*The Teaching of Mathematics by Women in 19th Century Britain*(B.Sc. Project, University of St Andrews, 2016):

Examinations were a means of raising standards in girls' education, both demonstrating their capabilities and formalising their education. Thorough examinations by external examiners were held every year at Cheltenham. Beale insisted on their value for children, she commented, "they give them a sense of security, they show to themselves whether they really did know what they thought they did, and they teach them to feel that vague half-knowledge is worthless, since it will stand no fair test." However, many still objected and would not allow their daughters to attend, so Beale had the Council pass a resolution that those who absented from examinations would have to leave the College. The fact Beale was willing to lose pupils over the examinations emphasises the value she placed on them. The formal admission of girls to the Cambridge Local Examinations took place in 1865; although Beale did not accept the examination at Cheltenham as its arrangements did not fall within those of the College year, she did recognise that with these examinations new impetus had been given to the progress of education and she saw it her duty as a teacher to guide this movement.

Looking over the Oxford local examination papers from Cheltenham provides interesting reading. In 1858 the first arithmetical calculation set before the first class reads, "reduce 14 to an improper fraction having denominator 9; and 34 to be an improper fraction having denominator 91." We can observe from these papers how the mathematical curriculum has evolved. This simple sum is made more difficult by the need to understand the definition of an improper fraction and the wordy nature of the question. Straightforward calculations are made more complex by the choice of numbers, for example "if a yard of lace cost £1^{29}/_{96}, what will 16^{11}/_{25} yards cost?" The arithmetic paper also asks for definitions of a circle, sphere, square and cube, which we would now consider to be a geometry question. The 1858 examiner T Marshall stated that, "the standard of Arithmetic appears hardly equal to that attained in other subjects: nor was the accuracy of some of the work so remarkable", demonstrating the low mathematical ability of the pupils in the early days at Cheltenham. From the 1867 examination report, we have evidence that the situation in arithmetic was much improved with little incorrect work being sent in by pupils. However, there were still some difficulties with principles such as vulgar fractions, for example explaining how ^{5}/_{16} is reduced to a decimal fraction, and under what circumstances the decimal will recur. Without a calculator, pupils today would not be expected to convert such complicated fractions to decimals.

In former years the College had stood alone in submitting the work done to the Council. However, from 1869 comparison was now possible; this was due to the fact that five examiners of the London University and three Assistant Commissioners, Mr Bryce, Mr Fitch and Mr Fearon had undertaken portions of examinations as part of the SIC investigation into the quality of the education of girls. Of the Mathematics, Mr Fitch, H.M.'s Inspector of Schools, Examiner to the University of London and Assistant Commissioner Schools' Inquiry, said, "of the 3 or 4 pupils to whom I have assigned the highest number of marks, it is only fair to say that I have rarely read Mathematical Papers showing a sounder knowledge of principles, or more methodically arranged." In 1869, in Euclid the pupils, "obtained the good average of 62^{5}/_{7} marks in this difficult subject, which affords an excellent test of good teaching." One pupil obtained ninety-eight marks out of one hundred and another two years younger did nearly as well. The College was excelling with its mathematics teaching but it is important to remember improvement was still necessary. In his report on the algebra examination, Mr Fearon, H.M.'s Inspector of Schools and Assistant Commissioner Schools' Inquiry Commission, commented, "I do not say that, in breadth or depth, it reaches the standard of an ideal education for girls; but looking to what my experience on the Schools' Inquiry Commission shows me to be the actual condition and results of teaching in schools such as these, the Council may well be satisfied."

Praise for the mathematical teaching at Cheltenham is common throughout the examination reports, in 1871, Mr Fitch noted, "the papers on Euclid and Algebra are of considerable merit, and indicate that the teaching of those subjects has been very thorough and intelligent." Similarly, in 1876, Mr Magnus, Life Governor of the University College, London, remarked, "I found evidence of thoughtful training and of well-directed efforts to teach the pupils to think for themselves and to rely on a knowledge of principles rather than of mere rules. This fact reflects great credit on the teachers of the College." This highlights the effectiveness of the mathematical teaching methods employed by Beale and her staff. In the 1873 report, talking of geometry, the examiner declared, "I have never seen a better set of papers from students of the same age on this subject."

The 1878 algebra paper is most similar to what schoolchildren nowadays are being taught, question 4, for example, reads, "multiply 2*a*^{3} - 3*a*^{2b} + *ab*^{2} - *b*^{3} by *a*^{2} - *ab* - 2*b*^{3} and verify your result when *a* = -3 and *b* = -^{3}/_{2}." However, Mr Magnus commented of the 1876 paper that there is a lack of care and accuracy shown by too many pupils in the algebra examination. The 1867 algebra paper includes a question on simultaneous equations but there are also more complicated questions such as, "find two numbers, of which one is to the other as three to four, and of which the sum is to the sum of their squares as 7 to 50." Algebra questions set in schools nowadays would not require as much thinking as this does. Arithmetic and geometric progressions also feature heavily, question 13 in the 1875 paper asks, "deduce an expression for the sum of n terms of a geometric series." Factors are called common measures and question 8 of the same paper instructs the pupil to, "find the greatest common measure of 3*a*^{3} - 3*a*^{2}*b* + *ab*^{2} - *b*^{3} and 4*a*^{2} - 5*ab* + *b*^{2} ".

Examination questions make it clear that the pupils in class I were expected to have memorised Euclid Books I-IV. Question 10 from the 1867 paper reads, "demonstrate Prop. XII. in Euclid's Second Book" and question 11 states "The sum of the squares of any two lines, is equal to twice the square of half their difference, with what proposition of Euclid is this statement identical? State it in his words, and demonstrate it." We can also observe from the papers the move away from Euclid towards more practical geometry. Mr Magnus commented, in the 1876 report that, "the adoption of the Syllabus, instead of Euclid, seems to have worked well, and should certainly be continued." Questions in the 1873 geometry paper read, for example, "explain the process by which in Book II, Euclid shows all rectilineal figures admit of quadration" and "demonstrate Prop. XLVII. Book I., and its converse", whereas there is no mention of propositions in the 1878 paper, questions four and five for example, read, "construct a triangle having given two sides and an angle opposite one of them" and "find the locus of a point which is equally distant from two straight lines." We can see that this geometry bears more similarities to that taught in schools today; focusing more on the structures of mathematics rather than on the routines and techniques present in Euclidean geometry.

Not only were pupils excelling at Cheltenham but many also went on to pass the general examination of the University of London. The 1875 examination report reads, "since the first opening of this examination, six years ago, fifty-seven have passed, and of these, twenty-two were pupils prepared here." This once again is testament to the quality of teaching at Cheltenham; the remarkable exam success is highlighted by the general report of the Oxford Local Examinations from August 1897, which demonstrates that learning by rote and lack of understanding still prevailed in too many other schools. It reads, "in many other cases candidates who wrote out correctly all propositions for the first six books sent up attempts at problems that can only be described as grotesque, and showed their complete failure to understand the subject, giving the unpleasing impression that all they knew was learned by heart."