WitCH 105: Flippin’ Ridiculous

It’s hard to be believe we haven’t already done this one, since it’s been irritating us for years. But in any case it really irritated us again yesterday during a tute, and so here we are.

The following introduction and example, chosen largely but not entirely at random, is from Cambridge’s Specialist Mathematics 3 & 4.

UPDATE (08/07/23)


Commenters have pointed out a number of irritants and more serious issues, and there are others, which we’ll get to in Part 3. The line by line issues are secondary, however. The primary problem, as commenter anonymouskouri was getting at, is that the entire approach is fundamentally screwy. This screwiness turns out to be way more than an issue of mathematical integrity. It actively confuses the hell out of students.1

First, some scene-setting background. VCE Specialist Mathematics includes the study of Differential Equations, but such a title is way overselling the breadth of the topic. Excepting for those to be solved by direct antidifferentiation the only DEs that are required to be solved are separable, that is are of the form,

\color{blue} \boldsymbol{(1) \qquad  \dfrac{{\rm d}y}{{\rm d}x} = f(x)g(y)\, .}

Yes, there’s Euler’s method and slope fields and some simple modelling. But in terms of solving DEs, separable is all there is, and they are easy. Allowing ourselves a little black magic for the moment, we cross multiply to get the x‘s on one side and the y‘s on the other:

\color{blue}\boldsymbol{(2) \qquad \dfrac1{g(y)}{\rm d}y = f(x){\rm d}x \, .}

Then, we integrate:

\color{blue}\boldsymbol{(3) \qquad\bigintsss \hspace{-3pt} \frac1{g(y)}\,{\rm d}y \, = \, \bigintsss\hspace{-3pt} f(x)\,{\rm d}x \, .}}  }

That’s it. Everything else, the actual integration and solving for y and so forth, is simply detail.

Now, before getting on with hammering Cambridge, let’s demystify the black magic at least a little. Of course students are taught to pretend to never to think of dy/dx as a fraction, and they would/should never write the middle line (2): they only think the middle line (without telling anyone), and then they write the integral equation (3). But (3) is easy to justify.

Starting with (1), we can divide both sides by g(y) and integrate with respect to x. That gives,

\color{blue}\boldsymbol{(2') \qquad \bigintsss \hspace{-3pt} \frac1{g(y)}\frac{{\rm d}y}{{\rm d}x}\,{\rm d}x \, = \, \bigintsss \hspace{-3pt} f(x)\,{\rm d}x \, .}

But now the LHS of (2′) is exactly set up for integration by substitution, with the (as yet unknown) function y being the new “variable”. (Think of the substitution being u = y, if that helps clarify what’s going on.) The result of applying substitution to the left side of (2′) gives us exactly (3).2

Now, one may argue that we have simply replaced the black magic of separation with the black magic of substitution. That’s not really the case: the only step in going from (2′) to (3) is exactly applying integration by substitution (albeit not in the function notation that makes the legitimacy clearer). Of course trying to think why substitution works this way is not at all obvious.3 But in any case, the magic of separation is, at worst, no blacker than the magic of substitution. And the solving of DE’s in Specialist Mathematics contains nothing else of particular note.

1. This WitCH was motivated generally by seeing years of confused students, and it was specifically motivated by a tutorial the previous day. That tutorial was for a very good student, who is at a top notch school, who has a top notch Specialist teacher, and nonetheless the student had essentially no clue what was going on with differential equations. 

2. Cambridge later provides this computation, using (2′) to justify (3), but they do so without even a hint that integration by substitution is the key ingredient. Which is stupid. The entire Cambridge chapter on DEs is a WitCH-infested mess.

3. In the comments we’ve promised to do a post on the black magic of integration by substitution. We’ll get to that as soon as we can.



Now, to the poisonous heart of the WitCH. Cambridge begins the DEs chapter with “verifying” solutions,4 after which they take care of the trivial cases, y’ = f(x) and y” = f(x). Then, it’s on to separable DEs.

