The CAS Betrayal

This post will take the form of Betrayal, with a sequence of five stories going backwards in time.

STORY 5

Last year, I was asked by an acquaintance, let’s call him Rob, to take a look at the draft of a mathematics article he was writing. Rob’s article was in rough form but it was interesting, a nice application of trigonometry and calculus, suitable and good reading for a strong senior school student. One line, however, grabbed my attention. Having wound up with a vicious trig integral, Rob confidently proclaimed,

“This is definitely a case for CAS”.

It wasn’t.

The integral in question was of the form

    \[\boldsymbol{I = \int\limits^{\pi}_0 F(x,b)\, {\rm d}x\,,}\]

with b ≥ 0 a real parameter. The parameter made the integral undeniably vicious, perhaps “solvable” in terms of gamma functions and the like via complex techniques, but not doable in any manner that would be clarifying for the intended reader. All that mattered in the context, however, was to determine the sign of I for b positive. The sign of I, however however, could not be read off by an easy inspection. Since the interest was in perturbing from b = 0, however however however, all that really mattered was the sign of I for b positive and close to 0. And that provided the non-CAS “in” to solving the problem.

The point was to think of the integral \boldsymbol{I(b) = \int \! F(b)} as a function of b. (Of course F still also depended upon x, and was integrated with respect to x, but that could be left implicit.) Now, \boldsymbol{F(0)} was simple enough that it was immediate that I(0) = 0. Moreover, by “differentiating under the integral sign”,* we had

    \[\boldsymbol{I'(b) = \int F'(b)\,,}\]

and in particular \boldsymbol{I'(0) = \int \! F'(0)}.** Then, F'(0) was simple enough to easily conclude I'(0) > 0. Since also I(0) = 0, it followed that I(b) > 0 for small positive b, the problem was solved, and the CAS went back in its case.

*) The technique is slightly beyond high school, but so was Rob’s article. 

**) Yeah, yeah, those should be partial derivatives inside the integrals. But if a student is happy to integrate with respect to x while keeping b constant, they can just as happily differentiate with respect to b while keeping x constant.

 

STORY 4

About five years ago, I was asked to tutor a Year 12 Specialist Mathematics student, let’s call him James. It was pretty silly, since James was smart and studious, and attended a very good private school, let’s call it Scotch College. James needed no help whatsoever. But, James’s mother insisted, and so we kept up the meetings. Given James needed no help with Specialist, we typically talked about mathematics instead. It was fun.

During the year, James showed me a number of his Specialist SACs and SAC-preps. Scotch’s stuff was good. The SACs et al were based on natural scenarios, with well-constructed (and error-free) questions, and with an emphasis on algebraic rather than numerical framing. Nonetheless, and inevitably, the SACs also contained plenty of questions that were intended to be done with CAS.

Typical was an assignment problem that required maximising a quantity something like

    \[\boldsymbol{f(\theta)=\frac{a}{b + \sqrt{c+d\sin \theta \cos \theta}}\, ,}\]

with a, b, c and d suitable and fixed (e.g all positive and c > d). I can’t quite remember the form, but I can remember the message. James and, it seems, every one of his classmates saw the messy function, contemplated the messy derivative to come, noted the 2 Marks or whatever on offer, and decided it was a case for CAS.

It wasn’t.

This happened repeatedly throughout the year. James would show me some function he had needed to manipulate or maximise or whatever, and if there was a hint of hard algebra or calculus, he would have reached for the CAS. The notion that a calculation might be easily handled or avoided with a little thought or a delicate touch was too rarely considered. Specialist discouraged such notions, even at Scotch.

 

STORY 3

In 2009, the Mathologer and I wrote a (not very good) opinion piece for The Age, decrying the state of Australian maths ed. Our piece included a two-sentence slap at CAS, which spurred a vocal CAS fan, let’s call him Victor, to email us. Victor was friendly, agreeing with most that we wrote, but Victor also claimed that research showed that the use of CAS strengthens algebraic skills, and that CAS allows the students to explore and to focus on understanding, and so on.

