38 Replies to “WitCH 30: Absolute Zero”

    1. Hi Glen. Can you spell out why? Sure, there are plenty of reasons to weep, and I had thought of making this one a PoSWW. But I don’t think the stupidity is necessarily so obvious. Certainly not enough to stop three million people watching the thing.

      1. The whole lesson is fundamentally flawed in (and this is impressive) multiple ways. At many times in the video I was shocked. I find it difficult to decide which is the worst offense.

        I’ve decided they fit into two categories: lesson design, and execution.

        Lesson design:

        The basic premise is to understand the limit of f:U \rightarrow \R given by

        f(x,y) = x^y

        at (0,0), where U \subset \R^2. (Note that the nature of U and any kind of limit is going to be clunky at best. Trying to understand the “maximal domain” of f would be very difficult at this level, and would be difficult to get right. More on this later.)

        The apparent goal in doing this is to work out what value “0^0” should have. The appeal is to a kind of “logic” or procedure that will “convince” us of what is correct.

        This is a terrible, awful idea for a number of reasons:

        The topic relies on the notion of a limit, which is explicitly not dealt with — ignored. (Actually the truth is worse, the teacher in the video says that limits will be dealt with later and that this is an example of one, which, as presented, is absolutely false.)
        Furthermore the topic relies on the notion of a limit in TWO dimensions, which is a topic that is barely understood by students at second year in a mathematics degree at university. Note that our mathematics teachers rarely take this topic and those that do struggle enormously.
        The function in question (x,y) -> x^y is very complicated and in particular has complex behaviour around the origin. Missing from the lesson is a graph of the function, or any talk of when it is or isn’t defined. If the teacher had an idea about this, then they could perhaps understand that their approach and explanation is fundamentally flawed.

        The lesson is designed so that the origin is approached in three ways: along each axis from the positive direction, and then along the diagonal from the positive quadrant. If the students or the teacher had an understanding of limits in two dimensions they would be able to see that this does not prove that the limit at zero is what they claim it to be. In fact, if they did the one-variable limit correctly along these three (straight) paths, they would see that there are different answers. If they had covered limits, they would know that if a limit exists it must be unique. This is a fundamental property and should indicate something, at least to the teacher.

        Even without knowing about 2D limits, the fact that along two curves the limit is different should indicate that considering another particular curve approaching the origin is not fundamentally meaningful.

        This is my last point:

        The function f DOES NOT HAVE A LIMIT AT THE ORIGIN.

        So: the plan is bad in multiple ways, but for me fundamentally it is teaching incorrect intuition at a vulnerable stage which at best contributes only future confusion for the students.

        Now onto my other big problem.

        Execution:

        Even with a horrible lesson plan, I can envisage a lesson where I could make this work. Basically, you give an idea of this function being a weird beasty around the origin, show some pictures, and zoom in a bit. Look at all those spikes. Talk about how we might want to give ourselves the challenge of working out the value at zero from the definition of the function. Quickly decide that the definition of the function is a very difficult task that we can’t really do much with, and so restrict the domain. Talk about the limits along the axes — they are not equal. So, not only are all the quadrants with negatives out, but the axes are out too. Can we make sense with what we have left?

        I would talk about different rays approaching the origin and how we can work out the limit along these rays, from the definition of the function. NO CALCULATORS. The calculator work in this lesson is mind-blowingly terrible for learning. (As an aside, the talk about “Who has a REAL calculator?” gave me chills. “No phones” is code for you-gotta-pay-the-calculator-companies.)

        In the end, we can summarise what we have learned about the behaviour of the weird and wonderful f around the origin. Basically, I’d end with the idea that it actually is mostly a convention, one that is often actually true in a sense that means something (and has already been discussed). It is a choice that we make, an informed choice — but there could have been others. After all, the function f is not defined at the origin.

        It would take some management but I’ve worked and done lessons like this at local schools and it can work quite well. However, I think the idea is very flawed and I wouldn’t teach this if given a choice.

        But the teacher in this video did not do any of that. Worse: at NO POINT was I convinced that the teacher had any idea of what was going on. Instead the focus was on calculator work. Students are given no understanding of the actual issues at work here by the teacher.

        The teacher is acting as if the whole process, the lesson design and the basic idea, actually make sense. The teacher is buzzing along, carrying out the button-mashing on the calculators blissfully ignorant that what he is teaching is fundamentally flawed. All of those poor students have been misled by the teacher following this plan, and it is maddening. In the end I am deeply saddened that this is how our kids are taught. We all know that great tragedies in education happen all the time and all around us, but somehow seeing the depth of ignorance right in my face like that was just shocking. This is a MATH TEACHER. Their whole career, their work, is to teach mathematics. But they do not even sit back to think if what they are teaching makes sense.