As indicated by the excerpt above, Cambridge begins by considering DEs of the form

\color{blue} \boldsymbol{(1') \qquad  \frac{{\rm d}y}{{\rm d}x} = g(y)\, .}

Of course, (1′) is a special case of (1), so its solution and the method of obtaining it are also special, but not notably simpler, cases:

\color{blue}\boldsymbol{(3') \qquad\bigintsss \hspace{-3pt} \frac1{g(y)}\,{\rm d}y \, = \, \bigintsss\hspace{-3pt} 1 \,{\rm d}x \, = x .}

As such, there is no particular need to introduce (1′) as a first and special type. Nonetheless, it’s a standard and reasonable enough approach. What is not remotely reasonable, however, is to consider (1′) by also introducing a gratuitously different, and woefully unexplained, method of solution. There is simply no need for the method Cambridge employs and it is actively, hugely confusing to do so.

What Cambridge is applying is the inverse function theorem, but without a hint that this is what they are doing,5 or explaining at all what follows. It is all utterly pointless, since substitution is conceptually easier, and is required anyway to solve the general type, (1). Just to be absolutely clear, Cambridge‘s method of solving (1’) is legal, or at least legalisable. What it is not, is sane.

The absurd cherry on top is that when Cambridge finally gets around to general separable DEs they do not even hint that (1′) is separable, that it is a special case of (1). With the exception of the quick and inevitably ignored note in the excerpt above, (1′) is considered to be an entirely different form of DE, up to and including in the chapter summary:

An entire goddam page when one equation would suffice. Pure madness.

4. As VCE teachers and students are all too aware, one has to pay careful attention to VCAA’s prissy and perverted language. In particular, in VCE “verify” does not (usually) mean the same as “show that”; see here, for one of a thousand examples. This language perversion also screws up kids doing DEs, who commonly enough over-cautiously interpret “verify” to mean “solve”.

5. The inverse function theorem is never mentioned in VCE. It is always simply assumed that inverse functions can be differentiated by means of the chain rule.



To finish off, we’ll list the many line by line irritations. Thanks again to the commenters who worked hard to make sense of this nonsense.

(a) “Identity” is a very weird name for the formula for the derivative of an inverse.

(b) Cambridge‘s use of pronouns sucks: “this becomes” should be “the differential equation becomes”.

(c) No explanation is given for why dx/dy = 1/g leads to x = ∫1/g. Yes, it’s just definition, but in the context this will be far from obvious to most VCE students and teachers.

(d) “Differential equations can be constructed from statements” is one of the all-time great stupid sentences.

(e) As indicated, the example begins Cambridge‘s discussion of modelling. Why on Earth would one begin modelling with an entirely unmotivated and thoroughly implausible population model? Why would one begin with a “statement”, rather than an explicitly expressed model, with the parameter explicitly given?

(f) Even given the set up, the use of the \boldsymbol{\propto} “proportional to” symbol is pointless and distracting.

(g) “population-time graph” needs an n-dash, not an m-dash.

(h) In VCE it is standard for “increasing” to not mean strictly increasing; thus, the conclusion that k > 0 is questionable. There is also no particular benefit or extra naturalness in assuming the population is increasing. To do so, and thus to have to fuss about k > 0, is just muddying the message of this very first modelling example.

(i) Cambridge‘s use of pronouns still sucks: “Note that it is of the form” should be “Note that the differential equation is of the form”.

(j) Flipping the derivative has encouraged the needless and annoying fiddling with a 1/k.

(k) “Therefore” dots suck.

(l) Flipping the derivative has effectively turned the +c into a -c. These things matter. (In a moment c is gonna be negative.)

(m) It is not pointed out that, since t ≥ 0, we also require c ≤ 0.

(n) “Therefore” dots still suck.

(o) The graph is wrong, falsely suggesting that the turning point should be on the P axis.

(p) No consideration is given for the possibility of the solution P = 0, or the means of obtaining it, or the fact that this solution is not contained in the “general” solution given, or that the initial condition P(0) = 0 leads to two entirely different solutions, which kinda sucks for a model. One might wish to rule all this out with the “increasing” business, but it still has to be explicit and argued.

