This question is motivated by student responses to homework and quiz problems I have recently posed in an undergraduate real analysis course. I will share some examples and observations first, to present the main ideas, before posing a question.

  1. A homework problem asked students to consider an arbitrary set $A\subseteq\mathbb{R}$ that is nonempty and bounded below. From that set, define $B=\{b\in\mathbb{R}\mid b\text{ is a lower bound for }A\}$. The students were asked to prove that $\sup(B)$ must be equal to $\inf(A)$. I noticed two kinds of mistakes. They were not widespread, but they were common enough that I made comments about them during class time.

    • Proof by example, such as: "If $A=(2,\infty)$ then $B$ must be $(-\infty,2]$ and notice that $\inf(A)=2$ and $\sup(B)=2$." This is a gross misunderstanding of arbitrary versus specific.
    • Referring to "$b$" without instantiation, such as: "$b$ is a lower bound for $A$ so all $a$s are upper bounds for $B$." This contains a kernel of truth (indeed, every element of $B$ is a lower bound for $A$) but it demonstrates a misunderstanding of a universal assertion versus a specific instance.
  2. A quiz problem asked the following: "Suppose $A\subseteq\mathbb{R}$ is nonempty and bounded above. Let $s=\sup(A)$, and let $\varepsilon > 0$ be arbitrary. What can we say about the interval $(s-\varepsilon,s]$? How many elements of $A$ lie in that interval? None? 1? At least 1? Infinitely many? What can we say with certainty? Consider illustrating with examples."

    • I expected all students to at least point out that there cannot be zero elements of $A$ in the interval $(s-\varepsilon,s]$; if so, that would violate the fact that $s$ is the least upper bound for $A$.
    • I expected the better students to further point out that $s$ may be the only element of $A$ in that interval. For example, if $A=\{2\}$ is a singleton, then this is true regardless of $\varepsilon$.
    • I also expected the better students to further point out that there may be infinitely many elements of $A$ in that interval. For example, if $A=[0,2]$, then this is true regardless of $\varepsilon$.
    • Instead, more than half of the class submitted an answer that amounted to nothing more than saying "there are infinitely many real numbers in any non-trivial interval." Zero consideration or even literal mention of that arbitrary set $A$. They treated this as if the original question was: "Does an interval have infinitely many elements?"

This worries me for several reasons. One of the main reasons is that I think I have an "expert blind spot" regarding the ability to consider and work with an arbitrary mathematical object because that practice is so natural to me. I'm truly baffled why a student would consider an example as a reasonable solution to problem #1, yet I'm sure there was a time in my educational career where I would have easily made the same error without yet understanding the big issue.

But problem #2 worries me the most. Is there something about the way I posed the question that obscured its intent so much that students could have genuinely thought I was cryptically assessing them on their knowledge of the infinitude of the reals? I think that they lack strong skills in mentally working with an arbitrary set, that they need to consider specific examples to make mental progress, and that they are not yet in the habit of responding to a question like this by creating their own examples to consider while reflecting on the problem.

Question: How can I, as an instructor, help students to work with problems like the examples above, knowing that the students do not have good, practical habits for working with arbitrary objects? Beyond simply telling them to do so in the moment, how can I encourage them to create and test examples for themselves, to not "over/under assume" about what they're given, or otherwise just to correctly interpret given information about an arbitrary object?

Upon reflection, I seem to have addressed this in the past by just trying to "model good behavior" when presenting in-class examples and when working with students one-on-one. However, I wonder whether there are particular activities I could use, or problem types to assign and assess, that would better promote the kind of thinking and behavior that I want my students to develop.

A good answer will contain suggestions for activities or assessment tools. If there is any research about this, I would love to know about it, as well. If it helps narrow the scope, I am especially interested in undergraduate math majors learning to write proofs in their advanced courses (e.g. real analysis, linear algebra, etc.).

Meta comment: I could not find a good tag that properly encapsulates the main issue of this question. Would "abstraction" be a reasonable tag to create? This question is mostly about a student's ability to abstract from specific cases to general concepts, and I'm sure there are and will be other questions related to that ability.

Followup comment: I want to give a shoutout to #2 above as a really good problem to give students when learning about suprema before learning about sequences. Today, in my class, we proved the Monotone Convergence Theorem by defining the proposed limit to be the supremum of the set of the terms of the sequence. Given an arbitrary $\varepsilon>0$, I asked the students: "What can we say about the interval $(s-\varepsilon,s]$?" Some students laughed upon realizing this is precisely the quiz question from a few weeks ago and said, "There must be at least one element in there!"

