WitCH 36: Sub Standard

This WitCH is a companion to our previous, MitPY post, and is a little different from most of our WitCHes. Typically in a WitCH the sin is unarguable, and it is only the egregiousness of the sin that is up for debate. In this case, however, there is room for disagreement, along with some blatant sinning. It comes, predictably, from Cambridge’s Specialist Mathematics 3 & 4 (2020).

87 Replies to “WitCH 36: Sub Standard”

  1. Just to kick things off, the first thing I noticed is they chucked in the u’s without dealing with the original dx on that last image there (example 12). So I would say that integral equates to simply \frac{(2-u)}{u^2}x ,according to them. (I don’t know any LaTeX, so if someone could make that look pretty, be my guest.) (In fact, I’ll have a go myself.) And just noticed now they asked for “AN” antiderivatve, but what was answered is “THE” antiderivative, which is opposite to what I learnt in school that can lose you marks in exams. I guess technically you could say that “integral +c” is an antiderivative, but I was told horror stories about particuarly anal markers so I thought I’d point it out. And those “therefore dots” are weird. Haven’t seen them since year 8.

    Hi there, just an edit here to say that I believe the time mentioned above is adding an hour to the actual time, I wrote this circa 2AM, not 3AM. If that’s just an error on my machine then maybe Mr. Hyde can remove this small section here. Okay, it’s bedtime, bye for now.

    Just before this edit timer runs out, one of the tags for this WitCH is ‘trigonometry’ which I assume is an error. Or I’m probably out of my depth and just floundering here. Okay I’m gone for good this time.

    1. Thanks very much, Craig, and thanks for spotting the cut-and-paste “trig” error: fixed. Yes, the (bunch of u’s)dx line is confusing as all Hell, and what annoyed commenter David on the other post. There is an argument that the line has the meaning intended, but there’s no argument about whether the exposition is competent.

      Your point about the text asking for “an antiderivative” is less fundamental but perfectly valid. And, yes, given the nitpicking aspect of VCE grading, the point is important.

      As for the “therefore dots”, I loathe the things. They are standard in VCE, however, a plague of annoying little gnats.

  2. Probably not a huge issue (except to VCAA examiners) but when the text writes f(u) where u=g(x) they are talking about a composite function.

    Should there not be some justification that the composite function actually exists without domain restrictions?

    1. Hi, RF. That’s pretty funny. I guess in a technical VCE sense you’re correct, although it would drive everybody nuts for no gain.

    2. Is there any particular reason that composite functions are treated differently from the sum/diff/product/quotient of two functions, where if domain is not given it is implied, whereas
      for composite it basically has to be defined (unless it is all real)?

      One could say that f(x) = sqroot(x2-1) is undefined, because it is composite and x2-1 is sometimes negative.

      What if a student answers a question this way on SAC Exam?

      1. You mean any good reason? No. It is complete lunacy.

        Your example is not quite correct, but your pointing out the issue is. In VCE, the function f(x) =\sqrt{x^2-1} is defined, with implicit/maximal domain |x| ≥ 1. However, if you let \latex g(x) = x with domain \Bbb R, then in VCE the composition f \circ g is not defined.

        In practice, I have not seen this nonsense tested or nitpicked in VCE SACs or exams, and in fact the exams have screwed up on this point on occasion, with no one having ever noticed. But, others here are better qualified to say whether this is an active issue, or just dormant dumbness.

          1. Thanks for the reminder, RF. There’s a lot not to like about that question ….

            1) No domains for f or g are given, so one must assume a maximal domain of f and a maximal domain for g such that f(g(x)) is defined. In which case

            2) Part (b) should come before part (a). In which case

            3) Part (b) should ask for the maximal domain of g such that f(g(x)) is defined. In which case

            4) The preamble of the question should be

            “Let the functions f:  D1 \rightarrow R, f(x) = .... and g: D2 \rightarrow R, g(x) = .... where D2 is the maximal domain of g such that the function f(g(x)) exists.”

            In the above, D1 would be explicitly stated eg. [-3, 3]. Although personally I’d prefer something like [-1, 2] for reasons suggested below.

            What the question encourages (and the Assessment Report validates) is to get the maximal domain directly from the rule f(g(x)) rather than by considering D2 (to use my notation) such that the range of g is a subset of the given domain of f. I have seen so many students do the former even when D1 is explicitly given and is NOT the maximal domain of f …. This question just reinforces such ill-conceived thinking in students (and most likely misleads many inexperienced teachers. But then again, what else is new ….?).

            1. Thanks, JF. The obvious way out of this mess is, as banacek suggests, to always just take a composition to be defined on its maximal domain. For the life of me, I don’t understand why this approach isn’t taken. But this nonsense goes back, at least, to the 70s. I suspect it is an overly cute “pure” aspect inserted by the Monash Uni gang of the day. They were responsible for a hell of a lot of good, but there was a “New Math” aspect to them that didn’t help matters.

            2. Nothing is ever really new in VCE Mathematics.

              The very existence of this question though means that VCE methods teachers MUST teach restricting domains to compose functions AND IT SUCKS.

              1. What makes you say that RF (the second part of your comment)?

                I would have thought that it’s quite reasonable to expect Methods students to know that the maximal domain of \sqrt{9-x^2} is not the same as the maximal domain of 9 - x^2 (or insert your preferred example involving division by zero, logarithms, … ). What am I missing?

                1. SRK, the problem is as indicated in my reply above to Craig. In VCE, a composition function doesn’t have a maximal domain: it is either defined or it isn’t. Unless, that is, VCAA forgets this, as in in the exam question RF cited.

                  1. Well yes I appreciate that VCAA’s take on this issue is completely confused and confusing for students (and for teachers)

                    I have not taught Year 12 Methods, but when function composition has come up in 12 Specialist (which it generally does in the context of inverse or reciprocal circular functions) I just encourage students to use logic.

                    This is more a question for RF, but is the point that, in Methods, if f(x) = \sqrt{x} and g(x) = 9 - x^2, we must say that f(g(x)) is undefined, but there exists some restriction of g, call it g_r such that f(g_r(x)) is defined and has domain [-3,3]?

                    If that is what RF is getting at then now I completely understand why that sucks.

