Posted by: groupact | March 10, 2013

Definitions and Subtleties

Ben Blum-Smith has a great post about honoring kids’ dissatisfaction. He notes that there are instances where questions or concepts spark heated discussion and kids aren’t satisfied with the mathematical resolution. His draws on two examples, “Is 1 prime?” and “Does 0.999… equal 1?”. He writes:

But in both situations, this would be to dishonor everyone’s dissatisfaction. It is so vital that we honor it. Everybody, school-aged through grown-up, is constantly walking away from math thinking “I don’t get it.” This is a useless perspective. Never let them say they don’t get it. What they should be thinking is that they don’t buy it.

And they shouldn’t! If it wasn’t already clear that I think the above “proof” that 0.999…=1 is bullsh*t, let me make it clear. I think that argument, presented as proof, is dishonest.

In both cases (that 1 is not prime, and that 0.999…=1) there are subtle issues at play that get into advanced mathematics and so kids are right to be dissatisfied with either an invalid proof or, in the case of 1 not being prime, an appeal to a definition that doesn’t explain why the definition should be written so as to exclude 1.

I agree wholeheartedly with Ben, but I would like to explore these issues further as I see the two examples that Ben gives as fundamentally different situations. In the case of whether 1 is a prime, Ben is correct that an appeal to the definition given by Wikipedia or a textbook does nothing to resolve the issue at hand. The real question is WHY do we define primes in such a way that 1 is not included. Before that can be addressed, though, it is important to have a discussion about definitions. Students do not generally view mathematical definitions as something over which WE (the mathematical world) have control. That is most students will not even realize that a CHOICE was made to exclude 1 from the definition. Once we can accept that, then it makes sense to talk about WHY the choice was made. Sometimes some of the reasons for a choice in a definition or convention might depend on an understanding of more advanced mathematics and it is okay to explain that to students, but as much as possible we should try to provide some understanding of the choice, either by appealing to secondary reasons or simplified versions of the deeper reasons. In the case of 1 not being a prime, Ben discusses (especially in the comments) how one key reason for the choice is a desire to have unique factorization, but that requires some time to discuss why we should care about unique factorization. He gives a great example of counting factors and another commenter, Sue, asks students to think about what would happen to our factor trees if we considered 1 a prime. More can be said on this subject, but the point I want to make today is that the most important thing to understand is that a CHOICE was made and that there are reasons for that choice.

 

Other questions that deal with this issue of how we should define things (and that could spark heated discussions) include, “Is a square a rectangle?”, “Is an equilateral triangle isosceles?”, “Is infinity a number?”, “Is a line parallel to iself?” In the first two cases I would suspect any mathematician would make the choice of YES. In the case of a line being parallel to itself I would certainly define parallel lines that way, but I could see others defining parallel so that a line is not parallel to itself. It is a case where there are arguments in favor of either definition. Some of those arguments (like the notion of an equivalence class, or non-Euclidean geometries) might depend on more advanced concepts, but students can realize that some choices there are no clear answers. In the case of infinity being a number we have a situation where the answer depends on the context. In the context of K-12 mathematics it is probably best to not consider it to be a number, but there are other contexts in higher math where we do want to consider infinity (and perhaps negative infinity) in our extended number system, and yet other contexts where we want to consider various levels of infinity!

The case of 0.999…=1, I see as being fundamentally different. I see two significant issues here. One is that the “proof” of it often given in middle school is fundamentally flawed because of the subtle notions of convergent series (as explained well by Ben). I don’t have a problem with sometimes using an argument in a K-12 class that relies on some subtle points that are beyond the scope of the class. Almost any proof involving area falls into this category. In more advanced mathematics one comes to realize that the whole notion of area and measure is not as clear cut as it would seem. (There are unmeasurable sets!) We often make the assumption in K-12 that we can cut shapes up, move them around, put them back together, and keep the same area. This is a non-trivial assumption, but I don’t mind using it in the classroom when necessary to show for example that the area of a parallelogram is equal to its base times its height. Perhaps part of the reason I’m willing to use it is that kids aren’t dissatisfied by the argument. The assumption seems reasonable. (Although for this reason and others I prefer that students focus on a proof of the Pythagorean Theorem that relies on similar triangles, and not rely on cutting up square).

