In the midst of the upheaval due to the Coronavirus, students and faculty are transitioning to new virtual classrooms. Many of us haven’t chosen to learn or teach, but here we are, making the best of this new reality.

In this post, I describe some guidelines that may help students manage the transition to online learning as smoothly as possible. Instructors can support students by helping them to learn online, and I encourage instructors reading this to pass it along to your students. I offer these suggestions with a caveat: Some of these ideas may not be feasible for everyone, and that’s ok. We all have unique living, learning, and life situations, and what works for one of us may not work for others. Take what you can, and leave the rest. Keep realistic expectations of yourself, understanding that these circumstances are less than ideal. While the suggestions in this post are directed toward students, I also offer “teaching tips” to help instructors support their students.

__Teaching tip__: Remember the diversity of your students and keep equity issues in mind. Not everyone will have reliable internet access, computer access, time, or a quiet place to study. Students may be caring for children or sick family members, may be sick themselves, may need to work, and may be facing any number of other challenges and stressors.

**Above all else, take care of your physical and mental health. **Self-care isn’t self-indulgent. If your physical or mental health falters, the rest of your work will suffer. Time invested in healthy habits pays off.

*Create healthy habits* for sleep, eating, exercise, and hydration. These basics will help you focus and manage stress, and also have many long-term health benefits. The hardest part of self-care for many of us is getting enough sleep, but this is one of the most important things you can do for yourself.

*Focus on the positive*. While this may sound simplistic in these challenging times, finding ways to laugh, watch funny videos, sing, dance, or do whatever will help you unwind and give you a much-needed break from school and other pressures. Limit the amount of time you spend surfing about how bad things are. If you choose to follow the news, identify a trusted news source, and tune to it judiciously.

*If you struggle with anxiety or depression*, identify resources *in advance *and keep the information handy. This will make it easier to access support if you need it. Do a little research *today *to identify resources for counseling and crisis management available online or by phone. Your college or university has not shut down, only moved to a new normal, and most essential services are still available. You can also identify support groups in the community or online. There are several hotlines and other resources for students in crisis. The counseling center can help you find one to have on hand just in case you need it.

*Create a specific, intentional routine*. Have a structured plan for your days. Consistency in your schedule can help you reduce anxiety and stress.

__Teaching tip__*: **Understand the stress that students are experiencing in this transition. Remain flexible with deadlines, incompletes, extensions, and grades.*

**Make good use of resources***.* What arrangements has your college or university made for access to libraries, advising, and technical support? How can you contact each of your professors? If they have virtual office hours, learn the times and how to reach them. Is there an online forum for asking questions? If you know in advance how to access these resources, it will be easier to seek help when you need it.

__Teaching tips__*: Know which resources are available to students for advising, counseling, library use, crisis intervention, tutoring. Help students connect with these and also with external resources, such as **Virtual Nerd**, **Math is Fun**, **Khan Academy**, or **Math Forum**. Make sure students know when and how they can reach you: Establish virtual office hours and give students other ways to contact you for help. *

**Find a physical workspace that can help you focus and be productive **with as few distractions as possible. Depending on your living situation, this may be particularly challenging, but develop the best plan possible in the circumstances. Think about what kind of environment helps you study, and try to re-create that. Having a regular, designated space helps signal your mind that it’s study time. Aim for adequate lighting, access to your books, calculator, laptop, and other supplies. If you need background noise, or need to block outside noise, consider a white noise app.

*Talk with people around you about what you need***, **including parents, siblings, children, roommates, friends, bosses, and anyone else who has expectations of you or who impacts your daily life. Discuss your educational workload and what you need to meet it. Especially if there are other people in your living space who are studying or working from home, negotiate how you can have the best access possible to the space, technology, time, and quiet you need to meet your responsibilities. You may not be able to achieve the ideal, but planning and talking in advance will get you as close as possible.

**Make and maintain personal connections **to combat isolation. Personal connections can support your physical and mental health and can help you learn. Form a study group, use social media, connect through discussion boards or a Facebook group. Schedule video calls with friends and family. If you find it helpful to study with others, try a virtual or phone-based study session with your group. Keep in touch with instructors and classmates through virtual office hours or study groups so that you can stay up on your coursework.

