An Educational Conjecture

OK, time for a competition (And, no, we haven’t forgotten our previous competition, which we shall revive at some stage.)

Here is a conjecture:

Every new idea in modern mathematics education is either trivial or false.

Can we prove this conjecture? Of course not. Not, at least, if we do not Learn More about it and without reading thousands of pages of educational gobbledegook. (Which. We. Will. Not. Do.) Is the conjecture true? We don’t know. But we are not aware of any counterexample.

The competition, then, is to attempt to prove the conjecture false. The winner is the commenter(s) who comes up with and argues most persuasively for a counterexample: the new idea in modern mathematics education that is true and the least trivial. The winner(s) will receive a signed copy of the number one best seller,* A Dingo Ate My Math Book.

Just a few notes on the parameters:

  • By “idea”, we mean any claim about or approach to teaching or learning mathematics.
  • By “new” we mean something other than dressing up a traditional idea in new clothing.
  • By “modern”, we mean from the last fifty years or so, back to about 1970.
  • By ”least trivial”, we mean something of genuine value, least trivial to mathematics education. So, deep ideas in neuroscience, for example, will score little if the subsequent application to mathematics education is trivial.
  • By “true” we mean true.
  • Suggestions, which can be made in the comments below, need not be long, specific or heavily documented. We will reply politely to any suggestion (and other are welcome to reply), querying and critiquing. Further argument and evidence can then be provided.
  • Will we be fair? Probably not. But, we’ll honestly try.
  • Multiple entries are permitted, and there may be multiple winners.

Go for it. We’re genuinely curious about what the responses may be.

*) In Polster and Ross households.

UPDATE (17/4)

Just a few (?) words about this competition, and this blog.

The competition is, of course, a challenge: put up or shut up. If a reader cannot propose and defend one single idea of modern mathematics education then that reader should perhaps stop imagining that any such idea exists. And, if such an idea does not appear to exist, the reader should consider what that suggests about the mountains of Wow produced by the maths education industry, and what it suggests of the shovelers creating these mountains.

Now, what could or should one expect the response to be to such a challenge on an aggressively antiestablishment blog such as this? This blog has a decently large (but not huge) readership, although we can only determine the nature of the readership from the minority who comment, which is obviously a very biased sample. Still, it is probably reasonable to place readers of this blog into three camps:

  1. There are the fellow travellers: like thinkers and “Marty fans”.
  2. There are puzzled and/or annoyed teachers, who smell that there is something wrong with their teaching world, while still maintaining some faith in the orthodoxy. They may appreciate some of the specific critiques on this blog, while not buying the overall message of contempt.
  3. There are the Marty haters, people who are convinced that Marty is an asshole or a nutcase,** who loathe this “nasty” blog, but who visit occasionally in order to feel superior.

This competition is primarily directed at members of Group 3, those who create and promote and value modern maths education. Again, it is a challenge: put up or shut up. If such a person cannot defend such ideas outside of the comfort of their cult, then there is no reason for anyone else to take them seriously.

Do we expect responses from members of Group 3? No, of course not. They regard debating on a blog such as this as beneath them. But the challenge is there, and it will remain there.

What about Group 2? Here, we’re guessing there are some thoughts of possibly defensible ideas, but there is probably some nervousness in proposing them. Such ideas will of course be critiqued (that’s the whole point), and strongly. So, we totally understand any such trepidation, although it is misguided. This blog is scathing of bad ideas, but it is respectful to all good faith commenters, which has been pretty much everyone.

Group 1 can take care of itself, of course, although its members could be more actively critical of this blog …

**) Both are true.

UPDATE (12/05)

OK, it seems like a good time to begin rounding this off. So, who is the winner? We’re not convinced anybody “won” in the sense that anyone has suggested a significant counterexample to our conjecture. None of the suggestions compares, for example, to the elephant truth that mathematics teachers need to understand mathematics a hell of a lot better than they do. None of the suggestions deals with the fundamental flaws of modern education, with, in particular, the deification of technology and the demonisation of discipline. We’re convinced as much as ever that modern educational “research” is fundamentally useless, when it is not actively destructive.

Still, if the suggestions below are minor in effect, some are good and sensible. We have thoughts on a winner, but we thought to let readers have a shot at it first. So, if you have an opinion on the best response to our challenge, please indicate below. We’ll consider, and we’ll announce our winner later in the week.

Update (29/07/21)

We’ve finally ended this. The winner is student-teacher (wst). See here for details.

170 Replies to “An Educational Conjecture”

  1. Just to clarify, is time table drilling on computers (e.g. Mathletics) a traditional idea dressed up in new clothing? My gut says yes but my gut isn’t handing out (nice) prizes.

    1. Hi, Craig. Obviously drilling times tables is not new (although the idiocy of not drilling them is). But, yes, the use of computers for drilling is on its face new, at least how I’ve defined “new” and “modern”. The question is, then, is such computer drilling (in general or specifically for, e.g., mathletics) significantly different and, if so, is it significantly more valuable in some manner?

      When I made up the conjecture, which I’ve presented in talks, I was thinking more of theories of teaching and learning. But I deliberately broadened it here to include applications and techniques. So, your suggestion is definitely a contender (and more so if you argue strongly for it). And I’m genuinely open to the idea of drilling with computers being valuable, although I’d need to be convinced.

      1. It’s not just times table. Kahn Academy has required scores/mastery to “level up”. Many other computer-based homework systems are similar. They are basically updated Skinner teaching machines:

        Personally, I think you could do the same with pen/paper. But the computer does kind of provide a structure and an adaptive algorithm and spiral reviews and the like (and excess problems).

  2. I feel like the froggy calculator pic is trolling unsuspecting readers ;).

    Seriously though, I feel like old ideas tend to become new again, and it is going to be VERY hard to argue that any “modern” idea in math ed is actually new and good. For example, the Socratic method, if you don’t call it that, is very new for a lot of people in maths ed.

    So my submission is: Socratic method :D.

    1. Yes, Glen, I agree. Note that “idea” in the expansive, “tool/technique” sense that Craig took it to be raises more possibilities, and that is also worth considering. But yes, in terms of fundamental approaches, I find it difficult to see what anybody might come up with.

      But that’s not my fault, and that’s really the point. Teachers, especially trainee-teachers, are constantly bombarded by New! Improved! maths ed wonder cures, and I question whether one single ounce has both value and originality, when it is not actively destructive.

      1. Well, agreed.

        For instance, one thing that I am told is new in my own teaching is the use of social constructivism, active learning, and reflective writing. However, I don’t argue for these as fitting your conjecture, because I am certain they are far from new!

        Only the jargon is new.

          1. The jargon is blech, but the practice is fine — it’s just peer learning. I’m pretty sure Pythagoras was doing that.

            1. Yes, of course. But I can’t stomach the repackaging of traditional practice and sense as if it’s some brilliant insight.

  3. Just on another note: Is the AMS link you included the best way to buy ADAMMB? If you have a thousand copies lying around (as authors sometimes do), I’d be just as happy directing my $40 direct to you Marty, so you get all that, rather than the diluted 23c the author normally gets after everyone else has had a bite of the pie. If I can get a copy direct from you, how would I?

    1. Hi, Wally. A nice thought, even if it means robbing Burkard of 11.5 cents. As it happens I don’t have “author’s copies” per se, but Burkard and I bought a pile together at a (relatively) low price, for presents and prizes and the like. I’m happy to sell you a copy at whatever it cost me (I’d have to look), plus postage. (Or, you can wait until the plague is over to meet up.) Just email me at any of my ten addresses.

  4. Marty, because I’m a sucker for punishment (just look at where I have worked in the last 20 years…) is there a bonus if we can prove your conjecture?

    And by prove I mean in the legal sense (beyond reasonable doubt) as compared to the true, mathematical sense which, as Douglas Adams once said, “is of course impossible.”

    1. RF, there are two things you should never try to prove: The impossible and the obvious. So, good luck with the latter ….

    2. Hmm. Interesting thought, RF. I’m willing to entertain a “proof”, as long as you don’t raise specific ideas/theories/approaches: that’s for genuine advocates to do.

      1. A very smart person (who since finishing their PhD in Mathematics works for Google making more money that many people can calculate…

        …a very smart person I had the pleasure of knowing once told me that Mathematics is essentially doing one of three things:

        Proving things that are obvious (you only need four colours to colour a map to make it look nice)
        Proving things that are not so obvious (Fermat’s last theorem)
        Proving true things which are obviously false (the South pole is actually the “top” of the world).

        And I have no plans to raise specifics. That would require research and I’m not into that anymore.

  5. Suppose, Marty, that your conjecture stands, and the prize goes unclaimed. Certainly we have not circumscribed all new ideas for modern mathematics education. So here is a corollary challenge: Describe a new idea for mathematics education which fits all the stated criteria. Joys.

    1. Hi, SP. People are of course welcome to suggest their own new ideas, and in other fora plenty do: pretty much every maths ed conference will offer various new and improved snake oils. But I think the search for such new ideas makes about as much sense as trying to invent a new, improved wheel.

  6. This is OT, but related to the previous “competition” to which Marty linked in the first post. I’m replying here since this is the (far) more recent post.

    Does anyone know /why/ the NSW curriculum was so severely degraded? The HSC maths curriculum was definitely the strongest in Australia. So why, when there was a push to standardise curricula across Australia, does it appear that the rest of the states should (more or less) adopt Victoria’s curriculum, rather than the rest of the states adopting NSW’s?

    Maybe difficult for people to answer, but hopefully someone who reads this blog is plugged in enough to provide some insight.

