We’ll have more on ACARA’s great devotion to teaching algorithms in the near future, but this’ll do for now:

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# PoSWW 33: Algorithmic Unthinking

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30 Replies to “PoSWW 33: Algorithmic Unthinking”

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We’ll have more on ACARA’s great devotion to teaching algorithms in the near future, but this’ll do for now:

Hmmpphh. Then why can’t these so-called ‘everyday algorithmics’ be made explicit and emphasised, rather than vague (but in vogue) buzz words like ‘algorithmics’ be thrown around like rice at a wedding. I suppose the former don’t sound impressive enough. ACARA (and VCAA) clearly think that ‘management speak’ makes their curriculum look clever and fit for 21st century learning.

The gobbledygook ACARA, VCAA, DET etc. come out with makes SloMo sound clear and precise by comparison.

A nitpick, true, but I wonder why they chose to capitalise mathematics.

The whole thing is a mess. What does it actually mean? Even tweets should measure up to some standard.

I take it that they’re using “algorithm” in the loosest possible sense purely for the sake of seeming “”relevant””. Perhaps they think the ability to following instructions can be deemed the result of “algorithms”. It follows from their argument, that the instruction of toilet training, texting, and triple jump are all examples of algorithmic teaching. (And these examples are arguably more “algorithmic” in nature than spelling.)

Look, I’d be happy if the Australian curriculum actually taught algorithmic thinking — the ability to use rigourous, explicit thinking processes is good.

But this sort of slop…

To McSplutterwit: I have strong views about the overuse of capital letters. I notice that “Principal” is often written with a capital letter, but “teacher” is not. When I worked in a hospital I used to get into trouble for writing “surgeon”. I used a guide provided by the Economist magazine. Accordingly, I would now write “the king”, or “the pope”, or “the prime minister”. I would write “King Charles” and “Pope Francis”, but never “Prime Minister Albanese” because prime minister is not a title – it’s a job, like bus driver. We would not say “Bus Driver Smith”. The problem is that most writers don’t use a guide: they just make it up as they go. Of course different guides might give different advice but the main point is that one should use a guide for consistency.

I’d say that the author wrote “Mathematics” to highlight it over spelling, or, more likely, the author did not give it any thought at all. I doubt that the author followed a guide.

Thanks, Terry. I also puzzle about capitals. In this case, I assume “Mathematics” came consciously or unconsciously from “Mathematics Curriculum”, where ACARA tends to use capitals. It doesn’t make the stupid sentence any less stupid.

I use capitals as a sign of respect. I will/usually refer to Colleagues, Mathematics Teachers etc. I’d probably/usually refer to a Bus Driver, Accountant etc, again as a sign of respect towards the people whose job that is. I don’t care if this is a shooting offence.

I have no problem with ACARA referring to Mathematics, essentially using a proper noun for an important part of human knowledge and endeavour. But there’s plenty of other (bad) grammar I’m happy to see slapped down.

It’s a great pity that ACARA cannot follow the algorithm for writing a decent curriculum.

PS – I’d probably refer to a lawyer (half-joking) or a politician (not joking) …

PPS – I wonder why algorithm is favoured over “procedure”, coding over programming etc It seems that every so often we need to have fancy new buzz words that re-brand basic concepts and ideas. To make things look new and exciting …? Evolution of language …? This is not restricted to education, I have seen it for over 35 years in the fitness industry (for example). Can anyone locate a defining moment from which the word algorithm became so pervasive? In Victoria, was it with the introduction of VCE Algorithmics?

The problem with using capitals as a sign of respect is that the use of capitals then becomes a subjective matter.

“acara” it is.

LOVE IT!

TM, respect is often a subjective thing. I think Marty’s comment says it all.

JF – I cannot refer to the exact date, time nor place, but I have seen it slowly but surely seeping into the nomenclature for a very long time.

I first remember hearing of a friend’s son showing us his grade 2 homework which was a sheet titled “algorithms”. It was adding and subtracting. Maybe occasionally with the two procedures mixed in the one problem.

