Science Fiction, Double Feature

No, we’re not intending to go into this. Our concern is with the parallel nonsense of indigenous mathematics, about which we’ll be writing in the near future. But this stuff has to be noted, and someone should be hammering it. Aren’t there some active scientists around who are just a little perturbed?

For those interested, the philosopher Massimo Pigliucci has written on one encounter with this kind of thing. And yes, of course we could have made it a triple feature, or more; we just couldn’t resist the title. ACARA’s inevitably stunning contribution, from 2018, can be found here and here.

ps Merry Christmas, or Bah Humbug, or something. Stay safe.

UPDATE (27/12/21)

We weren’t intending this post to trigger a discussion of “Aboriginal mathematics”, but if it does, it does. And, given it does, it seems reasonable to remind readers that the Best ATSI Elaboration competition is still wide open.

61 Replies to “Science Fiction, Double Feature”

  1. It is statements like:

    1) ” Indigenise mathematical practice at Universities, both in education and research.”

    2) “The elaborations acknowledge that Aboriginal Peoples and Torres Strait Islander Peoples have worked scientifically for millennia and continue to contribute to contemporary science. They are scientifically rigorous, demonstrating how Indigenous history, culture, knowledge and understanding can be incorporated into teaching core scientific concepts.”
    [Statement by ACARA without any supporting evidence]

    3) “Students can explore connections between representations of number and pattern and how they relate to aspects of counting and relationships of Aboriginal and Torres Strait Islander cultures. Students can investigate time, place, relationships and measurement concepts within Aboriginal and Torres Strait Islander contexts. Through the application and evaluation of statistical data, students can deepen their understanding of the lives of Aboriginal and Torres Strait Islander Peoples.”

    that, to me, summarise what’s really going on.

    There’s an agenda, best paraphrased from statement 1), to Indigenise mathematical teaching. To me, this is the real issue. And many of those involved are pushing this agenda by claiming that something called indigenous mathematics exists. This claim has been carefully researched and found to have no basis in fact (see attached). The conclusion of this scholarship is an inconvenient truth and is extremely disagreeable to those pushing this agenda. Their rebuttal consists of unfounded assertions and personal attacks. No EVIDENCE is given. Mathematics education has been politicised.

    Examples of how to indigenise mathematics does not constitute evidence for the existence of indigenous mathematics. btw I’m not saying for one minute that some part(s) of mathematics shouldn’t be indigenised if it serves a valid educational – NOT political – purpose. What I’m saying is that valid examples of how to indigenise mathematics does not constitute an indigenous mathematical heritage.

    I really liked the article by Massimo. His conclusion:

    “In a nutshell, it was clear to me that the positive claims made by supporters of Indigenous science reduce to an attempt to introduce what to me clearly qualifies as pseudoscience in the university curriculum. When they experience some pushback, however, they shift to a position that is entirely unobjectionable — like bringing students to nature walks or teaching them about the medicinal properties of the local flora. But such unobjectionable proposals seem to be obviously designed as Trojan horses to get the real crazy stuff in by way of a secondary entrance. Once a university hires an Indigenous scholar to teach Indigenous “science” there is very little oversight over what, exactly, the fellow will be teaching in the classroom. And the problem with Trojan horses, even when they are so obvious to spot, is that they tend to work — just ask Odysseus.”

    Right on! This is a very common technique used by many agenda pushers (and sellers of snake-oil and their gullible dupes) – it’s hard to object to the unobjectionable …

    One last thing – the cheap shot … Here’s a quote from the Monash University Handbook for SCI2030 – Indigenous science: Science through the eyes of Australia’s First Peoples:

    “As such you will leave the unit with a deeper appreciation of different traditions and knowledges and better prepared to take on \displaystyle knew knowledge, and appreciate different views and approaches towards science.”
    (my emphasis on “knew”) I feel the credibility quotient dropping …

    CommsDeakin

  2. It is instructive to study how mathematics has been used in different cultures. Indigenous mathematics in Australia is relatively difficult to study when compared, say, with the mathematics of ancient Greece. However, it behoves Australian scholars who are so inclined to take up the challenges that this field offers.

