Mathematics Teachers and Their Culture of Mediocrity

Let’s see. Who haven’t we insulted recently? Ah, yes: teachers.

OK, so the title of this post could have been a tad less inflammatory. I’m serious, however, so what the hell. We’ll begin with a story.

Some years back, a teacher friend of mine was on a review panel for AMSI’s ICE-EM textbooks. The writing and reviewing of the ICE-EM texts involved academic mathematicians as well as teachers, and my friend’s panel included such a mathematician. In one meeting, my friend voiced his criticism of a passage of the ICE-EM text, suggesting that it was overly pedantic. The mathematician pushed back: “Well, as a teacher, you may not appreciate the importance of rigour.”

My friend would have been furious if he had not been so bemused. He knew that this wasn’t a question of rigour but rather of self-indulgent, fusspot pedantry. So, he simply wrote off the mathematician as a lordly clown. Correctly. The ICE-EM texts are too often pedantic for no great purpose and, as it happens, my friend is a stronger mathematician than the mathematician, which all made ridiculous the mathematician’s put-down. But of course the mathematician’s put-down was arrogant and classless and rude, whatever the context and whoever the participants.

At this stage, the reader may well be confused: wasn’t this supposed to be a teacher-bashing post? We’ll get to it, but mathematics education has plenty of worthy targets and there’s always time for everyone. (I have a mathematician-bashing post in the wings, just waiting for the cue.) I often ponder whether (a) teachers or (b) mathematicians are to be blamed more for the current maths ed swampland. The answer is, of course, (c) maths ed academics, followed closely by (d) education bureaucrats and (e) principals and their deputy henchmen. But I digress. Back to (a) and (b).

Despite the mathematician’s classlessness and specific wrongness, she of course had a point. Academic mathematicians, as a group, understand mathematics at a deeper level than teachers and they are more attuned to the rigour essential to mathematical foundations and mathematical proof. These same mathematicians may also overestimate the value of this understanding and this rigour in school mathematics but, nonetheless, there is unarguably something there. Which is not quite the point of this post.

As a mathematician who stumbled into the maths ed world, of course the mathematician-teacher divide has alway been in my mind. Such thoughts became stronger about a decade ago, when I began to look more closely at Victorian senior mathematics. With the 2022 exams debacle, these thoughts moved front and centre, where they have remained for the 2023 debacle sequel and Bennett and Deloitte. How could so many writers and vetters and reviewers give the tick to so many so bad exam questions?

To state the obvious, for the hundredth time, the writers and vetters and reviewers were not mathematically qualified for the job. There is no other explanation for the absolute howlers that have continually occurred, for years and probably for decades. But I now think there is more to it than that. I think there is an aspect in which the very strongest mathematics teachers, although they are much more likely to avoid the howlers, would still struggle with the job. I am not arguing that such teachers should not be so employed. But I think there is a hurdle for even these strongest teachers to climb and I think they, and everyone, need to be aware of it.

There is a sense in which the exam howlers have distracted everyone from more systemic issues. Yes, that distraction was largely Burkard’s and my doing, and it was a conscious strategy. Notwithstanding the Deloitte dissembling, the 2022 howlers are undeniable howlers, making them easy to hammer and making it easy to round up 70+ mathematicians to join in the hammering. This was critical in the play of things, and the mountaintop howlers did their job. But it is important to look more broadly, at the lesser hills and at the broader plateau of awfulness.

Consider, as one chosen out of a hundred, the following question, from the 2021 Specialist Exam 2:

This question, WitCHed here, is in a sense the origin story of the current exam saga. It was this question above all that made me decide to go for a front-on confrontation with VCAA, pressganging recruiting Burkard and arranging for us to harangue VCAA on this question and others. That was mid-2022, and it set the stage for the next attack in November 2022 when, inevitably, VCAA again screwed up the exams.

