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.
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.
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!"
