Algebra in the Higher School Certificate

At the Annual Meeting of the Mathematical Association on 3 January, 1939, Bill Ferrar gave the talk Algebra in the Higher School Certificate. His talk was printed in The Mathematical Gazette 23 (254) (1939), 144-149. We give some extracts from Ferrar's talk below.

Algebra in the Higher School Certificate, by W L Ferrar.

First, in order that my later criticisms may appear in their right perspective, let me say at once that, as far as my own experience goes, the mathematical work in the Higher School Certificate is good, sometimes amazingly good. With that said, I shall not speak further of what I think to be good in our algebra teaching, but only of what I think is taught less well than it might be taught.

I shall speak to three main headings: "Limits and convergence", "Proof in algebra" and the "Syllabus".

Limits and Convergence.

I find the place accorded in algebra to "limits and convergence" to be far too prominent; and the timing of the introduction of the more abstract notions of convergence I find to be premature. The convergence of sequences and series is a subject whose difficulties are, in my opinion, wholly unsuited to the minds of young people of sixteen or seventeen. I have no experience of teaching such minds, but I have a long experience of examining them and a lively recollection of my own early uncertainties as an undergraduate.

I am not suggesting that the word convergence be banished from the vocabulary of the school algebra course: but I am suggesting that a lengthy consideration of the convergence of series ought not to be attempted until the last year at school, or even later.

One of the joys of the mathematician, and especially of the young mathematician, is the exercise of precise thinking on topics that are hard enough to keep the mind on the stretch, but are not so difficult that the mind is defeated. When we insist on abstract questions of convergence before the mind is sufficiently ripe to appreciate them the young mind is befogged. It is fatally easy, in the cause of truth and with the highest motives, to rob our pupils of the joy of working precisely on something clearly understood and to offer them instead the discomfort of grappling with difficulties that are but dimly apprehended.

Let us ask ourselves in what sense we need limits and convergence in an algebra course. (I am not here concerned with the label you attach to this particular work; whether you call it algebra or analysis makes no difference to the point at issue.) I think a fair answer would be "We need limits for $(1 + x)^m, \log (1 + x)$, and $e^x$. Perhaps also for other special series whose 'sum to infinity' can be readily calculated". But none of these, even at the distinction level of Higher Certificate work, calls for consideration of general sequences $(a_n)$ or series $\sum u_n$.

Such general sequences and series raise the abstract problem, "Under what circumstances does a particular type of sequence converge?", whereas the binomial, exponential, and logarithmic series face us with concrete problems, perfectly capable of solution without appeal to general theories. Everyone will be familiar with a proof of the theorem

$\log(1 + x) = x -\large\frac {x^{2}} 2\normalsize + \large\frac{x^{3}} 3\normalsize ... ( -1 < x ≤ 1)$ that makes no appeal to general theory. I refer to the proof that integrates
$\Large\frac{1-t^{n}}{1-t}\normalsize = 1 + t + t^{2} + ... + t^{n-1}$
and uses inequalities between integrals to deal with the integral of $t^{n}/(1-t)$. This proof has been widely known for at least thirty years and very few modern texts use any other. There are similar proofs, also now in print, which establish the binomial and exponential series by the use of inequalities between integrals. None of these require any more about limits than is contained in the notion of a sequence that tends to zero and of the definition of a sum of an infinite series as the limit of the sum of its first $n$ terms. None of these proofs deals with general series or with theories about undefined monotonic sequences; they deal with quite definite concrete problems.

In the differential calculus the procedure of considering numerous particular examples first and general theories second is so well established that no one questions it. We teach the idea of a limit and the particular facts
$\Large\frac {d(x^{2})}{dx}\normalsize = 2x, \Large\frac {d(\sin x)}{dx}\normalsize = \cos x$.
to very young pupils. Only when their minds are thoroughly accustomed to a multitude of particular cases do we call upon them to consider problems about "functions that have a differential coefficient"; and even then it is only the very best pupils who have the least appreciation of a purely arithmetical approach. Yet in dealing with sequences and series we nod condescendingly to one or two sequences that obviously tend to zero and then plunge right away into monotonic sequences and series of positive terms. I am certain that half the minds which have been jumped through that series of hoops have no idea of what it was all about.

It is not for nothing that teachers and examiners bemoan the fact that pupils rush on every occasion to use $\Large\frac {u_n}{u_{n+1}}$. Why do our pupils do this? I am convinced that it is because $\Large\frac {u_n}{u_{n+1}}$ is the first thing in the maze of uncertainty and hesitation that they understand at all. If we will insist on putting the abstract difficulties of convergence too soon, and before the mind is sufficiently mature to comprehend them, then we must be prepared for the muddle and uncertainty that results.

It is easy to see why "limits and convergence" came to be placed so early in the algebra scheme. In the days of its introduction to our school algebra courses, the definition of convergence was wrong, but it was easy to understand. The schoolboy of that day saw nothing wrong with it; neither did his master: nor for that matter did any but the few research mathematicians whose interests were inclined that way. For the schoolboy of forty to fifty years ago
$u_{1} + u_{2} + ...$
was convergent if the sum of its first $n$ terms stayed finite and did not shoot off to infinity. That concept was easy and quite fittingly, it appeared reasonably early in the algebra books. But what we have done is to replace the old, easy, wrong definition by a new, difficult, right definition and still to insist on its old early place in the scheme of things. And the failure of set after set of young pupils, during the last twenty or thirty years, to get more than muddle and nervousness out of the new definition has not showed us our mistake: we have merely blamed the pupils.