      1. Marty, no. I’m not in the loop enough to know what is going on what that kind of thing. Of course I can speculate about their motivations / competence, but that doesn’t really help me understand.

        1. SRK, my point is that it’s not speculation. Sure, we don’t know the specifics, but we do know the active forces, which are at work everywhere. The exact same idiocy is being enacted upon the IB.

          1. I guess I would like to know more about the specifics.

            Perhaps your point is that the specifics aren’t important because the problem is so widely spread that what happened to the HSC was inevitable…

            1. Hi, SRK. Yes, that’s basically my point, although there’s a little more. Appreciating the general forces of dumbness is more important than any specific application of those forces. But of course you also have every right to be interested in and concerned about specific applications.

              But but, one can get usually a good sense of the specific from the general. Why is NSW screwing up their curriculum? Because the gullible decision makers trust the sanctioned education experts who are, in the main, idiots. So, you can still ask which gullible decision makers were swayed by which idiotic argument put forward by which idiot. But these specifics matter much less, and I think are much less interesting, than the universally perverted manner in which such decisions are made.

    1. Hi Steve. An interesting starting point, although I’ll admit GEB was too long-winded for me. How does (or could) Hofstadter’s (or anyone’s) notion of “cognitive function” change for the better how one can teach mathematics? More directly, how does/could the (as yet unstated) “link” between logic and art and music and mathematics change teaching for the better?

      I’m open to ideas, but I’d like terms to be at least approximately defined, so the ideas can be more solid and substantial.

  7. Oh, I’ll bite, but I’m just procrastinating on an essay which thinks that pedagogical content knowledge can come by diffusion and sitting in a classroom.

    My idea? Math anxiety is real, but it doesn’t come from fear of maths, but lack of understanding and skill in maths and gaps in their mathematical ‘floor’. (I’ll receive my award for proving fire is still hot later).

    Specifically, that lack of understanding and skill is caused by task avoidance reinforced by whatever the corollary of luddite is to mathematics for both parents and -some- teachers (and society), and students do not learn basic skills, such as times tables, meaning that when it comes time to do -some- mathematics later on in life, they reach cognitive overload quickly- because their brain is panicking trying to figure out how to add fractions etc.

    Also, giving students the skills to overcome those difficulties is differentiation- removing the barrier to learning and engagement.

    On a realistic note, I do want to experiment with somehow using abacus’ as a tool to combat dyscalculia, but I’m not amazing with one myself. Based on research implying that the areas of the brain involved in manipulating an abacus (even after the point where it’s visualised muscle memory) are different from the areas of the brain affected by dyscalculia. Inference, not tested.

  8. Hmm. Stephen, are you procrastinating writing that essay, or procrastinating reading it?

    As for your first idea, yep, it sounds pretty much like proving fire is hot to me. (Which is not as trivial as it sounds: math anxiety people seem hellbent on proving that fire is cold.) I think you’re making a very important point in your second and third paragraphs, but you’re not doing more than critiquing standard maths anxiety nonsense, correct?

    I’m sorry, but I don’t understand your fourth paragraph.

    I’m interested, if you wish to expand upon your fifth paragraph. Is there a formal (or semi-formal) definition of “dyscalculia”? How does the research indicate that these areas of the brain differ (and do you have references)? And, most importantly, how might any such insight be a benefit to the treatment (?) of dyscalculia? Is there any evidence as yet for any such benefit?

    Of course you’re not claiming to be an expert on any of this, and I’m not trying to pretend that you are. But any fleshing out of your conjecturing would be interesting.

  9. Marti,

    Yes it is certainly long winded and most of the ideas are not new but I notice Martin Gardiner’s review was very complimentary in the above link.

    I was thinking simply of the connection between creativity and the imaginative side of the brain which often occurs in mathematics , music,logic and art . Personally I like visual solutions and ‘ proof without words ‘ which often get the teaching ideas across quite nicely. Eg Mr Polster’s book QED

    Steve R

    1. Hi, Steve. I remember the book was very popular when it came out, and I guess the ideas were new to a lot of people. At the time I was an undergrad, immersed in pure maths and I had studied a lot of formal logic, and sufficient music. (In those barbaric times, logic was still considered a valuable subject for maths student.) So, I had little interest in Hofstadter, but I can appreciate the reasons for the interest.

      In terms of my conjecture/challenge, I don’t know of any implication/suggestion from GEB for maths ed that was particularly new, even if it was a force for these ideas.

      More interesting is your note about visual solutions, PWoW and QED and the like, which might falsify my conjecture, although I don’t think so. There has definitely been an increasing use of pictorial/geometric justification in undergraduate education. Probably less so at the school level, which has resorted to pictures for decades or more, but still some increase here as well. Now, the extent of that change, and whether that change has been for the good, I think that’s a very tricky question. Probably worth its own blog post.

      1. Hi,

        the Blue1Brown series is another good example of animated visual solutions used to cover a range of secondary , undergraduate and research topics .

        Grant Sanderson and friends in my opinion have done an excellent job and even give a plug to the Mathologer in the second link.

        a couple of examples

        Introduction to Calculus –Lesson of 1 of the series

        Solving one of the harder William Putnam questions on a tetrahedron in a sphere

        Steve R

        1. Hi, Steve. It seems to me you’re now considering “visual proof” in a fundamentally different sense. in any case, I don’t like 3B1B and I don’t like Mathologer. I don’t think either in sum is a positive to mathematics education.

          1. Marti,

            Ok I will apply Voltaire’s principle in this case…

            Perhaps the first video is a little long but I liked the visual neatness of the 2nd proof using random diameters on a circle and a coin toss to pick a random point and then generalising the proof to 3 dimensions .

            Steve R

            BTW I can vaguely remember struggling to get my head around epsilon delta proofs of limits before the Internet was invented . To me iatbtgatbtime it seemed a little dodgy to reverse engineer the delta values.

  10. I guess I’ll have a crack and hope I don’t get cracked too hard.

    Human Memory: A Proposed System and its Control Processes, R.C.Atkinson and R.M.Shiffrin. (1968).

    Its about how working memory process information into long term memory. The big take away was cognitive load and information that was meaningful to the learner was able to be stored in long term memory.

    I’ll given an example of how relates to maths.

    I had to teach integration to a group of students who didn’t find mathematics the easiest subject.
    I also had less time than usual to teach it (this included integrating non-polynomial functions like exponentials and trig functions) due to misjudged time leading up to an assessment – had to condense the topics without loosing rigour.

    So what I did was start with the concept of the anti derivative and focus on the procedures. I created meaning at this stage but relating it to the previous topic we had done about calculus and rates of change (e.g. finding displacement given velocity). When through integrating various functions, essentially it was always applying the concept of the reverse chain rule. The repetition helped students remember but it was not brain dead repetition of 50 question with little though; it was repetition with meaningful change to highlight the connections between the different functions.

    Then I went into the area under the curve concept of integration (1st and 2nd fundamental theorem of calculus). With the procedures already understood, there was less cognitive load on the working memory for when we approached the concept of definite integrals, area between curves and all that. I could also link together applying the different functions to similar questions (more repetition).

    I ended up completing the topic before the assessment and my students did well in there assessment. So I’ve fund what I’ve learning about memory to help me teach more efficiently.

    1. Hi, Potii, and I appreciate your bravery. I was assuming at some point someone would bring up cognitive load theory.

      I think the idea of CL is extremely important in maths ed, and all learning, but is any of that really new? That is, I don’t care if the science guys think they understand CL better now, what I care about is whether and how CL has changed maths education.

      It seems to me that maths teaching as far back as I know — and discounting modern lunacy — has emphasised the importance of having facts and techniques at your mental fingertips, so you can focus upon what is novel about the problem in front of you. Any old textbook I can think of has dozens of problems on any topic, usually very carefully chosen, and with progressively building depth and novelty. And, old textbooks typically don’t tell you to look at Example 5 to do Exercises 3-7: rather you were expected to develop and to retain a set of techniques, a set of weapons to apply to any new problem. What else is drilling multiplication tables about? And on and on.

      So, maybe I’m caricaturing the (critical) message of CL to education, or the newness of that message, but if so I’m not sure how. If cognitive load theory is now especially important in maths education it seems to me that that’s not because it is new, but because it has been forgotten.

      1. You make a fair point. I haven’t been around as long as others, so not sure how things used to be taught or what was known.

        1. Thanks, Potii, I completely understand and that’s really the point: the societal amnesia, the forgetting of a properly mathematical culture. The corollary of that is that garbage like NAPLAN and PISA, and their idiot supporters, are framed as “conservative” or “traditional”, whereas in fact they are just variations of the same modern disease. In the main, attackers of the “traditional” have no clue whatsoever of the tradition they are attacking.

          1. The problem runs deep here. In Education, most journals won’t allow (very small number of exceptions) references to papers more than ten years old. This means that they continually re-invent things as a matter of course. It is built into and exposed in the rules they do their research by.

            1. Screw them. I understand the corrupting pressure of “publish or perish”. But if in fact there is nothing of substance to publish then the whole discipline can and should sink, and we’d all be the better for it. If every maths ed journal on the planet went out of business tomorrow what would be lost? Seriously. Anything?

            2. Glen, I should add that any journal with a 10-year policy such as you indicate is actively poisonous and should be blacklisted by every maths ed author. Can you supply links to the policies of such maths ed journals? Email if you prefer.

              1. Hear hear!

                Imagine what would happen if this was policy for scientific journals. It’s proof positive of a culture that is rotten to the core.

