A Tale of Three Cities

This is kind of a WitCH, but there’s plenty of non-crap as well. It is also properly a tale of two states and a planet, but the title was difficult to resist.

A few months ago a teacher-reader sent me three recent assessment questions from three different sources: NSW’s HSC Extension 2 Exam, the World’s IB Higher Level Analysis Exam, and Victoria’s VCE Specialist Mathematics Sample Assessment Tasks (SACs). All three questions involve integrating functions of the form \boldsymbol{x^ne^{kx}}, and the teacher-reader had suggested that there was a smell of plagiarism about it all. We weren’t convinced of that; the functions are very natural fodder for senior mathematics questions, and if there is arguably some borrowing, it is definitely not a Harvard standard of plagiarism. Indeed, the three questions are notably different in, um, style. That seemed worthy of a post.

Here are the three questions. Readers can have at it.

 

HSC Extension 2  2022 

(Here, with scientific calculator, about 15 minutes)

 

IB Higher Level Analysis 2023

(No link, with graphics calculator, about 27 minutes)

 

VCE Sample Application Task 2023

(Here (Word, idiots), with CAS calculator, God knows)

13 Replies to “A Tale of Three Cities”

  1. (Admittedly I haven’t taught IB since before they changed the names of the subjects) in the IB question, parts (a), (b), (e) and (f) look pretty good; seem to be testing appropriate skills.

    The NSW question had me wondering what part (a) looked like (spoiler: it is not related to part (b) in any way that I can see) and it seems to take an interesting direction in the final step that I would have guessed was actually from an IB paper had I not known otherwise.

    The Vic “exploration”… yes, well… I think I can see what the intention is/was here but that doesn’t mean it will achieve that purpose.

    1. Further spoiler: The letter parts on a NSW exam question are typically not related. (I’ve never understood why they don’t just make them separate questions by number).

  2. At the risk of poking the bear (*), I question how this VCAA Sample application task satisfies ALL of VCAA’s assessment standards and requirements. In particular:

    1) How is it 4-6 hours in duration?
    2) How does it NOT consist “mainly of structured exam-style questions”?
    3) Where are the “open-ended” questions? (Oh, my mistake. I see it – Component 1 part (d))

    2) and 3) are favourite phrases used in VCAA audits to criticise and fail a SAC as being non-compliant.

    I also wonder:

    4) What does the magical word “Determine” mean? Does it mean do all the working ‘by hand’? (In which case, where’s the mandated use of CAS technology – Outcome 3?) Or, without any marks to guide us, does it mean we can determine answers simply by using a CAS and then just write those answers down?

    5) What are “small values of n” and what exactly does “Describe the graphs” mean?

    6) What the hell does Component 1 part (e) mean?

    I wonder whether someone swapping the functions around in Components 2 and 3 would pass an audit?
    If the HSC Extension 2 2023 and IB Higher Level Analysis 2023 questions were incorporated, marks for each part were shown and some re-structuring and re-wording applied, it would be a pretty good sample task.

    There is so much more that could be said, but I just can’t be bothered.

    * Apologies if this is not the sort of comment you’re looking for. I can’t resist dynamiting fish in a bucket.

  3. OK. I’ve thought about this a bit some more and I think the NSW question achieves what the IB question tries to do but then goes off on a different path.

    The NSW question starts with a function that is probably not commonly known but can be graphed without a calculator at least in the n=1 case.

    The reduction formula is not too difficult, provided you choose the parts correctly (which I would assume was obvious for most NSW Extn 2 students).

    The final 3 marks are not easy to obtain, require some proper algebra and therefore the question is a nice one.

    Does the involvement of CAS ruin the question? For an answer, see the Victorian version.

    1. (Replying to one’s own comment is perhaps not ideal, so consider this more of an “added thought”)

      Perhaps these three questions would make a good case study in how “technology” impacts mathematics.

      Of course, we would need more than one set of examples. I’ll start looking.

      1. It’s an interesting question. I’m not sure the technology per se is what muddies the IB question and destroys the VCE question, but it feels relevant, if only to the way the authors frame the questions.

  4. For the HSC Extension 2 2023 and IB Higher Level Analysis 2023 questions, it might depend on the definition of “Show”. However, the marks provide a very useful guide that a detailed ‘by hand’ solution is undoubtedly required. ‘Technology’ is then relegated to checking purposes.

    As for the VCAA Sample application task, I’ve already given my self-indulgent thoughts.

  5. I agree that these are not plagiarised – it’s a pretty standard integral to investigate and they are quite different “approaches”. Also, this is not a like-to-like comparison. The HSC and IB questions are from exams, so are necessarily fairly tight. The suggested SAC from VCAA is arguably intentionally vague so schools do not just use it without modification [although a fully detailed exemplar SAC with a couple of student solutions and marking scheme from VCAA would be nice to see!]. Too many comments below… sorry.