  3. Second and third images: There’s not much point in immediately stating that f(u) = e^u when this only becomes clear after the substitution is done. Better to just cut it out altogether, IMO, so that students don’t get the notion that they need to somehow immediately know it to be the case.

    Last four images: Students are asked to find an antiderivative, but the final answer in each example is an equivalence class of antiderivatives. Wouldn’t it be better to just directly have the examples ask “Evaluate the indefinite integral ∫3xe^x^2” and “Evaluate the indefinite integral ∫(2x+1)/(1-2x)^2”?

    Also, the workings in both examples take some large steps that might be confusing for the less-able students.

    1. Thanks, edderiofer. I agree about the function business. They seem to want f(u) there as some kind of symbol of integrity, but it’s not doing anything but adding to the confusion. if you’re gonna do function notation, that is most definitely not the way to do it.

      Your point about an/the antiderivative is the same as Craig’s, and is correct. I’m not sure there are two too (doh!) few steps in the working, and I was thinking in the other direction, but others are probably better placed to judge that aspect.

      1. For students first seeing integration by substitution, I would certainly present at least as many steps as this. Probably the only thing I would add is an initial line \int 3x e^{x^2}dx = \frac{3}{2} \int 2xe^{x^2}dx before replacing the x^2 with u.

        Of course, I would hope that with a bit of practice, students would be much more efficient. After writing u = x^2 and \frac{du}{dx}=2x, the last three lines is what I would expect a reasonable student to write down as sufficient working.

        But on the broader issue of how many steps to show when presenting worked examples, I wonder if there’s a segue here into the discussion about “cognitive load” occurring in another topic….

      1. When finding an anti-derivative (or derivative), domains are typically not required by VCAA unless explicitly asked for (but I agree that it’s good form to give them).

        In fact, should a student do so and make a mistake with the domain, VCAA will penalise that student even though the domain wasn’t required (over-engagement with the question and making a mistake in the process). This of course creates ambiguity and stress for both students and teachers. For example, when giving a final answer, what final form is required? Should denominators be rationalised etc. On the one hand, if you don’t ‘over-engage’, it might turn out that VCAA wanted you to. On the other hand, if you do ‘over-engage’, you might make a mistake and it might turn out that VCAA didn’t require you too – you lose a mark for trying to do doing something you never had to do.

        It’s a little known fact that in Specialist Maths, when asked for the equation of a line, in the absence of what form to give it in, VCAA require the y = mx + c form. They do not accept the perfectly good ‘slope-point form’ y – y1 = m(x – x1). Crazy. So here is an example of where you do have to ‘over-engage’ when it is not obvious. If you did not know this, you would obviously be encouraging your students NOT to ‘over-engage’ and risk making a careless mistake and hence unnecessarily losing a mark …. Note: You will only know this if you are either an assessor or you know an assessor who understands the manifest unfairness in keeping something like this a secret.

        Thinking about the domain of an anti-derivative becomes important when definite integrals are required. Unsurprisingly, VCAA has form for not doing this – presenting improper integrals (not on the course) for students to calculate in several exams.

        1. Hello John friend.

          Can’t agree with you more – you are absolutely right.
          Many VCAA questions lacked sufficient vetting, as in Spesh.

          For instance, you mentioned ‘improper integral’ being present in past exams – 2010, Q14.
          They were ignoring the fact that under a square root everything must be positive unless over C.

          As for the form of line, I think you are referring to 2016 Spesh exam 1, the line perpendicular to the curve at its y-intercept.

          ‘Avoid over-engagement’ should be taken as a suggestion made to low-mid range students which could save them some hard-earned marks, while creating some ‘convenience’ for exam 1 assessors, more of a method focus.
          But I really like your point: ” On the one hand, if you don’t ‘over-engage’, it might turn out that VCAA wanted you to. On the other hand, if you do ‘over-engage’, you might make a mistake and it might turn out that VCAA didn’t require you too – you lose a mark for trying to do doing something you never had to do.”

          Isn’t this a real life lesson? Opportunity costs. You take some risks, you earn the money, or you don’t.

          P.S.: I really appreciated for those exam solutions written by you (if you are the same John friend I am talking about who authored those suggested solution with great details and rigorous, in-depth discussions)


          1. Thanks for your comments and kind feedback, P.N. I probably am the one you’re talking about …. Unfortunately, the days of detailed solutions are over (as you might have noticed) – apparently a new paradigm is better.

            Re: “Opportunity costs. You take some risks, you earn the money, or you don’t”. I totally agree.

            Re: “Avoid ‘over-engagement’ should be taken as a suggestion made to low-mid range students which could save them some hard-earned marks.” Very excellent advice!

              1. Yes Marty. We all know!

                It’s so doomed, isn’t it?

                However, Small things will add up.

                At least (luckily) we’ve got a significant number of industrious classroom practitioners who worked to their best of abilities, attempting to give kids best they could.

                Of course, maintaining the rigorousness of Maths learning is very crucial, I agree. Especially to later studies.

                Abusing the operational convenience of u substitution is not very healthy, though it could serve the purpose and earn some coins for now (or probably not sometimes) because there are a lot of teachers who disagree with du=g’(x)dx operations or the similar sorts. Without understanding the nature of linear approximation (and linearity) it is dangerous for kids to play it that way.

                Let’s just admit it is a possible hedging strategy for some kids to earn their certificates, for now!

            1. Hello again John friend,

              Recently delved into some fancy SACs authored by you. Extremely intensive. However, Must say well crafted by such a master, because you have embedded many decent advanced maths knowledge well with VCAA maths. Excuse my bad analogy: what students have learnt in VCE are just raw material, like M8 steak. The fancy maths, such as complex integration with Euler formula, are perfect seasonings. Altogether they beautifully yielded very colorful flavours…Besides, the sense of humor and language used in those SACs are indeed intriguing! Even my wife likes it.

              Naively speaking, if only you were my Spesh teacher 12 yrs ago!

              You (as well as many other stakeholders) might have noticed the change for VCE maths in 2020, which was fresher released a couple of hours ago. Finally they made some decisions! Hopefully it would lessen some burdens from our kids and many teacher colleagues under such a difficult circumstance. They removed statistics from Spesh, sadly. However 22% for App task? For real? That’s too much!