87 Replies to “WitCH 105: Flippin’ Ridiculous”

  1. “Using the identity…” hmmm… yes… I guess the authors just know that \frac{dy}{dx}=0 is not something to worry about, for reasons not given.

    Although, technically that means it is not an identity, so…

    1. To be semi-fair, the authors consider the issue of dy/dx = 0 a little while after the intro above, and a little while before the example above.

  2. The restriction P neq 0 is not stated and investigated during the solution process, the equilibrium solution P = 0 is not contained in the general solution and is therefore lost.

    1. Yes, which was what RF was getting at. There’s plenty more, and P = 0 was not the main reason I posted it.

  3. The graph appears to have the vertex at t=0 which means that c=0, which means the P intercept should be 0. If we assume that the initial population is positive then the initial gradient should be positive and the turning point is to the left; c<0. Am I just being picky?

    1. Nicely spotted like a leopard. But shouldnt we assume ‘initial population is positive and increasing’? Then c < 0 must follow. I have questions:
      "the appropriate" or "an appropriate" differential equation?
      Is it only the solution when P is in terms of t?
      Who else hates therefore dots?

      1. If the solution in a textbook uses any such assumptions, even if natural, it should be made explicit.

            1. …anything turning there, at the starting point (end point, left end point)?
              Sorry, too careless reading.
              TP is theoritically on (c, 0) which is not on the graph.
              Left end point looks way too much like 0 gradient
              I would have the graph extended (say , dotted line) all the way to (c,0). Clear +ve grad at t=0
              I dont see anything else…only
              dy/dx = 1 / (dx/dy) taken for granted. Some danger here

              1. ??? The solution is a chunk of a parabola, which has a turning point, which tom chose, I think idiosyncratically, to call the vertex.

                They’re just words.

              2. Good points about how the graph should have been drawn. And make it clear that the solution can not be extended for t < c

  4. A quick note.

    People have made some very good points so far, and there’s plenty more nitpicking to be done. (Personally, I find the m-dash in “population–time graph” extraordinarily annoying.) However, no one has gotten remotely close to why I posted this WitCH.

    I think there is a sermon brewing.

        1. I have often heard American commentators talk about a car attaining a high “rate of speed”.

      1. What Anonymouskouri said. But I’ll update the post later today, with why this stuff pisses me off so much.

  5. Integration of the reciprocal of the function g(y) is not what integration by separation of variables is about, nor is it a special case thereof. A function of x on the left hand side would need to be integrated w.r.t. x for separation of variables to warrant a mention (1 is not a function of x, it is a constant). Standard integration is not a special case of separation of variables. Actually, does even a special case exist.that can still be considered separation of variables? No. Either it is separation of variables or it isn’t. This isn’t.

  6. The actual full solution to the given differential equaltion is

    P(t) = 0 for t ≤ c ,
    P(t) = k^2/4·(t-c)^2 for t > c ,

    with c in (–∞,+∞].

    (The population can spontaneously arise from zero at any random moment of time and then increase quadratically, or stay zero forever.)

        1. And shouldnt the restrictions on c be the other way around (since the answer has (t – c)^2)? And Im not sure how the population can spontaneously arise from zero unless its a dust of the ground moment followed by a rib.

          1. The c are fine but Im not so sure about P = 0 for t leq c and the spontaneous discontinuous arising of a non-zero population. We prefer c < 0, having some initial population at t = 0 and knowing nothing prior to t = 0.

            1. There is nothing “discontinuous”. The piecewise defined function P(t) is everywhere continuous and differentiable, in particular, P'(c)=0, and it satisfies the differential equation everywhere.

              If you want to talk about “population”, then, well, a population described by this differential equation can arise from nothing at any moment of time. If you think that this is not what normal populations usually do, then there is something wrong with the “modelling”. Or maybe it’s a model of a population created by god. Or whatever.

  7. Question (for anyone): is it acceptable to treat the derivative \frac{dy}{dx} like a fraction?

    Is it acceptable in some circumstances but not others?

    It seems that a lot of these textbook sections assume that \frac{dy}{dx} is a fraction, but… I don’t know, it just feels like there might need to be a bit of caution sometimes, but I cannot immediately think of a reason why.

