ACARA Crash 18: Errors in the Draft Mathematics Curriculum

The following is a list of errors – and possible/arguable errors – in the draft mathematics curriculum. Commenters are invited and encouraged to suggest additions, and deletions.

By “error” we mean a statement or instruction that is factually wrong or that makes no logical/mathematical/everyday sense. Some of the listed “errors” are clear-cut, while others are less so. Of course the fact that a statement/instruction made no sense to us does not prove that it makes no sense; we’ve attempted to be fair, being tough on the improper use of technical terms while giving weird phrasings a good-faith pondering in context. Nonetheless, there may well be reasonable interpretations that we have missed. (Of course phrasing that is difficult to interpret has no place in a curriculum document, but that is a separate category of sin.) As well, it is not always clear whether to characterise a statement as an error or simply a really dumb idea, but we’ve tried to stick pretty closely to “error”, leaving the noting of really dumb ideas to our other ninety-eight posts.

The list follows. The majority are elaborations. There are a few content descriptors, for which associated elaborations are indicated by a further indentation. Again, commenters are encouraged both to suggest additions to the list, and to argue for deletions from the list.

Year 1 Number

using part-part-whole reasoning to partition the 365 days of the year into the seasons of an Aboriginal and Torres Strait Islander seasonal calendar, saying how many days are in each of the seasons (AC9M1N02_E5)

sharing out a set of $10 notes between three people and skip counting by tens to ensure they have equal amounts (AC9M1N05_E4)

Year 1 Statistics

investigating the difference between a question that is statistical, for example, ‘Are most flowers red?’ and a question that isn’t, for example, ‘Are there flowers that are red?’ (AC9M1ST01_E1)

Year 2 Number

exploring a base 5 number system used by Aboriginal and Torres Strait Islander Peoples and the many ways whole numbers in the base 10 system can be rearranged and partitioned to this base 5 system, finding similarities and differences with the base 10 number system (AC9M2N02_E5)

Year 2 Probability

predicting and testing what would happen if ten names were put in a box, drawn out and then replaced after each selection, discussing how even though there is a chance all names will be drawn, there is also a chance not all names will be drawn from the box (AC9M2P01_E4)

discussing how even when you are sure something will happen there is always a chance that it won’t happen, for example, discussing that although yesterday was a regular school day, you didn’t have school as the temperature was too hot and school was cancelled (AC9M2P01_E6)

sorting a set of event cards according to whether they are impossible, certain to happen or uncertain, discussing how impossible events cannot happen and certain events must happen, so they are not affected by chance whereas uncertain events are affected by chance (AC9M2P01_E7)

using the language of chance to predict colours of cubes as they are drawn from a bag and using reasoning to predict the subsequent cubes as they are drawn, for example, the next cube could be green or red, but it is impossible that it is yellow as there are no yellow cubes in the bag (AC9M2P01_E8)

Year 3 Number

recognising that 10 000 is equivalent to 10 thousands, 100 hundreds, 1000 tens and 10 000 ones (AC9M3N02_E1)

exploring hunting circles used by Aboriginal and Torres Strait Islander Peoples to catch prey to investigate and represent different models of unit fractions based on different numbers of hunters between 1 and 5

Year 3 Algebra

exploring and identifying similarities in growing patterns formed by doubling and halving and identifying these onCountry/Place including branching in trees and river systems (AC9M3A01_E3)

following an algorithm consisting of a flowchart with a series of instructions and decisions to determine whether a number is even or odd; using the algorithm to determine if a set of numbers are divisible by two (AC9M3A04_E1)

Year 3 Space

walking around the playground or taking a nature walk to explore their environment looking for examples of symmetry, such as, school buildings, trees and using a tablet, capture images to bring back to the classroom and share (AC9M3SP03_E1)

Year 4 Number

using the four operations with pairs of odd or even numbers or one odd and one even number, then using the relationships established to check the accuracy of calculations (AC9M4N04_E3)