I replied to Victor, focussing on his claim about the research on CAS and algebraic skills. I indicated that I would be interested to read such research, and that if he suggested “one or two” references then I would read them. I made it explicit that I did not want ten references thrown at me. Victor somehow interpreted this as an onerous restriction to not provide more than ten references, but restrained himself and sent six. I read none.

 

STORY 2

In 2008, a teacher friend, let’s call him Fred, emailed a pro-CAS professor, let’s call her August. Fred asked August for the strongest academic references demonstrating the pedagogical value of CAS. This prompted August to email one of her minions, noting that “We do need a good answer to this question”. August’s email, however, was sent to Fred by mistake. Fred then quickly received a second email from August, indicating that it would “take a bit of time” to gather the references. If August and her minion ever compiled their “good answer”, they never got back to Fred to let him know.

 

STORY 1

Around 2005, soon after I started pondering maths education, someone handed me a paper with the title The Case for CAS. That was the first time I was touched by this stuff, and I remember my reaction: “Uh oh”.

74 Replies to “The CAS Betrayal”

  1. Underselling CAS a little in Story 4 Marty.
    If there was a hint of technology-active, I would keep in my hand the CAS.
    Not much is given to the beauty in VCE mathematics.

      1. I mean CAS is often abused far more usually, at least by me, than the story would suggest. Of course, maybe that particular school has a particular culture among students around CAS.

        1. Oh I see. Sure. The point of the story is that a good question from a good school given to a very smart kid still succumbs to a culture of CAS thoughtlessness.

  2. Story 4 is depressing.
    Since c > d, it is seen by inspection that the function is a maximum when \displaystyle \sin(\theta) \cos(\theta) = -1.
    1 mark for the observation, 1 mark for the answer.

    And if you want to know the values of \displaystyle \theta that give this maximum value, you just solve \displaystyle \sin(2 \theta) = -\frac{1}{2}.

    I always make a point of doing a couple of examples like this in class (such as finding the maximum and minimum value of \displaystyle \cos^{2}(\theta) - \cos(\theta) + 4) when teaching the trigonometry topic to remind students that you don’t use an elephant gun to kill a fly.

    In class I always ask students to find the maximum and minimum values of something like \displaystyle \frac{3}{7 - 2\sin^{2}(\theta)}.

    Re: “Nonetheless, and inevitably, the SACs also contained plenty of questions that were intended to be done with CAS.”
    It is inevitable because VCAA mandates that 30% (give or take) of every SAC must require the use of a CAS.

    Story 5 reminded me a lot of Feynman’s integration trick* of differentiation under the integral sign.

    * https://www.cantorsparadise.com/richard-feynmans-integral-trick-e7afae85e25c

    1. Of course I know why it was inevitable. I wasn’t criticising Scotch. And if the story is depressing, the story at pretty much every other school is way more depressing.

      CAS poisons everything.

      1. That’s why I said the story was depressing. If it had been Highschool High I’d have been somewhat less depressed.

        PS – James’ mother

  3. My first encounter with CAS was on a teaching placement, observing a dedicated and competent Mathematical Methods teacher say to Year 12 students, “if you are asked to find the tangent to a function, don’t even THINK about differentiating! You don’t have time. Use your CAS.” (There is a TangentLine command. Apparently it’s quicker.)

      1. She seemed to know what she was doing (would have no trouble differentiating herself) and was dedicated to helping her students achieve good results. It struck me as perhaps a case of needing to choose between good pedagogy and preparing students to do well at exams.

        1. In other words an ability and willingness to teach mathematics is entirely irrelevant to being a competent Mathematical Methods teacher.

          1. Yeah, maybe. I felt glad I wasn’t the one having to say that, having to make that choice. It did make me wonder if all I knew was irrelevant in VCE, since perhaps my instincts are wrong for it. (I would never think to use a CAS straight away.)

              1. Prove to me that “… an ability and willingness to teach mathematics is entirely irrelevant to being a competent Mathematical Methods teacher.”

                But I’m a good sport – I’ll meet you half-way. I’ll prove necessary but not sufficient using proof by contradiction.

