Tuesday, September 12, 2017

Language matters: 5 Ways Your Language Can Improve Your Classroom Climate

Rachel Levy, Contributing Editor, Harvey Mudd College

The language we use in our classes extends beyond mathematical content. We communicate subtle (and not so subtle) messages about who belongs in the classroom and in our profession. Signals transmit through our level of enthusiasm, quizzical looks at incorrect or convoluted statements, and focus of our attention through eye contact, time to speak, and personal interactions. To avoid perpetuating our unconscious biases through language, we can recognize them and find ways to reduce their impact on our students.

Even when we are careful, at some point our language will likely cause unintended ouch for one of our students. Hopefully we can create feedback mechanisms and classroom environments where they can let us know. But as long as we give grades and write letters of recommendation, the power dynamic in the classroom is unavoidable. This may make it hard for students to speak up and let us know. We hold the responsibility to create a welcoming environment for all of our students.

Here are five ways you can modify your language to improve your classroom climate:

1. Convey explicitly in your syllabus that you believe that mathematics belongs to everyone and that everyone can be a math doer. Share with your students that making and discussing mistakes are a normal part of learning (and being human). See this tool for surveying your syllabus and course design for examples of inclusive syllabus language.

2. Be intentional about encouraging questions. Pay attention to which students in the class feel empowered to speak and provide a variety of ways for students to communicate with us and with each other. Many of the Teaching Tidbits have concrete suggestions, such as ways to engage your students through reflective writing; your responses to incorrect answers; office hours; and inquiry-based learning.

3. When they suggest an answer to a question, ask students to justify that answer, whether is it right or wrong. For example, let students know if they don’t provide a justification you will ask “And why would you say that?” This is a technique common in Russian pedagogy. It allows you to better see how your students are thinking and where they might have gone awry. Students may also sort out their own errors as they argue their point.

4. Avoid perpetuating mathematical language that fails to acknowledge the challenge of learning, such as "clearly, " "only” and “obviously.” These words tend to cue the audience that the speaker thinks the work is trivial. The problem is that even when ideas are taught well, they may not be at all simple for new learners. They also may carry an underlying assumption that all students have had access to the same prerequisite information. Since students enter with a range of previously acquired knowledge and experience, it can be more welcoming to say “the rest requires algebra” instead of “the rest is *just* algebra.”

5. Aim to use inclusive and unbiased language. For example, privately request students’ preferred pronouns and preferred names and use them. Pay attention to how you use humor, encouragement and analogies while teaching. Small comments can have a big positive impact. For example, “When a mathematician approaches this problem, she…”. or “When you explain it like that, you are really thinking like a mathematician.”

Unintended ‘ouch’ happens. What one person finds funny, another finds offensive. What one person finds welcoming, another finds off putting. We are not perfect and we can’t please everyone all the time. But my hope is that when we establish a constructive classroom climate with opportunities for feedback, students will let us know when they experience an ouch because of some way we communicated.

Related Links Karp, Alexander, and Bruce Ramon Vogeli. Russian mathematics education: Programs and practices. Vol. 2. World Scientific, 2011.

Tuesday, August 29, 2017

6 Ways to Upend the Focus on Good Grades

By Chad Topaz, Williams College; and Jude Hidgon, Bennington College (guest bloggers)

As a math educator, there is a good chance this thought has crossed your mind: “All my students care about is grades. They don’t seem to care about learning the material.”

In the parlance of educational psychology, this complaint suggests a tension between mastery goal orientation (e.g., “I want to understand the material”) and performance goal orientation (e.g., “I want to get a good grade”). Research suggests that learning is optimized when learners have high levels of both mastery and performance orientations. However, the structure of higher education arguably stresses performance over mastery. We assign grades, grades get used by educational institutions and by society, and there’s little wonder that students care about the grade more than mastering the material.

Grades remain relevant, and we certainly do not advocate their abolition. But it is important to balance an attention to grades with emphasis on mastering course material. Here are six strategies we have implemented that we have found to support this balance.

1. Design a multifaceted assessment scheme. For the sake of example, consider two different schemes for determining a student’s final grade.