              Hope you are doing well. Take care.

              P. N.

                    1. Particularly in Spesh, without the prob and stat this year, I really wish there would be some nice questions with more depths…such as good DE questions and calculus questions back in the old time…
                      Seemingly the end of year methods exams could share a lot of similarities with 1995-1999CAT exams and some 2000-2005 pilot exams…

              1. Ha. Well I did try. I’m glad to get some feedback that my efforts weren’t in vain. Thankyou.

                Yep, 22% for an App Task is too much. That’s as much weight as Exam 1! And 34% for two internally assessed tasks. Way too much. I worry that the stakes are even higher now for the kids – at least with 3 tasks they had more wriggle room to redeem themselves if they messed one up ….

                (Personally, I’d like to see SACs completely abolished – a discussion for another day).

  4. Not exactly on the point (indeed, not remotely on the point) but anyway here is an example of integration by parts.

    J = \int \displaystyle{\frac{1}{x}}\, dx  = \int \displaystyle{\frac{1}{x}} \cdot 1\, dx  = \displaystyle{\frac{1}{x}} x - \int x \left(\displaystyle{\frac{-1}{x^2}}\right) \, dx  = 1 + \int \displaystyle{\frac{1}{x}}\, dx = 1 + J.

    So J=1+J and therefore 0=1. QED

      1. Source: Maxwell, E. A. (1959), Fallacies in mathematics, Cambridge University Press

        Also many other interesting facts are proved.

    1. OK, at the risk of spoiling all the fun, I’ll point out the fallacy in the proof:

      An arbitrary constant of integration is missing. It should read:

      J = 1 + J + C

      which then mistake perfect sense since C must be taken as -1.

      I like to get my specialist maths students to solve \int 2 \sin(x) \cos(x) dx in two different ways but not including using the double angle formula and so ‘prove’ that 1 = 0 ….

  5. Re: Example 10, for example. I’m just gonna show what I’d teach my kids and damn the torpedos:

    Let I = \int 3x e^{x^2} dx.

    Substitute u = x^2 \implies \frac{du}{dx} = 2x \implies dx = \frac{du}{2x}:

    I = \int 3x e^u \frac{du}{2x} = \int \frac{3}{2} e^u du = \frac{3}{2} e^u + C.

    Back-substitute u = x^2:

    I = \frac{3}{2} e^{x^2} + C.

    Let the stoning begin ….

        1. Less queasy, but not much less. I prefer to keep my u’s on one side and my x’s on the other. Unless the integral is so hard I’m genuinely exploring, to see what might happen.

          1. Yeah, I forgot that it was u dx.

            Just to be clear though, are you OK with breaking up a fraction such as du/dx if an integral sign appears in doing so?

            1. Hi, RF. Yes. I’m happy with du = 2x dx along the way to doing an integral, although I think one has to argue for it (as David on the MitPY post). I’ll try to get to updating this witCH and the associated MitPY, just a couple posts I really want to do first.

  6. Does anyone really have fun integrating 3x\,e^{x^2} in this substitution way? All such procedures demand familiarity with integration, and by the time you’re familiar enough to think about them, you are surely familiar enough to look at 3x\,e^{x^2} and immediately see that a trial solution is e^{x^2}. Differentiate that to give 2x\,e^{x^2}; realise you’re out by a constant factor, then go back and multiply your trial solution by 3/2; that will multiply the derivative by the same amount. Done—in a few seconds with nothing written down. What about the other one, (2x+1)/(1-2x)^2? Here we might want to write things down. Look at the numerator and guess \ln (1-2x)^2. Differentiate that, scale it, and compare with where you want to end up. You’ll need to add something whose derivative is 2/(1-2x)^2. That is 1/(1-2x), found in one step by starting with a guess, 1/(1-2x). Am I doing a lot of guessing here? Maybe no more than guessing an appropriate u. And anyway, really tough examples might need to be broken up into simpler parts, which is essentially what I’m doing here.

    The calculational dead ends that result from starting with an inappropriate guess for u aren’t presented in the answers above, and that only makes them look shorter and sweeter than what they probably were in raw form when originally worked out. In integration, the end justifies the means, and as long you can differentiate your result to get back what you started with, you’re done. The “guessing” process I mentioned here has that check-by-differentiating built in.

    I think that if students are told that it’s okay to start with a guess and just differentiate it to see how well they did, and then refine that, it would make the whole thing more exciting, like a game to get better at. And the less refining they have to do, the more they might feel like they’re learning something. And starting with a guess is probably how most “serious research” is done anyway. So it’s a good lesson for them.

    (By the way, did anyone notice the lack of parentheses around 2u^{-2} - u^{-1} in the answer to Example 12?)

    1. Thanks, Don. I’m not quite sure what you’re arguing. For your example, are you unhappy with students/teachers setting \latex u = x^2 in some form and then working from there? How would you suggest approaching something like \int e^{\sqrt{x}}\,{\rm d}x ?

      1. Marty, I’m not sure your example makes the point that I think you’re implying. I think it actually makes Don’s point!

        After a false start (see 2 below), there is a ‘guess’ – not too unobvious to think of – that might be tried. This is consistent with Don’s approach of “start with a guess and just differentiate it to see how well they did, and then refine that”.

        The particular guess I have in mind works very nicely and makes short work of the integral. On the other hand:

        1) Making the obvious substitution results in an integral that then requires integration by parts – beyond the scope of the course (for some stupid reason).

        2) Assuming an obvious ‘guess’ for the answer leads to a differential equation that must be solved using the ‘integrating factor’ method – beyond the scope of the course (again, for some stupid reason). But, looping back to the start of my comment …

        However, having said this I must say that I disagree with Don’s “start with a guess and just differentiate it to see how well they did, and then refine that”. I think this is very inefficient for all but simple cases. Games can be fun, but when you’re in an exam and only have 6 minutes to solve an integral worth 4 marks, a student foolish enough to play this game will be destroyed. Game Over.

        I think an example that makes this point is an integral that appeared on the 2017 NHT Specialist Exam 1 (Q11): Students had to integrate \displaystyle \sqrt{1 - \cos(\theta). There is a simple substitution that perpetrates this integral almost immediately. I’d hate to be playing guessing games.

