Further Exams and Further Errors

By overwhelming demand,* we have decided, much belatedly, to put up a post for discussion of the 2021 Further Mathematics exams. We have no particular plans to update this post, although we will do so if anything of interest arises. We’ll just note the two excerpts below, from Exam 2, the first of which is discussed here, at 5:30. Thanks to Simon and SRK to bringing these to our attention.**

*) From people named Terry.

**) There’s also a third question from Exam 1 that puzzles us, but we’re waiting to hear back from Further Gurus before posting on it.

UPDATE (24/12/21)

The exams are now available, here and here.

27 Replies to “Further Exams and Further Errors”

  1. Part (f) is overly wordy, but nothing I haven’t come to expect from VCAA exams (Specialist too has become excessively wordy in Paper 2 section B lately).

    As for the matrices, yep… a lot of rubbish there, but I doubt the typical Further student would be worried by it. To be clear: it is horrible Mathematics, but fairly standard for what I would expect on a VCAA Further exam. You learn to read questions a bit differently and move on with it all.

    Worried by the question that followed, yes, but by the question itself? Probably not.

  2. The annoying thing with the javelin question is that it’s not clear whether outliers are strictly greater than or greater than or equal to the upper fence. The wording makes it feel like it includes the fence.

    Not sure what the query is with the send/receive question.

    1. Thanks, Alex. One might define “outlier” to include values right on the border, but the accepted definition, both in VCE and generally, excludes such values.

      1. Don’t think I’ve dealt with a question where a value is directly on the fence, and just assumed with the ‘less than’ and ‘greater than’ to not include it. It’s one of those things that would be good to have clear in the study design but isn’t.

        1. Thanks, Alex. I’m generally open-minded about how much detail the study design should include, although the current study design is pretty useless. In this case, I think the definition of outlier is generally clear enough and accepted enough.

          The problem here is not with the definition but with the exam question: purely and simply, the exam question is screwed. There is no correct answer. VCAA should award the 2 marks for students who give the fence value, not because it is correct but because it is the least incorrect.

          1. Terry, this is Further. Why do you ask?

            But in answer, I don’t know and it’s a good question. The one aspect that seems natural is that “outlier” somehow means “not in the general group”, and so the border values should not be outliers. That is, the Further Exam writers have no excuse for this screw-up.

            1. If an outlier is “not in the general group” what would be an outlier of a random sample drawn from a standard Normal distribution?

              1. To be honest, I don’t care at all. But, I paid my dime, so I’ll go for the ride.

                Does one talk about outliers of “continuous” distributions? Also, even if “outlier” has a precise definition, it is obviously more qualitative in spirit. Is fact that the notion is less or more helpful in this or that scenario a cause of concern?

  3. Apparently the expected answer to the javelin question was Upper Fence + 0.01, since students were expected to be prompted by the table of data on a previous page having the javelin throw lengths recorded to 2 decimal places.

    Apparently an answer like “anything greater than … ” also received full marks.

  4. Given the content (broad ideas, not the specific outcome statements), Further Mathematics really could be a good subject.

    Unfortunately, exam questions like these are so common I don’t consider them to be outliers.

    Practice the exams, study the VCAA solutions, pass and move on…

    Occasionally there comes a student who questions the accuracy of these things. Tell them to focus on maximising their score and to not think quite so much and they will be fine.

    Sad. True.

  5. Is M^2 incorrect? I find that (M^2)_{2,1} = \sum_{k=1}^{5} (M)_{2,k} (M)_{k,1} = 1. I also find that (M^2)_{3,2} = 2. More generally, I believe that M^2 should be:

        \[\left( \begin{array}{ccccc}  0 & 0 & 1 & 2 & 0 \\  1 & 1 & 0 & 1 & 0 \\  0 & 2 & 0 & 0 & 1 \\  0 & 0 & 1 & 1 & 0 \\  0 & 1 & 0 & 0 & 0 \\ \end{array} \right)\]

    If so, how on earth did this happen?

      1. It’s really inexcusable sloppiness. How ironic that for a CAS enabled exam, that the result of matrix multiplication was not checked over. Then again, so are “almost”-probability density functions, so I shouldn’t be surprised. As has been pointed out in other threads, something like this would have to pass vetters and the question maker, which I find disturbing.