    “To what extent should findings in this field influence the school curriculum?” is a different question.

    BTW, I note that CSIRO has been involved in studying Indigenous calendars: https://www.csiro.au/en/research/natural-environment/land/about-the-calendars/ngurrungurrudjba

    1. Terry, let’s leave aside for now the “different question”, and consider the first question you raise. You write,

      “Indigenous mathematics in Australia is …”

      To what “indigenous mathematics” mathematics are you referring? Are you considering the concept of seasons of the year as mathematics, or science?

      1. The study of Indigenous mathematics means, to me, the study of how Indigenous Australians traditionally thought about concepts that we would regard as mathematical.

        Measuring time might be classified as a scientific topic or a topic in applied mathematics – with cultural, political, and religious influences.

        Once I wrote this (probably in 2017): “There are many different ways in which we can stimulate the interest of our students in learning about Indigenous Australians. This could be achieved in art classes, or history classes, or geography classes. Who could fail to be interested in the latest discoveries by archaeologists in Australia recently reported in the journal Nature? Their discoveries showed that humans arrived in Australia at least 65,000 years ago.

        All it takes is a spark to light the fire of interest in a student. And the spark could happen in a mathematics class too.”

        I found this book to be a useful reference: Harris, P 1991, Mathematics in a cultural context: Aboriginal perspectives on space, time and money. Deakin University Press, Geelong.

        1. Terry,

          1) Can you provide any example of “how Indigenous Australians traditionally thought about ideas that we would regard as mathematical”?

          2) In what sense was there “the measuring of time” in traditional Aboriginal culture?

          3) Stimulating an interest in Indigenous Australians is fine and good, even in maths and science classes, although that’s a stretch. But making stuff up stimulates contempt, not interest.

          4) 65,000 is highly contested. But of course it is standard to take an inflated figure in this debate and declare it to be a minimum.

          1. Marty, Thanks for your questions. I’ll take them in turn.

            1) The book by Harris provided details. I borrowed it from a library many years ago and I don’t have it handy. But I did write this in my notes.

            “In Aboriginal cultures, measuring kinship would be regarded as the most fundamental mathematical task (Harris 1991, p. 19). This is the first shock to a non-Indigenous Australian mathematics student. Why is measuring kinship so overwhelmingly important? This question provides a hook to capture the interest of the students in Indigenous culture.”

            A mathematician who has worked extensively in the NT later confirmed this view to me.

            2) According to CSIRO’s work, traditional Aboriginal cultures developed calendars; these measure time.

            3) Teachers should never make things up.

            As I said before, studying these topics is worthwhile, especially for Australian scholars who have the ability and inclination. But because these topics are so difficult to study, we should be careful in introducing findings into the school curriculum.

            4) The 65,000 figure I obtained from: Clarkson, C Jacobs Z, Ben Marwick B, Richard Fullagar R, et al. 2017, ‘Human occupation of northern Australia by 65,000 years ago’ Nature, Volume 547, Issue 7663, pp. 306-310. I regard Nature as a reliable source.

            1. 1) is nonsense. Mike Deakin discusses this at length. 2) That’s a damn weak notion of “calendar” 3) You’re coming damn close to making things up. 4) Whatever Nature says, the figure is contested.

              Terry, you’re working hard here. Way too hard. Be honest.

              1. I think we would all agree that “Stimulating an interest in Indigenous Australians is fine and good, even in maths and science classes …” [Marty] provided “it serves a valid educational – NOT political – purpose.” [Friend].

                The ridiculous thing is that there doesn’t have to be an indigenous mathematics culture in order to do this. So I don’t understand why it’s pushed so hard. Saying something is true doesn’t make it true.

