Arithmetic, CAS and the Dangers of Obsession

Over the summer, my family and I took a holiday, driving around Tasmania. It was great, of course, albeit the ferry over was a bad mistake.¹ Our first stop was Stanley, a gorgeous, windswept town, featuring a mini Ayers Rock, and penguins that could be viewed unobscured by an MCG-sized stadium or a crazed horde of fellow tourists. Also in Stanley, as it happened, I found a book.

As do all of Tasmanian’s little towns, Stanley had an antique store. Stanley’s was small and amateurish and friendly, containing odd bod stuff, evidently from locals who had cleaned out the shed (or had died). Having decided against purchasing the thesis on mathematics education (sic), the chatty proprietor pointed out an old book on arithmetic. She charged me the exorbitant fee of $5.

Intermediate Certificate Arithmetic was a Form 4 (Year 10) textbook written by Robert Wilson, an Assistant Master at Melbourne’s Scotch College. It was first published in 1934 and was squarely aimed at the Intermediate Examination, then required to pass Form 4.² There is nothing too special about ICA – I own a number of such old Australian texts – but it happened to be the text I was browsing while sipping beer and waiting for penguins, and there are a number of very interesting things to note about it.

The first, and somewhat misleading, thing to note is that the text contains some nice mathematics. Primes and highest common factors and so forth are there, with significantly more depth and challenge than provided by modern texts. The Euclidean algorithm is there, albeit only as an unexplained and untitled recipe. Recurring decimals are nicely converted to fractional form, essentially using the x trick (but without the x). That section of the text culminates with a very good exercise, that one wishes all modern texts contained:

Beyond some nice mathematics, ICA is written in a straight-forward, serviceable manner. The exposition is dry as dust but everything is stated clearly and succinctly; there is zero doubt of what is expected of the student and there is sensible guidance on how to do it. To this extent, Wilson’s text is strong, random evidence in support of Tony Gardiner’s sermon on the superiority of past texts.

In other ways, however, ICA is very painful to read. It is mathematics education from a much tougher and much more utilitarian age. Many, many students would have left school after Form 4, and a decent handle on annoying arithmetic, more often than not framed in crazy imperial units, was typically an essential professional skill and an essential life skill. Although some attention is paid to purer aspects of number, there is a strong emphasis on (someone’s) real-world practice. From the Preface:

In Chapter XII a section has been devoted to practical work in stocks and shares. Tables of the rates of brokers’ charges on the Melbourne tock Exchange are given and the student is required to work problems of a practical type by referring to these tables.

Then immediately follows another world entirely:

In Chapter XXI a description of the actual methods employed by painters in papering rooms is given, and the problems in “papering rooms” must be worked according to the rules laid down in this description. 

This real-worldness was apparently insufficient. The Preface to the 1938 edition, which is the edition I found in Stanley, notes,

In order to meet the suggestions made in the 1935 syllabus of the Melbourne University, several chapters have been added; these chapters deal chiefly with the following topics: Federal and State Taxes, Municipal Rates and Taxes, Time-Payment Systems, Co-operative Systems, Money Borrowings and Cash Orders, etc.

Of course, something had to make way:

To make room for these, several deletions have been made, these comprising topics not prescribed for the Intermediate Examination, e.g., Recurring Decimals, Mensuration of Frustrums of Cones, Pyramids, etc.

One need not be a great fan of cones and pyramids to be disheartened by their replacements.³

Which is not to imply that this very specific real-world emphasis was, for the time, necessarily wrong. True, it smells wrong. There is no question, for example, that ACARA’s current push to reform their own crappy K-2 curriculum by emphasising “consumer and financial literacy” is blinkered nonsense.⁴ Generally, one wishes the teaching of skills and knowledge to be unencumbered by distracting detail and as readily transferrable as possible; demanding the exact arithmetical methods of paper rollers doesn’t sound like a great idea. But it was a very different world and we should be slow to judge.

Nonetheless, necessary or not, it is all disheartening, a reminder of how narrow life was for so many.

Given current circumstances, it is also notable that in the 1930s the University of Melbourne was the unquestionable ruler of the manor. UoM set the syllabus, then set the exams in synch, and then schools and teachers and students were required to keep in step. IAC is unapologetically focussed on UoM’s exams, with “a very large number” of exercises taken from past UoM (and Scotch) exams, and a few entire exams included as an appendix. I have not looked closely (and haven’t checked for errors), but the UoM exams seemed mostly serviceable, reasonable tests of functional arithmetic. There are very few questions on pure number, but of course that was not the purpose of the subject.

This was all very interesting, at least to me, but it was something else that first grabbed my attention. Flipping through randomly, I came across some miscellaneous exercises on calculations with logarithms:

In the pre-calculator age, logarithms were of course a brilliant and essential tool for the performing of heavy arithmetic. They still have value, and are undervalued, for the insight they give to inverses, but it is now natural enough (no pun intended) that logarithms are mostly considered in school for a much more specific application in calculus. So it goes.⁵ In the 1930s, however, it was essential for an ending subject on arithmetic to provide a grounding in the use of logarithms.