              2. The one time I was co-author on an article submitted to IJMEST, this was the reason for rejection. However I do not find this rule anywhere on the journal’s homepage. I suspect they don’t publicise these things. When I complained to my colleagues in education, they said it is annoying but standard. Thus my opinion above is (in)formed…

                One of my best TAs (in the last 8 years) is currently doing a masters in education. She is running into this issue as well when publishing.

                1. Thanks, Glen. I have no idea what IJMEST is, but I’ll find it and enquire. If your TA, or anyone, can help you add to the list of purportedly poisonous journals, please do so. I’ll check out any journal suggested to have such an insane policy.

                    1. Thanks very much, JF. Simply checking on a year’s worth of references of Journal X for various values of X is on my to do list. (A long list …) But apart from the hundreds of smoking not-guns one many not find, if there is a formal Y-year policy along the lines Glen suggests, that should be discernible in stated policies or written correspondence. I intend to find out.

                      The editorial to which you link, however, is puzzling (as well as nauseating). I’m less than a fan of either of the writers (and I believe the feeling is mutual), but both were formally trained as mathematicians. As such, I find it difficult to believe that either would condone or enforce a Y-year rule.

  11. I’ve been thinking more about this — the conjecture, that is, and what it (if true) is saying about maths ed research in general.

    I actually don’t think it invalidates the field. I know, unpopular opinion. But hear me out.

    Good research in maths ed, if the conjecture holds, is then either:

    1) Rehashing in a modern context, rediscovering, etc, previous known good ideas; or
    2) Statistical evidence for these good ideas, with good analysis backing it up, further enforcing that they are actually good.

    (There can be a lot of bad research too, but that is true of any field, including math.)

    I think the value of both (1) and (2) is pretty high. I can certainly see why good researchers persist in writing things both of the form (1) and (2).

    1. Hmm.

      Glen, you wont be surprised that I’ll be looking to take a whack at this. First, I have a question. Are you suggesting research of type (1) or (2) does currently exist, or just that it might exist (i.e. beyond your knowledge of the maths ed world, and/or in a better world). If the former, could you provide some examples?

      1. Yep, I expect you to!

        I’m claiming might, but I wouldn’t be shocked to find does. As to proof for does, I don’t have any at hand, but, I present the following argument for your enjoyment.

        Suppose, there exist good ideas \mathcal{G}. Take any g\in \mathcal{G}. At some time, say t_G, the idea g was first exposed in the literature. Grab your nearest time machine and go back to that year plus one. Then, at that time, there was an example that invalidated the conjecture. It would also be an instance of does in the above.

        In order to contradict this argument, you would need to show that no good ideas exist (preposterous!) or that any that do exist are not exposed in the literature (seems unreasonable).

        OK, I’m ready for the stones :D.

        1. Hi, Glen, sorry for the delay and I’m still not properly replying yet. Just briefly for now, here’s why I think your time machine musings miss the point.

          First, my conjecture is most definitely not about maths ed research/ideas/orthodoxy in all places at all times: it is about now.

          Secondly, I expressly wrote that proving the conjecture was effectively impossible; the whole point was to challenge those who disbelieve the conjecture to present anything that they consider falsifies it. I was deliberately trying to put the burden of proof back on those who claim modern maths ed has any merit. So, yeah, i can’t prove the non-existence of g, but that’s not my job.

          For a similar reason, I’m less interested in the “might exist” version of your argument about good maths research than a “does exist” version. It’s kind of like “principled Republican” or “caring Liberal”: yeah, you can kinda sorta imagine that a few might exist, but it seems better to wait for a purported example to appear, and then ponder the specifics of that example.

          I’ll reply more directly to your argument tomorrow.

    2. Hi, Glen. Sorry to be so slow to respond (and for the eventual response to be so brief).

      Whether or not my conjecture is true, there has clearly been the forgetting of (or deliberate erasing of) good (read “essential”) ideas from traditional teaching. So, yes, there is then obviously at least some value in (1) and (2). One might argue, for example, people pushing cognitive load theory is in that category. Still, I have reservations:

      a) Rediscovering is one thing, and reinventing is another. They are different, and the former has an important message that the latter lacks.

      b) I’m much less convinced in the value of (2). I don’t think it’s easy, in the main I don’t think it’s necessary, and I think playing on that field makes it easy for simple reason to be lost in a field of muddy study.

      c) Is that the most modern maths ed research has to offer? Undoing the screw-ups of all the other modern maths ed research?

      d) Your “all fields have bad research” is a pretty weak both-sideism.

      1. Don’t worry, busy times.

        My root comment here is assuming the conjecture is true, which necessarily answers your part (d).(Of course, then I present an argument for why the conjecture can’t really have been always true, because then there wouldn’t have been good ideas. If it wasn’t always true, then why should we think it will be true into the future? It’s just an argument.)

        I agree wholeheartedly about (a). I wish people would be more cognisant of past advances. This is something we do really well in maths that other fields struggle with.

        For (b): Well, (2) is essentially IMO applied statistics, and I think statistics is quite valuable (no matter the application). It isn’t really education research, however many academics in education schools around the world end up writing many, many papers on this. (Some only on this.)

        And, just to be clear, I really am quite heavily saddened by the vast majority of research that I see in maths ed, so I’m not very keen on arguing too hard. I suppose my overall point here that I’m driving at in this sub-thread is thus: bashing all of maths ed seems unfair. There are some good people trying to survive in there, and I think we should instead bash specific shit research, specific shit conferences, or even specific prolific (“famous”) shit researchers. The good people can do research of (1) and (2) until they get a good idea and the mafia in charge let them publish it.

        The good people may be heavily influenced by the environment in which they find themselves. I want them the engage in things counter to the main, like this blog, things that get to the heart of what it means to teach mathematics well (and not well). I think if they see “all maths ed research is shit” and then just move on, that’s a missed opportunity.

        1. Thanks, Glen. I think you’re clutching at straws a little here. I’d think it a lot less if you were arguing the “does” version, rather than the “might” version. Show me specific researchers and specific research, and I’ll happily (and hopefully fairly) argue the merits.

          As for statistical “evidence”, I will continue to claim that in in certain areas and for certain questions, its reliability and its importance is over-rated. I am happy to be out on this limb.

          Do I really think all maths ed research is bad? No. Do I think maths ed research is generally and fundamentally screwed up? Yes.

  12. Many improvements in mathematics education come from improvements in education generally, which have been adopted in the mathematics classroom.

    One example is abolishing corporal punishment – which happened in the last 50 years. I recall that the Bracks government put the final nail in the coffin of corporal punishment in Victoria. (I guess that this decision was not based on any research and so does not qualify for the competition.)

    Another example is the emphasis on learning by Indigenous students, and students from various cultural backgrounds which is also recent. I found this useful in my pre-service teaching.

    Even NAPLAN has its uses. The results make it abundantly clear that the variation of ability in a single class can be wide, very wide. Now this may be obvious to an experienced teacher. However, as one who has worked in hospital statistics, I know that the value of research in health care is that, often – but not always – it validates the views of the people who work on the wards.

    NAPLAN results help to prepare those new to teaching, like me, for the reality of the classroom.

    Developments in IT have had a huge impact on teaching, especially mathematics, and this impact will only increase. Our task is to maximize the benefit of these developments. It’s important to get this right, not necessarily be first in using it. (Which university was the first university in Australia to introduce white-boards? Who knows – and who cares?) I can easily imagine robots being used in the classroom.

    I don’t know the meaning of “constructivism” but I gather that this was invented more than 50 years ago. However, I have found that it is useful to have some theoretical basis for what I do in the classroom. (Of course, as the adage goes, in theory, there is no difference between theory and practice – but in practice there is.)

    Education is a social science, and research in the social sciences is quite different from research in mathematics.

    It just occurred to me about this 50 year limit – 50 years ago, I was 22 years old! Seems like yesterday.

    1. Geez, Terry, that’s your argument? The great benefit of education research is that we now understand beating up kids isn’t great for their learning?

      I’ll address the rest of your comment with (slightly) more seriousness when I get a moment.

      1. I did say that the decision to abolish corporal punishment was probably not based on any research, and hence my comment does not qualify for a prize in this competition.

        In 2020 Australia, abolishing corporal punishment might sound like a no-brainer. However, as far as I can tell, many countries, including Australia, abolished it only very recently (well after I had left school !), and, in some countries, the practice still exists.

        Educational reform is a slow process.

        1. Yes, Terry, I know you weren’t entering the competition with that comment. But to point to the end of the strap as an “improvement in maths education”? Yeah, I guess. But then so is the elimination of polio.

    2. I agree. I think there is educational research in general that might be useful for understanding students. Also you might be surprised at how many Australians think smacking kids is a good idea:

      In the same vein, an idea I’ve found informative is the Dual Iceberg Model for bilingual language proficiency by Jim Cummins (from the 1970s): basically the idea is that there is common element of academic language proficiency that overlaps between languages. It can be used to justify allowing students opportunities to maintain and develop proficiency in their first languages. I’ve heard that it is backed up with proper research showing that students with academic language skills in their first language acquire them easier in other languages. (I think some people used to think one language competes with the other and you had to stamp the first one out!)

      This is very much in the vein of something that I could maybe use to make the case in favour of what I already believed in though. Some Australian teachers are very ENGLISH ONLY about things, and I never liked that. When I was in high school, I had a physics teacher who used to enforce this quite sternly and wouldn’t let students speak to each other about physics in Chinese, but didn’t have any problem with students talking about whatever in English.

