Purposes of School Algebra

8/10/2019 Purposes of School Algebra

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JOURNAL OF MATHEMATICAL BEHAVIOR

14,

41-73 (1995)

Purpose in School Algebra

ALAN BELL

The Uni versity of Nottingham

Algebra is commonly regarded as a problem area for many students, which has led

teachers in schools to question its necessity, whereas lecturers of tertiary-level courses

continue to complain of incoming students lack of algebraic skill.

In considering how this situation should be addressed, I need first to clarify the aims

and objectives of the school algebra course. I then

review

the research evidence on

students performance and proceed to

some

suggestions for curriculum modifications that

might be helpful.

AIMS OF SCHOOL ALGEBRA

I give a few examples of tasks to focus the discussion.

Most

of these are from

examination papers for the most able 25% of 16-year-olds. Which of these would

you regard as representing the aims of a school algebra course?

Example 1:

(a) Show that the sum of a number of four digits and the number formed by

reversing the digits is always divisible by 11.

(b) The greatest and least of four consecutive numbers are multiplied together; so

also are the middle pair. Show that the difference of the two products is always 2.

Example 2:

(a) A boat rows a certain distance upstream at 2 mph, stops for an hour, and

returns at 4 mph. The total time is 3.5 hours. What is the distance each way?

(b) Same questions with speeds 2.6 mph and 4.1 mph.

Example 3:

(a) The cost per hour of running a ship is a fixed amount of La together with a

variable amount of fbV3 which depends upon the speed

V

miles per hour) of the

ship. The total cost of a journey of d miles at a uniform speed of V miles per hour is

fC. Prove that C = d a/V + bV2).

Given that a = 3.3, V = 15, d = 3000, and C = 1300, calculate b correct to two

significant figures.

Correspondence and requests for reprints should be sent to Alan Bell, Shell Centre for

Mathematical Education, The University of Nottingham, Nottingham NC7 2RD, United Kingdom.

41


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Example 4:

Show that, if 2s = a +

b + c,

(a) s(s - a) + (s -

b) s - c) = bc

(b)

s2 -

(s -

a)2

b+c

=-

s - c)~ - s - b)2 b - c

Example 5:

PIZZA PRICES

Size

Mini

Small

Medium

Large

Family

Diameter of

Pizza Plate

20 cm

25 cm

21.5 cm

30.5 cm

38 cm

cost

$ 4.00

$ 5.00

$ 7.50

$ 8.60

$10.50

Explore the relations between diameter and cost, and discuss which size is good

value.

The first of these examples requires the problem to be represented alge-

braically, using knowledge of place value, as

1000~ + 1OOb+ 10~ + d + IOOOd + 100~ + lob + a)

which leads, in two steps, to

1001 (a + 6) + 110

b + c)

which is easily verified to be a multiple of 11.

The second part of the question involves expressing the consecutive numbers

as n, IZ + 1, II + 2, n + 3, leading to

n (n + 3) - (n + l)(n + 2)

which on multiplication shows that the difference is 2, independent of n.

This problem thus demands formulation in algebraic terms, some manipula-

tion, and finally interpretation of the conclusion. This is a typical example of the

use of algebra to establish generalizations-except that the full process includes

the discovery of the generalization, not simply its proof.


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BELL

compared with the three modes of algebraic use illustrated by Examples 1, 2, and

5 and which we shall take as our curriculum framework:

Generalizing.

Forming and solving equations.

Working with functions and formulae.

Students need to become competent in handling these processes, not just in

the symbol manipulations represented by Example 4. Moreover, it appears that

conceptual obstacles arise less frequently and are easier to overcome when the

work is embedded in appropriate meaningful activities of these kinds. This claim

will be discussed more fully in the last main section of this article. In the next

section, I review the evidence from research regarding the nature and extent of

difficulties that students experience in the learning of algebra in existing courses.

I first look at research indicating how far students are able to handle the three

general processes previously listed-generalizing, forming and solving equa-

tions, and working with functions and formulae. Then I report some work that

cuts across the three main categories.

RESEARCH ON GENERAL ALGEBRAIC STRATEGIES

Generalizing

Generalizing was the focus of a study by Lee and Wheeler (1987). It concerned

the algebraic thinking of 15-year-old students, focusing on their conceptions of

generalization and justification. Three hundred and fifty four students were

tested, using 12 problems, and 25 individual interviews were subsequently con-

ducted. The problems demanded the use of algebraic language in reasoning about

situations.

The first problem used was:

Is the statement

Say how you know.

2.x+ I

1

2x+1+7=8

definitely true?

possibly true?

never true?

