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. 9
^{1/2}= √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 9
^{1/2}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.

Can you provide the name of a textbook or two that you consider ok – not necessarily part of the Australian curriculum? I’d like to do maths but don’t want to start from behind.

Mark, I’m probably not much help, but if you indicate the level of maths you’re interested in, I’ll suggest what I can.

OK. The first thing that jumped out at me was the phrase “irrational AND cannot be expressed as a fraction” as though these are two sets which just happen to overlap for surds. I think you will find very few teachers who actually use these explanatory pages in textbooks for any purpose at all. If textbooks were reduced to exercise sets (or half exercise sets since most teachers set the left hand side…) they would be cheaper, lighter and… not going to happen I know.

There is clearly a lot of crap going on here, but this one phrase really grated against me.

That’s pretty funny. I hadn’t even notice that one. But yes, the “and” is the kind of careless (or clueless) phrasing that makes the sentence close to meaningless.

As to your suggestion that most teachers (and presumably most students) don’t refer to the explanatory pages, a couple of thoughts come to mind. First, even if true, that doesn’t mean that the teachers are aware of the mathematical flaws or that they are using anything that is better. Secondly, the fact that the explanations in the current textbooks suck doesn’t mean that they have to suck. Thirdly, if the explanations are redundant this indicates that the exercises, and by extension the curriculum, are too shallow.

I accept points 1 and 2. Point 3 is questionable (briefly) because a) *some/most* teachers use other sources for skill practice, b) *some/most* teachers use other sources for the explanations they give to classes and c) I have suspicions (unprovable) that textbook writers in the main look to other, popular textbooks for guidance about what to include and what depth to chart rather than go to the ACARA/VCAA curriculum documents which can be difficult to follow (and sometimes appear a bit contradictory when compared to the November examinations – but I digress).

Yeah, possibly. But Groucho springs to mind.

Without going into too much detail (wanting to give someone else a chance to do some whacking) the whole concept of *showing* fractional powers can be written as integers with radical signs is backwards.

Radicals, especially square roots, are well known to students long before they learn index laws.

If the sentence was written the other way around: “…this shows that radicals can be represented as a fractional power…” it almost works.

Some genuinely good Mathematics can come about in this topic by taking things students already know and using this to build new knowledge. Starting with something they don’t know and concluding something they do is a bit like saying a purple cow is proof that swans are black because something that is not a swan is not black.

Yep, you’ve isolated the major (but not the only) crime, though you’re massively understating the severity of the crime.

I was told last week by management that I was to make every attempt to “calm the waters” when issues were detected… true story.

Maybe I’m just a slow learner.

Teacher, let me try: Since (-3)*(-3)=9 and sqrt(9)*sqrt(9)=9 then sqrt(9)=-3. That’s your reasoning, isn’t it? [Does that mean 3=-3?]

Thanks, DF. That’s very interesting, and I like it. You haven’t indicated the major crime, kind of like nailing Al Capone on tax evasion. But you’ve elegantly identified a logical crime.

Of course you’re right that this ground is fertile (fertilizer-filled) beyond belief. My earlier concern was the strong suggestion that every function is one-to-one: both arguments are structured as, “Since f(x)=f(y) then x=y”. This is a crime against mathematics, no doubt, but I constructed my response to spotlight the extremely serious crime against education. What happens to the reputation and health of our subject when an invisible booby-trap like this one blows up on a student with initiative, perhaps on their first attempt to understand a “proof” by generalizing or adapting it?

But what about the surds? For starters, this is arcane terminology that adds a barrier to dialogue for no corresponding benefit. We should use plain English wherever possible. So let’s start: how about the phrase “infinite and non-recurring (with no pattern)”? This facile juxtaposition insinuates that “recurring” and “with a pattern” are the same thing. But they’re not: 0.101001000100001… is a decimal with a pattern (count the run-length of 0’s), but it’s non-recurrent, hence irrational. (For most online dictionaries, “surd” just means irrational; a minority add the further condition that it be expressible as some fractional power of a natural number.)

What kind of flaming genius invites innocent children to guess whether a number is rational or not using a calculator!? I want to know what the class would come up with for a number whose “index form” is (1/167)^1: that’s a repeating decimal of period 166, according to Wolfram Alpha and the enlightening discussion at https://math.stackexchange.com/questions/377683/length-of-period-of-decimal-expansion-of-a-fraction . No calculator I have ever seen will reveal that pattern.