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    $\begingroup$ The test-question in #2 does seem open-ended and ambiguous and I'd be prone to tightening that up (others will differ). However, I'm nigh-terrorized by the proof-by-example in #1; that says someone doesn't even get the idea of proofs -- clearly unprepared for an analysis course. Does your institution have a intro-proofs-transition course, or does it lack that? $\endgroup$ – Daniel R. Collins Feb 14 '18 at 1:54
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    $\begingroup$ @DanielR.Collins I agree with tighting up number question 2. You could ask: Is it possible for $A \cap (s-\epsilon,s]$ to have $0$, $1$, $2$, or infinitely many elements? If the possibility is allowed, illustrate with an example. If is is impossible, explain why you think so. $\endgroup$ – Steven Gubkin Feb 14 '18 at 14:15
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    $\begingroup$ @DanielR.Collins: I agree that #2 was challenging and open-ended, but I hope it is not considered ambiguous, although I do like Steven Gubkin's wording better and will use that next time. $\endgroup$ – Brendan W. Sullivan Feb 14 '18 at 17:13
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    $\begingroup$ @DanielR.Collins: Yes, we have a "Discrete Methods" course that teaches proof-writing and logic. And I taught that particular student last year in that course and tried to nip that habit in the bud back then. I don't know why it persists still. $\endgroup$ – Brendan W. Sullivan Feb 14 '18 at 17:14
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    $\begingroup$ I think the word “let” is being used in two subtly different ways in the passage “Let s=sup(A), and let ε>0 be arbitrary.” This may be related to the students’ difficulties. $\endgroup$ – James Propp Feb 14 '18 at 23:27

I'll focus on question 2 from a perspective of "maybe the right thing to think about is: what happens in the students' minds while they read this question?"

When you say "Suppose $A⊆R$ is nonempty and bounded above," the following thing happens in my mind before I continue reading:

enter image description here

This picture takes up very little working memory for me because I am familiar with how I can mess with it. This may not happen in their minds. They may think of $A$ as a set that follows arcane rules. After all, the set is arbitrary. It may feel pointless to think about the set at this point. Have the students practiced how to read a quantified phrase like this? Maybe you are assuming they know this from a previous course.

Then you say "Let $s=sup(A)$, and let $ε>0$ be arbitrary."

My new picture is this, and it's important that even though $\epsilon$ is a real number, I haven't placed it on the real line, because the use of the letter $\epsilon$ implies to an expert that this would be the wrong way to think about this specific real number.

enter image description here

Then you say "What can we say about the interval $(s−ε,s]$?"

We are deep into multiple objects at this point. A weaker student's picture may look as poor as this right now:

enter image description here

The student is suspicious that they have missed something (and they have; they didn't notice that the way $s$ was introduced is important) but their mental picture is highly deficient, so they said the best thing they could with what they were working with: "there are infinitely many real numbers in any non-trivial interval."

Basically I agree those answers are terrible. But investigating more explicitly with the students what should happen in their brains when they read a sentence with multiple quantifiers would be the in-class exercise I suggest. They may not realize that every time a variable is quantified, they should now be able to think about it or even have a good schematic picture of it.

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    $\begingroup$ This is a wonderful explanation. (+1) That said, I might complain that the colors your chose are red and green, which are the most likely to cause trouble for people who are colorblind. $\endgroup$ – Xander Henderson Feb 15 '18 at 1:47
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    $\begingroup$ I have known some students for whom even thinking about the question in the way you've suggested would be too taxing for their working memory. These are a minority, but a substantial enough minority to consider. Do you have any suggestions for working with these students? $\endgroup$ – Alexander Woo Feb 15 '18 at 4:27
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    $\begingroup$ @AlexanderWoo I guess I do not know any other way to think about the question. Instructors with more experience teaching abstract math courses could maybe talk more about it. I'm mostly coming from the perspective of teaching about $\epsilon$ and $\delta$, where instructing students to draw pictures of things as they gradually appear can help a lot. The technique seemed relevant here as well. My best guess is they should use paper (literally draw some sample set "A"s before even finishing reading the problem) to offset limited working memory. $\endgroup$ – Chris Cunningham Feb 15 '18 at 13:54

Can all of your students correctly explain what the supremum and infimum of a set are? If not, they will be unable to reason about these concepts. It would be like expecting a student to do something with the order of an element in an abstract (arbitrary) group if they can't define what the order if an element is (e.g., thinking "order of $g$ is $n$" means only that $g^n = 1$).

  • $\begingroup$ I thought so! They all can correctly identify the inf and sup of an example set, e.g. a homework problem: $\{\frac{m}{m+n}\mid m,n\in\mathbb{N}\}$. And they've certainly had enough time to digest the concepts. That quiz question #2 was given after 3 class meetings about the Axiom of Completeness and its consequences. $\endgroup$ – Brendan W. Sullivan Feb 15 '18 at 19:42

Believing that they can infer that something is true for all real numbers based on a few examples points to lack of understanding of basic methods of proof, specifically how we go about making generalizations in mathematics.

May I humbly suggest my interactive, self-study tutorial in the basic methods of proof and accompanying proof checker that provides students with immediate feedback. Not a single invalid line of proof will be accepted by the software.