The mountaintop howler in this question is, of course, that no “relationship” between z2 and z3 can be deduced because nothing precludes z2 and z3 being any real numbers. This howler, still unacknowledged in the exam report and now reproduced in VCE texts, is the most important aspect. But it is important to note, and why I reproduced the question here, that everything about this question is off. All the framing and phrasing is a little off, or worse. Readers can go to the full hammering at the WitCH, but to take one small but stark aspect: why are the coefficients of the polynomial α, β and γ? The Greeks are conventionally used for the roots of polynomials, so why steal them to be coefficients and instead use the muddying subscripted z guys for the roots? This is not from a lack of mathematical insight or from unfamiliarity with conventions. The answer is because the writers and vetters and reviewers didn’t care enough to think about it. They simply agreed, “That’ll do” and moved on.

Is this a hanging offence? No. But it is a whipping offence, and so, so many VCE exam questions are whipping offences. So many exam questions are composed and phrased with a “that’ll do” attitude. And this is the point.

Mathematicians, by the nature of their profession, must be careful. When trying to write a paper for publication they must be sure that the details are correct, and they must write in a manner so that the referees will also be sure that the details are correct. Which means writing to be clear as possible. Of course it is not pure and perfect: some mathematicians are more diligent than others and some journals are fussier than others; even diligent mathematicians can make big mistakes; and many mathematicians do not carry over their care for research into a care for teaching and writing lecture notes and exams. But it is still a fundamental truth, that being a research mathematician requires attention to detail, a duty to be correct and clear.

The same is simply not true for mathematics teachers. Teachers are trained to declare “that’ll do”. They are required to declare “that’ll do”, on a daily basis. At 10 at night with a test to give the next morning, of course they’ll simply grab the test from two years ago, and of course they’re not going to check it enough, if at all, to notice the error that was there two years ago, and two years prior to that, and so on back to Adam.

Teachers have no choice but to function in a “that’ll do” manner. That is the mediocrity of the culture of teaching and it is unavoidable. There is no other way to survive, and it does well enough. If there’s an error on the maths test, it’s no big deal: you simply adjust the grading as best you can, in an ad hoc but reasonable manner. No one is at risk of losing a sheep station.

If I had been less trolling, I would have titled the post “Mathematics Teaching and Their Necessary Culture of Mediocrity.” But it is still mediocrity. It is still a culture of “that’ll do”. Which is exactly what you require as a teacher and it is exactly what you do not want when writing or vetting a high stakes exam to be taken by thousands of students.

When writing a big exam, the question to be asked is not,

Will this do?

The question to be asked is,

Is this perfect?

Of course perfection is unachievable and there will always be subjective judgments and trade-offs. But it is the North Star and it means that that every question must be scrutinised to death, down to the last variable and to the last comma. In VCE mathematics, nothing remotely like this has been happening.

Properly strong mathematics teachers can be a proper part of VCE exam writing and vetting. It is arguable whether they should be, but they certainly can be. But these teachers, even the strongest of them, must be aware of the culture of mediocrity that is their regular world, they must be conscious of their teacherly instincts. If a teacher, if anyone, is not sweating bullets during the exam production process, if they are not fussing over every single idea and every single mark and every single word, then they have no proper role in the process.

29 Replies to “Mathematics Teachers and Their Culture of Mediocrity”

  1. Hi Marty – I know you are being provocative, but I would like to suggest a clarification.

    The headline “Mathematics Teaching and Their Necessary Culture of Mediocrity.” would still be misleading. In relation to strict mathematics, yes, in relation to pedagogy, no. Teaching is a messy business (as is pretty much anything involving real human beings). In order to do high quality teaching, we must make subject and many other compromises. But the teaching itself is not mediocre.

    A teacher’s job is to maximise learning. This doesn’t just involve the necessary compromises due to workload and the industrial nature of our schools, it also relates to pedagogy. If I am too rigorous at the wrong points, my students will go down rabbit holes. We may acknowledge things aren’t quite right, but at that stage of the student’s learning, that might be what is called for. I choose what I am rigorous about on the basis of pedagogy.

    I agree with what you are saying about preparing exams needing mathematician input. I don’t know the balance but teacher involvement is essential as they know how the students have been taught, the traps they may fall into etc etc.

    There are necessary and creative tensions in the whole process and going too far one way or the other (as currently) is a disaster.