Like Number 8, I must try to stop now to leave room for others.

Thanks again, DF. I like the expression “flaming genius”, though i like Samantha Bee’s language even more. I’ll keep my reply short. First, of course I agree entirely that the invertible-reversible flaws in the text’s purported proofs are genuine mathematical crimes. Still, it’s not the fundamental crime here. I might make a separate post soon. Secondly, yes, the testing of irrationality with a calculator is utter bullshit. It’s also very common bullshit in Australia. Finally, yes, the word “surd” is arcane to the point of meaninglessness.

Would it be better to make the link between fractional indices and roots by getting the student to follow a pattern? For example, halving indices so you get 3^4 = 81, 3^2 = 9, 3^1 = 3, 3^(1/2) = ? with the link between the answers being you are finding the square root each time to continue the pattern. Then you can generalise it for x and other fractional indices.

Thanks, Potii. There’s obviously something there, but for now I won’t reply in detail.

AMSI (OZ not US) (http://amsi.org.au/teacher_modules/Indices_and_logarithms.html) makes the claim that a(1/2) can be DEFINED as sqrt(a) because if you square a^(1/2) you obtain by a previous index law a^(2/2)=a^1 which has previously been defined as a.

A similar argument holds for other fractional indices.

However, this is given as a DEFINITION, rather than a concept requiring proof (and certainly no evidence of any calculators in sight!)

This definition breaks down however once you start wandering around in complex-space, but the concept of a PRINCIPAL nth-root is probably not in the lexicon of this particular textbook series.

Jesus, that’s one appalling webpage. Who does AMSI think they’re serving with such long-winded, horribly formatted crap? Did you have to sweep the dust and brush the spiders away to find the pertinent line?

But AMSI is also correct, and the fundamental point is the point you highlight: 9

^{1/2}= √9 because that is how 9^{1/2}is defined. Done. (The principal root stuff is not relevant here, though an ignorance of the issue is at the heart of Witch 2.)Thanks very much, Number 8. That effectively answers Witch 1. Once we get a free moment, we’ll update this post above with a summary of all the crap. And, eventually, we’ll also write a separate post on index laws.

Well, a lot of the criminality has already been exposed so there’s not a lot left to say. (My first point would have been that 9^1/2 should have been defined to be sqrt{9}, then justified as given in the textbook (but the other way around like a previous colleague has mentioned) but that thunder’s been stolen).

But it should be noted that every number including irrational numbers can be written as a fraction. eg. pi = pi/1. (Maybe it has been said, I didn’t see it but then again quite often I don’t see the cheese in the fridge either). What the textbook fails to say (perhaps implicitly assuming) is that the fraction must have an integer numerator and denominator for it to represent a rational number. This lack of qualifier really annoys me. (It’s like saying that a transcendental number is not a root of a polynomial rather than qualifying the polynomial to have integer coefficients). But maybe the authors have a narrower definition of fraction than I do.

Re: Using the calculator. A wasted opportunity to discuss the difference between a conjecture and a proof. The fact that sqrt[2] does not appear to have a period according to the calculator suggests a conjecture. How can the conjecture be proved ….? (or disproved). I don’t think proof by contradiction is beyond the average student at this level and would surely appeal to the better students. Although it’s not relevant to indices, the textbook opens this can of worms so surely could include a brief side note along these lines. A nitpick is that the table does not specify how many decimal places (presumably as many as the scientific calculator can give).

I don’t mind the word ‘surd’ but it should be defined after a discussion of irrational numbers (maybe that discussion has occurred earlier in the textbook) so that it can be defined as an irrational root as opposed to a rational root. It would have been nice for the textbook to say in passing that ‘surds’ are not the only type of irrational number.

And on the subject of ‘pattern’, the textbook could have included a small note on continued fractions for the better student. Fascinating for that student that sqrt{2} is irrational but has a continued fraction with such a nice pattern …. The fact that it’s infinite is another pointer to sqrt[2] being irrational …. (but for all I know this stuff is discussed earlier in the textbook).

Thanks, John. I agree with pretty much everything, though I think it’s a little nitpicky to ask for an explicit definition of “fraction” here. In particular, yes, some continued fractions in textbooks and classrooms would be lovely to see.