When Should We Teach Calculus?

When Should We Teach Calculus?Author(s): A. OrtonSource: Mathematics in School, Vol. 14, No. 2 (Mar., 1985), pp. 11-15Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213968 .

Accessed: 22/04/2014 08:22

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 5.198.113.170 on Tue, 22 Apr 2014 08:22:50 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=mathas

http://www.jstor.org/stable/30213968?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


HOCULD

HOCULD HOCULD HOCULD

HOCULD

by A. Orton Centre for Studies in Science and Mathematics Education

University of Leeds

Although there is widespread agreement about the importance of calculus as a part of mathematics, there is no general agreement about the best time to introduce students and pupils to calculus and about how much we should aim to achieve in the years of compulsory schooling. The current situation is certainly very confused. New Ordinary level syllabuses, operative from Summer 1985, may have excluded calculus when previously at least one option included it. Existing 16+ or CSE syllabuses may be offering the opportunity to teach calculus whilst Ordinary level syllabuses available in the same school may not. The issue of calculus before the 6th form is discussed in this article.

Why was Calculus Introduced? The introduction of calculus into examinations around the age of 16 resulted directly from the recommendations of the Jeffery Report of 1944 on School Certificate Mathematics. Reports, however, rarely consist of original ideas and often merely endorse and give authority to views which have been around for some time. It seems that this was the case with the Jeffery Report's recommendations concerning calculus.

In 1932 the mathematician and philosopher Whitehead' wrote:

"I cannot help thinking that the final review of this topic (functionality) might well take the form of a study of some of the main ideas of the dzfferential calculus applied to simple curves. There is nothing particularly difficult about the conception of a rate of change; and the dziferentiation of a few powers of x, such as x2, x3, etc., could easily be effected; perhaps by the aid of geometry even sin x and cos x could be differentiated."

Whitehead's comment about there being nothing particularly difficult with rate of change is interesting. Taking the entire ability range into account, research evidence does not support his optimistic assumption. His comment may, however, have been acceptable at the time, considering the pupils for whom academic education was then provided.

A further statement about calculus was made by the eminent mathematics educator Nunn2:

"When we consider the position of the differential and integral calculus we have to protest against a tradition which forbids all but the exceptional pupils to become acquainted with the most powerful and attractive of mathematical methods ... the mischief is the result of a technical elaboration which, though essential to the historical development of the calculus, has had the effect of making the really simple ideas upon which it is built inaccessible to the ordinary boy or girl at school. In this instance the history of the subject suggests a remedy for a state of things which is generally regarded as unsatisfactory. The calculus began, in the writings of John Wallis and others, merely as a special kind of algebraic argument which might be introduced at any appropriate point and without the apparatus of a technical notation."

In proposing calculus as a topic worth introducing to all pupils, however, Nunn left no doubt that he acknowledged the inherent difficulties of calculus. Once again, as with Whitehead, we may query what range of pupils he meant by "all pupils".

The Report of the Board of Education3, commonly referred to as the Spens Report, included a clear recommen- dation about the teaching of calculus in schools:

"We hold that the ideas of calculus, both differential and integral, should be reached through the graph and through the course in algebraical methods before the majority of pupils leave school."

It was, perhaps, inevitable that the Jeffery Report also faced up to the calculus issue. This report was the outcome of a 'Conference of Representatives of Examining Bodies and Teachers' Associations' and it is interesting that the largest group of representatives of the various bodies concerned was from the Mathematical Association. The initiative for the conference came from the Cambridge Local Examinations Syndicate and the meeting was to consider School Certificate Mathematics with an alternative syllabus in geometry as the major intention. Other issues considered included the existing division of mathematics into three separate and distinct branches and the examination of pupils by three papers corresponding to these three

Mathematics in School, March 1985 1 1



branches. The conference prepared and submitted for the consideration of the existing Examination Boards an entire alternative syllabus in mathematics. The suggestions of the Report concerning calculus were cautious and the difficulties were acknowledged in this extract from a reprint in a handbook of the, then, Incorporated Association of Assistant Masters4:

" The pitfalls are obvious, for there are few branches of the subject which can more easily afford the opportunity for blind manipulation of a notation or the mechanical application of rules. But these pitfalls can be avoided and the intrinsic importance of the subject is very great. The transition from the static mathematics of the formula, which enables one quantity to be calculated when another is known, to the dynamic mathematics of the function, which considers how one thing changes with another, is one of the chief ways in which mathematics has adapted itself to the consideration of practical problems. The ideas underlying differentiation and integration are not difficult to grasp if presented in appropriately simple circumstances. They grow naturally and easily out of the consideration of graphs which rightly now occupy so important a place in the elementary teaching of the subject."

