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 math**ematics 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.

I’m missing some context. Who are Clements’ maths ed friends? Or do you just mean that Clements was arguing in a direction compatible with maths ed people of the time or soon to come?

The musical interlude was wonderful.

No particular friends. I changed it to “colleagues”, which is more accurate, although “friends” has a more desirably snide tone.

O’Mara’s treatment of the train problem is not “cookbook” in the sense that you use cues and keywords to apply blindly to some formula. He gives an explanation that the reader still has to take the trouble to understand; and if the reader does understand, then she is better equipped to tackle similar problems.

A good final examination confers a number of benefits, some of which are realized before the candidates even write it. While teaching is necessarily linear — you treat topics in some order — learning is nonlinear — understanding is a cumulative process that occurs over time with exposure to the topics and their sequels. In preparing for an examination, a student has to review a whole body of material and in retrospect is equipped to see what is actually going on, sometimes wondering why something seemed so difficult the first time around.

Of course, examinations can also be “gamed”, but this is not necessarily a bad thing. My generation was pretty good at gaming examinations and I have been amazed that modern students seem to have no sense of what might be on an exam, which to them seems something dropped out of the sky like lava from newly live volcano. A student who has familiarized himself with the material will realize that there are only so many things that can be reasonably on an examination (and might be even able to construct possible questions), and from looking at previous examinations will get a sense of the “signatures” of the examiners.

There are two caveats. One is how to handle situations when a student has a bad day on an important examination. In Ontario, when there were provincial Grade 13 examinations, when a teacher registered a student for the exam, he provided a “confidential mark” or an estimate of how a student might reasonably be expected to do. If these confidential marks are generally reliable, then one can be used to grant a pass to a student whose raw examination mark might be a failure. (Another way to do this is to look at overall performance on a number of papers, so a student is not sunk by a particular one.)

The second caveat is the students have to be honest with themselves as to the state of their knowledge. One can get even an A by going through the motions without having mastered the material; the purpose of the exam is to test basic knowledge and procedures, not to throw curves that will arouse censure when there are unreasonably many failures that have be “fixed”. Today, with grade inflation, too many students are getting 90s and then being completely sideswiped when they hit the realities of a serious university mathematics course because they take the results too seriously.

Thanks, Ed. Just to be clear, my title and my suggestion that the train example was “mindless stuff” were sarcasm.

That’s the thing I found in general in O’Mara’s text, that it wasn’t close to cookbook, that Clements was broadly condemning about very little.

O’Marawas “gaming” the exams in the sense that the content and the techniques had a stated focus on the exams, and that many examples and exercises were taken from past exams. But a student preparing for the 1903 exams usingO’Marawould still have learned many coherent topics in a properly deep and solid manner.I took the Victorian senior exams, then called HSC, in 1977, and of course it was all “gamed” in the way O’Mara advocates and that you describe. Our texts, on Pure Mathematics and Applied Mathematics, included many past exam questions, and the final assault on the subject consisted of working through every past exam paper that we could find. We learned the nature of the exam questions in broad brush. But

onlyin broad brush, because that was all that was possible. The subjects were simply too deep and too varied in the possibility of questions to do more than learn well the required techniques and the general style of questions for which these techniques could be applied.I really like reading Clements, but he can be absolutely infuriating in his blind spots: in his blindness to past good, and especially in his blindness to modern madness. Clements wrote

Mathematics for the Minorityā and note the cutely populist title ā in 1989. By 1989, Victoria’s long educational decline, due in large part to Clements’ more stupid mates, was well under way and undeniable. To take just one example of every, by 1989 Pure Mathematics and Applied Mathematics had been replaced by Mathematics A and Mathematics B: meaningless titles, which almost guarantee meaningless content. The 1989 exams are notably worse than in 1977; the 1989 textbooks, by the very same authors, are notably worse than in 1977. But Clements only has eyes for the real and, much more, imagined, sins of the past.And now, of course, it is utter madness. The textbooks now, at all levels, with or without exams in sight, are so robotic, so thought-killing, it beggars belief. Given a choice, I would much prefer to teach senior school students from

O’Marathan from any standard Victorian text. Even with all the division of decimals and the imperial units and the other century-old eccentricities, I’d chooseO’Mara. In a heartbeat.Very interesting read, Marty. Thanks for posting.

Some related points.

Hence my use of books by Turnbull at school.

At the tertiary level, I recall a professor at University of Sydney arguing that the main problem with universities was that they offered degrees.

When I was an undergraduate, we were given the mid-year exam in politics on day 1 of the academic year.

What is the possible point of an argument like that?

Feel free to delete it .

Feel free to explain it.

I was offering a few examples to illustrate that, over the years, there have been many different ideas about how to present material to students and assess what they have learned as a result. And I imagine that there will be more changes to come. So far, I have been able to find opportunities to let my ideas come through in my teaching – in some form or other. Whether I can maintain this is another matter.

Certainly this blog helps me to think about how I teach students about mathematics, and for this I am grateful to Marty and all the correspondents.