                Step 1: Assume necessary and sufficient.

                Step 2: Show that this assumption leads to either a contradiction or error.

                Now it’s your turn – show me that there is actually no contradiction or error (and hence prove me wrong). (And no monkey business).

                  1. But you made the antecedent claim:

                    “… an ability and willingness to teach mathematics is entirely irrelevant to being a competent Mathematical Methods teacher.”

                    Age before beauty, Marty.

                    (I thought I said no monkey business. That includes no deflecting).

                    1. Oho ho! So that’s the way it is?

                      Axiom: To be a good teacher of Maths Methods (or of anything) you must have mastery of the content.

                      So we only need to decide whether or not a good Maths Methods teacher (or any teacher) must also be able to help a student pass a mickey mouse examination. This requires more than mastery of content. It also requires teaching the student stratagem.

                      Do you agree or disagree that a good Maths Methods teacher must also be able to teach stratagem for passing a mickey mouse exam?

                    2. John, I fundamentally disagree with your axiom. That’s the whole point.

                      Being a “competent Mathematical Methods teacher” is solely about getting your students the highest grades possible. That is obviously the framing that wst and I were using, and the framing that *everybody* uses. And for that framing, a teacher’s solid understanding of the mathematics is a second order concern, maybe third order.

                      I admit I was overstating it. I acknowledge the second/third order benefit of a teacher’s mathematical understanding to being “competent”, but it is not first order. Not even close.

                    3. “Being a “competent Mathematical Methods teacher” is solely about getting your students the highest grades possible.”

                      But can that be done without a solid understanding of the mathematics? If it can, then the whole system is totally f%$ked. Which I suppose is your point. I prefer to think that the f%!k-up is not total, and that a teacher with a solid understanding of the mathematics plus ‘street smarts’ will be the more ‘competent’ Maths Methods teacher under your definition (at least for the stronger students). And that such a teacher can generate greater passion for mathematics. (I don’t see how you can have any passion with no understanding, and – some – students do respond to passion).

                    4. Yes, the system is pretty much totally, um, screwed.

                      I acknowledged that a Methods teacher’s solid understanding of mathematics is relevant, but I honestly believe it is only relevant as a second or third order concern. Same for Specialist, although maybe closer to 2nd than 3rd.

          2. I really don’t like this definition of “competent math methods teacher”.

            I don’t hate exams. But it is a big problem in my opinion that so much of mathematics teaching is exam focused. They should focus on teaching mathematics, that’s the important work that needs to be done.

            1. Glen, I’m fine with the definition. That is the nature of high stakes exams. The issue is whether or not the exam/subject demands the students learn any mathematics.

            2. Perhaps we should refine the wording a bit to competent Unit 3&4 Mathematical Methods teacher?

              A unit 1&2 teacher (which is Year 11 for most but not all students, some start a year early) is not judged on their exam results anywhere near the same extent, because the exams at Units 1&2 are internal, not set by VCAA.

              Marty is (unfortunate as it may well be) 100% correct in this matter: a Unit 3&4 Mathematical Methods teacher is and will forever be judged by the results obtained by their students. Of course there are HUGE issues with this, which have all been documented and subsequently ignored by more than one principal/department head.

              Such is the nature of the system.

                1. Sure. Not all 3&4 teachers teach 1&2 and v-v for a variety of reasons.

                  It is a trial run for the student more than for the teacher in many cases.

    1. Also, Franz Lemmermeyer’s story reminded me of how in General/Further Mathematics, it seems that teachers similarly encourage students to use Solve. Even to solve simple linear equations, it seems (from my limited observations) normal for teachers to encourage students to use ‘Solve’ in order to avoid any risk of making a mistake on an exam. I don’t like it. Why spend three and a half years teaching students basic algebra, only to then make them unlearn it in Year 11 and 12?

      1. A teacher spending “three and a half years teaching students basic algebra” doesn’t imply that her students learned any basic algebra.

        1. I meant the school system as a whole. We introduce students to linear equations in the second half of Year 7. They revisit them each year in Years 8, 9, 10. Then in Year 11, we act like it’s too hard and they need CAS now?

