Tuesday, April 25, 2017

Read the 3 Most Popular Teaching Tidbits Posts of the Year

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

As the academic year comes to a close, Teaching Tidbits is headed for summer vacation. We hope you enjoyed the inaugural year of the blog and found it useful for your classroom. As you prepare for next year’s classes, be sure to read our posts from the last year, particularly our three most popular posts:
While the blog is on summer holiday, we encourage you to seek out other sources of good teaching tips, like attending the ‘Encouraging Effective Teaching Innovation’ contributed paper session at this year’s MAA MathFest in Chicago this summer. We also welcome your suggestions of topics or ideas for future posts by contacting: teachingtidbits@maa.org. Enjoy your summer and see you in the fall.

-The Teaching Tidbits Team

Tuesday, April 11, 2017

Want to Give Your Teaching Style a Makeover This Summer? Here’s How.

By Dana ErnstContributing Editor, Northern Arizona University


Active learning is all the rage these days, and with good reason. As teachers embrace active learning, students are building problem solving skills that promote analysis and evaluation of the content they are given in the classroom. Read on to learn how active learning can give your teaching style a makeover.

Active learning has been gaining traction over the past few years, aided in part by public approval from several entities, including the 15-member society presidents of the Conference Board of the Mathematical Sciences in 2016. Active learning comes in several shapes and sizes, and inquiry-based learning (IBL) is just one of many forms. In particular, the IBL community has grown up to be very active and supportive in the past few decades. Loosely speaking, IBL is a pedagogical framework characterized by two essential principles:

  • students deeply engage in meaningful problems, and 
  • students collaboratively process ideas. 
According to education research, these "twin pillars" of IBL are at the core of most IBL implementations.

Here I have summarized a few resources for learning more about IBL and active learning, and how to get started. This list is certainly not exhaustive and is not intended to be a "how-to guide.”

Workshops and conferences
  • Head to Chicago for MAA MathFest in July and attend a number of sessions dedicated to active learning and/or IBL. 
  • Inquiry-Based Learning Conference: As the name implies, this annual summer conference is devoted to IBL. It's also my favorite conference. It's inspiring to be surrounded by so many educators that are devoted to engaging and empowering students. The conference is also run in conjunction with MAA MathFest, so participants can get even more out of this double meeting.
  • IBL Workshops: The NSF-sponsored IBL Workshops are practical, hands-on, and interactive workshops for college math instructors interested in teaching via IBL or hybrid IBL. There are three workshops offered during the summer of 2017: 
    • DePaul University, Chicago Illinois: June 20-23, 2017 
    • Cal Poly State University, San Luis Obispo, California: June 27-30, 2017 
    • Nazareth College, Upstate New York: July 18-21, 2017
Summer Reading List
Other resources 
  • Math Ed Matters: This MAA-sponsored column explores topics and current events related to undergraduate mathematics education. Posts will aim to inspire, provoke deep thought, and provide ideas for the mathematics—and mathematics education—classroom. Most of the posts address IBL in some way. 
  • IBL SIGMAA: There is a newly-formed Special Interest Group of the MAA (SIGMAA) devoted to IBL. 
  • The IBL Blog by Stan Yoshinobu (Cal Poly): This blog focuses on promoting the use of IBL methods in the classroom at the college, secondary and elementary school levels. 
  • #mathchat: This is active Twitter hashtag that is used by teachers, educators, students, or anyone else interested in math and math education to highlight conversations related to math education.


Tuesday, March 28, 2017

3 Ways to Engage Your Students in Reflective Writing

Rachel Levy, Contributing Editor, Harvey Mudd College


Contemplation and reflective writing can be powerful tools for teaching and learning. Students benefit from considering the way that they learn and do mathematics (in addition to thinking directly about the subject matter). This intellectual activity is often called metacognition. Written reflections can also help professors get to know their students, both personally and mathematically.

Three ways I engage my students in reflective writing:

  1. Have students write periodically in a physical journal. Assignments could be very general, such as “How’s it going in this class?” to more structured prompts, such as “Describe your process for solving one of the homework problems you found challenging” or “Name three strategies you employ when you get stuck on a problem.” When the journal is a physical book, I collect and return the posts with a smiley face, sticker or small comment so students know I looked. I used to use the old fashioned bluebooks created to administer exams because they only cost $0.10. You could use an online submission process. Paper is nice, because students seem more likely to doodle fun pictures.

  2. Ask students to answer a question or two (for credit) at the end of a quiz or exam. I like this approach because it communicates that I value the writing and I will already be in “grading” mode when I look at the result. On the downside, students might be more stressed and less attentive to the task during a quiz. Francis Su has outlined his approach to reflective exam questions in a previous Teaching Tidbits post.

  3. Direct students to complete an “exit ticket” or “minute paper” at the end of class. A prompt might ask what the student found most interesting or confusing that day. Sometimes I encourage students to pose a “what if” question. You could use slips of paper or a web form for these end of class questions. Web forms can make it easier to skim and manage comments from a large class.
Keep reading for more sample questions.

Connectedness Often Translates to Engagement
The more you know about your students, the easier it can be to choose a combination of strategies that promote teaching, learning, transfer and affective gains.

In their reflective writing, my students have shared their hobbies, preferences/likes/dislikes, hopes and dreams, difficulties and triumphs in the course, questions about the subject matter, personal challenges, undiagnosed or unreported learning disabilities and general feedback on their experience in the course. I often indirectly learn about my students’ preparation for the course, attitude, culture, maturity, life pressures and personal goals.

A big caveat: some faculty do not want to know these kinds of things about their students. It is a personal choice, of course, and faculty should be aware that they are opening the door to some potentially heavy topics. Some students will want to share very personal information. Others will not. With this in mind, I try to ask relatively unobtrusive questions (such as the ones above) that students can answer many ways. Even the question, “How’s it going in this class?” has started conversations leading to decades-long connections with former students.

I recommend searching on the terms “math” and “metacognition” for related reading opportunities. Start with the reference linked at the end of this post.

Sample Questions
These questions are from my Spring 2016 differential equations course in-class quizzes.

  • What is something that you do that gives you joy and rejuvenates you? Try to think of something that you don’t judge yourself about - something that makes you happy whether or not you do it “well.” 
  • When I encounter mathematics that challenges me, I use these strategies to get unstuck (circle the letters of everything you try): (a) go to office hours (b) sleep on it (c) go to peer tutoring (d) look online (e) read a textbook (f) take a break (g) go over my notes (h) eat/drink a snack (i) watch a DE video (j) ask a friend (k) other: 
  • If you had a magic wand and could change one thing about our college, what would you change? 
  • What’s something you are looking forward to this summer? (Write something or draw something.)
When my colleague and I forgot to put a journal question on one quiz we were surprised that some of our students wrote their own questions and answered them!

Related Links:

Schoenfeld, A. H. (1987). What's all the fuss about metacognition? In A. H. Schoenfeld (Ed.), Cognitive Science and Mathematics Education (pp. 189-215). Hillsdale, NJ: Lawrence Erlbaum Associates.

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/