The general conclusion here was definitely true, but for two distinct

reasons-either the 2~s were canceled, or cross-multiplying led to an equation

which was then solved, or at least seen to be solvable. Those who canceled were

asked to check by putting a numerical value for n, but no one was able to use this

to recognize that the equation is true for x = 0 but for no other value. Hence, the

concept of an equation-like statement being true for some values of x, but not for


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About half of the students tested were able to express consecutive numbers as a,

a + 1 and to obtain 2a + 1; they then tried various substitutions and manipula-

tions, but few could give a general argument to show that this sum must be odd.

In another question, fewer still could explain why x(x + 1) is even.

Thus, the central purpose of algebra was perceived by these students as the

performance of some manipulation; its use as a mode of expression of some

generalizations, allowing discussion of the conditions of their truth, or as the

basis of an argument, was absent.

Another study of typical 15-year-old students ability to explain and justify

generalizations also showed a low level of application of algebra. One of the

tasks required deriving and explaining the fact that if the same number is added

to 10, and subtracted from 10, the sum of the two results is always 20. Only 3 of

4 1 students even attempted to represent the situation algebraically; the remainder

used verbal explanations, often failing to distinguish data from conclusion (Bell,

1976).

Forming and Solving Equations

The forming of equations was the subject of a study by Galvin and Bell (1977).

Here the evidence showed a strong tendency for students to write the arithmetical

calculations required to solve the problem. Writing an equation to represent the

problem, and then working with the equation instead of with the original prob-

lem, was a major obstacle. Perhaps this is the most important difference between

arithmetic and algebra. Galvin and Bell stated:

It appeared that equations represented a quite distinctive form of expression which

was unlikely to be adopted by the pupils spontaneously unless they both recognised

that this expression led to the possibility of algorithms for solution, and also that

they had had some practice in the solution

of such equations resulting in the

solution of the corresponding problem, that is they needed to know that equations

could be manipulated and solved in order to decide that it was worthwhile trying to

formulate the problem in this way. (p. ii)

The other question that arose was whether it is more advantageous to use

suggestive letters, such as h for head and b for body, or conventional ones, like x

and y. It appeared that the use of suggestive letters facilitated the initial writing of

equations, but was less helpful at the subsequent stage of manipulation. In

proceeding from h = b/2 + 4 and b = h + 4 to the combined equation b = b/2

+ 4 + 4, it is necessary to stop thinking about h as head, b as body, and to think

of them as symbols which are manipulated and moved around according to

algebraic rules.

Functions and Formulae

Parts of the Lee and Wheeler (1987) study touched on functions in looking for

the ability to recognize and express a functional relationship in sequences of


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PURPOSE IN SCHOOL ALGEBRA

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numbers, in dot patterns, and in tables of values. They remarked that though

seeing a pattern was not a problem, students lacked flexibility in generating

sufficient possible patterns, in selecting useful ones, and in checking their validi-

ty (p. 146).

The generation of possible variables and relations in practical situations-

mathematical modeling-is a strong interest among curriculum developers, but

not very much research exists documenting students difficulties in perceiving the

relevance of algebra to such situations or their ability to see how to make

appropriate applications. However, material designed to display and evaluate

these modeling abilities was developed by Treilibs, Burkhardt, and Low (1980).

Their problems included one about the advice to give the management of a large

supermarket which is trying to estimate how many of the checkout tills should be

operating at any given time. As well as tests of the students ability to solve the

complete problems, subtests also dealt with generating variables, selecting vari-

ables, generating relationships, selecting relationships, and specifying questions.

On a sample of able 17-year-olds, all the tests (except selecting relationships) had

low correlations with mathematical attainment as measured by school tests and

teacher ratings, indicating that the development of the ability to apply algebra in

this way is a neglected area in the school curriculum.

Conceptual Obstacles

Firth (1975) showed that many lCyear-old students responded to a request to

add 15 to X by asserting that they could not do so until they knew the value of x;

they could not accept the unclosed x + 15 as an answer. However, subsequent

research has shown that this need not be a serious problem, given experience of

working in suitable contexts. For example, Sutherland (1992) showed that lo-

year-old pupils working with a computer spreadsheet can use such expressions;

see also the later part of this article. Another item investigated in Bell and Low

(1982), following Collis (1975), is

ll-6=0-11

This showed two kinds of error. At one level, the response was 5, indicating a

fixation on the early notion that = means makes. This directional approach

to the equals sign has been shown to persist even at college level (Kieran, 1981;

Mevarech & Yitschak, 1983; both quoted in Kieran, 1990). At another level, the

response was

6 showing the persistence of an assumption of commutativity

of subtraction probably deriving from experience with natural numbers. Each of

these errors has been shown to belong to a substantial complex of misconcep-

tions .

