely and
concisely. A written account renders the thinking open to check
and scrutiny by others. It leads the way for manipulation of
expressions a key component of algebraic activity.
Although our ultimate aim with this sort of work is for
children to be able to use full algebraic symbolism
and natural way, it should be recognised that this
time and practice. At first children will
themselves totally in words for example:
in a meaningful
will take some
usually express
"To get the next number you add four to the last number in
·the sequence".
or: "To find the perimeter of a rectangle you double the length
and double the width and then add them together".
If they are unable to express themselves totally in words,
then it is probably because they have not really understood the
generality that they are,trying to express. The longer and more
complicated the expression the more likely the student is to want
to abbreviate or use symbols. The first symbols that they are
likely to use will be numbers in figures rather than words because
these are as natural to children as words, and the basic symbols of
arithmetic +,-,X'7'~ giving rise to expressions which combine words
and symbols. For example:
New number ~ 4 X old number
Page 19
Chapter 3
or: Perimeter = 4 X length of a side
What distinguishes these kind of expressions from fully
verbal ones is the way that particular variables are given single
words or short phrases, which are combined by the symbols of basic
arithmetic.
The teacher should take the opportunity to point out any
ambiguities in student's expressions which may arise from incorrect
ordering of operations. For example:
Perimeter = 2 X length + width
requires that the adding be done before the multiplying. A child
may be tempted to write:
Perimeter length + width X 2
He has invented his own notation, ie. perform arithmetic
operations in the order in which they are written. The teacher
should use the opportunity to show the conventional notation. The
operation which causes the most problem is division. Children are •
used to the symbol~, and not to the convention of putting one
number over another separated by a straight line. The increasing • use of calculators with the key labelled with the symbol--.-, and
the decreasing importance of fractions in the mathematics syllabus
will probably make matters worse. In the early stages the teacher
should accept the conventions of the child so long as they are
unambiguous. But the teacher should gradually steer the child
towards conventional notation particularly if the child can see the
flaws in his own notation or can recognise the advantages of
convential notation.
Page 20
Children
quantities. They
example:
can use various
may have met some
0+ 17 32
or 2 X 0 = 17
or 31 X 6 =?
devices to
in their
Chapter 3
record unknown
textbooks. For
TheeJ, 0 and? are being used to represent, not general
variables, but specific unknown numbers to be worked out. There is
little to be gained at this stage from insisting that pupils use
letters for such unknowns.
The change from totally verbal expressions to mixtures of
verbal and symbolic expressions is easily made and easy to justify
in terms of precision and succinctness. The change to full
symbolism is more difficult. The ease and clarity from using
mixtures of words and symbols may well outweigh the advantages, in
terms of conciseness, of a totally symbolic expression. Whenever
generality is being recorded whether in words, symbols or a
combination of both, algebra is at work. Children are thinking
algebraically. Any notation is acceptable if it is understood by
everyone and no-one is slowed down by it. They will only employ
succinct symbols successfully themselves when they perceive a need,
and when the symbols are supported by rich associations and
meanings. The major advantage offered by full symbolism will be
appreciated when re-arrangement and manipulation of expressions is
required.
Page 21
Chapter 4
Manipulation
For too long the teaching of algebra has concentrated on
manipulative skills to the exclusion of all else. Children are
given drill exercises in "collecting like terms", "removing
brackets", "solving equations", "factorising tl, "cross-multiplying"
etc. where the uses of these skills are never made apparent. The
only reward for the child is the right answer ie. an algebraic
expression identical to the one in the back of the textbook, or the
one arrived at by the teacher. Understanding or meaning is not
stressed or even tested. The justification for learning these
skills will only come eventually to the more successful pupils who
continue their studies of mathematics to a higher level.For the
rest, the impression is given that algebra is the application of
apparently arbitrary rules to meaningless expressions of numbers,
letters and symbols.
Ideally our starting point should be problem solving. The
need for algebra to solve particular or general problems. The use
of algebra to prove generalities; to record relationships in
concise formulas. The value of rearranging awkward expressions into
a simpler form should be appreciated through practical
applications. The value of changing the subject of a formula should
arise naturally not as an artificially imposed exercise. The need
to solve equations will be apparent when a given problem is
expressed in an equation by the child.
"The
There is nothing new in
Teaching of Algebra in
the above ideas. In the report
Schools 1933", prepared by the
Mathematical Association, it was noted:
question
"The application of arithmetic and
or field of thought to which it is
also algebra
applicable
to any
normally
Page 22
Chapter 4
involves two distinct processes:
(a) Analysing the situation and expressing in words or
in symbols the calculations which will be necessary;
(b) Performing those calculations for the particular
values of the numbers involved.
Between these there may and generally will intervene:
(c) such transformation of the original expression as
will bring most clearly into prominence the result required or will
reduce as far as possible the labour of affecting the actual
calculations.
Of these three, clearly the most fundamental is (a): this
always involves real thought and realisation of the facts: (b) is
merely direct numerical calculation : (c) represents the mechanical
or manipulative side of the subject. .
It follows that, while in practice (b) is indispensable
and (c) very useful, neither is of any use at all without (a).
Hence the fundamental necessity in the teaching of algebra is to
give training in analysis and expression".
A rather long quotation, but bear with me, we are getting
to the point. The report goes on:
"It does not necessarily follow that this, though the most
important is also the first thing to be done, and until
comparatively recent years the general practice was to start with
(c) ie. with the manipulation of algebraic symbols".
Remember, this was written in 1933, and yet it is still
Page 23
Chapter 4
common in schools to see algebra taught in the following sequence:
substitution, collecting terms, use of brackets, factorisation,
solving equations and finally problem solving.
Have the lessons of the past really been learned? Of
course algebra is not the only subject to have undergone a change
of teaching method. In geometry teaching, the definitions and
propositions of Euclids Elements was the starting point. The
ability to do "riders" or to solve geometric problems seemed added
as an after thought and not fundamental to the justification for
the study of geometry.
Similar changes have taken place in other subjects apart
from mathematics. Witness the emphasis on expression in English at
the expense of strict grammatical accuracy; on verbal communication
in foreign languages; on sheer enjoyment of music at the expense of
the ability to read a score. The list goes on.
Of course such changes of fashion are not without their
critics, and give rise to questions of falling standards (the Great
Debate of the 1970's and now the imposition of compulsory testing
of basic skills in Kenneth Bakers Education Bill) •. It is important
to achieve the right balance. Basic skills must not be unduly
depreciated. But there can be little doubt that a change in
teaching style can greatly increase the interest generated in
pupils, particularly in the early stages of learning a subject.
For children to operate successfully at rearranging
algebraic expressions they must first be competent at arithmetic
expressions. Recent research into "child-methods" suggest that the
intuitive methods that children use to solve easy questions do not
always generalise to harder questions and may lead to a lack of
rigour in what is written down. Children interpret what is written
Page 24
Chapter 4
down in terms of "common-sense" within the context of the question
(see K.M.Hart (1981) [24] and L.R.Booth (1981) [6] ).
So for example in answer to the following problem (taken
from CSMS Number Operations Investigation):
A bar of chocolate can be broken
into 12 squares. There are 3
squares in a row. How do you work
out how many rows there are?
12-;" 3 3x4 12x3 3-12
6+6 12--;"3 12-3 3.!..12 •
Intuitively children realise the problem is asking how
many 3's make 12. So they see no ambiguity in writing 3-;-12 or
12 -;- 3 as the answer since the meaning is determined by the context
of the question.
Alternatively they may apply the rule that "you always
divide the large number by the small one (or subtract the small
number from the large number)". The vast majority of contexts in
which children will be required to divide or subtract will support
this rule. Consequently the order in which the operation is written
is irrelevant so long as this rule is applied.
Perhaps other children interpret 3 -712 as 3 divided into
12 confusing the expression with the 3 I 12 form of writing a
division. Similarly 4-17 is interpreted as take 4 from 17.
Another consequence of childrens intuitive approach is
that they fail to realise the necessity for brackets since the
order of operations is obvious from the context of a problem. (see
L.R.Booth (1982) [8] "Ordering your operations").
Page 25
Faced with an expression such as:
3 + 4 x 5
Firstly they may not realise that
operations are carried out will produce a
they believe that:
(3 + 4) x 5 = 3 + (4 x 5)
Chapter 4
the order in which the
different answer. ie.
Secondly they may apply the rule that operations are
carried out in the order in which they appear.
ie. 3+4x5 means (3+4)x5
and 3x4+5 means (3x4)+5
Or thirdly, since the order is apparent from the context
of the problem, it is not necessary to record the order by the use
of brackets.
Childrens intuitive methods give
arithmetic problems despite the fact
them the right answer in
that they record their
expressions incorrectly. However the same is not true for algebraic
problems. Care has to be taken with the order conventions and the
rules governing the use of brackets. Unless the child sees the need
for such notions in arithmetic expressions, it is unlikely that he
will use correct conventions in algebra. So firstly the teacher
must address the problem of drawing the childs attention to the
need for an ordering convention in arithmetic.
Calculators can be very useful here. It always amazes
Page 26
Chapter 4
young children that pressing identically the same sequence of
buttons on two different calculators can produce different answers.
For example when finding the average of the three numbers 7, 8, and
10 the sequence of buttons:
produces the answer 18.3333 on a scientific calculator, but 8.3333
on a non-scientific calculator. Which is correct and why? What has
the "incorrect" calculator actually worked out? The calculator also
frees the teacher from the need to work with "simple" numbers. For
example if instead of the above question concerning the chocolate
bar wit? 12 pieces, we set the following:
A gardener has 391 daffodils.
These are to be planted in 23
flowerbeds. Each flowerbed is to
have the same number of daffodils.
How do you work out how many
daffodils will be planted in each
flowerbed?
391 - 23
23 - 391
391 + 23
23 x 17
23..!.-391 •
23 + 23
391; 23
391 x 23
23 x 391
The child does not immediately "know" the answer and will
be expected to use a calculator. The correct key sequence is vital.
So 23~391 will clearly give an incorrect anSwer. The correct key
sequence is 391·+23. So that is the correct written expression.
Unfortunately calculators are not the magic teaching aid and can in
fact add to a childs confusion. For example to work out 3~we must
Page 27
Chapter 4
press the key sequence:
In some older calculators it is necessary to find the
square root first.
To work out 6sin45 we must press:
Some calculators will also accept:
But on the other hand some will not. Small wonder that
children are confused.
Also, in written expressions we do not use the sign
so the average of 7, 8 and 10 would be written as:
and not as:
7 + 8 + 10
3
(7 + 8 + 10)';-3
The former does not lend itself to a calculator key
sequence whereas the latter does, provided that you are using a
calculator with bracket keys. At least the calculator does
highlight the need for correct order of operations, but may not
help in teaching accepted notation.
Page 28
Chapter 4
Computer notation may be even more confusing.
Multiplication represented by a star ",* I', and division by a
stroke "/" and indices shown with a small arrow
expression: " '" ". The
12 x 8.2.
3 + 4
would have to be written 12*Bt2/(3+4), which is hardly helpful.