Shortly after the Jeffery Report was published a new system of examining was considered, and, in due course, the School Certificate was replaced by the General Certificate of Education (GCE) Ordinary level. The "alternative" syllabus in mathematics at that level incorporated the Jeffery proposals concerning calculus. What had been pro- posed as an option in School Certificate mathematics became compulsory in the "alternative" Ordinary level mathematics syllabus. Those of us who were teaching mathematics in schools before "modern mathematics" will recall the choice between the old syllabus tested by three separate papers in Arithmetic, Algebra and Geometry and the new syllabus tested by two papers of assorted questions including calculus. This calculus content has survived until the present day. At the moment, however, it is being removed in the introduction of certain of the most recent syllabuses.

How did "Modern Mathematics" Affect Calculus? This same calculus content also appeared in some of the Projects which grew up in the 1960s and was retained in the establishment of Examination Board syllabuses in "modern mathematics". The Midlands Mathematical Experiment (MME) included precisely the same amount of calculus, and still offers an Ordinary level examination including calculus. Yet the modern project which captured most attention and whose materials are now ubiquitous, the School Mathematics Project (SMP), rejected the formalisation of the introduction of calculus. It all depends, of course, on what you mean by the introduction of calculus. It is very difficult to say where and when, in a mathematics course, one begins to anticipate calculus. The SMP, however, certainly deferred the study of differentiation and integration until the sixth form. We may read in the SMP Report of 19635:

" The pupil learns at a fairly early stage how to draw graphs of functions and how to find rates of change and areas under graphs by drawing. At the age of 16 or so the more formal language of the calculus is introduced and Stage B of the pupil's calculus education begins."

Subsequently, however, other mathematics educators have spoken out in favour of some calculus for all able pupils. Marjoram6 wrote:

" There are those who feel that the basic ideas are so difficult that the calculus should be postponed for as long as possible. This is a mistaken policy. It withholds from the pupil (and his science teachers) a valuable tool which is essential for sixth form work in science. It is also unnecessary, for although such ideas as 'infinity' and 'tending to a limit' are

difficult ones requiring time and patience, much preparatory work can be started long before the sixth form ..."

Shuard and Neill7 developed the same point in greater detail:

"Some teachers who argue against teaching integration and

differentiation to the more able pupils before the sixth form argue this on the grounds that an introduction to an important topic shortly before an examination may lead to a hurried approach in which a proper understanding takes second place to the learning of rules to answer examination questions. Other teachers argue that it is very important that able pupils whose major interests lie outside mathematics, and whose formal mathematical education will end at age sixteen, should see something of the power of the calculus ...

The authors of this book take the second view. They believe that mathematics is a national resource and that it should be developed to the maximum potential. The more numerate the decision makers, the more likely the decisions are to be well informed. It is because government, industry and business often communicate quantifiable ideas by graphs and because a knowledge of calculus helps to read and understand those graphs that calculus assumes its position of importance. We cannot as a nation afford not to develop this mathematical skill in our pupils."

To teach or not to teach calculus for Ordinary level appears to be one of those issues which will not go away. In order to help us to make up our minds about the issues we ought to look into the difficulties and we ought to consider the teaching implications.

Is Calculus so Difficult? Views on the difficulties associated with introducing calculus have been well-documented. In the introduction to his book Caunts wrote:

"Considerable space has been devoted to explanations and illustrations of the meanings of 'limits' and 'continuous functions' for I am convinced that, unless the student has clear ideas on these points, it is impossible for him to grasp the true meaning of a differential coefficient, although he may be able to acquire a certain amount of facility in the use of it."

The above quotation refers to the distinction between processes in calculus, which can be carried out merely by the application of remembered rules or algorithms, and ideas or concepts which underlie the processes and which may be extremely difficult to understand, particularly if insufficient time is spent on background work.