Year 4 Algebra

identifying and describing fractal patterns on-Country/Place, creating number sequences using technology to describe them and exploring their connections to Aboriginal and Torres Strait Islander Peoples’ artworks (AC9M4A01_E5)

find unknown values in equivalent number sentences applying an understanding of associative and commutative properties of addition and the inverse property of addition and subtraction (AC9M4A02)

using balance scales and informal uniform units to create addition (or subtraction) number sentences showing equivalence, for example, 7 + 8 = 6 + 9, and to find unknowns in equivalent number sentences, for example, 6 + 8 = + 10 (AC9M4A02_E2)

constructing equivalent number sentences involving addition, demonstrate an understanding of the associative property and explaining what they did, for example, ‘If I start with 3 + 8, I know 8 = 3 + 5 and so 3 + 8 = 3 + 3 + 5 then 3 + 8 = 6 + 5 as I can add the 3 + 3 first’ (AC9M4A02_E3)

exploring the fact that addition is associative but subtraction is not using physical or virtual materials, for example, using counters to show that 3 + 2 + 5 = (3 + 2 ) + 5, or 3 + 2 + 5 = 3 + (2 + 5) as they both equal 10 but (10 – 3) – 2 = 5 and 10 – (3–2) = 9 so it is not the same, and 2 + 5 = 5 + 2 but 5 – 2 does not equal 2 – 5 because 5 is greater than 2 and you cannot take more than you have (AC9M4A02_E4)

Year 4 Probability

explaining why the chance of a new baby being either a boy or a girl does not depend on the gender of the previous baby (AC9M4P02_E2)

Year 5 Number

apply knowledge of factors and multiples to compare and order fractions with the same and related denominators (including numbers greater than one) and represent them on number lines explaining any equivalences and the order (AC9M5N04)

using an understanding of factors and multiples and equivalence to explore efficient methods for the location of fractions with related denominators on parallel lines, for example, explaining on parallel number lines that \frac28 is located at the same position on a parallel number line as \frac14 because \frac14 is equivalent to \frac28 (AC9M5N04_E4)

Year 5 Algebra

find unknown values in equivalent number sentences involving multiplication and division applying an understanding of the associative, distributive, commutative and inverse properties, using factors and multiples. Identify and use equivalent number sentences involving multiplication and division to form numerical equations (AC9M5A02)

forming numerical equations using equivalent number sentences, for example, given that 3 x 5 = 15 and that 30 ÷ 2 = 15 then 3 x 5 = 30 ÷ 2 (AC9M5A02_E2)

constructing equivalent number sentences involving multiplication to form a numerical equation, applying knowledge of factors, multiples and the associative property, for example, using 3 x 4 = 12 and 2 x 2 = 4 then 3 x 4 = 3 x (2 x 2) and 3 x 4 = (3 x 2) x 2 so 3 x 4 = 6 x 2 (AC9M5A02_E6)

Year 5 Measurement

using a physical or a virtual geoboard app to explore the relationship between area and perimeter and solve problems, for example, investigating what is the largest and what is the smallest area that has the same perimeter (AC9M5M02_E2)

solving measurement problems, for example, ‘What area can be painted when you have one 10 L can and one 4 L can of the same paint?’ by modelling and calculating the possible dimensions of the walls and providing a range of solutions (AC9M5M02_E4)

Year 6 Number

exploring boundaries on-Country/Place including the boundary between saltwater and freshwater, dry and wet seasons, hot and cold country, identifying ‘0’ as the boundary line and connecting this to positive and negative integers on a number line (AC9M6N01_E6)

understanding that a prime number has two unique factors of one and itself and hence 1 is not a prime number (AC9M6N02_E1)

understanding that if a number is divisible by a composite number then it is also divisible by the prime factors of that number, for example, 216 is divisible by 8 because the number represented by the last three digits is divisible by 8, and hence 216 is also divisible by 2 and 4, using this to generate algorithms to explore (AC9M6N02_E5)

using an understanding of prime and composite numbers to explore efficient methods for determining the lowest common denominator for fractions with related denominators (AC9M6N06_E2)

Year 6 Algebra

recognise and distinguish between patterns growing additively and multiplicatively and connect patterns in one context to a pattern of the same form in another context (AC9M6A02)

investigating the number of regions created by successive folds of a sheet of paper, one-fold – two regions, two folds – four regions, three folds – six regions and describing the pattern using everyday language (AC9M6A02_E3)

investigating patterns on-Country/Place and describing their sequence using a rule to continue the sequence such as Fibonacci patterns in shells and in flowers (AC9M6A02_E4)

solving two numbers sentences to show that they are true, for example, list possible combinations of whole numbers that makes this statement true 6 + 4 x 8 = 6 x ? + ? (AC9M6A03_E3)

Year 6 Measurement

establish the formula forthearea ofarectangleand use to solve practical problems (AC9M6M02)

investigating the connection between perimeter and area for fixed area or fixed perimeter, for example, in situations involving determining the maximum area enclosed by a specific length of fencing or the minimum amount of fencing required to enclose a specific area (AC9M6M02_E3)