                To those with the agenda, I say: Just come up with some apting VALID contexts and examples. NOT gratuitous bullshit examples like the ones found in the ACARA Daft Curriculum that are simply intended to satisfy a political agenda. Be upfront with the reasons, don’t make false claims and clutch at straws.

                By the way, someone creative enough could probably make assertions and claims about the mathematical heritage of bees (assertions based on https://www.science.org/doi/10.1126/sciadv.aav0961), or termites (assertions based on https://www.sciencedaily.com/releases/2021/01/210119194326.htm) , or Homo Neanderthalensis (https://www.inverse.com/science/neanderthal-math-study), or … (Maybe there’s even a few PhD’s in it, not to mention an academic career …?)

                @Terry:
                1) https://www.theaustralian.com.au/commentary/opinion/teaching-children-aboriginal-kinship-in-maths-does-not-add-up/news-story/bf5904d6ac6db0003425497d645968f8

                2) I assume you mean what’s found here: https://www.csiro.au/en/research/natural-environment/land/about-the-calendars
                CSIRO could probably work with birds to develop a calendar representing seasonal ecological knowledge … I don’t see this representing a mathematical heritage of birds …

                3) Many maths teachers make stuff up, they just don’t know that they’re doing it. (“A graph can never cross an asymptote”, “x^2 + 1 = 0 has no solution” etc etc). But the *deliberate* fabrication of bullshit is particularly odious.

                4)
                (i) You mean \displaystyle modern humans.
                (ii) https://australian.museum/learn/science/human-evolution/the-spread-of-people-to-australia/

                1. Thanks, John. The Warren Mundine opinion piece you link to in regard to (1) is interesting (and Murdoch, paywalled). I don’t entirely agree with Mundine, but he calls out very well the misplacement and the misguided emphasis and the tokenism. A brief discussion of the piece can be found here, which also has an interesting reference on the evolution of aboriginal number systems, here.

              2. The books by Mike Deakin and by John Crossley, which are on my shelves, on the history of the concept of number, suggested to me that the study mathematics of Aboriginal cultures is a difficult, contested, area. Deakin points us to the literature on linguistics as a starting point; this sounds sensible – although I have not pursued it.

                Let me draw an analogy. Western scholars have long been familiar with the mathematics of ancient Greece. Not so with the mathematics of ancient China. Western scholars in the history of mathematics could not read the materials – a sweeping statement but broadly true. They were better with ancient Greek, Latin, German and French. Things have changed (thanks to Needham, Crossley, et al.), and as a result we have a better appreciation of mathematical thinking in ancient China.

                Similarly with the study of the mathematics in ancient Australia. However, this topic is on our doorstep. If Australian scholars don’t come to grips with it, who will?

                1. But Terry, what if there’s no mathematics in ancient Australia to come to grips with …? You accept that “the study [of] mathematics of Aboriginal cultures is a difficult, contested, area”. You are *conjecturing* that “mathematics in ancient Australia” existed. Which is more honest at least than the agenda pushers, who state *as fact* (with no evidence) that an indigenous mathematics culture exists.

                  With regards to the linguistics hypothesis:
                  1) Mayan mathematics seems to be pretty well understood …
                  https://www.storyofmathematics.com/mayan.html
                  https://www.math.tamu.edu/~roquesol/M629_6.html

                  2) As does Inca mathematics …
                  https://second.wiki/wiki/matemc3a1tica_incaica

                  1. I don’t know if Terry is conjecturing. He seems to be endorsing the claims that having kinship systems amounts to mathematical thought, and that having the concept of seasons amounts to having calendars and thus the measurement of time. Terry just doesn’t seem to be willing to attempt to substantiate these tendentious, if not outright absurd, claims. That is, Terry is, once again, trolling.

                  2. Thanks for your comment and the links. I am not an expert in this field, and so I cannot assess how what we might regard as mathematical concepts were understood in ancient Australia. My comments have been based on reading and conversations.

                    Even if we did discover “there’s no mathematics in ancient Australia to come to grips with”, this would be an interesting discovery in itself.