            1. Sometimes I do get the sense that school mathematics is like a train that just keeps going even if most of the passengers have fallen off.

          1. Is that right? I’ve got stage 3 (year 5 and 6) students who have linear equations (pairs of simultaneous equations) down pat. I thought that was normal. The optional maths tests have lots of word-problem versions of these things, even in stage 2. It helps to treat them formally if the students will be tested on it.

  4. Nice set of stories. Of course CAS is awful and I’m so happy that it isn’t in NSW…

    2005 was the year I randomly decided to take a subject in measure theory, despite being a computer science graduate.

    Does CAS help with programming? Makes me wonder if it would fit better in an IT subject. Might be easier to move the problem somewhere else compared to obliterating it entirely.

    1. My experience with Classpad, including its programming features (some of the fruit of which I have published at https://nhtnhanbn.github.io/whitecas/), is that it is not much like other sorts of programming that I have done. Partly this is because the language and interface that Casio has is very painful to use, so I would not recommend Classpad programming as an educational experience for any IT subject that aims to be an IT subject and not a Casio Classpad programming subject. Having said that, I believe that TI CAS calculators have a small range of options that would be more tolerable to play with.

    2. Absolutely not. There’s good programming, then there’s this. Being taught a particular language that has no uses in the real world and is so obscure would undermine one’s understanding of writing code. There is a place for CAS…. in university, where you may want to solve a large system of equations, do some computational group theory, or implement some numerical methods, but that’s distinct in its own way from what a regular programming unit does. Showing hs students a domain-specific language (for math) is a poor introduction to programming (IMO).

      1. “Being taught a particular language that has no uses in the real world and is so obscure would undermine one’s understanding of writing code.”

        It undermines everything.

        Given we’re stuck with CAS in secondary school mathematics (not to mention the pseudocode fetish that has infiltrated mathematics like an intrusion of cockroaches) until one of us becomes Education Minister, it is mind-bogglingly stupid that we’re stuck with the handheld CAS calculator. If we have to have shit imposed on us, at least let it be the least bad shit (apologies to Marty for any ugh!, having deviating from his main point. I think we all agree that there should be none of this shit anywhere near a secondary school mathematics classroom).

      2. I object to your third sentence.. You can easily change it to be as such:

        “Being taught algebra by hand, which is usually automated by spreadsheets in the real world undermines one’s ability to effectively use technology.”

        Something’s pedagogical value is usually irrelevant to how useful it is in the real world. No one would consider using scratch “in the real world”, but it’s much more useful than say, Python, for teaching children the basics of programming.

        The real reasons that the Ti-nspire CAS is a bad introduction to programming, is because the default programming language is very limited, and because the interface for writing programs (the mini-keyboard and the various other buttons) is vastly inferior to a desktop keyboard. Not because it’s not useful in the real world or whatever.

        Also, I still doubt that the calculator is that useful in university, as you’d probably use an actual CAS (as in a computer program, not a calculator) tied to some programming language of some kind.

        1. See my comment below. Yes, we’re talking about actual CAS in university. The calculator becomes pretty useless within a year or two. And that’s what my point was with TI-Nspires language, it’s very limited in use and capability. It being worthless in actual, large-scale applications is just an added bonus.

        1. When I use the word “CAS”, I use it to mean the class of languages (or libraries) whose primary function is mathematics. i.e Sympy, MATLAB, Mathematica, Julia, Maple, SageMath, etc. I know that the word CAS has become synonymous with three dingy calculators with antiquated software but I should clarify.

  5. Having lectured pre-service maths teachers at ACU who exhibited extremely bad anxiety upon hearing my exam (for an equivalent-in-difficulty-to-first-year-undergraduate-subject-at-Monash/UniMelb) was to be calculator/CAS-free, to me it’s like “Well the best way not to get hooked on heroin is to NEVER try it in the first place.”

    Never mind that I purposely wrote the exam such that virtually none of the questions could’ve benefited from having one anyway.