When the ground has been prepared by particular examples; when the mind is ripe to receive abstract ideas; when, for the average young specialist, the last year at school has arrived: then is plenty time enough to begin on general sequences $(a_{n})$ and series $\Sigma u_{n}$. As an examiner I have always taken a definite view on this point. I ask freely, from a finite series lead, for proofs of such things as
$\Large\frac 1 {1 -x}\normalsize = 1 + x + ... + x^{n-1} + ... (-1 < x < 1)$
or
$\sum\Large\frac 1 {n(n+1)}\normalsize = 1$,
but I never ask schoolboys questions about sequences $(a_{n})$ or series $\sum u_{n}$ unless the syllabus and established custom of the examination compel me to do so.

Proof in algebra.

In geometry the young mathematician learns to state his hypotheses, to say clearly what he has to prove, and to think out the steps whereby he may arrive at his proof. Algebra too is an excellent ground for the practice of precise reasoning from hypothesis to conclusion. But this side of algebra seems to have been much neglected and, in some measure, it is easy to see how this has come about. The early stages of algebra are dominated by the logic of each particular step; why you may add 3 to each side of an equation; why and how you remove or put in a pair of brackets. I have taught that sort of algebra and I know that the logic of each step is quite enough for any teacher to cope with in those early stages.

When, however, the early stages are past and the mathematical sixth is reached, there seems to be little reason for the apparent neglect of logical form in algebraic work. As an examiner I have often been dismayed by its complete absence. A candidate often appears to be able to do the manipulations that connect A and B but whether he is assuming A and proving B or doing the converse is a matter for the examiner to elucidate from his own inner consciousness; the candidate gives him no help. In geometry, the distinction between theorem and converse is an elementary point that is grasped by all save the stupid; in algebra, one can search script after script for the distinction between the two; one is pleased to find the odd candidate, or set of candidates, whose work insists on the distinction. I do not say that all candidates are badly taught on this matter, but I do say that there is a considerable falling away from the high standard of our geometrical teaching.

I have often thought that it would be an excellent thing if our algebra books were set out as a series of theorems and exercises thereon; the numbering of theorems and the consequent task of seeing to it that we did not use Theorem 29 to prove Theorem 26 would, I am certain, lead to a greater clarity.

Another detail that seems to call for remark is the proof by induction. The exercise of setting down an induction proof in convincing form is one that calls for a firm grasp of the underlying principle. It is such an important method of proof that it is worth a little more attention than it gets. The all too common statement that induction can prove only stated theorems seems to me a travesty of the truth. The method is, in its origin, one that guessed the answer first and proved it afterwards, and I think something could be made of this in teaching. Without wishing to stress this side of the matter, I should like to stress the importance of clarity in writing out a proof by induction. It is only too frequently that I am presented with proofs by induction that would convince nobody, least of all the presenter.

To round off this rather disjointed series of remarks about proof in algebra, may I give a concrete example by discussing, quite briefly, elementary difference equations. I shall treat the matter only in outline and not as I should set it out for a book or a note- taking lecture. Whether or not difference equations, divorced as they sometimes are from their context of generating functions and recurring series, ought to form part of the school syllabus, I do not wish to prejudge. But if they are to be taught, then the form of their solution ought to be dealt with in a logical manner.
...

The Syllabus.

In raising this question before the Association I wish to recall the preliminary appeals by Mr Parsons and Mr Langford at the Nottingham meeting of the British Association. My appeal is supplementary to theirs and it is made to the officials of our own Association. It is to ask them to publish the findings of a committee on the two queries

1. What subjects are suitable?

2. What subjects are not suitable?

- and of the two I stress the latter as the more important - for the higher divisions of school algebra. I do not ask for a cut-and-dried syllabus, in which each separate item is meticulously laid down: such I hate. I do not ask for an examination syllabus. I ask for a syllabus of work suitable for schools.

I should hope that the textbook of the future and the examiner of the future would be guided by such a syllabus, though not necessarily agreeing with it at all points.

The present-day insistence on the examination I find to be most disquieting. At the School Certificate stage I can see the causes of this insistence: mostly, they are outside the scope of the examination itself and are concerned with the certificate as a passport to remunerative employment. But there is no such outside influence to overstress the importance of the examination beyond the School Certificate stage. If we allow the Higher Certificate to become a serious menace to the freedom of teaching, then it is our own fault.

We pay far too much attention to the examination and, as an Association, I think we should express our views on the teaching of mathematics and on what parts of the subject should be taught in schools. The examination is not likely to ignore topics that are given prominence in our teaching, nor to insist on topics that cannot be studied in modern textbooks. As one who has always tried to examine so that there shall be a minimum of interference with the teachers' methods and no insistence on set routines, I find the view that the teacher must always follow the examiner very exasperating.

However, I will not detain you longer. Let me say merely that I want a schedule of study suitable for the top forms of schools. In algebra there is so much that once was studied at school and is studied there no longer; so much that used not to be studied at school and now claims to be a part of every young mathematician's equipment. It is time that the situation was reviewed and an expert opinion published.

Last Updated September 2020