{"id":3076,"date":"2020-05-20T12:41:47","date_gmt":"2020-05-20T16:41:47","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=3076"},"modified":"2021-10-27T19:33:12","modified_gmt":"2021-10-27T23:33:12","slug":"the-things-in-proofs-are-weird-a-thought-on-student-difficulties","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2020\/05\/20\/the-things-in-proofs-are-weird-a-thought-on-student-difficulties\/","title":{"rendered":"The things in proofs are weird: a thought on student difficulties"},"content":{"rendered":"

By Ben Blum-Smith, Contributing Editor<\/a><\/i><\/p>\n

“The difficulty… is to manage to think in a completely astonished and disconcerted way about things you thought you had always understood.” \u2015 Pierre Bourdieu, Language and Symbolic Power<\/i>, p. 207<\/p><\/blockquote>\n

Proof is the central epistemological method of pure mathematics, and the practice most unique to it among the disciplines. Reading and writing proofs are essential skills (the<\/i> essential skills?) for many working mathematicians.<\/p>\n

That said, students learning these skills, especially for the first time, find them extremely hard<\/i>.[1<\/a>]<\/sup><\/p>\n

Why? What’s in the way? And what are the processes by which students effectively gain these skills?<\/p>\n

These questions have been discussed extensively by researchers and teachers alike,[2<\/a>]<\/sup> and they have personally fascinated me for most of my twenty years in mathematics education.<\/p>\n

In this blog post I’d like to examine one little corner of this jigsaw puzzle.<\/p>\n

<\/p>\n

Imported vs. enculturated<\/b><\/h4>\n

To frame the inquiry, I posit that there are imported<\/i> and enculturated<\/i> capacities involved in reading and writing proofs. Teachers face corresponding challenges when teaching students about proof.<\/p>\n

Capacities that are imported<\/i> into the domain of proof-writing are those that students can access independently of whether they have any mathematics training in school or contact with the mathematical community, let alone specific attention to proof.[3<\/a>]<\/sup> Capacities that are enculturated<\/i> are those that students do not typically develop without some encounter with the mathematics community, whether through reading, schooling, math circles, or otherwise. Examples of imported capacities are the student’s capacity to reason, and fluency in the language of instruction. Enculturated capacities include, for example, knowledge of specific patterns of reasoning common to mathematics writing but rare outside it, such as the elegant complex of ideas behind the phrase, “without loss of generality, we can suppose….”<\/p>\n

For imported capacities involved in proof, the teaching challenge is to create conditions that cause students to actually access those capacities while reading and writing proofs.<\/p>\n

For enculturated capacities, the prima facie teaching challenge is to inculcate them, i.e., to cause the capacities to be developed in the first place. But there is also a prior, less obvious challenge: we have to know they’re there. Since many instructors are already very well-enculturated, our culture is not always fully visible to us. If we can’t see what we’re doing, it’s harder to show students how to do it. (This challenge has the same character as that mentioned by Pierre Bourdieu in the epigraph, although he was writing about sociology.)<\/p>\n

When my personal obsession with student difficulty with proofs first developed, I focused on imported capacities. I had many experiences in which students whom I knew to be capable of very cogent reasoning produced illogical work on proof assignments. It seemed to me that the instructional context had somehow severed the connection between the students’ reasoning capacities and what I was asking them to do. I became very curious about why this was happening, i.e., what types of instructional design choices led to this severing, and even more curious about what types of choices could reverse it.<\/p>\n

My main conclusion, based primarily on experience in my own and others’ classrooms, and substantially catalyzed by reading Paul Lockhart’s celebrated Lament<\/i><\/a> and Patricio Herbst’s thought-provoking article<\/a> on the contradictory demands of proof teaching, was this: It benefits students, when first learning proof, to have some legitimate uncertainty and suspense regarding what to believe, and to keep the processes of reading and writing proofs as closely tied as possible to the process of deciding what to believe.[4<\/a>]<\/sup><\/p>\n