__Teaching tip__*: Promote community through interactive class activities and authentic communication with students. Recognize and disrupt any racist, xenophobic, or other controversial issues that may arise in your classes. **Making uncomfortable conversations productive* *can help our students and colleagues move forward with greater understanding and inclusion.*

**Find out what is expected of you. **A chart or list of requirements for all of your courses might help you stay organized. Identify changes to course requirements, assignments, and due dates. Record the tools you need to access each course and where they are, including any chats, discussion forums, or other means to ask questions and communicate. Identify if you need to be online at specific times, how you are expected to submit assignments, and the schedule and process for quizzes, exams and other assignments. These details may be different for different courses, so a system for keeping track of what you need to do, and where, how and when, will really help you as you move through the rest of the semester. Ask lots of questions, but also be patient with your instructors! Many of them are new to this online environment as well, and they are figuring all of this out as quickly as they can.

*Get comfortable with the technology in advance*. Test computers, internet connections, software, webcams, headsets, mobile devices, and microphones. Mobile devices may be convenient, but may not always provide all the functionality you need. Know where to get technical support in the event that you need it. *If you don’t have access to something you need, contact your professors as soon as possible*. They may be able to provide alternative assignments or different ways to access the course material.

__Teaching tip__*: **Communicate expectations clearly and often. Use multiple modalities like email, announcements, texts, small peer-support groups, and other means. Offer students multiple ways to participate. Some students have more limited resources than you expect and may need alternative ways to meet course requirements.*

**Use good study habits. **Take notes and study just as you normally would. Close distracting tabs and apps. Humans are not as good at multitasking as you may think! Set yourself up to be able to focus on your work. Limit social media. Give your fullest attention to all course activities (discussions, group work, watching videos).

__Teaching tip__*: Provide students with individualized support and feedback.*

*Manage your time and stay organized. *Setting a schedule can help provide structure, keep you motivated, and prevent you from falling behind. Make sure your schedule is realistic and achievable. Record due dates, exams, and other assignments. Set aside specific times to study, and strive to keep balance by including the things that are most important to you. Allocate sufficient time for self-care. Schedule breaks. Do you need one-on-one time with your family? Build it into your schedule on a regular basis. If unexpected circumstances arise that interfere with your schoolwork, communicate with your professors as soon as you can.

__Teaching tip__*: **Communicate weekly about what is due, when, what tools should be used, and how work should be submitted. Establish weekly routines and rhythms to help you and your students keep on track. *

**Engage actively in online group work.** Group work and collaboration will look a little different online, but is still important to learning and creates a much-needed sense of community. Meet regularly with your team. Check in regularly with a quick text on your group chat. Identify a purpose for each meeting to help your team stay focused and on track. Take notes in a shared document so you can all contribute and keep track of progress. Meet by video when you can, to help you communicate more clearly and stay connected to each other. If someone has been absent from your group meetings, check in on them to make sure they’re ok.

**Remember, as time goes by we will all adjust. **This crisis has disrupted travel plans, ended sports seasons prematurely, confounded important projects, separated friends and family, and overall is a big strain on us all. Remember to take good care of yourself. You will find your way. We all will find our way as we settle in to the new normal. *Until then, take a deep breath, do your best, get some rest, and wash your hands.*

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Many of us are experiencing stress as schools, colleges and universities move instruction out of the classroom. Fortunately, even if distance learning is new to you, it isn’t new, and there is a lot of wisdom to draw on.

This document describes some practical strategies that will hopefully get you started, along with some helpful web-based resources. From there, you can do a deeper dive by accessing the open community on MAA Connect called “Online Teaching and Distance Learning.” MAA members can log in with their member credentials, and anyone who creates a free profile can join this group. This is an extensive platform to exchange ideas with other faculty and to access resources and advice for developing your courses. The STEM faculty blundering through remote teaching in a pandemic FaceBook page is another great place for faculty to share ideas and figure all this out together.

