Monday, March 13, 2017

5 Reflective Exam Questions That Will Make You Excited About Grading

Francis Su, Guest Blogger, Harvey Mudd College
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“To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.” -Henri Poincare, Science and Hypothesis 
Do your exams accurately represent what you value in your course? Only after many years of teaching did I begin to ask that question.

For instance, one of the goals for my upper division courses is for students to be able to articulate what mathematicians do. Another goal I have is for students to learn to generate their own questions for further investigation. Even though I might have seen a student exhibit such skills in the occasional conversation, the tools that I greatly valued were not showing up regularly in how I evaluated student progress.

Why Use Reflective Exam Questions 
To give students opportunities to demonstrate these reflective skills, I began to assign reflection exercises as exam questions. There are certainly other ways to elicit such information--for instance, you could assign research papers or reflection journals--but I was interested in something that wouldn’t be additional work. Putting a question on an exam was a simple way to signal to students that I cared about their ability to process and reflect on what they were learning, in addition to the mathematical reasoning I expected them to demonstrate.

What I didn’t anticipate was the benefit reflective exam questions would have for me!

First of all, these questions made exams much more interesting to grade. (If you know me, you know that while I love teaching, I have never enjoyed grading. The monotony!) Now I say to myself: ‘if you grade all the other questions, then you get to read the reflections!’ Without reflective questions, the exams show very little of my students’ personalities. Having reflective questions helps me see the unique ways my students are thinking and feeling, and that gives me joy.

The second reason for adding reflective questions to my exams is that I often learn things from my student responses that help me become a better teacher. Sometimes students will explain an idea in a way that I had not considered. For instance, in reflecting about the importance of definitions in mathematics, one student described a definition as a choice of what conversation you are going to have with the material. That’s a metaphor that I now use in my own teaching!

Assigning Points to Reflective Exam Responses
My advertised grading system for such questions is simple: give me a thoughtful answer, and you’ll get full points. Less thoughtful responses get slightly fewer points, but students rarely fail to give thoughtful answers. That also makes my heart happy! Depending on the question, you may wish to give your students the question in advance, so they will have time to think of thoughtful answers and they can reflect on it as they study for their exam.

Below are five examples of questions I have used in the past, and some actual responses I have received.

  1. What three theorems did you most enjoy from the course, and why? Choose one theorem of moderate difficulty and reconstruct its proof.

    I like this question, because the answers often surprise me. What I think is interesting is not always what they think is interesting.

    One student responded: The moment in class where I was truly blown away was when we applied Van Kampen’s Theorem on the torus to derive its fundamental group...the simplicity of its application is a moment I will never forget. 

  2. Formulate a research question related to the course material that you would like to answer. (You do not have to answer the question. Just ask a good question whose answer is unknown to you, and doesn’t have an obvious answer based on what you know from the course.)

    One student responded: Is there a classification theorem for 3-manifolds? (This came after we had discussed the classification of surfaces.)

    The main value of this question is that you signal to students that you value question-asking and conjecture-making. But students often rediscover questions of historical significance that lead to important conjectures or theorems. In such cases I have an opportunity to affirm the student’s intuition for asking a good question, as well as to answer it.

  3. Reflect on your overall experience in this class by describing an interesting idea that you learned, why it was interesting, and what it tells you about doing or creating mathematics.

    One student responded: One interesting thing I learned from the class was the equivalence of open-cover compactness and subsequential limit compactness. Both of the definitions are quite abstract, but both end up being extremely significant in their consequences. I think this seeming disconnect between definition and consequence emphasizes the importance of definitions in mathematics. Definitions essentially frame the type of conversation you are going to have--some definitions that seem different produce conversations with similar results. Many definitions lead to conversations with results that are hard to predict.
    What a thoughtful answer! I learned a new way to explain the importance of definitions from this response.

  4. How did the ideas of this course enlarge your sense of what it means to do mathematics?

    One student responded: This class gave me a much better understanding of what it means to do mathematics than I had in the past. Most of our problem sets in other classes were applying theorems that we learned in class, and the problems were roughly of comparable difficulty. However, with this class, we did much of the learning on our own, through results that we proved. In addition, some of the problems were relatively straightforward, but there were several very challenging problems, where my group didn’t even have a clear idea where to start. This seems much more realistic to the life of a mathematician, where problems don’t present themselves in homogeneous sets.