In my tutorial, I introduce students to the following basic methods of proof with simple, to-the-point worked examples plus exercises with hints and full solutions:

  1. Direct Proof
  2. Proof by Contradiction
  3. Proof by Cases
  4. Proof of Biconditionals
  5. Manipulating Quantifiers -- including making universal and existential generalizations.
  6. Proof by Induction
  7. Very Elementary Set Theory (not ZFC)
  8. Very Elementary Number Theory

Visit my homepage for more information, testimonials, and a free, fully-functional download including the above mentioned tutorial.

  • $\begingroup$ This looks interesting and I will explore in more detail later. Thanks for sharing. $\endgroup$ – Brendan W. Sullivan Feb 14 '18 at 22:53

If you buy into the notion of "learning outcomes" over "content," then maybe "develop ability to work with arbitrary mathematical objects" is one of the learning outcomes of this class. Your colleagues in the department will understand this, even if administrator-level assessment gurus don't.

In my mind this developmental learning goal is one of the reasons we teach analysis. It's just messy and hard, and your question accents this. This is a challenging and important class. Some obvious mechanical checks: does the class have a helpful prerequisite, for example a course in linear algebra with some abstraction and where some proof is introduced, and is class size reasonable? Instructor time with individual students is crucial in this course, so I don't think class size should exceed 15. Something like 10 students is better. Then the usual list of strategies should be employed depending on instructor judgement: students presenting solutions/proofs to one another in class, structured study groups, rewriting proofs and problem solutions after initial review by instructor, etc.

  • $\begingroup$ Class size is 12. Problem #1 in OP was assigned for homework, I wrote comments in response to student submissions, and gave them 5 days thereafter to edit and resubmit for full credit. The "proof by example" was a student's edited submission :-( $\endgroup$ – Brendan W. Sullivan Feb 14 '18 at 19:08

Suppose the conversation begins with the secret goal of tricking students into discovering sup for themselves, to own the idea, and apply it usefully at will.

A good 6th grader might understand a number as a thing that has a decimal representation, possibly with infinitely many digits to the right of the decimal place, but only finitely many to the left.

S/he can already calculate a bit, and perhaps grasps, at least implicitly, that subtraction is trying to measure the geometric distance between two numbers.

Now majoring in math, perhaps our students have at least a very basic grasp of the basic basics of sets. A set of real numbers partners with a bunch of dots on the real line.

Does every finite set of numbers have a largest element?

( Of course! Just write them all down and pick the biggest one, or pick the rightmost dot in the picture!)

Does every infinite set of numbers have a largest element?

( Of course not! 1,2,3,...)

Nice! But that's too easy! What if all the numbers in the set are at most 10?

(Wheels turning....Ah! The set 9, 9.9 , 9.99,... has no largest element!)

Great! So what's going on here? How can we talk about the ``largest element'' when it might not belong to the set?

(Now you have tricked them into caring about the idea of sup... they are ready to try to parse the official definition...and they can begin to teach themselves... and each other.... )

The mathematical upshot is that students can already "see" that the least upper bound axiom should true.

( Indeed there is something like an algorithm for "finding" sup of a set with an upper bound that a 6th grader could in principle parse 1) Start with the original numbers, for each number throw away everything to the right of the decimal and pick the biggest number among the new numbers 2) Start with the original numbers and throw away everything to the right of the 1/10 place, and pick the biggest new number. Of course the sequence we're building is piling up on the sup).

The previous paragraph thinks it's a computer program that can "calculate" sup(A) for any bounded set. This could serve as a crutch for students who are used to thinking of mathematics as all about calculation.

In general `Getting good at proofs'' is hard, a special case of which is related to the question at hand ``getting good at exercises related to material involving inf and sup."

``Getting good at proofs'' is hard for most of us. Here's one reason, perhaps a perpetual reason, why.

Given some mathematical assumptions and hoped for conclusion, there is a human tendency to try and "classify" every instance of the hypotheses and then "check" every case. If the starting data is a set of real numbers, that could be a lot of "classifying" and "checking".

But there is an antidote. I remember once as an undergraduate, after failing miserably to really understand and especially classify the hypotheses, I was startled to see that one could still reach the desired conclusion, just by using the definitions and assumptions in the hypothesis!

There was no need to try to imagine what every possible case ``looks like''.

Having said that, it is of course an extremely useful tactic to try and conjur up specific examples which satisfy a given hypotheses --- it can make it much easier to see the REASON a given conclusion might hold more generally.

False starts, gratuitous detours, leaps of faith posing as logic, will happen to all students -- especially the good students, since they are at least trying.

In general mastery of abstraction is hard, even when the material is ultimately elementary. Yoneda Lemma anyone?

The important thing for the instructor is to be ceaselessly positive,encouraging, and tolerant, and to be able to see where/why a given student might be trapped, and offer a way out, at least pointing towards a door.

The important thing for the student is to never...ever give up.


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