        It is just so sad. Thus, my crying.

        1. There are some formatting oddities in my comment — missing numbers before paragraphs for example. I’m not sure what I’m doing wrong in my commenting but I’ll try to keep it in mind. Maybe in the future I’ll write.

          ONE. blah blah

          TWO. blah blah blah

          1. Sorry, Glen, but i have to correct you. This is not “a math teacher”, this is THE math teacher. This is the teaching hero to whom all other teachers must bow.

            As for your criticisms, I agree almost totally (and there were a couple aspects you didn’t mention that really, really pissed me off). The only sense in which I think you may have been a little hard is on the level of mathematics to be presented. At the start of the video Woo talks about the previous lesson (which, by the way, is worse), which starts with the index law a^m ÷ a^n = a^(m-n) and gets to a^0 = 1. From that, one can reasonably assume that the class is lower level (Year 9?) or weak upper level. Given the thin background and/or weak students, one shouldn’t try to do too much. Of course that argues for doing less a lot more carefully, not for making shit up.

            ps Sorry about the formatting issue. (Note you can edit your comment up to a half hour after initial posting.) There are symbols that WordPress doesn’t see to like. Possibly if I could figure out the right setting or plug-in, i could fix it. But …

  1. I apologise in advance for any engineers or ex-engineers I may offend with my coming comments.

    But for now I will just leave it as…

    …WHY?

  2. Hi Marty, bit off topic but I went down a small rabbit hole of clicking on the teaching tag, which somehow led me to your page on qedcat.com about greedy pig and I went to click the link to the one about tipping competitions (http://www.qedcat.com/misc/81.html) but it leads to a 404. Any way to see that old page? Thanks.

  3. Just because a function approaches a value doesn’t mean the function is defined for that value.

    However, for many applications of Mathematics (Engineers for example) near enough is good enough.

    Except in this case, it isn’t.

    1. Hi RF, without wanting to put you on the spot, how might you deal with 0^0 in class differently? I think it’s genuinely tricky to deal with this both clearly and correctly with students, so feel free to take the fifth.

      1. I casually asked colleague A yesterday, “what is 0 to the power of 0?” he sensed it was a trick question and replied, “what do you want it to be?” I said “one half, because that will annoy two groups of people at once.” Not sure he got the sarcasm.

        But in answer to your question, I think it is an interesting thing to look at graphically, because the function x^x is not too difficult to graph over the domain (0,1] and you can then ask interesting questions such as “where is the TP” very few students would guess 1/e at first, but it makes for an interesting discussion.

        But, to actually answer your question, I would take a pragmatic, philosophical approach and say that two index laws are in disagreement here, and since neither has priority over the other, the question of 0^0 remains indeterminable, thus undefined. Some examples of rational functions such as (x^2-1)/(x-1) at x=1 may also be worth looking at for this very purpose.

  4. How I would structure the lesson would depend on the specific aim of the lesson. Possible aims might be as follows: (1) to explore complexities associated with indices; (2) to explore the meaning of 0^0; (3) to introduce the pupils to the concept of a limit.

    I don’t know whether pupils care about the aim of the lesson. However, this video shows me its importance.

  5. It seems from the beginning of the video, Mr Woo has introduced his pupils to indices in the previous lesson. He recalls the definition of a^m as a x a x…(m times).

    Rather than jumping into 0^0, a simpler problem is to consider the meaning of 8^{2.5}. Of course we can work it out on our calculator, but what does it mean? Following on from a^m, I could write 8^{2.5} a product of two and a half eights on the board like this: 8 x 8 x o.

    One might get to argue that 8^0.5 = \sqrt{8} and hence 8^{2.5} = 64 x \sqrt{8}.

    But what does 8^{\pi} mean? Again, this can be done on the calculator – but what does it mean?

    For a Year 9 class, it may be sufficient to ask these questions without providing an answer. It’s complicated.

      1. Terry/Marty/anyone: can an idea such as 2^Pi really be taught without students first understanding indices as the inverse of a log function, defined as a definite integral? Genuinely curious. It has been a long time since I’ve studied pure Mathematics and I’m really struggling to fill in a few gaps such as this!

        1. Hi, RF. The short answer is “no”. You can readily define 2^r if r is rational (though the textbooks tend to stuff it up). But you need to use either logarithms or limits if you want to consider r irrational. Of course, what this means is that the graph of 2^x in Year 11-ish is pretty fictional.