        1. VCAA does not care. The teachers do not care. The students do not care. No one cares. ER Love cared, and he is dead.

  6. Teachers cared about the angle between two dodecagons (50 cent pieces) a few years ago, to the point it made the popular press.

    I’m not a Further Mathematics teacher this year (or any year since 2004) so will admit to not actually reading the papers, but the silence from the FM teachers I know is a bit strange on this matter.

    Students don’t care – that is mostly true. They care to a point which varies from student to student. Trust in VCAA is something they have little choice in.

    Do teachers care? One would hope so. Do they though? Maybe the question is “do they know HOW to care?” which could equally be asked of VCAA.

      1. Is this because a meaningless exam has become the accepted normal for pretty much the entire time a lot of teachers have now been teaching?

        Genuine question.

        Had another look at the matrix question on the further paper – I would be very interested to know how many students (suspect very close to zero) actually squared M. This probably allows VCAA to say there was no harm done.

        Not defending the question, just making a prediction.

          1. When the organisation that sets the exam also sets the rules for how schools are allowed to query results or parts of the exam, can the ritual genuinely be disturbed?

            Without a nicely worded letter to the newspapers, of course.

  7. Thanks Marty for starting this discussion. I care about Further Mathematics and Foundation Mathematics, even though I am not teaching these subjects at present, and not likely to be in the near future because I will be happily employed in a Year 7-10 school for 2022. I care because the majority of students who enrol in VCE mathematics take Further Mathematics, and they deserve to be taught a curriculum designed by, and presented by, people with mathematical expertise. At present they are the Cinderellas of VCE mathematics. (Current trends lead me to a question: do mathematics teachers need to know anything about calculus? As far as I can see, many – perhaps most – mathematics teachers never get to teach calculus in their entire career.)

    I have not seen the Further Mathematics examination but I did watch a chap going through his solutions on youtube. I am always on the lookout for the problems on transition matrices. Readers of this blog will have heard me bang on about this problem before. Basically my issue is this. You need the Markov condition to solve these problems. Without it, there is not enough information to solve the problems. However Markov is not mentioned in the main text books, and he does not get a mention in the study design or examinations. VCAA has brought Markov back into the new study design which has not yet been implemented. I imagine that lots of Terrys wrote to VCAA on this issue.

    So I was curious to know what the examiners would do in 2020 when (i) it is clear that the error has been recognised but (ii) not yet rectified in the curriculum. From what I recall of the video, the examiners said, in effect, “Just do it.” Bless their sox.

    Thanks again Marty. You provide a valuable service to mathematics education in Australia.

    1. “… the majority of students who enrol in VCE mathematics take Further Mathematics, and they deserve to be taught a curriculum designed by, and presented by, people with mathematical expertise.”

      Are you suggesting by omission that MM and SM students are currently taught a curriculum designed by, and presented by, people with mathematical expertise? If so, I want a case of whatever you’re drinking.

      1. I am suggesting that Further Mathematics is often taught by people who do not have sufficient expertise. This is my observation. Teaching students about elementary ideas often requires more advanced knowledge than is commonly assumed. (I recall meeting a teacher who was trained as a primary teacher who was asked to teach mathematics to Year 7 and 8 students. She just managed with Year 7, but when it came to Year 8 she was out of her depth. She broke down in tears when discussing this with the principal.)

        I admit that I don’t know who designs the curriculum. It appears that, provided that the problems can be solved with a CAS calculator, it’s fair game to be put in the syllabus irrespective of the mathematical basis of the work. This is the essential problem with teaching statistics in schools.

        I’ll go back to what I was drinking.

        1. “Teaching students about elementary ideas often requires more advanced knowledge than is commonly assumed.”

          OK, I see your point, and I agree entirely. It is why primary school maths is such a mess, and why I’ve started to hammer more and more on it here. But, the bad teaching of Further is the least concerning effect of a very general failing. (The bad teaching of MM and SM has largely different causes.)

          Your story of the teacher is terrible, but not really evidence of anything except an idiotic Principal.

          1. I told you in an email about my discussion with other teachers about the laws of indices. One memorable response was from a very experienced teacher. He had always told his students that 0 is not a number – it’s a placeholder. Is there no limit to the depth of misunderstanding?

            I have often thought about retraining as a primary teacher. Many years ago, I met a chap with a PhD who did this. I wondered why he did it. But maybe now I understand.

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