    1. Re: “… pre-service maths teachers at ACU … exhibited extremely bad anxiety upon hearing my exam (for an equivalent-in-difficulty-to-first-year-undergraduate-subject-at-Monash/UniMelb) was to be calculator/CAS-free”

      How depressing, Matt. This attitude/mind-set is a huge part of the problem. No doubt they got through the exam (more to say below) and then happily returned to their supplier.

      So how did they go on the exam? Did you have to grade on a curve? I’ll assume everyone passed because:
      1) No-one is allowed to fail (university policy) therefore you graded on a curve.

      2) Some nevertheless failed and were given a make-up exam (university policy) that was identical to the one they failed in order to pass. Which you were told had to be graded on a curve.

      Did any students experience a change of mind-set as a result of sitting this exam? (I will assume no, they all went back to their supplier as soon as possible).

      Maybe you should have let them use the CAS if “… virtually none of the questions could’ve benefited from having one anyway” – just to teach the most important lesson of all.

      They’ll be in a classroom the following year telling students of their harrowing experience and then happily start teaching how to press buttons. The future of mathematics education is bleak. I doubt there will be any mathematics subjects, except in name only, in the future. Maybe the cockroaches will do better when they get their turn with the planet.

  6. As for story 1: A few years back I wrote an article on the sorry state of education science in Germany. I mentioned the case of the University of Münster, who received, starting in 1997, funds from TI (547.000 DM in 1997) and at the same time started publishing material praising CASs and graphic calculators. In 2004, graphic calculators became compulsory in our state (Baden-Württemberg) for students from grade 7 on; with 40.000 students buying the TI 93 each year for about 100 Euro, this turned out to be a very good investment.

    Of course the article was not accepted for publication.

    In April 2021 I was approached by a few guys who were programming machines. They had to solve the equation

        \[x = O -  r \cdot \cos(\alpha + \beta) - \sqrt{\ell^2 - (r \sin(\alpha + \beta) - b)^2}\]

    for \beta. Actually, they were in possession of such a formula that was provided by some CAS, but it returned incorrect values. After some effort I managed to solve this equation; it turned out that there were several choices of inverse functions to make, and that the CAS had not made the right ones.

    1. Hi Franz.

      Re: “there were choices of inverse functions to make, and that the CAS had not made the right ones.”

      I love it. I always make sure that the same thing happens in any SAC I write. For example:

      Finding the value of \displaystyle f^{-1}(a) is easily done by solving \displaystyle a = f(x) under a suitable restriction on x). But nearly every student insists on using the CAS to first find an inverse function \displaystyle f^{-1}(x) (eg. productlog) and then substituting \displaystyle x = a into this inverse function. Of course, the question is deliberately cooked so that the CAS uses the wrong branch (*) and hence give the wrong answer. What’s really funny (in a black comedy way) is that most students happily use the unfamiliar function spat out by the CAS (eg. the productlog function) when they have absolutely no knowledge or understanding of the function. They don’t have a clue what they’re doing. No mathematical common sense, they just completely outsource their thinking to CAS code.

      It’s like those fools who use a Sat Nav and end up driving into a lake because “that’s what the Sat Nav told me to do”.

      * Actually the CAS tells the students exactly what branch it’s using but the students don’t realise it’s the wrong branch because they don’t understand the output. They just blindly copy the output and feel smug that the question was so simple.

      1. The thing I worry about with this strategy is the end-game. Basically, you trick students who don’t know what they are doing into getting the wrong answer.

        But if the students do understand how to read what the CAS is saying, they won’t get it wrong. Does that teach them mathematics?

        Something I’ve wondered about is if we should put any effort into teaching students why they should avoid reaching for the technology, be it CAS, Wolfram Alpha, google or whatever. And if the answer is yes, then when should that happen?

        1. Re: “… you trick students who don’t know what they are doing into getting the wrong answer”

          Yes, Glen. I freely and gladly admit it. I do. I prefer this to the alternative, namely, tricking students who don’t know what they are doing into getting the \displaystyle right answer.