    Hence, asking the question.

    1. It’s as least as acceptable as applying the inverse function theorem without ever noting the inverse function theorem is being applied, or that, as a theorem, it requires a proof.

      I’m busy now but I will try to update the Witch tonight, and I’ll address such points.

    2. I’d love to hear an answer to this! I feel it’s dodgy – something about the parts being limit to zero? But as it ‘works’ in the Specialist course particularly with setting up related rates questions it’s an easy way for kids to think about how to set up the rates correctly. The Edrolo video guy teaches substitution by rearranging du/dx for dx ( if u=X^2 ending up with dx=du/2x for example) which I’m really not sure about but students like – it makes it more formulaic to do.

      1. Hi ,Amber. Justifying the du = du/dx dx black magic for substitution is a related but separate issue. As it happens, I spent some time recently thinking how to present substitution, and figured it out as best I could, at least for myself. I’ll talk about the black magic for separable DEs when I update this post. I’ll try to write a separate post soon, on the substitution black magic.

          1. A little external pressure could balance the internal pressure, and I’d be in a rare equilibrium.

    3. Hi RF. No, IMO it is not acceptable. It is acceptable to prove certain notations and manipulations carefully, and then use them. If proof is inaccessible, then precise statements that are true can be argued and sketches presented. Once students have been through this, they won’t have any desire to treat dy/dx as a fraction, because they will understand how silly that is.

      In fact the notation dy/dx itself for the derivative is problematic because it suggests that the label “x” is somehow special for the function y. This leads to many misunderstandings.

      The most I would permit that would be as a mnemonic device to help students remember.

    4. dy/dx is not a fraction but treating it like one can be a useful heuristic (black magic in Martys words) for doing certain things eg related rates. Provided its made very clear that whats being done is a heuristic (black magic).
      I think it works because if you take a first principles approach, before taking the limit of stuff –> 0 you actually do have fractions.

    5. Ever since Abraham Robinsohn put infinitesimals on a rigorous basis, it is considered acceptable to regard dy/dx as a fraction. Infinitesimals, like dx, are an extension of the real numbers (yet again). So the authors of this question could argue that we can separate variables as Alasdair has written, dP/\sqrt{P} = k\, dt. However, to justify that by mentioning some advanced mathematics that the students will never meet seems a stretch. Better to take the usual route, as Anonymous writes, to tell the student that dy/dx is not a fraction, it just works like a fraction. But cross your fingers while you say it.

      1. tom is correct – and indeed this comes to the heart of what I was saying about “hand-waving” at school level. Pretty much as tom says, dy/dx is not a fraction, but you can treat it as though it was one. Confusing for students.

        The trouble with Robinson’s infinitesimals is that their definition is quite involved: you define the hyper-reals as Cauchy sequences of real numbers, with two such sequences being equivalent if the indices of equal values form a set which is a member of a non-principal ultrafilter over the integers. This is a highly non-constructive approach, and in fact the Weierstrass \epsilon\delta approach is in some ways more direct. Errett Bishop’s constructive mathematics – which aimed to provide “numerical meaning”, mainly by not allowing the law of the excluded middle – was very opposed to infinitesimals. I like the theory of infinitesimals, but I also have a lot of sympathy for Bishop’s philosophy. Anyway, this is irrelevant to the discussion at hand. I just wanted to show off a bit by sounding off about infinitesimals.

          1. The problem may then be pedagogical. I don’t know if DEs can given any sort of rigorous argument at a school level. In which case you pile error on error in the hope that, as they say, “two wrongs make a right”. In fact I’m not wholly out of sympathy with this, I think there is a place for sloppy reasoning to introduce students to an idea; rigour being applied later (if they ever do a formal DE subject in their university studies).

            1. Thank, Alsadair. I think we’re arguing, or agreeing on, two slightly different issues. One issue is the legitimisation of differentials, the other is how to present such arguments in high school.

              On the first issue, I was simply pointing out that I don’t think Robinson’s stuff is functionally relevant, for mathematicians anyone. I’ve never heard an applied mathematician, while flinging differentials around, declare “I can do this because of Robinson”.