    From this response, I could see that my student was able to articulate what mathematicians do. Goal accomplished.

  5. I have emphasized the importance of struggling in mathematics: that it’s normal and part of the process of learning. Describe an instance, so far in this course, where you struggled with a problem or concept, and initially had the wrong idea, but then later realized your error. In this instance, in what ways was a struggle or mistake valuable to your eventual understanding?

    This is one of the best reflective questions I have used. The prompt helps my students recall specific struggles that have helped them, and reinforces a theme I have emphasized in class. If you’d like to see a wonderful response to this question, you might enjoy my MAA FOCUS magazine article “The Value of Struggle.”

    I’m sure you can think of other reflective questions that can advance your goals for your courses. Student reflections will help your students grow as learners and will help you grow as a teacher too.

Tuesday, February 28, 2017

Engage Your Students in 60 Seconds or Less

By Lew Ludwig (Editor-in-Chief), Denison University


Ever feel like the teacher from the movie "Ferris Bueller's Day Off,” asking a question and just getting silence back? We’ve all had those moments in the classroom. You pose a well-crafted question to the class, and no one responds.

 Several years ago, I watched Dr. Michael Starbird of the University of Texas – Austin employ a simple technique that has forever changed my own teaching. After you pose your question to the class, pause, then state a slightly rephrased version of the same question. After this, ask your students to take two minutes to discuss with a nearby neighbor.

I saw Dr. Starbird use this technique at a national convention with 300 attendees in the room. After two minutes, the room of strangers was vibrating with engaging discussion. Dr. Starbird could then point to a person and ask, “What did your neighbor say?” Not only did this technique prompt active discussion and engagement, but avoids the risk of embarrassment when putting someone on the spot.

For those familiar with this technique, it is a variation on the Think-Pair-Share model that can help learners of all ages. In this method, the students might first reflect individually on a question, maybe for several minutes writing notes or solving a math problem (Think). Next, the students would turn to a nearby neighbor to discuss their work (Pair). Finally, the instructor calls on students to report (Share).

I have slightly modified this technique to assure a varied discussion: every week I randomly assign students to a pair. These pairs have to physically sit next to each other for that week. When I ask the class to discuss something with a neighbor, they know exactly where to turn.

Does it work? First, the weekly pairing creates a notable community within the classroom as students get to better know each other over the course of the semester. This is very apparent by mid-semester when I walk into the classroom to see students actually chatting with one another as opposed to being absorbed in their own thoughts. Second, students often highlight this technique in the course evaluations. They appreciate the opportunity to test out ideas in a low-stakes environment. Lastly, the class discussion is much richer. Since we have a variety of ideas and viewpoints being shared, the discussion goes much deeper and broader than when only one student answers my well-crafted question.

The “talk to a neighbor” technique is extremely easy to employ, and the time required is very flexible. Sometimes I will give them as little as 30 seconds to compare ideas while longer exercises might call for as much as ten minutes. I circulate the room nudging discussions and gauging understanding. In a given 50-minute class period I use this technique three to ten times, depending on the topic and the mood of the class. I highly recommend you give it a try in your next class. You will be amazed at how quickly your students catch on and become engaged in their learning.

Recommended app: Poll Everywhere

This semester, Teaching Tidbits will focus on useful and engaging apps. A helpful app for the "talk to your neighbor” technique is Poll Everywhere. Give students a link to respond to your question in the app via mobile phone, Twitter, or web browser. Responses are posted online or in a Powerpoint presentation.

This app has many of the same advantages of "talk to your neighbor" but functions more like a clicker system because students can provide their responses anonymously. This makes it a valuable tool for formative assessment and quick checks on difficulty level, pacing and retention.

The free version allows for up to 40 users with a variety of question types and displays. Upgrades allow for more users and additional reporting and tracking options.

Pose a question, visualize responses via Poll Everywhere, and let the discussions begin!