There are many excercises available to the teacher which
stress the importance of brackets and the order of operations. Here
are some suggestions:
How can you make all the numbers from 1 to 10 using four
4's and the normal operations of arithmetic? For example:
4 x 4 ";-(4 + 4) = 2
Replace the stars by an arithmetic sign to make the
following statements true. You may need to use brackets:
8 :f: 2 :f: 3 = 30
7;t 2 ;( 9 = 1 etc.
Page 29
Chapter 4
How many different ways can you put in brackets and supply
the missing numbers to complete the statement:
8 = 24 /4+ 2 x •• ?.
For example: 8 = (24 / 4 + 2) x 1
or: 8 = 24 / «4 + 2) x;t)
Alter the following statements to make them correct using
the least number of alterations. What alterations are possible?
13 + 45 = 57
14 x 7 91
How many ways can you find to insert operations (and
brackets) to make a true statement from:
5 4 3 2 1 = 15
Find a short way to work out these sums (you should be
able to do them in your head). Explain how you arrived at the
answer.
58 x 13 + 42 x 13
75 + 38 + 25 =
37 x 6 + 37 x 4 =
25 x 44 = etc.
Page 30
Chapter 4
Children are using the laws of arithmetic: associative,
distributive, commutative, in a natural way. The conventions of
algebra are a consequence of the laws of arithmetic, not the other
way round. Thus the reason for omitting the multiplication sign so
that:
a x b + c becomes ab + c
is that the multiplication is done first, so a and bare
associated together. The reason that Sa + 4a can be written as 9a
is a consequence of the distributive law seen to work in
arithmetic. Removing brackets so that a(b + c) = ab + ac follows on
from long multiplication or from seeing a short way of working out
sums, for example 8 x 51, in the form:
8 x 51 = 8 x (50 + 1) = 8 x 50 + 8 x 1 etc.
Children perform these sort of mental calculations without
realising the rules that they are following. The algebra teacher
should ensure that these rules are made explicit by recording them,
using letters for numbers.
The importance of order of operations becomes apparent
when solving equations. The use of flow-charts to solve equations,
widely used by the early SMP courses, has the advantage of
stressing order. By this technique, the equation 3x + 7 = 22
translates into the flow chart:
3x
x -11-_X_3-J1----~~ +7
3x+7
1---....;)~22
Page 31
Chapter 4
Reversing the flow chart involves reversing the direction
of flow, and also "inverting" the operation, giving the following
"reverse" flow chart:
5 15
x~-:- 3 ~1(----IL---7---,~~---- 22
Thus the solution to the equation is x = 5.
There are many reasons why this method of solving
equations has been widely dropped from text books (although I
suspect many teachers still use it). For a start the technique only
works for a limited range of equations. Put an x term on the right
the method fails. hand side (such
"Inverting" the
example:
as 3x + operation
7 - x = 4
7
is
= 22 x) and
not always straight forward for
The operation is "take from 7". What is the inverse of
this operation? Or in the equation:
12 6
x
The operation is "divided into 12" whose inverse operation
is also "divided into 12".
In a later SMP book (SMP New Book 4 Part 1 C.U.P (1982)
Chapter 2). Ive see the following example:
5 + 6 = 9
t
Page 32
<
The function t~5 + 6 corresponds to the
t
following flow chart:
t ~ 7 Find the -.-:, Multiply ~ Add ~
reciprocal by 6 5
Reversing this gives:
Find the ~ Divide ~ Subtract
reciprocal by 6 5
So we use these steps to solve the equation:
subtract 5 6 = 4
t
divide by 6
Find the reciprocal t = 3 = l~
2
5 + 6
t
./
I"'-
Chapter 4
The flow chart is. now representing a function, and the
reverse flow chart is the inverse function. The problem of "divides
into" is now dealt with by multiplying by the reciprocal. The
technique is still rather questionable. Are we inverting
Page 33
Chapter 4
operations, (the inverse of add is to take away etc.) or inverting
elements, ( the inverse of add 5 is add -5) or inverting functions?
The method is designed to ensure that the student uses the
fact that the inverse function of the composite function "f
followed by g" is "g-I followed by f- I ". Reversing the flow chart
automatically reverses the order. But when children do without the
flow chart will they still realise this? In any event the
construction of the flow chart requires a sound comprehension of
functions and their composition in the first place. To be fair to
the SMP writers, I think it was always intended that the child
would eventually do without the flow chart and work directly with
the equation as the above example shows. The method does have a
certain visual appeal, but too many disadvantages.
The vast majority of children can solve simple equations
in their heads. In fact, I include equations in mental arithmetic
ex ercises. For example, I would include such questions as:
"I am thinking of a number, I double it and get 24. What
number was I thinking of?"
Here there is only one operation namely doubling. The
child must realise that the opposite of doubling is halving and so
you half the twenty four. When more than one operation is included
in the question, then the order becomes important. For example:
"I am thinking of a number. I add 6, then I half the
result. The answer is 30. What number was I thinking of?"
Page 34
Chapter 4
Written equations can also be solved by children in their
heads (provided the numbers are not too difficult) using the
following sort of logic:
To solve: 2x + 5 13
I am adding 5 to something to get 13, and 8 + 5 is 13,
so 2x = 8.
I am doubling something to get 8, and double 4 is 8,
so x = 4.
They are using the fact that if x is a solution to the
equation:
2x + 5 = 13
Then it is also a solution to 2x = 8.
The idea of the equation as a balance reinforces this
method.
2x + 5 13
\. ) i
\. !
2x + 5 is in one pan of the scales, 13 is in the other.
Take the 5 away from the 2x + 5 and the balance is upset, 5 must be
taken from the 13 to maintain the balance. This is a better visual
image for the child of what an equation is than the flow chart.
Unfortunately, it may encourage the child to think of the x as a
"thing" rather than as a number (the way children misinterpret the
Page 35
Chapter 4
meaning of letters is dealt with more thoroughly in the next
chapter), but the idea of maintaining equality by performing the
same operation to both sides of an equation is an important
concept. You can double both sides, you can take five from both
sides, etc. and equality is still maintained, the value of x is
unaltered. So the idea is to isolate the x on one side of the
. balance, and whatever number is on the other side is the solution
to the equation. Not all equations lend themselves to this method.
For example:
3 = 12
x
Putting 3 divided by x into a scale pan does not make much
sense. A negative number sits rather incongruously into a scale pan
as well. For example:
6 - 2x = -20
Once the idea of applying the same operation to both sides
of an equation is appreciated by the students, we can do without
the scales and just operate on the equation. The question is then
what operation to apply. This brings us to the idea of inverse
elements.
When solving: 2x + 5 13 we take 5 (or add -5) to both
sides because -5 is the inverse element of 5 under the operation of
addition. Giving 2x = 8. We then half both sides (or multiply both
sides byt) because -{ is the inverse element of 2 under the
operation of multiplication.
-5 is the inverse of 5 because 5 + -5 = 0 and 0 is the
additive identity.
Page 36
:i is the inverse of 2 because
multiplicative identity.
2 x.J.. = 2.
Chapter 4
1 and one is the
In fact the teacher would not be as formal as this, but
these concepts are implicit in the method of solution. An
understanding of inverse elements is necessary if the correct
solution is to be found.
This method of solving linear equations has the advantage
of always working. For example we can solve:
2x + 5 = 13 + x
by recognising that -x is the inverse element of x. So we add -x to
both sides.
Or, for the equation: 6 = 12
x
we multiply both sides by x because x is the multiplicative inverse
of 1
x
Page 37
Chapter 4
This method also lends itself to other algebras. For
example Boolean Algebra and Matrix
equation Ax = B by multiplying both
(if it exists) giving:
Algebra. We solve the matrix -. sides by the inverse matrix A
x = A-I B
But this method is rather abstract and must be seen by the
teacher as the culmination of time spent working with equations in
many different ways.
For some reason, children who are quite adept at solving
linear equations often find literal equations, or changing the
subject of a formula surprisingly difficult. I suppose it is
because they do not really understand what it is that they are
trying to work out. With an equation you find a value for the
letter which "works lt, ie. which satisfies the equation. But a
rearranged formula may not appear to have worked anything out. It
is important that the formulas that children are expected to
rearrange should be known and recognised by them, and the
rearrangement should be applicable to problem solving. Another
difficulty for children is that they may not recognise the
equivalence of different forms of the same answer.
Page 38
by:
follows:
Chapter 4
For example: The total surface area of a cylinder is given
A 2
2iTr + 2 "TT rh
"2-Adding -2 "TT r to both sides gives:
2 A - 2 iT r = 2 rr rh
Di viding by 2 TT r gives
h = A - 2"1Tr'l.
2iT r
Alternatively the original formula could be factorised as
A = 2.".r er + h)
Di viding by 2 rr r gives
A = r + h
2 rrr
Subtracting r gives:
h A - r
21Tr
Will children realise that these two formulas are the
same? Finding an answer in the back of the text book which is
different to the answer which they have, is never reassuring for
children. It may well be an interesting exercise to confirm that
Page 39
Chapter 4
several different expressions are in fact identically equal. Work
on this should perhaps be done before attempting rearranging
formulas. This requires such skills as factorising, working with
algebraic fractions, etc. As always children must be capable of the
necessary arithmetic before the algebra is attempted.
Page 40
Chapter 5
Childrens Errors and Misconceptions
The research programme "Strategies and Errors in Secondary
Mathematics" (SEMS.) was a sequel to the "Concepts in Secondary
Mathematics and Science" (CSMS.) programme in which problems most
commonly experienced by many secondary school pupils were examined.
Several areas of mathematics were investigated: Ratio, Algebra,
Graphs, Measurement, Fractions.
The algebra investigation was resticted to the elementary
algebra of generalised arithmetic. The use of letters for numbers
and the writing of statements using letters, numbers and operations
in the accepted convention. The algebra of solving equations,
factorising or simplifying rational or higher order expressions was
not included.
The findings of the CSMS programme were published in 1981
(see "Childrens Understanding of Mathematics: 11-16", ed. K.M.Hart
[27] chapter 8). The later SESM programme published its report in
1984 (see "Algebra: Childrens Strategies and Errors", L.R.Booth
[10]) •
The first
letters in algebra.
problem identified was
Kucheman identified
interpreting and using letters
the interpretation
six different ways
of
of
(l)Letter Evaluated: where children avoid having to operate
on a specific unknown by giving the unknown a value.
(2)Letter not used: where children ignore the letter, or
acknowledge its existance but without giving it a meaning.
Page 41
Chapter 5
(3)Letter as Object: refering to the use of a letter as a
shorthand for an object (eg. a stands for apple) or for a quantity
(eg. I stands for length) rather than as an unknown or general
number.
(4)Letter as Specific Unknown: where the child regards a
letter as a specific but unknown number, and can operate upon it
directly.
(5)Letter as Generalised Number: The letter is able to take
more than one value. Essentially the letter is seen to take on
several values in turn rather than representing a set of values
simultaneously.
(6)Letter used as a Variable: where the letter is seen as
representing a range of unspecified values. Implicit in this
interpretation of letters as being variable is an understanding of
the way in which a dependant variable behaves with respect to
another variable. Kucheman defines it this way "Letters are used as
variables when a second (or higher) order relationship is
established between them".
