ACARA Crash 8: Multiple Contusions

OK, roll out the barrel, grab the gun: it’s time for the fish. Somehow we thought this one would take work but, really, there’s nothing to say.

It has obviously occurred to ACARA that the benefits of their Glorious Revolution may not be readily apparent to us mathematical peasants. And, one of the things we peasants tend to worry about are the multiplication tables. It is therefore no great surprise that ACARA has addressed this issue in their FAQ:

When and where are the single-digit multiplication facts (timetables) covered in the proposed F–10 Australian Curriculum: Mathematics?

These are explicitly covered at Year 4 in both the achievement standard and content descriptions for the number strand. Work on developing knowledge of addition and multiplication facts and related subtraction and division facts, and fluency with these, takes place throughout the primary years through explicit reference to using number facts when operating, modelling and solving related problems.

Nothing spells sincerity like getting the name wrong.* It’s also very reassuring to hear the kids will be “developing knowledge of … multiplication facts”. It’d of course be plain foolish to grab something huge like 6 x 3 all at once. In Year 4. And, how again will the kids “develop” this knowledge? Oh yeah, “when operating, modelling and solving related problems”. It should work a treat.

That’s the sales pitch. That’s ACARA’s conscious attempt to reassure us peasants that everything’s fine with the “timetables”. How’s it working? Feeling good? Wanna feel worse?

What follows is the relevant part of the Year Achievement Standards, and the Content-Elaboration for “multiplication facts” in Year 4 Algebra.

ACHIEVEMENT STANDARD

By the end of Year 4, students … model situations, including financial contexts, and use … multiplication facts to … multiply and divide numbers efficiently. … They identify patterns in the multiplication facts and use their knowledge of these patterns in efficient strategies for mental calculations. 

CONTENT

recognise, recall and explain patterns in basic multiplication facts up to 10 x 10 and related division facts. Extend and apply these patterns to develop increasingly efficient mental strategies for computation with larger numbers

ELABORATIONS

using arrays on grid paper or created with blocks/counters to develop and explain patterns in the basic multiplication facts; using the arrays to explain the related division facts

using materials or diagrams to develop and record multiplication strategies such as skip counting, doubling, commutativity, and adding one more group to a known fact

using known multiplication facts for 2, 3, 5 and 10 to establish multiplication facts for 4, 6 ,7 ,8 and 9 in different ways, for example, using multiples of ten to establish the multiples 9 as ‘to multiply a number by 9 you multiply by 10 then take the number away’; 9 x 4 = 10 x 4 – 4 , 40 – 4 = 36 or using multiple of three as ‘to multiply a number by 9 you multiply by 3, and then multiply the result by 3 again’

using the materials or diagrams to develop and explain division strategies, such as halving, using the inverse relationship to turn division into a multiplication

using known multiplication facts up to 10 x 10 to establish related division facts

 

Alternatively, the kids could just learn the damn things. Starting in, oh, maybe Year 1? But what would we peasants know.

 

*) It has since been semi-corrected to “times-tables”.

15 Replies to “ACARA Crash 8: Multiple Contusions”

  1. Note: in the following rant I’m writing TIMES TABLES in capitals first of all because that’s what I’m talking about, and also as a kind of embittered shout. However, I doubt that any amount of shouting will get through to ACARA.

    I’m only aware of two “multiplication facts”: that multiplication is commutative, and distributive over addition. In other words, that the integers are a commutative ring (or if you like, that the natural numbers including zero are a commutative semiring). To call a statement like 2\times 3 =6 a “multiplication fact” is stupid, pointless, and dare I say it, another example of ACARA’s intellectual dishonesty.