      1. Thanks for the comment especially about the use of several languages in a classroom. This raise many interesting questions and issues.

      2. Thanks, S-T. I’m curious about bilingual education, if only because my primary school daughters attend a language immersion school. I’m open to the idea that there may be benefits beyond the obvious, and that some understanding of this may be new. I’d need to think about (or be convinced of) this, however. In particular, I’m not sure how directly it relates to the topic of this post. I’ll happily look at any article you want to throw at me. I will also note this study; I haven’t read the study, so in no way do I vouch for it or reject it.

        As for some people still believing that it’s ok to whack kids, colour me surprised. There’s a whole “civilised” country that has declared cold-blooded murder to be neither cruel nor unusual. Go figure.

        1. Hi. Yes, I don’t think I will make an entry in the competition (because I just don’t know much). I have lots of questions about mathematics education though, and most of them are to do with understanding people, how they think, what they want, (how many people have trouble with abstraction really and why?), what is the point of all this anyway? what are the political forces pushing us one way or another with the curriculum?

          I wouldn’t mind if the research which answered these questions was an application of trivial ideas to the current context. In fact that would make it more useful because I might understand it and be able to explain the case for using it.

          You wouldn’t think you’d need research evidence to say “hitting kids is not good”, but then you do.
          You wouldn’t think you’d need research evidence to say “people who speak languages other than English shouldn’t be treated like they’re inferior”, but then you do.

          I imagine you can think of a maths-related example. 🙂

          1. Thanks, S-T, and a few comments in reply.

            I don’t believe, and indeed I know that it is false, that you “don’t know much”. At minimum you know the right questions, and the right kind of questions, to ask. What is not so obvious is whether those questions can be answered in a substantive manner by education research that will help (and not hurt) in real life teaching, beyond what an intelligent and empathetic teacher learns from their practice. And what is very very very not obvious is whether a Grand Theory of Learning or a Neurowhatever Brain Study helps one iota.

            As for your decision to not enter the competition, that is a very interesting choice. Perhaps you’re just aware that the prize sucks, but let’s assume otherwise. You are a smart and questioning student-teacher, who is currently being bombarded with this stuff. Is there really nothing that has been thrown at you that you might contemplate has some worth? Nothing that you might conjecture falsifies my conjecture? I’m not pushing, just noting.

            As for the necessity of research to say, for example, hitting kids is not a good idea, I’m sorry but I think you’re wrong. I doubt that any such research had any influence on the lessening of such practices.

            1. In my teaching degree, I mostly just encountered the opinions of one academic about mathematics education. I only had one lecturer on the topic and their assignments were geared toward supporting their point of view, which I didn’t find very helpful. It wasn’t even so much that I disagreed with them but that they seemed to live in an entirely different world to me, and they were very satisfied there, but it didn’t match with anything I knew.

              So I don’t feel like I have delved far into the world of mathematics education research yet.

              ETA: With regard to the hitting: yes, I guess you are probably right.

              1. Thanks, S-T. It sounds like you learned a lot from that maths ed lecturer, in an anti-manner.

                I guess it also points out a problem, or at least a confusion with my challenge: what counts as maths ed research. Of course there is specific Boalerisms and the like, but most of it is more general, and then is expected to apply and be applied to maths ed. So, Boalerism is built on growth mindset, but not every Grand Education Theory has such a clearly promoted mathematical instantiation.

                And, research and hitting, yes I’m afraid I probably am right. I think it’s a natural desire to try to prove that people/society shouldn’t be assholes, but I can’t see that such proof is ever effective.

    3. Hi, Terry. Sorry to be slow to respond. I’ve already responded about whacking kids. Briefly on your other points.

      a) I agree, the modern concern (in Australia) to better include and cater for kids from marginalised groups is real and important. I don’t see that this has occurred because of any research or accumulating evidence, rather than from an overall cultural change.

      b) NAPLAN is a disaster. Whatever minimal benefits a teacher might obtain are trivial in comparison to the “numeracy” poison it has injected into primary school teaching.

      c) IT is a disaster. Yes, it is important to get it right, and we haven’t come close.

      d) You can claim that constructivism, or some other “theoretical basis”, helps you. Unless you indicate what and how, I won’t comment. Except:

      Terry, you are a great teacher, and you were a great teacher 25 years ago, before you started reading all this nonsense. Try to not rob yourself of the credit.

    1. Thanks, bosjo. An interesting suggestion, and one that leads to other interesting suggestions. Wästlund’s proof of the Basel formula is indeed beautiful and (to the best of my knowledge) novel, and 3B1B’s video has exposed his proof to a couple million people.

      As you suggest, Wästlund’s proof isn’t an “idea” in mathematics education, but one can think about the idea behind the idea: finding an enlightening/engaging/unifying proof or insight to a known truth. That is not new, and it is not new to (at least try to) incorporate it into mathematics education. Mathematicians have been proposing this kind of material for decades, or longer. Hilbert and Cohn-Vossen’s 1932 Geometry and the Imagination (Anschauliche Geometrie) is full of Ahas. (John Conway was of course a genius at this, before Trump killed him.)

      One could argue that including such one-off aha material in a more systemic manner is (or would be) new to maths ed. One would also have to argue that this is (would be) a good thing. I doubt it.

      The other aspect, which is clearly new, is the rise of popular mathematics in video form, such as 3B1B/Numberphile/Mathologer. Again, one then has to question that, to the extent that school (and even university) mathematics mimics or uses such material, whether that is a good thing. Again, unless it is in very small and carefully selected doses, I doubt it. (Not an argument, but to get a sense of my perspective, I did not like the 3B1B Basel video.)

      1. I’m interested to know what about the 3B1B video you didn’t like. Did he get something wrong or is it just the way he presented the proof, or something else? Not trying to defend him, just wondering what grains of salt that should be taken when watching videos like that.

        1. Craig, I’ll answer, but may I first put the question back on to you? What did you like about the video? What did you get out of it?

          1. Well I’d say that it was a well presented video that went through each step in (some) detail and made it (somewhat) easy to understand. What I got out of the video was seeing that proof for the first (and only) time, and I guess it was just a nice video to watch that connected different parts of maths with each other.

            1. Thanks, Craig. I know that most mathsy people really like 3B1B videos. Or at least they claim they like them: I’m not entirely convinced.

              For me, the videos are like Merchant Ivory movies: the effect (and, it feels to me, the intention) seems to be more that they be admired rather than enjoyed. So, yes, 3B1B videos are wonderfully produced to the microscopic level, they’re beautiful to look at, and they tell a story in a careful and accurate manner. But, for me, they’re also humourless and boring. I’m also not convinced that the 3B1B videos, or videos by the other big hitters, are all that educational, and they are arguably anti-educational.

              On the Basel video, I watched it at the recommendation of an extension (AP) student, after my class had proved the Basel formula in an assignment. I hadn’t seen Wästlund’s proof and I thought it was fascinating. For about three minutes. Then there was the work of going through the details, and at that stage I got bored. I just wanted the argument written down, so I could go through in a proper manner, at the pace my brain was able to go.

              I don’t think video is an effective way of presenting such an argument, except maybe for mathsy people who have already done enough of the hard yards. I do think it’s an effective way of having lots of people tricking themselves into thinking they know a proof when they don’t.

              1. Yeah, no, I must be an unusual mathsy person then. I don’t likee 3B1B, I don’t like most math youtubers. 3B1B, Khan Academy, ugh. Mathologer is good though. But I think his videos are quite different in style.

                I don’t hate math on youtube though. I think it can be good, just at a different level. For example, I watched Leon Simon give GMT lectures a year or two ago on youtube, and I’ve seen plenty of decent seminars that I would have otherwise been unable to if it weren’t for youtube. Banff (is not youtube but still) have videos of most of their meetings, I like those too.

                In the days of COVID, I’m also making my own videos. But, they are simple recordings of live lectures. And the production quality is TERRIBLE. I only care about teaching my students the content, and making sure they understand. I think there is a certain value in the videos I make, but only for my students. I’m not sure they would be helpful for the general public (and anyway I’m not allowed to put them on youtube publicly, so :P).

                My productivity is actually pretty high (doing anything at all) when there is a talk on in the background. Years of training :).

              2. Finding myself defending the use of videos in maths education is actually quite amusing. To be able to appreciate that, you’ll have to know that I haven’t had a TV since 1985, haven’t watched a film since 1990, and generally hate when things start moving and “sounding” on a web page…

                I completely agree that a video is not the proper medium for understanding a mathematical proof, and can even take your argument a bit further by claiming that “understanding” is a process of stepping through a series of “propositions”, and understanding how to climb from one to the next. This is far better done on paper (or an equivalent medium), where it is easy to keep both propositions in view; in a video, the propositions are usually shown one at a time, so even if you stop the video, it is not easy to grasp the actual steps from proposition to proposition.

                However, I am not convinced that ordinary people really believe they “know” a proof seen on a video; at least I don’t. In my opinion, the value of 3B1B, mathologer and similar video channels is what I would call “Gardner value” — a light-hearted quick glimps of what higher maths is and what it can do, just as Martin Gardner did in his classic columns in Scientific American. I see these video channels as heirs to Gardner more than new educational tools, and the importance of that should not be underestimated.

                1. Thanks, bosjo. Yes, these video guys are the direct descendants of Gardner. The first problem is, they do less than Gardner, while creating the illusion of doing more, which, knowingly or otherwise, tricks people. The second problem is that they are not an addition to Gardner, they are a replacement for him. They are teaching people to not read.

