Tuesday, November 7, 2017

The Role of Failure and Struggle in the Mathematics Classroom

By Dana Ernst, Contributing Editor, Northern Arizona University


The purpose of this post is to generate discussion and to get us thinking deeply about our teaching practices. This post introduces teaching concepts that researchers and educators developed to promote student success.

In an attempt to understand who is successful and why, psychologist Angela Duckworth has spent years studying groups of people in a variety of challenging situations. Over and over again, one characteristic surfaced as a significant predictor of success: grit. According to Duckworth, grit is passion and perseverance for long-term goals. One of the big open questions in cognitive psychology is how to develop grit.

Growth Mindset
Research suggests that one of the key ingredients to fostering grit is adopting a growth mindset. Psychologist Carol Dweck defines growth mindset as the view that intelligence and abilities can be developed with effort. In contrast, a fixed mindset is the belief that one's talent, intelligence, and abilities are fixed traits with little room for improvement. Dweck has found that:
 "People’s theories about their own intelligence had a significant impact on their motivation, effort, and approach to challenges. Those who believe their abilities are malleable are more likely to embrace challenges and persist despite failure." 
 The claim is that those with a fixed mindset will tend to avoid challenges, while those with a growth mindset will embrace challenges. Research has shown that effort has the potential to physically alter our brains, strengthening neural pathways and essentially making one smarter.

A growth mindset increases potential. Teachers who have a growth mindset about their students can help share this perspective and reap its benefits. Views like "some people just aren't good at mathematics," or "not everyone is cut out to be a math major" exhibit a fixed mindset and may severely limit the potential of our students.

Productive Failure
At the core of the growth mindset paradigm is the notion of productive failure. While mistakes and failure are part of the learning process, productive failures provide an opportunity to learn and grow. According to Manu Kapur, productive failure activates parts of the brain that trigger deeper learning. Unfortunately, failure and mistakes are stigmatized in our culture, especially in many mathematics classrooms.

One of the key ingredients to a successful active learning classroom is getting the students on board. One approach is to explain the concept of productive failure to our students and go out of our way to point out when we see it happen in class (this is a part of being more transparent in the classroom). Some teachers even make productive failure part of the course grade.

If we want students to feel comfortable taking risks, making mistakes, and failing, then we need to provide an environment where this type of behavior is encouraged. I'm not suggesting that we lower standards (or allow wrong or inadequate answers to be sufficient), but give space for tinkering and failure along the way. Our assessment and grading practices need to jive with our philosophy of productive failure. I think that it is important to build opportunities for failure into our everyday structure.

Productive Struggle
Due to the negative connotation of the word "failure,” I started referring to productive failure as productive struggle, thinking that these two phrases meant the same thing. However, over time, I have come to view these two concepts as related but not identical. Let's do a little experiment. Take a minute to look at the following list of word pairs, but do not write anything down.


Now, without looking at the list of words, write down as many pairs as you can. You do not need to remember where any missing letters were nor which column a pair was in. Next, looking at the table below, count how many pairs you found in column A versus column B.


This list of words is from The Talent Code by Daniel Coyle (although I've rearranged the order a bit). According to Coyle, studies show that on average people remember three times as many pairs in column B, the one with missing letters. The claim is that a moment of struggle (cognitive demand) makes all the difference. I regularly utilize this exercise on the first day of class as part of my Setting the Stage activity, the purpose of which is to get students on board with an active learning approach. The point of the exercise is to have students experience in a simple, yet profound way, the value of productive struggle.

This exercise does an excellent job of distinguishing the difference between productive failure and productive struggle. The key to success here is the struggle, not failure. This exercise also helps pinpoint the "productive" part. Imagine we removed most of the letters from the words in column B. As more letters are removed, the less productive one will be in figuring out what the pairs are. It is important to keep this in mind when designing tasks for our students to engage in. There is a "Goldilocks zone" (related to the zone of proximal development), where struggle will be most productive. How much scaffolding we should provide our students will depend greatly on who our students are.

The Goldilocks Zone
The following images, taken from Patrick Rault, provide an analogy for productive struggle and productive failure.


All three images are meant to represent a challenge or task. In Figure A, all the obstacles have been removed; struggle is minimized, and failure is unlikely. There is no opportunity to develop grit, and we have provided too much scaffolding for our students. This is what happens when teachers demonstrate a certain type of problem and then ask students to do problems using the demonstrated method. Certainly a useful skill, but we want more.