    1. Thanks, JJ. Of course I was being provocative. And, yes, I was referring to the mediocrity, or this’ll-do-ness, in regard to the mathematics tests and so forth. But I’m not sure why the this’ll-do-ness doesn’t apply more generally to teaching. You guys have all manners of constraints and barriers to get in the way of good teaching. To acknowledge these barriers, and the plain exhausting nature of being a teacher, is not to suggest teachers do not care about good teaching, or that they do not teach very well very often.

      To give the one pertinent example, the vast majority of mathematics teachers could most certainly do with a stronger mathematics background, not just theoretically/ideally but for the subjects they are actually teaching. How is these teachers accepting their current background not this’ll-doing, even if necessarily so?

      Teacher involvement was not traditionally considered essential to preparing statewide exams. What is different now? Or were the traditional guys wrong?

      1. Marty, I think you have this one pretty right.

        The step from mainly focussing on research at uni to being a teacher did involve a lot of learning how to let things be good enough and not burn through too much time. Both in terms of the resources created (which we really don’t have enough time for, especially in govnmt schools), but also in terms of depth of understanding of content.
        Of course, better and more easily adaptable shared/commercial resources should be able to reduce the teacher burden. The amount of recreated resources around the world is staggering.

        I’ve got a fairly strong maths and physics background – but there are things I have to teach (astronomy in physics, statistics in maths, design and security in software dev) in which I have very little post secondary formal education. I do try to learn, but I rarely get the time to get a satisfactory depth of understanding and my list of things to learn only gets longer…

        1. Thanks, Simon. Of course a sane and detailed curriculum and decent textbooks would help no end. I don’t think they would change the fundamental this’ll do nature of teaching. But at least you guys could occasionally get to bed at 10 rather than 12.

      2. Hi Marty – I agree that modern teaching is insanely overworked, underpaid and challenging. Why anyone with sufficient maths would want to teach is a good question in the modern world, when there are so many other far better paid easier jobs….. I think it’s so far from ideal and so full of compromise that there are many many ways it is this’ll do.

        I suppose the point I was making is that my gut says that even in a much better education system (read 50 years ago in many ways), there are still compromises to be made with mathematical rigour in favour of pedagogical rigour. ie learning should be the priority and if that involves a (conscious) compromise with mathematical rigour, so be it (probably with a throwaway remark that it’s not quite right but close enough).

        Re the exams – I am not sure, but to me logically it seems that the exams should be informed by how students are actually being taught and what seems to work best with them, rather than just mathematical rigour. In my experience, multiple perspectives in dialogue often provide the best outcome. eg I see questions in textbooks that are a long way from kids’ experience and background knowledge and it gets in the way of properly assessing their level of maths.

        1. Thanks, JJ, but I think “mathematical rigour” is a red herring, twice.

          First, re the teaching I think you’re basically agreeing with me. If teaching is “insanely overworked”, and it is, and always has been, then of course there are compromises. Of course there is a (necessary) culture of that’ll do. These compromises have nothing to do with rigour versus pedagogy. They have to do with compromises between, e.g., meeting with your students for extra help versus writing a new test and/or checking thoroughly the old test.

          Secondly, “rigour” is the wrong word for the exams, except for “rigorously checking”. What is required is accuracy and coherence.

          Your argument for the involvement of teachers in the writing of exams is a hell of a lot of cart, and not a horse in sight.

          Just to be clear, the traditionalists would of course had had contact with teachers about what was being taught and how, and what happened with students and so forth. Of course any system would involve talking to teachers (and hopefully students). But that is different from teachers being part of a writing and vetting team.

          When I began writing this post, I was thoughtlessly in favour of teachers being part of writing and vetting. By the time I had finished the post, I was in two minds. Now, the more I’m being asked to think about this, the more I’m against it.

          Apart from starting to lean against teacher involvement simply on principle, there is also a practical consideration: how many teachers in Victoria are strong enough for the job? Five? Ten? It’s not a lot.

          1. Hi Marty – I essentially do agree with you – except over probably too fine points.

            My position on teacher involvement in exams is purely guesswork. My radical suggestion is that I would ask people who have done it about the process (obviously not in Vic and probably overseas) and see what worked best.

