Tuesday, December 12, 2017

Use the MAA Instructional Practices Guide to Maximize Student Engagement with Math

By guest bloggers Martha Abell, Georgia Southern University, and Linda Braddy, Tarrant County College

If you enjoy Teaching Tidbits, be sure to get a free electronic copy of the forthcoming Mathematical Association of America Instructional Practices Guide, which is chock full of great ideas to help you and your students. The guide will be available as a free download on the MAA website on December 20. The MAA IP Guide is intended for all instructors of mathematics:
  • New graduate teaching assistants 
  • Experienced senior instructors 
  • Contingent faculty member at a two-year institution 
  • New faculty member at a doctoral-granting institution 
  • Instructors who want to transform their own teaching 
  • Mathematicians delivering professional development to colleagues
The guide boldly responds to challenges articulated in the 2014 INGenIOuS report and the 2016 Common Vision report, which both call for transformation within the mathematical sciences community toward deeper, more meaningful learning experiences. The MAA IP Guide was written by faculty from a wide range of institutions and professional associations. It supports the use of evidence-based instructional strategies to actively engage students in the learning process by providing “how to” suggestions. The supporting research focuses on effective teaching, deep student learning and student engagement with the mathematics both inside and outside the classroom.

The content organization is based on three interconnected foundations of effective teaching: classroom practices, assessment practices, and course design practices. It also addresses two key cross-cutting themes, technology and equity, that permeate all three practice areas.

The Classroom Practices chapter provides multiple entry points to help instructors implement practices that foster student engagement. Topics include building community within the classroom, using collaborative learning strategies, developing persistence in problem-solving, and selecting appropriate tasks.

The Assessment Practices chapter offers guiding principles to assess student learning through both summative and formative assessments. Most instructors routinely employ summative assessments such as quizzes and exams, but some may not be as familiar with formative assessment practices which can inform their decisions during the course regarding “next steps” in instruction based on students’ current needs.

The Design Practices chapter guides instruction planning and revision to maximize student learning. Each chapter includes vignettes, practical tips, and references to research-based studies that support the effectiveness of the practices. Each chapter also offers strategies for trying new instructional methods and avoiding common pitfalls.

The MAA IP Guide will be a topic of discussion at the Joint Mathematics Meetings in San Diego in January 2018, most notably at the MAA Invited Paper Session on this topic, Thursday from 8:00-10:50. The writing team is excited to present this resource to the community in support of deep, meaningful experiences for all instructors and students. We promote the use of engaging instructional practices in your own department by sharing this resource with your colleagues!



Tuesday, November 28, 2017

“I’m Worried About My Grade.” How to Pre-empt the End of Semester Panic

By Julie M. Phelps, Contributing Editor, Valencia College


As the end of the semester nears, educators brace for the inevitable student questions about their final grades. If you are anything like me, this can be distracting from my goal of the class: to teach mathematics and make a difference in student lives. Sometimes I find myself dreading the last couple weeks because many students are stressed and solely focused on the final grade, not on learning ways to utilize mathematics in their major.

This year I decided to try something new and pre-empt student questions about their grades by having them reflect on their class work and engagement. To do so, I showed a YouTube video called “I am worried about my grade” to my College Algebra students at the beginning of the semester (just before the first test). While very basic, the video goes over the many ways we educators evaluate student performance, and how we make time for students to approach us outside of class.

After showing the video, I asked the student to reflect on it and write a brief essay about what they saw in the video. Here are the themes from their essays that you can use to set course expectations early in the semester (and pre-empt the end of semester panic):

1. I need to be a responsible college student/adult.
  • Ask for help/ask questions/make sure I understand/go to tutoring. 
  • Always try to improve your work/use the resources. 
  • Turn in assignments on time/don’t procrastinate. 
  • Come to class/participate. 
  • Passing the class is up to the student. 
  • Take your education seriously/have good work ethic/don’t be lazy/take the initiative/have a good attitude. 
  • Learn from your mistakes. 
  • Manage your time. 
  • Taking this class is not about a grade but about learning mathematics.
2. There are plenty of opportunities to get help.
  • Go to office hours.
  • When the professor takes time to be flexible within your schedule, make sure to show up.
  • Go to tutoring.
3. The professor is there to help!
  • Establish a connection with the professor.
  • "Professors have a lot of students, so if they offer to help...accept it!"

Three notable quotes from student essays on being responsible and engaged students:
  • “OMG… I’m the bear… you’re talking directly to me aren’t you? Don’t drop me… I get it! I need to make a schedule, study, and ask for help immediately. Thanks for the wake-up call.”
  • “By showing me this example, Dr. Phelps showed me that grades aren’t given, they’re earned. After realizing this, I’m going to utilize every resource that is available to me to show that I have the potential to get passing grade and higher, and that I have the capacity to be good at math and enjoy the work that I do in the class and in the other math classes.”
  • “The video that we watched in class shows in a very droll way a conversation between a student and a teacher. I identify his [the students’] attitude as a student from high school or a middle school, since they only want to pass the year, not to learn things for the future, they can’t see the utility of their learning.”
Because of this activity, I am now able to focus on what the end-of-term should be about: teaching mathematics for long-term use and retention. At the end of the semester, we educators only have a few more classes to make an impact on our students, and our students only have a few more class meetings to get the most out of the semester-long experience. I encourage you to have this conversation with your students before the end of the semester so everyone can have productive class time.


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.