A few days ago, the Sydney Morning Herald had an article on “the hardest question” from this year’s NSW Extension 2 Exam. The question is worth 5 marks, which equates to 9 minutes of a 3 hour exam (accessible here). The question, and another question (6-ish minutes), which apparently came along for the ride, are posted below. Readers’ homework exercise is to Compare and Contrast. Continue reading “How the Other Maths Lives”→
The following is an article by Tony Gardiner, originally published in 1996 in the Mathematical Gazette. It is reproduced here with the kind permission of Tony, the Chief Editor of the Mathematical Gazette and the Mathematical Association. The original article is available on JSTOR, here (via an educational library), and we’ve also separately posted the problems in Tony’s article here.
This is a story from long, long ago. It is about Mr. McRae, who was our grade 4 teacher, at Macleod State School. We have written about Macleod before, and we have written, briefly, about Mr. McRae before, in regard to the moon landing:
I still have vivid-grainy memories of watching Armstrong’s first steps. A random few students from each class in Macleod State School were selected to go to the library to watch the event on the school’s one TV. I was not one of the lucky few. But Mr. Macrae, our wonderful Grade 4 teacher, just declared “Bugger it!”, determined which student in our class lived closest to the school, and sent out a posse to haul back the kid’s 2-ton TV. We then all watched the moon landing, enthralled and eternally grateful to Mr. Macrae.
He was that kind of guy. No-nonsense and intelligent and cultured.
The year he taught us, Mr. McRae was new to Macleod. He had just appeared on the playground before the first class of the year, tall and commanding. Rumour had it that he had played Under 19s for the Richmond Football Club, making Mr. McRae just shy of a Greek god. (The actual Greek god was, of course, Carl Ditterich.) He was a standard and excellent teacher. Firm, disciplined and disciplining, but kind, and with a calm and intelligent air of bemusement. He was the boss, but a thoughtful and unpredictable boss. Hence, our class getting to watch the moon landing. And, how else to explain the boxing match?
One day, Mr. McRae inadvertently started a harmless play-scuffle between two students. He then decided the dispute should be settled by a proper boxing match in front of the class. Once, of course, a kid had been sent home to fetch a couple pairs of boxing gloves. We can’t remember whether we lost, although we remember we didn’t win. In any case, neither of us had a clue how to box, and so the match was followed by Mr. McRae giving the class an impromptu lesson on technique. This was, to explain it a little, the era of Lionel Rose and Johnny Famechon and TV Ringside.
That’s all by way of background. The story we want to tell is of a mathematics lesson.
One Friday afternoon, Mr. McRae introduced his grade 4 class to Pythagoras’s theorem. Or, at least, to Pythagorean triples; we can’t specifically remember the triangles, or anything, but undoubtedly made an appearance. Why he showed us this, God only knows, but Mr. McRae ended the class with a challenge: find more triples. Our memory is that the specific challenge was to find a certain number of triples, maybe three, maybe five.
We have no idea what Mr. McRae hoped to achieve with this challenge, but we remember pondering, aimlessly, hoping to find triples. Eventually, by smart persistence and dumb luck, we stumbled upon the trick: doubling a triple gives a new triple. So, , and so on. With this kid-Eureka insight, we then happily spent the week-end doubling away.
Come Monday morning, Mr. McRae asked for the class’s triples. We proudly went to the blackboard and wrote up our largest creation. By memory, it was something in the millions. So,
or thereabouts. And then Mr. McRae uttered the fateful words:
“Let’s check it!”
There were the inevitable groans from the class, and the little Archimedes hero of the story was more popular than ever. But, Mr. McRae was the boss, and so we all set down to multiplying, including Mr. McRae himself. And, ten or so minutes later, the class collectively started to conclude … the equation was wrong. Yep, Little Archimedes had stuffed up. Which led to more fateful words:
“Let’s find the mistake!”