        1. As I wrote, I’m not sure what point Don is making. I know what point I’m making, however, and I think maybe you just disagree. My point is that at times you want to, or need to, perform a substitution without knowing how or whether it will work. I think this point is illustrated very well by \int e^{\sqrt{x}}.

          As for integration being a guessing game, well, it is a guessing game. Yes, one can learn many types, new tricks and develop a better sense for the game. But it’s still fundamentally a game.

          1. Edit: Sorry, John Friend is accidently Annonymous – I forgot I deleted all my cookies, cache etc. because of a problem created by a DEET migration. John Friend says:

            OK, I understood your point, up to a point.

            I thought Don was saying that substitution should be avoided, and that integration should be treated like a guessing game where you guess the answer, see if it works and then refine.

            This is very different to guessing a technique such as substitution and seeing if it works. So I thought the point you were making (which I agree with) is that “at times you want to, or need to, perform a substitution.” Rather than exclusively guessing the answer etc.

            I misunderstood the subtler point you were making – that a substitution is often made “without knowing how or whether it will work.” I also agree with this, and I agree that your example illustrates this very nicely.

            But I still claim that your example illustrates Don’s point better than the point that “at times you want to, or need to, perform a substitution.” Which was my point! I think the example I gave leaves Don very little wriggle room and I look forward to his/her response.

            I totally agree with you that integration is fundamentally a guessing game. Your comment that “one can learn many types, new tricks and develop a better sense for the game” is also correct. But I contend that at the VCE level (which is the context of the discussion in this thread), the game is completely rigged – the capable student is not guessing, s/he knows exactly what technique to use and that it will work.

            The deeper discussion in my opinion should be the fact that VCE encourages the complete myth that most integrals can be done, you just have to be clever enough. This is disgraceful. The deeper discussion should be whether or not basic theorems (no proofs, just theorems) for proving particular classes of indefinite integrals to be non-elementary should be included or at least commented on. The application of such theorems is within the scope of the current Specialist Maths course. Such a discussion might also include examples of where the indefinite integral \displaystyle \int f(x) \, dx cannot be done but the definite integral \displaystyle \int_{a}^{b} f(x) \, dx can be done.

            1. Thanks, JF. Do you want me to change the author of the last two “anonymous” comments to John Friend?

              Of course you are correct, that integration in VCE is paper-thin, and so mechanical and meaningless. I don’t think it is obvious how or whether calculus should be done in senior school, but it is obvious that VCE is just about as bad as one can make it.

      2. My point is that integrating 3x\,e^{x^2} is a classic example of a task that is done trivially with a guess, and yet I don’t see anything in the textbook discussion that suggests that “guessing plus iterative refinement” is a valid way to do an integral. Of course, maybe the book does suggest precisely that elsewhere.

        Okay, 3x\,e^{x^2} might have been given merely as a simple demonstration of the substitution technique. And that’s the question here. Is making a substitution being sold as -the- method for solving this integral, when the integral is clearly extremely guessable? That is, is any room being left in the curriculum for pupils to guess anything? I hope so; after all, guessing is built in to some of our most basic maths. If I ask you “What is 97 divided by 13?”, I’ll bet that you’ll guess an answer based on your previous experience of 2-digit numbers, check it with a multiplication, decide if the remainder is greater than 13, then—if necessary—refine your guess. We should be encouraged to do similar with integrals. Many of the integrals that I’ve done over and over again in four decades of integration have been standard ones that are easily tackled by a guess plus iterative refinement. It’s a handy technique that I’ve probably never observed in others whom I’ve asked “What is this integral?” (although I live in the world of physics, not pure maths). Ask someone to integrate \sin 2\theta, and there’s a good bet they’ll furrow their brow as they try to remember an appropriate rule, instead of just guessing \cos 2\theta, then quickly seeing that this differentiates to -2\sin 2\theta, followed by a quick “divide by -2” correction to arrive at the result of -1/2\,\cos 2\theta. Maybe school pupils will feel that they must tackle that integral with the substitution u=2\theta.

        Of course, I’m not saying guessing will always work, and I certainly appreciate substitution. When I first learned substitution in my school years, I would spend hours doing complicated integrals of my own choosing that involved several levels of substitutions within substitutions. So I know all about the niceties of the technique! Even so, many (most?) of the standard integrals that we meet time and time again can be done with a guess, and so I think that this is a technique that should be given some air time.

        And since I’m being challenged to integrate e^{\sqrt{x}} by guessing, let’s give it a go. A first guess is e^{\sqrt{x}}. Differentiate to give e^{\sqrt{x}}/(2\sqrt{x}\,). (I do that on paper by drawing a down-pointing arrow below the e^{\sqrt{x}}: what’s above is my suggested integral, and what’s below is its derivative.) Hey, we’re out by a factor of 2\sqrt{x}, so multiply the original guess by 2\sqrt{x}. Differentiate again to give e^{\sqrt{x}} + e^{\sqrt{x}}/\sqrt{x}. Nearly there; but didn’t we see almost that extra term just a moment ago with the first guess? So subtract e^{\sqrt{x}} and differentiate; no, inspection shows that we should’ve subtracted 2e^{\sqrt{x}}. And now we’re done: the answer is 2\sqrt{x}\,e^{\sqrt{x}} - 2e^{\sqrt{x}}, or 2e^{\sqrt{x}}(\sqrt{x}-1). Sure, I could’ve done the same job neatly by guessing u = \sqrt{x}, but that’s not my point. My point is that I think “guessing plus iterative refinement” should be encouraged. It’s how many unrelated problems are tackled, both in maths and in other subjects. It’s a good skill, but unfortunately the word “guessing” has bad connotations. Substitution requires guessing too, but I’m not sure that anyone uses that label when they teach it.

        1. It’s a pleasure to see a virtuoso at work, Don. I did warn Marty that his example would only make your point lol! (But I misunderstood half of Marty’s point).

          I really do like what you’ve posted – and I agree that it should be a string that students are taught to add to their bow. I actually do teach this, but only very superficially (like integrating \displaystyle \sqrt{2-3x }) where you’re simply trying to get the ‘coefficient out the front’.

          Marty’s example is interesting because making a substitution will not work for a Yr 12 student. But making an intelligent guess and then intelligently refining it will work. However, the exam strongly discourages this sort of thinking – it encourages rote learnt procedures that are certain to work for a capable student.