          In some sense this is ok at the school level, and it’s unavoidable. Whenever you get into the real numbers at school, you’re gonna have to cheat somewhere: they’re just too hard. But, it’d be nice if the curriculum and texts and exams encouraged a little more honesty about the cheating.

    1. Sigh. Yeah, I know. Lily Serna tends to say silly things at times, and Burkard and I actually gave her a quick whack about golden ratio stuff in one of our columns. But a fly-by repeat of a standard myth is not nearly as foolish or as damaging as a three-million-viewed false sermon by The Great Master. I don’t think Lily is generally held up or, more importantly, holds herself up as God’s gift to maths or maths ed.

  6. Here is a collection of opinions about the topic which seem to argue that the natural choice for 0^0 is 1:

    https://www.maa.org/book/export/html/116806

    I’m not saying these opinions are correct, quite the opposite, but the phrase “If you are dealing with limits, then 0^0 is an indeterminate form, but if you are dealing with ordinary algebra, then 0^0 = 1.” makes me wonder, what do they consider “ordinary” algebra?

    1. Thanks, RF. A nice page. Yes, probably the most natural choice is 0^0=1. But the key word there is choice. The moment you slide into “proving” it, you’ve lost sight of mathematics. (Which is one reason why Cambridge’s “Essentials” texts are much less than essential.)

  7. OK, a question then (for anyone): at Year 9 level in a typical school it is probably OK to say 0^0 is not defined because two index laws contradict each other for this special case. But then later (say in Year 12 Specialist or in the first year of a pure Mathematics degree) it becomes perhaps(?) useful to think of 0^0 as being defined as equal to 1. Are there any really serious issues with this if we are only working in R^2 space, not R^3 space?

    I’m racking my brain trying to think through the practical applications, but my Mathematical knowledge is sadly lacking!

    1. Hi Red Five!

      I think there seems to be a lot of confusion about exponentiation. In particular, the index “laws”. These are not physical laws, they are consequences of our definitions. They can be thought of as emergent properties of exponentiation.

      Importantly, there is a logical ordering of events here: first, we define exponentiation. Then, we discover the laws.

      It can’t go the other way around. So, it isn’t coherent to use the “index laws” to motivate a definition (or not) of 0^0. The basic reason for this (as I touched on in my rant up above) is because we can’t sensibly give a value to the function (x,y) -> x^y in a neighbourhood of (0,0), and even if we did (perhaps by making infinitely many choices, or restricting its domain), the limit at the origin would not exist. Unless we cut out the x=0 and y=0 axes from its domain, which is such a crippling decision that the exponentiation object would become useless.

      I’m sorry I must confess that I don’t understand the R^2 and R^3 question.

      For practical applications, yes there are many. My preferred construction (not useful in high school though, sorry about that) is to define the exponential via its Taylor series, or alternatively as the unique solution to a differential equation posed on the whole real line. Then I use the exponential and this definition to derive the index laws and all properties of the logarithm. I like using the differential equations approach since this makes it incredibly obvious how fundamental exponentiation is to SOLVING differential equations (this goes back to Euler) and therefore applications include all the typical applications of differential equations solved in this way — everything from logistics, predator-prey, to population growth and time of death.

      Cheers
      Glen

    2. Hi, RF. You are correct, there is simply no point (not counting IB) in defining 0^0 in school, and it need never be defined. Once you get to power series, however, it is useful to define 0^0 = 1. That then means, for example, the power series e^x = 1 + x + x^2/2! + \dots can be written as \sum_{n=0}^\infty x^n/n!, even for x = 0.

      1. Thanks everyone – I’m starting to realise the problem here is the curriculum (although textbooks have managed to add a new level of confusion in many cases) and more specifically the language – “law” is perhaps an unfortunate term for these things.

        Would it then be acceptable (at high school level) to say, for example, that the index laws we teach are a very useful short-hand notation with associated implications (that we call the index laws) but when we start to examine them more generally, zero power, irrational powers, that the problem becomes beyond the scope of high school mathematics and largely leave it there?

        Curious (as always) about what others think.

        And yes, I agree that the whole graph y=a^x becomes a bit problematic but by this time the natural curiosity seems to have been beaten out of most students and they have learned to push buttons to solve such problems.

        1. Hi, RF. Good questions again. I’ll repeat much of the following as a post at some point.

          You are exactly correct: the word “law” here is disastrous. It suggests that these identities come down like commandments from God, before which All must bow. And, although for once button-pushing is not the primary source of the evil, the Magic idiotic box reinforces the idea that nothing can be understood, that the mathematical gods work in wondrous ways, and that’s it.