                    My point about ancient languages in Australia is that they are not so well known among scholars who are interested in mathematics. Certainly there is a considerable body of knowledge about the languages themselves among experts in linguistics.

                    I’ll sign out for now on this topic. Thanks for listening.

                    1. Terry, no one is contesting your point about languages and the pointers they contain to historic and prehistoric notions of number. You were making other points as well, which were contested. If you are not now interested in defending those points, that’s fine.

                      The attempt to understand Australian indigenous culture, current and historic and prehistoric, is obviously valuable, and the basis of a valuable cross-curriculum priority in the Curriculum. But, again, making stuff up does nothing for such endeavours except to undermine them.

                2. Terry, I’ve now read Mike Deakin’s excellent book. It is a fascinating and careful study of linguistic clues to the precursors of our base ten number system. I think it is also fair to say that Deakin’s book makes no claim of, and provides no evidence for, traditional Aboriginal mathematics except in the most trivial sense.

                  I will read Crossley next.

  3. I have reached my decision at this point of time.
    It is too early on the evidence to include Aboriginal and Torres Strait Islanders to this extent in the Mathematics Curriculum.
    I have learnt of the tactics that are being used without compelling supporting evidence.
    Primary school mathematics has suffered enough without another issue that will not improve students’ mathematical skills. More time will be taken away from developing the skills.

    This issue belongs predominantly in Integrated Curriculum/SOSE.

    1. Thanks, John. I largely agree that the study of Aboriginal culture belongs in Social Studies, or whatever that stuff is called these days. Still, in principle I’m fine with and I like ATSI studies as a cross-curriculum priority, and I’m fine with (a few well-chosen) ATSI priorities in the mathematics curriculum.

      What I’m not fine with is making stuff up. Which is what is happening. Everybody who has not drunk the Kool Aid knows this, but they are too polite or too intimidated to say so. It is embarrassing, especially for the screamingly silent Big Shots. It is also destructive for the proper study and appreciation of ATSI culture, for the general intellectual culture, and for anyone who wants to see the end of this psychopathically awful ScoMoFo government. Any movement that turns that asshole Tudge into the Good Guy should be seriously questioning its credentials and its strategies.

      1. JH, a very reasonable decision. The key of course is making a decision based on the “EVIDENCE”. To the agenda pushers I ask again – Show us the EVIDENCE! Education should NOT be informed by political pressures, wishful thinking and lies.

        Yes, Marty. The Big Shots aren’t making much noise at all. I think they’re happy to bury their heads in the sand (or cucumber sandwiches), hoping it will all go away. Perish the thought any of them would give an informed, objective and decisive statement to the media. The Big Shots are beholden to the political correctness and timidity that their Institutes have chosen. They are Sweden. It would be really nice to see a few give up their memberships on principle and make a statement to mainstream media that calls out all the bullshit and make-believe.

        It would seem that a few indiscrete movements have turned Tudge into the Gone Guy. To be replaced by … the Gormless Guy …?

  4. Marty, JF, Terry and others:

    Thanks. Seriously. I think I’ve learned more from reading the posts, comments and associated links on this blog than any “professional development”. If only ACARA and other “professional” bodies would engage in debate the way you all have, we might not be in the position we are…

    The underlying point made much more generally “first, tell no lies” seems as relevant here as anywhere.

    So, although I cannot contribute to the debate in any way (total lack of knowledge), my thanks for those who could and did.

    1. Of course, we should not tell lies. However, some teachers may tell their students things which are not true as a result of their own lack of knowledge. This can arise particularly in the humanities which, in Victoria in years 7-10, encompasses history, civics, geography, business studies, and economics. In such situations, teachers probably regard the textbook as the definitive source of truth.

        1. There are UNintentional lies told as a result of a lack of knowledge, and there are intentional lies told in order to support an agenda.
          This blog is clearly discussing the latter.

          1. @JF: My understanding is that a lie is intentional by definition. But I get your meaning.