          Glen, there is zero, nada, bupkis, zilch chance that \displaystyle any Maths Methods student would look at Mathematica output such as

          -ProductLog[-(2x)]

          and think, hmmm … I better check I don’t want the branch corresponding to

          -ProductLog[-1, -(2x)].

          Nor should they have cause to, if they’re using their mathematical understanding rather than outsourcing their thinking and ignorantly deferring to the almighty Mathematica output. I’ll bet London to a brick that there’s no chance a student using a handheld CAS calculator would get so easily (and willingly) stooged. (So much for the famous Mathematica advantage). And there’s countless functions to choose from that will do this.

          Re: “Something I’ve wondered about is if we should put any effort into teaching students why they should avoid reaching for the technology, be it CAS, Wolfram Alpha, google or whatever. And if the answer is yes, then when should that happen?”

          Well, I like to think that ‘tricking’ students in the way I’ve described is a step in the right direction. I think Franz’s anecdote could also be used in a similar way. Let them use the technology and get stooged by its output. Only the truly stupid won’t learn the lesson sooner or later.

            1. Marty, do you think that teaching mathematics will also teach students that they should avoid reaching for the technology?

              I like to think that I teach mathematics in my classes. I also have to teach how to use the CAS technology. Students reaching for the technology continually wins. Particularly when students are not even using pen and paper to write notes.

              I think the only way your step in the other direction could work is if there is no technology to reach to in the first place.

              Personally – The only way to learn not to touch the fire is to get burnt by it.

              1. Ugh! I’m not taking any part in this ridiculous discussion.

                The point of this post is that CAS poisons everything. I couldn’t give a stuff how to teach people to use or not use the damn things. You have to worry about this, but I don’t. I just want the damn things out of the classroom, and Kaye Stacey and David Leigh-Lancaster on an iceberg.

                1. Of course you do. We all do. And for all we know, at least one of them is on an iceberg right now (wishful thinking). Although I’d prefer they were both somewhere warmer. Much, much warmer. Together with all their friends.

                  Marty, we understand the point of your post. But what comments do you expect to get? Comments of support? Of course we support – strongly – the point your post makes. We are every bit as frustrated as you are. Maybe more so. We see it day after day after day. And just when we think we’re making progress, a new year starts and it’s groundhog day. Year after year. Comments are going to head in the natural direction of trying to mitigate the poison. Develop immunity.

                  Things are getting worse, not better. Mathematics curricula is becoming infested with pseudocode and technology and inquiry-based learning. Soon mathematics will be a subject in name only. Unless the pendulum starts swinging the other way. What’s needed for that to happen? Vent? Engage in a war of attrition against CAS? Guerilla warfare? Refusal to use it? Silent quitting? Gain positions of power and influence and get rid of it? Teachers en masse can’t even bother caring about errors on exams. They certainly don’t care about getting rid of CAS, particularly since more and more of them depend on the CAS themselves and would never be able to teach without it. Even the IB has succumbed.

                  So, what to do?

          1. But don’t you worry that students will take away from this the message that “I need to learn how to better use CAS”? Just “I am bad at CAS” and no inkling of “CAS is a problem”.

            1. Nope. I sleep very well at night.

              Students clearly see that it’s not about learning how to get better at using CAS. Or being bad at CAS. Because it’s not actually a CAS thing. It’s pretty clear that to know how to interpret the ProductLog output, you have to know advanced level maths. Which is contextually futile to attempt because there’s always another ‘ProductLog’ ready to bite you in the arse. Easier to forget about learning how to interpret esoteric (for them) CAS output and start learning how to apply mathematical understanding.

              The lesson that ignorant/lazy reaching for CAS is a problem gets learnt. Except by those with no pain receptors. They just keep wanting to touch the fire.

  7. Does anyone here teach the IB Diploma Mathematics?

    I got out before that became CAS-ified. Does anyone have a sense of why CAS was introduced into the IB?

    (I’m assuming it was – there was a lot of discussion about it around 2010ish but from memory we were all teaching with the non-CAS TI machines. Same price for less features…)

  8. In the Australian Mathematics Competitions, calculators are permitted only for papers at middle and upper primary levels, not secondary levels.

