ACARA Hates Algorithms

Yep, ACARA hates algorithms. That may come as a surprise, since ACARA evidently loves the word “algorithm”; it appears fifty-seven times in the draft curriculum. They love something. But it’s not algorithms.

Not counting little kids pretending to be robots, ACARA’s algorithming properly gets going in Year 3. There, the Level Description has students

begin to apply their understanding of algorithms to experiment, explore and investigate mathematical relationships and recognise patterns

Applying an “understanding of algorithms” to experiment, explore and investigate. Sure they will. And what is this supposed to achieve? Here’s the Achievement Standard: 

They create and use algorithms to investigate the properties of odd and even numbers and to identify patterns and develop facts for single-digit multiplication of two, three, five and ten.

Sounds great. Specifically, here is the accompanying content:

describe, follow and create algorithms involving a sequence of steps and decisions to investigate numbers including odd and even numbers and multiples of 2, 3, 5 and 10 using computational thinking to recognise, describe and explain emerging patterns

A quick reminder: ACARA regards the above “content” as essential, unlike, for example, the concepts of lowest common multiple and greatest common divisor. But, back to ACARA’s love-hate relationship with algorithms. The content is still buzzy nothing, so here are the associated elaborations to clarify:

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 [sic] divisible by two

using an algorithm designed to determine whether a number is a multiple of 2, 5 or 10 to explore a set of numbers, identifying and discussing emerging patterns

exploring algorithms used for repeated addition, comparing and describing what is happening, and using them to establish the multiplication facts for two, three, five and ten, for example, following the sequence of steps, the decisions being made and the resulting solution, recognising and generalising any emerging patterns

creating an algorithm as a set of instructions that a classmate can follow to generate multiples of 3 using the rule ‘to multiply by 3 you double the number and add on one more of the number’, for example, for 3 threes you double 3 and add on 3 to get 9, for 3 fours you double 4 and add one more 4 to get 12…

creating a sorting algorithm that will sort a collection 5 cent and 10 cent coins and provide the total value of the collection applying knowledge of multiples of 5 and 10

Everyone’s picked out their favourite? OK, let’s review.

These “algorithms”, to extent that they are anything, are obscene in their pointlessness and triviality. The proper purpose of an algorithm is to provide a reliable and efficient method to perform some interesting or valuable task that would otherwise be cumbersome or practically impossible. The above “algorithms”, however, do the exact opposite, transforming simple and interesting observations, and essential and insightful calculations, into tedious and aimless mechanics.

Notwithstanding the nonsense attempts above, it is possible to teach interesting and important algorithms in school, although this is a very new idea. It began only 800 or so years ago, and was popularised by that radical fellow, Fibonacci. He probably sold it as 12th Century Skills or something.

We’re referring to, of course, the traditional written algorithms for integer arithmetic. These algorithms actually do something. ACARA hates them.

How do we know ACARA hates the traditional algorithms? First of all: well, duh. That’s what ACARA is. ACARA idolises understanding, loathes doing, and is incapable of seeing that the latter is critical to the former. Secondly, and conclusively, one simply has to look at the treatment of, or, more accurately, the obliteration of, the traditional algorithms in the draft curriculum. Which we now do.

 

ADDITION AND SUBTRACTION

Yes, yes, these are different operations, they have different algorithms and they should be considered separately. The draft curriculum never does. Which suggests the amount of attention to subtle detail we should expect to see.

After some low-level fiddling in the early years, something akin to flimsy method is suggested in Year 3, although the Level Description is just ad-man pitch:

experience the power of being able to manipulate numbers using [a] range of strategies that are based on knowledge of single-digit addition facts and their understanding of place value/base 10, partitioning and regrouping

The Achievement Standard is slightly more specific on how students will “experience the power”:

Students … develop additive strategies for modelling and solving problems involving two-digit and three-digit numbers.

“develop additive strategies for …”. Do you ever get the feeling that ACARA just doesn’t get it? They’re not getting it continues with the content:

model situations and solve problems (including representing money in different ways) involving addition and subtraction of two-digit and three-digit numbers, …

No, we haven’t made a mistake. That’s how the content for the addition of two-digit and three-digit numbers begins, complete with financial fetishisation. The content continues:

applying knowledge of partitioning, place value and basic facts. Explain results in terms of the situation

What, then, is the “range of strategies”? Ignoring the money or whatever, how are students eventually supposed to add or subtract the numbers? The elaborations offer nothing solid. The single elaboration that contains even a suggestion of written method and which, being an elaboration, is optional, is

using partitioning and part-part-whole models to mentally solve addition or subtraction problems, making informal written ‘jottings’ to keep track of the numbers, and the inverse relationship between addition and subtraction as needed

“Informal written ‘jottings’ “, which are optional. That’s it for Year 3 written addition and subtraction. It is mandated that Year 3 kids uses “algorithms” to “investigate” odd and even numbers, but setting up the standard addition and subtraction of numbers doesn’t rate a mention.

