WitCH 45: Learning from Our Betters

OK, we’ll get back into this slowly, and let others do the work. (Yes, at some point soon we’ll write about the seventy million knuckle-draggers who voted for Trump.)

A couple days ago The Conversation published a math ed article by some familiar maths ed folk:

Fewer Australians are taking advanced maths in Year 12. We can learn from countries doing it better.

To be fair, and making our way past the pithy title, we’re not sure the article is crap: we’re just not sure what it is. See how you go.

37 Replies to “WitCH 45: Learning from Our Betters”

      1. In these debates, it is usually assumed that you need calculus to enter particular courses such as engineering. The new admission requirements at RMIT suggest that RMIT does not agree with this assumption. This change may flow through to universities that are nervous about losing students to RMIT – especially international students.

        There is an analogy with respect to teaching foreign languages at university. When I finished school, you could not study a foreign language at university unless you had completed the equivalent of Year 12 in the language. Now you can start from scratch in studying a language at university.

        Similarly, students at RMIT will be able to study engineering without prior exposure to calculus. I imagine that RMIT will restructure the engineering courses to enable students to do this. And the change will make Further Mathematics an even more attractive option at VCE.

  1. There’s a simple and efficient way of doing it: Make sure that the kids learn the times tables and the algorithms for multiplication and long division, and stop dumbing down the curriculum, in particular in algebra and elementary geometry. Math is fun if you can compute for three lines without making four mistakes, and it is not if you can’t.

    1. Thanks Franz, I completely agree with your opening premise.

      So… why is it not being done?

      Apart from schools not being able to find teachers who actually know what Mathematics is, of course.

      And primary schools who think it is totally fine to say (sometimes very publicly): “we don’t believe in teaching times tables”. (*dies a little bit more inside*)

      1. Why should they learn to spell, count, read warning labels on medicine bottles…?

        There is a certain level of basic skill which, unless learned to the point where it does not require thought, impedes progress (and in some cases, basic safety as an independent member of society). Times tables are probably the best (but far from the only) example of a relatively simple skill, the benefits of which far outweigh any “philosophical objection” a primary school or primary teacher (include some Year 7/8 in that definition) may hold.

        And yes, Postman, wonderful (and totally terrifying in some ways).

        Another favorite of mine is J Abner Peddiwell: “The Sabretooth Curriculum” in which “educational experts” debate the need to teach sabretooth tiger scaring once the ice-age comes and there are no more sabretooth tigers on which to practice scaring and other wonderful trivialities such as how many tigers someone must scare before they are allowed to teach others…

        1. Thanks for this Red Five. I was not asking why should students learn to spell, count, or read. I was asking why should students memorise multiplication tables. I can think of situations where memorising certain facts is necessary (e.g. learning a foreign language). What I am after is a compelling argument that students should memorise multiplication tables.

            1. Not sure what that means; I assume that it has something to do with the Billy Goats Gruff – but I can’t see the connection.

          1. Fair question, but you brought it up, so before I answer again (with a more specific opinion – that will be an opinion and little else) – perhaps riddle me this…

            Why should students *not* memorize multiplication tables?

            1. Of course, students are free to memorize whatever they please. I envied my grandmother who could recite the alphabet backwards.

              I am not arguing that students should not memorize multiplication tables. I am asking why should they? (In the Catholic tradition, the role of the Devil’s Advocate was to make sure that the arguments for canonisation of a person were sound rather than to argue against canonisation.)

              In the many discussions of this issue on this website, it has often been stated that students should memorize the multiplication tables. But I do not recall seeing any justification for this.

              I am not opposed to memorization per se, as I indicated above. However, we should be selective in what we choose to memorize. There should be some value in the exercise. One might argue that memorizing multiplication tables (i) facilitates learning more advanced ideas, or (ii) helps students to develop memory skills, or (iii) helps us to better understand the number system – although these arguments need elaboration in order to convince me.

              From my experience, I’d say that many students in VCE mathematics have little recollection of multiplication tables. If they have been drilled in them in primary school, then the drilling did not last long. I assume that many of these students will go on to successful, happy lives without ever being able to, or needing to, recall that 7×8=56.

              I strive to memorize as little as possible in mathematics. I could not even tell you the sine or cosine of the angles 30 degrees or 60 degrees because I know how to find these quickly.