In Figure B the rocks are slippery, and the path is not clear; the risk of failure is high, and most attempts to cross the river will involve a tremendous amount of struggle. Whether the failure and struggle are productive depends on how agile the person is crossing the river. For many, the risks may be too high. Perhaps only a few make it across. This is what happens when we only provide students with tasks that are mostly beyond their current reach.

Figure C is meant to represent the Goldilocks zone. The path across the river isn't trivial, but hopefully the risk of failure and amount of struggle isn't too great. As students become more proficient at crossing the river, the rocks in Figure C should be moved further apart, and potentially introduce a maze of rocks for students to navigate. Ultimately, we want students to have developed enough grit to strive for crossing the river in Figure B. It's possible that eventually the stepping stones in Figure B become the new Goldilocks zone.

Stan Yoshinobu says that the right question to ask is "how are my students intelligent," as opposed to "how intelligent are my students?" What the Goldilocks zone looks like for one group of students may be wildly different for another group of students. In fact, the ideal set of stepping stones is different for each student. This may make the task of providing this experience for our students sound too daunting, but we would be lying to ourselves if we thought the problem went away by not providing the opportunity. Being forced to confront these issues is a feature, not a bug.

Here are some questions to ponder. We welcome respectful dialogue in the comments section.

  • To what extent is grit necessary or sufficient for a student (or even a research mathematician) to be successful in mathematics? 
  • How do we go about fostering a growth mindset and altering a fixed mindset? 
  • How do we provide a classroom environment where risk-taking is encouraged, mistakes and failures are valued as part of the learning process, and high standards are maintained? 
  • How can we identify productive versus nonproductive failure and struggle? In particular, how can we locate the Goldilocks zone for a given set of students?















Tuesday, October 24, 2017

How Transparency Improves Learning

By Darryl Yong, guest blogger, Harvey Mudd College


When we clearly communicate to students the rationale behind our instructional choices, they are more likely to do what we intend, be more motivated to learn, and be more successful. It is an idea that is so simple and obvious and yet often overlooked.

Recent research suggests that being more transparent with our students can improve their learning. In one study, conducted at the University of Nevada, Las Vegas (UNLV), first-year students who took introductory-level courses from instructors trained to be more transparent were more likely to enroll the subsequent year (a 90 percent retention rate compared with the prevailing 74 percent rate for first-time, full-time, first-year students).

Mary-Ann Winkelmes and her colleagues at UNLV have developed a useful framework for making our teaching more transparent.
  • Be more transparent about the purpose of your course content and activities. What knowledge and skills will students get out of the course and how do those things connect to their lived experiences and personal goals? (Examples: If you have students give oral presentations to the class, explain what they will gain by honing that skill. Connect what students are learning in your class to what they have learned and will learn in future courses.)

  • Be more transparent about the tasks that students have to complete. What is the first step that students need to take on an assignment? How can students get themselves unstuck? How can students complete those tasks to get the most out of them? How can students complete those tasks efficiently? (Examples: Describe common errors that students tend to make and how to avoid them. Require students to visit your office hours or your school’s drop-in tutoring center at least once, so that they become familiar with how to get help on their work.)

  • Be more transparent about the criteria for success in your class. What do you expect good work in your class to look like? What does bad work look like? (Example: I give students this handout by MAA Past President Francis Su with annotated examples of good and bad homework problem write-ups. Francis gives this version to students in lower division courses.
One interesting finding from the work by Winkelmes and her colleagues is that while all students had improved learning outcomes in more transparent classes, the effect was greater for underrepresented students.

Why might that be? One possible explanation is that transparency creates a more level playing field for everyone. Let’s face it: students don’t walk into our classes equally prepared to learn. For example, first-generation students tend not to ask for help because they’ve not learned how, or they have coped for so long on their own that they are ashamed to ask for help. And we, because of our mathematical skill, experience, and wisdom, tend to leave a lot of things unsaid because we take them for granted. Those two things combine to confer advantages to students who have mastered the hidden curriculum of our institutions.

The key to being more transparent is to learn to see your classroom from each student’s vantage point. What would they find bewildering or frustrating or alienating? One of the best ways to do this is to ask your non-STEM colleagues to look at your syllabus and assignments. Ask them what questions and frustrations they would have.