The so-called students and professors problem (Clement, 1982) has re-

ceived much attention in the research literature over the last 10 years. In the


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original study, several groups of university engineering students were presented

with the problem:

Write an equation using the variables S and P to represent the following statement:

There are six times as many students as professors at this university. Use S for the

number of students and P for the number of professors.

It was found that 37% of the students answered incorrectly, and of these, 68%

represented the problem as 6S = P. A related result was reported by Mevarech

and Yitschak (1983), who found that 38% of the 150 college students they tested

answered that, given the equation 3k = m, k is greater than m.

MacGregor (1991) also studied this phenomenon. In a test based on her work, I

found some 40% of secondary-school students selecting the reversed response to a

similar additive question (a smaller minority selected the commuted 6 - m = k :

I have m dollars and you have k dollars. I have $6 more than you. Which

equation must be true?

6k = m 6m = k k+6=m m+6=k 6-m=k

The first and most obvious explanation offered for these errors is that students

tend to transliterate the verbal form: six students to every professor + 6s = P.

However, subsequent research showed that though this misconception governed

some cases, the error still occurred in translating from tabular or diagrammatic

representations into algebra. In these cases, the error appears to relate to an

associative link perceived in the statement 6S = P, the larger number S being

associated with 6. Wollman (1983) and MacGregor (1990) separately conducted

teaching experiments that show that improvements results from generating num-

bers to fit the equation, reflecting on which is the larger quantity, and expressing

the relationship verbally in different ways.

The incorrect direct transliteration of verbal statements into algebra, where

3a + 4b could be derived from 3 apples and 4 bananas (irreverently known as

fruit-salad algebra), has been shown to be a frequent source of error (Galvin &

Bell, 1977; Kiichemann, 1981), and is of course encouraged by some algebra

texts. In Kiichemanns test, only 10% of 14-year-olds correctly symbolized the

following problem, writing 5b + 6r = 90, whereas 17% gave instead b + r =

90:

Blue pencils cost 5 pence each and red pencils cost 6 pence each. I buy some blue

and some red pencils and altogether it costs me 90 pence.

If b is the number of blue pencils bought and if r is the number of red pencils

bought, what can you write down about b and r?


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Misconceptions regarding the commutativity of the subtraction and (partic-

ularly) the division operations, and their notations, have been widely docu-

mented (Bell, Fischbein, & Greer, 1984; Bell, Greer, Grimison, & Mangan,

1989; Brown, 1981). At this point, I will only remark that whereas in an arithme-

tic problem any confusion or error in the order of a division operation, for

example, using 7.2 + 3 for 3 + 7.2, may be detected from the size of the answer;

in considering the rearrangements of D = S X T to T = DIS or SID, the error is

less easy to spot. In a question asking which of six such rearrangements of

formulae were also true, only 6% of secondary-school students chose the correct

two (Bell & Onslow, 1987).

Breakdowns in the equation concept at a higher level have been studied by

Filloy and Rojano (1985; see also Kieran, 1990). They showed that, whereas

equations containing a single occurrence of the variable (such as 3x - 5 = 22)

can often be solved intuitively, or by backtracking or unpacking, those in which

the variable appears twice, on both sides, as in examples of the type 3x - 5 = 2~

+ 22 or, indeed, 3x - 5 = 22 - 2x, need a more detached concept of equation.

Some algebraic errors appear initially to be more technical failures, or memo-

ry lapses. Examples of such errors are adding fractions by adding numerators and

denominators, false canceling of part of a term, manipulating directed numbers

according to the implicit rule drop sign while operating (e.g., - 11 - 6 =

-( 11 - 6) = -5), and treating exponents as multipliers (3* = 6). These all arise

from a failure to make important conceptual distinctions, and hence need to be

treated by an appeal to meanings rather than to the simple reassertion of a correct

rule.

Some Survey Results

Some specific data follow from the British national survey of 15-year-olds (Fox-

man, Martini, Tuson, & Cresswell, 1980-1982).

I. Evaluating d3 when

d =

3 (44% correct; 16% gave 9; 25% other; 15% no

answer).

2. (a) I am x years old. Peter is two years older. What is his age? (49% gave

x + 2; 10% gave 2.x)

(b) Similar question but with 1 more, 3 less, twice (all about 45% correct).

(c) Represent the number which is n bigger than 3 (27% correct; 24%

n > 3).

