Tag: irrationals

What is this Crap Here?

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May 29, 2018

6:03 pm

19 Comments on What is this Crap Here?

WitCH

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OK, Dear Reader, you’ve got work to do.

So far on this blog we haven’t attacked textbooks much at all. That’s because Australian maths texts are, in the main, well-written and mathematically sound.

Yep, just kidding. Of course the texts are pretty much universally and uniformly awful. Choosing a random page from almost any text, one is pretty much guaranteed to find something ranging from annoying to excruciating. But, the very extent of the awfulness makes it difficult and time-consuming and tiring to grasp and to critique any one specific piece of the awful puzzle.

The Evil Mathologer, however, has come up with a very good idea: just post a screenshot of a particularly awful piece of text, and leave others to think and to write about it. So, here we go.

Our first WitCH sample, below, comes courtesy of the Evil Mathologer and is from Cambridge Essentials, Year 9 (2018). You, Dear Reader, are free to simply admire the awfulness. You may, however, go further, and what you might do depends upon who you are:

  • If you believe you can pinpoint the awfulness in the excerpt then feel free to spell it out in the comments, in small or great detail. You could also offer suggestions on how the ideas could have been presented correctly and coherently. You are also free to ponder how this nonsense came to be, what a teacher or student should do if they have to deal with this nonsense, whether we can stop such nonsense,* and so on.
  • If you don’t know or, worse, don’t believe the excerpt below is awful then you should quickly find someone to explain to you why it is.

Here it is. Enjoy. (Updated below.)

* We can’t.

Update

Following on from the comments, here is a summary of the issues with the page above. We also hope to post generally on index laws in the near future.

  • The major crime is that the initial proof is ass-backwards. 91/2 = √9 by definition, and that’s it. It is then a consequence of such definitions that the index laws continue to hold for fractional indices.
  • Beginning with 91/2 is pedagogically weird, since it simplifies to 3, clouding the issue.
  • The phrasing “∛5 is irrational and [sic] cannot be expressed as a fraction” is off-key.
  • The expression “with no repeated pattern” is vague and confusing.
  • The term “surd” is common but is close to meaningless.
  • Exploring irrationality with a calculator is non-sensical and derails meaningful exploration.
  • Overall, the page is long, cluttered and clumsy (and wrong). It is a pretty safe bet that few teachers and fewer students ever attempt to read it.

Obtuse Triangles

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June 25, 2017

1:00 pm

9 Comments on Obtuse Triangles

Australian Curriculum

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Whatever the merits of undertaking a line by line critique of the Australian Curriculum, it would take a long time, it would be boring and it would probably overshadow the large, systemic problems. (Also, no one in power would take any notice, though that has never really slowed us down.) Still, the details should not be ignored, and we’ll consider here one of the gems of Homer Simpson cluelessness.

In 2010, Burkard Polster and I wrote an Age newspaper column about a draft of the Australian Curriculum. We focused on one line of the draft, an “elaboration” of Pythagoras’s Theorem:

recognising that right-angled triangle calculations may generate results that can be integral, fractional or irrational numbers known as surds

Though much can be said about this line, the most important thing to say is that it is wrong. Seven years later, the line is still in the Australian Curriculum, essentially unaltered, and it is still wrong.

OK, perhaps the line isn’t wrong. Depending upon one’s reading, it could instead be meaningless. Or trivial. But that’s it: wrong and meaningless and trivial are the only options.

The weird grammar and punctuation is standard for the Australian Curriculum. It takes a special lack of effort, however, to produce phrases such as “right-angled triangle calculations” and “generate results”. Any student who offered up such vague nonsense in an essay would know to expect big red strokes and a lousy grade. Still, we can take a guess at the intended meaning.

Pythagoras’s Theorem can naturally be introduced with 3-4-5 triangles and the like, with integer sidelengths. How does one then obtain irrational numbers? Well, “triangle calculations” on the triangle below can definitely “generate” irrational “results”:

Yeah, yeah, \pi is not a “surd”.  But of course we can replace each \pi by √7 or 1/7 or whatever, and get sidelengths of any type we want. These are hardly “triangle calculations”, however, and it makes the elaboration utterly trivial: fractions “generate” fractions, and irrationals “generate” irrationals. Well, um, wow.

We assume that the point of the elaboration is that if two sides of a right-angled triangle are integral then the third side “generated” need not be. So, the Curriculum writers presumably had in mind 1-1-√2 triangles and the like, where integers unavoidably lead us into the world of irrationals. Fair enough. But how, then, can we similarly obtain the promised (non-integral) fractional sidelengths? The answer is that we cannot.

It is of course notable that two sides of a right-angled triangle can be integral with the third side irrational. It is also notable, however, that two integral sides cannot result in the third side being a non-integral fraction. This is not difficult to prove, and makes a nice little exercise; the reader is invited to give a proof in the comments. The reader may also wish to forward their proof to ACARA, the producers of the Australian Curriculum.

How does such nonsense make it into a national curriculum? How does it then remain there, effectively unaltered, for seven years? True, our 2010 column wasn’t on the front of the New York Times. But still, in seven years did no one at ACARA ever get word of our criticism? Did no one else ever question the elaboration to anyone at ACARA?

But perhaps ACARA did become aware of our or others’ criticism, reread the elaboration, and decided “Yep, it’s just what we want”. It’s a depressing thought, but this seems as likely an explanation as any.