Being more transparent with our students is not the same as coddling them. There are certain aspects of your class that is designed to engage your students in a productive struggle: that challenging proof, difficult derivation, or multi-step computation. There are other aspects of your class that you probably don’t want to cause struggle, like: what you mean when you say you want their work to be “rigorous,” how to find more example problems when the ones in the book just aren’t working for them, or whether they truly belong in your class because they couldn’t follow a calculation that you said was “obvious.”

Even if you already say things to your students during class to be more transparent, it is also important to write them down on your syllabus, assignments, and handouts. Why? Students are far too likely to miss important details that are just spoken during class instead of being written down. Also, English language learners and students with learning disabilities will appreciate having the information presented to them in multiple ways.

The amount of transparency that you provide to students depends on their maturity and the level of the course. There are times when you don’t want to be explicit about everything. For example, you don’t want to constrain their creativity by priming them with examples, you want them to struggle with figuring out what the first step should be, or you want them to be more independent in their learning. However, even then you can be transparent about your intentional vagueness. For example: “I have given you problems that may have extraneous information or missing information (like the thermal diffusivity of steel) that you will need to look up. I’m doing this to help you acclimate to what it will be like to solve problems in an industrial engineering environment.”

A few more suggestions on how to be transparent in your mathematics classroom:
  • The verb “simplify” is ambiguous and overused in mathematics. The context of a calculation determines the form of the answer that is “simplest,” but students often don’t have that intuitive sense when they’re new to the subject. Either help them develop that intuition to know what is “simplest” given the context of the material or problem or be more specific (e.g., say “express the answer as a single fraction with denominator factored as much as possible”).

  • At what point are students allowed to use tools like Wolfram Alpha, Symbolab, or Maple? If those tools are allowed, explain how to use those tools in a way that maximizes their learning.

  • Share your rationale for how you’ve chosen to assess their learning. (That is, why a final project? Why a written exam? Why is the exam timed or untimed?) Have compelling reasons for those choices that connect with your learning outcomes. Collect strategies that successful students have used to prepare for your assessments and share them with your students. If you assign writing in your class and will use a rubric to assess it, then share that rubric with your students. Consider sharing examples of good writing, annotated in a way that refers to your rubric.

  • If you teach via inquiry, explain to students why. This wonderful activity by Dana Ernst will help students understand how inquiry-based learning can help them develop independence, curiosity, and persistence.

  • In my partial differential equations class, I start every class by highlighting a current area of research and the people who are doing it. I acknowledge our field’s lack of inclusivity in the past and that I intentionally showcase women and people of color in these highlights to engage in counter-stereotyping.
Unspoken expectations set us up for disappointment and others for frustration. I know that I could be more transparent in my teaching and I invite you to comment below with your strategies and ideas for promoting transparency.


Tuesday, October 10, 2017

5 Benefits to Having Students Grade Their Own Homework

By Rejoice Mudzimiri, Contributing Editor, University of Washington Bothell


Do you have a hard time keeping up with your grading? Do you have to cut back on your homework assignments to make grading manageable? Have you ever considered making your students grade their own homework? Well, if you answered yes to any of these questions, this post is for you! Having students grade their own homework is valuable, saves teachers time, and enhances student learning. I had always hesitated to have my students grade their work, however, when I could not keep up with my grading, I decided to give it a shot. I wish I had considered doing this sooner!

How to let students grade their own work?
Please note that I do not let my students grade all their homework. Personally, I grade every other homework assignment, starting with the first one, so that they get used to my grading style. There is more than one way you can have students grade their work. Some instructors, like Nelta M. Edwards, hand out a key at the beginning of the lesson on the day the homework is due and let the students grade themselves with the key. If a student is absent, they do not get credit.

I go over all the homework problems with my students and then let them assign themselves points depending on what they missed. Then I collect the homework to check on their grading and enter grades. The first time I tried this, I was surprised by how many points my students took off their work. They graded harsher than I would have. Also, they were surprisingly honest about what they did wrong.

Benefits of Having Students Grade their Own Homework
There are several benefits to letting students grade their homework, and the following are my top five:
  1. Helps student reflection. When I grade my students’ homework, they seem to care more about their grade than what they did wrong. They would not even bother trying to do corrections on their own. However, when I have them grade, they do their corrections as we are going over the homework. This is a valuable learning experience that gives them an opportunity to reflect on their own thinking.

  2. Offers immediate and relevant feedback. Students value identifying their own mistakes shortly after making them. When students grade their work, they get immediate feedback on what exactly they missed, rather than waiting for the instructor days after their homework was turned in for grading.