But that does not mean that it was essential to teach anything that might be done with logarithms, and the later exercises above seem to me to be plain nuts, on any terms. It is difficult to imagine, for instance, a bank teller or a paper roller ever having required the fifth root of a cube divided by a square root. Sure, there is some benefit, or at least the temptation, to demonstrate and teach the power of mathematical tools beyond immediate functionality. But such lessons still require some respect for practicality, or elegance, or something. Heavy computation for its own sake is pointless and painful.

Why did Wilson include such nutty computations? Undoubtedly because the University of Melbourne exams contained, at least on occasion, similarly nutty questions. The strong majority of the UoM exam questions in ICE, including most of the questions requiring logarithms, are solidly sensible, but one, and only one, exam question is Wilsonian:

Wilson’s text includes many more such exercises, including a number on Scotch exams that the text reproduces. Was this a case of Wilson being spooked by one over-egged UoM question, resulting in a little avalanche of nonsense? It doesn’t seem quite that way. Reviewing a few other UoM exams of the period, and comparing dates, Wilson’s text definitely contains more such weirdness, but there are other UoM questions that are at least borderline silly. Nonetheless, whether inspired by UoM, and/or driven by Scotch, the resulting emphasis, particularly in an age and for a subject demanding practicality, was an emphasis everyone could have well done without.

The multiplication here is from a Grade 5 student’s 1863 workbook, which was reproduced in Ken Clements‘ Mathematics for the Minority. There is something absolutely beautiful about the multiplication, but of course Clements’ purpose was to note the utter pointlessness of such an exercise or probably, such copying from the board, and the instruction of which it was part, and of course Clements was correct. Even noting the illustration of the elegance and the power of the algorithm, the example is utterly pointless, if only because most mortals have a snowflake’s chance in hell of completing such an example successfully.⁶

Wilson’s logarithm obsessing brought to mind the multiplication obsession from Clements’ book, and both brought to mind the current CAS obsession. In all three cases, and whatever sensible prehistory, the question strayed far from “What is this tool/technique good for?” to “How can we test further?” The possibility that the tool or technique had reached or overreached its natural limit and natural use (if any), and that it was subsequently only perverting education rather than assisting it, is not even contemplated.

Of course I want CAS removed entirely. To me it is obvious that, whatever minor but proper uses may exist, the overwhelming effect of CAS is to pervert the learning and the testing of school mathematics. But, even if one disagrees with that, one has to be an obsessive denialist to avoid acknowledging how thoroughly crazy VCE CAS exams have become, and thus how thoroughly crazy VCE teaching has become.

It is not by a longshot the worst example, but VCAA’s new-found gridlines and error bars for graphing are probably the most tellingly mad of the CAS perversions. It is mad as Wilson’s algorithms and as Clements’ mad multiplication. I feel as sorry for current VCE kids, futilely trying to glean VCAA’s secret and insane desires, as I do for 1930s kids and for whatever poor 19th century kid attempted or copied Clements’ multiplication. VCAA’s assessment of graphing has no greater meaning.

Demanding such aspects of graphing serves absolutely zero proper purpose. It only exists because CAS encourages it, and because VCAA, still determined to maintain their wading pool shallow curriculum, is desperate to find something new to test. Which then carries over to non-CAS exams. VCAA’s clowns are like the Chinese students moving their abacus fingers even when the abacus is not there.

Fifty years ago, Stephen Jay Gould wrote a great article about the extinct Irish elk and its massive, seemingly counterproductive, antlers.⁷ In the early days of evolutionary theory, the Irish elk was used to argue for orthogenesis, the theory that evolutionary change is driven by internal forces rather than by the external environment promoting natural selection. Thus, it was argued, the Irish elk became extinct because the unstoppable growth of its antlers led it to doom. This was successfully argued against; the Irish elk’s crazy antlers were indeed created by natural selection, even if the manner in which that occurred is not so obvious or entirely clear.

Unlike the Irish elk, the evolution of mathematics education seems to be determined by orthogenesis, and the crazy-seeming antlers that it can create can be genuinely crazy. Internal forces will act and external reality be damned. It is then not survival of the fittest, it is the survival of the most arrogant.

CAS must go. As must the arrogant obsessives who are unable to question CAS, who are unable to admit the crazy-antler truth that CAS is currently serving no proper purpose.⁸

 

1. Even the cattle on board were complaining about the low class.

2. There were seemingly at least two “standards” of intermediate arithmetic, and there were other Form 4 mathematics exams. I am unclear on the overall structure, and what was mandatory and what was elective.

3. As noted, the 1938 text contains a section on recurring decimals. I am not sure what aspect of the topic may have been deleted.

4. As usual, Tony Gardiner said it short and well. 

5. Burkard and I wrote a column on the deafening silence that greeted the 400th anniversary of logarithms’ birth.

6. In this instance, the student, or teacher, did not.       

7. “The Misnamed, Mistreated and Misunderstood Irish Elk”, published in Gould’s first collection, Ever Since Darwin.    

8. Readers have until 2 April to complete VCAA’s survey.