New Cur 26: Algorithmic Sinking

One of the shiny new expressions of mathematics education is algorithmic thinking or, if one prefers, computational thinking. It is 10% shrewd rebranding and 90% poisonous snake oil. The new Australian Curriculum is full of it.

The ACARA document About the learning area (Word, idiots) lists “computational thinking” as a “key proficiency”:

Students develop computational thinking through the application of its various components: decomposition, abstraction, pattern recognition, use of models and simulations, algorithms and generalisation.

This is insidious bait and switch, introducing such “thinking” as motherhood mathematical thought but then strongly pushing a particular, and a particularly ineffective and ugly, style of school mathematics and the approach to learning it. Which the About document begins to make clear:

The capacity to purposefully select and effectively use the functionality of a digital device, platform, software or digital resource is a key aspect of computational thinking in the Mathematics curriculum.

The heart of all this, the brave new push, is algorithms:

An algorithm is a precise description of the steps and decisions needed to solve a problem or a set of rules to follow in order to accomplish a task. … As students develop a conceptual understanding of how an algorithm works and fluency with using algorithms appropriately, they can reason and solve problems using algorithms as part of a computational thinking process.

ACARA could do with a writing algorithm, but let’s continue. And, admittedly, algorithms can be very useful:

In Music, students can apply knowledge of patterns and algorithms when composing.

We’re sure the music teachers will be absolutely thrilled with this suggestion.

Systematic methods are of course fundamental to school mathematics, and there are critically important and (once) very familiar algorithms that we discuss below, but the role and value of algorithms can easily be overestimated. Children are not robots, at least not after Foundation. It is thus surprising that a curriculum otherwise dripping in the glorious goals of “reasoning” and “understanding” is so routinely robotic about how kids are supposed to achieve these goals.

The Curriculum first refers to algorithms in Foundation, with kids pretending to be robots, moving left and right and so on as instructed (AC9M1SP02), which is fine. It is soon not fine.

The Curriculum contains the term “algorithm” and its variants 57 times, almost never with clear and good intent. Disregarding the Foundation robots, the algorithming begins in Year 3, as announced in the Level description and Achievement standards:

[Students] begin to apply their understanding of algorithms and technology to experiment with numbers and recognise patterns 

They create algorithms to investigate numbers and explore simple patterns.  

These Year 3 kids’ understanding of “algorithms” and “technology” will be very limited, but more fundamentally their understanding of and comfort with numbers will be limited. The very worst way to attain any such number sense, and to “recognise” and to “explore” patterns, is to watch a machine do the work or to try to think (?) like a machine. This worstness is then spelled out in a dedicated content descriptor and accompanying elaborations, and in a stray and bizarre “algebra” elaboration:

follow and create algorithms involving a sequence of steps and decisions to investigate numbers; describe any emerging patterns (AC9M3N07)

following or creating an algorithm to generate number patterns formed by doubling and halving using technology to assist where appropriate; identifying and describing emerging patterns

following or creating an algorithm that determines whether a given number is a multiple of 2, 5 or 10, identifying and discussing 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 of 5 cent and 10 cent coins and providing the total value of the collection by applying knowledge of multiples of 5 and 10

recall and demonstrate proficiency with multiplication facts for 3, 4, 5 and 10; extend and apply facts to develop the related division facts (AC9M3A03)

systematically exploring algorithms used for repeated addition, comparing and describing what is happening, and using them to establish the multiplication facts for  3, 4, 5 and 10; for example, following the sequence of steps, the decisions being made and the resulting solution, recognising and generalising any emerging patterns

This stuff may not be worthless but it is damn close. Other than some devious rebranding, we could not find any such content descriptors or elaborations that weren’t a substantial or total waste of very precious time. A number of the elaborations were perversions of traditional and good investigations, of primes and factors and so forth; the perverted forms will, of course, win out.

And what of the algorithms, the traditional written methods for arithmetic? The Curriculum is such an incoherent lump of God Knows What, it is difficult to tell. But we looked hard, and we could locate exactly two vague, apologetic, crowded out, optional elaborations:

using and choosing efficient calculation strategies for addition and subtraction problems involving larger numbers; for example, place value partitioning, inverse relationship, compatible numbers, jump strategies, bridging tens, splitting one or more numbers, extensions to basic facts, algorithms and digital tools where appropriate (AC9M4N06)

using an array 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 (AC9M5N06)

That’s it. That’s what we could find. That’s all we could find. Except, there was also an oddly optimistic instruction in the Year 7 Level description:

[Students] extend their understanding of the integer and rational number systems; strengthen their fluency with mental calculation, written algorithms and digital tools; and routinely consider the reasonableness of results in context 

When and how the students were to have previously attained any fluency, or even knowledge of, arithmetic algorithms to strengthen is not made clear. How the students are then to “strengthen” this imagined fluency in Year 7 is also not made clear.

Just in case anyone is in doubt where we are, let’s take a quick look at Singapore. Remember them? The country that we kind of, sort of teach maths as well as? The Singapore 2103 primary curriculum uses the term “algorithm” 26 times, and every single time it is in explicit reference to learning or practising the standard arithmetic algorithms. There is not a single reference to “algorithmic thinking” or “computational thinking”. The new primary curriculum (up to Year 3) and the secondary curriculum discuss the role of algorithms – but not “algorithmic/computational thinking” – in the make-up of mathematics, but again all references to “algorithm” in the content are explicitly to the traditional algorithms. Of course it’s only Singapore. What would they know?

The creators of this poison are clueless and thoughtless. Robots, as it were. Terminators.

5 Replies to “New Cur 26: Algorithmic Sinking”

  1. I think that choosing calculators (“purposefully select and effectively use the functionality of a digital device, platform, software or digital resource”) is not really mathematics. I studied mathematics and don’t feel like I have any particular expertise in this skill. I often read complaints that there are not enough teachers who studied mathematics. But then in the mathematics curriculum, there’s this as well. If they keep adding such stuff into the mathematics curriculum and taking mathematics out, what are teachers supposed to have studied to prepare them to teach it in a meaningful way?

  2. Computing has always been part and parcel of mathematics: counting rods, abacus, log tables, etc. In recent years, the power of calculating machines has increased remarkably. The capability of CAS calculators is astonishing. Now not all aspects of mathematics need to be taught in schools. But I don’t see how we can ignore the role of computing in mathematics. The question for me is: How can we maximise the benefit of computers in the mathematics classroom?

    1. I have absolutely no idea what point you are trying to make, or how in any remote sense it excuses ACARA’s idiocy. Which was the point of the post.

    2. Which aspects of mathematics would you say don’t need to be taught in schools anymore? (Many aspects of mathematics are not taught in schools, but I guess you mean there are some things that were previously taught and don’t need to be?)

      I worry that some things we used to do, though perhaps seemingly mundane and time-wasting, maybe had an accidental effect of developing number sense and understanding of geometry. If we move too fast to remove things, then it might have unintended effects?

      (For example, when I was at school, we sometimes had to draw grids by hand. While annoying, I think that in this way, by measuring out perhaps 8mm or 15mm squares, we reinforced our understanding of times tables and numbers. Now students can just print a grid at will, or teachers do it for them, and many have very shaky number sense. Could such things be related? There’s just so much lost repetition of basic facts because things are automatic. Students don’t get as much opportunity to learn the details that big ideas are made of.

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