Years later, I began to see it in Year 7 textbooks and colleagues began to use the phrase a bit too comfortably for my liking. If only I knew then what it would become…

Why it started and why it was allowed to survive are usually two very different things.

What is this supposed algorithm for spelling? “To spell ‘algorithm’, first write the letter ‘a’, then write the letter ‘l’, …”?

Teacher: How do you spell alligator?

Student: a – l -e – g – a – t – e – r

Teacher: That’s not correct. It should be a-l-l-i-g-a-t-o-r

Student: But you asked me: How do *I* spell alligator?

Mathematics is an algorithm now, is it? I raise my eyebrow to that.

Let me go out on a limb. I suggest that nearly all mathematics at school is about algorithms, or algorithmic approaches to solving mathematical problems. It gets more so the closer you get to VCE. Here is one of my favorite examples.

Find the inverse of the function .

Step 1: Interchange and .

What’s your point?

Terry, the VCAA exam reports explicitly encourage the ‘interchange x and y’ crap. (For example, see 2017 MM Exam 2 Report Q4 (b), 2016 MM Exam 1 Q5(b)(i)).

So of course many teachers brainlessly teach this as the ‘first step’. Most would not know any better. Monkey see, monkey do.

Now see what happens when the question is “Find the inverse of the function g(t) = 3t + 1.”

Step 1: Interchange y and x.

That’s what students get trained to do by VCAA and teachers!! ‘Step 1’ of this ‘algorithm’ is a gigantic irritation of mine.

Of course “nearly all mathematics at school is about algorithms, or algorithmic approaches”! The same can be said for university undergraduate mathematics. But the ‘algorithms’ have to be used with care and the ‘approach’ is often far from obvious. And the algorithms are known. Every time an axiom or theorem (or lemma or corollary etc) is used, one could argue that the usage is an ‘algorithmic approach’. One could argue the Euclid’s The Elements consists of 13 books of ‘algorithms and algorithmic approaches’. Interpret ‘algorithm’ broadly enough and one might argue that any proof in mathematics is simply algorithms and algorithmic approaches. (Algorithms proved the four-colour theorem).

As the mathematician sang in the Wizard of Oz:

I could while away the hours

Conferrin’ with the flowers,

Consulting with the rain;

And my head I’d be a scratchin’

While my thoughts are busy hatchin’

If I only had an algorithm.

By the way Terry. You used the word “nearly”, implying not all. So what part of mathematics at school do you think is NOT about algorithms, or algorithmic approaches to solving mathematical problems?

Back to to my irritation. In my opinion, here is how the ‘algorithm’ for finding the inverse should be taught:

“Find the inverse of the function g(t) = 3t + 1.”

‘Algorithm 1’:

Step 1:

Let be the inverse.

By definition, :

.

‘Algorithm 2’:

Step 1:

Let t = 3y + 1 where by definition.

‘Algorithm 2’ wall papers over the understanding made explicit in ‘Algorithm 1’ but it’s better than the ‘swap x and y’ crap.

Not everything has to be about VCAA.

@RF: Re inverse functions

I called “inverting x and y” my favourite algorithm with tongue in cheek. In fact I had never heard of this before embarking on teaching high school mathematics.

However, I recall attending a meeting of examiners where I asked the question, “If I had solved the problem on the exam without swapping x and y, would I have lost marks?” The examiner’s reply was “Probably”. The teacher’s dilemma.

Thanks, Terry.

I recall you sharing this anecdote elsewhere, and I recall my outrage at the idiotic answer. A pity the ‘examiner’ wasn’t pressed further on this (although it would probably have pushed them far, far outside their competence zone). Anyway …

In your earlier comment, you used the word “nearly”, implying not all. So what part of mathematics at school do you think is NOT “about algorithms, or algorithmic approaches to solving mathematical problems”?

@JF: When I wrote “nearly all mathematics at school is about algorithms”, I was just trying to cover myself. However, your question made me think more about that – and I am grateful for your prompt.

