ACARA Crash 10: Dividing is Conquered

This Crash is a companion to, and overlaps with, the previous Crash, on multiplication. It is from Year 5 and Year 6 Number. and is, as near as we can tell, the sum of the instruction on techniques of division for F-6.

ACHIEVEMENT STANDARD (YEAR 5)

They apply knowledge of multiplication facts and efficient strategies to … divide by single-digit numbers, interpreting any remainder in the context of the problem.

CONTENT (YEAR 5)

choose efficient strategies to represent and solve division problems, using basic facts, place value, the inverse relationship between multiplication and division and digital tools where appropriate. Interpret any remainder according to the context and express results as a mixed fraction or decimal

ELABORATIONS

developing and choosing efficient strategies and using appropriate digital technologies to solve multiplicative problems involving multiplication of large numbers by one- and two-digit numbers

solving multiplication problems such as 253 x 4 using a doubling strategy, for example, 253 + 253 = 506, 506 + 506 = 1012

solving multiplication problems like 15 x 16 by thinking of factors of both numbers, 15 = 3 x 5, 16 = 2 x 8; rearranging the factors to make the calculation easier, 5 x 2 = 10, 3 x 8 = 24, 10 x 24 = 240

using an array model to show place value partitioning to solve multiplication, such as 324 x 8, thinking 300 x 8 = 2400, 20 x 8 = 160, 4 x 8 = 32 then adding the parts, 2400 + 160 + 32 = 2592; connecting the parts of the array to a standard written algorithm

investigating the use of digital tools to solve multiplicative situations managed by First Nations Ranger Groups and other groups to care for Country/Place including population growth of native and feral animals such as comparing rabbits or cane toads with platypus or koalas, or the monitoring of water volume usage in communities

LEVEL DESCRIPTION (YEAR 6)

use all four arithmetic operations with natural numbers of any size

ACHIEVEMENT STANDARD (YEAR 6)

Students apply knowledge of place value, multiplication and addition facts to operate with decimals.

CONTENT (YEAR 6)

apply knowledge of place value and multiplication facts to multiply and divide decimals by natural numbers using efficient strategies and appropriate digital tools. Use estimation and rounding to check the reasonableness of answers

ELABORATIONS

applying place value knowledge such as the value of numbers is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a natural number by a decimal of at least tenths

applying and explaining estimation strategies to multiplicative (multiplication and division) situations involving a natural number that is multiplied or divided by a decimal to at least tenths before calculating answers or when the situation requires just an estimation

deciding to use a calculator in situations that explore multiplication and division of natural numbers being multiplied or divided by a decimal including beyond hundredths

explaining the effect of multiplying or dividing a decimal by 10, 100, 1000… in terms of place value and not the decimal point shifting

25 Replies to “ACARA Crash 10: Dividing is Conquered”

  1. I can’t believe they put “deciding to use a calculator [because you can’t do basic maths]” in the maths curriculum.

    I suspect that some schools perhaps don’t even teach the traditional (long or short) division algorithm anymore. Could that be right?

    For instance, recently a Year 7 student volunteered to do a division problem that came up (two-digit number divided by one-digit). I held my marker over the usual position waiting for her to tell me the first digit of the answer. But this method was foreign to her, so she took over and did it using repeated subtraction. She said they “didn’t do the short way” at her primary school (it’s only a short logical step away from what she was doing anyway). I’m inclined to believe her because she is a very attentive, enthusiastic student. And she did get the right answer, eventually.

    How can students choose efficient strategies that they haven’t been taught?

  2. Quite true.

    Also, what does “in context” mean? Surely a remainder is a remainder regardless of the context?

    1. I guess they mean with problems like “101 people are travelling by van. Each van can seat 8 people. How many vans do they need?” Or: “Nanna has 22 eggs and needs 4 eggs to make a pavlova. How many whole pavlovas can she make?” Depending on the context, you need to round up or down?

    2. Context is different when sharing balloons vs sharing pizza. Eg You can have a fractional amount of pizza but not a fractional amount of balloon.