            One thing I do not appear to agree with is ‘If teaching is “insanely overworked”, and it is, and always has been.’ Of course it all depends on expectations, which are now skyhigh and totally impossible to reach. However teaching used to be more manageable. My essential argument is that teacher ratios and contact hours are about the same as they were 40 years ago, but admin, emails and so many other things have gone on top of that. In admin jobs, that is absorbed into the workload which adjusts. But it doesn’t work that way in teaching.

            1. Thanks, JJ. Yes, we’re mostly agreeing.

              Of course the job of teaching has gotten much worse, because of the insane amount of admin and pastoral care twaddle and the like now loaded onto you guys. But the job of teaching was always a pretty tiring gig. It’s just that time now spent doing worthless crap used to be spent on teaching better. But yes, probably worse now.

              In any case, this only reinforces my argument, that teachers must learn to survive in a this’ll do manner. Not just in regard to test production (which was the original point), but in regard to every aspect of the teaching.

        2. Re: “Re the exams – I am not sure, but to me logically it seems that the exams should be informed by how students are actually being taught and what seems to work best with them, rather than just mathematical rigour.”

          It is impossible to know what and how students are being taught. There is a curriculum (that lacks many important details) but there is no doubt that students get taught from this curriculum to different depths. But the curriculum is the only common factor and so the exam must assess from it.

      3. Whatever culture exists has been significantly diluted by the number of out-of-field mathematics teachers in lower and middle secondary school. The same problem exists, albeit to a lesser extent and in subtly different ways (*) at the VCE level.

        As for “the vast majority of mathematics teachers could most certainly do with a stronger mathematics background”, that will never happen. It is not considered important professional development by the leadership at most schools. And the teachers who do seek it are usually the ones who least need it. Most think that attending a VCAA session on new Study Design content or attending a mathematics education conference is sufficient. Such VCAA sessions would be useful if they were run by people with genuine mathematics credentials and the materials provided were written, or at least proof read, by people with genuine mathematics credentials. And such mathematics conferences would be useful if there was genuine mathematics content (I have not heard of genuine mathematics content at any maths education conference for nearly 10 years now).

        You may refer to this culture as necessary. I strongly disagree. It’s a culture created by apathy and inertia. On a related note, I know many teachers who can’t be bothered submitting feedback about the Study Design. Their reason? “Why bother. VCAA won’t listen so what’s the point.” That’s apathy in action.

        There is clear evidence that an influential group of people have fooled the VCAA for many years with their so-called mathematics credentials. As for the origins of this influence, I believe it started with the introduction of the CAS calculator back in the 1990’s. This is where much of the mediocrity began, and it is teacher apathy and ignorance that allowed it to grow and become embedded.

        If you wanted to stick the troll pin where the truth hurts, I think a more apt title for your blog is “Teachers and their culture of apathy”. Mediocrity is merely a symptom.

        * Competent lower and middle school specialist mathematics teachers that are required to teach outside of their competency at the VCE level. There are also teachers who present as competent at the VCE level that are anything but (they are usually CAS-calculator button pushers). There is direct evidence for this in most third party trial exams, as well as many past VCAA mathematics exams.

        1. No, it’s not apathy; it’s mediocrity. You’re missing the point.

          Your first paragraph is an important point but entirely off my point. You should go to work for AMSI.

          Your second paragraph is confused. They don’t have to be “mathematics conferences”, and they shouldn’t be. There is nothing intrinsically wrong with a conference on mathematics teaching or mathematics education or whatever. They are only bad because VCAA long ago lost the plot, and MAV long ago lost the plot, and MERGA long ago lost the plot, and AAMT never had a plot. Yes, I cannot see it becoming any different, at least in the longish short run. But a new study design and the elimination of CAS could take away a significant amount of motivation to present and to swallow the techno-garbage.

          Your third paragraph makes no sense.

          Your fourth paragraph is simply wrong. If any teachers gives me a test or SAC that they have written, I will be able to look at it for ten minutes and make a half dozen improvements, if not correcting subtle or blatant errors. That’s the difference.

