bluuebunny/crossref_metadata_2025_split huggingface dataset:
Statistics as Applied to Educational Questions
Authors: Meldrum H. J. 
Published: January 2007

It is becoming more widely recognised each year, among leaders of educational thought, that in the field of educational science, where mere opinion holds sway, this opinion should be replaced by conclusions scientifically obtained from carefully collected data.In the teaching of mathematics we have such questions as:(i) What is the most economical length of drill periods in first-year algebra ?(ii) What is the best method of procedure for acquiring skill in the mechanical work in algebra ?(iii) To what extent does mechanical skill in algebra help the pupil intelligently to use algebraic processes, such as the equation, the graph and the formula ?(iv) What proportion of pupils benefits by a certain type of course and up to a given standard ?(v) The results of (iv) would be a factor in deciding whether all secondary school pupils should take mathematics throughout their course.
The Calculus as an Item in School Mathematics
Authors: Jackson C.S. 
Published: January 2007

Few subjects deserve attention better than the problem of teaching higher mathematics to the average student.Any man who does work in the world will some day, even though he may be a classical scholar investigating variants in manuscripts, be the worse for not knowing more mathematics ; and if he is not too closely limited by his environment he may even realise this fact. A goodly proportion of the work of the world must be done by quite ordinary folk. Something will be gained if the average student is helped even a little by being enabled to use the calculus and to think of its machinery as an engineer thinks of a drill, and not as a West Coast native thinks of a Ju-ju.An illustration may be taken from the Astronomer Royal's recent text-book. A star of the third magnitude is very much brighter than a star of the fourth magnitude. But the fourth-magnitude stars are more numerous than the third-magnitude stars, so that in fact we receive more light from the fourth-magnitude than from the third-magnitude stars. To raise all fourth-magnitude stars to the third magnitude would increase our light more than to raise all third-magnitude stars to the second magnitude.
Duration of Day
Authors: Bell A. H. 
Published: January 2007

The following is a simple method of establishing the formula for the determination of the duration of day, at any latitude, for a known declination of the sun.terminator or limit of the sun's illumination of the earth (at right angles to the line showing the direction of the sun's rays), ά is the point at which the terminator, άO'c', cuts the elevation of the line of latitude λ.Referring to the plan, α and α are the points in plan corresponding to ά in elevation ; 0 is the plan of the polar axis.
Presidential Address on the Theory of Proportion
Authors: Hill M. J. M. 
Published: January 2007

I desire in the first place to express my thanks to the members of the London Branch of the Mathematical Association for the honour they have done me in electing me to the office of president. I esteem it a privilege to take part in the efforts the Association is making to bring about improvements in the methods of teaching Mathematics.In what position does the work of the Association now stand ? Is it in fact in the position described by Sir J. J. Thomson in his address to the Association of Public School Science Masters? He is reported to have said that he had come to the conclusion that if you have intelligent masters and small classes it does not matter much what theory of education you adopt, and if you have not these, well, it does not much matter either.
Methods of Learning Geometrical Theorems
Authors: Siddons A.W. 
Published: January 2007

FIRST of all I should like to say that I am not responsible for the subject of this paper. Some member of the Association proposed that a discussion should be held on this subject and the Committee that arranges the programme for this meeting asked me to read a paper.This Association was founded a little over sixty years ago, mainly by schoolmasters, and its main object was to advance mathematical knowledge by improving the teaching of mathematics in schools, so that I am only too glad to welcome any discussion on actual school work, and I hope that what I have to say may provoke a discussion to which many of you will contribute.After promising to read the paper I sat down to consider what ground it should cover. It seems to me that the subject covers nearly the whole ground of geometry teaching up to the School Certificate stage, so the subject is wide.It will not be uninteresting to look at the past for a few minutes. I started teaching in the last century, and there was much teaching at that time that had made little advance since the date of the foundation of the A.I.G.T. But for a moment I will go back still earlier and tell you two stories of the middle of the last century
A New Attempt to Prove the Parallel Postulate
Authors: Ward D.R. 
Published: January 2007

Periodically until the end of time there will appear published works purporting to prove the parallel postulate, and it has fallen to the writer's lot to read what is perhaps the most recent of these. While the exact mathematical value of such attempts can always be gauged a priori, it is useful to put on record any new method followed, if only to warn future generations of parallel-postulate-provers against trying it. But such attempts are not without all value , some readers may find it a congenial occupation to spend a little time in discovering the inherent fallacy , alternatively, the task may be given to a student as a profitable and novel exercise in geometryFor these reasons, then, the following résumé of this most recent attempt is given here. And in order that the full interest of discovering the fallacy may not be denied to the reader or the luckless student to whom he delegates the task, it has been thought better to give only the proof as the author gives it, and not to point out where the fallacy lies. The writer has, however, allowed himself to make one or two comments on certain noteworthy points.
The Methods of Integration of the Differential Equation: P dx + Q dy + R dz=0
Authors: Underwood F. 
Published: January 2007