  3. Reduces instructor grading time. Perhaps an important benefit for instructors, having students grade their homework could reduce their own time spent on grading. If you decide to have your students grade every other homework assignment, that is a 50 percent reduction in your grading time. Since most of our precalculus and calculus classes tend to have high enrollments, a 50 percent reduction in your homework grading is a welcome relief.

  4. Shifts attention away from grades. In addition to the 6 Ways to Upend the Focus on Good Grades, having students grade their homework also refocuses their attention away from grades. Instead, they focus more on why they got the problems wrong, thereby allowing them to take responsibility for their own learning. It also eliminates the need for any grade-related discussions with students as they know exactly how they were graded. According to Edwards, having students grade their own work “alleviates student anxiety and, subsequently, eases student-teacher conflict by demystifying the grading process and making students feel that they have control over their own evaluation.” When my students ask questions while grading, they are usually more concerned about how many points should be taken off for certain kinds of errors.

  5. Provides students with another learning opportunity. Having students grade their own work can help provide them with another opportunity to learn concepts they might have missed. Sadler and Good looked at the correlation of grades by comparing self- and peer-grading with the test grades that a seventh-grade science teacher assigned to 101 students in four classes. They also measured the impact on learning by analyzing students’ performance on an unannounced second administration of the test a week after self- or peer-grading. They noted that “students who graded their peers’ tests did not gain significantly more than a control group of students who did not correct any papers but simply took the same test again,” however “those students who corrected their own tests improved dramatically.”

References
Edwards, N. M. (2007). Student Self-Grading in Social Statistics. College Teaching, 55 (2), 72-76.

Sadler & Good (2006). The Impact of Self- and Peer Grading on Student Learning. Educational Assessment, 11(1), 1-31.

Weimer Maryellen (2009). Benefits of a Student Self Grading Model. Faculty Focus. https://www.facultyfocus.com/articles/educational-assessment/benefits-of-a-student-self-grading-model/.

Tuesday, September 26, 2017

5 Ways to go Beyond Recitation

By E. Fuller, WVU Mathematics (guest blogger)


Students at almost every institution of higher education will encounter a recitation as part of their mathematics class at some point, part of the class time set aside to repeat foundational mathematical equations. Graduate teaching assistants (GTAs) are frequently called on to lead these smaller groups of students through the basics of finding the roots of a quadratic equation or computing derivatives using the chain rule. Recitation time is often left for practice of the techniques students learn in lecture. But what if we could do more during this class time? What would that look like?

Here are a few approaches you can take to change your students’ experiences during recitation.

1) Focus on getting students to do the work instead of doing it for them. Homework problems are great and it’s sometimes easiest for us to go to recitation prepared to work out many variations of problems we‘ve done ahead of time. The problem is that we already know how to do them. We are better served, as are the students, by providing the space to let them work through the content with guidance. This is perhaps the easiest way to stay true to the content of the class while creating student-focused time. Use inquiry and questioning to get students to tell you how to do the problems instead of the other way around.

2) Incorporate group work into your sessions. Build teams and leverage peer instruction (a method that allows students quick to understand a method or solution to help his or her peers through the problem) so that they can become teachers themselves. Empowering students is always a good thing.

3) Get students to communicate what they understand to each other and to the class. Research shows that students need to explain what they understand to really master a topic. This practice forces them to rethink concepts as they try to convey knowledge to someone else. Writing prompts such as ‘Explain why this procedure works…’ or ‘Evaluate this solution and determine if there are errors’ force students to think through ideas and develop reasoning to support conclusions.

4) Have students relate mathematics to their own experiences. To develop a connection with mathematical ideas, students can investigate how mathematics is related to their futures or how multiple levels of mathematics show up in their day to day experiences. Connecting ideas like contour maps to real world activities like hiking can bring even more advanced concepts into life.

5) Cultivate an environment where failure is ok and experimentation is encouraged. Students need to learn that trying is important even if it doesn’t lead to the (correct) answer the first time. Making your classroom safe for exploring ideas (even incorrect ones) helps support a growth mindset among the students, especially important if the classroom is student-centered and they are doing and explaining the mathematics that is happening.

It’s important to keep in mind that you can start small - you don’t need to do these things in every meeting. You can pick some manageable topics to try something new with and build from there. It can be hard work and takes time and practice, but your students will benefit from it, and you will find that those recitation sessions can lay the groundwork for some pretty amazing mathematical discoveries for the students.











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!

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.