- In the current situation you don’t need to become an expert in online course delivery. Your course won’t be perfect, and it won’t be the same as it was in the classroom, and that is ok. Give yourself permission to just do the best you can do. Many of your students don’t have much experience learning online, and they are adjusting too. Set realistic expectations for yourself, your course, and your students.
- Keep flexibility and empathy in the forefront. Some students may not have ideal learning or internet environments—they have family responsibilities, lack privacy or quiet space, have unreliable internet access, are in other time zones, need to be online at a library or other public space, or have any number of distractions or obstacles.
- To the extent that your institution allows it, be particularly flexible with deadlines, independent study, and extended incompletes. Focus on what your students need in order to learn, rather than on structure or deadlines. We are in an unusual situation and this flexibility will make it easier on all of you.
- Online learning is one specific means of distance learning, but it is not the only one. Long ago distance learning took place by snail mail. I’m not necessarily advocating that approach, but cite this as a reminder to think outside the box.
- There are many tools available: Zoom, Skype, Dropbox, Blackboard, Canvas, Slack, VoiceThread, email, online chats, video chats, MS teams, Google docs, and many others. Investigate if there are particular platforms or tools that your institution already uses. Even if you are not familiar with them, your students might be and your institution is more likely to offer support for those platforms. Coordinating with your colleagues to use similar tools will allow you to support one another. Keep it simple.
- Does your institution have an office that supports online learning? They’ll be overloaded now, but make sure to check their website to see what they have to offer. Remember, even if you haven’t taught online before, others have, and they will hopefully share their expertise.
- Unless your institution requires it, it is ok to build the course week by week and adjust as you go. You don’t need to have it all figured out in advance.
- Prioritize the learning goals for your course. What are the most important things for your students to come away with? Focus on those, and build up the rest if you can.
- As with any teaching, focus on what you want students to
*learn*rather than what you need to*teach*. - Be transparent. Communicate very clearly with students about expectations, and about why you are making decisions and setting priorities. You can also be open about how you’re all in this together. This will reduce anxiety for everyone.
- Every week, provide a list of deliverables: read this, start this, submit that.
*What*should they be working on,*where*, and*with whom*?*Where*and*how*should their work be submitted? - I always remind students at the start of an online course that even if they are in the course platform frequently, if they don’t speak up and participate, no one knows they’re there. The same applies to instructors—communicate often.
- Are other faculty teaching sections of the same course? It might help to work together.
- Be aware of FERPA and accessibility requirements. Your campus may have resources to guide you with this. If needed, there are services online that will caption or transcribe videos.
- If you use video, use small chunks, no more than about 5 minutes at a time. This allows students to stay focused. Also think about the easy parts of production quality—adjust the lighting to be clear, don’t move around too much, and avoid other distractions that are within your control (until your cat runs across your keyboard or your toddler comes into the room). It might help to use the microphone on your earbuds rather than your computer.
- Test all technology. Do microphones, electronic whiteboards, video cameras work? Can the online platform handle the number of students who will login? But also remember that technical snafus happen. Communicate with your students with humor and you can all figure it out as you go along.
- Consider a combination of synchronous (everyone there at the same time) and asynchronous instruction. Synchronous instruction allows for more direct interaction, but it can be challenging for everyone to make timing work in the new normal. There are many ways to engage students asynchronously instead. Asynchronous instruction is easier on everyone. If you decide to ask everyone to be there at the same time, use the same time slot that your class was originally scheduled for.
- To keep students engaged, have frequent, small assignments with clearly-communicated due dates, and create learning activities that require students to interact at specified intervals. If you are requiring them to engage in collaboration or discussion, communicate clearly about how often you expect them to be online. I usually tell my students that they should be online 2-3 hours per week, on a minimum of 3 separate days.
- Establish netiquette rules up front. Be clear about expectations for respectful and professional communication. “Tone” can easily be misunderstood online.
- Encourage your students to compose their work in Word, LaTeX, by hand, or whatever works in your context,
*before*posting it online. Composing responses directly in a chat box leads to less effective communication. - Students can solve problems on paper, scan or take a picture of the solution, and upload it somewhere. Let them know that you can’t grade it if the image is unclear. Avoid using email for submitting assignments—it gets messy quickly—but provide a clear alternative.
- Ask open-ended questions. Discussion happens when there is struggle or debate, which doesn’t happen easily with yes-no questions. Ask students to interact about
*how*or*why*, not*what*or*whether*. - Some universities offer online tutoring, writing, or other forms of support. Check what is available to your students. You can also refer students to websites like Virtual Nerd, Math is Fun, Khan Academy, or Math Forum. If you hold virtual office hours or offer extra help, try to work with several students at a time so they can support one another and you can use your time effectively.
- There are applets available online for students to create and manipulate graphs. One of my favorites is Desmos, but there are many others.
- Some textbook publishers have online test banks. Google forms has a feature for creating tests and quizzes in an easy-to-use form, and for multiple-choice and fill-in responses, the quizzes can self-grade. In addition, some companies and organizations are providing access to resources during this current crisis. Even if these are not your ideal choice for assessment, you may be able to make them work for you in order to complete this semester. As my advisor would tell me when I was writing my dissertation, “‘Better’ is the enemy of ‘good enough’ .”
- Let your students know that many internet providers are offering free internet service for a fixed period. That does not mean it will be easy for everyone, but it should help many students. Some students will still need to be online in a library or other public space.
- Many group and interactive activities can be adapted to an online setting. For group work, develop ways that all students are held accountable to their group. Assigning group grades is one option (this also reduces your grading load). Some instructors require students to make individual submissions of assignments, and then assign everyone in the group the lowest grade; this is great motivation for them to make sure their group-mates understand what they’re doing.
- No matter what you do to defeat cheating, someone will find a way to work around it. One option is to require students to find a proctor of their own (often a librarian or local teacher) who will attest to their independent work, but this may be challenging for those who are self-isolating or maintaining the recommended social distancing. You could also assume that all work is open book and open notes and design assignments accordingly.
- Talk openly with your students about what they need to know in order to be ready for next semester’s courses. If they don’t do the work honestly, they are really cheating themselves. One contributor to a discussion about online learning hosted by MAA’s Rachael Levy suggested, “I think it’s worth asking what kind of deception we’re trying to prevent. Do we want to keep our students from deceiving us on assessments, or themselves? Because I think the second one is the real danger, and might be better to address directly.”
- Here are a few activities that have worked well with my online students:
- Students write or type the solution to a problem (or problems) using words and sentences and a minimum of mathematical notation. This is challenging! But it also requires a different level of understanding of the mathematics. Communicating about mathematics deepens understanding.
- Students solve a problem and share it online with a partner, a group, or the rest of the class. They provide feedback on a specific number of others’ solutions (for example, if they are in groups of 6, I might require them to comment on 2 other solutions) within a few days. Then they submit a written solution to the problem, using someone else’s strategy. This requires them to communicate with their classmates enough to understand someone else’s reasoning.
- Give them personalized problems to solve. For example, in one assignment about compound interest with regular savings contributions, each student submitted calculations for their personal retirement savings plan. They identified how many years they had until retirement, how much money they wanted to have access to each month, and then calculated their required monthly contributions. This way, everyone could help one another because they were working on the same task, but each had to do independent calculations.