    The HSC question sequence (note: it’s from 2022, not 2023) is particularly tidy with a nice induction step. It has an unstated connection to the Gamma function [made more obvious in the IB question and might make a nice SAC focus], but it’s integrated only up to 1 which leads to a nice squeeze theorem derivation of the formula for e.
    [Ext2 papers almost always have a reduction formula obtained from IBPs and connected to some other concept/topic. I ran though a few as exam prep / integration challenges for my SM students last year
    – 2022 you had above
    – 2021 was integral of log(x)^n, connected to a volume of rotation – an ok question, I made an indefinite integral version of it for my students
    – 2020 connected a couple of different integral representations of the diagonal beta function B(n, n) to derive a nice inequality.
    – 2023 was basically just a rearrangement of 2020 but without the inequality… a bit disappointing!]

    The IB question from the newish (2021-2028) Maths: Analysis and Applications course is just the standard Gamma function (shifted), but feels a bit weird. It first gets the student to briefly explore the integrand (defined as f(x) and f_n(x) in the first two lines!), which is good. Then there is a “show that” step requiring IBP to integrate x e^{-x}. Then there is some faffing around with l’Hopital which is seemingly just a consequence of writing the integral from 0 to b in a inefficient and unnatural form, especially as the required hint for the natural limit is given later in the question. Then students use their graphing calculator to (easily) evaluate the integral for a few n and conjecture that the integral equals the factorial. And prove it by induction using IBP for the step.
    I’m not a fan of this question for the l’Hopital and as it is not obvious that IBP is required or what the conjecture should be. If you miss either (the former is more obvious is you have a recurrence relation like HSC and VCE), then you drop quite a few marks. Not nice in an exam!

    The VCAA example SAC is a bit clumsy.
    – Component 1 gets students to explore the integrand (for n=k=1) and then part c [differentiating wrt x] loosely guides them to an integration by recognition approach [standard maths methods skill]. Typical VCAA suggested structure of start with a specific case, explore more cases, and generalise. Which can be good in some situations, but is not universally the best approach!
    The k parameter here does not do much, basically just affects some scaling. Missed opportunity to look at differentiating wrt it to get the recurrence relations without IBP.
    – Component 2 starts with a clunky request to obtain the IBP formula from the product rule and then a clunky request to find the recurrence relation. Note the k has been dropped. Missed opportunity to find and prove by induction the general form the integral for integer n. Or to *do* anything other “describe” the shape of the integral for various n. Feels very methods-y, just doing algebra and describing graphs!
    – Component 3 needs unpacking to be useful, but is a nice integral to explore. The k is just an overall scale factor, really not needed. Most students would need a hint on the other term in the IBP (1 = dx/dx).

    1. Thanks, Simon. Date error corrected.

      Yes, it is not quite “like with like”, but it’s not my fault that VCE SACs are not remotely like sane mathematical assessment. That the SAC questions are “intentionally vague” does not mean that they are excusably vague. They are not; they are garbage.

    2. The reason the sample SAC from VCAA is “arguably intentionally vague” is NOT “so schools do not just use it without modification.” Read any audit report for a ‘failed’ SAC – This is what VCAA wants schools to do (but note my earlier self-indulgent questions), including not including marks for each part.

      At the risk of being misunderstood as being self-indulgent, in my view there can be no fair marking scheme if the marks per part are not shown for each question. VCAA has consistently stated that if writing lines and marks are shown for each part of a question, the question is nothing more than a structured exam-style question. I have not heard any rational argument that supports this statement, nor any argument for why exam-style questions are a bad thing.

      The ‘thinking’ behind the sample SAC is here: https://www.vcaa.vic.edu.au/news-and-events/professional-learning/VCE/Pages/VCESpecialistMathematics.aspx

      1. Ignoring your continued whining about my slap, I agree. The SAC vagueness is not a bug, it’s a feature.

        Anyone who cannot recognise the fundamental incoherence of the VCE SAC question, made particularly stark by the HSC and IB questions, should be permitted no role in maths ed. Which implies that at least one person in VCAA should be permitted no role in maths ed. Most likely, all of them.

  6. Interesting comment about IBP – I didn’t think of that. For those wanting to be brought quickly up to speed:

    – Integration by parts was brought into the SM3&4 curriculum for first examinations in 2023. When it was last in the SM curriculum… maybe before the subject was actually called “Specialist”.

    – International Baccalaureate HL (it looks like there are now two slightly different HL Mathematics subjects) has had integration by parts in the syllabus for as long as I can recall. It looks like NSW has been similar, so perhaps this is as good an explanation as any for the appearance of a similar-looking question.

    – For what it is worth, I think technology (or lack thereof) must have been in the mind of the question setter(s) from the outset and therefore had an immediate impact on the direction taken by the question/exploration.

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