In this article an attempt has been made to indicate the ordinary methods of integration of the total differential equationThen the condition for integrability of (1) , i.e, the condition that (1) may possess a single integral equivalent of the form ϕ(x, y, z)= c, is
On the Necessity of Placing the Sign × before the Multiplicand
Authors: Ganguli Saradakanta 
Published: January 2007

Among many English writers of text-books on arithmetic and algebra there appears to exist a state of confusion regarding the use of the sign of multiplication. Here are three instances. Godfrey and Price’s Arithmetic (1915) contains the following statement:“The sign × means multiplied by; thus 3 ft. × 7 = 21 ft., and (7 × 3) ft. = 21 ft.; but 7 × 3 ft. is unmeaning ” (p. 13). Yet, on page 147, we find the expressionsin which multipliers precede the corresponding multiplicands.
On Certain Transformations in Euclidean Spaces
Authors: Maxwell E. A. 
Published: January 2007

This paper attempts to illustrate the method of representation, which is so powerful a weapon of modern geometry, in terms of elementary Higher Certificate mathematics. The results in their general form are familiar to students of geometry; but the special problem given here has several interesting features of its own.No originality is claimed for this paper, though I do not know of any other account of the work. It seems, however, worth while to give a brief summary, as it covers a great deal of the ground with which schoolboys should be familiar.
Rules for Bordered Magic Squares
Authors: Travers J. 
Published: January 2007

Rouse Ball in his Mathematical Recreations tells us that there are no rules for the construction of bordered squares ; I have made the following rules, which are now published for the first time as far as I am aware.It is remarkable that the rules for odd bordered squares are apparently not known as they are so simple. We use two diagrams, one a skeleton to make the method clear, and the other completed to show the full square.Rule I. Place the letter M at the (n +1 )th square as shown in Fig. 1, where n denotes the order of the square (here n = 9).Rule II. Now, commencing at the second column, write the numbers 1, 2, 3, … up to and under M. Then write the next number in the diagonal at the bottom right-hand corner ; the next number in the same column and in the second row, and continue until we arrive at the square above M.
The Cissoid of Diocles
Authors: McCarthy J. P 
Published: January 2007

The construction, presumably used by Diocles, who flourished in the second century B.C., for the cissoid is as follows.Let AB, CD be perpendicular diameters of a circle. Let E, F be points on the quadrants BD, BC such that the arcs BE, BF are equal. Draw EG, FH perpendicular to CD. Join CE and let P be the point of intersection of CE and FH The cissoid is the locus of P corresponding to points E, F on the quadrants BD, BC such that the arcs BE, BF are equal.
Some Aspects of the Development of Modern Statistical Method
Authors: Irwin J. O. 
Published: January 2007

There are two main roots to the tree of modern statistical method : the longer one takes us back to the mathematical investigations into games of chance of the seventeenth and eighteenth century mathematicians, the other starts with the statistical researches of Quetelet and Galton.Of the former we are told by Poisson “that a problem, relative to games of chance, proposed to an austere Jansenist by a man of the world, was the origin of the calculus of probabilities”.
On the Limit of the Ratio of Sin X to X
Authors: Goodstein R. L. 
Published: January 2007

The fallacies in the traditional approach to the problem of evaluatingin a pre-calculus course have been well brought out in an interesting article by Mr. Turton where the limit is obtained by means of polynomial approximations to the area of a circle. On re-reading the article recently it occurred to me that, at very little extra cost in effort and depth of ideas, a rigorous evaluation of the limit was possible which was completely free from appeals to the physico-spatial properties of geometrical figures and assumptions of symmetry and continuity which Mr. Turton makes.
Hill's Differential Equation
Authors: McLachlan N. W. 
Published: January 2007

1. We deal with the equationwhere ψ(ωt)is continuous, periodic in t with period 2π/ω a, k2 are real and positive, and
A stable (bounded) solution was given by Erdélyi in 1934 employing a complicated transformation and integral equations.We shall obtain this solution by an elementary method, thereby reducing the analysis to one-third of its former length.
When is a Parallelogram not a Parallelogram?
Authors: Gant P. 
Published: January 2007

Is a quadrilateral in which one pair of opposite sides are equal and one pair of opposite angles are equal necessarily a parallelogram? This question arose in class recently and led me to discover much that I did not know about an elementary topic. Possibly others may be interested in the results of my investigation.The following ‘proof’, for which I am indebted to my colleague, E. H. Lockwood, answers the question in the affirmative. The subsequent paragraph contradicts this.
Two Tripos Questions
Authors: Watson G. N. 
Published: January 2007