              On the second issue, I substantially agree with you, and I think the (understandable) desire to nail things down can over-pure things and screw up the practical teaching. But I think we can do better than what is done. I’m writing an update of this post as we speak (and plan a separate post on substitution).

  8. This whole business always annoys the absolute shit out of me. How are students expected to make sense of this. This completely glosses over the main course while pretending it doesn’t exist: implicit and inverse functions. When a student notices that something is missing, what happens? Probably they are made to feel stupid.

    1. I think if students are explicitly told that its black magic that gives correct results but its not valid mathematics but rigor will be taught later at a higher level then I think its OK.

      1. I agree that if it is made explicit what is missing (I think it would be better to call it a mathematical miracle, black magic has some kind of sinister implication to my ears…) then that is totally fine.

          1. Keep using it before the woke folk cancel it. On the topic of black magic, a fun game with small kids is to introduce them to the blackhole number – 4. I like the term voodoo mathematics – and the woke folk cant cancel it, not any time soon anyway. But I think the two terms might mean different things. I think study scores get calculated using voodoo mathematics rather than black magic. Either way its a dark art. Avada Kedavra.

                1. I honestly don’t know. There’s obviously a systemic looniness to the modern left (?) that needs capturing. But I cringe every time I hear or read “woke”. It’s just too lazy and broad-brush, too Credlin-ish.

  9. Good points have been made, and I can’t at the moment think of any others – except my own aversion to the “proportional to” symbol which seems utterly pointless. There are number of pedagogical issues here though, mainly about what the authors are trying to achieve. DEs are tricky, slippery things, and at high school level the only way to “solve” them is by a lot of hand-waving, and assumptions. This particular problem boils down, mathematically, to a single integration. I’m not sure what is achieved by dressing up an integration problem as a DE. The problem would be far better phrased as an IVP, maybe with k specified so that the students could obtain a specific function as their answer.

    I notice also in the first “therefore” line (and the use of the three dots is another pet aversion of mine), in order to obtain the integral on the right they’ve had to “multiply” out by dP. As you do. In other words they’re treating this DE in terms of separation of variables, so all the nonsense about dy/dx = 1/(dx/dy) is totally unnecessary (as well as not being an identity, as several have noted).

    By this means, you could go from

    dP/dt = k sqrt(P)


    dP/sqrt(P) = k dt

    and have done with it.

    (Sorry about my ASCII maths, I’ve had issues with LaTeX here before…)

    The final outcome is that except for a very tiny handful of excellent students who may just possibly learn something, everybody else will be bamboozled, confused and annoyed, adding to the myriad of school leavers for whom mathematics was a mysterious awfulness of which they’re happy to have left for ever.

  10. OK, sorry for the delay, but I’ve now updated the post with my thoughts, including many of the points commenters have made.

  11. Just for interest sake, I had a look at the current IB Higher Level curriculum (and some recent exam Qs) on DEs.

    OK, the IB HL course is a lot more “pure” than Specialist in VCE, but still… reduction formulas and homogeneous forms (the IB examiners use the substitution y=v x with v being the variable of choice for some reason). It is done in a lot more depth and, I assume, taught with a lot more rigour.

    The IB papers, for reasons that might make sense to someone more familiar with the course that I am, do not seem to test separation of variables that much which is interesting.

    So, it can be done better and is done better. VCAA, if you’re listening… there are role-models out there – seek them out, PLEASE!

  12. How should differential equations be taught in VCE (and in university), if they should be taught at all?

    I’ve gone through the traditional Specialist maths -> Linear algebra -> Calc 2 -> Real Analysis pathway finishing real analysis last semester, and I don’t think I really have any actual understanding of differential equations other than a random assortment of general techniques.

    I understand that there’s an explicit “Differential Equations” subject as well which I’m guessing is the comprehensive course I’m looking for.

    1. Hi ana,*

      I don’t think there’s one correct answer, either at school or at uni, but every Australian answer I’ve seen recently is pretty obviously incorrect.

