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.