    1. Hi Terry. My son just did one in stage 2 (middle primary) and there were no calculators allowed. Indeed, he does not own a calculator, they don’t use them in the school at all. So I think if there were such competitions and calculator use was helpful/allowed, it would be an issue.

        1. I introduced and administered the AMC for my daughters’ primary school this year. I noticed and simply ignored the permission for students to use calculators.

          1. I tried to introduce AMC at my kids school.
            Joked that I will regender kids (opposite sex school nearby had run it).
            Even coined “AMC: snub at your own peril” for maths teachers.
            To no avail.
            Most AMC problems are fun, challenge, can be used throughout the year in class,
            some mathematics kids will not be exposed to otherwise (bar a few other good comps).
            No Maths Challenge fYA? How do you expect your students to reason in senior years?

                1. Oh, whoops. Yep, I misread.

                  Yeah, it’s a tricky one. I try to tread gently with my kids’ school, since there’s never a shortage of parents whining for/about something, valuable or otherwise. I was lucky to have a natural opportunity to suggest introducing the AMC.

                  1. I actually offered to organise/conduct it.
                    I guess a bit of diplomacy does not hurt. So many fragile egos around these days…

                    1. Yes. An offer to implement X is readily, and maybe not unreasonably, interpreted as a criticism that X is not already there.

  9. A substantial part of each lesson of VCE mathematics is devoted to the “CAS skills” that can be noted down into the summary book and deployed in SACs and exam to trivialise the topic of the day. As if it were a determining factor in mathematical success. I know of some teachers who explicitly tell students to avoid thinking in the multiple choice questions as much as possible, rely on the calculator to do practically everything.

  10. Another problem that I see with CAS is that if you have a student who is vision-impaired, making them use a fiddly CAS calculator in VCE adds an additional burden. A scientific calculator has ready alternatives, but a CAS calculator is so weird and specific that it doesn’t.

  11. Regarding Story 5, unless that would divulge Rob’s identity (which I don’t aim at): did he acknowledge your contribution to his paper in the final form – assuming it was not condemned to the desk drawer? And do you think there was a sort of moral imperative in the decision he took (if applicable)?

    1. Thanks, Christian. For other reasons, Rob decided not to polish and publish his article. Which, including or not my de-CASification, I think was unfortunate. It was a worthwhile article.

      It was all amicable. I wasn’t critical of Rob for having not contemplated my CAS-duck, and I didn’t mean Story 5 to be read that way. What’s a standard technique for a mathematician needn’t be on the radar of a teacher-ish person. But what grabbed me was Rob’s “definitely a case for CAS”.

      One of the critical undertakings of senior mathematics teaching, including the guiding of senior teachers, is to get students and teachers to realise that mathematics is powerful and clever, and *lots* more techniques can be on their radars if they read more and practise more and ponder more. CAS poisons this in a double-barrelled way (to mix the metaphor). First, so so so much time is spent now in teaching and learning Stupid CAS Tricks, there is little time or concern for treating any topic in any reasonable depth. Secondly, when the inevitably shallow techniques learned turn out to be insufficient to tackle a tricker problem, the knee-jerk reaction, ALWAYS, is not to go deeper into the maths, but to simply give up and reach for the CAS.

      It is truly, unbelievably, mind-numbingly awful, and the fact that the VCAA is led by CAS-insaned half-wits is just so, so, so damn depressing. They are way too stupid and ignorant to contemplate how stupid and ignorant they are.

      1. Thanks, Marty. It is nice to have such a sum-up of the evil consequences of CAS. I had previously thought that knee-jerks rather originate from hammers – though not those you’d apply to calculators!

        Actually, I think that I may have caused you to misunderstand; with the “decision that he [Rob] took”, I meant his acknowledgment of your contribution to his paper in a final short note therein. Of course this question is void now, given that Rob decided not to publish his work. I would think that de-CASification could warrant an acknowledgement if the mathematical (non-CAS) argument was important; maybe a differentiation under the integral sign wouldn’t be enough. But this may not belong here.

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