Tuesday, February 14, 2017

They’re in My Office - Now What? 3 Tips for Productive Office Hours

By Jessica DeshlerContributing Editor, West Virginia University


For students, office hours can be an opportunity to catch up or gain additional insight to coursework that challenges them. Recently, we posted about “5 Successful Ways to Get Students to Office Hours,” but what do we do when they come? How do we make the most out of that time so that it’s productive for faculty and students? Here are a few tips you can try to help your students during office hours:
  1. Tell them your expectations. Let students know early in the semester what your expectations are for office hours. Do you expect them to bring their attempts at working out problems with them to see you? Do they have to keep and bring a journal? What type of pre-meeting preparation do you require of them? Ideally, these expectations should be outlined in the syllabus and during your first class. Be specific and repeat your expectations throughout the semester.
  2. Once you have expectations set, stick to them. Reserve the right to reschedule a meeting if a student shows up completely unprepared to engage in a productive conversation. This is fair to students who have put in the expected effort ahead of time and come to your office with specific questions. Specifically, if students have not done the readings or assignments, or have missed class, give them an additional “assignment” of reviewing another student’s class notes before coming back to you to ask for clarification (and bringing those notes with them so you know they did it - it was an assignment after all).
  3. Let them do the work. Much like class time, the time spent in office hours is most effective if your students spend it working through the mathematics instead of watching you do problems on the board. Your goal as an instructor is not to show them how to solve questions, but to teach them how to go about solving questions and how to think while problem solving. Leading students through the work is incredibly valuable. Questions like “How would you get started on this one?” and “What have you tried so far?” are ways to help students talk to you about their troubles in working through problems.  
Recommended tools: Check out these two tools to help you schedule office hours: youcanbook.me and a cool little setting in Google calendar. Both of these allow you to set aside specific blocks of time in your calendar for students, and allows them to book just part of that time. They’ll get reminders (always helpful for students) and you’ll know that they’re actually going to show up!

Tuesday, January 31, 2017

5 Successful Ways to Get Students to Office Hours

By Rejoice Mudzimiri, Contributing Editor, University of Washington Bothell


Are you tired of sitting alone during your office hours waiting for students to show up? I used to feel the same way, until this past fall quarter when my students came to office hours in better numbers than ever. What changed?

Studies show that office hour visits are positively correlated to academic performance (Guerrero & Rod, 2013). More so, they are an important opportunity for faculty-student communication and interaction. So how can you get students to attend your office hours?
  1. Timing. When it comes to office hours, timing is everything. The best ways to schedule office hours include:

    • Avoiding conflicts with other classes. If office hours are scheduled during times that most students have classes, chances are very few students will be able to find time to attend your office hours. As part of scheduling your office hours, you should find out peak times when most of your students are available.

    • Eliciting student input. One of my colleagues who has had success with getting students to come for office hours administers a survey the first day of class to elicit student input on the times that work best for them. She then schedules her office hours depending on the times that most of the students are available. Using Poll Everywhere provides a quick way to survey students on their preferred times.

  2. Location. Some students are not comfortable with meeting in their instructor’s office. Alternative office hour locations include:

    • Public Places. Holding office hours in public areas such as a student lounge or conference room may be more relaxing for students. A colleague of mine holds her office hours in our “foyer” because her office is very far away from the building where she teaches and this makes it convenient for students to attend her office hours. I hold my office hours in the conference room right next to my office - with big white boards - that provide a lot of working space for a number of students. This allows me to accommodate more students at the same time. Students worked either individually or in groups during my office hours.

    • Virtual Office Hours. Holding some of your office hours online gives your students flexibility. One study on undergraduate millennial students’ perceptions of office hours suggests that they preferred virtual communication with their professor over face-to-face. Another colleague holds evening virtual office hours using the app Canvas Conference. In addition to audio, video and screen sharing options, this conference feature allows students to upload PDF files of their work so that the instructor can give feedback while talking with them. My colleague allows his students to schedule one-on-one appointments with him for these virtual sessions.

      Interestingly, some students reach out for help during these virtual sessions who never attend traditional office hours. The main advantage of virtual office hours is that office hours can be scheduled at flexible times such as the evenings or weekends. To avoid back and forth emails on availability and double booking, an instructor can use free online resources such as https://appoint.ly or https://youcanbook.me/. Both of these add the appointments directly to your calendar.

  3. Make Homework Assignments Due During Office Hours. Two of my colleagues who have had a good office hour turnout have their students turn in their written homework during office hours. They do this strategically so that students must attend office hours and can get help with their homework.

  4. Educating Students about the Benefits of Office Hours. Some students don’t attend office hours because they do not know what the purpose of this time. In addition to having office hours listed on the course syllabus and announcing them regularly in class, instructors need to educate students about them. Let students know what office hours are for and the kind of things they can expect or benefit from taking advantage of them.