              1. In my 40+ years of interacting with people in the realm of physics and maths, I have seen a strong correlation between co-workers’ levels of memorised physics/maths data and my ability to talk with them easily on any physics/maths topic. To take the example of angles: whether I’m talking simple geometry or not-so-simple radar signal processing (with its complex numbers and Argand diagrams), if I sketch a simple example by drawing a 30°-60°-90° triangle and appeal to some easy-numbers intuition to make a point, I know the discussion will be slowed just that little bit if the guy I’m talking to doesn’t immediately make the connection that e.g. sin 30° = 1/2. It’ll be slowed just a tiny bit; but there are a hundred little such connections that work together to create a smooth, productive, and fast conversation. Multiplying simple numbers quickly is part of this. Each of those facts and connections, in itself, might seem pointless. But line them up, and their aggregate will make what would otherwise be an arduous interaction into something that flows smoothly.

                Sure, we can have a conversation about the above radar signal processing without the other guy knowing what 8 x 7 is; but I can be confident that he then also won’t know a miriad of other things, making the interaction tiring or tedious. In contrast, -he- won’t be aware that there is any problem in the communication, since it will be reduced to the rate that he can comfortably handle. He will never be aware of how much faster and smoother things would go if he had taken the small time to memorise a few apparently useless facts. It’s like people who live in awful-looking houses: they have the nicest view in the street, because all -they- see are nice houses. They don’t know why their neighbours are so unhappy.

                Memorising such things as multiplication tables takes, what, a handful of neurons? I am always amazed at how many people devote what must surely be many, many more neurons on footy factoids such as who won some championship, and yet they never memorise a few facts that will be around forever, and will help them talk so much more fluently to others in their chosen field.

                On that note, I’m not a school teacher, but if a school kid asked me “Why do we have to learn matrices: when am I ever going to use those in my life?”, I’d reply “I don’t hear you complaining about cross-country running in your phys-ed classes. When will you ever need -that-? And yet cross-country running is good for you, because it builds strength and coordination, both mental and physical. Sure, you don’t always enjoy it, especially when you have the stitch. But still you know it’s good for you. Treat matrices and times tables in the same way. Although -you- might never notice the increase in mental suppleness that they give you, other people will”.

                1. Thanks, Don. I think that’s the point. The difference between “quick” and “immediate” doesn’t matter (much) if it happens once, but it’s critical if it happens five times or ten times, or a hundred times. No one can think about big things if they’re constantly tripping on little things.

        2. Thanks, RF. I’d never head of Peddiwell’s book. It sounds more cheap shots-ish than Postman, and more palatable to those promoting “21st century education” and the like, but I’ll check it out.

        3. Thanks for letting me know about the Saber-tooth curriculum. This is hilarious – he ridicules every idea that “education scientists” have come up with in the last 30 years, and the book is from 1939.

      2. Without times tables, no fractions.
        Without fractions, no algebra.
        Without algebra, no calculus.

        Or: Why do we teach kids to walk when we know they’ll be driving Porsches when they’re grown up?

        It honestly breaks my heart when I get classes in which half the students confuse products with sums. It all begins with confusing 2*3 and 2+3. And this is only possible when you actually have to think when you hear 2*3.

        1. This reminds me of the old saying:

          For want of a nail, the shoe was lost.
          For want of a shoe, the horse was lost.
          For want of a horse, the rider was lost.
          For want of a rider, the battle was lost.
          For want of a battle, the kingdom was lost.

          Which can be re-framed as:

          For want of numbers, counting was lost.
          For want of counting, addition was lost.
          For want of addition, multiplication was lost.
          For want of multiplication, division was lost.
          For want of division, fractions were lost.
          For want of fractions, algebra was lost.
          For want of algebra, calculus was lost.
          For want of calculus, mathematics was lost.
          For want of mathematics, science was lost.
          For want of science, engineering was lost.
          For want of engineering, civilisation was lost.

          I’m an electrical engineer so, I suppose, that makes me biased. 🙂

              1. This is from her publication “gravity and grace”
                (see https://boxes.nyc3.digitaloceanspaces.com/wp-content/uploads/Gravity-and-Grace.pdf ):

                “Money, mechanization, algebra. The three monsters of contemporary civilization. Complete analogy.

                Algebra and money are essentially levellers, the first intellectually, the second effectively.”

                I read the whole chapter, but I am none the wiser. If money ever was a leveller, it failed badly. And whatever the three monsters of contemporary civilization are, few people on this planet save a few teenagers would include algebra.