          Although practice and experience makes such thinking routine, the cognitive effort and time investment for an average Yr 12 student compared to choosing and executing a substitution makes it a very low-percentage option for most students (particularly in an exam). The required thinking process will be beyond most students – it will just look like black magic. And it assumes that most students can accurately and efficiently differentiate! (a dangerous assumption!)

          Plus, students have to show ‘working’ – can you imagine a student who can think this way trying to explain it on paper. VCAA would be too dumb to understand what the student was trying to explain (and it’s not the easiest thing to try and explain on paper anyway). Creativity is not encouraged in the VCE – ‘recipe-based’ thinking and button pushing is the rigueur du jour.

          Remember that smart kid in the Day After Tomorrow who failed his calculus test even though he got every answer right: https://www.moviequotedb.com/movies/day-after-tomorrow-the/quote_33978.html

          (Yep, I know it’s just a movie but I wonder how many times this does happen in real life).

          I’d love to see how you would use this approach in the example I mentioned in my earlier comment: \displaystyle \sqrt{1 - \cos(\theta) (from 2017 NHT Specialist Exam 1 (Q11). (However, I will understand if your response is that like all techniques, some work better than others and this example is one of those times when substitution works best).

          Re: “(although I live in the world of physics, not pure maths).” I wonder how many students – physics or maths – get taught ‘differentiating under the integral sign’ (a technique popularised by Feynman) for solving certain types of definite integrals?

          1. JF, would you mind explaining a bit more about that 2017 NHT Question, and in particular why you think a “substitution works best”? Perhaps I am misunderstanding exactly what you mean by a substitution, but I would have thought this one is most easily done using 1-\cos(\theta)=2\sin ^2 \left(\frac{\theta}{2}\right), and if I recall correctly, the limits of integration only required the positive root. Of course, noticing that the double angle formula could be used is a different matter.

            1. Hi SRK. I simply noticed that if you made the substitution \theta = 2t then you got a double angle and from there a wonderful simplification using a cosine double angle formula jumps out at you.

              It would have been good to get student performance data on this question, but the NHT Specialist Maths Examination Reports are among the laziest reports I’ve ever seen – they are not reports, simply a collection of answers from the ‘back of the book’ (so to speak).

              What might be very interesting is how students handle \displaystyle \int \sqrt{1 - \sin(\theta)} \, d\theta. There are two simple substitutions that make short work of it …

          2. Thanks for the compliment, John. I didn’t see any obvious first guess for \sqrt{1-\cos\theta}. We can call in the half-angle formula, or use the substitution u=\cos\theta, which takes a little more work. Both approaches require us to pay attention to relevant signs, and both require simple integrals that can be done by guessing. Again, my point is only that I think students should be encouraged to guess. I agree that guessing is not a straightforward thing to examine. But that doesn’t mean we should teach only techniques that can be examined. Maybe students need permission, so to speak, to let their minds roam when attempting to solve a problem. It’s a good life lesson, after all.

            I’d like to think that university students in physics do get taught differentiating under the integral sign, but I guess that most don’t these days. Or, if they do, they get taught it in a course on statistical mechanics, and courses on that subject have a reputation for being wind-blown dry (except, I like to think, the one I gave a decade ago). The students have long gone numb by the time they are taught that technique.

        2. Thanks, Don and JF. I’ll figure out my thoughts and will reply tomorrow.

          Also, JF, like SRK, I was puzzled why you were arguing for substitution for \sqrt{1-\cos \theta}, rather than just using the double angle formula. The substitution wasn’t obvious to me, and it wasn’t obvious to go to substitution as a first resort, although admittedly it works nicely.

  7. OK, sorry to be slow to have gotten back to this. (And, I do really want to update the MitPY and this post, and be done with them.)

    Don and JF, I agree with the value of guessing and, if need be, re-guessing, and so on. But I also think there is value in written method, whether or not this is part of some form of “guessing”. I’m lazy, and it doesn’t take much integration fiddliness for me to be more comfortable with employing a substitution, even a trivial scaling. So, yes, I’d probably “guess” the antiderivative of 3xe^{x^2}. But for \latex \int 3xe^{5x^2}, say? Probably not, even though I know I could.

    More to the point, this post is about VCE and I think it is perfectly fine for VCE students to want employ simple substitutions to clarify or solve an integration. This is the first time they’ve seen substitution and, because of the CAS-idiocy and the over-teaching of special cases, they don’t get nearly enough practice. Consequently, and anyway, what is obvious for Don and JF and Marty is in fact a “guess” for many Year 12 students. Many, many specialist students will think of the substitution u = x^2 for \int 3xe^{x^2} as a guess, rather than the obvious method/key. Even if it’s reasonably obvious, many will also be insufficiently practised in the mechanics and the consequences to want to perform some mental version of it.

    Of course being OK with students spelling out such details is not the same as teachers expecting it or VCE graders demanding it. Textbooks will always have a tricky balancing act, since they will want clear method with many steps for weaker students, particularly early on, but they should also want to be pushing the deeper conceptual sense of the process. And, yes, all the VCE textbooks suffer from pushing too many steps and too little thought. But it’s difficult.

    None of this excuses the cartoon treatment of integration in VCE, and there’s more room to argue about what is and should be done at university. But I’m fine with gentle substitution as the norm in VCE.

    Don, your “guessing” of \int e^{\sqrt{x}} was impressive, and a little weird. Of course, you’re not really arguing against substitution in this case but, to hammer the point, what would you do with \int e^{x^2}?

    Quickly on differentiation under the integral sign, my experience is that most upper level maths undergrads have picked it up somewhere along the way, although it’s not clear where. What none of them seem to have picked up, however, is that the technique can give the incorrect answer.

    Finally, a little note on \int \sqrt{1-\cos \theta}. Like SRK, I was puzzled by JF’s comment, since the substitution \theta = 2t seemed unnecessary, at least as a first step. (I was forgetting, of course, that VCE students’ trig skills also suck.) Puzzling over this, it turns out u = 1 + \cos \theta works nicely.

