As flagged in a recent post, Western Australia’s SCSA is undertaking a review of and public consultation on the syllabi for two of their senior mathematics courses: Mathematics Methods and Mathematics Applications. The draft syllabi, prepared by a “working party” of unknown but we-can-take-a-very-good-guess constitution, are available here.
We have had no solid time to look but the following, from the draft MM12 syllabus, caught our eye:
But, sure, we don’t need no mathematicians.
*************************************************************
UPDATE (13/09/26)
I think for teachers this one can be a bit like the gorilla on the basketball court: one is so busy pondering the detail that one can miss the obvious. For mathematicians, the nonsense above is blinding, but it need not be blinding for others. In that sense, this post is similar to a notable WitCH (and, see this post). With hindsight, I probably should have made this one a WitCH as well.
I will try to lay it all out clearly without writing a lecture on the theory of caclulus.
The (second) fundamental theorem of calculus states that
(*) 
This will obviously only be true for suitable functions. So, what do we mean by “suitable”? There are less and more sophisticated/rigorous approaches. The unsophisticated approach is:
All functions we’re gonna try this on in school mathematics are fine.
Well, not hyperbolae and other asymptote-ish guys, but the standard functions are fine.
Others may disagree but I think this is a perfectly reasonable approach at high school. The kids are not gonna have to contemplate unsuitable functions unless we’re stupid enough to throw one at them, so why bother fussing it? Indeed, this is exactly the approach that SCSA takes in their current curriculum:
• 3.2.17 develop the formula 
and use it to calculate definite integrals
I can see no practical value in being more detailed that this statement.
Of course, there is mathematical and aesthetic value in introducing some more solid sense, in treating a theorem as a theorem. The problem is, if a theorem is tricky then there is a good chance that the resulting statements will either being incomprehensible or wrong: in their new, draft curriculum, SCSA has achieved both.
The SCSA statement, that the equation (*) is true “for any function f(x) continuous in the domain a ≤ x ≤ b” is simply false. A function being continuous does not promise us anything about the derivative of the function. The standard example is f(x) = |x|, which is nicely continuous but is not differentiable at x = 0.
Now, f(x) = |x| is not bad enough to falsify SCSA’s statement, since things are going wrong at just one point, and not badly wrong there. Such examples, however, can be much much much worse: it is possible for a function f to be continuous everywhere but for the derivative to not make sense anywhere (see here and here). That is, the derivative function can simply fail to exist. And, if a derivative function does not exist then it is damn hard to integrate it.
Are high school students going to run into such crazy functions? No, of course not. So, we can read SCSA’s statement to only apply to “suitable” functions. But we could have done that in the first place, just as we used to, and not have wasted our time with SCSA’s new-found but fake rigour.
But SCSA’s fake rigour is not simply a waste of time: it is also badly misleading. It is encouraging students to think about continuity and differentiability – topics that at least in some sense are part of the the curriculum – in a fundamentally flawed manner.
There is more that we could say here. We could take good stab at what SCSA thought they were writing. And, we could give a statement of the theorem that is correct and that works pretty well for high school. But it’s late, and perhaps that stuff is better left for a separate not-PoSWW post. Or, people can comment.
Looks like they need to smooth things out because they are on the cusp of something bad.
Jesus. Thank you for your contribution, Potii …
Couldn’t help it, my apologies.
Oh dear. I wish I could say I was surprised! You are correct – this sort of thing reinforces the need to involve mathematicians. In the meantime, I hope you tell them!
“I wish I were surprised.”
(Come for the maths, stay for the jokes, tolerate the grammatical pedantry.)
Thank you, Anna, and welcome to the blog. The story of me, and bigger shot mathematicians, attempting and failing to teach the SCSA some mathematics is long and hilarious and mostly untold. (Whether or not the story gets told depends upon how the next few months play out.)
Yes, I will tell the SCSA about the clanger above, and probably/hopefully/springs-eternally someone already has. But the point of the above clanger is not the clanger, which, to be fair, can be seen as a bit of a gotcha. The point is that the SCSA, and seemingly all but one such Australian education authority, is methodically tone deaf even when avoiding blatant error. For the obvious reason.