When learning about logarithms last year, my students were asked to write an essay on “Seismology and logarithms” as a piece of formal assessment. A rubric provided guidance on what was expected.

This term, my Year 9 students will be learning about indices. In the first week, they will be given this problem: “What are taxicab numbers? Give a brief written answer mentioning what these numbers have to do with indices, and why mathematicians are interested in them. (Hint: Google)”.

At a lower level, I have asked students a question about applied mathematics: “What is the price of keeping a dog for a year?”

I do not regard these examples as exercises in applying an algorithm.

Thanks again for your prompt. Best wishes for 2023.

Hi Terry. Thanks for your follow-up.

Your three examples are assessment tasks – specifically assessment tasks. You are an anomaly – in a good way! – and your approach would not be typical. I should have qualified my question with “at a school”. Even so …

1) “Seismology and logarithms”. Maybe there’s no algorithms to see here. Unless your students do this task at home in which case … ChatGPT (an AI program that can write essays or complete tasks such as a student’s homework). Also, the technology underlying seismographs is based on algorithms. The conversion of ‘energy’ to the Richter scale is an algorithm.

2) Taxicabs numbers. That’s a good task. Maybe there’s no obvious algorithms here … However, “The taxicab numbers subsequent to 1729 were found with the help of computers.” And the computer code is based on algorithms …

And ” such numbers exist for all positive integers n, and their proof is easily converted into a program to generate such numbers. However, the proof makes no claims at all about whether the thus-generated numbers are the smallest possible and so it cannot be used to find the actual value of Ta(n).”

I’m not sure that many Yr 9 students will understand why mathematicians are interested in them. I anticipate that ‘copy and paste’ – the good friend of most students – will be assisting without giving too much understanding:

https://www.sciencedaily.com/releases/2015/10/151014132457.htm

But it’s a good thing to ask and anticipates the “Who cares?” question.

3) “What is the price of keeping a dog for a year?” That’s a total algorithm, and I anticipate spreadsheets will be the order of the day.

Nah. The fact that answering a question involves algorithm doesn’t imply that answering the question is algorithmic.

Hi JF,

If you’re going to go down this road, you really should define precisely what you mean by algorithm and algorithmic approach. Because I suspect if you try to make the definition too broad, you’ll be including quite a lot of structured thinking, which is one definition of mathematics (“the creation and study of structured thought”) that I’ve heard before.

Instead of just throwing shade, I’ll help with a suggestion. How about we call something an algorithm if it is “a set of rules that can be followed by a computer”. I think if we do that, then e.g. long division is an algorithm, but choosing an approach to solve an addition problem is not.

Anyway, maybe people disagree with these definitions!

Hi Glen.

Indeed. I completely agree that “you really should define precisely what you mean by algorithm and algorithmic approach.” I confess to having been mischievous.

But I’m not going to provide a precise definition. It’s not my job. It’s the job of the muppets (like, say, vcaa) who decide that littering its curriculum with words like “algorithm” etc. is a good idea. Otherwise what you suspect is indeed exactly the case. Everything in secondary school mathematics reduces to an algorithmic approach.

I’m not sure about your suggested definition (particularly since AI is just a fancy computer) but I understand the sentiment behind it. I think that formulating an appropriate definition is very tricky and requires very careful thought. Certainly much more thought than vcaa has shown. In which case we might wonder: What’s the bloody point in what vcaa has done!?

vcaa has no idea. If it did, we would have all appropriate resources, written to a competent standard, by now. We don’t even have the new exam formula sheets.

John, you’re arguing and disagreeing with Terry. You don’t get to say “not my job”.

OK. Algorithm: A process or set of rules to be followed in a problem-solving operation.

So what’s the point in acara, vcaa and their ilk littering their mathematics curricula with it? Everything in secondary school mathematics follows an algorithm.

None. But you’re changing the topic.

@JF: You are correct in saying that my school is not a “typical school”; 49% of students come from families in the lowest 25% Socio-Educational Advantage in Australia.