It’s NHT, and I don’t teach for an inner sanctum edu-empire, so I don’t really care. But, others might, so here you go. Continue reading “The 2025 NHT Exam Discussion Post”
Month: May 2025
The Return of A, B and C
Clearly, readers got weary of the TNDOT exams (and I still have some tidying up to do). There is also still a bit to say about the Two Ladies Problem. So, I’m reluctant to keep going. But, Red Five has thrown the problem below at me.
The problem comes from an 1858 Cambridge exam, for “students who are not more than 18 years of age” and “who are not members of the university”, which presumably amounted to an entrance exam for senior school students. This particular exam was on “Arithmetic, Algebra, Trigonometry and Land-surveying”. Go figure.
Here is the problem.
A, B and C rent a field for £180. A puts in 12 horses for 5 months, and 45 sheep for 3 months. B puts in 15 oxen for 6 months, and 54 sheep for 2 months. C puts in 6 horses and 47 oxen for 3 months. Now 4 horses and 3 sheep together eat as much as 5 oxen and 1 horse; also two oxen eat as much as 7 sheep; how much of the rent should A, B and C pay respectively?
Ochre Education
I had a great title all lined up for this post: MediOchre. But Ochre turns out to be better than that, dammit, so I’ll play it straight. Continue reading “Ochre Education”
Should We Form a New Mathematics Association?
And, prior to that, are we a “we”?
Long ago, before it became a gutless and utterly predictable rag, Crikey did valuable work by annoying a lot of people. Crikey‘s subscriptions were never high and the politicians and big media tried to pretend that the upstart mag didn’t matter, but it was never convincing. Crikey was smart and feisty, beholden to no one and with enough of an appreciative following to matter. The powers that be read it, even if they pretended otherwise.
Continue reading “Should We Form a New Mathematics Association?”
Ed Barbeau’s “Two Ladies” Problem
I’m a bit flat out, and working on another post, but here’s a quick one. It is a lovely problem told to me by Ed Barbeau, who credits it to Andrei Toom:
Two ladies, A and B, live in separate towns. At the same time one day, each sets out on foot for the town of the other, walking at her own constant rate. They pass at noon. A reaches her destination at 4 pm, while B reaches her destination at 9 pm. At what time did they set out?
UPDATE (29/5)
This seems an easy problem to get tangled, although it’s not too bad if you set up the diagram as above. Then, the ratios of times are the same for each part of the trip (because the ratios of speeds are the same). That leads to the equation
T/9 = 4/T
So, T2 = 36, and thus T = 6 (and also T = –6, if we permit time travel).
I just wanted to point out an alternative, very natural approach that one of my students thought up. (He framed it as a guess-and-check thing, but it is natural to frame it as a proper algebraic solution.) What the student did was consider the fractions of the trips, for each lady, in terms of T. There are various ways to do this, and here is one way.
Consider Lady A on the the first part of her trip. She covers T/(T + 4) of the trip. Similarly, Lady B covers T/(T + 9) of the trip. But together these fractions add up to the whole trip. So,
T/(T + 4) + T/(T + 9) = 1
Then, solve away: multiply both sides by (T + 4)(T + 9), expand and cancel, and we end up with T2 = 36, as before.
Of course, this is a lot messier. (The mess is lessened by adding the second halves of the trips.) Nonetheless, this seems a conceptually simpler approach, even if it is more difficult to think of.
How to Improve a School – Part 1
Teach some mathematics.
Here endeth the series. Continue reading “How to Improve a School – Part 1”
Tony Gardiner’s Manifesto
A couple months ago, the comments on Post 1000 led to a discussion of manifestos on mathematic education. That inspired occasional commenter Solange Amato to link to a wonderful presentation by the irreplaceable Tony Gardiner. Gardiner’s talk was part of a conference, Thinking about Mathematics, held in Portugal in 2017; for a little about the conference, see here and here.
(UPDATE 23/11/25 Commenter JJ has emailed us, noting Anna Stokke’s podcast-interview with Portuguese mathematician, Nuno Crato. Crato was also Portugal’s Minister for Education from 2011 to 2015. It’s not exactly clear how this relates to Gardiner’s appearance in 2017, but it undoubtedly relates somehow.)
The video of the talk follows, and below that is my transcription (aided by auto-subs). The text is essentially verbatim, although I’ve tidied a sentence or two. I have not included Tony’s replies to a couple comments/questions at the end, although these are also of interest. Continue reading “Tony Gardiner’s Manifesto”
Learn 21st Century Mathematics With Justin Lanier
No, this is not that 21st century skills crap that people keep trying to sell us. It is learning something about 21st century mathematics, as invented/discovered by 21st century mathematicians.
Justin Lanier is an American (forgive him), a former schoolteacher and now a postdoc at the University of Sydney. For the last couple of years, Justin has organised a Zoom program, 21st century mathematics. The goal of the program is to enable school teachers, and pretty much anyone, to listen to and virtual-meet and learn from mathematicians involved in the business of proving theorems. This year, Justin’s program will take place in July. Continue reading “Learn 21st Century Mathematics With Justin Lanier”
What is a Script?
There is a famous story about the great comic playwright George Kaufman, co-writer of the books for the Marx Brothers’ two Broadway shows and then the screenplays for three of their great films. Kaufman was supposedly chatting with someone backstage during a performance of The Cocoanuts, when he abruptly stopped the conversation. Kaufmann apologised:
“Excuse me, I thought I heard one of my original lines.”
Not everyone is inhibited by a script. Then again, not everyone is granted or simply demands as much license as a Groucho or a Chico, and not everyone is the master of improv to so justify that license.