But maybe ACARA is just a little slow. So to speak. It’s gotta get more solid and substantial in later years, right? Right?

Year 4 begins with more ad-manning, where students will supposedly

draw on their facility with patterns in whole number facts … to deepen their appreciation of how numbers work

Then, the Achievement Standard:

They model situations, including financial contexts, and use addition … facts to add and subtract four-digit numbers

Yeah, again with the modelling, and again with the finance. Nothing will make ACARA stop worrying their dumb bones. But, at least we’re up to four-digit numbers. Does this then require different content? Of course it does:

model situations (including financial contexts) and solve problems involving addition and subtraction of numbers to at least 10 000, by formulating expressions and choosing efficient strategies, including digital tools where appropriate. Justify choices and explain results in terms of the situation

See? It’s harder now, so the resourceful tykes are “formulating expressions”, and they’re “choosing efficient strategies, including [using] digital tools”. Quite the achievement.

Still, after the mandated “digital tools”, it is possible that something of substance appears in the (optional) elaborations. Of course, it does not. The first elaboration gives us nothing:

representing a range of additive situations involving larger numbers using materials, part-whole diagrams and/or a bar model, and writing addition and/or subtraction number sentence [sic], based on whether a part or the whole is missing; explaining how each number in their number sentence is connected to the situation

The second elaboration is worse than nothing:

choosing between a mental calculation or a calculator to solve addition or subtraction problems, using a calculator when the numbers are difficult or unfriendly and a mental calculation when the numbers can be connected to a familiar mental calculation strategy; reflecting on their answer in relation to the context to ensure it makes sense.

There’s your choice: a mental calculation if the problem is easy and otherwise lunge for the calculator. The third elaboration hammers home the stupidity:

choosing an efficient mental calculation strategy, for example, place value partitioning, inverse relationship, compatible numbers, bridging tens, splitting one or more numbers, extensions to basic facts etc, and using informal written jottings to add or subtract larger ‘friendly’ numbers

There, again, the “informal written jottings”, if the numbers are ” ‘friendly’ “. At this stage, the fourth and final elaboration, which is just more finance tripe, is relatively painless.

To sum up, the kids have completed Year 4 and there has not been a hint, even in the optional elaborations, of the traditional written algorithms for addition or subtraction. Not one single word. Of course the remaining six years might contain something of substance, and of course they do not. We’ll return to that after considering the other arithmetic operations.

 

MULTIPLICATION

The draft curriculum only has kids learn(ish) their tables in Year 4, So, it is no surprise that Year 4 multiplication is a mish-mosh of little tricks and “groups” and “array diagrams” and “written jottings”. Mr. McRae will just have to wait.

The Year 5 Level Description does not even contain the ad-man stuff, but the Achievement Standard is genuinely, remarkably straight-forward:

They apply knowledge of multiplication facts and efficient strategies to multiply large numbers by one-digit and two-digit numbers …

It’s not enough for Mr. McRae, and it’s a year late, but it’s starting off better than addition-subtraction. So, on to the content:

choose efficient strategies to represent and solve problems involving multiplication of large numbers by one-digit or two-digit numbers using basic facts, place value, properties of operations and digital tools where appropriate, explaining the reasonableness of the answer

Once again with the “digital tools”, and to “represent and solve problems”. Well, it was good while it lasted. But, to be fair, there is one (misplaced) elaboration that is suggestive of a written algorithm, and which even uses the phrase:

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

So, finally, the suggestion of an algorithm. In an optional elaboration. The illustrative example being multiplying by a single digit. With the resulting addition having no carries. Hooray?

Yes, there is the mention of an algorithm, but it is obviously Potemkin. There is no suggestion of actually mastering an algorithm. And, the indefinite article is telling. There is only one “standard written algorithm” for multiplication; the stupid box game, or whatever ACARA has in mind, does not qualify. Don’t believe us? Ask Mr. McRae.

 

DIVISION

It will come as no surprise that the treatment of division is woeful. It may be a surprise, however, that there is no treatment of division whatsover.

Really? Even for ACARA and the appalling draft, it seems difficult to believe. But, for the life of us, we could not find anything other than single-digit trivia and a couple of low-level tricks. Nothing even pretending to play the role of short division or – stop laughing – long division.

Supposedly, whatever passes for a method for division is taught in Year 5. The Achievement Standard gives,

They apply knowledge of multiplication facts and efficient strategies to … divide by single-digit numbers

So, dividing by single-digits is there? Well, no. Here is the corresponding content:

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

One again, the tell-tale “digital tools”. There is also no mention of general single-digit division, nor any suggestion of the size of the numbers being divided. There are then just two associated (and misplaced) elaborations:

interpreting and solving everyday division problems such as, ‘How many buses are needed if there are 436 passengers, and each bus carries 50 people?’, deciding whether to round up or down in order to accommodate the remainder

solving division problems mentally like 72 divided by 9, 72 ÷ 9, by thinking, ‘how many 9 makes 72’, ? x 9 = 72 or ‘share 72 equally 9 ways’

That’s it.