    1. Marty, the imaginary error function is an obvious first guess. Then you simply have to scale it. lol! Play fair! VCAA’s integrals can always be done using elementary functions (but not always within the scope of the course, and the Study Design contains nothing to discourage the misconception that in fact all integrals can always be done).

      Re: Your little note. And you wonder about my li’l old substitution! Sheesh. (But of course you’ve hit the nail on the head regarding trig skills – spotting a ‘half-angle’ formula is above the pay-scale of many Specialist students. My substitution simply accepts this and slaps them in the face with the obvious).

      u = 1 - \cos (\theta) works almost as nicely, and is probably a more obvious choice. u^2 = 1 - \cos (\theta) also works nicely. But in all three substitutions:

      1) I think you have to worry about the signs of the various square roots you meet along the way.

      2) ‘Nicely’ still requires getting \sin (\theta) in terms of u, which might prove a bridge too far for many students.

      3) If it was a definite integral (which it was on the NHT Exam – the lower and upper integral terminals were \displaystyle \frac{2 \pi}{3} and 2 \pi respectively) then these substitutions might work much less nicely.

      1. JF, I was playing fair. There is zero probability that Don would be unaware that \int e^{x^2} cannot be integrated in elementary terms.

        But for me that’s the point. I have no more intuition for \latex \int e^{\sqrt{x}}. So unless I happen to know an integral is impossible, it seems much more, um, methodical, to try various substitutions, including in each case the obvious one, to see what comes out.

        1. Indeed. The pity is that students do not get exposed to integrals like this in the context of understanding that some integrals just cannot be done in elementary terms regardless of how clever you are.

          On a related note, how many students are aware of the existence of formulae for solving cubic and quartic equations but that it’s impossible to solve in general quintic or higher? VCAA’s study design reminds me of Leunig’s “The Way Life Is Supposed to Be” cartoon.

    2. Of course there is value in the written method, even though it starts with a “u = \dots” guess too, albeit probably an obvious one. But consider proof by induction, which I presume is still taught at school. We present a statement, something maybe completely weird looking, and say “We’re going to prove it by induction”. Where did it come from? Was it generated by some other method and hence is certainly true, and we’re just using induction for the exercise? Maybe, but I’d like to think that the conjecture was generated by an educated guess, and now we’re going to prove it by induction. The end justifies the means. Same with guessing an integral—but call it “conjecturing” instead, because “guessing” has a bad connotation of being lost in a wilderness, whereas “conjecturing” has a suitably mathematical sound. We conjecture an answer in some dark and mysterious way, and then attempt to prove it by differentiating. I imagine that many pupils wonder about induction, asking themselves “That’s all very fine, but where did you get the original conjecture from?”

      Of course, conjecturing is not mindless. It’s an art, and part of the art is knowing when to stop. If you ask me to integrate e^{x^2}, and supposing I don’t know anything about erf or erfi, I’ll try a guess, and on seeing that the result only grows more complicated when I iterate, I’ll stop right there and try a substitution. (In fact, iterating from a straightforward guess looks to be generating an asymptotic series; useful to investigate, but not at school.) I don’t expect school pupils to know how far to push a guess. So, I would certainly say “If it doesn’t work in a step or two, turn to substitution”.

      But the fact is that most of the integrals I’ve ever done in my day job have been do-able with a guess. We’re not talking integrating e^{\sqrt{x}} or e^{x^2}. Just the simple functions that turn up again and again. If I’m in the middle of a Fourier expansion and I have to integrate a sinusoid, I’m going to make a guess that is refined and proved in 5 seconds of scribbling in the corner of a page devoted to such scribbles on my desk. Chances are, that’ll be sufficient. I’m well aware that this is not easily examinable. If I were a school teacher, to a pupil who was clearly headed down the maths road, I’d say “-I- wouldn’t do this particular example in the very powerful substitution way that I’m teaching you. That’d be cracking a peanut with a hammer. I’d do it more simply; but okay, I have some experience in these things that you haven’t had time to acquire yet. One day you’ll realise that you too have acquired some experience, and then you’ll do it more simply”. As I said before, I’m continually amazed that the physicists and engineers I deal with don’t have a ready ability to guess fairly straightforward integrals. I suspect this is not because they don’t know how to guess; rather, I think it’s because no one ever gave them permission to guess. Or to conjecture? That’s a frighteningly pure-mathematical word!

      Marty, about your comment on differentiation under the integral sign, “my experience is that most upper level maths undergrads have picked it up somewhere along the way”, I think you might’ve misunderstood what John spoke of. He referred to the way in which one might evaluate \int_0^\infty x^{10} \exp -x^2\,\textrm{d}x. Here, the “Feynman approach” is to begin with \int_0^\infty \exp -ax^2\,\textrm{d}x = 1/2\,\sqrt{\pi/a} for at least a positive, and then differentiate both sides five times with respect to a. At the end, set a=1. I once read a web story about a lecturer who had done this in an integral that had \pi in place of a from the outset. That lecturer had effectively set a=\pi at the end by starting with \pi in the role of a and “differentiating with respect to \pi“.

      And on a last note, it’s nice to hear John say that an imaginary error function is an obvious first guess. Gaussian integrals turn up again and again in physics, and so it has proved useful to me many, many times to memorise \int \exp (-ax^2 + bx)\,\textrm{d}x = 1/2\sqrt{\pi/a}\,\exp[b^2/(4a)]\,\mathrm{erf}[\sqrt{a}x-b/(2\sqrt{a})]. I don’t know if anyone else memorises that one, even though it turns up so frequently. I just know that if ask a fellow physicist “What is \int_0^\infty \exp (-2x^2+3x+4)\,\textrm{d}x?”, they’ll reply “I’ve seen this before. Isn’t it some trick about switching to polar coordinates? Do you complete the square?” Sure it is, but that doesn’t help them evaluate it.

      1. Hi Don. I hope you plan to follow Marty’s blogs and comment regularly.

        Thankyou for clarifyng the ‘Feynman approach’ – that’s exactly what I was referring to (for the benefit of the non-physicists, Feynman learnt this approach at high school from the book Advanced Calculus (by Woods) given to him by his high school physics teacher Mr. Bader).