3. (a) What is x if 2x + 7 = 45? (73% correct)

(b) 2.x + 3x + 4x = 2.x + 21, what is x? (50% correct)

(c) If

A = L

X

B

tells us how to work out

A,

what formula tells us how to

work out L? (39% correct)

Any manipulations beyond these levels tended to have 30% facility with some

25% or more omissions.


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5 BELL

AN APPROACH TO THE ALGEBRA CURRICULUM

With this background of research on the current state of understanding and

misunderstanding of algebra, I offer suggestions for a curriculum and a pedagogy

aimed at dealing with these problems. In general, the approach advocated is to

learn the algebraic language in a way similar to that in which the mother tongue

is learned; that is, by using it in order to communicate, with oneself and with

others, in the course of activities defined by the three main modes of activity

already described: (a) generalizing, (b) forming and solving equations, and (c)

working with functions and formulae.

As I have indicated, each of these strands embodies its own characteristic

purposes, and its own set of general strategies and procedures for achieving

them. There is, of course, some overlap, but fluent use of the language and

methods of algebra certainly includes the ability to handle each of these types of

situation confidently. One might make an analogy with the various uses of

language for description, instruction, expression, discussion, persuasion, and so

on. (The now common recognition of language competence as including speak-

ing, listening, reading, and writing can also suggest illuminating ways of think-

ing about mathematical activity.)

I also propose a pedagogy consisting of an alternation between broad activ-

ities embodying the main purposes of algebraic activity (and in particular, of

these three strands), and more focused activities aimed at acquiring particular

concepts or skills. I have called this

diagnostic teaching.

These focused activ-

ities, consisting mainly of provoked cognitive conflicts and intensive discussions

of critical conceptual points, have been shown to be strikingly more effective for

longer term retention than more usual methods (Bell, 1993a, 1993b; Bell,

Onslow, Pratt, Purdy, & Swan, 1985). In this article I consider broad activities

aimed at developing strategic abilities to handle the three strands we have been

discussing. My claim is that these provide a context in which specific misconcep-

tions can be more easily dealt with.

Generalizing

What needs to be learned about generalizing is the process of exploring a given

situation for patterns and relationships, organizing the data systematically, recog-

nizing the relations and expressing them verbally and symbolically, and seeking

explanation and appropriate kinds of justification or proof according to level.

Examples include, at the earlier stages, relations such as these illustrated by

9 = 4 + 5 = 3 + 6 = 2 + 7; 9 + 5 = 14 ==> 19 + 5 = 24; the digit patterns in

5, 10, 15,20 . . . and in 9, 18,27,36 . . . ; various patterns in the addition and

multiplication tables; Pascals triangle; and spiral patterns. The same process

applies to the study of the number field itself, with the recognition and expres-

sion of the general laws of associativity, commutativity, inverses, and so on.

Divisibility properties, such as that the sum of any three consecutive numbers

is divisible by 3, provide good examples of the power of algebra to prove general


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properties. Here the method of expressing the number following n as IE + 1 will

arise, but this does not appear to be a serious obstacle when it is dealt with in

context. Similarly, expressing the number with digits x and y as 10x + y is

needed for a certain class of problems.

Harder generalizations will appear in problems such as that of determining the

number of games needed if each of, say, 22 teams is to play against each other

once. Trial of a few simple cases, then systematic organization of the trial data

and a search for pattern follow.

I now give some fuller examples related to the table of counting numbers 1 to

100, the multiplication table, and the calendar.

Days and Dates.

The generalizations in Days and Dates (see Figure 1)

appear not to have been previously noticed by many 1 I-tol2-year-olds, so mak-

DAYS AND DATES

L

Today is the 3rd of October and its my

birthday on the 23rd. Today is a

Monday. So my birthday will be on a

Sunday

c

Great for a party

Peter

f course. If you go 20 days on, its always one day earlier in

1 Is Rachel right? Explain.

2 What happens if you go forward

(a) 10 days; (b) M days; (c) 40 days; (d) 100 days?

3 Explain the rule which governs these problems.

Figure 1.


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LINE PATTERNS

Take a copy of the Multiplication Square.

Find the box showing

14

0

21

28

Notice that 14 + 28 = 42

and2x21 =42

Try this other box in the same vertical line

28

0

35

42

Is it still true that T + B = 2M?

Try a few more, in the same line.

Try some similar boxes in other lines.

Try horizontal boxes also: Does L

+ R =

2M?

left right middle

Try lines of 4 and 5 numbers. Write what you find.

Use letters if you wish.

Figure 2A.

ing this link with a possible everyday situation is a good way of encouraging

awareness of mathematical relations in the real world.