Scheme I: two midterm exams worth 30% each and a final exam worth 40%.

Scheme II: 10 homework assignments worth 2% each, daily informal reflective writing assignments worth 20% total, 5 unit tests worth 8% each, and a final exam worth 20%. In this second scheme, the high frequency of assessments renders each one lower stakes.

Additionally, students benefit from frequent, ongoing assessment because it provides feedback on their learning and more opportunities to correct misconceptions as they are forming, rather than once they are baked into students’ brains. Frequent assessments need not be labor-intensive. When available, TA’s could perform the grading, and/or an instructor could use online assessments with automatic grading as needed. For items such as informal writing assignments, one could also use a low-labor point allocation system, for instance, 0 = assignment not turned in; 1 = assignment turned in but of low quality; 2 = assignment completed at a satisfactory level.

2. Use a drops policy. For example, if there are 12 weekly homework assignments during the term, tell students that the lowest two homework grades are automatically dropped. However, insist that they benefit from this privilege only if they turn in all assignments completed fully. This system provides a sense of security to students that they can struggle in good faith and perhaps not “get it” right away, but it also discourages them from simply not doing the work.

3. Allow corrections. Consider letting students make corrections to quizzes and exams to earn back half of the points they missed. This opportunity encourages students to think about the errors they made, and emphasizes that the midterm is a learning opportunity.

For added learning opportunities, ask students to add to their corrections a discussion of what they got wrong, why their corrected answers or analyses are better than their original, and how they will integrate what they’ve learned to avoid similar errors in the future. This procedure models good scientific inquiry (when we get something wrong, it’s a beginning, not an ending) and tells the student that what you value is that they’ve learned, not that they are perfect. If written corrections are too onerous, an instructor or TA could allow students to perform corrections oral exam style during a designated time period.

4. Structure grades as formative feedback. Don’t merely give numerical grades. A numerical grade says to a student “the number or letter assigned to you is the most important thing.” On the other hand, written feedback can correct mathematical misconceptions, and it provides a metacognitive moment in which students can reflect on their level of understanding of the material. We encourage you, however, not to just tell students the correct answer; instead, point to errors, make suggestions, and then encourage students to correct their own work. To reduce any additional grading load, consider leveraging technology to offer audio feedback. If giving feedback to every student individually is not feasible, hold a single class meeting or discussion section to go over common mistakes, and require students who want to drop their lowest grade (strategy 1 above) or to submit corrections (strategy 3 above) to attend in order to be eligible.

5. Make mathematics verbal. Ask learners to discuss what they understand and don’t understand in words, not just in calculations. Encourage learners to consider the application of your course concepts to real-world scenarios, to other courses, or to the students’ lives outside of school. Find ways to encourage student self-talk (written or oral) about what they are learning. This can be done via a simple blog post, or as a “think, pair, share” opportunity at the start of class. Invigilate these sessions through a Socratic approach; randomly call on two groups from a larger class to report on what they discussed to help ensure that they focus on the task at hand. These activities encourage metacognition, which has been correlated with mastery goal orientation.

6. Discourage (or even abolish) discussions about grades or points. We believe that it is our responsibility as instructors to constantly direct our students’ focus to be on learning. In our syllabi, we use a statement like this, and we stick by it, referring students back to the policy as needed: “The purpose of grades is to provide formative feedback that aids your learning. I keep course grades in the online gradebook, so you can always check them there. But what matters is learning. I’ll enthusiastically talk to you about your learning anytime and I encourage discussions in which we go over the work you have completed. These conversations let me hear about your challenges and questions, and provide important learning opportunities. However, my rule is that we shouldn’t talk (and especially haggle) about the points or letter grade assigned unless I have made a clerical error.”

We recognize that some of the suggestions above create more work for instructors; we have offered a few options for helping to reduce this additional load in instructional settings with large class sizes but without the benefit of TAs or grading support. We’ve found that having more (but lower-stakes) graded items, allowing students to make corrections and giving more corrective feedback all contribute to an environment where the focus is on learning. Just as we would tell our students that more time studying will help their learning, we believe more time spent on these grading activities will help strengthen the learning-focused culture we want.

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!