Year 7 Number

choosing an appropriate numerical representation for a problem so that efficient computations can be made, such as 12.5%, \frac18 , 0.125 or \frac{25} {1000} (AC9M7N06_E2)

using the digits 0 to 9 as many times as you want to find a value that is 50% of one number and 75% of another using two-digit numbers (AC9M7N06_E7)

Year 7 Measurement

(added 09/08/21) using established formulas to solve practical problems involving the area of triangles, parallelograms and rectangles, for example, estimating the cost of materials needed to make shade sails based on a price per metre (AC9M7M01_E3)

establish the formula for the volume of a prism. Use formulas and appropriate units to solve problems involving the volume of prisms including rectangular and triangular prisms (AC9M7M02)

Year 7 Space 

investigating which lengths of strips can make triangles and quadrilaterals and contrasting the rigidity of triangles with the flexibility of quadrilaterals (AC9M7SP02_E1)

exploring the conjecture that the area of a shape is the product of the average of the lengths of a pair of parallel sides and the distance between them (AC9M7SP02_E7)

Year 8 Number

investigate the Golden ratio as applied to art, flowers (seeds) and architecture (AC9M8N01_E3)

using expressions such as \frac{3^4 }{3^4} = 1, and 3^{4-4} = 3^0 to illustrate the convention that for any natural number n, n^0 = 1, for example 100^0 = 1 (AC9M8N02_E4)

Year 8 Algebra

exploring the meaning behind the components of linear equations using patterning connected to processes (AC9M8A02_E1)

graphing the linear relationship ax + b = c and discussing for what values of 𝑥 is ax + b < c and ax + b > c using substitution to verify solutions (AC9M8A02_E4)

developing an algorithm on-Country/Place for the solution of a linear equation of the form ax + b = c (AC9M8A02_E6)

Year 8 Measurement 

investigating the circumference of a circle as a scaling of its radius or diameter and deduce that the area of a circle is between two radius squares and four radius squares (AC9M8M03_E3)

exploring the relationship between the squares of sides of different types of triangles; right-angled, acute or obtuse, and hence identify Pythagorean triples (AC9M8M07_E3)

recognising that right-angled triangle calculations may generate results that can be integers, fractions or irrational numbers (AC9M8M07_E4)

Year 8 Space

using the enlargement transformation to explain similarity and develop the conditions for triangles to be similar (AC9M8SP03_E4)

developing and evaluating algorithms or flowcharts as to their accuracy for classifying similar versus congruent triangles (AC9M8SP04_E3)

Year 8 Probability

using digital tools to conduct probability simulations to determine in the long run if events are complementary (AC9M8P01_E3)

applying the probability of complementary events to situations such as getting a specific novelty toy in a supermarket promotion (AC9M8P01_E5)

use observations and design and conduct experiments and simulations to explore and identify complementary and mutually exclusive events (AC9M8P03)

understanding that two events are complementary when one event occurs if and only if the other does not (AC9M8P03_E2)

Year 9 Algebra

simplifying and evaluating numerical expressions, involving both positive and negative integer exponents, explaining why, for example, 5^{-3} = \frac1{5^3} = ( \frac15 )^3 = \frac1{125} (AC9M9A01_E2)

choosing efficient strategies such as estimating and order of operation and applying them to exponent laws of numerical expressions with positive and negative integer exponents (AC9M9A01_E4)

recognising the relationship between expansion and factorisation and identifying algebraic factors in algebraic expressions including the use of digital tools to systematically explore factorisation from x^2 + bx + c where one of b or c is fixed and the other coefficient is systematically varied (AC9M9A02_E3)

graph simple non-linear relations using graphing software where appropriate and solve linear and quadratic equations involving a single variable graphically, numerically and algebraically using inverse operations and digital tools as appropriate (AC9M9A04)

using graphs to determine the solutions to linear and quadratic equations (AC9M9A04_E2)

representing and solving linear and quadratic equations algebraically using a sequence of inverse operations and comparing these to graphical solutions (AC9M9A04_E3)

exploring quadratic functions through hunting techniques of Aboriginal and Torres Strait Islander Peoples by increasing the number of hunters to increase the area/circumference to catch more prey (AC9M9A05_E3)

Year 9 Measurement

exploring different scales in fractals on-Country/Place, expressing their different attributes using scientific notation (AC9M9M03_E3)

Year 9 Space

establishing and experimenting with the algorithm for determining the sum of the angles in an n-sided polygon (AC9M9SP04_E1)

investigating visual proofs of spatial theorems and design an algorithm to produce a visual proof (AC9M9SP04_E4)