Line Patterns and Corners and Middles. Line Patterns and Comers and

Middles (see Figures 2a & 2b) have proved to be good ways of getting pupils to

use algebraic language in situations where it forms a natural means of communi-

cation. Note that opportunities for checking, and understanding the possibility of

relations being true always, sometimes, or never, are built in also.

Some more extended individual explorations of these situations are shown in

Figure 3. These are from 13- and 14-year-old& but the examples could be used

earlier with faster pupils.

In Figure 3, Julia compares the sums of numbers in two similar L-shaped

boxes in the calendar. The difference in the numerical case is 44. We might


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CORNERS AND MIDDLES

I I6 24 32 46 46 56(641 72 60 I6 90

9 I6 27 36 49 54 63172111 90 99 106

10 26 JO 46 90:bO 7OllOi 90,100 110.120

11 22133 44~53~66(77~66~99~ilO~I~~32

12 24136 46 (CO/ 721 64/96/106~110(132]144

Cut

out a frame to put

over this table.

B-A=C-B

(the B number - the A number) = (the C number - the B number)

Check that this is true in both the boxes shown

Check other places on the table.

So

B - A = C - B is

always true

.

Consider

A t D = F + I

This is never true. Check this.

. Consider E = 2A

This is sometimes true. Check this.

.

With your partner, make a list of 6 relations and challenge

another pair to decide whether they are always, never or

sometimes true. But sure of your answers first.

Try to get some of each type.

Figure

2%

expect another numerical example at this point, but instead she introduces n for

the number in the central cell, and obtains the difference as (4x + 22) -

(4x - 22) = 44, showing that the pattern is true everywhere on the square.

Students were encouraged to experiment and find their own patterns for


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BELL

Julia is exploring patterns found by making L shaped boxes around some numbers

an old calendar. Notice how she has gene Aised the pattern with algebra in two

dBerent ways. This

shows that she has

real&d that

letters can stand for generalise

kt7t$. tX*\ t k+Z = 4xtkY

Xi \ Y+xt\ S+x+ntx*- J +A $+a

iul ia SwZSfu& haded the bracketted

eqressions, by thinking of their global

mcuring(~LmoreChanSr,taLB22leuthan4r).Somepupiistuddfficultierhere,

whichwerediswsedinWopa&Sons.

As WeU as thinking about meanings, pupil

experimmted by substituting numbers such as 10,3, KtO.... for x.

Figure 3

different shapes on the calendar, and to include some which did not give a regular

pattern. The second example (see Figure 4) is one in which the nongenerality of

the pattern is shown by the algebra. The difference x - 3 means, as the student


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PURPOS IN S HOOL

ALGEBRA

55

says, the answer isnt always 13, but, in the case I took, x - 3 = 13. So the

difference will vary, depending on where the shape is.

These nongeneral relations show more vividly than the general ones how the

use of a letter as a generalized number demonstrates the generality or non-

generality of the proposed pattern.

A more ambitious pattern, using multiplication, is produced by James (see

Figure 5).


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BELL

A

B

C D

2

5

1 a

1

6

2 7

3

4

0 9

0

7

3

6

Figure 6

can be generated. In this way, a very important idea is met in a quite elementary

setting and an intuitive familiarity with the possible transformations of a given

additive relationship is built up. Later, a systematic look at such a set leads to the

identification of formal rules that summarize the possible transformations of the

given additive relationship.

Forming and Solving Equations

The earliest examples may be missing number problems such as 8 + ? = 11 or

? - 7 = 5. The corresponding verbal or situational problems at this stage are not

normally solved by representing the problem symbolically and then transforming

it, but by a mental transformation into the solution form. Thus, a problem asking

how many marbles Jane had at first if she gave away 7 and had 5 left, would be

answered directly by adding 7 and 5. As the work progresses to more complex

problems, involving first one operation, then several, the same tension arises

between the questions which can be resolved mentally and those which require

symbolic representation by one or more equations which are then solved by


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59

manipulation. Learning the solution methods for the different types of equation is

important here.

At a later stage still, this process is central to design problems where there are

sizes or quantities to be chosen and constraints to be met. An example is the

problem of designing a box of a given shape to have a given volume. A more

complex case is the general transportation problem where various quantities of

goods are to be carried as cheaply as possible from a number of sources to a

number of destinations, the cost of each journey being known. This involves

forming and solving a set of linear equations. The same general process is at the

core of many of the problems and puzzles that have fascinated humankind

throughout the history of mathematics. Here are two early ones:

Baker 1568

AD:

One man demaunded of another in the morning what oclok it was y other made

him this answr; yf you doo adde (sayth hee) the r/4 of the howers which be past

midnight with the % of the howers which are to come until noone, you shall haue

the just hower, that is to saye, you shall know what oclok it is.