The Theorem that, if y3 + 3xy + 2x3 = 0, then

has been the topic of four recent notes in the Gazette, namely 1542 (not 1546) by E. H. N., 1610 by Mr. F. Bowman, 1684 by Mr. F. M. Goldner, and 1801 by Mr. H. V, Lowry. It seems worth while to supplement these notes by some comments which were for the most part suggested by the concluding lines of the note by E. H. N.The writers of the four notes have looked at the theorem from the point of view of a candidate in an examination ; if it were set in a Higher Certificate Distinction Paper, I have little doubt that a fair proportion of the candidates would recognise that the cubic is a unicursal cubic, they would write x/y = u (or make some equivalent substitution), they would express x and y as rational functions of u, and they would realise that the question must inevitably come out by calculating accurately the differential coefficients of y with respect to x and then substituting in the left-hand side of the differential equation. In fact, they would reproduce the substance of the work of E. H. N. with the omission of his refinement of taking the reciprocal of y as an auxiliary variable to abbreviate the analysis.
Mathematics and the Child. III. The Use of Mistakes in the Teaching of Mathematics
Authors: Gattegno C. 
Published: January 2007

To the mathematician, at least in the field of elementary mathematics, it is immediately apparent what is true and what is false, and for the teacher of mathematics it is the truth, the correct answer, the right reasoning, that are all-important.In this article, I propose to examine the other side of the picture, to consider the mistakes made by our pupils and to see how they can be turned to advantage in our work as teachers.
Perspective Triads
Authors: Andress W.R. , Saddler W. , Sawyer W.W. 
Published: January 2007

Consider two triads of points, a, b, c, a', b', c' in a plane ; then, using homogeneous coordinates and regarding the points as specified by the corresponding vectors, we may writeand writing b × c = α, c × a = β, a × b = γ then a × a' = m1γ − n1β, b × b' = n2α − l2γ, c × c' = l3β − m3α.
Dirichlet
Authors: Davenport H. 
Published: January 2007

This year has brought the centenary of Dirichlet’s death (5 May 1859), and it is fitting that we should recall some of his achievements and his part in the development of mathematics.Gustav Peter Lejeune Dirichlet was born on 13 February .1805, and was the son of the postmaster at D?ren, near Cologne. Among the schools he attended Was a Gymnasium at Cologne, Where one of his teachers was the physicist Ohm. When the time came for University study in .1822, Dirichlet persuaded his parents to let him go to Paris, which Was then the world centre of mathematics. Laplace and Legendre Were still alive and active, Fourier Was at the height of his career, With a group of brilliant young men round him, and Cauchy had already begun his massive development of the theory of flmctions.
An Operational Method for Determining the Series Solution of a Linear Differential Equation of Rank Two
Authors: Sarginson K. 
Published: January 2007

The equation considered is of the typeand f(δ) and g(δ) are polynomials in δ.There are three cases to consider These are:(I) The roots off f(z) = 0 are distinct and no two differ by an integer.(II) Some of the roots of f(z) = 0 are co-incident.(III) There exists an integral difference between some of the roots of f(z) = 0.
Sur des Points de Gergonne et de Nagel d’un Tétraèdre
Authors: Thébault Victor 
Published: January 2007

Introduction. De nombreuses analogies entre les propriétés du triangle et du tétraèdre ont déjà été signalées qui concernent des points, des droites, des cercles et des sphères remarquables associés à ces deux configurations.Nous nous proposons de définir des points d’un tétraèdre qui possèdent des propriétés comparables é celles des points de Gergonne et de Nagel d’un triangle. Ceux-ci pouvant recevoir plus d’une définition, d’aprhs les propriétés qu’ils possèdent, plusieurs points pourront leur correspondre dans le tétraèdre comme nous le montrerons pour le point de Nagel.
Some Characteristic Properties of the Circle
Authors: Rosenbaum R.A. 
Published: January 2007

There is a considerable literature on characteristic properties of the circle; a bibliography is listed in Bonnesen-Fenchel. It is proposed here to present some simple characteristic properties of types which are, in a sense, related to one another, as will be seen in what follows.Principal Definitions and Theorems. The discussion will be restricted to smooth, convex, closed curves. Relative to such a curve, a point, P, will be called(i) a π-point, if the product of the segments of all chords through P is constant;(ii) an α-point, if, for every chord through P, the angles, on the same side of the chord, between the chord and the tangents to the curve at the extremities of the chord are equal ; and(iii) a β-point, if the perpendicular bisectors of all chords through P are concurrent.
Maximum Length Decimals
Authors: Collings S. N. 
Published: January 2007