I stand by this conclusion, and more broadly, by the view that the core of teaching proof is about empowering students to harness their imported capacities (in the above sense) to the task, rather than learning something wholly new. That said, in the last few years I’ve become equally fascinated by the challenges of enculturation that are part of teaching proof reading and writing. If I’m honest, my zealotry regarding the importance of imported capacities blinded me to importance of the enculturated ones.<\/p>\n

What I want to do in the remainder of this blog post is to propose that a particular feature of proof writing is an enculturated capacity. It’s a feature I didn’t even fully notice until fairly recently, because it’s such a second-nature part of mathematical communication. I offer this proposal in the spirit of the quote by sociologist\/anthropologist Pierre Bourdieu in the epigraph: to think in a completely astonished and disconcerted way about something we thought we already understood. Naming it as enculturated has the ultimate goal of supporting an inquiry into how students can be explicitly taught how to do it, though this goes beyond my present scope.<\/p>\n

The things in proofs are weird<\/b><\/h4>\n

I recently encountered an article by Kristen Lew and Juan Pablo Mej\u00eda-Ramos, in which they compared undergraduate students’ and mathematicians’ judgements regarding unconventional language used by students in written proofs.[5<\/a>]<\/sup> One of their findings was that, in their words, “… students did not fully understand the nuances involved in how mathematicians introduce objects in proofs.” (2019, p. 121)<\/p>\n

The hypothesis I would like to advance in this post is offered as an explanation for this finding, as well as for a host of student difficulties I’ve witnessed over the years:<\/p>\n

The way we conceptualize the objects in proofs is an enculturated capacity.<\/i><\/p>\n

These objects are weird<\/i>. In particular, the sense in which they exist, what they are<\/i>, is weird. They have a different ontology than other kinds of objects, even the objects in other kinds of mathematical work. An aspect of learning how to read and write proofs is getting accustomed to working with objects possessing this alternative ontology.[6<\/a>]<\/sup> If this is true, then it makes sense that undergraduates don’t quite have their heads wrapped around the way that mathematicians summon these things into being.<\/p>\n

The place where this is easiest to see is in proofs by contradiction. When you read a proof by contradiction, you are spending time with objects that you expect will eventually be revealed never to have existed<\/i>, and you expect this revelation to furthermore tell you that it was impossible that they had ever existed<\/i>. That’s bizarro science fiction on its face.<\/p>\n

But it’s also true, more subtly perhaps, of objects appearing in pretty much any other type of proof. To illustrate: suppose a proof begins,<\/p>\n

Let \"\Lambda\" be a lattice in the real vector space \"\mathbb{R}^n\", and let \"v\in be a nonzero vector of minimal (Euclidean) length in \"\Lambda\"…<\/p><\/blockquote>\n

Question. What kind of a thing is \"v\"?<\/i><\/p>\n

[The camera pans back to reveal this question has been asked by a short babyfaced man wearing a baseball cap, by the name of Lou Costello. His interlocutor is a taller, debonair fellow with a blazer and pocket square, answering to Bud Abbott.]<\/p>\n

Abbott: It’s a vector in \"\mathbb{R}^n\".
\nCostello: Which vector?
\nAbbott: Well, it’s not any particular<\/i> vector. It depends on \"\Lambda\".
\nCostello: You just said it was a particular vector and now it’s not a particular vector?
\nAbbott: No, well, yes, it’s some vector, so in that sense it’s a particular vector, but I can’t tell you which one, so in that sense it’s no particular vector.
\nCostello: You can’t tell me which one?
\nAbbott: No.
\nCostello: Why not?
\nAbbott: Because it depends on \"\Lambda\". It’s one of the vectors that’s minimal in length among nonzero vectors in \"\Lambda\".
\nCostello: Which vector?<\/i>
\nAbbott: No particular<\/i> vector.
\nCostello: But is it some vector?
\nAbbott: Naturally!<\/p>\n