The first three categories indicate a low level of response
of which perhaps the most common was the use of letters as objects.
There were several items on the CSMS algebra test where this usage
occurs, the following being typical:
Cabbages cost 8 pence each and
turnips cost 6 pence each. If c
stands for the number of cabbages
bought and t stands for the
number of turnips bought, what
does 8c+6t stand for?
Page 42
Chapter 5
Of 1000 fourteen year olds, less than 4% answered the item
correctly, while 52% interpreted 8c+6t as eight cabbages and six
turnips. they are clearly assuming that c and t are abbreviations
for cabbage and turnip respectively.
Often such an interpretation can lead to a correct answer.
For example in the APU survey [2] we have item A8:
A bar of chocolate costs x pence
and a packet of crisps costs y
pence. What is the cost of 2 bars
of chocolate and 3 packets of
crisps?
In this case 59% of 15 year olds tested gave the correct
response of 2x + 3y (or 3y + 2x) but one wonders how many were
thinking of this expression as meaning 2 bars and 3 packets, rather
than the correct interpretation. Namely 2 bars at x pence each plus
3 packets at y pence each. Many children are even taught to think
of letters as objects when learning such skills as "collecting like
terms". Interpreting:
Sa + 2b + 2a
As 5 apples add 2 bananas add 2 apples leads to the
correct simplification 7a + 2b. But such interpretations can lead
to problems. From my experience the majority of 13 year olds would
say that 3a - 2a = la and they are very reluctant to drop the one,
(3 apples take away 2 apples equals one apple, it does not equal
apple), and if the one is not there the confusion is compounded.
Page 43
Chapter 5
9a - a 9
If you start with 9a and take away the a you are obviously
left with the 9. If the problem is written as 9a - la they would be
more likely to get the correct simplification.
An alarming number of school text books provide examples
of letters as objects (see "Object Lessons in Algebra", D. Kucheman
(1982), [28]). Sometimes by analogy:
"The sum of 5 apples and 8 apples is 13 apples and in the
same way Sa + 8a = l3a" (General Mathematics 1 (1956)).
The words "in the same way" are the only indication that
we are dealing with an analogy and that Sa really means 5
multiplied by the number a. Some text books are even more extreme.
A World of Mathematics 4 (1971) asks children to "write an
algebraic statement to represent each of the following" by letting
"a stand for apple" etc. and there follows a series of pictures of
rows of assorted apples, bananas and coconuts (a's, b's and c's
presumably).
In SMP 7-13, (1979) we see the following:
U 0 4 _e"" "M ."le '"'' 4,"
They are then asked to re-write statements like "Five
apples cost 20p". Here not only are letters used as objects, but
pictures of objects are used for numbers.
SMP 11-16 has studiously avoided the use of letters to
represent objects but a lot of teaching material still in use in
our schools does encourage this misconception. An even more common
Page 44
(and more subtle) misuse of letters in
representation of numerical quantities. For
formula:
2. A =1\ r
Chapter 5
algebra is their
example: In the
What does the r represent? Is it an abbreviation for the
word radius? Does it stand for a line stretching from the centre to
the edge of a circle? Is it the length of such a line? Or is it the
number of units of length of such a line? The mathematician knows
that the last interpretation is correct, but how many
schoolchildren know that? More alarmingly how many teachers of
school mathematics are aware of the subtle difference between the
last two answers? In fact the physics teacher would be quite happy
to use a letter to stand for a quantity (see J. Ling, (1977),
[29], "Maths Across the Curriculum", pp14).
In SMP 7-13 we see the
Srn
Perimeter = 2L + 2W
= 2(5) + 2(3)
16m.
following:
In the diagram it is not clear if Land Ware representing
words, objects, or quantities (certainly not numbers). In the
formula they are correctly replaced by numbers, but then finally
Page 45
Chapter 5
represent quantities again 2L + 2W = l6m.
Does it really matter? Are we being too pedantic insisting
that "letters stand for numbers and not numbers of things (or the
things themselves)"?
Certainly textbooks should avoid such errors and teachers
should be more aware of the way children and they themselves use
letters in algebra. Children will think of letters as objects or
numerical quantities no matter what we do and in many cases this
really will not matter.
If a child writes "let a stand for the cost of an apple"
and solving a problem writes the answer:
a =-10 pence
Then strictly speaking we should insist that this is
incorrect use of letters in algebra. He should wri te "let a stand_
for the number of pence that an apple costs" with the solution a =
10 leading to the conclusion that an apple costs 10 pence. An
insistence that children always define the units of quantity (for
example "let the length of the rectangle be x metres and not just
x") and always give their solution in words (Answer: the length of
the rectangle is 15 metres and not x = 15 metres), will at least
improve the presentation of written algebra even if children still
think of letters as representing quantities. The use of neutral
letters (using x for the number of apples, and not a) might have
some effect in preventing children from thinking of letters as
objects. Teachers should seek out demonstrations of the inadequacy
of using letters as objects in the hope that children will
eventually see that it should be abandoned.
Page 46
Chapter 5
Apart from the incorrect interpretation of the use of
letters in algebra, other sources of error identified in the CSMS
team were incorrect notation when writing an answer down and use of
an inappropriate method for arithmetic. For example the following
question is taken from the SESM test:
West Ham scores x goals and Manchester
United scores y goals. What can you write
for the number of goals scored
altogether?
Given simple numbers such as West Ham scored 2 and
Manchester United scored 3. The child would immediately answer 5.
He is unlikely to record the anSwer in the form 2 + 3 since the
answer 5 is intuitively obvious. The child did not have to reflect
upon what operation to carry out or how to record it. With larger
numbers, West Ham 27, Manchester United 79 the child is made more
aware of the operation to carry out, but is more likely to record
the sum as:
7 9
+ 2 7
A form which does not lend itself to the correct algebraic
expression x + y.
The child who performs multiplication by repeated addition
may have difficulty even understanding an algebraic expression
involving multiplication.
Page 47
For example:
7
A child
perimeter
by working
instead of
Sx7.
Chapter S
may find the
of this shape
out 7+7+7+7+7
working out
He will find the correct solution 3S for this arithmetic
problem, but is unlikely to write the correct expression for the
algebraic problem:
There are n sides
altogether all of length 2.
What can we write for the
perimeter?
Repeated addition n times cannot be easily recorded in
algebra. These examples are taken from the SESM algebra tests and
it is interesting to note the absence of units or any indications
that the numbers represent lengths in standard units. Children are
expected to automatically interpret the numbers in the diagrams as
representing a number of standard units of length. A mathematician
would view numbers as being dimension-less and yet in these
examples (and regrettably in many textbooks) numbers are clearly
being used as quantities ie. being the product of number and unit.
It is clear from these examples that children's
non-recognition of structure and methods in arithmetic will have a
considerable effect on their performance in algebra. Repeated
addition for multiplication, repeated subtraction for division,
failure to realise that subtraction is the inverse operation to
addition etc. will lead to failure. Children who do not realise
Page 48
Chapter 5
that, for example: "What must be added to 7 to get lO?" is in fact
10 take away 7, will be unable to answer the corresponding algebra
question: "What must be added to x to get y?".
Children experienced in arithmetic are used to finding an
"answer", usually a single number. The answer in algebra is not of
this form. It is usually an algebraic expression of method. For
example, in the West Ham problem the answer is x+y. Children are
reluctant to write that as an answer. Their response is either that
the problem can not be done unless you know the values of x and y,
or they will conjoin the letters giving the answer as xy. They
perhaps are so used to seeing the plus sign (+) as something that
has to be done, and if they don't do it then their answer is not
complete. They would view the multiplication sign (x) in the same
way, but because we leave the sign out in algebra, they feel that
they have actually done something when they replace axb by ab. Not
only must children accept the idea of an unevaluated operation,
they must hold the answer in suspension, using brackets, and
continue operating upon it. As in the follo;ling example:
What expression tells you the
area of this rectangle?
5
I e r 2
[Again note the absence of units in the SESM tests.)
The child must accept e+2 as the correct expression for the
length of the rectangle and then operate upon it by multiplying by
Page 49
Chapter 5
5. If the child conjoins the addition he could produce the answers
10e or 5xe2 or e2x5 or elO. Failure to use brackets produces the
answers 5xe+2 or 5e+2 or e+2x5. The latter answer being more common
because the child is applying the rule "perform the operations in
the order in which they are written". I have already mentioned the
order of operations and use of brackets in previous a chapter.
Misuse of brackets is a common error in childrens algebra, but
there are other sources of error in the use of correct convention
and notation.
Although algebra may be regarded (at least in part) as a
generalisation of arithmetic statements, the conventions used
differ from one domain to the other. In arithmetic conjoining can
denote addition, as in mixed fractions. For example 3~ means 3 +1; Or place value, which is in effect a type of addition.
For example 65 means 6 tens + 5 units.
Whereas in algebra conjoining denotes multiplication (eg.
2a = 2 x a). It is hardly surprising that when asked, for example,
to find the value of 2a where a=3 the child produces the answer 23
or 5.
When letters appear in arithmetic they usually denote the
units of a quantity (m for meters, p for pence, etc.). We get
expressions such as:
100cm lm
Page 50
Chapter 5
In algebra we would use m to represent the number of
meters, and c to represent the number of centimeters. The above
relationship would be written:
lOOm = c
ie. the number of centimeters is 100 times the number of meters.
This is completely the opposite way round to the arithmetic
expression. The algebra statement is a more dynamic expression.
This difference in usage is bound to contribute to childrens
confusion. This is not only a difficulty for children. The
following problem was given to 150 second year engineering students
(see J.Clement, (1981), [13], (1982), [14]).
Write an equation using the variables 5
and P to represent the following
statement: "There are six times as many
students as there are professors at this
university. " students and
professors.
Answers like 65 = P,
shorthand for 'student' and
Use 5 for the number of
P for the number of
where 5 and P are simply used as a . .
'professor' were gLven by 25~ of the
sample. They are giving an arithmetic interpretation in the form of
"six students are equivalent to one professor" instead of the more
dynamic algebraic interpretation that "six times the number of
professors gives the number of students".
Another source of error in notation is use of indices.
Confusion between squaring and doubling, cubing and tripling etc.
Children may not recognise that m + m + m + m is the same as 4
times m or 4m, whereas m x ID x m x m is different and is written m~
Page 51
Chapter 5
The fact that 21 is the same as double 2 reassures the child that 32.
must be 6; a false generalisation.
surprisingly large number of children make
From my experience a
the mistake that 1'2. = 2
confirming even further the incorrect generalisation that squaring
is the same as doubling.
