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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

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[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