Costello: You said it depends on \"\Lambda\". What’s \"\Lambda\"?
\nAbbott: A lattice in $\\mathbb{R}^n$.
\nCostello: Which lattice?
\nAbbott: Any lattice.
\nCostello: Why won’t you say which<\/i> lattice?
\nAbbott: Because I’m trying to prove something about all lattices<\/i>.
\nCostello: You mean to say \"\Lambda\" is every lattice???<\/i>
\nAbbott: No, it’s just one lattice.
\nCostello: Which one?!<\/i><\/p>\n

For any readers unfamiliar with the allusion here, it is to “Who’s on First?”<\/i>, legendary comedy duo Abbott & Costello’s<\/a> signature routine.[7<\/a>]<\/sup> What’s relevant to the present discussion is that the skit is based on Costello asking Abbott a sequence of questions about a situation to which he is an outsider and Abbott is an insider. Costello becomes increasingly frustrated by Abbott’s answers, which make perfect sense from inside the situation, but seem singularly unhelpful from the outside. Abbott for his part maintains patience but is so internal to his situation\u2014enculturated, as it were\u2014that he doesn’t address, or even seem to perceive, the ways he could be misunderstood by an outsider.[8<\/a>]<\/sup><\/p>\n

My goal with this little literary exercise has been to dramatize the strangeness of the “arbitrary, but fixed” nature of the objects in proofs. Most things we name, outside of proof-writing, don’t have this character. Either they’re singular or plural; one or many; specific or general; not both. Every so often, we speak of a singular that represents a collective (“the average household”, “a typical spring day”), or that is constituted from a collective (“the nation”), but these are still ultimately singular. They are not under the same burden as mathematical proof objects, to be able to stand in for any member of a class. Proof objects aren’t representative members of classes but universal<\/i> members. This makes them fundamentally unspecified, even while we imagine and write about them as concrete things.<\/p>\n

There’s an additional strangeness: proof objects, and the classes of which they are the universal members, are themselves often constituted in relation to other proof objects. We get chains, often very long, where each link adds a new layer of remove from true specificity, but we still treat each link in the chain, including the final one, as something concrete. I was trying to hint at this by posing the question “what is it?” about \"v\", rather than \"\Lambda\", in the example above. As consternated as Costello is by \"\Lambda\", \"v\" is doubtless more confounding.<\/p>\n

I think there are at least two distinct aspects of this that students new to proof do not usually do on their own without some kind of enculturation process. In the first place, the initial move of dealing with everything in a class of objects simultaneously by postulating a “single universal representative” of that class just isn’t automatic. This is a tool mathematical culture has developed. Students need to be trained, or to otherwise catch on, that a good approach to proving a statement of the form “For all real numbers…” might begin, “Let \"x\" be a real number.”[9<\/a>]<\/sup><\/p>\n

But secondly, when we work with these objects, their “arbitrary, but fixed” character forces us to hold them in a different way, mentally, than we hold the objects of our daily lives, or even the mathematical objects of concrete calculations. When you read, “Let \"f\" be a smooth function \"\mathbb{R}\rightarrow\mathbb{R}\",” what do you imagine? A graph? Some symbols? How does your mental apparatus store and track the critical piece of information that \"f\" can be any<\/i> smooth function on \"\mathbb{R}\"? Reflecting on my own process, I think what I do in this case is to imagine a vague visual image of a smooth graph, but it is “decorated”\u2014in a semantic, not a visual, way\u2014by information about which features are constitutive and which could easily have been different. The local maxima and minima I happen to be imagining are stored as unimportant features while the smoothness is essential. Likewise, when I wrote, “Let \"\Lambda\" be a lattice in the real vector space \"\mathbb{R}^n\",” what did you imagine? Was there a visual? If so, what did you see? I imagined a triclinic lattice<\/a> in 3-space. But again, it was somehow semantically “decorated” by information about which features were constitutive vs. contingent. That I was in 3 dimensions was contingent, but the periodicity of the pattern of points I imagined was constitutive. I’m positing that students new to proof do not usually already know how to mentally “decorate” objects in this way.[10<\/a>]<\/sup><\/p>\n

Examples<\/b><\/h4>\n

Here are some specific instances of student struggle that seem to me to be illuminated by the ideas above.<\/p>\n