In the Report produced by the Incorporated Association of Assistant Masters4 we find:

"Of all branches of mathematics calculus requires the most careful introduction and development. Its ideas are so novel that there must be no attempt to rush the early stages, and it must be emphasized that the initial approach should not be stereotyped, but varied according to the capabilities of the pupils being taught. With a good class not only will progress be quicker, but the calculus notation can be introduced at an earlier stage than with others. Unfortunately, the intrinsic

12 Mathematics in School, March 1985



difficulties of calculus are met at the start, and it is impossible to deal with them in any rigorous sense at this level. For example, although some idea of the meaning of 'limit' is essential, we cannot attempt any study of the theory of limits. On the other hand we must, at all costs, avoid letting pupils perform operations of differentiation and integration mechanically, without any understanding of what they are doing. It should be our aim to develop the subject so as to give them, at any rate, an intelligent comprehension of the processes they are using."

More recently Matthews, in Servais and Varga9, summed the issue up more briefly:

"'Teaching calculus at school involves compromise: the mathematician demands rigour, the pupil demands motivation. Concepts must therefore be introduced intuitively and the need for refinement of ideas then gradually makes itself felt."

The Report of the Mathematical Association'o, still an important and useful reference for teachers, included the following statement:

" There is no part of mathematics for which the methods of approach and development are more important than the calculus, partly on account of the novelty of the notation, but

chiefly on account of intrinsic difficulties. These occur at the start, or more acutely at the start than at any later stage. For this reason the early development must be gradual: any rushing of the introduction will lead to chaos; and it will be found that pupils who have learned to apply processes mechanically are mystified about the principles, and are therefore liable to serious error in any matter that is slightly outside the usual routine."

Research recently carried out by the writer11,12 has revealed a great deal of detail about the difficulties experienced by pupils and students. Naturally, test items involving the nub of the concepts of integration and differentiation were found to be very difficult. Other very difficult items included the meaning of symbols and many of the aspects of rate of change. Whitehead's view, expressed earlier, cannot be supported by the research evidence. Only a very few students revealed the sort of relational understanding of the ideas of differentiation and integration which teachers of calculus have to achieve in order to teach intelligently.

Should Calculus be Taught Before the 6th Form?

The critical issues have long been known and debated. Is relational understanding achievable? If not, are we justified in teaching for instrumental understanding? In this branch of mathematics, as perhaps in others, is instrumental understanding for many pupils the first step in the progression to relational understanding? But only a few pupils will con- tinue with calculus beyond Ordinary level, so can we justify teaching only the rules and algorithms? Are there ways of promoting an understanding of elementary calculus which avoid all of the "pitfalls"?

It may be supposition, but it has always seemed to me that, although the decisions concerning calculus at Ordi- nary level made by the SMP and the MME appeared to be opposed, the same concerns about how calculus should be introduced were felt by both groups. It has seemed to me that the SMP decision may have been to try to ensure that essential background mathematics was included (pre- calculus, to steal and perhaps re-define an American term),

but at the same time to prevent an instrumental approach to differentiation and integration. The MME decision was to include differentiation and integration, but to try to ensure that they were not taught merely as rules. Study of MME textbooks reveals attempts to lead up to calculus relation- ally. Unfortunately, some teachers, in their interpretation of the examination interests of their pupils, may choose to by- pass exploratory and investigatory approaches even if they are included in textbooks, because they are not examined. In any case, it is difficult to capture investigatory ideas in the cold print of a textbook.

Some pupils are capable of making enormous progress towards calculus before the age of 16. The Schools Council publication "Mathematics in Primary Schools"'13 records exploratory and investigatory work carried out by 10 year olds which led to gradients and areas of curves. Formal teaching did not play a part. Unintelligible notation did not play a part. Yet the essential ideas, the really important aspects, emerged naturally to some pupils. There are pupils for whom it might be considered wrong to deny them the opportunity to study limits and rate of change leading to calculus through appropriate methods. There are many other pupils for whom ideas underlying calculus will be unattainable by the age of 16. What is required is what is always needed, a flexible approach which allows all pupils to study mathematics which matches their level of understanding and attainment.