            2. Sorry Craig, just to reply directly. I think one issue here is in your comment: …went through each step in detail and made it easy to understand.

              In my experience, there is a certain difficulty that some things possess — many mathematical things — and you just won’t get it straight away. Like if you have to climb a 100m hill, you’re not going to just hop over it. It won’t happen. You have to climb it (or not). People can encourage you, but if someone is “making it easy” then I think they are just tricking you.

  13. The issue of teaching out-of-field is a recognised problem in Australian schools. The issue was first raised in a research report by Paul Weldon (2016) “Out of field teaching in Australian secondary schools” from ACER. AMSI has been pushing this issue. The federal government report “Shifting the Dial” (2017) contains this quote.

    “Teaching out of field should be addressed through targeted professional development of existing teachers willing to acquire the relevant knowledge. Teacher salary differentials should also be used to overcome subject-based teacher shortages.” (p. 92)

    Click to access productivity-review.pdf

    It could be the time that the recommendations might be put into action.

    This is an example of educational research that, if implemented, would have an impact on the mathematics classrooms in Australia. As I said before, progress is slow.

    1. Yes, RF, that’s one point (but not the only point). The idea that one needs research to point out the extent of and the problem with out-of-field maths teaching is utterly absurd.

      1. I suppose my gripe with this date is: if in 2016 such an idea was considered “new”, what have the maths ed researchers been doing/thinking pre-2016?

  14. Again, from my experience in health care, one aspect of research is that it often – but not always – verifies what people on the ground have observed. Arguments for change can then be based on research findings rather than personal opinion. Sometimes – and I say sometimes – people on the ground have impressions that are wrong.

    Since Weldon (2016), little has been done. Weldon’s work was based on a survey of teachers in 2013; as far as I know this was, and still is, the only source of such data. Even the definition is debatable. One might propose several definitions of teaching out-of-field, all of which would lead to different estimates of the extent of the problem. And even then, where would you get new data from? Is teaching out-of-field increasing or decreasing? How does it affect different disciplines? Is it worse in the bush than in the city? What are the views of principals?

    So it’s not so easy to measure the extent of the problem.

    I have asked state and federal politicians about this issue. Some could not see a problem because teachers are trained to teach anything. Some gave more thoughtful replies. Some didn’t reply.

    Just to clarify things, I taught out-of-field last year; way, way, out-of-field. I spent more time preparing the classes in that subject than all my mathematical subjects. I enjoyed it; I like to think I did a good job; but I would not be inclined to repeat the experience. At the end of the last class, one student said to me “I think that you are a pretty good teacher … but I don’t like your rule about music.” (I don’t allow students to listen to music in class.)

    1. God, Terry. I still haven’t had time to get to your previous comment, but let’s deal with this.

      I understand that politicians and the general public typically have to have the bleeding obvious hammered into them with statistical evidence or whatnot before anything will change. I also understand that sometimes the “obvious” is not true, or at least more nuanced and more difficult to quantify than those close to the action may believe. O-O-F teaching isn’t remotely as nuanced as you seem to be contemplating but, yes, to the extent that ACER/AMSI are doing the hammering on O-O-F teaching, that is prima facie a public service. At least it is if they don’t stuff it up.

      But I get sick to death of people whining on and on and about out of field teaching as if it’s the only problem with Australia’s maths education. It is isn’t. There are other, much greater problems to be concerned about. Problems that AMSI ignores, and problems that ACER defends, promotes and profits from.

  15. Every day I look for teaching jobs in the secondary mathematics (this is the only subject that I am qualified to teach). Some positions are for teaching mathematics + something else. I see some odd combinations such as mathematics + PE, mathematics + psychology, mathematics + English, mathematics + humanities. Or today, there is a job for a teacher in mathematics + science + English (Additional Language). Do principals really expect to find well qualified teachers in these combinations?

    1. I have heard that sometimes schools have a candidate already in mind for a position, but are still required to advertise it. So they advertise it with the existing candidate’s precise combination.

      1. As someone who has written these advertisements on more than one occasion, that is true SOME of the time. About half the time someone resigns/retires and the principal wants a direct replacement for them because the timetable has already been done or some other excuse.

        Tends to happen a lot more in the public school system. By which I mean I’ve never seen it in a private or Catholic school setting (not saying it doesn’t happen, just that I do know it happens in public schools).

    2. s-t’s suggestion that

      “… sometimes schools have a candidate already in mind for a position, but are still required to advertise it. So they advertise it with the existing candidate’s precise combination”

      could be true in your case, TM (unfortunately, it is true in many cases, particularly in the State System, and causes great despair and time-wastage for the unaware external applicant).

      However, I think it’s more likely that the positions you’re looking at are for teaching at Years 7 – 10 level. Because, as we all know, anyone can teach maths at those levels, you don’t need to be a specialist maths teacher (ahem unlike Trump, I’ll declare my sarcasm now rather than attempt to use it as a feeble excuse down the track for a dickheaded moronic ignorant comment).

      Typically, when a specialist maths teacher is wanted, the ad will ask for a maths teacher [ for whatever year level] and will often include a statement like “The ability to teach ….. would be an advantage”

  16. I decided I was unfair to my teaching degree. There was one idea that I learned that I think is perhaps new, true, and non-trivial.

    This is the idea that when writing problems and assessments, teachers should be aware of common arithmetical misconceptions and strategically include questions that allow the teacher to diagnose them.

    For instance, in relation to misconceptions about decimals, see articles by Kaye Stacey and Vicki Steinle where they claim (based on research) that some students will think that 0.15 > 0.8 because they have a ‘longer is larger’ misconception. These students may have formulated a ‘rule’ that works enough of the time to mask their complete misunderstanding of decimal notation, allowing them to pass into adulthood still not understanding it (or knowing that they don’t).

    Is it new?
    I searched (“misconception*” and “mathematic*” and “student*”) in my library database and the earliest I found advocating for being systematically strategic in this way was an article called “Diagnostic Models for Procedural Bugs in Basic Mathematical Skills” by Brown and Burton (1978).

    Is it trivial?
    I’m not sure it is, because it seems falsifiable. For example, I wondered if these shortcuts in thinking are actually stable within individuals. Maybe students would devise a rule in the moment (because they are overwhelmed?) and delude themselves that it makes sense, only to realize the next day that it is incorrect? Personally, I’ve managed to trick myself sometimes (not recently in arithmetic but other things) and then realize an obvious error when I go for a walk or something. I wouldn’t call that a misconception but rather an error in logic.

    However, it seems like individuals do maintain persistent misconceptions:

    1. Hi S-T!

      If I understand you correctly, I think this idea may have been exposed by Polya. I don’t have them handy, so I’m working from my shoddy memory here, but I believe it may be in:

      “Mathematics and Plausible Reasoning”, by Polya (note there are two volumes).

      This is a pedagogy text.

      Another great contributor to the written description of mathematics teaching is Klein, and I am reminded of (in connection with another recent discussion here, but may also be relevant to this one)

      “Elementary Mathematics from an Advanced Standpoint”, by Klein (note again two volumes).

      I believe the dates on these are 1948 and 1954, although certainly they were “standing on shoulders”. It is more a reflection of my limited knowledge than a claim that in e.g. the 19th century the whole gamut of ideas around teaching mathematics did not exist.


      1. Thank you. I don’t have access to the book by Polya either right now. But an ebook version of the book by Klein is available through my library so I’m going to read it. Wow! That’s the kind of thing I think would make for good teacher education.

        1. Before you graduate, go and get a copy of Polya’s “How to solve it.”

          Even better, get your students to read it (or a summary at least, there are several online) and then remind them of step 1 constantly…

          1. Thanks. Yes, I have that one already. 🙂

            Also, I dream about having a classroom maths library one day.

    2. Thanks again for your posts S-T; they are interesting.

      In a Year 11 class recently I found that several students write the remainder after division as a decimal; e.g. \displaystyle{\frac{7}{3}} = 2.1.

      In another Year 11 class, I found that several students believed that 8-2+6 = 0. Not just one student – several of them. Their argument was that according to BODMAS, you add before you subtract. This is what they been told by their teachers in earlier years. I suggested that they work it out on their calculator and they got a different answer. “That’s because the calculator does not use BODMAS” was their reply. Finally one student piped up “Whatever” – what a put down! One student suggested: “Look Terry, you are not making any headway on this. You need a new approach.” I agreed.

      When I think about all the money spent on the education of these students up to year 11, I wonder if it was worthwhile.

      1. Calculator :(.

        I’d take that golden opportunity to explain to them the inherent ambiguities behind BODMAS, maybe throw in a reference or two to popular twitter memes that do the rounds (only GENIUS-level brains can understand this!! can you??). Should be a valuable learning experience for them!

            1. Yes, Terry, you’re correct. The example you put up is beyond standard BODMAS nonsense. So, why do you think current Year 11 students so, so suck at arithmetic (and algebra)?

              1. Come on Marty, we all know the answer to that one… CAS.
                When teachers have to spend time teaching button pushing, some things need to be assumed, or at least occupy less time in class. When I last taught methods year 11 I agonised that so many students didn’t know how to find the equation of a line given two points… it was (and still is) depressing that so few “colleagues” see it as an issue.

                1. Well, yes, my question was mostly rhetorical. But although CAS is undoubtedly a disaster, I think the real damage comes much earlier.

        1. Oh, S-T, please stop apologising. Your comments are very thoughtful and very intelligent. They add hugely to the discussion here, and on the other post. I’m not shooting at you: I’m shooting at the awfulness of your teaching degree. (And you know me: you know the way I shoot.)

