Tips and Tricks for Specialist Mathematics

I just received the following email from Mystery Student, Alex:

Hi marty,

I’m currently taking Spec 3&4 and just had a couple of questions reading this post.

For testing linear dependence, you recommended using a ‘3×3 determinant’. I was just a bit confused, and I’m always looking for areas to improve my knowledge, blah blah blah.

Do you have any other areas that make questions more efficient that are glossed over by VCAA or textbooks?

Thanks a bunch 🙂

I answered Alex briefly on the determinant question, but there are obviously readers much better informed than me about helpful tips and tricks for Specialist. And, in any case, such questions are best replied to by the crowd.

So, please make your suggestions in the comments below, including answering Alex’s specific question.

If the post takes off then I’ll perhaps try to categorise and summarise the suggestions in updates to the post. Also, if people think a companion Methods Tips post is worthwhile I’m happy to do that (although the worth of that is less obvious to me).

Quick Notes on the Herald Sun’s Exam Errors Article

There is report today in The Herald Sun (Murdoch, paywalled), titled,

Mistake-riddled VCE maths exams robbing students

Regular readers will know pretty much the lay of the land. However, there may be some non-regular readers in the next few days. So, a few clarifying remarks are probably worthwhile. (This is quick: I’ll adjust as I can through the day.)

First of all, without reflecting at all on the accuracy or the merits of the report, I want to make clear that I had no role in the creation of the report. 

Secondly, at one point the report makes quick reference to this blog:

A Bad Mathematics blog run by a professional mathematician with a PhD in maths has identified more than 90 serious problems with specialist maths exams and 77 in maths methods, including sample exams and Northern Hemisphere exams going back to 2006.

More specifically, this appears to refer to the Specialist and Methods (and there’s also Further) error list posts (and the subsequent links included there). The report refers to “serious errors”. Without rejecting that language, the language I use on these posts is of “major” and “minor” errors:

To be as clear as possible, by “error”, we mean a definite mistake, something more directly wrong than pointlessness or poor wording or stupid modelling. The mistake can be intrinsic to the question, or in the solution as indicated in the examination report; examples of the latter could include an insufficient or incomplete solution, or a solution that goes beyond the curriculum. Minor errors are still errors and will be listed.

With each error, we shall also indicate whether the error is (in our opinion) major or minor, and we’ll indicate whether the examination report acknowledges the error, updating as appropriate. Of course there will be judgment calls, and we’re the boss. But, we’ll happily argue the tosses in the comments.

In recording and characterising such errors, I have made no attempt to determine or guess the effect of such errors on students’ scores. That seems to me to be a very difficult thing to do, for anyone.

Thirdly the report refers specifically to three questions in error on the 2022 Specialist Exam 2. That exam is discussed generally here. (The other 2022 exams are discussed here and here and here and here and here.) The specific questions are discussed here and here and here. These three questions (and others on the 2022 exams) appear to me to be unquestionably in error.

Fourthly, and finally for now, for me the prevalence of errors on the VCE exams is simply the tip of the iceberg. The many posts on this blog concerning VCE and VCAA indicate my more general concerns with VCE mathematics. (My broader maths ed concerns are probably best captured by this post.)

That’s it for now. I’ll update this post if something occurs to me, or if someone suggests in the comments that I somehow should.

Integration By Substitution

This is just a straight post, framed around answering the question:

How does one introduce-explain integration by substitution to high school students?

That is the question, but I’ll declare from the outset that I cannot answer it. What I will do is explain as clearly as I can why integration by substitution works in the form(s) in which we use it. It is then up to the teacher to decide how much of this “why” message, if any, is required or helpful for their students. (It is not at all clear to me that delving into the proper “why” of substitution will have much meaning or benefit for more than a few school students.)

The post was motivated by a related request on a recent WitCH. Also, having pondered and hunted through the blog, I notice that frequent commenter SRK made a similar request long ago, and there was a related WitCH. The extensive discussion on those posts may be of interest.

THE BASIC MEANINGS

Just so were all on the same page, the only thing we’re considering on this post are antiderivatives: there is no calculation of areas, no fundamental theorem of calculus. I shall use the term “integration” and integral notation because it is common to do so, but the word and notation properly refer to the summing up of bits, which is not what we’re doing here.

So, the function \boldsymbol{F} is an antiderivative of \boldsymbol{f} if \boldsymbol{F' = f}. We then use the integral sign to represent the general antiderivative:

    \[\boldsymbol{\int f \ = \ \int f(x)\, {\rm d}x \ = \ F + c\,.}\]

(Just as a function may be referred to as f(x) or simply as f, the dx notation in integrals is optional, and I’ll use it or not as seems to be clearer.)

An alternative name for this general antiderivative is indefinite integral. Then, the definite integral indicates for us the evaluation of the antiderivative at the “endpoints”:

    \[\boldsymbol{\int\limits_a^b f = F(b) - F(a)\,.}\]

Again, there is no “integration” here, no computation of areas. It is almost solely definition and notation. The only substantive point is to recognise that any two antiderivatives of \boldsymbol{f} differ by a constant, which is intuitive but takes a proof. Then, this +c, whatever it is, cancels out in the evaluation of the definite integral, implying it doesn’t matter which antiderivative we happened to choose.