    As an international graduate student, I did not know anything about office hours since I grew up in a school system where lectures were accompanied by one-hour tutorials. When I struggled with my math class, one of my friends suggested that I visit my professor during office hours and that was the end of all my struggles! My professor had assumed that as graduate students, we would know about this already.

  5. Make Office Hour Visits an Assignment. Gooblar suggests actually making office hour visits one of the course assignments, because giving students feedback face-to-face is easier than written comments. Gooblar believes that “if you make them [students] come in once, they may start dropping by on their own.” I usually encourage students who did not do well on an exam to make an appointment with me to go over the exam. As they go over the exam with me, I sometimes give them points back as they explain to me their thinking. In addition, we talk about what they could do differently next time and how I could be of help to them. A fellow editor, Jessica Deshler, sometimes makes visiting the tutoring center a course assignment so that students become more comfortable with seeking help and talking about homework problems.

Related Links

Edwards, J. T. (2009). Undergraduate Millennial Students’ Perceptions of Virtual Office Hours. International Journal of Instructional Technology and Distance Learning. 6(4) Retrieved from http://www.itdl.org/Journal/Apr_09/article05.htm

Gooblar, D. (2015). "Make your Office Hours a Requirement." Retrieved from https://chroniclevitae.com/news/1167-make-your-office-hours-a-requirement

Griffin, W., Cohen, S. D., Berndtson, R., Burson, K.M., Camper, M., Chen, Y ,Margaret Austin Smith, M. A. (2014). 62 Starting the Conversation: An Exploratory Study of Factors That Influence Student Office Hour Use. College Teaching. 62(3), 94-99

Guerro, M. & Rod, A. B. (2013). Engaging in Office Hours: A Study of Student-Faculty Interaction and Academic Performance. Journal of Political Science Education. 9(4), 403-416.

Weimer, M (2015). Office Hour Redux. Faculty Focus. Retrieved from http://www.facultyfocus.com/articles/teaching-professor-blog/office-hours-redux/





Tuesday, January 17, 2017

5 Free Apps to Use in the Classroom

By Julie Phelps, Contributing Editor, Valencia College


Students have smartphones and want to use them all the time! My solution: well, if you can’t beat them, join them. With that in mind, I began looking for apps that can help facilitate learning. I want my students to use electronic devices for learning ‘good’ and not as the ‘evil’ learning detractors that we educators often perceive them to be. Here are some of my favorites to use in the classroom, plus one to keep an eye out for on your algebra students’ screens.
  1. Kahoot! is a game-based classroom response system that uses quizzing to present content and generate discussion. The game can be displayed on a shared screen. Students can join the game on their own smart device/computer as long as they have a browser and a good internet connection.

  2. Quizizz is game-based tool similar to Kahoot!. With Quizizz you can randomize the questions to allow students to go at their own pace. The game also displays the correct answer when they make the wrong choice.

  3. Socrative allows the educator to initiate formative assessments for students. The educator can ask open-ended questions and vote on the results in addition to the multiple choice and true/false questions. The drawback is that the tool is only free up to 50 users.

  4. Evernote is a tool for both educators and students to capture and share notes across technology platforms. The notes are searchable and can be text, images, video, audio and/or handwritten. There are other apps that do the same thing, but many of those do not communicate across platforms and they are not free.

  5. Desmos Graphing Calculator is a web-app interactive, easy to use calculator. A slider tool animates the graph to demonstrate transformations and supports the founders belief that people learn by doing. The embedded tools are intuitive. Zooming and points of interest can be found by just touching the screen.
Bonus links!

Beta (testing version): Classkick is a new web-based free app. The educator can create a class assignment for everyone to access either to work on individually or in small groups. The app can monitor student progress as they work through the question in real-time and can give feedback to each student individually.

And one to watch out for in algebra-related courses: Photomath, a camera calculator that is also a free app that does exactly what it sounds like. Just point your camera toward an algebraic math problem (type or hand-written) and Photomath will tell you the result with detailed step-by-step instructions.

Tuesday, December 6, 2016

Taming the Test

By Lew Ludwig (Editor-in-Chief), Denison University



I usually give three to four tests during the semester, and I was puzzled why the first test always had the lowest average test score. After all, this should have been the “easier” material. After some reflection, I found that students were not accustomed to taking my tests.