        I hadn’t heard the ‘a = pi’ story before (or maybe I have and just forgotten it over the years), it’s very funny. It reminds me of another unrelated story (probably somewhat exaggerated):

        The great British mathematician G. H. Hardy was giving a lecture during which he said “It is obvious that…”, paused at length in thought, and then excused himself from the lecture temporarily. Upon his return some fifteen minutes later he said “Yes, it is obvious that….” and continued the lecture.

        And yes indeed, those Gaussian integrals are everywhere (especially in Stat Mech!) It’s quite amazing.

        1. Thanks John. I know nothing of the details of current school curricula, and hence much of what is written in this blog is a little beyond my ken.

          Mind you, I’m surprised that something (induction?) can be in a syllabus, yet not taught. But okay, I went to school in a different era (1970s). When I was 8 we all learned base 2, base 8, and modulo arithmetic. When I was 9, our textbook contained a proof that a circle’s area is \pi r^2. It was based on cutting the circle into wedges and placing them in an alternating pattern, and then noting that as the wedges were made smaller (so that their number got larger), their arrangement tended towards a rectangle with side lengths r and \pi r. It was an early introduction to calculus; but I bet they don’t do -that- anymore. I don’t know what my classmates thought of such things, but I just took them as given, with no need to get stressed about or to ask why I was learning them. I do think, though, that the curriculum should lose some of its obsession with the geometry of triangles and circles (if indeed it hasn’t already). Okay, there are some nice theorems there, but there are plenty of other nice things to be studied too.

          I’ve heard the story about Hardy; not that I could’ve told you it was Hardy.

          On Marty’s note below that the differentiation under the integral sign is not always valid, I presume the exceptions lie in areas lacking uniform convergence. I agree that it’s always important to be aware of such possibilities for things to go awry.

          1. Re: “I know nothing of the details of current school curricula, and hence much of what is written in this blog is a little beyond my ken”.

            Don’t let that stop you! (It doesn’t stop VCAA). Perhaps be a lurker and chime in when the occasion arises. Your re-collections of early 70’s maths classes were really interesting – I enjoyed them. Did you go to school in Victoria? Or elsewhere?

            Re: Convergence. More or less. And Marty’s note is fair enough. Yes, you definitely need to know the restrictions under which something is valid. But one could apply his note to, say, finding the limit of a composite function (in particular, the ‘trick’ used to find limits of the indeterminant form \displaystyle 1^\infty). Or the Conjugate Root Theorem (if we want an example closer to Specialist Maths) ….

            I suppose the difference is that ‘differentiating under the integral sign’ is usually taught purely as ‘trick’ (when taught at all) (and you learn lots of great tricks in physics) rather than a mathematical theorem, and that counter-examples (where the restrictions fail) are typically not given to show that it cannot always be validly used.

            A good review of ‘differentiating under the integral sign’ (for those who are interested) which includes counter-examples is given here:

            Click to access diffunderint.pdf

            1. Thanks, JF. Very nice article. And yes, my point is that DUTI is taught only as a trick. It’s less than not taught as a theorem: it’s taught as if no theorem need be considered.

            2. John, to answer your question of which school I went to. My schooling was in Auckland, as was my university education. I do wonder about how much of what I learned is no longer taught, both at school and in university. A friend proudly shows me an “advanced” maths textbook that his primary-school son is learning maths from. I look at the book and comment politely on it, while quietly being struck by how un-advanced it is, compared to the maths that my primary school taught us as a matter of course when we were that kid’s age. But this book has lots of colours and photos, and apparently that counts for quite a lot nowadays.

                1. Marty, I suspect it’s because Don is a gentleman (and would be too gentlemanly to say so), unlike most of the trouble-making riff-raff who post comments (case-in-point).

                  I don’t know much about Aukland’s current education system, and even less about what it was in the 70’s. Except to say that it appears superior to Victoria, both then and now. Possibly you were at an exceptional school. Certainly, my maths memories of the 70’s in Victoria don’t much match with yours – we played a lot with cuisine rods.

                  I agree that most early secondary school maths books nowadays have far too much colour, photos and pretty pictures. As well as far too many contrived ‘real life’ (ie. artificial) questions. These books would be at least 50% slimmer if this stuff was removed. I assume it’s included to make the maths more ‘attractive’ and ‘relevant’ to students. It’s a real shame that the maths is not considered attractive and interesting enough to stand on its own two feet. But that’s what happens when you have a boring as bat-shit curriculum. How many years in a row can you include descriptive statistics, basic probability etc. before it’s called torture?

                  I’ve recently had reason to work through a long list of Yr 10 questions from Russia, all to be solved ‘by hand’. I’m happy to say that I think I could scrape by teaching at that level. But only barely. It’s astounding what those Russian students are studying, and more astounding to wonder what they have already studied. In fairness, these questions came from what is probably one of Russia’s top ‘technical’ schools. But still …. there is nowhere in Australia that comes close to matching it.

                  Here are two of the easier questions (remember, this is Yr 10):

                  1) Solve \displaystyle \sqrt{3} \sin (x) - \cos (x) = \sqrt{2 - \cos (2x) - \sqrt{3} \sin (2x)}.

                  2) Find the values of a for which \displaystyle x^4 - ax^3 +(3- 2a) x^2 + ax + 1 = 0 has positive and negative solutions.

                  (By the way, I can supply my answers, but I don’t guarantee they are correct (in the sense of being a maximal solution set)).

                  I’m certain the first question would be considered too difficult for a Specialist Maths Exam 1, and the second would be considered too difficult even with ‘scaffolding’ for a Methods Exam 1. Nevertheless, I’ve written a scaffolded version of the second question for a Methods Exam 2 trial paper and will be very interested in the feedback I get from the vettor(s).

                  It appears that the { } and [ ] functions are also on the course, with questions asking to sketch the graph of y = {sin (x)}, for example.

                  1. Is there a freely accessible source for these JF?

                    I’m genuinely intrigued by the Russian curriculum given these examples!

                    That said, looking at some HSC papers from the 1970s compared to today’s VCAA exams is sobering at best.

                    1. Hi RF. No, there’s no public source – I received them via private communication. But if Marty wanted to create a post “From Russia With Love” for example I could post a few more and some interesting discussion might be had …. The two I’ve already posted, and the two below, could be included.