            In this sense, I have never come across a teacher who lies in teaching his or her students.

            I am not saying that it never happens; it’s just that I cannot recall encountering an instance of this.

            1. Terry, I believe lying, including by omission, is plenty common, and that wilful ignorance is extremely common. You may not want to characterise statements based upon the latter as “lying”, but there can be a reckless indifference to truth which the innocent-sounding “may tell their students things which are not true” does not remotely convey.

                1. They certainly have a reckless disregard for the truth.

                  If you agree with this obvious statement then we can discuss whether, on occasion, this reckless disregard amounts to lying. If you don’t agree with this obvious statement then I don’t think I’ll bother.

      1. Humanities and Mathematics are, to my mind, to be kept in very different camps when it comes to education. But that is a different story – and I want to stick to Mathematics if that is OK.

        A line I perhaps over-use in Year 9 when teaching, say quadratic equation solving and equations that have 0, 1 or 2 real solutions is “I’m not telling you the whole truth here…”

        When a teacher is not aware of what the truth actually is (and then, of course will assume the textbook to be completely correct, and the VCAA exams to be equally without fault…) we have a problem. It would be very easy to blame the number of teachers who are teaching “out of their area of qualification” but I think the issue is much, much deeper than that.

        And, to briefly touch on the humanities… I suspect the issue is also much broader.

        1. @RF: I agree with you. It can happen in mathematics. I have seen several textbooks that define a simple random sample as one in which every member of the population has the same chance of being selected. Teachers will pass this definition on to their students, and students will be expected to know this for the examination.

          I pointed out the difficulty with this definition to a teacher. Suppose that I have a population of 10 people (5 men and 5 women). I want to choose a sample of 5 from this population. I toss a coin: Heads I pick all the men, Tails I pick all the women. Every individual has the same chance of being selected (0.5) – but it’s not what we would regard as a simple random sample.

          The teacher replied “But this definition is in the text book!”

          1. Yep. And if the teacher who disagrees with the textbook happens to be a graduate in a school that doesn’t have a great record in attracting teachers with a solid mathematics (tertiary) education they will have more than a few issues convincing their (on paper) superiors of the issue.

            It might just be me, but these definition errors seem particularly common in Probability and Statistics chapters. All levels as well, not just VCE.

            1. I totally agree, RF. I had one or two battles early in my teaching career and the experienced teachers were not pleased. (“I’ve got 30 years experience teaching this!!” Actually, no. They had 30 times 1 year of experience).

              I don’t think it’s by chance (ha ha) that these definition errors seem particularly common in Probability and Statistics chapters. It can be very difficult to give definitions that are technically correct AND clear. The foundations of many things taught in probability and statistics are often very advanced. A lot of what gets taught is dumbed down and diluted. And it’s not as simple to say “I’m not telling you the whole truth here…” as it is with quadratic equations. (I don’t have a problem referring to solutions that are not real and so we ignore them at this level).

              I’m guessing many textbook writers copy from ‘reputable’ sources (that are sometimes unreliable *), the error gets propagated and becomes ‘fact’. I’d be very surprised if many writers researched what they wrote before they wrote it (some probably think there’s no need because of who they are *snort*). It’s only when publishers are called out on errors that things – sometimes – get changed (I’ve successfully done this a few times *)

              In my experience many teachers struggle to teach probability and statistics. They often say that they dislike the topic (usually because they get confused). The wording of questions often exacerbates (or is perhaps the main cause) of this. I’ve found that many teachers are particularly confused with conditional probability – they try to use formulas without any real understanding of what they’re doing. This works well for them in simple questions where the data is spoon-fed, but does not work so well when you get questions about false positives, DNA matching etc. I think many teachers use the text book to ‘learn’ most of what they teach. (I guarantee this will happen with the the new content in the 2023 Stupid Design).