Another difference in notation exists between algebra and • arithmetic in the recording of division. In algebra the sign~ is
never used. Do children realise that writing one letter over
another with a line between them means division? Work with
fractions in arithmetic may help, but on the other hand do children
recognise ~ as 2 divided by 3 or merely as a way of recording a
specific proportion of a whole? Top heavy fractions are dealt with
by division, but children are made to feel that top heavy fractions
are somehow "wrong" or inappropriate and this may add to their X unease at recording 1i when the value of x is likely to be larger
than y. Addition and subtraction in arithmetic is recorded with one
number being written underneath the other whereas division is
recorded by writing one number alongside the other in one of these
two ways: 36; 4 or 4 136':'"
In algebra it is completely the other way round. Division
with one letter above another(~) addition and subtraction with one
letter alongside another (x - y). I am not suggesting that we
change our notation. It is too well established and does have
certain advantages.
difficulties children
But teachers should
experience. Children
be aware of the
should be made more
familiar with recording division as one number over another in
arithmetic. Using brackets to record parenthesis in an arithmetic
expression so that they
They
are familiar with brackets before meeting
them in algebra. could be
operations in arithmetic and not just
encouraged
give the
to record
answer t so
their
that
recording of method is seen as an acceptable sort of answer.
Page 52
Chapter 5
Teachers should emphasise the similarities between algebraic and
arithmetic notation to make the change from one domain to the other
more natural and meaningful.
It has been suggested (see K.F.Collis, (1969), [16]) that
errors in algebra may be related to a child's cognitive
development. A child's ability to record generalisation using
symbols being indicative of a change from the concrete to the
formal operational stage (in the Piagetian sense). A concrete -
operational child's thinking is restricted to the immediate
problem. The child operates in terms of the particular situation
and does not obtain a content - free understanding of the structure
of the solution. Certainly there is widespread evidence of children
using their own informal methods in elementary mathematics even
though formal methods are taught to them (see H. Ginsburg, (1975),
[22], S. Plunkett, (1979),[32]). As a consequence the concrete
operational thinker has difficulty working in an abstract
mathematical system as required when working algebraically. The
idea of clearly defined cognitive stages in a childs development is
not without its critics (see G. Brown, and C. Desforges, (1979),
[11]). The fact of cognitive development does not in itself ensure
the growth of understanding of algebra, although the attainment of
a particular level of cognitive maturity is certainly necessary for
instruction to be effective.
Page 53
Chapter 6
Other Algebras in School Mathematics.
A basic concept included in all courses in modern
mathematics is that of the set. It became very fashionable in the
late 1960's and early 1970's to teach children about sets from a
very early age. In the eyes of many, sets became synonymous with
modern maths. Unfortunately, much of the work was of a trivial
nature (see the chapter on sets in "Modern Maths for School Book
1") which children, in my experience, raced through without really
appreciating the underlying algebraic structure.
It is hard to justify the teaching of the algebraic
notation of sets to all children. The following questions on set
notation is taken from a CSE examination paper (EMEG. 1987 Syllabus
1 Paper 2a)
P,Q,and R are sub-sets of t where
f = {3,4,5,6, 7,8,9}
P = [3,4,53 ' Q = [3,6,8,9} and
R = { 4,6,8 J (a) List the elements of
(i) Q" R,
(ii) Q' ,
(iii) PI" Q(\R,
(iv) PvQ.
(b) Find the value of n(Q) + n(R).
Page 54
Chapter 6
It is a question solely on the correct usage of symbols.
Bearing in mind that this examination is set for children of
'average' ability, one can only imagine the hours of class time
necessary to get a class of youngsters to a level of familiarity
with the symbols that they are able to competently complete the
question. I find this hard to justify. An understanding of the
concept of sets is important. It is fundamental to many other
branches of mathematics. In a subject such as functions, the
domain, co-domain and range are sets. The range being a subset of
the co-domain. In probability we meet the sample space (the set of
possible outcomes). We have sets of solutions to equations and
inequations. A locus in geometry is a set of points satisfying
certain conditions or restrictions. Linear programming deals with
intersecting regions which are themselves sets of points.
The algebra of sets is very abstract, but does provide
important examples of the laws of operations applied to structures
other than numbers.
Commutative Law: AnB = B"A, AuB = BUA etc.
Of particular interest is the distributive law, where union
is distributive over intersection.
Au(BnC) = (AVB)('\(AUC)
and intersection is distributive over union.
AI"\(BUC) = (ArlB)U (Af'C)
Identity elements are the empty set for union, and the
universal set for intersection.
Page 55
Chapter 6
For the child capable of appreciating it. The algebra of
sets can provide rewarding and aesthetically pleasing course of
study, giving a deeper understanding of algebraic processes. For
many it will remain a jumble of confusing symbols.
The use of set notation should arise naturally for all
children in the course of other work. Teachers should talk of sets
of solutions etc •. Intersection, union and complement should be
part
they
of normal vocabulary during maths lessons, so that even if
do get the symbols U and n confused, they have an
understanding of the concepts and are able to use them to solve
logical problems.
For example, when studying factors and highest common
factors, the use of set notation is appropriate and a Venn diagram
helpful:
A = {factors of 18} = [1,2,3,6,9,18}
B = [factors of 24} = [1'2,3'4,6,8'12'24~
4-
8
A"B = [commom factors of 18 and 24J = [l,2,3,6J
The same is true of common multiples although now we have
infinite sets and a Venn Diagram is perhaps not so useful. The
concept of intersection is fundamen'tal to the problem.
Page 56
Chapter 6
The rules governing combined probabilities should be seen
as a consequence of set algebra:
probability of event A.2.L event B
= probability of event A + probability of event B
- probability of A and B
In set notation this corresponds to the formula:
n (A u B) = n(A) + n(B) - n(A()B).
A Venn Diagram makes this relationship clear. Venn diagrams
may also be useful in other subjects. For example the following
diagram shows the relationship
the Biology teacher:
between animals and may be of use to
BIRJ>S
MftM MAcS
FISH
AM f?HI 6",
t<ePTILES
C.OL]) e,LOoDc l> PrN I (';fA L S
INSEc-, S
Page 57
Chapter 6
There is a danger of course that the non-mathematician may
use set notation wrongly as in the following example:
What could possibly be the Universal Set for this Venn
Diagram? Is a solution both a salt and water at the same time?
The more opportunities that the teacher uses to introduce
set notation in a natural and meaningful way, the better the childs
understanding. Eventually the child will appreciate its structure
and as a consequence, have a deeper understanding of algebra.
The idea of a function is one of the basic ideas of
mathematics. It is only during the present century that general
agreement has been reached as to what precisely constitutes a
function. As recently as 1908, G.H. Hardy in his classical textbook
"Pure Mathematics" stated:
"All that is essential (to a function) is that there should
be some relation between x and y such that to some value of x at
any rate correspond values of y".
It is clear that Hardy is restricting functions to the
fields of numbers and is allowing "many-valued functions". The
fundamental feature is one of dependence ie the value of x. This is
certainly a feature of a function which should be appreciated by
school children.
Page 58
Chapter 6
Our modern definition of a function identifies three basic
features:
(1) a starting set A which is called the domain of the
function (only elements of this set A are processed).
(2) a target set B called the codomain of the function
(this set contains the processed elements, but may contain other
objects as well).
(3) an association, rule or process by which there is
assigned, to each element x in the starting set A, a single element
y of the target set B.
These three parts are emphasised in the current notation:
f A~B
Neither the domain nor the codomain are necessarily
numbers. The rule or process is not necessarily defined by a
formula, but it must apply to all members of the domain and the
image f(x) in the codomain must be unique. The diagram shown below
illustrates the fundamental features of a function and could be
used with secondary school pupils. 'DOMAIN
~1Jf\c.t;o" 01"
rule P
COJlOMAloV
B
,
~-~....---t-f- t(x) ~ •
Page 59
Chapter 6
It does clearly rely upon the child having an understanding
of sets and Venn diagrams.
The idea of functional dependence will be appreciated by
children at an early age by simple examples.
The bigger the block of chocolate, the more it costs.
The older you are, the later you are allowed to stay up
etc.
As the child progresses through secondary school the
functional dependence becomes more complicated and is not
restricted to mathematics lessons. In science the child learns that
the pressure exerted by a given volume of gas depends on its
temperature. In geography local time is dependent on longitude.
Climate is in some way dependent on latitude. In mathematics the
image of a point mapped by reflection depends upon the position of
the line of reflection: the area of a circle depends upon the
length of the radius. The mathematics teacher makes the features of
the function explicit: "What is the domain?" "I'hat is the
codomain?" "Can we write a formula for this dependence?"
A considerable problem for the mathematics teacher is one
of notation. The problem is compounded rather than solved by the
writers of many modern textbooks. For a mathematician, a function
has three parts. We wish to use a single letter to stand for a
function, and so the single letter represents the three parts as
follows: /
f : IR ~fR given by f(x) = xl.+ 1
Page 60
For a beginner this notation is
particularly when the domain and the codomain
obvious. It is tempting to speak of the function
the function f which maps x to x2+ 1. Mapping in
Chapter 6
rather unwieldy,
intuitively
1 rather than
this context being
synonymous with functional dependence, a meaning incidentally which
is not always given to the word mapping in current textbooks. We
would write:
f
By f(x) we mean that element of the range which is the
image of the element x of the domain under the function f. But we
may also write:
f(x) x2. + 1
as a definition of the function f itself. This abuse of
notation should be avoided by teachers and textbooks alike. A
consistent notation should be used which children can understand
and work with.
There are several ways of visually representing functions
for children. (see H. Shuard, and H. Neill, (1977), [35],
Mathematics Curriculum: From Graphs to Calculus, Chapter Four). The
basic arrow diagram (as shown above) has the advantages of being
simple to draw and understand. The domain and codomain need not be
sets of numbers, (eg sets of pupils mapped to modes of transport
etc.) More importantly, from the point of view of algebra, the
diagram brings out the essential properties of the function
(whether it is one to one, whether it has an inverse etc) and can
be used to show composite functions very easily. Of course such a
diagram has limitations, and so the teacher should only use it when
appropriate.
Page 61
Chapter 6
When the domain and codomain are sets of numbers the arrow
diagram can be adapted to show the order of the numbers by placing
them on number lines.
'I .~ '1 '" $ , , , 7 1 7
7 , , b '" S- S" 0; 'i
~ It ... It-
3 3 3 3
1 1-1-
) 1
0 ') () 0 0
:x >2x. X ) :x. -;- 3
This way of representing a function has all the advantages
of ordinary arrow diagrams but in addition the diagram helps
identify types of function (eg linear functions, functions which
preserve order, etc.)
It can, of course, only be symbolic unless the domain has a
finite number of members. Composite functions can again be easily
shown by joining one arrow diagram to another. The cartesian graph
is the most familiar of all illustrations of a function. The amount
of information that an experienced mathematician can glean just
from looking at a cartesian graph is an indication of the power of
such a representation of a function. It is the only representation
commonly used in schools in which the concept of a continuous
function is visually meaningful. At a more advanced level, the
Page 62
Chapter 6
derivative is meaningful as the gradient of the graph, and the
integral is meaningful as the area under the graph. It has its
disadvantages. The domain and range are sets of numbers, and pupils
may think that all functions have the set of real numbers as their
domain. The mapping or functional dependence is not explicitly
clear so careful teaching is needed for the essential features of a
function to be clear. It is also not good for showing composite
functions, one graph cannot be "added on" to another graph.
The last visual model for a function is the "function
machine" or "black box". These emphasise the processing aspect of a
function.
