Thanks for this comment; it advances my thinking. The example of “suppose I throw a stone into a window…” is illuminating.

You’ll be unsurprised that I disagree (rather strongly) that this “isn’t fundamentally different” from proof objects; I think the difference is fundamental enough to write a blog post about. But perhaps this type of hypothetical language is the (what I would call) imported capacity from which the enculturated capacity of thinking in proof objects is built! I look forward to thinking more about this.

Of course adding the words “arbitrary, but fixed” before an object in a proof changes nothing mathematically. Indeed, I don’t think I have ever seen this phrase in a proof. The phrase “arbitrary, but fixed” is used in the blog post (and in the article by Selden & Selden mentioned in note [9], and undoubtedly many other places) not as something you’d actually write in a proof but as a descriptor of the (ontological? psychological?) character of proof objects. Elsewhere, Susanna Epp uses the descriptor “generic particular” for the same phenomenon (I thank Japheth Wood for directing me to Dr. Epp’s writings and this phrase in particular).

The question is whether or not this “arbitrary but fixed”/”generic particular” character of proof objects is distinct from the character of the mental objects of every day life. The argument of the blog post is that it is very distinct, with the implication that it’s useful for educators to focus our attention on how students come to master the technique of working mentally with objects possessing this character.

You disagree that this involves learning anything new; if you’re right, then there’s nothing to see here. Your example does not convince me of this (more below). But the continuity you see, between proof objects and the objects of everyday speculative hypotheticals, strikes me as useful to the project I’m proposing, in at least two ways: (1) this continuity directs us toward a *place to look* for an understanding of the path students take toward competent handling of proof objects; and (2) it also gives a clue for how one might build on students’ (what I would call) imported capacities in taking them along this path.

Now for the substance of my disagreement. I think there is a great psychological distance between “suppose I throw a stone into a window” and “let v be a vector of minimal length in a lattice…,” and it goes far beyond the habit of naming. To me the key point is this: the stone and the window are not under a burden to be universal within the classes of stones and windows. Say the grownup says, “Don’t throw a stone into a window because it’ll break,” and the child replies, “What if it’s is made of bulletproof glass?” or better yet, “What if it’s the window of a giant’s house and it’s the size of Pluto?” In all likelihood, all involved will understand that the child is being cheeky. In the rare case they are being earnest, the grownup will at least understand they are having a different conversation than the intended one. In contrast, when the analogous thing happens in a proof context, then the audience is making a good point! And perhaps the hypotheses of the theorem, or the proof strategy, need to be adjusted in response.

This is happening because the “stone” and the “window” of the everyday hypothetical are not precisely defined categories. It’s okay to have the context cue us about what kind of stone, what kind of window. We don’t have to handle our conversation in a way that encompasses everything that could rightly be called a stone and everything that could rightly be called a window. Context plays an important role in mathematical communication too; still, we make a great effort to minimize that role, with our lovingly crafted definitions. Because of this, mental images of proof objects have a much brighter line delineating their constitutive attributes from their contingent ones. When I imagine a window, I wouldn’t even know where to begin in sorting out the part of my mental image that “makes it a window” from everything else in my mind’s eye. (Well, it’s a hole in a wall of a house. Well, but so is a door. Ok, well, its purpose is to look out of. Ok well but is it still a window if it’s too high for my head to reach? In any case, why a house? What about a train window? Ok fine, it’s a hole in the wall of a dwelling or vehicle that’s designed to let light in. Well, but is it still a window if it’s on a subway train that never gets ambient light? Etc.) When I imagine a vector of minimal length in a lattice, on the other hand, it’s very easy for me to identify what makes it a vector of minimal length. It’s the bright line separating the constitutive from the contingent that makes the proof object different.

]]>Seems like you already found your way to my related post (https://blogs.ams.org/matheducation/2020/05/20/the-things-in-proofs-are-weird-a-thought-on-student-difficulties/) on the strangeness of the generic particular! (I’ve never heard this phrase before, thank you for introducing it to me!)

I’m extremely interested in the question of how students develop an understanding of this idea and what steps instructors can take to support this development (cf. that other post). The quote from Susanna Epp you give directs students to the right path to take, but there’s also the question of how a student gets convinced that that’s the right path. The tack I took with Ricky, described in note [14], did make some progress with that particular learner, but this is just one trick that worked a little bit, in one context. I’m interested in developing a more comprehensive map of the landscape of learning (in the sense the phrase is used by Cathy Fosnot in her “Young Mathematicians at Work” books) involved in developing this understanding. Excited to be in conversation with you about this!

]]>Thanks for pointing out this reference! For interested readers, here’s the url for convenience:

https://condor.depaul.edu/~sepp/NCTM99Yrbk.pdf

Looks like it has a nice discussion of differences between the linguistic conventions of mathematical writing and everyday language.

