In 1973, the BBC televised The Ascent of Man, the brilliant series by Jacob Bronowski on the development of science and society. In his final episode, The Long Childhood, Bronowski sums up what he regards as special to being human, and the essence of a healthy scientific society:

If we are anything, we must be a democracy of the intellect. We must not perish by the distance between people and government, people and power, by which Babylon, and Egypt, and Rome failed. And that distance can only be … conflated, can only be closed, if knowledge sits here, and not up there.

That seems a hard lesson. After all, this is a world run by specialists. Isn’t that what we mean by a scientific society? No, it isn’t. A scientific society is one in which specialists can indeed do the things like making the electric light work. But it’s you, it’s I, who have to know how nature works, how electricity is one of her expressions, in the light, and in my brain.

And we are really here on a wonderful threshold of knowledge. The ascent of man is always teetering in the balance. There’s always a sense of uncertainty as to whether, when man lifts his foot for the next step, it’s really going to come down ahead. And what is ahead of us? At last, the bringing together of all that we’ve learnt in physics and in biology, towards an understanding of where we have come, what man is.

Knowledge is not a loose-leaf notebook of facts. Above all, it is a responsibility for the integrity of what we are, above all, of what we are as ethical creatures. You can’t possibly maintain that if you let other people run the world for you, while you yourself continue to live … out of a ragbag of morals that come from past beliefs. That’s really crucial today. You see, it’s pointless to advise people to learn differential equations, “You must do a course in electronics or in computer programming.” Of course not. And yet, fifty years from now, if an understanding of man’s origins, his evolution, his history, his progress, is not the commonplace of the schoolbooks, we shall not exist.

Bronowski spoke those words forty-seven years ago. Three more years.

This one is due to commenter P.N., who raised it on another post, and the glaring issue has been discussed there. Still, for the record it should be WitCHed, and we’ve also decided to expand the WitCHiness slightly (and could have expanded it further).

The following questions appeared on 2019 Specialist Mathematics NHT, Exam 2 (CAS). The questions are followed by sample Mathematica solutions (screenshot corrected, to include final comment) provided by VCAA (presumably in the main for VCE students doing the Mathematica version of Methods). The examination report provides answers, identical to those in the Mathematica solutions, but indicates nothing further.

UPDATE (05/07/20)

The obvious problem here, of course, is that the answer for Part (b), in both the examination report and VCAA’s Mathematica solutions, is flat out wrong: the function f_{k} will also fail to have a stationary point if k = -2 or k = 0. Nearly as bad, and plenty bad, the method in VCAA’s Mathematica solutions to Part (c) is fundamentally incomplete: for a (twice-differentiable) function f to have an inflection point at some a, it is necessary but not sufficient to have f’’(a) = 0.

That’s all pretty awful, but we believe there is worse here. The question is, how did the VCAA get it wrong? Errors can always occur, but why specifically did the error in Part (b) occur, and why, for a year and counting, wasn’t it caught? Why was a half-method suggested for Part (c), and why was this half-method presumably considered reasonable strategy for the exam? Partly, the explanation can go down to this being a question from NHT, about which, as far as we can tell, no one really gives a stuff. This VCAA screw-up, however, points to a deeper, systemic and much more important issue.

The first thing to note is that Mathematica got it wrong: the Solve function did not return the solution to the equation f_{k}‘ = 0. What does that imply for using Mathematica and other CAS software? It implies the user should be aware that the machine is not necessarily doing what the user might reasonably think it is doing. Which is a very, very stupid property of a black box: if Solve doesn’t mean “solve”, then what the hell does it mean? Now, as it happens, Mathematica’s/VCAA’s screw-up could have been avoided by using the function Reduce instead of Solve.* That would have saved VCAA’s solutions from being wrong, but not from being garbage.

Ask yourself, what is missing from VCAA’s solutions? Yes, yes, correct answers, but what else? This is it: there are no functions. There are no equations. There is nothing, nothing at all but an unreliable black box. Here we have a question about the derivatives of a function, but nowhere are those derivatives computed, displayed or contemplated in even the smallest sense.

For the NHT problem above, the massive elephant not in the room is an expression for the derivative function:

What do you see? Yep, if your algebraic sense hasn’t been totally destroyed by CAS, you see immediately that the values k = 0 and k = -2 are special, and that special behaviour is likely to occur. You’re aware of the function, alert to its properties, and you’re led back to the simplification of f_{k }for these special values. Then, either way or both, you are much, much less likely to screw up in the way the VCAA did.

And that always happens. A mathematician always gets a sense of solutions not just from the solution values, but also from the structure of the equations being solved. And all of this is invisible, is impossible, all of it is obliterated by VCAA’s nuclear weapon approach.

And that is insane. To expect, to effectively demand that students “solve” equations without ever seeing those equations, without an iota of concern for what the equations look like, what the equations might tell us, is mathematical and pedagogical insanity.

*) Thanks to our ex-student and friend and colleague Sai for explaining some of Mathematica’s subtleties. Readers will be learning more about Sai in the very near future.

This MitPY is a request from frequent commenter, Red Five:

I’d like to ask what others think of teaching (mostly linear) Diophantine equations in early secondary school. They are nowhere in the curriculum but seem to be everywhere in competitions, including the AMC junior papers on occasion. I don’t see any reason to not teach them (even as an extension idea) but others may have some insights into why it won’t work.

H. R. Currie and G. M. Currie, Open Access Journal

This one was brought to our attention by the Evil Mathologre. It is a tricky one, since it involves the work of a school student, and the student is in no way a target for our criticism. Out of such concerns, we haven’t made this post a WitCH; it should be considered in the same vein as this Maths Masters column.

As reported in Wagga’s Daily Advertiser a couple weeks ago, and as picked up by The Canberra Times, IB student Hugo Currie was given a “mathematics assignment” (presumably an Internal Assessment) on the golden ratio:

“… we had to investigate an element of the golden ratio in the built or natural environment so I decided to look at atomic structure …“.

Hugo considered the atomic mass number A (protons plus neutrons) of nuclides (isotopes), comparing A to the number N of neutrons and the number Z of protons. Of course, A = N + Z. Hugo then looked for “fibonacci nuclides”, nuclides for which the ratios A/N and N/Z are very good approximations to the golden ratio. He found a bunch, and suggested his results as a guide to hunting for new elements and nuclides. Hugo’s graphic above is a good illustrative summary of his investigation; the horizontal axis is N, the vertical axis is Z, and the black line indicates known stable nuclides.

OK, no big deal. From our perspective, having a class sent off to hunt for the golden ratio is asking for trouble, but it’s just an IA, and Hugo’s work seems interestingly exploratory-ish, in the manner the IB foolishly demands. But why did Hugo make the news, and what’s the problem?

Unsurprisingly but unfortunately, we can see little if anything research-worthy in the Curries’ paper, and we noticed a number of “Uh-oh”s. A fine IA, sure, but not a research paper, and not news.

We’ll leave it at that. Readers are free to hunt for the uh-ohs.