          As to the recognition of and concern for fundamental arithmetic misunderstandings, no that’s not new. But I’ll bet it’s hell of a lot more of a problem now than it was 50 years ago. Why might that be?

    3. Off the top of my head, I’d suggest that some students (of all ages) think 0.15 > 0.8 simply because 15 > 8.

      And again, off the top of my head, I’d think the ‘longer therefore larger’ hypothesis for the misconception might be easily tested by comparing

      0.15 with 0.800

      0.15 with 0.8

      0.15 with 0.08

      etc. and looking for a pattern in the answers.

      Maybe this has already been ‘researched’ – I don’t know, but I do know it would be an intuitive part of the discussion I had with students if I was teaching this stuff. It’s surely the job of pre-service teaching training to ensure they are familiar with the common misconceptions students have about a variety of mathematical topics.

      A big problem with teaching any subject is the misconceptions that students bring that you have to try and extinguish. These misconceptions come from somewhere and are usually the result of poor/lazy teaching (which includes the misconception being implicitly or explicitly taught by a teacher). Once entrenched, it takes real effort to extinguish a misconception held by a student.

      A particular bugbear of mine, whilst on the topic of misconceptions, is the teacher who reinforces to his/her students that a graph can never cross an asymptote. This is just either plain lazy teaching or ignorance.

      By the by, I would suggest that in response to Marty’s challenge, a significant positive change in modern mathematics teaching (at school) is perhaps the much greater explicit awareness and understanding of misconceptions that students can have, and the explicit teaching strategies that can be used to try and identify and eradicate them. (Yes, I know that experienced teachers have always known this through their accumulation of experience, but I think there is greater awareness in inexperienced teachers these days).

      (I’d suggest this is one of the greatest positive changes in modern science teaching)

      1. Thanks, JF. I’m willing to entertain the idea that the understanding of and handling of mathematical misconceptions has improved, but I’d have to be convinced. I’d also entertain the idea that the prevalence of such misconceptions has increased.

        1. I don’t have the time or energy to collect evidence and attempt convincing arguments that “the understanding of and handling of mathematical misconceptions has improved”, so I won’t even try.

          Maybe contributors with more recent experience with pre- and post-service teaching courses, are more widely read in educational research or are working closely with more novice teachers can jump in if they think the suggestion has legs. However …

          I agree with “the idea that the prevalence of such misconceptions has increased” and think that there are several (non-mutually exclusive) reasons for this. One of those reasons might be that if “the understanding of and handling of mathematical misconceptions has improved” (my emphasis), then this might be expected to lead to a perceived increase in prevalence …

          (In other words, the nature and extent of mathematical misconceptions is largely unchanged but greater awareness and understanding has led to a greater detection).

          Marty, I would include in the above ‘reason’ the suggestion that you’re possibly seeing more secondary school students in the last few years and it might therefore be expected that you would see a greater prevalence of misconceptions …?

          (And yes, I’m aware of my misconception that ellipses add gravitas to statements …)

          1. Thanks, JF. I am definitely willing to entertain the possibility that “misconceptions” are now better identified and addressed. I’m also willing to entertain the idea that there has been no general increase in “misconceptions”, and that my feeling that there has is just a consequence of my own blinkers. Indeed, I trust your instincts and observations on this as much or more than my own.

            But, two arguments for why misconceptions may have increased: 1) the decline of drilling and the practice of automatic skills; 2) calculators. If so, and if modern treatment of misconceptions has improved, this suggests to me the doctors treating the injury they created. Um, thank you?

            I’d still be curious about specific research, both now and in the past. (Not your job, unless you’re entering the competition.) For example, student-teacher’s citing of Stacey and Steinle. Was that really all that new or insightful? Did it change what teachers do, and if so, why?

            Perhaps, as you seem to suggest, such research (when promoted) is helpful for less “experienced” teachers. (Is that really the word you want?) Which raises the further question: are maths teachers less “experienced” now than in times past? It seems clear that the mathematics training of qualified maths teachers is generally poorer than it once was.

              1. Hi, RF. Not quite sure what you’re suggesting here. I accept JF has a sense that the awareness of and consideration of misconceptions has improved, and he’s no obligated to substantiate that. Then, nor is anyone obligated to take JF’s claim on face value.

                1. Yeah, I re-read that and realise my sarcasm was far too well disguised. If very few people read the published rubbish, the authors can keep on writing it, referencing each other and getting their publication count up while everyone in classrooms just does their own thing, totally ignorant of what “academics” say is happening in education.

                  My point was, in the main, IF JF is the person I think they are (its been a while, btw…) then they are a well educated teacher who frequently looks to go beyond “the box” of thought. So if they are unaware of something, maybe it is not worth looking for.

                  1. OK, whoops, sorry. I think I should look for a plug-in that by default inserts “(sarcasm)” at the end of every post and comment.

                  2. I’m flattered, RF. In reality however, there’s probably lots of things I’m not aware of that are worth looking for ….

                    However, I have found that teachers can become totally absorbed in the latest ‘shiny toy’ sold by the snake-oil salesmen – with little or no empirical evidence that it makes much difference. In fact, when challenged on producing the evidence, you just get fogged. So I usually, rightly or wrongly, just stick to my old-fashioned way of doing things and ignore all the faddish ‘best-practice’ crap that comes (with trumpets, dancing girls and ticker-tape) and goes (silently slinking into the night without a whimper).

                    By the way, I’m very happy to be the person you think I am (and quite possibly I am!) if it means taking credit for being “a well educated teacher who frequently looks to go beyond “the box” of thought”

                    (although I’m troubled by the lack of words such as ‘charismatic’ and ‘adonis-like’)

                    1. JF one question will answer it for the both of us. Did you bring an air horn to Latrobe University one day a few years ago while wearing a “Listen to John” T-Shirt? If so, I referenced your paper from that day in a publication in ASMJ a few years ago… some detective work should you choose to accept it…

                    2. It could have been me, RF ….

                      1) I do own a T-Shirt that says what it says and have worn it to some presentations.

                      2) I also have an air-horn which I freely use (including at my current school which has no bells – my not so silent prank this year).

                      3) And all my presentations were published in the MAV Conference Book (with the exception of one year (*2014) where I missed the cut-off date by a couple of days and an exception could not be made – The unpublished paper was called Probability and the Pell Equation – Experimental and Algebraic Approaches).

                      4) My papers were among the very few that were actually mathematical (most of the presentations were either commercial, CAS or ‘a new way to teach fractions’)

                      Can you give me a hint – what category did the quoted paper fall in? I’ll make it multiple choice (who doesn’t love a good multiple choice question haw-haw!):

                      A. Probability and Statistics (2007, 2009, 2010, *2014, 2015).
                      B. Integration (2011).
                      C. Mardens’ Theorem (2012).
                      D. Subsets of the Complex Plane (2004, 2005, 2008).
                      E. Polynomials (2013).
                      F. Inverses (2006).

                      Or better yet – what year? I guess the fact that it was at Latrobe rather than Monash eliminates 2004 (one of my favourite papers) …. The move to Latrobe still irks me to this day.

                      If I am who we both think I am, I’ll suggest it was either 2010 or 2015 (the only two papers of mine that contained some interesting and original calculations. The 2014 presentation wasn’t published – although I did hand out copies of the paper).

                      I was originally going to present in 2016 and beyond – 2016: The Many Ways of Proving the Sum of Two Normal Distributions is Normal – but lost interest for a number of reasons. These days I would not even consider presenting, given that you have to pay to present.

                      What ethical organisation sells what you give it, but charges you for giving it to them! Without Presenters the MAV Conference would simply be one big infomercial (which it more or less already is these days from what I’ve heard).

      2. Yes, I feel like as a new teacher, I really want a heads up on how students might be thinking in advance. Sure, I will ask them to explain things to me and figure it out from what they say, but it would be good to have a general vague idea of it beforehand.

        So I appreciated the research of Stacey & Steinle (knowing nothing else about them) because they tested some students on decimal problems, and then interviewed the ones who made consistent errors, and asked them to explain what they did. The uncovered specific patterns and various misconceptions associated with them (some which I wouldn’t necessarily think of) and that’s the kind of information I need.

        1. Thanks, S-T. I’m sceptical, but I’ll look at the paper and ponder this more.

          I’ll confess, I’m a lot more interested in Stacey’s experiment with the VCAA, to push CAS calculators on a generation of Victorian kids, to see just how much it could fuck up their education.

          1. Oh, I didn’t know that was her. I can understand now. There are so many reasons to not require CAS calculators in VCE. They’re a big reason why I feel that the subject is just not one I know. It’s not the subject I did in high school, which is not necessarily bad but it feels like they secretly deleted a subject.

            And it’s a barrier to students now because $200 dollars is not a small amount of money for a lot of people. One of the great things about maths to me when I was young was that you didn’t need a lot of equipment or money, just your mind and a pen and paper (I even made my own ruler out of cardboard in early high school to give you an idea). And if the person who corresponds to “me” in some sense was doing the VCE today, maybe they wouldn’t take maths because $200 for a calculator would seem like a major investment. I just think that’s wrong, and it makes me wonder if it’s intentional or blinkered and for what? To seem modern?

            1. Thanks, S-T. It’s beyond not just the subject you did in high school: it’s not a subject. And, CAS poisons a lot more than VCE. The poison trickles (?) down.

    4. Hi, S-T. I finally found some time to read the article by Stacey and Steinle. I’ll write a bit, but to summarise my reaction: meh.