 

And, that’s it generally. Yes, later years still refer to the arithmetic operations, on integers and more generally. In Year 6, for example, the Level Description includes

use all four arithmetic operations with natural numbers of any size

Then, in Year 7,

develop their understanding of integer and rational number systems and their fluency with mental calculation, written algorithms, and digital tools …

But, how? What algorithms? The draft curriculum includes a number of such descriptions and standards, up to and including Year 8. There is not, however, at least as far as we could tell, any general written methods. There is simply nothing of substance.

The pinnacle of this absence of method occurs in Year 8, with the following content:

recognise and investigate terminating and recurring decimals

And, how is the student to convert a fraction into a decimal?

using calculators to investigate fractions or computations involving division that result in terminating and recurring decimals

Calculators, and only calculators. It is not even considered an option for a Year 8 student to use written arithmetic to convert a fraction to a decimal.

Such is the inevitable product of ACARA’s hatred of algorithms.

28 Replies to “ACARA Hates Algorithms”

  1. I think I’m going to use the phrase “experience the power of…” more often in class. What a weird phrase to put into a curriculum document. Maybe He-Man was somehow involved.

    On a more serious note, I have no idea how ACARA expects student to be able to do harder maths in the later years if at the first sign of an unfriendly number students use a calculator. Where is the development of mental perseverance? How will students grapple with unfriendly situations? Could they even know what to put into a digital tool to solve such a situation. I guess it wont matter as a lot of the “modelling” I see in educational resources are just find the cylindrical object and put it in the circular whole kind of thing – there isn’t much critical thinking going on.

    1. “Maybe He-Man was somehow involved.” Indeed … By the power of Numbskulls.

      Actually I think Marty has it wrong … ACARA doesn't hate algorithms. ACARA doesn’t understand algorithms. It has little understanding of what an algorithm is, and even less understanding of how they’re used. A little knowledge is a dangerous thing, making ACARA very dangerous.

      And this is half the problem. The writers at ACARA only have a superficial understanding of what they’re writing about. They do not have expert understanding. It is clear that no-one at ACARA has the necessary expert understanding to write this curriculum.

      The other half of the problem is ACARA’s denial of the first half of the problem.

      1. The third half of the problem may well be that a robust curriculum is deemed beyond the capacity of teachers to understand and teach from. So a superficial curriculum is considered more desirable in order to save a lot of money on teacher training.

        A couple of decades ago, when the DEET decided it wanted to close a school and sell the land, it didn’t declare its intention. Instead, it appointed an incompetent principal to the school and waited for the inevitable to happen. Then the School Council votes to close the school because of insufficient enrolments and the desired outcome is achieved and the DEET is publicly distanced from the decision. The ACARA curriculum might be following a similar strategy.

          1. DEET sold the school to a re-developer for a lot of millions. Because of this, local schools these days are packed beyond capacity so that recess and lunchtimes have to be staggered, portable classrooms dominate what was once a decent oval etc.

            But my point is that DEET put incompetent dupes in charge, let nature take its course, and then ‘reluctantly’ shut and sold the school on the advice of School Councils. It’s a neat tactic when you want to make an unpopular/poor decision and want to avoid being seen responsible for that decision. What better way to ensure a hidden agenda on curriculum than to give responsibility for that curriculum to incompetent dupes and then let nature take its course. Of course, this is all speculation on my part and very hard to prove (until after the fact).

            1. Let me be more specific: what (in broad terms) is the end-goal here?

              Back to basics? Or, perhaps the QLD version: Forward to fundamentals (which makes me a bit nauseated even typing the phrase!)

      2. I think if their goal was to simply teach students how to do the algorithms, then that would not be so complicated, and the curriculum writers wouldn’t have to understand much other than a natural progression from simplest to most complicated cases? But their goal is to mix in “understanding” all along, and maybe this can make simply calculating things seem really complicated.

        It’s a bit like if in the most popular cookery books the recipe for “Chocolate cake”, instead of simple steps, was a twenty-page discussion about the chemical processes involved in baking, and different strategies that people might choose to use in order to achieve them, and then some suggested approaches and experiments to try out and a few pictures, before finally ending with instructions to “just buy one already made”. People would think that actually baking cakes was beyond their capabilities. I think that’s kind of what’s happening with arithmetic in Victoria?

        1. wst, the understanding fetish screws up the entire curriculum, but the airbrushing of the traditional algorithms seems even worse than that.