]]>I already commented on your epsilon-delta post that I’m a fan of Professor Susanna Epp’s writing on teaching and learning proof. She has a nice article titled “The Language of Quantification in Mathematics Instruction” on her webpage that offers similar ideas. ]]>

Method of Generalizing from the Generic Particular

To show that every element of a set satisfies a certain property, suppose x is a particular but arbitrarily chosen element of the set, and show that x satisfies the property.

(from Discrete Mathematics with Applications by Susanna S. Epp)

Anecdotally, I observed that my students with an understanding of “generic particular” were able to produce more coherent proofs of universally quantified statements, and to understand the proofs process in deeper way.

]]>The title of the text is “Modern Mathematics for Junior High School Book 2” the publisher was Silver Burkett. I believe that I understood the use of the projector as an attempt to model negative values of time in the d=rt formula, but there were the wake of the boat and the eyes of the pilot pointing in the real direction of time, and where were the models for the negative areas in the A=lw formula? This just did not seem like math. By the end of ninth grade, with the help of a very dedicated teacher and without ever hearing the terms group or integral domain I understood that the integers were a system which expanded the natural numbers in a manner that included an additive identity and inverses, was closed under addition, in which the non commutative operation of subtraction was not needed and that if the commutative properties of addition and multiplication along with the distributive property were to be preserved in the extension then a negative times a negative must be a positive.

]]>Thanks for your calling my attention to this book, and for your many years of serious engagement with elementary mathematics! I hope it’s apparent from the post that I’d like to see everyone (mathematicians, teachers, teacher educators, etc.) attend with loving care to the logical structure of elementary mathematics, and you’ve been doing that for a long time. Any bibliography of the kind I evaded in note [3] would include numerous of your publications, certainly including the book in question.

(One could be forgiven for the misimpression that you’re asserting that everything contained in the post is already in your book. It seems to me our treatments of theorem 2 of the post are essentially the same in their logical content [the equality of the quotative and partitive models of division is a consequence of commutativity], but our treatments of theorems 1 and 3 are very different. Your proof of theorem 3 is very nice!)

]]>Thanks for this. What was the text?! And maybe it’s a conversation for another place and time, but I’m now curious how you made sense of the idea that the boat analogy had anything to do with multiplication?

]]>Of course you’re right that it’s not a ÷ b that’s a binary operation, but just ÷. Perhaps I was cavalier in my “how could they be?” comment.

The magic you refer to, namely understanding the equivalence of interpreting the slash as a division vs. as a fraction, is the whole point of #3. That’s the theorem! If you like, you can forget that I brought the obelus into it at all. As you say, it’s possible to have the whole conversation without it. I included it both for the sake of the section 3 lede, and more importantly because without it, I’m forced to state the theorem as a/b = a/b, with one side denoting a fraction and the other a division. Written this way, the notation serves to hide the underlying theorem. The fact that the slash has both meanings packages the theorem into the notation, simultaneously encoding it and rendering it invisible.

(This is what I was referring to in the post when I said that using the slash to denote division is built on, but also elides, the theorem in question.)

There are other instances of this, where we package theorems into our notational conventions, and I think it often makes for “good notation” in the sense that the notation reflects the underlying mathematical situation. The first examples that come to mind for me are from calculus. Leibniz notation for the derivative, dy/dx, makes the chain rule look like an evident fact about fraction multiplication. It’s not actually a fraction multiplication, but we can get away with this because we can prove the chain rule in other ways. Meanwhile, definite and indefinite integrals are written with the same symbol even though modern math understands them as two completely different things: one is a limit of Riemann sums (or their Lebesgue analogues), while the other is an antiderivative. We can get away with this because of the Fundamental Theorem of Calculus.

At the same time that these notations are “good” for doing math with, I think they can also conceal interesting ideas from learners, precisely because they hide the theorem they encode. Reflecting on my own experience of learning calculus, I think it took me a long time to appreciate the details involved in proving the chain rule, precisely because the notation makes it look so obvious. It also took me a long time to appreciate the vastness of the conceptual difference between definite and indefinite integrals.

All this is to say: my purpose in writing #3 is to call attention to the theorem that reconciles the two interpretations of the slash (as fraction bar and as division). Because you acknowledge this as “magic”, I assume we agree that it’s a theorem. I don’t think this theorem gets its due, hence its inclusion here. Let me add a speculation: I suspect part of the reason it doesn’t get its due is that we use the same symbol for both meanings, a convention that encodes this theorem but in the same moment hides it from view.

The challenge of getting used to the fact that our symbols also have synonyms, such as the obelus vs. the slash (in its division meaning, not as a fraction bar), the times sign vs. juxtaposition for multiplication, and the fact that we sometimes write a calculation horizontally and at other times vertically, is also interesting and important, as you say. It’s not particularly the subject of this post, though — hopefully this digression has helped to isolate the point I’m trying to make.

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