Year 9 Statistics

(Added 09/08/21) investigating where would be the best location for a tropical fruit plantation by conducting a statical investigation comparing different variables such as the annual rainfall in various parts of Australia, Indonesia, New Guinea and Malaysia, land prices and associated farming costs (AC9M9ST04_E3)

Year 10 Space

using dynamic geometric software to investigate the shortest path that touches three sides of a rectangle, starting and finishing at the same point and proving that the path forms a parallelogram (AC9M10SP02_E3)

Year 10 Statistics

investigating the relationship between two variables of spear throwers used by Aboriginal Peoples by using data to construct scatterplots, make comparisons, and draw conclusions (AC9M10ST03_E4)

23 Replies to “ACARA Crash 18: Errors in the Draft Mathematics Curriculum”

  1. The errors in the probability/statistics sets of questions – which are the only ones I looked through – seem to have a widely varying degree of seriousness. I certainly second the Year 2 Probability AC9M2P01_E4. Blissfully ignoring the number of trials is unforgivable. More forgivable is perhaps the fact that the probability of not drawing all names from the box may be extremely small (e.g. two names in the box, 1000 trials = draws) and hard to simulate as is, simply because in practically all samples the event occurs zero times.

    Year 1 Statistics (AC9M1ST01_E1), “Are most flowers red?”, ignores the probability space from which the flowers are selected. The field out there? Or my book on the shelf? Such things can perhaps only be taught at uni level or shortly before, yet the question is perhaps justifiable at Year 1 level, with a grain of salt.

    1. Hi Christian!

      Oh, I wouldn’t say (re: AC9M1ST01_E1) that it is beyond Year 1 students to appreciate the difference between “Are most flowers in my house red?” and “Are most flowers at my school red?” (and even: “Are most flowers on my teacher’s hawaiian shirt red?”). I’d say not highlighting this in the curriculum document is a pretty big fail.

      1. Hi, Glen. Having “Year 1 Statistics” is a bigger fail. The question framed any way is ridiculous. Given, however, it’s Year 1, I’m not so fussed that the question doesn’t indicate the field/shirt to consider. As I wrote in my reply to Christian, I had a different issue with the elaboration.

    2. Thanks, Christian. I made no effort to restrict to “serious” errors, and for this list just considered an error to be an error. (The really serious issues with the draft are not the errors but the idiocies.)

      In regard to AC9M1ST01_E1 in Year 1 Statistics, my issue is “Are there flowers that are red?” not being a “statistical question”.

      With AC9M2P01_E4 in Year 2 Probability, I can’t even figure out what the question is supposed to be.

      1. Marty, understanding AC9M2P01_E4 is not the real shit here. I’m embarrassed to admit that I understand what they’re asking. Actually, they’re not asking anything per se. They’re presenting an ‘activity’ and the teacher has to decide what questions to ask:

        Activity: Imagine 10 balls, all of different colour, in a bag. You pull out a ball, note its colour and then put it back in the bag. You keep doing this again and again.

        Conclusion: There’s a finite chance that all colours get selected but there’s obviously also a chance that some colours will not be selected.

        But the conclusion is wrong if the number of selections is less than 10. Anyway …

        So the teacher has to decide things like how many selections should the students make etc. But I wonder if the teacher will understand the difference between making less than 10 selections and making 10 or more selections and that the conclusion is only true for 10 or more selections …?

        Be that as it may, the *real* erroneous shit here is NOT in figuring “out what the question is supposed to be” … This is all happening in Year 2, and being taught by Year 2 teachers. I strongly doubt anyone at that level will understand or even accept that the above statement is true once the the number of selections is greater than 9.

        So the *real* erroneous shit is the “predicting and testing what would happen”. I’ll tell you what happens: 25 students do it 100 times and they all find that every colour gets selected. They (and their teachers) conclude that the above statement is true for ‘small’ numbers of selections but not for ‘large’ numbers of selections. They conclude that if the number of selections is large enough, it is certain that every colour will get selected.

        “Discuss how even though there is a chance all [colours] will be drawn, there is also a chance not all [colours] will be drawn from the box”.

        The discussion will be that all colours get selected once the number of selections is large enough. This is the sort of diabolical bullshit activity that causes all sorts of misconceptions. And we all know how hard it is to extinguish a misconception once it takes hold. ACARA has no clue the *real* damage it’s bullshit will cause.

        On a 1 – 10 scale of serious erroneous bullshit, this is an 11 (thankyou, Spinal Tap).

        By the way, one could argue that it requires a certain level of sophistication to be able to prove that not every colour gets selected if the number of selections is less than the number of colours in the bag. I’m sure there’s some decent maths somewhere in this …!!