For example:
input" add 3
/L---_---'
output,
or
2-x~x r-
output· [
Composite and inverse functions are again very easily
shown. Use of a computer (the ultimate black box?) can help with
this conceptual model of a function. This work also ties up with
flow charts for solving equations.
Composition of functions is perhaps one of the first
operations for which the lack of commutativity proves a problem for
youngsters. In some respects the problem is again one of notation:
fog(x) means apply the function g first and then
apply the function f (even though we write the f before the g).
Many texts leave the sign 0 out writing fg(x). I feel that
initially it may be of some value to write it in the form f(g(x))
Page 63
Chapter 6
which appears cumbersome especially if more than two functions are
combined, but at least the order of operations is clearer. The
inverse of a composite function is the composition of the inverses
of the functions but in reverse order. This can be clearly shown
using arrow diagrams or function machines. This ties in well with
matrix transformations which are themselves functions of 2 (or 3)
dimensional space.
Matrices are usually a part of modern maths algebra courses
for secondary school pupils. The context in which they occur can be
rather trivial and appear pointless. For example, using matrices as
glorified shopping lists. See SMP Book E (1970):
Flat SA Flat SB Flat SC Flat SD
Gold Top 2 0 2 1
J Red Top 0 2 1 3
Silver Top 0 1 1 1
The above matrix shows the orders for milk for four
families living in a block of flats. Are we really suggesting that
a milkman uses matrices in his daily work? Clearly not. Using
matrix multiplication to calculate such information as: The milk
bill for each flat; The total amount of red top milk sold; etc
seems a cumbersome and inappropriate way of tackling a problem
which most children could simply work out in their head. Also
matrix algebra is not made apparent by such trivial examples.
A better way of introducing matrices is through geometric
transformation in two dimensional space.
Children will be used to coordinates from an early age, for
drawing graphs and plotting points (links here with science and
geography map references). They can observe and record the effects
Page 64
Chapter 6
on coordinates for a given transformation. For example under
rotation of 90 anti-clockwise with centre the origin, we have
that
(x,y)---70) (-y,x)
We are regarding the transformation as a function M say, so
:t 't M: (x,y)---? (x ,y )
in this example
M(x,y) = (-y,x)
Where the function is given by the following formulas:
l( x = Ox - ly
We combine these 2 formulas into one formula of the form:
The notation of matrices, and matrix multiplication is
introduced to children in a meaningful and relevent way.
Within the context of geometrical transformations we can
investigate combined transformations represented by matrix
multiplication. The question of what transformation returns a shape
to its original position leads to the idea of an inverse matrix. Of
Page 65
Chapter 6
course not all matrix transformations have inverses, the value of
the determinant is seen to be important. Consideration of the area
of transformed shapes also ties up with the value of the
determinant. At a more advanced level, consideration of invarient
lines leads us to investigate eigenvectors and eigenvalues. So you
see that the whole theory and techniques of matrices can be applied
to geometric transformations in a meaningful way.
Another common way for textbooks to introduce matrices is
with direct route matrices. (see Chapter 1 Matrices at Work:
networks in SMP Book F (1970». The following network:
A
B c
Can be represented by the following matrix:
A
from :( ~ C 0
to
B
1
o 1
C
D Children would find this fairly straightforward although a
route from A to A is rather difficult to conceive of, and the fact
that the route is "one stage" means that there is no direct route
from A to C because you must go via B. Addition of matrices becomes
a natural operation. Adding extra routes means you add the matrix
of the extra routes to the original matrix. Scalar multiplication
is also a fairly obvious operation. The main disadvantage of this
Page 66
Chapter 6
way of introducing matrices is that matrix multiplication is not
motivated in a natural way. Squaring the stage route matrix does
give the 2 stage route matrix, but the process of combining rows
and columns does seem rather like a "magic process" invented by the
teacher to give the answer he wants. Apart from which, only the
most trivial of networks can be managed by children because of the
sheer size of the matrices involved. A network with ten nodes gives
rise to a 10 by 10 route matrix (ie a matrix with 100 entries).
The study of route matrices does not lead naturally to an
understanding of matrix theory. It has its place in the study of
topology and graph theory ("graph" in this context meaning
network), but it is not surprising that it is becoming less common
to see route matrices in a secondary school maths syllabus.
We can not leave matrices without considering simultaneous
equations. Solving sets of simultaneous equations is at the heart
of matrix theory. Unfortunately the full power of the method is not
apparent when solving pairs of simultaneous equations in 2 unknown
(ie in the case of 2 by 2 matrices). Without using matrices, there
are three methods of solving simultaneous equations commonly taught
in schools.
Method 1 Graphical Solution.
The two equations are represented by straight lines. Each
line representing the solution set of each individual equation.
Where they meet gives the point which belongs to both solution
sets, so it gives the solution of the simultaneous equations. The
advantage of this method for use with school children is that the
solution will be meaningful. It may increase a childs understanding
of what a solution to a pair of simultaneous equations really is.
The special cases of the lines being parallel and hence no solution
Page 67
Chapter 6
existing, and an infinity of solutions for coincident lines, are
also made apparent. But where the method fails is when the two
equations do not have a "nice" solution (a non-integer solution for
example) although an approximation can be found. Also equations
that are "ill-conditioned" (ie the lines representing the equations
are almost parallel making them difficult to sketch and a slight
error in drawing leading to a large error in approximation).
Method 2 Substitution.
One unknown is made the subject of the first equation and
substituted into the second equation which then becomes an equation
in only one unknown. This method requires a high degree of skill in
manipulation; it can also be applied where one equation is linear
and one quadratic. It is not so good if there are more than 2
unknowns.
Method 3 Elimination.
This method is widely used in both traditional and modern
courses. On the whole I would say that teachers like it. It gives
quick results, or seems to. The algorithm is simple to describe,
and simple to drill, and it appears to teach an important skill
which is easily tested or examined. Of course teaching and drilling
an algorithm does not in itself give insight into a process, so it
is important that this method is seen in conjunction with the
graphical method or matrix method. A more thorough criticism of
this method is given by E.B.C. Thornton, (1970), [37].
Solving a pair of simultaneous equations
matrices does require a knowledge of the inverse
in 2 unknowns by
of a matrix.
Mathematically it is a far more sound method, but does seem rather
cumbersome. For large systems of equations the inverse matrix is
Page 68
Chapter 6
not as easily found as for the 2 by 2 matrix, and so the method of
elementary
used. This
row operations to get it into upper echelon form is
is clearly a method used by computers and is
unmanageable as a paper and pen exercise for schoolchildren. So
although solving systems of simultaneous equations is at the very
heart of matrix theory, the method does not lend itself to school
use except for the most advanced students, or for the most trivial
of equations for which an alternative method gives rise to the
solution far more efficiently.
It is in the context of simultaneous equations that
matrices are most widely used in such fields as statistics, control
theory, computing and mechanical vibrations; so this is surely how
matrices should be introduced. Since the work is too difficult or
unsuitable for the 11-16 age range it should perhaps be left out or
postponed at least until 'A' level courses.
Children who have familiarity with geometric
transformations will recognise reflections, rotations, enlargements
etc. and will have experience of combining transformations. They
will appreciate that order of operations is important. That some
transformations have inverses, some are self-inverse. The
symmetries of two dimensional shapes provide excellent examples of
a group structure.
For example: in the case of a rectangle the symmetries are:
X the reflection in a horizontal axis of symmetry
Y the reflection in a vertical axis of symmetry
H half turn about the centre of symmetry
I the identity transformation
Page 69
,
I H ~
I -~
~
'I
Chapter 6
--x)
giving the following combination table, where the operation is
"followed by"
I X Y H
I I X Y H
X X I H Y
Y Y H I X
H H Y X I
The lack of commutativity is not made apparant by this
example but does arise in the symmetries of an equilateral triangle
as follows:
I The identity element
RI Rotation of 120 anti-clockwise
R2 Rotation of 240 anti-clockwise
A Reflection in line a (see diagram)
B Reflection in line b ( " " )
C Reflection in line c ( " " )
Page 70
Chapter 6
Then R I followed by A ; C
Whereas A followed by R,; B
The combination table is seen not to be symmetrical.
I R, R'l. A B C
I I R, R2. A B C
R, R, R2 I C A B
R2 R2. I R, B C A
A A B C I R, R'2. B B C A Rl I R, C C A B R, R2. I
It is clearly closed, an identity element exists, every
element has as inverse element. Associativity is not so easily
shown, although the teacher should draw the childrens attention to
it by taking several examples as follows:
(R I followed by A) followed by R'2.; C followed by R2 ; B
R, followed by (A followed by R2 ) ; R, followed by C ; B
There are other equally effective ways of illustrating
group structure:
Page 71
Chapter 6
Modu10 arithmetic (called clock face arithmetic in some
the added pay-off of increasing chi1drens textbooks) has
understanding of numbers. In the Scottish Mathematics Group
Textbooks, modu10 arithmetic is first introduced using the idea of
a rotary heat control on an electric fire:
OFF 0 +
1 MOD4 0 1 2 3
0 0 1 2 3 HIGH3 I LoW 1 1 2 3 0
2 2 3 0 1
2. MED. 3 3 0 1 2
Modu10 arithmetic groups can also illustrate isomorphisms
between groups. For example the above group ([ 0,1,2,3} ,+MOD4)
is isomorphic to the group ([ 2,4,6,8} ,xMODlO).
x
MOD 10 2 4 6 8
2 4 8 2 6
4 8 6 4 2
6 2 4 6 8
8 6 2 8 4
Groups of permutation of n objects can be investigated by
children. Permutations of 3 objects is rather nicely introduced in
SMP book D (1970) using arrow cards to represent the 6 possible
permutations of 3 objects.
etc.
Page 72
chapter 6
Permutations are combined by putting one arrow card after
another. Again commutativity does not hold since the group is
isomorphic to the symmetry group for an equilateral triangle.
Also in SMP book D is an example of a group whose elements
are rectangles with holes cut in them, which are combined by
placing one shape over the other.
I am not suggesting that group theory should be a major
part of all childrens mathematical education. A couple of lessons
spent constructing any of the combination tables suggested above,
would be worthwhile. They give children a simple experience of
symbolisation and algebraic manipulation. For example solving
equations in modulo arithmetic.
x + 3 1 MOD4
Adding the inverse element of 3 to both sides gives
(x + 3) + 1 = 1 + 1
~ x + (3 + 1) = 1 + 1 associative law
~ x + 0 2 since 3 + 1 o MOD4
~ x = 2 since 0 is the identity
element for addition MOD4
The study of simple groups makes children more aware of the
laws of algebra. Such concepts as identity elements and inverse
elements become more meaningful, which increases a childs
understanding of number. Negative numbers being the inverse of
positive numbers under the operation of addition. Similarly for
Page 73
chapter 6
rational numbers and the operation of multiplication. The axiomatic
definition of a group is not appropriate for youngsters at this
stage. The essential features of a group are highlighted by the
combination table (apart from associativity), although when we come
to infinite groups a combination table is inappropriate and the
axioms must be more explicitly investigated.
Work with vectors can also give children an insight into an
algebraic structure other than generalised real number arithmetic.