The second (and more critical) issue with 0.999… is not just that the proof relies on some subtle and unstated assumptions, but that the whole definition of a non-terminating decimal is on shaky ground. Again, I’m sometimes willing to overlook this in the case of area because (1) the concept itself seems intuitive, and (2) I believe it is vital to have kids learning about and discussing the concept well before it can be formalized. I don’t see either of those conditions holding for a non-terminating decimal. The idea of an infinite sum does not seem intuitive to me, and nor do I see a great deal to be gained by introducing non-terminating decimals. (I stil think terminating decimals should be studied in middle school). I will follow up shortly with a post going into detail about why I think we would be better off delaying this topic, but for now I will just note that when we introduce definitions that rely on subtle concepts we should consider both how to introduce it and whether it is worth introducing it. Examples in this category include concepts such as π, e, circumference, and exponential functions. (In all of these cases I think the concepts should be introduced before they can be formalized, but we should acknowledge that there are subtle issues at play, and I’m still a little torn on e).

Ben has given me a lot to think about, but I’ll end this post by noting that the more that teachers are aware of the subtleties involved, the better they will be about leading interesting and worthwhile discussions about some of these issues raised in the classroom.

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Responses

  1. I really appreciate what you’ve written about area here, and my general inclination is to see nonterminating decimals the same way, in the sense that it’s useful for kids to get some familiarity with them before they are ready to engage all the subtleties. Except that the subtleties that have to be glossed over are less subtle. What I think is vitally important is that teachers know about these subtleties and be ready to acknowledge the validity of students’ doubts.

    In a way, I think that nonterminating decimals are inevitable in middle school, partly because they arise naturally from division by any number divisible by any prime other than 2,5. Also I think they’re a well-motivated preview of the idea of successively finer approximations, which will help people understand calculus later if they study it. But there’s no way the general issues of series convergence are going to be properly dealt with, so that means we as teachers have to be prepared to admit that we’re pulling a fast one. Even better, to cultivate kids’ sense that there is something fishy going on here, so that when 9 years later they take real analysis, they’ll actually be relieved to finally be getting the full story. (Or even better, take the kids who are pissed off about the bullshit in 0.3333… and start to do a little analysis with them in a free moment! Maybe wishful thinking…)

    The same issue comes up in calculus. From my experience teaching calculus at the high school level, I think that the \varepsilon-\delta definition of the limit is a bit much for a first calculus course at the high school level, especially if the kids don’t have prior experience with proof-based courses. (That definition has 3 nested quantifiers!) That said, that is the definition of the limit, so any other discussion of the limit is handwaving. What I regret when I look back on my calculus teaching in 2003-5 is not that I left out the \varepsilon-\delta limit, but the way I responded to kids who “didn’t buy” my more informal discussion. While I had studied real analysis, I didn’t yet understand it as a retrofitting of calculus’ logical back-end. Half-way through 2005 I read William Dunham’s beautiful book The Calculus Gallery and realized suddenly that when one of my seniors had complained after school the previous year that the whole thing was a paradox, he was actually echoing Bishop Berkeley’s 1735 complaint about the work of Newton and Leibniz, which type of thing eventually led to the development of real analysis in the 19th century. The big wish I have is that I had known enough then to be able to properly acknowledge the legitimacy of his complaint.

    • I still need to read The Calculus Galley; I’ve read most of Dunham’s other books. I agree that Berkeley’s criticism of Calculus at the time is a great example to illustrate honoring dissatisfaction. I promise more detail in subsequent posts on non-terminating decimals. My short response, though, is that while they arise from division of most of integers, that is ONLY if we insist on decimals. I would like to see us work more at developing comfort with fractions. What is 7 divided by 3? Well it’s a number that’s a little more than 2, but the nicest way to refer to that number is seven-thirds. What do we know about seven-thirds? We know it falls just to the right of 2 on the number line, but the most important thing we know about that number is that when we multiply it by 3, we get 7. So if we multiply it by 15 (5 times 3) we must get 5 times 7 or 35, etc. What if we wanted to get a decimal approximation? Well we could perform long division, but the process would never end, but the further we go on, the better approximation we would get. Even with terminating decimals, sometimes they don’t terminate for awhile and we might wish to stop before the end of the process if we’re satisfied with our approximation. In this manner, I think students would get a better sense of finer approximations AND would have more comfort later with transcendental functions. What is log base 2 of 7? Well it’s a number that’s a little less than 3, but the nicest way to refer to that number is as log base 2 of 7. Sure a calculator could get us a better decimal approximation, but a little less than 3 is good enough for most purposes and the most important thing about that number is that when we take 2 and raise it to that number, we get 7. So if we took 8 and raised it to that number we’d get 7 cubed or 243.

  2. […] Nicole (and Chris) that debatable questions can be quite compelling.  One source for that can be definitions.  For example, “Is a line parallel to itself?”  Students are shocked to learn that […]


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