          1. Yeah, but somehow I think they’ve airbrushed them out in a way that makes it seem like it would be very difficult to learn them? So then it seems like it would be mean to get children to do it.

            And also, then anyone who admits to finding the curriculum hard to understand can be bundled in as part of the “not enough appropriately trained teachers” problem that John Friend mentions above? They tell us they’ve made it easier to follow and “expert teachers” have no problem. Anyone who complains doesn’t have the correct expertise?

            1. You might well be onto something here, although I’m still not sure what the end-game of ACARA is here (and yes, I’m still operating under the illusion that there is one).

              1. Yeah, I have no idea about their intentions. But I see how there are there are the ingredients for frustration if someone tries to argue with the proponents of the curriculum, because they can always blame people who criticize it for not understanding it well enough, and blame teachers if it doesn’t work.

  2. Thank you Marty for clearly exposing another huge flaw in the draft curriculum. It becomes more and more obvious that the only thing ACARA understands or wants to understand is how to push their favourite teaching method onto the country: one that is hugely time-consuming and only works by accident.

    They certainly have no idea what the purpose or duty of primary school mathematics is.

    1. As a wise man once said, “primary school Mathematics is really important. It would be nice if there was some.” (Not an exact quote, but a memorable one).

            1. A very long 1-liner and I threw out the booklet it appeared in, so could not accurately quote the drunken uncle from the family Christmas party.

              Still one of my favourite PD events ever.

  3. Is all, or most, of what you have outlined above already being done in a large number of classrooms?
    Do materials already exist?

    If not, what are the ramifications of that?

    If so, where and where?

      1. Milton’s point seems relevant to school mathematics today. We spend years on teaching our students about mathematics and at the end, many of them have learnt so little. Many do not have a good gasp of basic arithmetic even after 11 years of schooling.

        This is acknowledged by VCAA in the proposal to introduce Foundation Mathematics 1,2 at Year 11. Students “often have not successfully engaged with the mathematics content or curriculum and are still struggling with learning the foundations of mathematics and numeracy. This challenge needs to be acknowledged and addressed, and in many cases some content taught again as if it is a new experience. “

        1. Again, I’ll bite…

          …how would you, someone who has practical experience and a proven academic ability (by virtue of being a published research author) address the challenge?

          I am but a lowly teacher and to me the issue looks a bit too big to fix in less than a decade, and even then it would require some serious investment of political will.

        2. In the old days, if a student “[did] not successfully [engage] with the mathematics content or curriculum and are still struggling with learning the foundations of mathematics and numeracy.” then that student would be kept down. Alternatively, there would be other pathways for that student.

          But what is this “foundations of mathematics and numeracy” that students have not learnt by Yr 11 …? And why do you think this has happened? (I’ll partly answer the latter – because students get promoted to the next year level each year whether they have passed or not).

        3. Milton…. come on Terry. I don’t think that is Milton’s point. From that tiny excerpt, it seems to me that he is saying (a) teaching is not effective; and (b) the content is very light.

          That’s out of context. IN context, Milton’s “On Education” is a meandering soup of thought bubbles. (Apologies to any Milton fans reading. I’m an uneducated philistine.) In my opinion. He writes, just above your quote:

          “The end then of Learning is to repair the ruines of our first Parents by regaining to know God aright, and out of that knowledge to love him, to imitate him, to be like him, as we may the neerest by possessing our souls of true vertue, which being united to the heavenly grace of faith makes up the highest perfection.”

          Uh huh. By the way… yes, Milton believed that Adam forfeited some pure knowledge of God by eating the apple from the tree of knowledge. Also, here “The end” is referring to the aim or goal.

          I’ll just also put the concluding paragraph here. I think it speaks for itself. ACARA and Milton might get along pretty well.

          “Now lastly for their Diet there cannot be much to say, save only that it would be best in the same House; for much time else would be lost abroad, and many ill habits got; and that it should be plain, healthful, and moderate I suppose is out of controversie. Thus Mr. Hartlib, you have a general view in writing, as your desire was, of that which at several times I had discourst with you concerning the best and Noblest way of Education; not beginning, as some have done from the Cradle, which yet might be worth many considerations, if brevity had not been my scope, many other circumstances also I could have mention’d, but this to such as have the worth in them to make trial, for light and direction may be enough. Only I believe that this is not a Bow for every man to shoot in that counts himself a Teacher; but will require sinews almost equal to those which Homer gave Ulysses, yet I am withall perswaded that it may prove much more easie in the assay, then it now seems at distance, and much more illustrious: howbeit not more difficult then I imagine, and that imagination presents me with nothing but very happy and very possible according to best wishes; if God have so decreed, and this age have spirit and capacity enough to apprehend.”

          1. I like “The end then of Learning is to repair the ruines of our first Parents…”

            Whenever I read that, I say to myself “That’s the big picture!”

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