Trying to understand when and how Victorian maths education went so wrong has led us down some pretty obscure rabbit holes. A very helpful guide to this warren is Ken Clements and, in particular, Clements’ interesting and interestingly slanted history of Victorian school mathematics, Mathematics for the Minority. Pictured above is one of the rabbits (so to speak).
Clements doesn’t have much good to say about Victorian school mathematics, at least until the maths ed guys took over, and he has nothing good to say about the University of Melbourne’s long influence on this schooling. But Clements is worth reading, even if you have to grit your teeth while doing so. He delves deep into history little covered elsewhere.
Clements doesn’t like exams, and in particular he doesn’t like matriculation exams. Exams affect the nature of schooling, of course, and Clements believes for the worse. To be fair, bad exams make a mess of everything, and Victoria had bad mathematics exams for a very long time (and does again). Victoria’s matriculation mathematics exams began in 1856, and for the first decades were very bad, a deadening mix of arithmetic, algebra and absurdly thoughtless Euclid.
As Melbourne and the University of Melbourne grew, the matriculation exams became more important. They began to serve the familiar double role: as university entrance exams; and as de facto school completion exams, including as official or unofficial entrance exams for the public service and other professions. As a consequence, at the turn of the century there were significant reforms to education, largely under the leadership of Frank Tate. This included reform of the school curriculum and secondary school exams, the reestablishment of a teacher training college and the introduction of public secondary schools. This atmosphere of reform also gave birth, in 1906, to the Melbourne Mathematical Society, then revived in 1924 as the once-purposeful Mathematical Association of Victoria.
With secondary education more broadly public, and with the established importance of secondary exams, there also grew an industry to study for these exams. Which brings us, finally, to our rabbit, Father Patrick O’Mara. Paddy, the reader will be surprised to hear, was Irish. A Jesuit, Paddy came to Australia in 1895, to teach mathematics at Xavier College, returning to Dublin in 1905.
In 1903, Paddy wrote a mathematics textbook, the real rabbit of the story. But first, a musical interlude, just because.
O’Mara’s textbook, Reasoned Methods in Arithmetic and Algebra for Matriculation Candidates, was very popular and for obvious reasons. As its title suggests, O’Mara’s text was unapologetically focussed on preparing for the matriculation exams, including many worked examples from previous exams. In essence, O’Mara’s text was the very first Cambridge Checkpoints.
Clements dislikes O’Mara’s book for the very same reasons it was so popular. Clements introduces and then quotes from O’Mara’s text:
The Victorian teachers were too occupied with preparing their students for forthcoming examinations to be concerned with the educational worth of what they were teaching. Thus, it was not surprising to find, in books written by Victorian mathematics teachers of the time, statements like that of Father P. J. O’Mara (1903, preface), who was an experienced mathematics teacher at Xavier College, Melbourne:
University professors and examiners may be reasonably considered the best judges of what is most important for, and most wanting in, would-be undergraduates. Hence the lines they have marked out in their examination papers deserve our very careful attention.
That the arithmetic and algebra papers set in the last decade in the Matriculation Examinations of the Melbourne and other Universities reflect a persistent need for careful attention to some points more than to others will be clear to anyone who even cursorily examines them.
That these points possess an intrinsic importance of their own will also be apparent. That, finally, the very peculiar type of paper, hitherto set, calls for more special attention to special questions than a text-book written to supply general needs can afford, will be felt by any master or student who has gone over the course.
Yes, heaven forbid university mathematicians might be considered the best judges of what is required for those entering university. And heaven forbid having a specific textbook designed for a specific purpose.
Clements continues, smugly referring to notably focal aspects of O’Mara’s text:
One has only to examine the style in which O’Mara’s book was written to realise why it quickly became a best seller. Throughout the book, explicit advice was offered to readers on what they should learn for matriculation mathematics examinations. Thus, on page 116 of the arithmetic section of the book, readers were told that train sums are frequently to be met with in examination papers’;
Some more detail. After this declaration, and after noting the “rates” of the trains have to be added or subtracted according to the directions of travel, O’Mara works through an example, taken from another text of the time:
Obviously, mindless stuff.
on page 117, they were informed that the solution of one example would suffice as an illustration of the ideas which were presented because ‘exercises on this point have hitherto been seldom given in the Melb. Matric’;
This criticism negates Clements’ first criticism, since the “one example” is the one example O’Mara provides of a “train problem”, demonstrating that O’Mara is hardly dwelling on the topic. It is also unclear what O’Mara means, since he also brought up the prevalence of train problems, only to then seem to claim otherwise. Whatever his intended meaning, O’Mara continues with “closely allied” rowing problems, and then time-ratio problems with clocks running at different speeds. None of this is bad, or badly presented.
on page 33 of the algebra section, Father O’Mara spoke of ‘a very important theorem’ even though he could not claim ‘its occurrence hitherto in the Melb. Matric.’;
So, what is the objection, precisely? For better or worse, O’Mara goes on to declare that, “due to the frequency at which it turns up on other university tests”, the theorem “warrants the most careful attention”.
O’Mara follows this proof with two nice examples, illustrating the use of the theorem, and then some very nice exercises, on the theorem and algebraic identities.
and on page 40 of the arithmetic section, readers were advised that they were now coming to ‘a class of questions which are of very frequent occurrence in the Melbourne Matric’.
The section to which Clements refers covers some problems on decimal arithmetic and imperial currency. Very painful, and of course standard and very necessary for the time.
Significantly, in the whole book there was no reference to the important reforms of the Mathematical Association in England in 1904 even so reviews of the book which appeared in the Australasian Schoolmaster (16 September 1903, p.52 and 21 September 1904, p.56) suggested that it contained ‘all that is newest and best’ in the subject and would be invaluable’ ‘especially for country teachers’.
Sure. O’Mara’s is not one of the great textbooks. It suffers from its time, and from the clunky curriculum still practised in Victoria. It doesn’t incorporate the reforms just then taking place in England. But still, all in all, O’Mara’s is a good book. The text may suffer from its time but it also benefits from its time. Some of the material on algebra, and on ratio and proportion is very good, and little now seen, at least in Australia.
O’Mara’s text may be exam focussed, but having such a focus is not necessarily a drawback, and it is not particularly so for O’Mara’s text. To the extent that an exam system tests well on good material, a focus on the exams is perfectly fine. Exams notwithstanding, even Victoria’s clunky exams of the time, O’Mara’s text still exhibits a general clarity and coherence. A clarity and coherence, one might add, seldom seen since Clements’ maths ed colleagues took over.
Just a coincidence, of course.