One of the most neglected ideas in mathematics teaching is limits. This may be because it is difficult to test limits in an external examination at 16+. This situation is not helpful to the introduction of calculus. There are, in fact, many ways in which an enlightened teacher can introduce ideas of limits throughout a child's mathematical education. Without such insertion of work on limits the introduction of calculus is made harder. The same is true of rate of change. Many formal aspects of rate of change are undoubtedly difficult for pupils but informal studies of graphs are possible and helpful in terms of the subsequent introduction of calculus. If calculus is removed from the options at 16 + there is even less chance that background studies of limits and rate of change will be included in the mathematics curriculum. Calculus may come as an even more nasty shock when it arrives. The right time to carry out investigatory work in pre-calculus is throughout the whole age- range. There may be inadequate motivation for teachers to allow this if calculus is removed as an option at 16 +.

How Should Calculus be Taught? The answer is clear throughout many of the quotations given earlier. We should not be happy with "blind manipulation of a notation or the mechanical application of rules". Pupils must be allowed to "draw graphs of functions and find rates of change and areas under graphs by drawing". Concepts "must be introduced intuitively" in the first instance. Two illustrations are given below. They both come from the stage where pupils are moving from explora- tions and investigations in rate of change and limit towards the ideas of differentiation and integration.

Consider the graphs of y= x2. The gradient or rate of change or derivative at a point, P, on the curve may be studied from gradients of secants PQ, where Q takes a variety of positions on the curve. This is not a new idea. What is relatively new is that the electronic calculator enables pupils to calculate differences and ratios more quickly and more accurately. If we take P = 1 and let Q take a variety of values greater than 1 but approaching 1 we produce the following table. [The difference between the x-coordinates of P and Q is called h and the corresponding difference in y-coordinates is called k.]

Mathematics in School, March 1985 13



y=2

01

03

0 '

x Px

x-coordinate y-coordinate h k Gradient k/h of O of O

2.00 4.0000 1.00 3.0000 3.00 1.90 3.6100 0.90 2.6100 2.90 1.80 3.2400 0.80 2.2400 2.80 1.70 2.8900 0.70 1.8900 2.70 1.60 2.5600 0.60 1.5600 2.60 1.50 2.2500 0.50 1.2500 2.50 1.40 1.9600 0.40 0.9600 2.40 1.30 1.6900 0.30 0.6900 2.30 1.20 1.4400 0.20 0.4400 2.20 1.10 1.2100 0.10 0.2100 2.10 1.09 1.1881 0.09 0.1881 2.09 1.08 1.1664 0.08 0.1664 2.08

1.07 1.1449 0.07 0.1449 2.07 1.06 1.1236 0.06 0.1236 2.06 1.05 1.1025 0.05 0.1025 2.05 1.04 1.0816 0.04 0.1806 2.04 1.03 1.0609 0.03 0.0609 2.03 1.02 1.0404 0.02 0.0404 2.02 1.01 1.0201 0.01 0.0201 2.01

For appropriate pupils, it is reasonable to enquire "What is happening to k/h as Q approaches P?", and to expect the response, "k/h is approaching 2". Having produced such a table once, in detail, similar tables for different values of P may be obtained more quickly and with less detail. Finally, a summary of results such as

x Gradient

1 2 2 4 3 6 4 8

is likely to suggest the formula for the gradient. The same procedure can be adopted for other curves, e.g.

y = x3, leading to the result

x Gradient

1 3=3x1 2 12=3x4

3 27 = 3 x 9 4 48=3x 16

The pattern is again obvious. In the same way the gradient formula for y = kx2 and for y = kx3 for different values of k may be investigated. Further, the gradient of, for example, y = x + x2 may be studied, leading to the table

x Gradient

1 5=3x1+2 2 16=3 x 4+4 3 33= 3 x 9+ 6 4 56= 3 x 16+8

By such means pupils may discover the gradient formulas for a variety of polynomial functions and thus also discover the general principle involved. No algebraic proof has so far been necessary, and this may be an adequate first introduction to differentiation. Elaborate symbolism might only serve to confuse the issue. Perhaps examination syllabuses have made the mistake of demanding the use of dy/dx too soon.