          Also the unwillingness of teachers to fill in the questionnaire is not evidence of apathy. If anything, it is evidence of learned helplessness. For decades, VCAA treated any expression of concern with disdain if not active hostility. Why the Hell should a teacher trust in VCAA’s good faith now? In fact, I believe that a teacher should now extend cautious trust in VCAA, at least to the extent of filling out the questionnaire. But I cannot be too critical of any teacher who sees no point.

          Your concluding sentence is wrong.

          1. “You should go to work for AMSI.” No need to get nasty.

            Paragraph 1: So you don’t think that’s a source of mediocrity? I think it is, and it’s a totally unnecessary source IF there is better policy around education.

            Paragraph 2: I agree with your suggestions. But how to solve “the vast majority of mathematics teachers could most certainly do with a stronger mathematics background”. Should those who are doing a M.Teach and wish to teach maths be required to have done several substantial maths units at a minimum of 2nd year level? Or should the M.Teach. be done concurrently with those units for those who lack them? Should current teachers of maths be required to enrol in such units?

            Paragraph 3:
            Q: Where does a lot of the mediocrity come from?
            A: The Study design and CAS.
            We agree. So what causes that answer?
            Do you disagree that there’s an influential group of people that have fooled the VCAA for years with their so-called mathematics credentials? Or do you agree but disagree that there’s evidence for this? Or do you disagree that such a group could be influential enough to create or greatly exacerbate the mediocrity? To clarify my paragraph:

            1. I believe that there is – or maybe was, post Bennett Report – an influential group of people that has fooled the VCAA for many years with their so-called mathematics credentials.

            2. I believe there is clear evidence for this.

            3. I believe that 1. caused or exacerbated the current culture of mediocrity. That includes influences on the CAS culture and the poor Study Design.

            Paragraph 4: Be careful what you offer. I don’t doubt you could look at a test or SAC for 10 minutes and make a half dozen improvements. Then you have to communicate those improvements – that takes longer but let’s say another 10 minutes. Then it might be necessary to provide some clarification or explanation for the improvements, let’s say another 10 minutes. So conservatively, I estimate 30 minutes per assessment (but I think for a SAC it would be much longer). Now let’s assume that x teachers per week take you up on your offer. So that’s 30x minutes per week. How large a value for x before the offer becomes unsustainable? (Or do you think that you’re safe because x will never get to double digits?).

            Generous as your offer is (and it is very, very generous), it’s not a sustainable solution.

            I’m sure I’ll get jumped on here, but I see “Learned helplessness” as a fancy way of saying apathy. To those teachers, I would present the following scenario:

            A student comes up and asks what should they do if they cannot get the answer to a multiple choice question. The teacher will – probably – say to guess. But the students says “What’s the point? I’ll probably guess wrong.” The teacher will – probably – say: “You probably will. But there’s a chance you’ll guess right. You’re guaranteed to get zero if you leave it blank. So take the chance to get the mark.”

            You’ll probably say I’m making no sense. I don’t care. What I hope is that it makes some sense to the teachers who moan and bitch and worry about, say, the infestation of pseudocode, but don’t think there’s any point in providing feedback.

            Concluding sentence: I respect your opinion but I disagree with it.

            1. This is weeds. You’re missing my point, in both the post and my reply.

              My point about reviewing a teacher’s test in ten minutes wasn’t to offer a solution, but to point out the issue: teachers do everything, necessarily, with a this’ll do attitude. This is fundamental to the teaching culture. You’re talking about other stuff.

  2. To paraphrase one of my best (non-mathematics) teachers, “I’m not paid enough to simultaneously deal with a bunch of shithead teenagers and also teach this subject properly.”

    As for whether teachers should be writing exams, I don’t think teachers should be anywhere near the exam writing process. Certain private schools (including my own) use knowledge of how VCE exams are written/marked from teachers involved in these processes to give students an unfair advantage.

    1. Thanks, Joe. The paraphrasing is very funny and very true.

      I wasn’t considering the insider knowledge argument against teachers cowriting exams, but it is obviously worth raising. I wonder if the advantage may be more theoretical than actual, but it’s not for me to judge.

      1. Well, when you have teachers saying stuff like “as someone who has marked VCE English exams in the past, students tend to do the best with ABC structure” or “for questions like these, make sure you have points X, Y, and Z in your answer” I think the advantage is pretty real.