Calandri 1491 AD:

A

lion can eat a sheep in one day, a leopard can eat it in two days, and a wolf in

three days.

How

long will it take all three?

The first of these leads to the equation

/4 h + % (12 - h) = h

and the somewhat unsatisfying solution

S1/17.

The second may be solved by

considering the reciprocals of the given rates, that is, a leopards daily meal is l/z

of a sheep, a wolfs is

V3

of a sheep, so in a day all three eat 15/6 sheep or one

sheep in VII day.

Bath-filling problems, and problems about electric resistances in parallel,

have the same structure: They require what we might call the reciprocal method

(or the harmonic method by analogy with the harmonic mean).

These two problems perhaps illustrate how algebraic symbolism becomes

advantageous for more complex problems. They also remind us of the motivating

power of a good puzzle, which we should aim to retain in school work.

Childrens first introduction to the notion of extracting an unknown number

from a statement expressing constraints may well be through missing-addend

questions. Later, steps toward more systematic equation forming and solving

may be taken by setting up diagrammatic situations containing hidden numbers.

These have the advantage that (a) they are self-checking and (b) the unknowns

have an obvious concrete existence, thus avoiding the difficulties associated with


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having to decide what quantity to denote by x, and how to translate from verbal

information into symbolic statements. Arithmagons provide one example and

others follow.

Other Examples of Diagrammatic Situations. Other examples of possible

situations are Pyramids and Number Routes. In Pyramids the construction

rule is that a lower number is the sum of the two adjacent ones above it (see

Figure 7).

The mode of writing the equation may be suggested by the teacher, but is

normally readily adopted. The collection of terms and the solution of the equa-

tion may similarly be the subject of discussion but are normally accepted as self-

evidently good by the pupils. By changing the construction rule from

A + B

to

A + 2B, and then to A - B, more difficult manipulations arise. These need

dealing with by focused discussions and with experiments showing the validity

of laws such as 2(A +

B) = 2.A + 2B

and

A - (B - C) = A - B + C.

Number Routes is another type of concrete setting for equations (see Figure

8). Number Routes can be made harder by, for example, having addition or

subtraction before multiplication, which gives rise to the need for the distributive

law, and by making longer paths and more complex networks. Students can be

asked to make up their own examples, pitching them at a level of difficulty

suitable for themselves.

Forming Equations From Verbal Problems. The research quoted previously

shows that many students regard the purpose of all questions in algebra as the

r ,

5

4

5

X

4

5+x

x+4

21

21

5+x+x+4=21

2x+9=21

2x=12

x=6

Put in 6 for x; it is seen to be correct

Figure 7


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Following this they were asked to work, in groups of three, at solving the next

question, taking each of the three piles in turn as X, and to compare their results.

On the following day, each group was asked to make up and solve three similar

problems, two easy and one hard, to be attempted by another group.

This led to a lot of insight into the way different assignments of x affected the

expressions, turning + into - and multiples into fractions. It also led to an

unexpected degree of richness in the problem statements. As well as four bean

bags,

and the numbers of pupils in three rival schools, we had A nuclear

scientist must complete 4 experiments to save the world, and has 23 days to do

them in. The first will take twice as long as the second. . .

To conclude, we may say that the main initial difficulties lay in expressing

relations such as Pile 3 has 15 more stones than Pile 2 when Pile 2 was x,

making Pile 3 x + 15; and more so when Pile 3 was x, needing a reversal to

make Pile 2 have X - 15; 10 less than x + 15 was a step up in difficulty.

However, although this was observed to be a serious obstacle for some students

in the early lessons, on being offered the answer they soon picked up the idea

and, in the school examination question on this work, no student failed to

formulate an equation, though there were some reversal mistakes.

Recognizing Equivalent Equations. The research quoted previously shows a

strong need for activities in which algebraic expressions and equations are treated

with common sense and normal understanding rather than by a number of special

rules that are liable to misinterpretation and not subject to normal critical apprais-

al. I sketch briefly one such activity.

Arithmagons provides another.

Giving Clues is another generic situation that leads naturally to the intuitive

recognition of equivalent equations. It also provides a body of experience on

which the study of sets of linear equations can be built.

(L - 3)/M = N

1

3M L= 1 2

4N M=2

3

These equations were put on the board; the teacher stated that he had chosen

three numbers, had denoted them by L, M, and N, and had written down these

three clues by which they might be found. The game was for each person to try to

find the numbers (by trial and error or by any other method). When a person had

found them and had checked that they satisfied all three equations, she was not to

say what the numbers were, but was to make up a new clue and offer it to be

added to the list.