These resources provide solid guidance to help you get started online:

Stanford University Teach Anywhere

The Chronicle of Higher Education: Move Online Now

Get Up and Running with Temporary Remote Teaching: A Plan for Instructors who Lecture

Move Your Course to Remote Delivery

Seven Practices for the Online Classroom

A mathematician describing his experience teaching online

Here you can find ideas for specific online learning activities

You might want to share this one with your students: Tips for students to participate in online group work and projects

YouTube playlist: Ideas for new online faculty

And everyone’s big concern, online assessment:

Self-grading quizzes in Google Forms

And make use of the two excellent resources cited at the top of this blog, MAA Connect and the STEM faculty blundering through remote teaching in a pandemic Facebook page.

This is by no means a comprehensive list of what to do or how to do it. There are as many ways to teach online as there are online instructors (maybe more). Feel free to share your ideas and questions in the comments so we can all help each other manage this transition and provide our students with quality learning opportunities.

Some of these ideas are drawn from resources developed by Open SUNY Center for Online Teaching Excellence of the State University of New York, and the Institute for Teaching, Learning, and Academic Leadership at the University at Albany.

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Ray Levy, Mathematical Association of America

*This is cross-posted in MathValues and Abbe Herzig has written a companion post. Additional resources and future meetings are also available here: https://tinyurl.com/OnlineTalkshop.*

In times of crisis we need community. With schools, colleges and universities mandating online teaching and learning in response to COVID-19, often with only a week of preparation time, people are scrambling for resources and information. Dr. Ray Levy, aås Deputy Executive Director of the MAA, asked an online group whether they would like a Zoom space to discuss online learning. With only 12 hours of notice, Dr. Jeneva Clark helped co-facilitate, and 36 people gathered. The next day, with only a few hours of notice Dr. Abbe Herzig and Dr. Yvonne Lai joined as co-facilitators, and 86 people gathered. Below is some of what we learned in the second meeting.