              * My favourite success was correcting a reputable textbook who had an example giving the ‘solution’ to
              |z – 2| + |z + 2| = 3, z in C
              as the set of points lying on a hyperbola.
              Their ‘solution’ was changed to the null set in subsequent editions.
              I actually found this sort of error being made by many teachers in various revision programs, much to their disbelief. They had no idea what an implied restriction was and had no geometric insight. They were blindly following an algebraic recipe. None had checked their ‘answer’. A simple counter-example did the trick … “So you’re saying that z = … satisfies the equation?! Let’s substitute and see … Uh oh …” Sometimes more than one counter-example was need to combat their absolute disbelief and chagrin as they spluttered “But I’ve taught this to students!” (As if that proclamation ameliorates the error).

              1. I did an experiment this year at a faculty meeting. I asked the Mathematics teachers present who could give me a simple statement of the law of large numbers.

                Since there were multiple teachers present who were “qualified” to teach Specialist I was hoping for multiple answers.

                It was at that moment I remembered why I don’t ask questions in meetings any more.

                  1. Unfortunately, many schools become the closure of otherwise open minds…

                    Although, having worked for both schools and a major university, I have to say that universities make schools look efficient.

                    Quite an achievement, really!

  5. At the end of the day, the message is that <insert your favorite minority> knew how to count and that Newton and Leibniz invented calculus. I fail how this is supposed to make <insert your favorite minority> feel any better. But there’s something seriously wrong when most students entering western universities these days know less about mathematics than some Babylonian scribes 4000 years ago.

    1. Thanks, Franz. I think I disagree on a couple points, but your comment seems to be missing some nouns. I’ll get you to suggest corrections before I reply.

    2. “But there’s something seriously wrong when most students entering western universities these days know less about mathematics than some Babylonian scribes 4000 years ago.”

      I don’t really see any problem with this. There’s no real need for an English teacher to know the quadratic formula. The real problem is when this is the case with students who are entering fields that heavily involve mathematics, which definitely is happening today. RMIT Engineering requires only further maths, and one is able to become a VCE mathematics teacher without taking Calculus II or linear algebra.

      1. Thanks, Anon. I agree that Franz’s point is debatable. The dumbing down of qualifications, however, for RMIT and VCE and pretty much everything, is not an argument for anything but the idiocy of our Age.

      2. @Anon: I agree with your sentiments.

        Let me add a supplementary comment. I estimate that, in Victoria, most mathematics teachers (i.e. those who spend most of their time teaching mathematics at secondary level), *never* teach calculus or linear algebra. However almost all of them have to teach arithmetic, algebra, mensuration, geometry, probability and statistics – and how to use a calculator and other technological resources effectively.

        Of course, a teacher should know much more about mathematics and statistics than what one has to teach, and I share your concerns. It is important that teachers are capable of keeping up with developments in the curriculum.

        1. Let’s see how many VCE teachers of Specialist Maths are capable of competently teaching Logic and Graph Theory in 2022 (in Units 1-2) and then succintly review it in 2023 (in Units 3-4) as well as teaching it in Units 1-2.

          One of the reasons that Specialist Maths teachers are sorry to see the removal of mechanics (maybe the main reason for many teachers) is that teaching this mechanics was as comfortable as wearing an old pair of slippers. Many Specialist teachers will be well out of their comfort zone teaching Logic and Graph Theory in 2022. I predict there will be a spike in retirements. Some teachers won’t even bother teaching it – particularly since the official implementation of the new Stupid Design is not until 2023. They will retire at the end of 2022 and let the new Specialist Units 1-2 material be someone else’s Specialist Units 3-4 problem in 2023.

          There will be many and huge problems caused by VCAA’s pig-headed and moronic decree that *everything* at the Units 1-2 level (including stand-alone topics that would be taught in less than 2 weeks and have no explicit overlap with Units 3-4) is examinable at the Units 3-4 level. I wonder if the new formula sheet will include formulae for sequences and series …? It’s a ticking time bomb for teachers. (I wonder how many NHT teachers expected the binomial distribution to be examined on the 2021 Specialist Exam 1 …?) Simply put, it’s classroom sabotage, a reprehensible Dullard legacy that I hope the new Mangler sensibly expunges.