          As to what we can do about it, yes there is a limit (pun semi-intended), but plenty can be said. First and foremost, it has to be made clear that all powers a^b are defined: first for b a natural number, then 0 and negative integers, then integer reciprocals, then rationals, and then we’re stuck. In particular, one never ever ever ever pretends to use the “laws” to prove what a^b means. (Have I mentioned non-Essential Cambridge lately?)

          Will kids in Year 9 and 10 and 11 and 12 get this? No and yes, and you have no choice. The intricate and intertwined nature of mathematical axiom and definition and proof is very difficult, and in total is beyond school maths, even if many individual parts can be stated clearly. That’s why so many textbook writers and teachers just give up on any explanation, even if they understand themselves (and many do not). They just get to the “laws” as quickly as possible, and away we fuckin’ go.

          But that is wrong. It is wrong in fact, it is wrong in spirit and it is wrong in effect. Yes, students have to be able to compute with the “laws”, without regard to the source and the meaning. But they also have to know what indices mean, to be able to effortlessly switch back and forth from roots and reciprocals. And many, many VCE students cannot. Many believe, for example, that a^{1/2} somehow means “a multiplied by itself one-half times”, and go nuts trying to make sense of the insensible. And why do they believe this? In large part, because they had a bad teacher who used a bad text to take a very bad shortcut.

          Finally, to the graph of 2^x, etc. Here, we’re pretty much stuck with cheating, and that cheating long predates the Magic idiot box (which makes the situation much worse, of course). You want the intuition of graphs as early as possible, and drawing an honest bunch of rational dots would be disastrous for that desired intuition. So, we cheat and draw a smooth line. But we can still be honest and say upfront that we’re cheating. And, if we are thoughtful and principled, we must say it.

          1. Are there any “good” textbooks (from the 1970s or so that might still be found) that have a decent treatment of index laws? I haven’t found anything satisfactory in the post 1990 offerings I have at home and work.

            1. Very good question, RF, and the surprising answer is “No, not really”. I’ll talk a bit about texts below, but also note that it doesn’t need a lot of text. It’s one simple message:

              Teacher: Why is a^0 =1?

              Students: Because we say so!

              Just keep hammering it, a reminder each time the identity (or a similar identity) arises. Students like it. They like the idea of humans having that (albeit limited) power to define things. And they get it.

              Sure, you can go further, and explain (and remind) that we define indices the way we do so the index “laws” remain true for new numbers. And that’s important, especially on first exposure, but it’s secondary. Primary is the fact that we’re defining, not deriving.

              Now to texts. The standard and very good 70s Year 11 text by Fitzpatrick and Watson gets it right, with a little wobbling. But lower level texts seem to do the fake proof stuff, just like today. Unfortunately, the evil David Treeby stole my mid-school texts of the era, so I can’t check much right now. But my memory is that even the great Bernie Fitzpatrick stuffed it up. Possibly 60s texts did it better, when maths education was infected with the New Math disease, but I’m not sure. More recently, the ICE-EM texts do it correctly.

              Of course, beyond the basic sense, one needs decent exercises on index laws (and everything) …

              1. OK. I have some pretty good Year 11 and 12 texts by F&B and also some US and British senior high school books that are quite good when a proof is required. Problem is (as for many things) in years 8, 9 especially, proof seems to be an alien idea. Not without reason, to a certain extent, but there is always going to be that one curious mind that I don’t want to be discouraging (or lying to).

                I like the ICE-EM texts a lot, especially for their size. I don’t see them in wide usage however.

        1. Hi Steve!

          I’m a little concerned by your statement “validate the power rule”. Surely when explaining to students how to calculate the derivative of x^n, you are careful about what x and n are, and state clearly the conventions in use.

          For instance, if n=0, you obtain d/dx x^n = 0 for x \ne 0. You don’t just blindly apply the formula.

          Also, for n=1, you obtain d/dx x^n = 1 for all x. Another example of not blindly applying the formula, unless we are very clear that in this formula only we are taking x^0 = 1 for all x. But, this just seems to be complicating the matter.

          Finally, for n < 0, we need to be careful about which x we are considering.

          Cheers
          Glen

        2. Thanks, Steve. That’s a very good example, similar to but more direct than my power series example. I agree with Glen, however. I’m uncomfortable with the word “validate” here.

            1. Hi Steve!

              I’m not going to touch L’Hopital’s :).

              The formula for differentiating powers can be shortened to include the case where the exponent is zero and this pairs nicely with a discussion about how we are using the zero power as shorthand for one in this case. Then you can say that actually in general the expression 0^0 can’t be assigned a meaningful value, which is why we were explicit in saying that for this formula specifically we are setting 0^0 = 1.

              Hope that helps.

              Cheers
              Glen

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