    What we are seeing here, though, is a fine example of ACARA taking an idea – problem solving – to ludicrous lengths. It’s vital that primary school children learn their TIMES TABLES for any number of reasons: it increases their number sense; it provides a foundation for further work; and it gives them useful tools in their mental mathematical toolbox. There is simply no short-cut here; TIMES TABLES have to be learned, by rote if need be, and when they’re well embedded they can be used. I don’t see any problem with rote learning, or repeated practice – after all, athletes, sportspeople, musicians all practice their techniques – so why not schoolchildren? I also think that TIMES TABLES should be learned up to 12\times 12; yes we have a metric measuring system, but there are still lots of 12’s about, in egg cartons, months of the year, hours in the day, American and historical measurements and so on.

    Also, the ACARA system promulgated in their elaboration seems like an absurd amount of busy work. First you learn your “multiplication facts” (I feel dirty writing this down in the ACARA context) for 2, 3, 5 and 10, and then apparently extend those to the other numbers less than 10. So to multiply, say 7 and 9, do you write this as (5+2)\times(10-1) and go through a long and complex rigmarole of expansion? At the age we’re talking about, surely it’s much easier simply to know that 7\times 9 = 63 and have done with it.

    The trouble with problem solving as a basis for learning is that for a given problem, everybody will take away different lessons from it. To ensure that everybody learns the same thing you simply have to be more structured. And on a side note, have you noticed that “problem solving” has now become a verb? Instead of students learning “how to solve problems”, they are supposed to be learning “how to problem-solve”. This outrageous mockery of good English makes me furious.

    It seems that mathematics in this country is heading into a sharp nose-dive if the policy makers and curriculum designers are themselves so benighted, ignorant, and incompetent.

    1. Hi AtA.

      I agree with all you say – it’s hard not to since it’s just plain common sense (but common sense seems to be out of fashion these days). The athlete analogy is a really good one – no-one is surprised when they hear that Roger Federa was practising one specific shot for three hours, for example. But apparently we have to be entertainers and learning has to be an exciting adventure. Apparently students do not have the concentration span (we can thank screens and digital technology for that, IF it’s true).

      I agree that the times tables should be learnt up to 12 \times 12 – I have a powerful memory of doing this in Grade 4. Personally, I would have students in Grade 6 memorising the perfect squares up to 20^2 = 400. I recall in Year 9 that our Science teacher made us learn the first 20 elements of the Periodic Table by writing them out 50 times. In Year 12 we had to memorise the first 36 elements of the Periodic Table.

      In Yr 9 students must memorise the quadratic formula (well, used to). In Maths Methods we insist that students memorise the sine, cosine and tan of the ‘special angles’. And yet memorisation and rote learning are considered dirty activities.

      As for clowns creating new jargon (such as “how to problem-solve”) to make their bullshit look and sound erudite, ACARA aren’t the first and won’t be the last. Which doesn’t excuse them. Some Svengali will have invented this, probably in his/her/their wanker PhD thesis in mathematics education. Bullshit politician-speak.

      (As an aside, who’s the moron who started the whole “learnings” bullshit. I enjoyed this: https://dailyreview.com.au/time-take-learnings-seriously/)

      But I will make one ‘correction’ to your comments.
      Re: “7 and 9, do you write this as (5+2)\times(10-1) and go through a long and complex rigmarole of expansion.
      I agree students should know 7 \times 9 = 63. What I think ACARA wants is students reasoning that 7 x 9 = 9 x 7 = 10 x 7 – 7 and 70 – 7 = 63. Which I don’t see as too much of a complex rigmarole. Having said this, I’m certainly not agreeing with ACARA and I re-iterate my total agreement that students should know 7 \times 9 = 63. What I would like is that students can do mental arithmetic such as 29 x 7 = 30 x 7 – 7 and 210 – 7 = 203, or 28 x 7 = 20 x 7 + 7 x 8 and 140 + 56 = 196.

  2. I recall a discussion with an expert on mathematics education.

    Me: Should students know 8 \times 7 by the end of grade 6?

    Him: Well….(long pause)…. they should be able to work it out. I mean, you just work out 4 \times 7 and double the answer.