  3. “solving multiplication problems such as 253 x 4 using a doubling strategy, for example, 253 + 253 = 506, 506 + 506 = 1012

    solving multiplication problems like 15 x 16 by thinking of factors of both numbers, 15 = 3 x 5, 16 = 2 x 8; rearranging the factors to make the calculation easier, 5 x 2 = 10, 3 x 8 = 24, 10 x 24 = 240”

    It’s all well and good to know some tricks like this for special cases (involving powers of 2, multiples of 10, etc.) but what’s a 10 year old to do when they have to compute 167 x 83?

    I guess the answer is:

    “using an array model to show place value partitioning to solve multiplication, such as 324 x 8, thinking 300 x 8 = 2400, 20 x 8 = 160, 4 x 8 = 32 then adding the parts, 2400 + 160 + 32 = 2592; connecting the parts of the array to a standard written algorithm”

    It’s hilarious that the curriculum writers can’t just bring themselves to say “use the standard written algorithm”, there always has to be a distracting and unnecessary sidetrack.

    (Actually, I suspect the REAL answer for how students should compute products of two/three digit primes is: use a calculator or “digital tool”)

    1. SRK, it’s pretty clear the writers don’t want students to use the standard written algorithm.

      1. I agree that’s the message being sent, but that (for me, at least) is what underlines the idiocy of the phrase “choose efficient strategies”. If the standard written algorithm is not an efficient (non-calculator) strategy for. eg. 167 x 83, then it’s not clear what teachers are meant to teach (assuming they are meant to teach efficient non-calculator strategies for such cases).

        1. The standard algorithm which has been taught for a very long time unfortunately is also very often taught without understanding. It may be efficient but it is not developing a fluency or understanding of number. Rather it rewards students (of which most teachers come from) who happily memorize a technique with little understanding of why it works. We dont need students to always use the most efficient method now, we use calculators to do that. What we do need is students to be creative and fluent with their use of number. While I don’t necessarily agree with ACARA’s wording I do believe they are attempting to emphasise number fluency and understanding. In regards to 167×83, regardless of how a students might find their answer they should understand an estimate around 150×100=15000 is on the right track. Followed by a discussion of whether the answer would be smaller or bigger than 15000. What skills students need to leave school with today is vastly different to even 20 years ago.

          1. Hi, Phil.

            It is simply not true that the skills, the education, students need now, in mathematics or in anything, varies dramatically from 20 years ago, or 50 years ago. Overwhelmingly, the “21st century skills” stuff is poisonous snake oil. RRR.

            Having said that, I agree that the role of the traditional algorithms is not obvious, and there is a debate to be had. But, on the flipside, if you want to do away with traditional algorithms, the onus is upon you to suggest an alternative to getting a decent sense of and facility with number and arithmetic. The current alternative, which I see pretty much everywhere, amounts to cheap tricks + games + calculators, and it is an unmitigated disaster. Students gain no fluency whatsoever, because nothing sufficiently solid is offered to gain fluency in.

          2. 167 \times 1 = 167 *
            167 \times 2 = 334 *
            167 \times 4 = 668
            167 \times 8 = 1336
            167 \times 16 = 2672 *
            167 \times 32 = 5344
            167 \times 64 = 10688 *
            Therefore, adding the * equations,
            167 \times 83 = 11861.

            Known in ancient Egypt more than 5000 years ago.

          3. Phil, I would largely echo Marty’s response, but I would also question the point – which is somewhat implied in your post – that these “fashionable” methods for teaching multiplication are resulting in a smaller proportion of students who “happily memorise a technique with little understanding of why it works”.

            From what I’ve heard, there’s still plenty of students doing that, it’s just that now they’re memorising horribly inefficient and clunky methods, like drawing rectangles dissected into smaller rectangles, or groupings of dots, or whatever.

            Yes, it is frustrating when one encounters a student who tries to do every question on autopilot and has no mental flexibility. But I would much prefer that this student can at least do the most routine of routine questions well, than not even being able to do that.

            1. I guess there is a certain philosophical debate to be had.

              Is it more important for students to follow a mechanical routine and get a correct answer they don’t trust or understand vs being able to get a correct answer in a less efficient way that they are confident with.

              No tricks games or parlour tricks involved. Just good old common sense and logic.

              The other question is should we be teaching two different subjects. Numeracy and Mathematics. The latter with a focus on efficiency, elegance, abstract problem solving. The former involving students feeling confident and trusting their ability to use basic numerical calculations in real world settings.