More groans, more multiplying, and eventually the error was found. By memory, after quite a few doubles, somewhere in the mid thousands. And, satisfied, Mr. McRae led the class on to whatever he had been planned for that day.
What is the moral? We have a reason for telling the story, beyond a simple tribute to a great, memorable teacher. We think there are morals there. We’ll leave it for the reader to ponder.
This is a brief post, intended as a place for discussion of the 2021 NHT Specialist Mathematics exams, here and here. See also the accompanying 2021 NHT Methods exams post, here.
We’ve made a few remarks, below, on specific questions, and see this WitCH, but we haven’t gone carefully through the exams. We’ll update with further remarks if and when commenters raise substantive issues with the questions, or either exam as a whole.
The exam reports are out, here and here. We haven’t looked and don’t intend to look, except for Q10 on Exam 1. The suggested solution is, well, rooted. See here.
A few months ago, frequent commenter Red Five noted a pretty much forgotten book, The Saber-Tooth Curriculum. Written in 1939 by the mythical J. Abner Peddiwell – the creation of education professor H. R. W. Benjamin – the book is a series of drunken lectures on the nature of education during the paleolithic era. That education supposedly included lessons such as saber-tooth-tiger-scaring-with-fire, long after saber-tooth tigers had disappeared, and so on.
The book is crazily satirical, happily takes shots at everybody, and it holds up well. Maybe not well enough to bother hunting out – it is difficult to sustain such a parody for 100+ pages – but The Saber-Tooth Curriculum is clever and pretty funny. Surprisingly so since humour, particularly topical humour, tends to date quickly.
Below is our favourite passage from the book, concerned with the establishment of university courses for teachers, and the introduction of professors of paleolithic education.
The crude, naive work of the education professors was regarded with contempt by the subject-matter specialists. It was inevitable that a man who who had devoted a lifetime of productive scholarship or systematic speculation to such a problem as The Mystical Element in Sputtering Firebrands as Applied to Tiger-Whiskers or Variations in Thumb-Holds for Grabbing Fish Headed Outward from the Grabber at an Angle of Forty-Five Degrees Plus or Minus Three should be contemptuous of pseudo scholars who were merely trying to show students how to teach.
The academic contempt for pedagogy had a good effect on the education professors. Stung by justified references to their low cultural status, they resolved to make their discipline respectable. With a magnificent display of energy and self-denial, they achieved this goal. First, they organized their subject systematically, breaking it down into respectably small units, erecting barriers to keep professors conventionally isolated from ideas outside their restricted areas, and demanding specialization and more specialization in order to achieve the narrow knowledge and broad ignorance which the paleolithic university demanded of its most truly distinguished faculty members.
Second, they required all members of their group to engage in scientific research in education by counting and measuring quantitatively everything related to education which could be counted and measured. It was here that the professors of education showed the greatest courage and ingenuity. They confronted almost insuperable obstacles in the fact that education dealt with the changing of human minds, a most complex phenomenon. The task of measuring a learning situation involving an unknown number of factors continually modifying each other at unknown rates of speed and with unknown effects was a tremendous one, but the professors did not hesitate to attack it.
Finally, the professors of education worked for academic respectability by making their subject hard to learn. This, too, was a difficult task, but they succeeded admirably by imitating the procedures of their academic colleagues. They organized their subject logically. This necessarily resulted in their giving the abstract and philosophical courses in education first, delaying all practical work in the subject until the student was thoroughly familiar with the accustomed verbalizations of the craft and, thereby, immunized against infection from new ideas. They adopted the lecture method almost exclusively and labored with success to make it an even duller instrument of instruction than it was in the fields of ichthyology, equinology and defense engineering. They developed a special terminology for their lectures until they were as difficult to understand as any in the strictly cultural fields.