    1. You’ve mentioned this before. In what possible sense of the word can this person be called an “expert”?

      1. Marty, all these idiots are called ‘experts’ because they have a PhD in [insert proper noun] Education.
        Apparently having a PhD that’s not worth the ink it takes to write ‘PhD’ makes a person an expert.
        It’s a sure bet that the svengali at ACARA pulling all the strings either has or is working towards a PhD in mathematics education …

      2. As far as I know, an “expert” is simply somebody who can out-shout everybody else, and who has such an over-inflated sense of their own self-importance that everybody else ends up agreeing with them. Unfortunately education is a field particularly rich in such experts, partly I imagine because there are so many different competing theories of education. (As Bertrand Russell once said, but in another context: “Since they all disagree, it is a matter of simple logic that at most one of them can be true”.)

        1. “An expert is someone who knows more and more about less and less until s/he knows absolutely everything about nothing” (Nicholas Butler).
          ACARA is packed to the rafters with experts – The Daft Curriculum is proof positive.

          (PS – I love the Bertrand Russell quote).

    2. Hmm … So calculate \displaystyle 28 \times 2 …The clowns at ACARA would probably think you wanted students to learn their 28-times tables …

      But I wonder what your ‘expert’ would have said about \displaystyle 9 \times 7. Perhaps an even longer pause, followed by “They should be able to work it out. I mean, you just work out \displaystyle 3 \times 7 and triple the answer.”

      Give me a break! Where do you draw the line? ACARA have drawn a line. The wrong line. Surely it’s far less cognitive load to just memorise the bloody things!

      I would not trust any ‘expert’ who has the word ‘education’ appended onto their qualification. I wouldn’t trust a single one of them to teach a fish to swim. The only thing they are expert in is self-promotion and bullshit. It’s all these ‘education experts’ that have created the mess of the last few decades. The educational landscape is littered with the results of their failed experiments. So what’s the response? Let’s ask more ‘education experts’ – this time at ACARA.
      FMD! (The second word is Me and the third word is Drunk).

        1. Do the decision makers realise the extent of lack of numerical skills among school students?

          What size shoe should a 12-year old wear? There is a view, with which I have some sympathy, that students should be expected to learn X when they are ready to learn X. “Stage, not age” is the mantra.

          1. Terry, you’re missing the point.

            That same mantra is at the core of this 50+ year old guide. BUT, also at the core of the same document is the mantra that students shouldn’t progress to Y until they have a decent mastery of X. And *that’s* what’s missing now. No one cares if anybody masters anything.

            Learning the multiplication tables up to 10 x 10 is in the current curriculum, and in a reasonably declarative manner: “recall multiplication facts up to 10 x 10 and related division facts”. Yes, it should be 12 x 12. Yes, it should be “multiplication/times tables”. Yes, it should be said stronger. But it is there, clear and clean.

            But do students now learn their tables? Properly? Not nearly enough, not nearly well enough. So why not? Because no one cares. Because no one tests for it. Because no one thinks it matters. Ticking boxes matters; mastery does not.

        2. Ha! I remember that! It’s odd alright (as in strange). Because they should be able to work it out. I mean, you just work out \displaystyle 3\frac{1}{2} \times 7 and double the answer … Boom boom.

          More seriously, I liked Lara’s answer for how she knew it.

          I’ll bet the ACARA clowns didn’t and probably still don’t like it because it’s absurd even for them to want reasoning such as 7 x 7 = 10 x 7 – 3 x 7 and 70 – 21 = 70 – 20 – 1 = 50 -1 = 49. But I think it would be too hard, even for de Carvalho, to advocate it’s exclusion. After all, 7 is the second prime number after 4 …

  3. A nostalgic aside. I started school at a local Catholic school starting in Prep – we called it something else, but that is beside the point. At the end of Year 2 my parents applied for me to go to my father’s old school (a Catholic boys’ school which was not so local). I was interviewed with my parents and the decision was that I would be accepted into grade 3 provided that I knew my times tables up to 13 before I started. Guess what I did that summer!

    1. I like your story. Often people say they want students to “think like mathematicians.” But I think it might be better to get them thinking and acting like mathematicians/statisticians did when they were the same age.

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