      As a first subject, DEs will tend to be, as you put it, a random assortment of general techniques. There’s not much getting around that, in the same way that integration naturally starts out as a bunch of techniques.

      For this reason, I think DEs as a named VCE subject is pretty much doomed to be stupid or trivial (and thus also stupid). Of course you’ll naturally run into specific DEs in VCE, such as population models and cooling and (once upon a time) spring equations. But I can’t see any value in pretending there’s time for any thorough or systematic treatment.

      In uni, such a techniques subject can be much more general and solid. But because uni is being watered down (because students entering uni now know bugger all, and the uni bosses don’t give a stuff), these subjects now tend to be token and slow and pointless.

      Beyond techniques, the theory and theorems and applications of DEs is really nice, and makes for a solid 2nd year (or advanced 1st year) subject, but I haven’t seen such a subject for a long time. You can get an idea of such a subject from books like Braun and Rabenstein.

      *) I really don’t understand why you anonymous guys don’t just choose names.

      1. * marty, if you can suggest a better name than Anonymouskouri then I’ll seriously consider it. Anonymouseketeer? Anonymouscouscous? Anonymoustache? Anonymoustafa? Anonymoussaka? Anonymousse? Anonymoose? 🙂

        1. There is no better name in history than Anonymouskouri. Although Anonymoose comes close …

          I know who *you* are (in a manner of speaking). It’s the multitude of Anonymouses (Anonymice?) that seems nuts.

          1. Anonymice? Ha, good one! (A rodent infestation?..?) My wife said that if she were to comment on your blog then she’d use Anonymoose. I encouraged her to do so but she said that her maths was shit and couldn’t possibly make a meaningful contribution. So I suggested that, in that case, she put in for a job at ACARA or VCAA to which she replied that, “due to the above” and “based on what I’ve heard”, she was “somewhat overqualified”. It’s good to have a laugh every so often. Baklava!

            1. Your wife’s maths being shit is not an issue: the blog is tolerant of and welcoming to people who acknowledge their maths is shit. What we’re intolerant of are people whose maths is shit while pretending otherwise. Which, come to think of it, probably means your wife is *not* qualified for VCAA or ACARA.

  13. A book that I enjoyed was Simmons’s “Differential Equations: With Applications and Historical Notes”.

    At a certain stage in life, when I was new to a mathematics department, I decided that I wanted to do research in DEs. I spoke to the head of the department and he told me to see Dr X. I went along and chatted to Dr X who asked me if I knew any physics. I said “No”. He told me that, in that case, I could not do work on DEs. So I went back to the head of department who said “Talk to Dr Y.” I went and had a talk to Dr Y who informed me that he worked on *functional* equations, not differential equations. I said, “That’ll do”, and off we went. It was fun.

  14. I have been waiting to learn what I wrote that was wrong. Starting to see some glimpses.

    My blurb on infinitesimals was in response to a tangential question asked by Red Five and Amber, as to the validity of interpreting \frac{dy}{dx} as a fraction. My response was it can be thus interpreted, but with considerable difficulty, and is best avoided. It seems that I was in error because this is “not functionally relevant”. How can an answer “yes” to a yes/no question be irrelevant? Ahah! Marty was thinking of the original theme of solving \frac{dy}{dx}= g(y), prompted I guess by my joking reference to the Cambridge authors being allowed to split dP/dt into differentials. {Thinks: In future I will denote my usual tangential wanderings by enclosing them in braces.}

    But there is more. I am wrong, it seems, because applied mathematicians do not argue for infinitesimals like I do. Actually there was at least one applied mathematician who did. His name was Robinsohn, who was working at Cranfield during the war, designing aircraft to repel the Nazis. {About as applied as you can get.} {I call him by his birth name to avoid confusion with Julia Robinson.} He was fascinated by Liebniz’s differentials and spent much of his later life in justifying their use. {I read biographies of mathematicians in preference to the hard slog of understanding their work.}

    Perhaps we not being too precious in demanding rigour in methods of solving differential equations. Heaviside wrote that (I paraphrase) it matters little how you get a solution, because you can check that solution directly. {But what if there are multiple solutions, as in this case? Would Heaviside have missed the solution P=0? Probably not, as it is the envelope of the family of solutions.}