The commutative law is nicely demonstrated by drawing:
c- Ct '"'"
..t '" As is the associative law:
Scalar multiplication is easily motivated whereby 2a is
seen to be in the same direction as a but twice as long. When
coordinate geometry is used a vector is defined to be a
displacement. It is usually specified by a number pair whose two
components are the changes in the x and the y coordinates of every
point undergoing the given displacement.
For example(_~ ) displaces the point (p,q) to the point
(p + 5 , q - 1)
Page 74
chapter 6
Vector addition is easily defined
and so is scalar multiplication
It is a straightforward process to deduce from these
definitions many of the laws of combination applied to vectors.
Associative law:
Distributive law:
=
There is a danger that over-reliance on the coordinate
system may encourage children to only recognise vectors as being
pairs of numbers written in a column (or triples in three
dimensions). It is important that they realise they are dealing
with "directed line segments" so that they can recognise, for
example, a vector difference in a diagram.
Page 75
chapter 6
This property as shown in the diagram would not be easily
apparent from consideration of number pairs.
There is also a danger involved in our notation. In the
statement:
The first plus sign has a completely different meaning to
the other two. It stands for the addition of vectors whereas the
other two plus signs stand for our normal number addition. In fact
in the statement
The first times sign has again a completely different
meaning to the other two. In the case of column vectors this is
unlikely to cause confusion, but for directed line segments it may.
Consider this triangle:
A
It would be false to say: c
AB + BC ; AC
because our normal notation is that AB, BC and AC are lengths, and
our plus sign is the addition of numbers.
Page 76
chapter 6
As" ~ --Whereas + BC = AC is correct.
~ -') -')
AB BC and AC are now no longer lengths, but
displacements (ie. their direction is being considered) and our
plus sign really means "followed by". A displacement from point A to point B followed by a displacement from point B to point C
results in a displacement from point A to point C.
Special attention is drawn to this meaning of the plus sign
in the Scottish Mathematics Group Textbooks (see Modern Maths for
School book 4 (1972) pp 90-93) where the notation used is to put a
circle around the plus sign.
-;. ---;. ---;. AB0BC = AC
Thus Qmeans "followed by". I am sure this notation if
consistently used would deter children from just adding lengths
when combining displacements. It seems strange that later on in the
same chapter they immediately drop the notation when using column
vectors. They give the following explanation:
"Since we get the answer by adding components it is usual
to think of adding translations in number pair form, and to use +
rather than G".
This must surely encourage children to think of column
vectors and their associated algebra as being alien to directed
line segments and their algebra. I feel that whatever notation is
adopted it should be consistently applied.
In Book 5 the circle notation is completely abandoned for
the following reasons:
Page 77
chapter 6
"We see ••••• that the operation denoted by G obeys the
commutative and associative laws in the same way as the operation
denoted by + • It seems reasonable to replace ~ by +, and to talk
of adding vectors. In some situations, however, it will be
necessary to make a clear distinction in our minds between vector
addition and numerical addition" (see Modern Maths for School Book
5 ppllO).
Actually it is a good job we do not replace all operations
which are commutative and associative with a plus sign or we could
have awful problems. At least the MMS books have made some attempt
to highlight the problems of the notation. Other schemes (SMP for
example) have no qualms in immediately using a plus sign to mean
followed by with no justification at all.
The other problem of notation is that of the vectors
themselves. Many texts use heavy type which is impossible for
children to reproduce in their work books. Notations recommended
for children are either a line or wiggle underneath a letter (thus
u or u) or line above, arrow above, or wiggle underneath a pair
~). Again the lack of
the fact that the textbooks
notation is to be different from the childs and the teachers, is
really not acceptable. Perhaps it is difficult for the publisher to
print the wiggle, but they really should make the effort.
'" of capital letters. (thus AB , AB or rv
standardisation and more importantly
I feel that a study of vectors should be included in the
maths course for the majority of children. They have important
applications in physics (force, velocity, acceleration etc.) and
even in chemistry when balancing chemical equations.
Page 78
chapter 6
For example finding values of x, y, z, and s to "balance"
this equation:
xZn + yHCl~ zZnCl + tH
involves solving the following vector equation:
where each vector represents a molecule. The three components of
the vector are the number of atoms of zinc, hydrogen and chlorine
contained in the corresponding molecule (see W.W. Sawyer, (1970),
[34], pp 113 - 117).
Vector notation can be taught to various levels of
mathematical maturity and develop skills in symbolism, geometry,
manipulation and understanding of laws and structures.
Page 79
Chapter 7
The Role of Proof in School Algebra.
Traditionally the idea of proof is linked with geometry.
Many of the more modern syllabuses in geometry have excluded the
idea of proof, although it is in fact compatible with any approach
to geometry. Proof is an important part of all branches of
mathematics. The mathematician should always be able to justify (or
prove) his methods and results. It is regrettable that proof has
almost disappeared from school mathematics. Although attempts have
been made to teach axiomatic systems in algebra, this approach has
not been widely accepted. But algebra can still be a medium for
teaching children the idea of proof. It may also have the advantage
of proving results that are not immediately obvious. This was often
not the case with Euclidean geometry where some of the results to
be proved appeared at least as obviously true as the axioms which
were supposed to be obviously true. For example it is not
immediately obvious that:
1 + 3 + 5 + 7 + •••• + (2n+l) is equal to n
By investigating the first few terms a child will probably
guess the result, but will surely recognise the need for a proof of
the general result.
This could be done visually by seeing how successive terms
give the next square number.
liJ' ~ •
+3 Co " 0 • 1\
+5 " 0 0 c:. 0
-tt " .. 0 \)
+9 " p
Page 80
Chapter 7
This is a kind of implied proof by induction ie. since the
next square number is always found by adding the next odd number to
the previous square number etc. It could be made more rigorous
(with more able or older classes) by considering the nth square
number, and how many dots must be added:
oE.('----n dots ___ ?»
extra n V dots i
n dots
~::::::::~::::::::::~~o~ extra one C / ~ dot
extra n dots ~
From the diagram 2n + 1 dots must be added to the nth
square to get the (n + l)th square.
Alternatively this formula could be proved by applying the
rules of arithmetic. Thus:
If S = 1 + 3 + 5 + 7 + ........ +(2n - 1)
Then rewriting the series the other way round (which is valid for
numbers addition) we get:
S = (2n - 1) + (2n - 3) + ....•... + 3 + 1
Adding the two series gives:
Page 81
Chapter 7
28 2n + 2n + ........... + 2n + 2n
There are n terms in the series so since multiplication is repeated
addition we get:
28 = n x 2n
Thus 8 = n2.
This algebraic proof uses the properties of addition,
multiplication and counting of numbers. Visual proofs are often
used in school text books. For example as an illustration of the
distributive law using areas of rectangles.
a b
"l. area=a area
a = ab
area
b area=ab = b2
(a + b) = a 1. + 2ab + b"
Care must be taken when proving results for infinite
series as the following example shows:
Page 82
Chapter 7
Consider the following infinite series which can be shown
to be convergent to ln2
=
=
=
=
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +
2 3 4 5 6 7 8
Rearranging the terms we get
1 - 1 - 1 + 1 - 1 - 1 + 1 - 1 - 1 +
2 4 3 6 8 5 10 12
h - 1\-~ +f~ -~\- 2. +(~ - 1~- ~ + \ 21 4 ~3 6·) 8 \s 10) 12
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 -- -
2 4 6 8 10 12 14 16 18
. . .
. . .
1 (1
- 1 + 1 - 1 + 1 - 1 + 1 - 1 + ••. ) 2 2 3 4 5 6 7 8
1 ln2
2
So we have proved that ln2=2ln2.
This paradox shows that infinite series have to be handled
with great care. A thorough study of the theory of series is
clearly beyond the scope of most secondary school pupils.
In the context of school algebra, proof does not mean the
sort of formal rigour beloved of the pure mathematician.
It is important that the child wants to know how or why
Page 83
Chapter 7
something works. The result should not be so intuitively obvious
that it is seen not to require justification. We must be clear
about the assumptions we make and ensure that they are both
reasonable and valid. The method of proof must be seen to be
legitimate and understandable by the pupil. A rigorous proof which
takes into account every subtlety will probably not impress a
pupil. With these limitations there is plenty of scope, at all
levels, to introduce proof into school algebra. Let us take some
examples:
"Think of a number" type problems often fascinate younger
pupils. Think of a number, double it, add ten, half the answer,
take away four, take away the number you first thought. of and the
answer is always one. Why? How does it work?
Write down a two digit number, reverse the digits to give a
different number, find the difference and add them together, the
result is always 99. Why?
Rules for combining odd and even numbers also provide scope
for proof. If we write an even number in the form 2n and an odd
number in the form 2n-1 (n a natural number). Then we can prove
such results as the product of 2 odds is always odd. We can explain
why the sum of 2 consecutive odd numbers must be a multiple of 4.
Similar results can be proved for other multiples.
If a number is divisible by 9, then the sum of its digits
is also divisible by 9. Why is this so?
There is a whole variety of number patterns whose results
and properties can be proved using algebra. (see J. Hunter, and M.
Cundy, (1978), [25], B. Bolt, (1985), [4] and [5], L.J. Motorhead,
(1985), [30]). The teacher should use such examples to motivate
Page 84
Chapter 7
youngsters to see a purpose and meaning to algebra. To prove a
pattern true for all numbers, then you just prove it for x because
x represents any number. Also with the decline of the use of proof
in geometry, it is important for children to experience the idea of
proof in other branches of their mathematics course.
Page 85
Chapter 8
Algebra and GCSE (MEG)
The GCSE syllabuses and techniques of assessment are
determined by the Examining Groups based on the National Criteria
first published in 1985. They have a considerable influence on the
content and pace of work in secondary classrooms. To a large extent
the school syllabus for the final couple of years at secondary
school is identical to the examination syllabus, such is the
influence they hold. Of particular interest to us here are the aims
and assessment objectives related to algebra. Aims are, by their
nature, rather vague so many of them could be interpreted as
referring to algebra. For example: All courses should enable pupils
to:
Aim 2.2 read mathematics, and write and talk about the subject
in a variety of ways;
2.5 solve problems, present the solutions
clearly, check and interpret the results;
2.10 develop the abilities to reason logically, to classify,
to generalise and to prove;
2.11 appreciate patterns and relationships in mathematics.
2.14 appreciate the inter dependance of different branches of
mathematics;
The above is just a selection, but in one way or another,
all could be said to be referring to algebra.
The assessment objectives are rather more specific and
clearly require algebraic skills. Again a selection will suffice to
Page 86
Chapter 8
give the general idea. The scheme of assessment should test the
ability of candidates to:
3.2 Set out mathematical work, including the solution of
problems, in a logical and clear form using appropriate symbols and
terminology;
3.9 Recognise patterns and structures in a
variety of situations, and form generalisations;
3.10 Interpret, transform and make appropriate use of
mathematical statements expressed in words or symbols;
3.12 Analyse a problem, select a suitable strategy and apply
an appropriate technique to obtain its solution;
3.13 Apply combinations of mathematical skills and
techniques in problem solving.
