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tl;dr: Some students expect to be told "what's on the test", to memorize and then move on. What can be done to change how they learn while teaching them what to learn?

Context: Introductory, foundational course for undergraduates, like College Algebra or Precalculus or even Calculus I, where most of the students are science majors and must complete some portion of the Calculus sequence for their department.

Observation: It seems as though many younger undergraduate students have difficulty learning mathematics because of the way they were taught in high-school, where the focus was more on rote, procedural knowledge and "teaching to the test". In college/university, the focus is much more on conceptual understanding, and it is assumed that students will work independently outside of class to enhance their understanding. In addition, students will be assessed on that conceptual understanding and will not merely be asked to regurgitate memorized formulas or solve problems identical to some model problem.

Main question: What can I do, as a teacher, to help students break this mold and get used to learning mathematics in a deeper way (and being assessed on that deeper understanding), while simultaneously actually teaching them the material?

(Note: MESE 2098 already addressed issues of students who feel like they already understand a course because they took it in high-school. I am not concerned here with those issues about students' content knowledge. I am focused on assessment, expectations, and learning behavior.)

Expectations: A good answer will share suggestions for activities or assessments or discussion topics to use that will teach students how to learn mathematics and teach them about the expectations of college-level mathematics. I'd prefer to keep any discussion about why this "high-school attitude" persists to the comments, unless you find such a discussion germane to your answer. I'm more interested in what can be done now.

Motivation: I have noticed this kind of behavior in some students in the past, but I recently dealt with a course where this attitude was widespread. Students complained to me (and even other teachers) about quizzes and exams that assessed their conceptual understanding. For example, after a unit on equations, functions, and graphs, a midterm exam included questions like the following. (Parentheticals are what I expected as an answer and to which I would give full credit, even with such brevity.)

If $f(x)$ is a function and I know its graph, explain how to find the graph of $-3\cdot f(x-2)$. ("Flip it across the $x$-axis, stretch it vertically by a factor of 3, shift to the right 2 units.") If $g(x)$ is a function, explain how to determine whether it has even symmetry, without knowing its graph. ("Determine whether $g(-x)=g(x)$ for every input $x$.") If $h(x)$ is a function and I know its graph, explain how to find the graph of $h^{-1}(x)$. ("Reflect it across the line $y=x$." Or, "take every point $(x,y)$ and swap the coordinates to be $(y,x)$." I would have even given extra credit if they added, "First use the Horizontal Line Test to see if it is 1-to-1 and therefore invertible.")

After this exam, I gave an (anonymous) survey to solicit feedback from students about how the course was going. Several students complained about those kinds of questions. The quote below is not the only one, but it exemplifies the issue behind my main question:

"It's really hard to explain how to do math problems or the concepts behind them on tests and quizzes. The time crunch makes it hard to think when you're ready to complete problems, not explain something."

Many other students said something to the effect of, "You should tell us what's gonna be on the tests and quizzes so we know what to study." This is despite the fact that I created an assignment containing lots of practice problems that reflect the content they should know, in addition to making a list of major topics. So, while part of me says, "Okay, I guess that this may be the first time you're being asked questions like this," another part of me says, "What do you want me to do, tell you exactly what I'm going to ask in advance?" And I fear that, yes, this is (almost) what they expect because it's what they are used to.

I did discuss this in class with the students: I tried to explain why conceptual understanding is essential, and I said that unless you can explain a concept to someone else then your actual understanding of that concept is superficial or fuzzy at best. However, I fear that this only demotivated the students, and that what they heard was not, "You need to be better about this," but rather, "You're not good and will never be good at this." I would like them to understand that conceptual understanding is important and I want them to strive for that deeper understanding. How can I help them see that as a goal, in the first place, and then guide them towards it?

Justification: I think this question belongs on MESE because I imagine this behavior is more prevalent in mathematics courses than in those of other subjects. I am sure that teachers in all disciplines lament poor background skills or learning habits of their students, but I doubt there are students in a History class complaining that they had to write an essay response on an exam.