STATEMENTS OF INTEGRATION BY SUBSTITUTION

Integration by substitution in indefinite form is standardly presented as,

    \[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x}\, {\rm d}x \ = \int f(u)\, {\rm d}u\,.}\]

Here, on the left hand side, u is some (differentiable) function of x. On the right side, there is a double think: we antidifferentiate thinking of u as any old variable, and then, when done, we think of u as the given function of x.

If we write u = g(x) explicitly as a function of x then substitution takes the form,

    \[\boldsymbol{\int f(g(x))g'(x) \, {\rm d}x \ = \int f(u)\, {\rm d}u\,.}\]

And, the definite integral version takes the form,

    \[\boldsymbol{\int\limits_a^b f(g(x))g'(x) \, {\rm d}x \ = \int\limits_{g(a)}^{g(b)} f(u)\, {\rm d}u\,.}\]

Note that the definite integral form requires no double-think: the u on the right hand side is simply a who-cares variable of integration. We can also do without x and u entirely, writing the definite integral equation more simply, more purely and less helpfully as

    \[\boldsymbol{\int\limits_a^b (f \circ g) g' \ = \ \int\limits_{g(a)}^{g(b)} f\,.}\]

Finally, a quick word on the intermediary, dodgy line:

    \[\boldsymbol{{\rm d}u = \frac{{\rm d}u}{{\rm d}x} {\rm d}x = g'(x) {\rm d}x\,.}\]

Whether or not one permits the dodgy line is really just a detail, since it is immediately followed by a non-dodgy line. It is, however, better to permit the dodgy line, because: (a) it works; (b) it helps; (c) it really annoys people who object to it.

JUSTIFICATION OF INTEGRATION BY SUBSTITUTION

At its heart, of course, integration by substitution is simply the chain rule in reverse. The chain rule for the composition \boldsymbol{F \circ g} is,

    \[\boldsymbol{(F \circ g)'(x) = F'(g(x)) g'(x)\,.}\]

The chain rule can then be written in antidifferentiation form as,

    \[\boldsymbol{\int F'(g(x)) g'(x)\, {\rm d}x \ = \ F(g(x)) + c\,.}\]

Or, with u = g(x), we can write the anti-chain rule as,

    \[\boldsymbol{\int F'(u) \frac{{\rm d}u}{{\rm d}x}\, {\rm d}x \ = \ F(u) + c\,.}\]

But now, given an integral of the form \boldsymbol{\int f(u) \frac{{\rm d}u}{{\rm d}x}}, it is easy to apply the anti-chain rule. All we need is to give a name to the antiderivative of f.

So, let’s write F for the (an) antiderivative of f: that is, F’ = f. Then, by the anti-chain rule,

    \[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \ \int F'(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \ F(u) + c \,.}\]

But also, just thinking of F as a straight antiderivative of f, we have,

    \[\boldsymbol{\int f(u) \, {\rm d}u \ =  \ F(u) + c \,.}\]

Combining the two lines, and keeping in mind we think of u = g(x) after antidifferentiating, we have integration by substitution:

    \[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \int f(u) \, {\rm d}u\,.}\]

The other forms of the formula can be thought of and derived similarly. For example, again setting F’ = f, the definite integral form can be justified as follows:

    \[\boldsymbol{\int\limits_a^b (f \circ g) g' \ = \ \int\limits_a^b (F'\circ g) g' \ = \  \int\limits_a^b (F\circ g)' \ = \ F(g(b)) - F(g(a)) \ = \  \int\limits_{g(a)}^{g(b)} F'\ = \ \int\limits_{g(a)}^{g(b)} f\,.}\]

CONCLUSION

Will this help? Probably not: the introduction of (and then disappearance of) the antiderivative F is not so easy to understand. So, it is not necessarily wrong to take a “looks kinda right” Leibniz shortcut, or to focus upon a specific chain rule or two. But, ideally, teachers should have some sense of why things are true, even if they then decide to not try to convey this sense to their students. And the sense, as best as I can express it, is the above.

Don’t Accept No Subsitutions

As a byproduct of the recent differential equations WitCH, I promised to write something on integration by substitution. I’ve been pondering that, and I will write it, soon. But there’s also a little VCAA story about substitution I’ve been thinking about posting for a long time, and it seems natural to do that first. In order to tell the VCAA story, I’ll also give a little history to frame the story. Because you young ‘un readers of this blog need some history. Continue reading “Don’t Accept No Subsitutions”

VCAA’s 2022 Exam Reports Are Up

Sorry, no energy for a joke title. The reports are here (Word, idiots):

Specialist Exam 1 report (and exam)

Specialist Exam 2 report (and exam)

Methods Exam 1 report (and exam)

Methods Exam 2 Report (and exam)

For background, readers may wish to go to the following blog posts:

Specialist Error List (and see here, here, here, here, here and here)

Methods Error List (and see here and here)

We’ll update the various posts soon(ish).

Continue reading “VCAA’s 2022 Exam Reports Are Up”