    Heaviside’s great protagonist, G. H. Hardy, would have us justify all steps as they occur, not on the basis of later checking. This is an argument about aesthetics; both approaches are ultimately rigorous. {I find myself preferring Hardy on odd numbered days. Because today is the 11th} I agree with Marty that the use of

        \[1  /\frac{dy}{dx} = \frac{dx}{dy} \qquad \qquad \text{(1)}\]

    to solve \frac{dy}{dx}= g(y) needs justification in the form of the inverse function theorem. And as this is all too difficult at school level {and undergraduate level?} it’s best to use the separation of variables approach instead. {My use of (1) for such differential equations is an example of the common teaching sin: just teaching the same way you were taught.}

    However, I also use (1) to obtain the derivative of some important functions, such as y=\arctan(x). I guess I could go the other way and start with \int \frac{1}{1+x^2} \,dx and substitute x = \tan{\theta} which will solve the integral as \arctan(x). Then the fundamental theorem of calculus gives the derivative \frac{d}{dx} \arctan(x) = \frac{1}{1+x^2}.

    This discussion of difficult results that are rightly glossed over in secondary education, invokes the elephant in the room: the existence of irrationals. For example, we assume that \sqrt{2} is a number and spend a lot of time teaching such things. {Why? because as the Greeks realised, otherwise there would be a gap in the number line, lines would cross without intersecting?} Am I right in suspecting that the theory of irrationals is not covered, even in a standard university pure maths course?

    1. Hi tom. {Your tangent brackets are distracting and annoying}

      What annoyed me about you Robinson comment was the line “Ever since Abraham Robinson put infinitesimals on a rigorous basis, it is considered acceptable to regard dy/dx as a fraction”. In practice, no one cares about Robinson and the rigorisation of infinitesimals. The answer to a yes/no question is functionally irrelevant if, correctly, no one cares whether the answer is yes or no.

      I’m not for a minute arguing that we be overly pure with school mathematics: I’m arguing the exact opposite, and the examples you note indicate exactly why such a pure approach, even if considered desirable, is doomed to failure. What I’m arguing for is: (1) good teaching; (2) honesty.

      The main point about the passage above is (1): it is horrible teaching. It is horrible teaching for many reasons, as indicated by commenters and in the update. But the central reason it’s horrible teaching is that the inverse method employed is unnecessary and fatally distracting from the general separation method to be done about two lessons later. (The original title of this post was “Flippin’ Pointless”.)

      Regarding (2) and inverse functions, yes the passage above is dishonest in this regard, but this is not nearly as great a sin as the (1)-sin. You are correct, of course, that we want derivatives of inverses in school maths, and we cannot properly justify it (even modulo the intuitive approach to limits done in schools). But, we can be honest about our cheating. I really think the Mathematic Oath is critical: first, tell no lies. It’s what separates us from the intellectual beasts. My heart sinks when I see inverse derivatives done autopilot via the chain rule, and I’ve never seen any different in a school text. I’m not suggesting a big song and dance: you don’t want legal/legalistic caveats to distract from the main game. But a word is needed, and that word is never included.

      This kind of dishonesty is now endemic in VCE, and it is as damaging as it is greasy. And, functionally, no one gives a stuff. When, for example, a text introduces \boldsymbol{i} as \boldsymbol{i=\sqrt{-1}}, I want to scream, and I want to grab a flamethrower. And you can point this out and point out the consequential idiocy, and still a new edition comes out and the crap is still there. Four maths PhDs listed on the cover, and still the text contains this crap. That is insane, and it is obscene.

      Finally, on irrationals. Of course we cannot say much other than “fills the gaps” in high school. At university now, I think it depends where. About twenty years ago, I undestroyed Monash’s real analysis subject, and I believe it is still undestroyed. About ten years ago, however, Melbourne Uni destroyed their real analysis; I don’t know if anybody has since undestroyed it.

      1. On irrationals, how do secondary teachers typically explain they are different from rationals? If any such teachers are still following this thread, please enlighten me.