                      A lot of the questions required the calculation of limits. Most required l’Hopital’s rule (for the indeterminant form \displaystyle \frac{0}{0}) or variations of it (eg. for the indeterminant form \displaystyle 1^{\infty}). For example

                      \displaystyle \lim_{x \rightarrow \infty} \left(1 + x + \sin (x) \right)^{1/x}.

                      Yes, using l’Hopital is mindless, but you still have to be able to differentiate using the product and chain rules.

                      Some of the limit questions had the indeterminant form \displaystyle \infty - \infty.

                      Some of the limit questions involved recurrence relations eg.

                      Find \displaystyle \lim_{x \rightarrow \infty} x_{n} where \displaystyle x_1 = \sqrt{2} and \displaystyle x_{n+1} = \sqrt{2 + x_n}.

                      And of course to find these limits rigorously requires our good friend induction!

                      It looks like standard limits such as \displaystyle \lim_{t \rightarrow \infty} \left(1 + \frac{1}{t} \right)^t = e, \displaystyle \lim_{x \rightarrow 0} \frac{\sin (x)}{x} = 1 etc. were assumed.

                      A high standard of algebraic manipulation was required – a standard I’d only expect to see set for capable 1st yr uni students.

                      All this at Yr 10 level (15-16 year old students). And representative of only a small sample of their course. It was a real eye opener – definitely 1st year uni level.

                    2. HI RF, and JF. I’m happy to set up such a post, although I’d like to have a sense of where the school is in the Russian system first. JF, maybe email me details (again).

                      Also, RF, I haven’t forgotten the MitPY post you requested. Will do ASAP, but am neck-deep in essays right now.

                    3. Erratum: The 1 + x + sin(x) limit I posted should be as x –> 0 NOT infinity. (Copy and paste fail on my part).

                2. Marty, my comments were polite because (a) I can’t rain on his parade, and (b) since I don’t know what is being taught nowadays, then for all I knew, the book really was advanced for its level.

                  To address John’s comment about exceptional schools, I presume the 1970s was a time when all schools were similar. Nowadays there seems to be a huge range of quality of curricula in schools. Maybe someone who knows the details of how the laws have changed can say something about that. For example, changes to laws governing whether one could attend an out-of-area school might have had a highly non-linear feedback on the quality of curricula and teaching.

                  1. Well, to play Devil’s advocate for a moment (or a lifetime):

                    a) Of course you can rain on his parade. You simply chose not to.

                    b) I don’t believe you. It sounds like you were convinced, and I’ll bet correctly, that the book was nonsense or simple, or both.

                    To be a little more serious, I’m not suggesting that you should have gone with “Mate, this is shit.” You don’t have to storm on his parade. But that doesn’t mean a light shower, and maybe suggesting darker clouds nearby, can’t be well-mannered and extremely helpful. As it is, you left him thinking, you believe incorrectly, that his kid is getting an “advanced” education? I honestly find this puzzling.

                    If my friend George raises a topic with me then I’ll try to evaluate the topic, evaluate my level of expertise, evaluate George’s investment in their opinion, and evaluate the importance to George of their opinion being queried or outright challenged. Then, subject to the perceived constraints, I’ll try to indicate, politely, what I honestly think. Sure, there’s lots of ways too screw that up, but i really don’t understand friendly conversation in any other way.

      2. Thanks, Don. I’m not disagreeing with the value of your approach, just the timing of the emphasis in teaching it. On differentiating under the integral, I knew what you and John meant, and my note about the differentiation not always being valid still applies. But, yes, it’s probably true that many fewer students know about it as a technique to evaluate definite integrals. As for induction, to my knowledge it has never been a part of the Victorian curriculum. (No, for whoever plans to object, Spec12 does not count.)

        1. And, as far as I know, proof by induction is only taught at Melb. Uni. after 1st year (in some pure maths subject, probably Analysis).

          I’m curious Marty, why does Specialist 1&2 not count? (It is part of the compulsory ‘Area of Study 2’ and therefore is meant to be taught).

            1. Good question. There’s no accountability so teacher’s will teach as much or as little of the “compulsory” sections of Specialist Units 1&2 as they want.

              I suspect anything that’s either not a pre-requisite for or a value-add to Specialist 3-4 won’t get taught. That would probably include proof and consequently proof by induction.

              Unfortunately, many teachers only teach what will be examined at the 3-4 level.

              Which makes me mad for many reasons, including the fact that the UMEP Maths course has a number of theorems that are nicely proved using induction, but I’ve discovered that not all UMEP students have done induction.

              However …. none of this invalidates the fact that induction is part of the VCE maths curriculum (irrespective of it being ignored). (I believe Michael Evans’ fingerprints are on proof at the VCE level).

              But I think we’re taking things a bit off-topic for this particular blog topic.

              1. Induction is part of the VCE curriculum in the same that sense that climate mitigation is part of Liberal-National policy.

                1. Marty, I’m not sure the analogy holds. In fairness to VCAA, it’s on the Study Design and been mandated for teaching. It’s the teachers that are at fault if it’s not being taught in schools.

                  Personally, I’d like to see it on the VicCuric at Yr 10 level or even earlier – it would be a powerful thing for students at those levels (conjecture from patterns and then proof by induction). But again, it would be the teachers that are to blame if it wasn’t taught (or if it was ‘taught’ to simply tick the box).

                  But I doubt it would be put at those lower levels – not because it would be too hard for students, but because it would be too hard for some teachers (you have to understand what you’re teaching, although some educators argue otherwise).

                  What would you do in VCAA’s shoes to make induction part of the VCE curriculum (and I’m not talking about wholesale reform, although probably that is the only solution)?

                  Would you move it to Units 3&4, hence making it externally examinable, and hence forcing recalcitrant teachers to teach it? (This might be the only sure-fire solution for a number of reasons). Or what you clearly state that compulsory Unit 1&2 content is examinable at the 3&4 level (which is what happens in Methods) and then follow through ….

                  I can imagine the outcry if Unit 1&2 stuff was on the VCAA exam (and yet that’s exactly what VCAA do with Methods!) As you would know, for example, there is nice proof by induction that could be done in trigonometry.

                  Is it worth creating a new blog post for this discussion?

                  1. The analogy isn’t perfect, but it’s close enough. I’ll set up a post for induction in a few minutes.

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