      Here’s a more detailed response:

      1) Being able to order decimals is important, and of course the inability to do so may indicate deeper problems and will likely result in further difficulties. The awareness of this issue is hardly new and I’ll bet goes back to roughly 24 hours after the introduction of decimals. I checked two old Year 7 texts, one from 1983 and one from 1966. Both had an excellent chapter on decimals and decimal arithmetic, and both had an excellent section with very good exercises devoted to decimal ordering. In particular, both texts were explicit and emphatic on the correct manner of determining the size of decimals, a message reinforced by other sections of the texts.

      2) Stacey and Steinle’s study considered more carefully various erroneous rules for ordering decimals, and investigated the stubbornness of these rules. Prima facie, the study is interesting, and here is why I’m not that interested:

      a) S & S’s test to students was introduced with the following instruction “For each pair of decimal numbers, circle the one which is LARGER”.

      God knows to what extent it affected the study, but I find that wording astonishingly confusing, and it it almost feels deliberately so. Firstly, “decimal numbers” isn’t even a thing: Jodie Foster made the same mistake in Contact. Secondly, it feels to me “greater” would have been much less dangerously ambiguous. In combination, the use of “decimal” and a physically large “LARGER” seems to me to have needlessly and dangerously conflated the representations of the numbers with the numbers, exactly the kind of error the study is meant to be investigating.

      b) Why does the study matter? What are the benefits of this study to a practising teacher? An observant teacher will be watchful for errors with decimals, and a decent textbook will implicitly test for such errors (which can then be explicitly corrected).

      3) The obvious response to (2) is that not all teachers are observant and not all textbooks are decent. So, yeah, I guess S & S’s study is of value to those unobservant teachers who will read and comprehend an educational study: all three of them. As for the textbooks, if they’re getting worse (and they are), it is primarily because they embody and reinforce a system perverted by calculator use, and by the denigration of proper practice and the understanding that such practice instills.

      In summary, if S & S is of non-trivial benefit it seems to me that is only because of the awfulness of modern maths education.

      1. Thank you for following up on that and for pointing out the serious issues with their question. To be honest, I’m not really inclined to go out of my way to defend them. It just seemed like maybe there was a chance it was worthwhile research.

        We spent a fair bit of time on misconceptions in my mathematics education subjects (not as much as on using technology, but still a lot). At the time I had a negative reaction to it, because it reminded me of similar preoccupations in language teaching (fear of teaching mistakes) that I had learned were not considered well-founded anymore; but other students seemed to appreciate it.

        The story I received about misconceptions research was that before this, people just thought students (a) had the right idea and simply made accidental mistakes; or (b) had no idea. But then the misconception researchers discovered option (c): students had strongly held wrong ideas. Such students present a more difficult challenge to teachers than those with no idea (that was a subplot in S & S’s study: some misconceptions being more stable). And the implications were that one needs to make a stronger impression on students to counteract wrong ideas than to simply teach new ones. The article doesn’t really cover how to go about it, but in one class we spent an hour constructing numbers with plastic tubes of different sizes and apparently that is best practice.

        By the way, I’ve been reading an old edition of Geometry by Jacobs after you recommended it, and it’s been instructive how he carefully presents ideas in great detail, including the linguistic aspects. I wonder sometimes how to make things clear without seeming to insult people by explaining things they already know. (That class with the plastic tubes felt insulting.) And I suspect the key might be in this precise attention to details (which you show as well).

        1. Hi, S-T. Some quick replies:

          a) Just to be clear I don’t know if the wording of the survey question is a serious; it is difficult to guess the magnitude of the effect that I’m suggesting. But the wording seems remarkably inept.

          b) You’re saying that “misconceptions research” is a general thing? My gut tells me this another case of the educational cure being worse than the disease.

          c) I got the S & S subplot about the stubbornness of some errors. I cannot see a systemic and stubborn error being eradicated by anything other than a teacher/tutor/parent/friend first asking one-on-one “Why did you do it that way?” That is always the first question one should ask. You always want to get as much as possible into the mind of the student. I’m not sure why this simple step, which amounts to nothing more than empathy, isn’t sufficient and a hell of a lot better than some cumbersome machinery.

          d) Who told you that tubes nonsense was “best practice”?

          e) Never use the expression “best practice” without some clear label of sarcasm.

          f) Yep, Jacobs is a great book. I don’t think it is proof that education was better then (although it was), but it is proof that at least some education of the era wasn’t the thoughtless exercising that is presented in the parodies.

          g) Students at all levels appreciate all manner of things. It proves nothing.

  17. Also, I thought I would add a book recommendation, because I think it’s in a similar vein to the Klein book (building connections back to the high school level). I’ve been reading a series called “A Mathematical Gift: the interplay between topology, functions, and algebra” by Ueno, Shiga, Morita, & Sunata. It is the book form of some lectures that they gave to high school students in Japan in the 1990s. (I borrowed quite a few books before the pandemic started and now my loans have been extended a lot.)

    I am reading it because I feel like my understanding of basic geometry is informed intuitively by the differential geometry and topology I learned at uni (not that I even learned that much), and wonder if some of the ideas can be communicated at the high school level: just the general idea of imagining yourself as a tiny creature moving around in a space, and the idea that you can do this rigorously. Of course, I’m not sure there would be any scope to actually do that, but I want to be ready anyway.

    1. Thanks S-T. (And sorry: somehow your comment ended up in the trash. Not sure what happened.)

      The book sounds very interesting. I’m much more concerned (pretty much only concerned) with whether students learn arithmetic and algebra, but of course the interconnectivity and self-reinforcement of mathematics is very powerful, and typically underemphasised. The (decimated) role of geometry and the (peculiar) role of topology in school education is interesting, but perhaps a discussion for another time.

      Some other like books you may want to consider are Geometry by Jacobs, The Shape of Space by Weeks (and the companion classroom materials), and Measurement by Lockhart.

      1. I meant to reply to Glen’s comment, so I deleted this and tried to copy and paste it there, but it didn’t work because it was detected as spam. I hope it doesn’t end up appearing several times now, because I tried again. I would normally apologise but…

        1. Ah, good. You’re improving at that not apologising thing! (I don’t think the duplicate comment will reappear, but easy for me to delete if it does.)

  18. Another area in which research in education impinges on mathematics teaching is the concept of inclusive education. Like other teachers, I have had students in my class who, for one reason or other, have “learning difficulties” – a polite term to cover a broad range of characteristics. How do I best help these students? The latest issue of Mathematics Education Research Journal is devoted to this issue. Maybe I can find some guidance in there.

    Perhaps one reason for the slow progress in mathematics education research – and education research in general – is that the problems are hard.

    1. Thanks, Terry. I still haven’t gotten to your long comment earlier on, which touched on a related point. Yes, the “inclusion” of people who have been traditionally ignored is new, developing and important. There is also plenty of nonsense written there, but I don’t think that invalidates the importance and difficulty of the area. Still, I’m not convinced that the “slow progress” is primarily due to intrinsic difficulty. And it doesn’t begin to explain the fast regress of maths education in general.

  19. Let me dare to ask Marty a question. What is your basis for saying that there is a “fast regress of maths education in general”?

      1. I realise that there are sometimes mistakes on VCE examinations (still in 2019); topics are not presented rigorously; students rely too heavily on calculators; some mathematics teachers are not well qualified; students don’t understand fractions even after many years of schooling; the curriculum is, to borrow a phrase from the US, a mile wide and an inch thick.

        Is this regressing?

        1. “Regress” as “returning to a former state”? No. I was just playing on your ludicrous implication that maths ed was progressing. It isn’t, and it’s not just because of a few fucking mistakes on exams, or a few phys ed teachers running maths classes. It’s the systemic perversion of education.

  20. Well, this competition has taken off with comments!

    An issue in education that concerns me is the prevalence of multiple choice questions in tests – of all sorts.

    Suppose that a student gives the answer (b) to Question 7 on a test. What does that tell me about the student’s learning?

    As far as I can see, there is only one advantage of multiple choice tests.

    Here is a paper on this topic.

    Hilton, P. (1993). The tyranny of tests. American Mathematical Monthly, 100 (4), 365-369.

    In the hope of winning the prize, I hereby nominate this paper.

    1. Thanks, Terry. I think you might have mentioned that paper on another post (which doesn’t make it ineligible as an entry). I’m no great fan of MCQs, although I don’t think I dislike them as much as some others.

      Are you specifically putting forward Hilton’s idea of partial credit MCQs as your entry? Or, are you claiming some general modern improvement on the role of MCQs?

      1. I don’t like MCQs; nor does Hilton. He accepts their inevitability and proposes a compromise which could be regarded as a modern improvement. So I am proposing Hilton’s paper because it is well-argued and suggests a small step forward in assessment in mathematics education.

        1. Yep, fair enough. I wouldn’t call the idea huge, and I don’t know that it is so easy to implement. But it is a good idea, and definitely not trivial.

          1. I think a good MCQ is valuable. By good, I mean that the question is non-trivial but doesn’t try to be too ‘fancy’, and every incorrect option follows logically from a common misconception, careless mistake or just ignorance (Option E – None of the above). It might be designed so that the correct option can only be efficiently found by eliminating four wrong options.

            But a good MCQ is hard to write, and if you have to write 10 of them, say … (which is why most MCQs are crap). I hate MCQs simply because I can write at least 20 good short answer questions in the time it takes to write 1 good MCQ.

            But I love MCQs when someone else has spent 10 hours writing good ones.