              1. Hi, Phil. For your first question, I just don’t believe that’s the choice. I agree that primary students can be taught, or will interpret, the algorithms in a thoughtless manner. But it just doesn’t take that much to understandardize the algorithms. It is not that big a deal. Plus, I don’t see the common sense you advocate actually used that commonly or that sensibly. Of course such sense is good, but at least as it is taught it doesn’t seem to stick for very many students. You seem to use “mechanical” as a pejorative, but there are benefits to thoughtless routine.

                For your second question, numeracy is not enough of anything to be a subject. (And Terry will be posting in five, four, three, two, one …). But I’d love it if Numeracy went somewhere else, and stopped polluting the teaching of arithmetic.

  4. This is a dumpster fire.

    I’m not even sure this comprises a curriculum, as a curriculum should specify what is to be taught and leave the pedagogy to the experts (teachers).

    Such brazen arrogance and idiocy. They probably think they’re being innovative.

    1. Unfortunately as far as expert Math teachers go there is a worldwide shortage. Many ‘math teachers’ need more guidance on the pedagogy.

      1. You say above: “What skills students need to leave school with today is vastly different to even 20 years ago.” If there is a shortage of maths teachers, I can’t imagine how bad the shortage of teachers who know whatever-it-is-that-they-need-to-know-these-days is.

        I know about mathematics, and think it’s a nice, useful, valuable thing to share with people of all ages. It helps me understand the world, even the modern world. I can only share what I know (although I keep learning more). And being able to calculate is the foundation for how I understand and learn mathematics. If you say, now that is irrelevant and people understand things differently these days, then what is a teacher to make of that? What do you think we do have to offer?

      2. Thanks, Phil.

        ACARA has said that they are not advocating pedagogy in the draft curriculum. Are you implying that they are lying?

        Independent of ACARA’s claims, to suggest that the draft curriculum works as pedagogical guide, or as anything, is hilarious.

        Notwithstanding that some teachers may need pedagogical guidance, some pedagogical strategies are way worse than others. Giving guidance is of no help to anyone unless it is good guidance.

        Notwithstanding, the shortage of “expert Math teachers”, the much larger problem is that many “expert Math teachers” are way too convinced that they are experts.

  5. Would it be preposterous to assume that some students pick the method 253 x 4 = (250+3) x 4, saving the effort of double doubling? And that they would pick the method 324 x 8 = (300 + 25 – 1) x 8? The “partitioning” nonsense may successfully stifle any thought of subtraction in contexts such as in the preceding example.

    And is the praise of the “doubling” strategy suggesting that students can just shift by one bit in their heads as computers do? I fear yes.

    This dumpster fire smells of plastic bags.

  6. Well, now we know what happened to the Year 5 Elaborations on the previous post. Looks like they’d accidentally swapped around the Elaborations for multiplication and division, resulting in this mess. Did they not proofread this document? Sloppy.

    Anyway, with regards to the “array method”: as far as I can tell, this *is* a standard algorithm, known as “grid multiplication”, taught in primary schools in the UK (and no doubt elsewhere too). If ACARA wants students to be able to connect it to a specific different multiplication algorithm, they should say what other algorithm(s) are expected.

    The Year 6 Content etc. looks identical to what was in the previous post.

    1. Hi edder.

      It’s possible I confused the cutting and pasting, but I don’t think so.

      I guess it’s a matter of opinion whether grid multiplication is “standard”, or maybe what “standard” means. Yes, the Year 6 content is identical to the previous post’s, because it is all I could find on either multiplication or division. I should have had the two posts as one, but realised too late.

    1. 167 \times 1 = 167 *
      167 \times 2 = 334 *
      167 \times 4 = 668
      167 \times 8 = 1336
      167 \times 16 = 2672 *
      167 \times 32 = 5344
      167 \times 64 = 10688 *
      Therefore, adding the * equations,
      167 \times 83 = 11861.

      Known in ancient Egypt more than 5000 years ago.

      Try some multiplications this way. It’s like a Philip Glass composition – repetitious but very satisfying.

      1. Terry, you’re like the happy dog who walks onto the field during an intense soccer match, and starts playing with the ball. May we have our ball back, please?

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