In addition as from 1991 two further assessment objectives
are included which can only be fully realised by assessing work
carried out by candidates outside the normal time - limited written
examinations. Namely:
3.16 Respond orally to questions about mathematics, discuss
mathematical ideas and carry out mental calculations;
3.17 Carry out practical and investigational work, and
undertake extended pieces of work.
The latter being the controversial course-work element of
the GCSE assessment. Most educators welcome course-work assessment.
It frees teachers from the straight jacket of a rigid examination
Page 87
Chapter 8
syllabus. A good teacher should be able to make his courses more
interesting and relevant and incorporate the interests of the
class. There is some anxiety on the part of many teachers who are
naturally concerned to ensure that their students achieve their
full potential and are unsure how to prepare them to complete a
coursework investigation. There is also the additional problem of
validity, how can you ensure that coursework is in fact completed
by the student? The various phases of training undertaken by the
examining groups and by local authorities are designed to alleviate
major misgivings, but I suspect that the scheme will be in
operation a couple of years before most teachers feel confident in
their new role.
range
The need for differentiated exams
of abilities to be assessed by GCSE
to cater for the wide
examinations has led
examining groups to adopt three levels of assessment. The
foundation level is an examination for which the great majority of
students should be awarded grades E, F, and G. The Intermediate
level is for students awarded grades C, D, or E. Students capable
of grades A or B should sit the Higher level examination.
The minimum syllabus content for both the foundation and
the intermediate level examinations have been supplied by the
D.E.S., whereas the additional material for the higher level
examination was left to the discretion of the Examining groups. The
advice was that:
"Examining Groups should have regard to the need both to enable
pupils to develop their powers of abstraction, generalisation and
proof and also to provide a firm foundation for future work in
mathematics at 'A' level and beyond".
Page 88
Chapter 8
References to "abstraction", "generalisation" and "proof It
clearly imply a greater algebraic content to be included in the
higher level course.
The items included in the foundation level list (a kind of
mathematical "CORE" curriculum) reflects quite closely the
recommendations of the Cockcroft Report, [17]. This committee of
inquiry produced a foundation list of mathematical topics (see
paragraph 458 pp 135-140). No specific mention was made of algebra
in that list although the use of formulas was implied in the
finding of areas and volumes, simple flow charts are mentioned, and
the substitution of numbers in simple formulas expressed in words.
Under the "use of calculator" heading comes the recommendation that
children should appreciate the need for careful ordering of
operations when using a calculator. The GCSE foundation topic list
does include the use of letters for generalised numbers, and the
substitution of numbers for letters as well as for words in
formulas. So it does go slightly further than the Cockcroft report
foundation list.
The list of topics to be included in the intermediate
level examination include all those in the foundation level
together with some additions. Again, just concentrating on algebra,
we have: Transformation of simple formulas; Basic arithmetic
processes expressed algebraically (ie construction of formulas);
Directed numbers; Use of brackets and extraction of common factors;
Positive and negative integral indices; Simple linear equations in
one unknown. In the section on graphs, is included: Constructing
tables of values for given functions which include expressions of
the form: ax + b, axl. ,-3: where a and b are integral constants.
Drawing and interpretation of related graphs; idea of gradient.
In comparison with the old CSE syllabus, this list is
Page 89
Chapter 8
remarkable for its omissions rather than its inclusions. Gone is
set notation. One would have thought that the concept of sets and
intersection etc would be of value when studying loci, probability
and graphs all of which are included. Presumably this is left to
the teachers discretion because no mention is made of sets, their
terminology or symbolisation. No mention is made of vectors or
matrices either. What is the thinking behind this decision? I have
stated elsewhere that matrix theory is not easy to apply at an
elementary school level, and will be a feature of very few adults
lives. (Certainly not at this level of mathematical attainment).
Whereas vectors I think should be included. Combining displacements
using bearings and coordinates can be easily motivated in school
and can provide youngsters with an experience of symbolism,
manipulation and algebraic structure.
The extra algebra to be covered for the higher level
examination includes:
l. 2 2 b'2. 1.. Factorisation ofax +bx+c, a x - y, ax+bx+ay+by
Expansion of products of the form (ax+by)(cx+dy)
Manipulation of fractions with numerical and linear
algebraic denominators.
ego x + x-4 or 1 • 2 , - • 3 2 x-2 x-3
Transformation of formulas including cases where the new
subject may appear more than once. Substitution into formulas which
may include positive and negative fractional indices. Fractional
equations with numerical and linear algebraic denominators.
ego 3 = 6
x-2
, or x + x-2 = 3
3 4
Page 90
Chapter 8
Transformation of formulas including cases where the new
subject may appear more than once.
Substitution into formulas which may include positive and
negative fractional indices.
Fractional equations with numerical and linear algebraic
denominators.
Simultaneous linear equations in two unknowns.
2 Quadratic equations ax + bx + c = 0 including cases where
the solutions are irrational.
Plotting and sketching graphs of cubics as well as
quadratics as well as rational functions of the form:
y = k (x = 0). X4
Interpretation of the constants m and c in the general
expression y = mx + c as gradient and intercept respectively.
Calculation of the area under a straight line graph, and
interpretation of the answer.
Formation, graphical representation and solution of linear
enequalities. (Although perhaps surprisingly linear programming
problems are excluded).
Pythagoras Theorem (This is of course included in the
intermediate level syllabus for numerical examples, but for the
higher level the syllabus specifically mentions that algebraic
examples will be included).
Page 91
Chapter 8
Also in the geometry section the combination of
transformation is included. Translations, described by column
vectors implies the need for vector addition. Clearly some algebra
of transformations will need to be covered by the teacher.
Vectors are included in the syllabus:
Idea of a vector as a translation represented by
(;) ,AB'>and a (heavy type, no wiggle).
So the notation to be used in the exam is spelled out. Text
book writers are clearly adopting this notation for new GCSE
textbooks. The difficulty of children being unable to reproduce the
heavy type notation in their workbooks is ignored.
Included is
magnitude of
addition of vectors, multiplication
vector ( Xy) as / x2+ y2.
Two dimensional work only at this stage.
by a scalar,
The representation of vectors by directed line segments.
The sum and difference of vectors and their use in expressing
vectors in terms of two co planar vectors.
Position vectors
Page 92
Chapter 8
Use of the results:
and a parallel to b
(ii) ha ; kb ) Either a is parallel to b, or h; 0 and k ; 0 """""''V -.., """'"
in deducing properties of equivalence, parellelism and incidence in
rectilinear figures.
Clearly the implication is to use vectors to prove
geometric results. Non trivial results are always difficult to
devise. Using vectors to prove that the diagonals of a
parallelogram bisect each other may appeal to the mathematician,
but may appear to be proving the obvious to most children, if that
is, they accept the validity of the proof anyway.
On the whole I am pleased that vectors are included in the
syllabus. It is not an easy topic for the teacher to put over. The
notation, the algebra, the concepts involved are all novel to the
student. Considerable familiarity will be necessary before vector
proofs of geometric properties will be understood, but I am
satisfied that with good teaching the competent higher level
candidate will benefit from studying vectors.
Finally matrices are included in the higher level
syllabus. Starting off with practical applications as stores of
information. Addition, subtraction, multiplication, of one matrix
by a scalar, and multiplication of compatible pairs of matrices.
Representation of transformations of the plane
(reflection, rotation, enlargement) given the matrix and finding
the matrix given the transformation using base vectors.
Page 93
Chapter 8
The algebra of 2 x 2 matrices is specifically required
including: the zero matrix, the identity matrix, inverses of
non-singular 2 x 2 matrices and use of determinant.
Even at the higher level the algebra of sets and boolean
algebra are not included so the only abstract algebra to be studied
is for vectors, transformations and matrices.
There is no indication in the syllabus of how certain
topics should be tackled. For instance whether to solve quadratic
equations by use of the formula or by completing the square. Or
which method to use when solving simultaneous equations. Since
inverses of 2 x 2 matrices is included, it may seem appropriate to
use matrices to solve simultaneous equations, but this is not
stipulated, and rightly so. The teacher should use the method he
feels to be appropriate for his class and the method he feels most
confident with himself.
Overall the syllabus is reduced in content from the
equivalent '0' level syllabus. This does seem to be in line with
the aims of the GCSE examination. A thorough understanding of these
basic skills is better than a scant knowledge of a more diverse
syllabus.
Of course the syllabus list is not the whole story.
Although it is rather early to make any definite conclusions, from
specimen papers available so far, there is evidence that some
attempt is being made to make algebra more meaningfully applied in
examination questions.
One effect of this
More practical and less abstract in nature.
may be to increase the number of "word
problems". These create linguistic problems for the student.
Firstly the student must understand the real situation described in
the written problem. Then he must understand the mathematical
Page 94
Chapter 8
relationships appropriate to the problem expressing them in precise
mathematical symbols. Only then can the appropriate process of
arithmetic or algebra be applied.
For example at Intermediate level we have the following
problem:
"Midland Motors hire out lorries at a basic charge of i 60 plus a
further charge of 60p per km travelled"
The question goes on to ask children the charge for
various distances travelled, as well as the distance travelled for
a given hire charge. They are also asked to draw a graph to show
how cost varies with distance, and finally:
"Write down a formula for C in terms of x".
This is the way I think an algebra problem should be
posed. Using realistic variables in a practical situation. I would
hope that in the future we will see less algebra set without any
context, purpose or meaning. Questions of the type:
Solve 3
x-7
= 2
have no reality for children.
Answer x
From 1991 onwards the GCSE assessment, at all levels, will
have a compulsory coursework component counting 30% towards the
final grade. lVithin this coursework component there will be a
mental test and four assignments. The assignments must be within
the following four categories:
Page 95
Chapter 8
(1) Practical Geometry
(2) Applications of Mathematics
(3) Statistics and/or probability
(4) Investigational work
Marks will be awarded for:
(A) Overall design and strategy
(B) Mathematical Content
(C) Accuracy
(D) Clarity of argument and presentation
(E) Oral skills
The assessment of oral skills is perhaps the most
controversial. This is to be carried out by the teacher in the
classroom situation or by personal interview between pupil and
teacher. Because of pressure of time and because it is unlikely
that maths teachers will be withdrawn from their timetable
commitment and replaced by a supply teacher, this assessment will
have to take place in a busy crowded classroom. It was originally
suggested that teachers should make tape recordings of oral
exchanges with pupils, but that requirement has now been withdrawn.
But a dated record of oral exchanges must be kept. How many
teachers have sufficient spare time to do that properly? The
purpose of the oral assessment is to measure a candidates ability
to communicate in mathematics by:
Page 96
Chapter 8
(i) responding directly to questions
(ii) discussing mathematical ideas
(iii) explaining mathematical arguments
The intention of the oral assessment is laudable, but
without the time, and with large classes to manage, I suspect that
the teacher will give a rather quick and entirely subjective mark.
This may not be a bad thing as an experienced and competent teacher
will know his pupils well and have talked to them over the course
of many weeks (even years). This is necessary in part to ensure
that the coursework is indeed the individual work of the candidate.