            1. Substitute ‘politics’ for ‘algebra’ and Ms. Weil might get my vote (whoever she is). Bear in mind that I’m part of the proletariat and so bourgeoisie philosophy doesn’t excite me.

  2. I read through one of the linked reports, the UK review of post-16 mathematics, to follow up on the article’s claim that in the UK three quarters of students who achieved “good marks” still chose not to study maths after 16 years old. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/630488/AS_review_report.pdf The relevant bits are from Chapter 5.

    I note that the report found that 75% of students who achieved at least a C in their GCSE did not continue studying maths. I don’t know that much about the GCSE – I gather it’s roughly equivalent to Year 10 in Australia – but I would be surprised if a C could reasonably be considered a “good” mark.

    The report also found that 79% of students achieving an A* (the highest grade) at GCSE continued with maths, and 52% of students achieving an A at GCSE continued with maths. Only 22% of students achieving a B continued with maths.

    The report also notes that the correlation between high prior achievement and post-16 participation in maths is stronger than in other subjects.

    1. Thanks, SRK. It seems to be a common trick now, to consider a link/reference to suffice as proof. Sort of a surreptitious Gish gallop. And, the trick almost always works, because people seldom chase up the “proof”. It smelled like the article was doing that, but i hadn’t gotten as far as doing any of the work to check. Glad you did.

  3. Here is something a bit crap “Australian maths curricula are ambitious, including a strong emphasis on calculus. And they have recently expanded the amount of statistics included at upper year levels.”

    In NSW they put more statistics (most of it a rehash of year 10 work) in Advanced and Extension 1 Maths at the expense of geometry. There are no stand alone geometry topics in senior maths and that’s crap and sad.

    1. Thanks, Potii. I’m probably an outlier in this company, in that I don’t care much about geometry in the school curriculum. But to lessen it to promote yet more stats is, of course, criminal.

      1. Potii – I am genuinely curious about the NSW curriculum as the exams I see (far earlier than VCAA publish theirs…) seem to have some quite good analytic geometry, especially the Extension 2 papers.

        I’m not arguing for more geometry in VCE Methods or Specialist; it just doesn’t fit with the functions/calculus course that is VCE in these two subjects (but neither does the stats we now have…) however I do believe spatial thinking is a wonderful part of Mathematics.

        1. The HSC papers generally tend to be decent. Though this year in the Advanced paper there was a questions that when on for a whole page of waffle involving “statistics” but was actually just a find the equation of the line question. I hope that it is not a sign of things to come for the new courses introduced last year.

      2. Marti,

        Good to see you bouncing back…
        I’d say your can be at least 6 standard deviations from the mean on high school statistics.

        As a statistician and retired actuary I’d like to see at least a little basic probability theory,combinatorics , discrete and continuous distributions and simple modelling theory covered in the math curriculum with less one mark button pressing questions

        BTW I see 538 got their polls more accurate this time on the lame duck POTUS

        Steve R

        1. Thanks, Steve. I’m not sure I’m exactly bouncing, but getting there.

          The Victorian curriculum I had in the 70s had a hell of a lot of probability, and a little statistics, and it was excellent. The incessant shoving of statistics down school students’ throats, however, is insane. It has no mathematical foundation or depth, serves no purpose, is batshit ugly and batshit boring. The ABS is largely responsible for this, and they should go fuck themselves.

          Re 538, Nate and his gang don’t do the polls: they just interpret them. His fault or not, I don’t know that Nate was more accurate this time, and they weren’t that inaccurate last time.

          1. Marti,

            Yes 538 does do intelligent weighted averaging of various polls of third parties but still relies on the soundness of the underlying samples of those selected

            They seem to have got most electoral college states accurate to around 3.5 % of the actual results so far.

            It seems crazy to me that the “first part the post system ” still applies in most states making the final result dependent on 5 or 6 swing states only. In 2016 Trump was less popular than Clinton but made a late run to snatch it in the blue wall states

            Steve R

            1. Steve, there is so much craziness in the American electoral system, and America generally, that it is impossible to know where to begin. It is not just that most governing processes are entirely ridiculous, it’s that the thinking about those processes, from pretty much everyone, is so perverted that there is no way out. The country is fucked, and I’m just thankful that my parents realised early that it was fucked, and abandoned ship for Australia.

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