Often, they were not aware of the format; I frequently use true/false questions that require a short argument. They were also not accustomed to the pace; like most college classes, in 14 weeks, we cover what most high school courses cover in 36 weeks.

To overcome this learning curve, I use an exercise I call “Test Tuesday” to encourage student success on tests in my classes. Every Tuesday when students arrive in class, I give them three or four questions from an old test and 10 minutes to complete so that they become familiar with the test format of my class in a low stakes environment. After the ten minutes are up, students share their work with a neighbor and I circulate to listen to these discussions. After about five minutes of paired discussion, we consider the questions as a class, focusing on the questions that caused the most difficulty.

The entire “Test Tuesday” process only takes about 20 minutes of class time. Even though the actual test questions are different than the “Test Tuesday” questions, I noticed that the average score on the first test increased by half a letter grade since enacting this exercise. Students speak highly of this activity on course evaluations as it gives them formative feedback in a supportive, low stakes environment. Moreover, recent research shows that a good way to retain information is to be tested on the material frequently. Not surprisingly, I have seen a small uptick in cumulative final exam grades since employing the “Test Tuesday” technique.

Related Links
Roediger, H. L., III, & Karpicke, J. D. (2006). The power of testing memory: Basic research and implications for educational practice. Perspectives on Psychological Science, 1, 181–210.

















Tuesday, November 15, 2016

Who generates the examples?

By Dana ErnstContributing Editor, Northern Arizona University

Have you ever had a student who could recite a definition or theorem word for word, but didn’t really know what it meant? Students often memorized a snippet of mathematical content without understanding where and how it applies.

According to Bloom's Taxonomy, these students have only reached the the first level in the taxonomy--recalling facts and basic concepts. Ideally, we want our students to reach higher levels in the taxonomy such as using information in new situations or producing new original work. In today’s world we need individuals that are capable of asking and exploring questions in contexts that do not yet exist and to be able to tackle problems they have never encountered.

The question is, can we, the instructors, work toward this?


There is a small change we can make to ensure students are progressing on Bloom’s scale and developing the habits of mind of a mathematician: encourage our students to generate examples and counterexamples. Requiring students to construct examples and counterexamples places them in situations where they must wrestle with definitions, concepts, and notation, which provides them with the opportunity to synthesize and analyze mathematical ideas. Below are a few examples from calculus, and additional examples can be found here.



Implementing these types of questions can be done in a number of different ways, but it is important to make it a regular thing. As a starting place, I encourage asking at least one question that requires each student to produce an example or a counterexample each day in class and on every homework assignment and exam. There is likely a limit, but in general, the more often students are asked these types of questions, the better.

Questions that ask students to generate examples are excellent for think-pair-share and small or large group discussions. My experience has been that the discussion surrounding student-generated examples is fruitful for the students and insightful for the teacher. Especially in the classroom, I encourage fellow instructors to allow the students a little room for making mistakes and to foster an environment where it’s okay to experiment in the hope that everyone can learn something from the conversation that follows.

It’s also important to vary the difficulty of the problems. Try not to give away which ones you think are “easy” versus “hard.”

As I've increased the number of opportunities for students to generate examples and counterexamples, I've witnessed an increase in student understanding of mathematical concepts and their interdependence. In particular, students seem to have a much deeper appreciation for the intricacies of key definitions and theorems.

I've included potential questions/problems in the context of calculus to give you a flavor of what such questions might look like, but we certainly we can employ the same strategy in other subject areas. Do you have a favorite question/problem that asks students to generate an example or counterexample? If so, share it in the comments.


Related Links
Dahlberg, R.P., Housman, D.L. Facilitating learning events through example generation. Educational Studies in Mathematics, 33(3), 283-299, 1997.

Hazzan, O., Zazkis, R. A perspective on ‘give an example’ tasks as opportunities to construct links among mathematical concepts. Focus on Learning Problems in Mathematics, 21(4), 1-14, 1999.

Katz, B., Thoren, E. Call for Papers for PRIMUS Special Issue on Teaching Inquiry. PRIMUS, 2014.

Watson, A., Mason, J. Student‐generated examples in the learning of mathematics. Canadian Journal of Science, Mathematics and Technology Education 2(2), 2002.