          1. And I wonder if the new formula sheet will include the axioms for a Boolean algebra … ? Or whether students will be expected to memorise them in case there’s a proof question on Boolean algebra. Or whether such questions will be included only on Exam 2 (Section A?) so that students can be expected to include these axioms in their bound reference.

            VCAA’s pig-headed and moronic decree that *everything* at the Units 1-2 level is examinable at the Units 3-4 level is a ham-fisted stick intended only to force teachers to exactly follow the Units 1-2 Stupid Design (*). I know where I’d like to shove that stick.

            Let’s hope this stick and other idiotic and reprehensible Dullard legacies that sabotage good classroom teaching and learning are expunged by the the new Mangler. And if the new Mangler’s hands are tied by pig-headed, moronic VCAA bureaucracy (exemplified by its refusal to implement the new Stupid Design over a 2 year period), then at least s/he gives timely and crystal clear advice that answers questions like the above.

            * I know that many teachers do not exactly follow the Stupid Design when teaching Specialist Units 1-2. And clearly so does VCAA. So I suppose in some ways we, the teachers, cop the blame for making this rod on our backs.

            1. On the topic of VCAA, with the New Year comes nominations for the Prick of the Year – the PotY – award. I’d have preferred the CotY award, but with a New Year resolution to keep my posts children friendly …

              In fact, VCAA are excluded from nominations, simply so that
              1) there will be more than one nominee, and
              2) other nominees have a non-zero probability of winning.

              In the spirit of my previous post, I nominate … Cambridge.

              The hard and digital copies of Cambridge Specialist Mathematics VCE Units 1&2 textbook are *incomplete*. Chapter 28 Logic is only available through the on-line Interactive Textbook, which is only available as a calendar-year subscription. The Interactive Textbook is “accessed online through a HOTmaths account using a unique 16-character code provided with the Print Textbook, or available for purchase separately as a digital-only option.”

              So unless a student buys a brand new “Print Textbook” or pays for an additional “digital-only option” (a “new level of digital support, powered by Cambridge HOTmaths” *snort*), that chapter is unavailable. And of course teachers have no access to it after the 1 year expiry either (and the 14 day free trial of Hotmaths *snort* does not include access to VCE ).

              Congratulations Cambridge. For this total scam and other notable DISservices (see relevant blogs) to VCE mathematics teachers, you are my nomination for the CotY … *ahem* I mean the PotY award. Never let money grubbing (which includes publishing new editions of textbooks with useless amendments, designed simply to subvert second-hand book sales) get in the way of decency and integrity. After all, you’re a commercial organisation whose \displaystyle raison \displaystyle d'etre is to make as much profit as possible, despite the Newspeak in your glossy marketing.

              I don’t want “A new level of digital support, powered by Cambridge HOTmaths” – I want a apting textbook that is COMPLETE and not missing chapters.

              1. John, just to clarify, is “Logic” one of the Specialist 12 topics that (presumably) will be formally introduced in 2023 but that VCAA advises be taught in 2022, in preparation for Specialist 34 in 2023?

                1. I quote from the implementation ‘advice’ given by your unfriendly neighbourhood spider-VCAA:

                  [Quote 1] “Implementation in 2022 of the following selection of topics from the current Specialist Mathematics Units 1 and 2 in 2022 will provide a suitable preparation for Specialist Mathematics Units 3 and 4 in 2023 …Unit 1 Logic and algebra”

                  And I quote from the Daft Stupid Design – Specialist Maths Units 3&4:

                  [Quote 2] “Proofs will involve concepts from across the areas of study of Specialist Mathematics Units 1–4 including: divisibility, inequalities, graph theory, BOOLEAN ALGEBRA; combinatorics, sequences and series including partial sums and partial products (students should be familiar with sum and product notations), complex numbers, matrices, vectors and calculus.”
                  (my capitalisation).