One point worth emphasizing is that online courses can look very different! Courses may have only text, audio, or video. They may be “synchronous” or “asynchronous”. These terms, originally from communications, to refer to how data is transferred. In online courses, this usually means whether everyone is online at the same time (synchronous) or online at different times (asynchronous). Often a course will contain multiple communication approaches.

Because many of us have not taught in these ways before, we discussed logistics and accessibility. Participants noted that FERPA (Family Educational Rights and Privacy Act) impacts sharing.

In this post, we discuss:

- Logistics of using Zoom for synchronous teaching
- Accessibility
- Writing on the board, online
- Ways to help students stay engaged
- Asynchronous teaching
- How to hold office hours
- Sharing recordings and FERPA
- Open questions

We are writing this because we believe that other mathematics instructors can benefit from the ideas shared by the participants. However, we are not speaking as experts and these are crowd sourced ideas rather than MAA or AMS recommendations! We are only speaking as peers who have had some experience, and who want to share key things that were helpful to us when we began teaching online.

First, many people talked about running courses on Zoom. This is partly because the MAA uses Zoom and we were using it for the meeting. We don’t know in the coming days how Zoom’s infrastructure will hold up with increased demand. However, if you are able to get Zoom or another platform to work for you, here are some things to consider:

- There are different grades of platforms, based on how much you pay. For instance, Zoom’s free version only allows for meetings of a maximum of 40 minutes. Their least expensive paid version (Pro) allows for 24 hour meetings, the option to record meetings. Webinars have some nice features for large groups, but it seems like only Meetings have the whiteboard and breakout room features.
- Make sure that you are in a well-lit room. If you are backlit, the camera software may result in a view of you where your face is completely in the shadows. This means that students who need to read lips won’t be able to. This also means that students will lose nonverbal communication cues.
- As well, practice keeping your head still. If you move around, the camera software can change what is light and dark, which can be distracting or result in backlighting.
- Practice logistics like muting microphones, sharing screens, recording, making sure you know how to see all participants, getting used to different “views” available, or grouping people.
- On Zoom, you can share the window for only one application or share your whole screen. This means that participants may not be able to see your mouse cursor, and it also means that they may or may not see information from any other windows. Watch out for privacy issues with email or pop up notifications.
- It can be helpful to work with a computer attached to an external monitor to give you more room to organize screens.

Once you are comfortable with a conferencing platform, here are some in-class tips:

- When starting class, you will need to allot extra time in the beginning as people “enter” the virtual room. Have some things for students to do during this time. For instance, Ray suggested that you can vary from journaling (“How’s the course going?”) to more personal (“What music should I listen to?” “What’s your favorite TV show?”) . It might be even more important in the next few weeks, as we isolate ourselves, to ask personal questions. Georgia Stuart suggested “quiz yourself” prompts, which could lead to students coming up with questions to ask you and help you see where students are. To show these prompts to students right away, you might put the prompt on a PDF or other kind of file, and screen share that file. (This is similar to the function of “warm up” problems that students are expected to do as they enter the room.)
- On Zoom, the chat box only shows you chats that you were present for! If a student logs in after 10 minutes, they will not see any chats that were sent in the 10 minutes before they arrived. So if there are important links or information to share with the class as a whole, you cannot rely on chat to do this; you need to announce to the class or put an announcement in your course management software.
- You might prepare some documents ahead of time to share, such as PowerPoint or Beamer slides, a google doc, or images. You can use these documents to have prompts for the class, or equations or diagrams. Test run any slides before using them, so you know what they will look like to you and your students. You might schedule a practice session with some friends where you take turn being the host or participant.

Across the group, we had experience with:

- Zoom white board
- Wacom tablet
- iPad with apps Notability, ExplainEverything, or Doceri
- Surface Pro tablet using PowerPoint annotation or Microsoft Whiteboard
- Webcam pointed at paper

Among these, none stood out as a crowd favorite; different instructors had different preferences based on what was easier personally or more familiar.