But. the need for a record of oral exchanges seems an unreasonable
imposition. Since validation is most difficult we should just
accept that this is a mark awarded at the discretion of the
teacher.
The category of coursework most applicable to the use of
algebra is investigationa1 work (although of course this is not
exclusively so). A lot of investigationa1 work in maths involves
discovering, recording, and validating generalisations. These are
processes at the very root of algebra. Sequences, number patterns,
geometrical patterns all lend themselves to the application of
algebra. It is in this category that a chi1ds experience of algebra
at a younger age should bear fruit.
Page 97
~hapter 9
Micro-Computers in the Teaching of Algebra
When using exercises in programming as a way of teaching
algebra on a computer, one inevitably comes upon the problem of
notation. In algebra we use single letters as variables. So abc in
algebra means "a times b times c", whereas abc in BASIC is read as
a single variable. This notation
programming so that "meaningful"
is
names
adopted for computer
can be assigned to
variables. Indeed it is considered good practice at programming to
use meaningful variables, so that "reading" a program is made
considerably easier.
TOTALCOST = NUMBEROFBOOKS'* PRICEOFBOOK
is the preferred way of writing a formula for use in a computer.
However if meaningful names are assigned to variables in programs
used as examples for children it may encourage the child to think
that the computer "understands" the formula in the same way as we
do. That the variables are "meaningful" to the computer. If we are
using exercises in computer programming to stimulate understanding
of algebra, then initially it may be better if we use single
letters as variables.
Operations are also recorded differently in BASIC than in
the conventional algebraic rotation. Multiplication cannot be
implied, but must be made explicit by using a star. In order that
an expression can be typed on a single line we cannot write
division as one number over another. We do not even use the
division sign ~ as used on a calculator, •
but use a stroke /. This
sign in some way implies one number over another, but there are
differences in its use.
Page 98
For a start more use must be made of brackets so:
a + x must be written (a + x)/(y + z)
y + z
Chapter 9
In the standard notation the long line implied the use of
brackets. Powers are written differently so x 2 is written x 1'2 in
BASIC. Perhaps the most iniquitous difficulty occurs with negative
numbers. The minus sign is read as the operation of subtraction
unless it is the first symbol in an expression:
So
So
completely.
infancy.
-31' 2 is read as (_3)2 which is + 9
o -31' 2 is read as 0 - (3)~ which is - 9
adding zero alters
Clearly programming
the value
languages
of
are
the
still
expression
in their
Not that our algebraic notation is without fault. The
expression 1 / x Y is ambiguous.
It could be interpreted as 1 ie. y
xy x
At least BASIC is consistent in regarding ~ and / as having equal
precedence and so reads from left to right. Thus the expression:
1 / OlfY is read as or y
x
Other factors which inhibit the use of computers to teach
algebra are: their poor notation for handling matrices, sets and
Page 99
Chapter 9
vectors and the limited character set of a typical micro-keyboard.
For a more thorough analysis of using BASIC notation to
teach algebra see D. Tall, (1983), [36], and R. Cooper, (1986),
[18] and [19].
Our current terminology and arithmetic notation have
evolved over centuries in a fairly illogical and haphazard way. For
example the reason we call xlto the power two, x squared, dates
from when such quantities were considered to be areas of squares.
Our notation is imprecise, ambiguous and ill defined. Attempts are
being made to develop executable notation for arithmetic (APL ,
AMPL) which is well defined and easily understood by man and
machine, but until such notation is in widespread use (which is not
in the forseeable future) notation will remain a problem.
I suspect that a programming language much closer to our
algebraic notation will be developed for use as a teaching aid for
algebra which will use single letters as variables, and insert
between letters. There will of course be problems. For example when
dealing with an expression such as 2SIN3x, the computer must not
put a star between the S, the I and the N. But such difficulties
are not insurmountable. The inconsistencies in our own notation
will also require some thought.
Doubtless a computer can be a useful tool in teaching
children the use of formulas. We use formulas to tell a computer
what to do, so exercises in programming clearly involve the use of
algebra. But I am not suggesting that we teach all children the
intricacies of computer programming. Of more relevance to the
school teacher is software available which can motivate
investigations leading to the use of algebra.
Page 100
Chapter 9
The biggest impact that computers have had on mathematical
education has been as a teaching aid or a learning resource. A
computer can be used as a visual aid in the same way as an OHP or
video recorder with the advantage that the teacher or learner can
interact with the machine. Computer graphics can provide animated
dynamic visual images. Computers can be used as a simulation device
for mathematical models or games of strategy. A games rules are
programmed in (it can never cheat). Children must solve problems
and devise strategies. Skills central to mathematical education.
Another effect of computers on algebra is the way they
effect what is being taught. Mathematics applied to commerce and
industry is now inseparable from computing. To the extent that
school mathematics should reflect contemporary "real" usage, then
techniques using computers should be introduced at school level.
Traditionally only simple linear and quadratic equations were
taught in school because they lent themselves to being solved by
pencil and paper algorithmic techniques. Using numerical methods
applicable to computers enables the student to solve a whole new
range of equations. There are many other areas of content which
become increasingly attractive for inclusion within the
mathematics curriculum as computers become more available in
classrooms. For example: modelling, number theory, numerical
methods, statistics and simulation are all potential candidates for
inclusion in school mathematics.
So far I have not mentioned hardware. British secondary
schools have on average only one micro for every sixty children.
That average hides a considerable variability, with TVEI schools
having many more than average. About 66% are located in designated
computer laboratories and the remainder are sited in a variety of
departments around schools with business studies and science the
most common. You begin to realise how few computers are actually
Page 101
Chapter 9
available for maths lessons. Add to this the fact that technical
support staff to assist with computing is virtually non-existent in
British schools. It is hardly surprising that in a recent survey
only 20% of headteachers think that computing has made a
significant contribution to teaching.
Page 102
Bibliography
Bibliography
[ 1] Anderson, J. , (1978), "The Mathermatics Curriculum:
Algebra", Published for the Schools Council (Blackie).
[2] A. P. U. , (1980), "Mathematical Development: Secondary Survey
Report No 1", (H.M.S.a.).
[3] Banwell, C.S., Saunders, K.D., and Tahta, D.G., (1972),
"Starting Points", (O.U.P.).
[4] Bolt, B., (1985), "Mathematical Activities", (C.U.P.).
[5] Bolt, B., (1985), "More Mathematical Activities", (C.U.P.).
[6] Booth, 1.R. , (1981) , "Child-Methods in Secondary
Mathematics", Educational Studies in Mathematics, 12, pp29-40.
[7] Booth, L.R. , (1982) , "Getting the Answer Wrong",
Mathematics in School, 11(2), pp4-6.
[8] Booth, L.R., (1982), "Ordering Your Operations",
Mathematics in School", 11 (3), pp5-6.
[9] Booth, L.R., (1982), "Sums and Brackets", Mathematics in
School, 11(5), pp4-6.
[10] Booth, L.R., (1984), "Algebra: Childrens Strategies and
Errors", A report of the Strategies and errors in Secondary
Mathematics Project, (NFER/Nelson).
[11] Brown, G., and Desforges, C., (1979) , "Piaget' s Theory: A
psychological critique", (Routledge & Kegan Paul).
Page 103
Bibliography
[12] Bufton, N., (1986), "Exploring Maths with a Computer",
Council for Educational Technology, (M.E.P.).
[l3] Clement, J., et a1., (1981), "Translation Difficulties in
Learning Algebra", American Mathematics Monthly, 88.
[l4] Clement, J. , (1982), "Algebra Word Problem Solutions:
Thought process underlying a common misconception", Journal for
Research in Mathematical Education, 13, pp16-30.
[15] Centre for Mathematics Education, (1985), "Routes to /
Roots of Algebra", (O.U.P.).
[16] Collis, K.F. , (1969) , "Concrete Operatioa1 and Formal
Operational Thinking in Mathematics", Australian Mathematics
Teacher, 25(3), pp77-84.
[17] Committee of Inquiry into the Teaching of Mathematics in
Schools, (1982), "Mathematics Counts", (The Cockcroft Report),
(H.M.S.O.) •
[18] Cooper, P., (1986), "Algebra with a Computer", Times
Educational Supplement(Extra), 9th May 1986.
[l9] Cooper, R., (1985), "Teaching Algebra with a Computer" ,
Polytechnic of the South Bank.
[20] Dienes, Z.P. , (1964) , "The Power of Mathematics",
(Hutchinson).
[21] Gattegno, C., (1983), "On Algebra", Mathematics Teaching,
lOS, pp34-36.
Page 104
Bibliography
[22] Ginsburg, H., (1975), "Young Children's Informal Knowledge
of Mathematics", Journal of Children's Mathematical Behavior, 1(3),
pp63-156.
[23] Hale, D., (1981), "Algebra Teaching Today", Mathematics in
School, 9(1), ppll-12.
[24] Hart, K.M., (1981), "Investigating Understanding", Times
Educational Supplement, 27th March 1981, pp45-46.
[25] Hunter, J., and Cundy, M., (1978), "The Mathematics
Curriculum: Number", Published for the Schools Council, (Blackie).
[26] Kuchemann, D. E. , (1978) , "Children's Understanding of
Mathematical Variables", Mathematics in School, 7(4), pp23-26.
[27] Kuchemann, D.E., (1981), Chapter 8 "Algebra", in Hart,
K.M. , (Ed), (1981) "Children's Understanding of Mathematics:
11-16", (J. Murray).
[28] Kuchemann, D.E., (1982), "Object Lessons in Mathematics",
Mathematics Teaching, 98, pp47-51.
[29] Ling, J., (1977), "The Mathematics Curriculum: Maths across
the Curriculum", Published for the Schools Council, (Blackie).
[30] Motterhead, L.J., (1985), "Investigations in Mathematics",
(Blackwell) •
[31] O'Shea, T., and Howe, J.A.M., (1980) "Teaching Maths
through LOGO Programming",
Page 105
Bibliography
[32] Plunkett, S., (1979), "Decomposition and all that Rot",
Mathematics in School, 8(3), pp2-5.
[33] School of Mathematics, University of Bath, "mathematics in
Employment 16-18".
[34] Sawyer, W.W., (1970), "The Search for Pattern", pp1l3-117,
(Penguin).
[35] Shuard, H., and Neill, H., (1977), "The Mathematics
Curriculum: From Graphs to Calculus", Published for the Schools
Council, (Blackie).
[36] Tall, D., (1983), "Introducing Algebra on the Computer:
Today and Tomorrow", Mathematics in School, 12(5), pp37-40.
[37] Thornton, E.B.C., (1970), "A Fresh Look at Simultaneous
Equations", Mathematics Teaching, No.52, pp44-47.
[38] Wheeler, R.F., (1981), "Rethinking Mathematical Concepts",
(Ellis Horwood).
[39] Wigley, A., et aI., (1978), "Algebra: Ideas and Materials
for Years 2-5 in the Secondary School", South Notts Project, Shell
Centre, University of Nottingham".
[ 40] Wood, M., (1978), "Formulae and an Initial Teaching
Algebra", Mathematics in School, 7 (1), pp27.
PLEASE NOTE: Standard School Mathematics Textbooks are not listed
in this bibliography
Page 106