In using an electronic calculator to introduce integration, a sum of rectangular areas has to be obtained several times over to get some idea of the sequence of sums approaching a limit, and the whole procedure takes much longer than calculating gradients. The example below illustrates this.

y A, y= X2

-

u. 1

14 Mathematics in School, March 1985



For the area under the curve y = x2 from x= 0 to x= 1 we take, say, 10 equal divisions of the x-range, calculate the heights of the rectangles and finally sum the areas of the rectangles. This may be carried out for rectangles both below and above the curve and the true area then lies between the two approximations obtained. The table of values summarises the procedure. The exact area can thus be seen to be between 0.285 and 0.385. Unfortunately this is quite a wide range, and it is not easy to deduce with conviction that the exact area is 0.3. A second attempt with x-values at intervals of 0.05 instead of 0.1 produces the two values 0.30875 and 0.35875. To use x-values at intervals of 0.01 would involve too many calculations to appeal to students.

x range Area of rectangle Area of rectangle below above

0.0-0.1 0.000 0.001 0.1-0.2 0.001 0.004 0.2-0.3 0.004 0.009 0.3-0.4 0.009 0.016 0.4-0.5 0.016 0.025 0.5-0.6 0.025 0.036 0.6-0.7 0.036 0.049 0.7-0.8 0.049 0.064 0.8-0.9 0.064 0.081 0.9-1.0 0.081 0.100

0.285 0.385

There is, however, an alternative approach, namely to use the trapezium rule. It is clearly more accurate to use the areas of trapezia than to use the areas of rectangles, as the diagram shows. Although the procedure is usually referred to as the trapezium rule, no additional rule is needed beyond the formula for the area of a trapezium which students should already know.

A,=xa

/

Using an electronic calculator the following results were obtained:

x-interval for Number of Sum of areas each trapezium trapezia of trapezia

0.2 5 0.340 0.1 10 0.3350 0.05 20 0.33375

Although a study of this nature, based on areas of trapezia, and using a variety of ranges of x and a variety of polynomial equations may lead to the discovery of a rule, the big disadvantage of the trapezium approach is that it does not lay the groundwork for a subsequent algebraic approach based on areas of rectangles. However, again, as with the approach to differentiation, the electronic calculator pre- sents us with a real opportunity to investigate areas under curves through simple arithmetic alone. Given the em- phasis on numerical mathematics to which the availability of cheap electronic calculators should lead, a numerical approach to elementary calculus might bring about a much greater understanding of what we are trying to do in calculus. Perhaps the real mischief has been to demand the use of S y dx too soon.

Even more important than how we lead up to differentiation and integration is probably what comes before, in earlier years. In the space available it is not possible to go into detail about all aspects of pre-calculus. We must avoid the "pitfalls", we must avoid producing pupils who have "learned to apply processes mechanically (and) are mystified about the principles". But this does not mean we need to deny all pupils the opportunity to study the beginnings of calculus at 16 +, whether within or without their formal examination syllabus. If a formal examination syllabus including calculus is required, they do still exist. But if we do choose a syllabus including calculus we must take into account that "of all branches of mathematics calculus requires the most careful introduction and development", and that "much preparatory work can be started long before the sixth form". The crucial issue is not when we should teach calculus, but how we should promote the understanding of calculus and pre-calculus according to the level of attainment of the pupil.

References

1. Whitehead, A. N. (1932) The aims of education, Williams and Norgate. 2. Nunn, T. P. (1927) The teaching of algebra, Longmans, Green & Co. 3. Board of Education (1938) Report of the Consultative Committee on

Secondary Education HMSO. 4. Incorporated Association of Assistant Masters (1957) The teaching of

mathematics, Cambridge University Press. 5. School of Mathematics Project (1963) Director's report, 1962/63, SMP. 6. Marjoram, D. T. E. (1974) Teaching mathematics, Heinemann. 7. Shuard, H. and Neill, H. (1977) The mathematics curriculum: from

graphs to calculus, Blackie. 8. Caunt, G. W. (1959) An introduction to infinitessimal calculus, Oxford

University Press. 9. Servais, W. and Varga, T. (1971) Teaching school mathematics,

Penguin-UNESCO. 10. Mathematical Association (1951) The teaching of calculus in schools,

Bell. 11. Orton, A. (1983) Students' understanding of integration, Educational

Studies in Mathematics 14. 12. Orton, A. (1983) Students' understanding of differentiation, Educa-

tional Studies in Mathematics 14. 13. Schools Council (1966) Mathematics in primary schools HMSO.

Mathematics in School, March 1985 15



Documents

When Should We Teach Calculus?