In his talk on Decimals at the Annual General Meeting of the Mathematical Association, 1962, (see the Mathematical Gazette, XLVII, 1963, p. 25), Mr. Green observed that the recurring decimal expression for 1/p where p was a prime other than 2 or 5 had the maximum possible length of p – 1 if and only if 10 was a primitive root of the modulus p. In practice there was maximum length forp = 17, 19, 23, 29, 47, 59, 61, 97but not forp = 13, 31, 37, 41, 43, 53, 67, 71, 73, 79, 83, 89,and he did not see any reason or pattern behind these figures.
Patterns in Group Tables
Authors: Francis F. H. 
Published: January 2007

When teaching elementary group theory, the ideas of coset, normal subgroup and quotient group are difficult to introduce.A possible approach is to make use of the patterns which appear on the multiplication of a group.It is easy to pick out from the multiplication table of a group the elements which form a subgroup. All that is needed is to find a set of elements which is closed under the operation.In the group of Table 1, clearly {I, C} form a subgroup.
Teaching Mathematics to Less Able Secondary Pupils
Authors: Backhouse J. K. 
Published: January 2007

The Mathematical Association’s News-letter No. 11, February
1967, contained this paragraph:
With so many schools going comprehensive, there are many experienced grammar-school
teachers who are wondering how they will get on with less able children. It might be helpful
if others who have been through this transition would share the results of their
experiences. Anyone who has something to pass on, no matter how small, is invited to write
to Mr. J. K. Backhouse, 44 Franklin Road, Headington, Oxford.
A New Triangle on the Election
Authors: Simons J. A. E. 
Published: January 2007

If we wish to plot the results of the election on a graph, constituency by constituency, we face the problem that a sheet of paper has two degrees of freedom, but there are three political parties. We may overcome this by using only the ratio of the votes obtained by each party. In any constituency,
let the Labour vote = xlet the Liberal vote = ylet the Conservative vote = z.
A Functional Equation For Implication
Authors: Goodstein R. L. 
Published: January 2007

Since all truth functions in two-valued logic may be represented by polynomials in the arithmetic of residues modulo 2, the question which R. Sibson raises in note 3127 may be expressed in the following form : Find all polynomials Φ(p, q) such that identically, where p & q is a commutative polynomial which vanishes only when p = q = 0, and p & p = p. Taking p = q = r in (1) we find
A Simple Visual Solution of the Technique of Undetermined Multipliers
Authors: Goodwin P. B. 
Published: January 2007

For many practical purposes, Lagrange’s undetermined multipliers can be used for the solution of problems where some function has to be maximised subject to a constraint.The technique described here arose from a consideration of one such case, where there was a choice of several types of investment, and, within each type, a different scale or level. With a limited total budget, it was necessary to decide how much to spend on each alternative.
Antelope
Authors: Gillings A. M. 
Published: January 2007

The name ANTELOPE stands for “A Nor (Transistorised Element) Logic Operations Patchboard Experiment.” It was originally designed by Mr. R. L. Lindsay, a mathematics master at Reading School, in the early part of 1965, to demonstrate how a complicated computer can be built up from a number of simple circuits. Mr. Lindsay left Reading School before it was fully operational for logic experiments, although it had been used for binary arithmetic.After a series of lessons on Symbolic Logic by Mr. J. Periton as part of the second year sixth form English course, two sixth formers developed a method of representing logical arguments by electronic circuits.
Combining Matrices
Authors: Fletcher T. J. 
Published: January 2007

[This is part of a lecture delivered at the Annual Conference of the Association on 31st March 1967. It contains an elementary teaching approach to matrices.]Let us look at this diagram and consider what it might depict. It might be a map of airline routes between airports in three countries; with two airports, x1 and x2, in the first country, three airports y1, y2 and y3, in the second country, and two, z1 and z2, in the third. The numbers beside the lines in the diagram might indicate the number of alternative flights (perhaps planes departing at different times) along the routes.
Some Results on k-maps
Authors: Meek B. L. 
Published: January 2007

For convenience we shall repeat here the definition by Hawkins, Hill, Reeve and Tyrrell (1, hereafter referred to as HHRT) of a k-map:
DefinitionA map is called a k-map if each of the following conditions is satisfied:
(i)It is a map on a sphere;(ii)Each face is a connected region whose boundary is a simple closed curve;(iii)Each vertex has at least three edges incident with it and each face has at least three edges on its boundary;(iv)Each face has exactly k edges, and therefore exactly k vertices.
Divisibility properties of binomial coefficients
Authors: McLean K. R. 
Published: January 2007

In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal’s triangle are reduced modulo m. For the case when m equals a prime number, p, the pattern formed by the zeros in the reduced triangle (corresponding to binomial coefficients divisible p) was completely described and the following result (stated by G. Gilbart-Smith) was proved:In the (n + l)th row of Pascal’s triangle, there are