Teaching undergraduates who expect a high-school-like learning environment

Question

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.)

1. 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.")
2. 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 $x$.")
3. 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.

Answer 1

I find class voting/clickers a good way to get students used to thinking about conceptual stuff. In fact, I often frame my lectures around three progressively more challenging conceptual questions. Sometimes I write my own, but here and here are some models of good, interesting questions.

For a good overview of the depth of learning that can come from applying this technique in the classroom, there's a nice video by Derek Bruff.

The typical cycle that I use is that students are asked thought provoking questions, asked to vote on an answer, asked to discuss the question with a partner and then asked to vote on the question again. Then, we discuss it as a class. These mini assessments, I find, help them get ready for the kinds of conceptual questions that I want to put on an exam.

Answer 2

Is there less proof of conceptual understanding if you've shown them each type of question beforehand? I have some questions for calculus I that I think address conceptual understanding, and we practice those sorts of questions before the test. Students may have understood the material, but just not known how to respond to those questions.

I also allow each test to be taken at least twice. My tests are split up into mini-tests which each center on one big idea. The ones I can easily make many versions of, they can retake in my office. For the others, we find a time they can all meet, outside of classtime.

I've laid it out in this blog post.

Answer 3

This may seem strange to offer an answer to my own question after just a week, but I did find a useful resource that may be helpful in addressing the issues described in my question. This week, I have been reading the recently-posted Instructional Practices Guide from the MAA (direct link to pdf). Throughout the document, they cite evidence-based research to advocate for active engagement in mathematics courses. A section in the Design Practices chapter addresses some challenges in designing and implementing such a classroom, and provides a link to an in-class activity to be done on the first day that may help students better understand the instructor's motivations and adjust their attitude towards the course. I will share excerpts from the MAA guide and the blog post they cite, both interspersed with and followed by some comments about how I think this may address my main question.

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DP.3. Challenges and opportunities

Successful implementation of engaging classroom components (a) necessitates a shift in the instructor’s role relative to traditional lecture, and (b) tends to alter the time and effort cost for students (Lee, Y., Rosenberg, J., Robinson, K., et al., 2016). In this section we discuss challenges and opportunities related to designing an active engagement environment.

[MAA, p. 137]

[...]

DP.3.2. Other Challenges

Buy-in

In order to successfully implement a course design, instructors must have the support of both administrators and students when employing a new instructional approach. Instructors should set realistic expectations and create an atmosphere where students can be successful. [...]

Most students will expect and be comfortable in a traditional mathematics classroom environment. They understand the traditional implicit contract between students and instructor: the instructor delivers clear lectures with appropriate examples, and the students listen to the lecture and do the assigned problems. Students may not hold up their end of the contract, but they understand the rules.

An instructor implementing new instructional approaches with different underlying agreements must make the new expectations explicit. Students will need time to come to terms with the intentional coherence of the methods and messages in the course. This process will require struggle and might elicit rebellion and frustration before students are comfortable with the revised contract. In order to facilitate this process an instructor can explain to students on the first day of class that this will not be a lecture-based class and ask them to read Dana Ernst’s blog post http://danaernst.com/setting-the-stage/. Using new instructional approaches may require new tools to help students succeed in the class such as offering additional office hours, meeting with struggling students early in the term, and coordinating tutoring opportunities.

[MAA, p. 140, emphasis mine]

This section resonated with my experiences last semester in some ways, especially the emphasized sentences above. However, I think the main message of the section applies to any instructional method that bucks a students' expectations, even if it's as simple as different kinds of homework or exam problems (per the examples in my question). That is, the suggestion of making expectations explicit applies to any kind of "new" activity or assessment that students may not be familiar or comfortable with.

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The linked blog post provides an example of an activity to engage with students and explain the motivation behind a course and its components, or to make more explicit some new expectations. The post describes how the author uses this activity to discuss with students why the course will follow an inquiry-based learning, but I believe this activity can be modified appropriately for other teaching styles and assessment methods.

Whenever I’m teaching via inquiry-based learning (IBL), it is important to get student buy-in. I often refer to this as “marketing IBL”. My typical approach to marketing involves having a dialogue with my students, where I ask them leading questions in the hope that at the end of our discussion the students will have told me that something like IBL is exactly what we should be doing.

Directions to the Students:

• Get in groups of size 3–4.
• Group members should introduce themselves.
• For each of the questions that follow, I will ask you to:
  1. Think about a possible answer on your own.
  2. Discuss your answers with the rest of your group.
  3. Share a summary of each group’s discussion.

Questions

1. What are the goals of a university education?
2. How does a person learn something new?
3. What do you reasonably expect to remember from your courses in 20 years?
4. What is the value of making mistakes in the learning process?
5. How do we create a safe environment where risk taking is encouraged and productive failure is valued?

http://danaernst.com/setting-the-stage/

The blog post contains a brief discussion of common answers to the questions and the author's experiences with it in the past, as well as a slide presentation to display in class during the activity.

As written, this activity may be more involved than is necessary to simply discuss with students that learning mathematics in a college/university environment is different than learning in high school. However, I do think a version of this activity can be very useful to engage students with the course and the instructor, to serve as a focal point in the future ("Remember when we discussed this?"), and to show the students that the instructor has (hopefully) good reasons for teaching the course and assessing students in the manner they have chosen.