It took some 5 min for the first person to find the numbers; after that clues


23/33


24/33

64

BELL

k'f~ ai(LCW fm,t :'sqw,bcwl

This is an interesting example, and

3 a@WOnd 2 Lorc &uS %,P 19 ,-.a+

successfully solved. EIut note the

2 b(inonaS

nd 1 CQ@X hW ~7 m jynj

attraction for adding all the equations,

1 camrot ond 2 Qrf/rs kol-~ jl ~0fia.

which might not have been helpful bad

mU MWj MjjWf dc.s4UIW Glut I-W-4.

the second

equation been different.

&+ Lb=39

Lb+ c ='+7

c +Ls=tS

U and

Y

This pupil seems

to be enjoying

himself, but has not noticed, so

far, that he has too few equations.

There is an opportunity here for

the teacher to discuss this point.

Figure 9.

value, showing the teacher what aspects of the situation have been recognized by

the pupils.

Working With Functions and Formulae

In work on functions and formulae, the typical situation is either a practical

situation that generates a sequence of numbers in some way, for example, a row

of squares made with matchsticks, or polygons of increasing size made on

pegboard, or sets of practical or experimental data, such as prices of a ferry

crossing for cars of different lengths. The usual task is to determine the rule

defining the function, to express it verbally (and later, symbolically) so as to be

able to interpolate and extrapolate from the data actually given to predict other

cases. The shape of the corresponding graph may be of interest and comparisons

may be made between the shape of the graphs obtained and those of known


25/33


65

functions. When an organized set of sequences is studied it becomes clear that

their difference patterns play an important part in recognizing and classifying the

types. Thus, one gathers knowledge of a variety of kinds of function; linear,

square, inverse, exponential, and wave (sine or cosine), together with the appro-

priate algebraic formulae. Equation solving to determine values comes in as a

part of this study.

Examples of such material will be discussed later. But, first, note that some

attention is also devoted to formula reading. I suggest some questions that can

form starting points for focused discussion on this point.

1. If three

numbers,

A, B,

and C are related so that

A X B = C,

in what other

ways can the relations be expressed?

2. If A = L X B, how do you get L if you know A and B?

3. If

V = w2h,

and h is doubled

while r is fixed, what happens to V?

Similarly, if r is doubled while h is fixed? And if r is doubled but V kept

fixed, what must happen to h?

4. Which of the following formulae do you think is most likely to be correct?

Underline your answer. Give reasons (k, n, and b are constants).

Force exerted on pedals when riding a bike with speed v mph.

F = kv F = kv2 F = k/v F = klv2

Pull of the earth on a satellite at height h.

P = kh P = kh2 F = k/h F = k/h2

Number of marbles of diameter d in a kilogram.

N = kd N = kld N = kld2 N = kid

Stopping distance of a car from a speed of v mph.

D = av + blv2 D = av + bv2 D = av

Among the broader activities on functions, the properties of sequences and

functions arising from practical or experimental data or from pure mathematical

situations are explored. Different ways of growing are compared and related to

difference patterns, graphs, and formulae; we may use the knowledge built up

both to understand better the way in which the situation works and to predict

values in particular cases. As before, the approach is essentially exploratory, the

students posing many of their own questions and checking the correctness and

completeness of their own findings.

The standard process for finding the expression for T,, (see Figure 10) some-

times causes difficulty. It can be eased if the reverse activity is experienced first,

that is, if various formulae are used to generate sequences, and the relation

between the type of formula and the difference pattern of the sequence becomes

familiar. The material (in Figure 11) has been used. A harder example is shown


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66

BELL

Making Seauences

cl *

d33

cfk,...

e

T

T

..

n 1

Tn I

d

2

4

3

3 .

9

5 . .

n 1 2 3

Tn 2 5 8

d 3 3

n 1 2 3

total 8 12 16

d 4 4 .

Using matchsticks, cubes, dot or square grids make a sequence of objects which grow in some regular

way. Identify the pattern along the sequence; consider successive differences. Find the rule which

takes you aaoss the sequence - from n to Tn.

Figure 10.

in the Bridge Cables problem, illustrated in Figure 12 (see p. 68). Reconciling

the different forms (see Figure 13, p. 69) gives rise to a need for the development

of manipulation. Similar treatments are possible in the previous easier cases. For

exponential functions, see the examples provided in Figure 14 (see p. 70).

In Warmsnug Double Glazing (see Figure 15, p. 71), just as in the previous

examples, a function has to be found from a set of data (from The Language of

Functions and Graphs, Swan, 1985).

