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.
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.
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’
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.