            1. Hi John — I came on here to say exactly this. As an example, in my analysis subject, it is standard to have (for each exam) 10 T/F questions, 10 MC questions, 6 short written questions and 3 longer questions. The T/F are worth half as much, but the rest are equally valued. (So say 1 mark each for the T/F, 20 marks for each of the other three sections, 70 marks total.)

              The MC questions after the exam generate so much discussion,. Yes they took an age to write, but they are really good in my experience at testing the knowledge of the student.

              So, a big agree from me.

              1. Thanks, Glen and JF. It’s interesting you both agree on the value of a good MCQ, and the difficulty of writing one. None of this argues against Terry’s competition entry, for Hilton’s suggestion of partial credit MCQs (where some answers are more wrong than others). It seems clear that this would increase the value of MCQs, as well as making them a hell of a lot more difficult to write.

            2. I don’t like MCQs. The fundamental purpose of assessment is to understand what stage students have reached in their learning.

              If my answer to Q4 is (c), what can you infer about my learning? I suggest that you can’t infer anything about my learning from my writing “(c)”.

              On the other hand if I write 2x+3=5 \implies 2x=2 \implies x=-1, then you can infer something about my learning.

              The views that good MCQs are hard to write, and many are not well written, give me more reason for not liking them. Time and time again, I see questions like this: What is the next number in the sequence 2,4,6,8,…? (I always suggest 42.)

              And MCQs are so prevalent – way beyond mathematics. Even the citizenship test in Australia is made up of MCQs. Take the practice test on-line:


              1. Hi Terry, I wouldn’t write a multiple choice question for your example. But if I did, my options would probably be:

                A. 4
                B. 2
                C. 1
                D. 0
                E. -1

                I’m sure you can see the common mistake that leads to each of the wrong options. Are you suggesting that nothing much can be inferred from the response of a student? I do agree with you that more (with less ambiguity) can be inferred if a short written response is required, but that would be the case for any question.

                Re: What is the next number in the sequence 2,4,6,8,…?

                I agree with you, but if the question was
                “What is the next number in the arithmetic sequence 2,4,6,8,…?”
                then I don’t have a problem.

                1. Taking your example, if I were a candidate, I might see that the answer is (c) – but then mark (b) accidentally. The examiner cannot see that I made a slip.

                  I used to play chess by post. You write down your move then post it off. I remember once, I decided to move my pawn to e5 which was an excellent move, but I wrote f5 which I did not realise until my opponent came back to me a week later with a crushing reply.

                  As for your second example, I reckon that test writers don’t include “arithmetic” because either they don’t realise that it is important, or they do not want to give the game away.

                  1. I accept your point that a student may know the correct answer but accidentally input the wrong answer. But I think if you have a series of questions such mistakes would average out for that student and the single question certainly averages out for the class.

                    Re: The second example. Two different types of stupidity on the part of the teacher. But remember, we’re talking about a good MCQ (which obviously neither teacher would have written) ….

                    1. A few other points occurred to me today.

                      First, a designer of a MCQ will often include an answer that one would arrived at by some common mistake. But what is the basis for inferring that a candidate who chooses that option made that mistake?

                      Second, to argue that mistakes average themselves out tells me even less about a student’s learning – which is the fundamental purpose of assessment.

                      Third, (and maybe this says something about me rather than MCQs), when I used to use them I found myself introducing tricky options that sounded quite plausible but were wrong. I didn’t like trying to trick students. I also thought that students who were not strong in English – and not only international students – were at a disadvantage. I read once that boys generally do significantly better than girls on MCQ tests; so I analysed the data from a large statistics class (c. 150) where I was using MCQ and sure enough my experience supported this theory.

                      Having arrived at this conclusion, I remember that I went to the pub for dinner and discussed it with a colleague. He told me that he used short answer questions instead; so I swapped over to them.

                    2. Hi Terry.

                      Re: “a designer of a MCQ will often include an answer that one would arrived at by some common mistake. But what is the basis for inferring that a candidate who chooses that option made that mistake?”

                      I agree that the answer might just be a guess, or some other obscure mistake has led to that answer. You never know for sure why the student chose the answer they did, but I think on the average it’s reasonable to assume that the ‘intended’ mistake was made.

                      Re: “to argue that mistakes average themselves out tells me even less about a student’s learning – which is the fundamental purpose of assessment.”

                      What I meant was that out of the whole class, for a given question I would not expect a lot of students to calculate an answer but by accident input a different one to the one they intended. So, such a mistake would average out for a given question …. (Not much help for whoever the individual student was but useful when looking at the class as a whole).

                      And I would not expect a student to consistently calculate an answer but by accident input a different one to the one they intended. If that happened, I’m sure that the student would notice when s/he got their test back – it might indicate some other more fundamental problem ….

                      Re: “tricky options that sounded quite plausible but were wrong. I didn’t like trying to trick students.”

                      If an option is plausible but wrong, then it’s most likely based on a misconception, in which case it’s your duty to try and ‘trick’ the student in order to probe their understanding …. What I don’t like are the tricky questions and options that VCAA often have on its exams – where the whole mess is just a straight up cluster-truck. (see 2019 Specialist Maths Exam 2 Section A Q12 for the most recent). Then again, such MCQs are not good MCQs.

                      And thinking about it, I suppose good is in the eye of the beholder. It’s rare a teacher will say they wrote a bad question. Their should be some criteria that the MCQ can be judged against.

              2. Hi Terry!

                If I were able to, I’d love to show you my Moodle quiz I just gave one of my classes. I think you’d be able to see immediately that i can infer quite a huge amount from their responses to this.

                Although it isn’t all traditional multiple choice, there is multiple choice in there and variants on that.

                Actually, with a bit of tweaking, I can probably get my quiz to print out with the correct answers indicated. (It’s all in LaTeX on the backend.) Would you be interested?


                    1. Thanks Glen.

                      Obviously a lot of effort has gone into preparing this, and it seems to fit the purpose of testing where the students are.

                      Let me say that I am always uncomfortable with the phrase “best fits” … in what sense?

                      An aside:

                      Teacher: How do you spell “alligator”?
                      Student: ALEGATER
                      T: That is not correct. The correct spelling is ALLIGATOR.
                      S: But you asked how do I spell “alligator”?

                      Thanks for sharing your work. Sharing is a real sign or a good teacher.

                    2. Yes, I hesitated with sharing it because of the classification questions! The issue is, that every linear PDE is also semilinear, also quasilinear, also fully nonlinear. So, I use the (explicitly taught) shorthand “best fit” to say “smallest”, in the sense that we say a PDE is semilinear only if it is not also linear (for example).

                      These kinds of questions have been asked on a number of assignments and tut sheets using this wording, so for the students it is OK. But looking at it out of context, it does appear bad!

          2. About the partial credit idea:

            I feel like a similar effect can be obtained by just writing standard (good) MCQs. When one of my colleagues tried partial credit MCQs (in a big first year subject, so high school + 1 year) there were a billion complaints about them being confused. In the end, I believe he had to just give everyone the full marks for those questions, which is silly.

            Now, does that mean partial credit MCQs are bad? I don’t think so. But, my colleague was pretty clear about the questions, and how he asked them, and IMO the questions were good. For the following year, we used a similar kind of question but made it not-partial-credit and there were no complaints about it at all.

            So I guess I’m not in love with the partial MCQ idea. I do like good MCQs though.

            1. Thanks, Glen. Very interesting. Maybe your colleague explained the game poorly, or maybe it was just too culturally strange (and the students too whiny …). Or, maybe partial credit MCQs are simply more trouble than htye’re worth.

            2. I have often heard it said that it is difficult to set good MCQs, and I have seen so many badly worded MCQs. This is one reason why I do not like them.

              Still, I hark back to my fundamental question: If I mark (c) as my answer to Q4, what does that tell you about my learning?

              MCQs have only one advantage.

              1. One isolated question – either you know how to solve a simple linear equation or you made a lucky guess. Several similar questions and you chose the correct option each time – most likely that you know how to solve simple linear equations.

                I’m not championing MCQ (and I hate their use in the VCAA exams), but I do believe that they can serve a useful purpose when written well and used appropriately.

                P.S. An MCQ often tells me a lot more about the learning of the people who wrote it, vetted it and wrote a solution for it:

                WitCH 37: A Foolproof Argument

                WitCH 35: Overly Resolute

                1. In my experience, tests that use MCQs, for assessment, usually do not have many similar questions. If you have a test that involves say 30 questions, you are not likely to use 5 of them to see if students can solve simple linear equations.

                  Interpreting the answer to a MCQ involves considerable guesswork on the part of the examiner. If the candidate gives the right answer, we assume that the candidate understands the material. If the candidate gives the wrong answer, we might assume that the candidate does not understand and has fallen into the set trap.

                  If I want to see whether students can solve simple linear equations (and I have done exactly this not so long ago), I would ask them to solve for x in, say, 2x + 1 = 5 and show any working.

  21. Let me ask a related question. What are some important research questions in mathematics education?

  22. One technology (but very low tech, paper based) that I like is programmed instruction.

    I have just found them to be incredibly time efficient when I needed to get horsed up on a topic fast.

    Regular text is fine otherwise…but I still sort of like the interaction of programmed texts. It can be sort of monastic and hard (to stick with a program) to do self instruction. So a programmed text is nice structure, feedback, hand holding.

    or… (non math, I have also used a programmed instruction book to rapidly learn accounting, that I needed as a prereq. and did not have time for a real course. For some reason programmed instruction was very popular for accounting teaching c. 1970.)

    Of course, this seems to make more sense for self instruction than for conventional classes.

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