CONCLUSION

I have illustrated and discussed tasks oriented toward fostering the broad aims of

school algebra and at several points I have indicated how particular aspects of


27/33

SEQUENCES I

LI ST OF FORMULAE

I . t (n) = 7n - 3 a. t(n) = 3. 7 - 0. 4n

2. t( _n)= 3nZ 9. t(n) f 2 + 3

3. t(n) = 10 x 2 10. t(n) = 4n

4. t(n) = 6 - 2n

li. t(n] =

n - Sn' + 17

c

3.

t(n) f 6

12. t(n) = (2 x n) + 6

6. t(n) = (n + 2](n - 1)

13. t(n) = n

+4n-6

7. t(n) = n 2 - 4n 14. t(n) = 327n - 3

.

Generate the sequences corresponding to each of these formulae. Find their difference patterns.

Relate the type of sequence - its way of growing - to the type of formula. Sketch and compare the

graphs. (Use graphic calculator

if available . Then try to reverse

process. Get the form of the

formula from the previous experimentation. Adjust the coefficients by putting in some values of n.

s, 2. 4. 6. 3, 10

SV. 3. 6. 12, 24, 48

sz 1, Ai, 9, 16, 2s

sir 4.

a,

14, 22. 32

S, 4, 7, 10. 13, 16

su 10, 3. 5. 4, 2

S. 4, 12. 20, 23, 36 517 1, 3, 6, 10, 15

Ss 6, 24, 54. 36. i3

jl8 25. 28 33

38. 43

Sr 2. 4, 3. 16. 32

SU 6,

a.

14. 26, 50

ST 1, 4, a, 4, 4

SlD 16. 13. 7, 4.

Sr 6. 16. 30. 48. 70

51 3. S.2, 7.4, 9.5. il.3

S*

3. 6. 9. 12. is

srr a, 1, 5, 5. 3

510 95. 39. 33. 77, il

Szr 4, 16. 37, 76. i39

511 1. 9, 2s. 49. 31

s:* 2. 3. l-8. 31. ZI

SIZ

2, 20. 38, 56. i4 Szs 57 4 39 7 122. 154.3, 166.6

SIP 3, 5, 9. TS. 23

Figure 11.

67


28/33

Suspension bridge ablu

When maldng a able for a suspensionbridge,

many s s are assembled nto a hexagonal

formationand then compacted together.

This

diagram illustrates size 9 able

made up of 61 strands.

How many strandsare needed for a size 10 cable?

How many for a site n cable?

Figure 12

68


29/33


1. Developing a sequence.

69

.

2 Looking for structure.

-1

1

n n-l

3n(n-I)+1

nCZ(n- l)Z+(n- 1)

There are many other ways of seeing the diagram.

For example, can you see how you can get

notation and of algebraic manipulation might be dealt with, by focused and

generally self-checking activities, in such a context. I have not addressed the

question of building up fluency, either with particular manipulative processes or

in the deployment of the three major strategies. I consider it to be important and

see it as a regular result of focused activities devoted to particular algebraic laws.

But the question of how to facilitate the development of fluency, from trying out

possible manipulative laws, including self-checking using numbers, to the auto-

matic symbol-moving procedures used by experts without loss of the link with

understanding, demands further research.


30/33

T T T

c G G c G cs c7 G

Note CJ

Cl c* =, =, c, c,

c, c,

Frequency

(HJ Z7 664

130.6 261.6 523.2

10464 2092.6 4185-6

(a&w=)

--

Row number

1 2 3 4

5

--

Numkrr of squares

46 24 12 6 3

--

--

X 1 2

3 4 5

--

Y 4 16

64- 256 1024

--

--

a 1 2

3 4 5

--

b 2 6

32 126 512

--

InternatIonal

aper Sizes

A

Area (I+)

Lengh (mm)

WidM (mm)

A0 1

1169

&l

A, 05

641

594

2 0.25125

42094

42097

A, 0062

297

210

A, 0031

210

148

A* O-016

146

105

AT O-006

105

74

A, 0004

74

52

A, o-002 52 37

A,, 0.001

37

26

Consider the relationship between fhe paper size number (n) and its area

figure 14.

70


31/33

WARMSNUG DOUBLE GLAZING

(The windows on this sheet are all

drawn to scale: 1 cm represents 1 foot).

How have Warmsnug arrived at

the prices shown on these windows?

*

Which window has been given

an incorrect price? How much

should it cost?

*

Explain your reasoning clearly.

D

Figure 15.


32/33

72

BELL

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Documents

Purposes of School Algebra