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I also envision that this activity could be modified to "teach" students about types of assessment they may encounter in the course and simultaneously serve as a tool for both the instructor and students to assess prerequisite knowledge. For example, let's take the context of a Calculus 1 course made of mostly 1st and 2nd year students. The instructor may be dealing with students who took an AP Calculus class in high school but performed poorly on the exam (or the school's placement assessment) as well as students who took a Precalculus course at the college the previous semester. The instructor can pose several mock exam questions to the students that assess similar background knowledge in different ways, such as:

1. Consider $f(x)=x^2$. Graph $f$. Then, simplify the expression $-3\cdot f(x-2)$ and graph that function, as well.
2. The image provided is the graph of a function $f$. On the axes provided, sketch a graph of $-3\cdot f(x-2)$.
3. If $f(x)$ is a function and I know its graph, explain how to find the graph of $−3\cdot f(x−2)$

Have students work to answer these three questions on their own for two minutes. Then, have them discuss their answers with their neighbors. As a class, discuss the correct answers. (You even wish to discuss with students the grading of incorrect or partially correct answers to these questions, if this comes up.) Then, have the students compare and contrast the question styles in small group discussions, leading to a larger discussion as a class. Explain to students how there can be various levels of understanding of a concept, and that we instructors care about many of them. Emphasize that students will be asked questions of each variety, just like the examples provided.

The actual examples chosen can be tailored to the course in question. Or, you could pick a very basic concept to ensure that all students will already understand the essential content, so that discussion can focus on the styles of assessment questions, instead.

I have not tried such an activity in my own courses, but I do plan on trying something like this in the near future. Perhaps, it can be followed up and revisited throughout the semester, with survey questions or brief personal reflection assignments to remind students about the importance of conceptual understanding.

Answer 4

"Explain _____" is an open-ended question. Someone who understands a concept and can easily explain it face-to-face, while being prompted with clarifying questions, might struggle when faced with a written question and a blank page. Knowing where to begin is half the battle. It's a form of writer's block.

Someone who has an insight might not recognize that a question is asking for a crystallized expression of that specific insight. As a student, I've had my fair share of short-form essay questions that were frustrating because, although my answer proved that I understood what I needed to understand, I had no way to tell if I was presenting the information in exactly the way the grader was looking for. (Especially questions that asked what are the steps in a process, where the division into steps is arbitrary.) Resistance to "explain _____" questions does not necessarily indicate resistance to conceptual learning.

The good news is that there are often ways to test someone's conceptual understanding without asking them to explain it ex nihilo. You just have to get them to apply their understanding in a way that reveals any gaps. And they have to understand and accept that they'll be tested on questions where there's no rote technique to arrive at the answer. This acceptance naturally follows if:

• The test/assignment is perceived as fair, i.e. there is an objectively true and uncontroversial answer and the student will get full credit for providing that answer.

• There are opportunities to practice for the types of questions that will be encountered on tests. Most importantly, the practice problems should reveal the exact conceptual weaknesses that the student needs to work on. This might require the teacher to review the student's practice work and point the student in the right direction.

So my answer is to find the middle ground. Teach to the test. Choose problems that play to the strengths of the test format and require understanding and insight. Use tests/assignments as diagnostics to identify conceptual gaps and motivate the student to fill in those gaps.

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For example, you mentioned: "If $h\left(x\right)$ is a function and I know its graph, explain how to find the graph of $h^{−1}\left(x\right)$."

Instead, try this: present three different graphs, and ask the student to draw approximately the inverse of each curve. The graphs are followed by a question: "How did you know what to draw?" This is an example of the more general questioning method of: instead of asking how someone would solve a problem, ask them to solve a problem and then ask how they did solve it.

Another thing you can do is pose problems that are trivial to solve if the student understands a concept, but would be impossible to solve (or impractical to solve within the time constraint) otherwise.

Example: say you want to test them on the concept that $\exp$ is an isomorphism from $\left(\mathbb{R},+\right)$ to $\left(\mathbb{R}_{>0},\times\right)$. You want to verify that the student not only has memorized the rule that $e^{x+y} = e^x e^y$, but also intuitively grasps that $\left(\mathbb{R},+\right)$ and $\left(\mathbb{R}^+,\times\right)$ are essentially the same structure. A problem could be:

Find the value of $e^1 \times e^2 \times \cdots \times e^{100}$.

The only feasible way to solve this on a test is to recognize that $e^1 \times e^2 \times \cdots \times e^{100} = e^{1+2+\cdots+100} = e^{100 \times 101 \div 2} = e^{5050}$.

Another problem:

The bottom half of this line is the standard real-valued number line. The top half of the line is empty.

Consider any two arrows that begin at the center and point either left or right. Their composition is the combined arrow you get when the first arrow begins at the center and the second arrow begins where the first arrow ends.

We want to fill in the top half with real numbers. If there's a point on the line equal to $x$ on the bottom half and $y$ on the top half, we call that point $\left(x, y\right)$.

If the first arrow stops at $\left(x_1, y_1\right)$ and the second stops at $\left(x_2, y_2\right)$, then we want their composition to stop at $\left(x_1 + x_2, y_1 y_2 \right)$.

We also want each top-half number to be greater than all top-half numbers to its left.

Is this possible? If not, then explain why. If it is possible, then describe how to fill in the top half and prove that it's correct. Which number is at the center?