Ok, thank you for the warning. I’ll try to be more careful in the future. We have similar issues (mathematicians being excluded or ignored when it comes to K-12 curriculum matters) here in Canada!
Of course, and everywhere. I’ll be writing about Country X in the next day or so.
I wondered about this definition from the WA draft on Mathematical Methods.
“A discrete random variable is an outcome of a chance experiment that can take on distinct values
only.”
Well, that’s not good.
Thanks, Terry. How would you word it to be comprehensible and (close enough to) correct? Without excusing SCSA (since Hell has not yet frozen over), it does not seem so easy.
I have often said to students at school, that a random variable is an unpredictable variable. (They seem to understand this.) Then I say that if the r.v. assumes only values which are whole numbers then we say that it is discrete. Then I say that if the r.v. represents measures on a continuous scale then we call it a continuous r.v. (This is true – not the whole story, but IMO, enough for school students.)
In the definition from WA, it seems to me that “distinct” is not the right word.
Furthermore, one might ask what is “chance”? Poincaré defines chance as an “assemblage of complex causes” (Poincaré, 1914. Science and Method. Pantianos Classics, p. 48).
In stuff that I have been reading lately, there is often reference to “finite probability”, by which, I think the authors mean “positive probability”.
Thanks, Terry.
I think your casual definition is pretty good. Certainly, “distinct” is very much the wrong word.
A syllabus definition of calculus tripping over its own smoothness is a special kind of irony. Comments sections catching what review panels miss is becoming the standard QC layer.
Relying upon public consultation to control the quality of an “expert panel” is a very special kind of lunacy.
Any competent teacher of this stuff would realise the basic error. Your point is obviously valid, but this particular example is more appropriately an illustration of the lack of competency of the ‘working party’ than the necessity to involve mathematicians. The big question is how such ‘working parties’ are chosen.
Thanks, Pitbg (and *great* name).
You are absolutely correct: any decent senior teacher, and most of the higher apes, would be able to spot this screw-up. So, yes, you are making a stronger point, that the teachers/ed-academics chosen for these “working parties” are chosen via mate-networks rather than because of strength. The exact same is certainly true in Victoria, with the insidious VCAA-MAV cabal of stupidity.
Still, just one solid mathematician can make up for a hell of a lot of idiot weight.
Gibberish.
I don’t even know what they *wanted* to say. Given f’ continuous, compute the integral? Try
. Or did the want to compute the integral of a derivative (assuming 
is continuous)? Then 
is given, so what exactly is the task of the student? Stand in awe before the fundamental theorem of calculus, I presume.
I can see, Marty, that others have looked at this bit of mathematics, the “Use the relationship … blah blah … and calculate …”, and immediately gasped in disbelief.
Let me confess ignorance, because it is probably more than forty years since I integrated anything, and then only cautiously.
So my request is that you, or someone else, please explain the mathematical reasons for this being a mathematical abomination.
I suspect the educational bureaucrats who devised this, and/or signed off on it, hadn’t worked with integral calculus for even longer than me, and glanced at it and thought it looked plausible.
Please rub my nose, and the noses of the educational bureaucrats (can we sat apparatchiks?), in the mathematical proof that this is garbage?
My understanding (as just a maths teacher) is that just because a function, f(x), is continuous it doesn’t mean you can find the derivative, f'(x), over the whole domain the function is defined for (e.g. corners or cusp). This is a result of how a derivative is defined (using limits). So when you are integrating the f'(x) over the interval a to b, there may be a value within that interval that is not defined because f(x) was not differentiable at that point. So there is gonna be problems with teachers who take this at face value and teacher inadequately and exam question writers who write broken questions.
Hi, John, and sorry to be slow to respond, and thank you to Potii.
Potii basically has it right, but I’ll hammer more nails more firmly into the coffin. It’s probably worth replying as an update to the post, which I’ll do tonight or tomorrow.
In rare act of conscientiousness, I’ve updated this PoSWW.