                  The Daft Stupid Design for Specialist Unit 3 contains “Area of Study 1 Discrete Mathematics” with the topic “Logic and proof”. I quote:

                  [Quote 3] “In this area of study students cover the development of mathematical argument and proof. This includes conjectures, connectives, quantifiers, examples and counter-examples, and techniques of proof including by mathematical induction. The concepts, skills and processes from this area of study are to be applied in the other areas of study.
                  This area of study includes:
                  conjecture – making a statement to be proved or disproved, such as ‘for all real numbers x and
                  y, |x + y| ≤ |x| + |y|’ and ‘every integer is the sum of the squares of two integers’
                  implications, equivalences and if and only if statements (necessary and sufficient conditions)
                  natural deduction and proof techniques: direct proofs using a sequence of direct implications, proof by cases, proof by contradiction, and proof by contrapositive quantifiers ‘for all’ and ‘there exists’, examples and counter-examples;
                  proof by mathematical induction.

                  Quote 2 should make all Specialist Maths 3&4 teachers very apprehensive, given the amount of content explicit only in Units 1&2 … And the word “including” is VCAA’s Hall Pass to include any other random shit from Specialist Units 1&2 (like circle theorems) in the exam. And who knows what they mean by “divisibility” … I assume this is a catch-all word for a routine type of ‘proof by induction’ question. But do students need to know gcd etc.

                  And who knows what proofs involving calculus will require. It will be the biggest joke of all time if differentiation from first principles is required given that VCAA completely erased it from Maths Methods. Is it another thing that will need to be taught in Specialist …? And of course VCAA has already said that any random shit from Maths Methods (like binomial random variables) can be examined.

                  Even when VCAA deigns to reveal the final Daft, there will be more questions than answers.

                  As I commented earlier, this idiocy “is a ham-fisted stick intended only to force teachers to exactly follow the Units 1-2 Stupid Design”

                  Boolean algebra is not explicitly mentioned in Quote 3. It’s only mentioned in Quote 2 which lists it as one of the concepts proof will involve.

                  The whole situation is a apting mess, thanks to VCAA’s total incompetence, ineptitude and pig-headness.

                  1. Mathematical induction is often referred to as the *principle* of mathematical induction. Peano calls it an *axiom*. Bertrand Russell calls it a *definition* (“Introduction to mathematical philosophy”, p. 21). In his book “Philosophical and mathematical logic” (p. 27), Harrie de Swart calls it a *theorem*.

                    What will VCAA call it?

                    1. Yes, most textbooks refer to the ‘Principle of Mathematical Induction’.

                      VCAA vacillates between ‘mathematical induction’ and ‘proof by mathematical induction’.

                      I would call it the ‘Axiom of Induction’ (and so side with Peano). I would say “Use the axiom of induction to prove …”
                      I doubt VCAA would ever say this, or thought about it, or considered it important.

    3. Did scribes really know ANY mathematics? I can copy lines from a book in a language I do not understand and, given enough time, will probably do a good enough job of it.

      There is a separate question here as well – just because something was “known” 4000 years ago does not mean it remains true today. Mathematics is not immune from this.

      Agree with the thrust of the argument: students don’t know enough when they enter universities (or graduate as teachers…)

      1. Perhaps Anon was saying that students entering universities today know less than the authors (rather than the scribes) of ancient Babylonian mathematical texts.

        I confess that I have no idea as to how mathematicians in ancient Egypt managed to express fractions of the form 2/n as sums of distinct unit fractions 1/m. They did not show their working!

        Most mathematics graduates have never read Euclid’s “Elements”. I am pleased to be able to say that in some of my classes in Australia and PNG my students read all of Book 1 of “Elements”.

    4. Thanks, Franz. I don’t think the message is remotely that Traditional Aboriginals, or whoever, simply knew how to count. The claim of a university subject titled “Indigenous Science” is much, much grander than that. I’ll leave others to continue to discuss what the “scribes” new, and how it does or should compare with current students.

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