Thus the subject of education became respectable. It had as great a variety of specialists as any field. Some of its professors tried to cover the whole area of the psychology of learning, it is true, but most of them confined their efforts to some more manageable topic like the psychology of learning the preliminary water approach in fish-grabbing. Its research workers were so completely scientific that they could take a large error in the measurement of what they thought maybe was learning in a particular situation and refine it statistically until it seemed to be almost smaller and certainly more respectable than before. Its professors could lecture on modern activity methods of instruction with a scholarly dullness unequalled even by professors of equicephalic anatomy. Their cultured colleagues who had once treated them with contempt were now forced to regard them with suspicious but respectful envy. They had arrived academically.
Porter is also a sadistically destructive asshole. As Minister for Social Services, he brought in the Turnbull Government’s psychopathic Robodebt scheme. It was evil, and obviously evil, from the start, bringing needless misery to thousands of the most vulnerable and powerless, and prompting God knows how many suicides. And it took four years of the fucking obvious, four years of senate reviews and legal challenges to get rid of the fucking thing. Without an apology. What a cunt leprous schlong.*
No one knows now – except Porter – and we’re extremely unlikely to find out. What would a review, any review, accomplish? What would it tell us other than, perhaps, something questionable happened 30+ years ago, when the guy was a teenager, merely a trainee sadist. Which we already know.
Why bother? Why not focus on Porter’s much more recent and much more provable awfulness? We know what Christian Porter is, and his disgusting colleagues know what he is. It has been, and still is, up to Morrison and his fellow thugs whether Porter should be in charge of the country’s law. And it’s now up to them if they want a show trial or a cover up. But no one else needs it. No one with an ounce of humanity can believe that Porter should be responsible for a dog, let alone a country.
And what if Morrison and his thugs don’t want an inquiry? Then so be it. Porter can stay, as a dead, festering lizard hanging around the neck of Morrison’s revolting, thoroughly evil government. And if that’s not enough to hang the whole fucking lot of them, then we’re fucked anyway.
Leave the loathsome cunt leprous schlong alone.*
*) We do our best to keep sanctimonious twats happy.
The 2019 TIMSS results are just about to be released, and the question is should we care? The answer is “Hell yes”.
TIMSS is an international maths and science test, given at the end of year 4 and year 8 (in October in the Southern Hemisphere). Unlike PISA, which, as we have noted, is a Pisa crap, TIMSS tests mathematics. TIMSS has some wordy scenario problems, but TIMSS also tests straight arithmetic and algebra, in a manner that PISA smugly and idiotically rejects.
The best guide to what TIMSS is testing, and to what Australian students don’t know and can’t do, are the released 2011 test items and country-by-country results, here and here. We’ll leave it for now for others to explore and to comment. Later, we’ll update the post with sample items, and once the 2019 results have appeared.
The report is out, with the ACER summary here, and the full report can be downloaded from here. The suggestion is that Australia’s year 8 (but not year 4) maths results have improved significantly from the (appalling) results of 2015 and earlier. If so, that is good, and very surprising.
For now, we’ll take the results at face value. We’ll update if (an attempt at) reading the report sheds any light.
FURTHER UPDATE (08/12/20)
OK, it starts to become clear. Table 9.5 on page 19 of the Australian Highlights indicates that year 8 maths in NSW improved dramatically from 2015, while the rest of the country stood still. This is consistent with our view of NSW as an educational Switzerland, to which everyone should flee. We’re not sure why NSW improved, and there’s plenty to try to figure out, but the mystery of “Australia’s” dramatic improvement in year 8 maths appears to be solved.
OK, no one is biting on the questions, so we’ll add a couple teasers. Here are the first two released mathematics questions from the 2011 year 8 TIMSS test:
1. Ann and Jenny divide 560 zeds between them. If Jenny gets 3/8 of the money, how many zeds will Ann get?
(The second question is multiple choice, with options 0.043, 0.1043, 0.403 and 0.43.)
To see the percentage of finishing year 8 students from each country who got these questions correct, you’ll have to go the document (pp 1-3).