        1. Sure, and on its face you have a strong argument. I’m just wondering if perhaps ABC and XYZ are not well known, and/or that it may not differ much from the olden days. I think it was always the case, or at least for many decades, that grading was done by teachers. So, the same insider knowledge is, and always was, available more generally.

        2. If VCAA released the actual marking schemes this insider knowledge would be less of a problem… the examiners’ reports are not enough.

          1. I honestly haven’t looked yet: what does Bennett recommend? I think he was pretty harsh on the exam reports, but I haven’t read the details.

            1. Recommendation 6 (a) from the Bennett Report is attached.
              The Minister has stated publicly that all recommendations will be accepted. The recommendation appears to only apply to the November Exams, not the NHT exams.

              1. Thanks for looking that up John – it is good news, and I hope VCAA applies it to more than just the maths exams!

  3. Hit the nail on the head Marty. This post is timely as I have been recently lamenting my colleagues’ dismissiveness of having mathematicians back involved in school education during discussions in the office. I worry that a culture of distrust has set it. Some good leadership is needed to sort this out.

    On a brighter note, I got a message from a student last week who has prepared critical feedback on the algorithmics content in maths methods to send to VCAA. Great seeing young people being thoughtful and trying their best to make things better.

    1. Thanks, Jay. Although it has to be said that posts such as this, and certainly titles such as this post’s, don’t help much with the trust thing.

      Many people, including many mathematicians, believe people like me, and me in particular, are doing more harm than good. I can understand their argument, although of course I do not agree.

      1. People do think that Marty but they aren’t right. If the state grows mathematics experts then doesn’t use them in their decision making on issues of mathematics then they aren’t doing their job right. And yes, us maths teachers aren’t mathematics experts.

        Maybe I just like being insulted but I read your article as an empathetic acknowledgement of the strain teachers are put under. Keep up the good work.

        1. Thanks, Jay. I wasn’t fishing for reassurance but I appreciate the comments. Of course my title was intended (mostly) as a stir, but there are a couple complicating aspects to this.

          First of all, even though my post was not primarily an attack on teachers, it’s not far removed from it. I try to not attack teachers because they get so much of the blame that should be assigned to (c), (d) and (e). And (f), parents. But I have my thoughts, and a small slide from this post would be more directly critical. You have me thinking of a second post, which i think I should probably do …

          Secondly, guys like me can try to contribute to the maths ed world without being so divisive. Burkard, for example, doesn’t encourage or attract the same negative reactions as me (although I think maths ed academics still dislike him). I’m more comfortable with my approach, for reasons I outlined here. But I think the majority of Australian mathematicians disagree, and regard me as an asshole. I weep at night, at the thought.

  4. Hi Marty. Firstly thanks for this post and for commenters JJ, Simon Joe John K and Jay for the interesting and informative replies. I have two questions,: do the exam vetters/reviewers actually sit down and write the exam under exam conditions? And can we see or at least know how they scored. Secondly, Marty, you were talking about a ‘perfect’ exam. Mathematical accuracy and rigour aside, just what in your opinion is it? Is it one where the “average” student can get a score of about 50% or 60% in the time available? Or is it something completely different?

    1. Thanks, Rob. I haven’t paid much attention to the vetting details, but I think I read somewhere (Deloitte?) that there is a final, “do the exam like a student” vetting step. I think both Deloitte and Bennett discuss at length what the process is, and how it should be reformed.

      My “is this perfect?” was intended to refer to the composing and the writing of an individual question, but yes, the same (unattainable) goal would apply to the exam as a whole. And, as a whole, the VCE mathematics exams are a disaster. The predominance of shallow 1-mark and 2-mark questions means that little of any proper depth is tested, and it exacerbates the awfulness of the awfully prissy grading.

      On your specific question, I think it is much better to have the “average” student be getting in the 40-ish range. This leaves more room to (fairly!) differentiate the top guys.

      1. The “final, “do the exam like a student” vetting step” is called a blind review. It is part of the VCAA process, as well as the process used by most third parties in the writing of their trial exams.

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