        1. Thanks, John. Of course I’m aware that the awfulness of the elaboration outweighs its errorness. But, in this post all I’m focussed on is the errors.

          1. Marty, it’s not awful. Awful simply does not do it justice.

            It’s insidious and diabolical. And it’s not an error. It’s an insidious and diabolical progenitor of errors. Diabolical errors in the thinking of millions of students and teachers across the country over the coming years.

      2. Thanks, Marty. I find the appellation of AC9M1ST01_E1 in Year 1 Statistics item, “Are there flowers that are red?”, as non-statistical question not that bad. I believe it depends again on the probability space – whether the maker who pours out flowers actually has red ones in their bag from which they could “in principle” choose, or not. Practical experience tells us that they do have, in which case you are right. How the urns (or “exalted trash cans”, as Rick Durrett calls them in one of his books) come into play with this interpretation will probably still be mysterious to most students.

        Perhaps some brand of (self-proclaimed) philosophers will rejoice in ACARA handing them a piece of maths ed.

        1. Thanks, Christian. Obviously the goodness or badness of the draft curriculum isn’t hinged on whether “Are there flowers that are red?” is or isn’t a “statistical question”. But if “statistical question” means anything, I assume it is a question that one answers (exactly or with some degree of uncertainty) by considering data from a given population. If so, how can the flowers question, however it is tightened to make sense, be other than statistical?

          1. Thanks, Marty. I think that specifying the “given population” is a crucial – and perhaps “the” crucial – part of tightening the question at hand to have it make sense. Mathematically, of course I can say in front of a field of sunflowers, the event of a flower being red has probability zero, just as I can say that the probability of a die (fair or not) yielding seven has probability zero. However, in general, to include all sorts of events which have probability zero, so that I can then statistically determine their probability (probably as zero), seems to me as opening a veritable Pandora’s box. I believe that the decision as to which (elementary) events should be part of the probability space has to do with prior beliefs of what could “in principle” happen. To reckon with finding a red flower is a priori not absurd; getting a seven on a die is.

            Perhaps someone with more knowledge of Bayesian statistics than myself will disagree with the above, but in my understanding, the Bayesian view is not taught.

  2. Hi Glen,

    Thank you for your response.

    I agree that even Year 1 students can grasp the “difference” between examining a house, a school yard, or a Hawaiian shirt. Not to say a single word about this is an omission in ACARA, sure. Appreciating such a difference per se is, IMHO, not equal to having a good grasp of what actually constitutes this difference in a reasonably rigorous abstract sense that goes beyond “of course a shirt is different from a house”. But perhaps I am setting the bar here rather high.

    As an afterthought, the vagueness of the ACARA formulation causes a further problem: I filled in the vagueness with the “field out there”. To count all flowers there could be tedious. (Well, if the red flowers are few and outnumbered then it may be easy, but that can hardly be taken as the norm and is a cheap shortcut anyway.) ACARA should thus better have said that the question is aiming at a genuinely “discrete” probability setting; after all, that is the only sort of probability of concern up to senior levels. Whether it would confuse students to say also, e.g., I meant the field as of YYYY-MM-DD – another day may give a different result –, I cannot judge.

  3. Year 7 measurement:

    “estimating the cost of materials needed to make shade sails based on a price per metre”
    Surely price per *square metre* …!

    (Unless each meter has a standard width as is the case with material bought for making clothing …?)

    1. TM, elaborating on your observation – Officially, it’s the Independent State of Papua New Guinea.

      For all of ACARA’s woke “connections to Aboriginal and Torres Strait Islander Peoples”, it can’t even get the name of the world’s third largest island country correct.

      Now, imagine the outcry if ACARA referenced the Wurundjeri, for example, and got the spelling incorrect …

      Marty, the name error might be the least of your concerns but it is symptomatic of the error-riddled sloppiness in general of this shitful, poor excuse for a curriculum document. All the document needs in order to complete its sloppiness is a map of Australia that omits Tasmania.

      1. JF is correct – in the same way that Australia is officially the Commonwealth of Australia. Still, I would write “Australia” and I would write “Papua New Guinea” and not regard this as sloppy.

        Click to access officialnamesofcountries.pdf

        BTW, I recall when “Commonwealth of” was officially dropped and then later officially reinstated.

  4. The rationale opens with this sentence. “Learning mathematics creates opportunities for, enriches and improves the lives of *all* Australians.” (My emphasis.) This is an exaggeration. Consider the hundreds of Australians who will die before their fifth birthday. How does learning mathematics create opportunities for, enrich and improve their lives?

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