        1. You’d better ask your question more clearly. But in sum: rationals are fractions and irrationals aren’t. Which only tells you what irrationals aren’t, not what they are. So, then you recite the magic incantation “infinite non-repeating decimals”. And then all is (pretended to be) well.

        2. I discuss the difference between rational and irrational numbers with Year 9 students. I use the explanation given by Marty above; after all, ir-rational means not rational. Students give me other examples of the ir- prefix. In the classes, I prove that \sqrt{2} is irrational, mostly to introduce that particular method of proof. My final line is that “You can spend a life-time studying irrational numbers” – and then we move on. On an examination, I asked students to explain the difference between rational and irrational numbers and pretty well all students gave the correct answer.

        3. Thanks guys.
          Is it clear to the students that rational numbers have a decimal representation that is repeated or terminated? If so then it is a simple matter to find irrational ones.

          1. It is clear that they have been told this is true. It is also clear that a maximum of 1% of them know why it is true.

            1. In my new life as a tutor, when I’m teaching a Year 4 or 5 student who knows their multiplication tables, I say “let’s do the hardest thing in Primary maths” and proceed to long division. At first we do integer division with remainder and later get to decimal answers. When the student discovers repeating decimals, I feel there is a quantum leap in both interest and confidence. As always, the most effective teaching is when the student works it out for themself.

  15. There is a common type of question that regularly appears on VCAA Specialist exams that requires using the integral solution. For example:

    Consider \displaystyle \frac{dy}{dx} = \frac{1}{\exp{(x^2)} + 1} where \displaystyle y = 1 when \displaystyle x= 2.

    Find, correct to four decimal places, the value of \displaystyle y when \displaystyle x = 6.

    Seque now to a similar type of VCAA question I have seen (not often, but often enough):

    \displaystyle \frac{dh}{dt} = \frac{3 - 2\sqrt{h}}{8h - 3h^2 } where \displaystyle h = 1 when \displaystyle t = 0.

    Find, correct to three decimal places, the value of \displaystyle t when \displaystyle h = 2.

    Is something flippin’ ridiculous needed to solve this question? In which case …

    1. Thanks, BiB. That’s interesting. I don’t know that I’ve seen such a question on a VCE exam (or textbook). Do you have a reference?

      Such a question is more directly solved by flipping. And in general, if the only DEs one is considering are of the form dh/dt = f(h), then flipping is quicker (whatever the integrity of the justification). That’s effectively what Fitzpatrick and Galbraith do, where they first get the solution via substitution, and then show the flipping (with poor justification).

      If such definite integral approximation questions (of the second type) arise, then the teaching has to take that into account. I would still not even consider introducing the solution of y’ = f(y) via flipping, and I doubt I’d introduce flipping at all, although the second is a judgment call.

      1. As recent as 2021 Exam 2 Question 3 part (c) (worth 3 marks, 77% of the state got 0).

        btw this from the Examination Report: “Obtaining this answer required an appreciation of the physical situation [1], sound calculus skills [2] and careful use of a CAS [3]. Relatively few students made a productive start [4].”

        [1] I don’t see how such “appreciation” is required.
        [2] I don’t see where “sound calculus skills” are needed.
        [3] As opposed to careless use of a CAS …? What care is needed, apart from the usual press the correct buttons and don’t misread or miscopy the output.
        [4] I’d love to know a reason for this. Are students not being taught the integral solution? Is it being taught simply as a formula to be regurgitated – students only know it in a very one dimensional way?

        1. Thanks, BiB. i hadn’t noted that aspect when looking at the question at the time.

          The “solving” of an initial value problem when you can’t actually do the integral is conceptually a bit tricky, and is something that is either taught or not. I am not at all surprised that the majority of students had no idea what to do with the exam question. (It was also very badly worded, which wouldn’t have helped.)

          I checked the latest edition of Cambridge, and I cannot see where they do any such question. They have 11H, where they numerically “solve” DE’s of the form dy/dx = f(x). But I cannot see any discussion of or examples of solving DEs of the form dy/dx = g(y). (I didn’t look hard.)

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