For students who need to read lips, you will want to make sure that you are in a room with good lighting, and to keep your head relatively still. As discussed above, this is a function of how Zoom and camera software operate together. Appropriate lighting serves everyone well since it can reduce eye strain.

For students who need captioning, options include using transcription services such as otter.ai (free), temi.com ($0.25/minute), or captioning services on YouTube. With all these solutions, you will need to edit the transcript manually, especially for correct technical language use; and you will also want to make sure to share video so as to be FERPA compliant. Because this was new territory to many of us, we were not as a group sure of what is FERPA compliant or not. In general, the more private, the better; and you may want to check with your institution. (If you do so, please share what you find with us!) Video captioning can also be helpful, even in a mathematics course, for students who are language-learners.

We did not talk about students with visual or other accessibility needs, but of course these are important considerations.

Learn about carrying out ADA requirements through your institution. Many institutions have support staff to help building and conducting online courses. The ADA’s website also has guidance, for instance on website accessibility.

Students who have been sent home may not have the same kind of time or space or bandwidth that they previously had. For instance, they may have additional household responsibilities, or they may not have reliable internet access. Here is a questionnaire for students, adapted from one developed by Christina Weaver:

- Where do you expect to be from now through [end of the semester]? (City, State, and time zone). If you expect to be in more than one city/state, please list that too.
- On a scale of 1 (really slow/unreliable) to 5 (really fast/reliable), how would you rate the internet connection that you expect to have while away?
- Do you expect to be available during all of our usual class times (keep in mind time zone)?
- Are there any online meeting software / video sharing apps (other than Canvas/Google) that you use and recommend? If so, tell me about them!
- What else do you want me to know? (You can tell me logistics here, or additional responsibilities, or anything you might be feeling right now.) I will not share this information with anyone else [at our college] without your permission. [Note from Ray: please be aware if you are a mandatory reporter that you may be obligated to report certain things.]

To help students stay engaged, Georgia recommended:

- Have regular due dates.
- Check in with individual students over the semester.
- Have a platform for students to talk to each other and answer each others? questions.
- Create learning activities that require students to interact at specified intervals. Be explicit with due dates.

These serve to help students’ executive functioning, the skills needed to plan, prioritize, focus, remember instructions, and handle multiple tasks. Georgia found this resource very helpful for thinking about teaching strategies: Universal Design for Learning’s resource on Executive Functioning in Online Environments.

For student chat platforms, Georgia has used Microsoft Teams and the paid version of Slack, and allows for picture sharing and video conversations. Ray recommended Piazza. Teams were provided through Georgia’s university, and Slack (paid) were deemed FERPA compliant by her university, but other universities may decide differently. You may need to consult with your university before using a new online tool that may expose your roster information.

You want to be careful in public spaces such as chats, to be kind and constructive. Ray and Georgia both shared stories of comments they thought were gentle, but chilled chat discussion to the point that students did not use it anymore. Both strongly urged us to stay away from language such as, “You forgot to ____” or “You didn’t _____”.But even comments such as “Did you remember to ___?” have chilled chat rooms.

A method that Georgia has found helpful is to chat with a student privately about it, and then ask that student they would be okay for her to respond publicly, or to delete the comment. When students are responding to each other’s questions (which can be a huge time-saver), setting expectations about kindness can be key.

In general, don’t ask yes/no questions nor instruct; ask open ended questions. Why did you do it like that? Can you explain to me why that works?

Ray recommended reading through chats at least before quizzes or exams, to see whether there are comments or concepts that you want to address. Although you want to read through the chats regularly, there isn’t a need to check it every second. Perhaps let students know that you will aim to check at particular times. You can think of this as part of your ‘office hours.’ Some people may prefer to disable this function because of the extra time consideration.

All the following advice comes from Georgia who has taught a completely asynchronous course, where students learned to program in R and learned concepts of statistics.

She recommends developing videos, no longer than 12-15 minutes long. This is really important, so that students can focus on only one major idea per video.

In her experience, students in asynchronous courses can get behind easily if assignments are too big. To help with this, she uses the strategies listed above for course engagement. Small chunks of work with very explicit deadlines can especially help.

Online courses can blur any work-life boundary, especially for instructors that aren’t used to teaching online. With Microsoft Teams, students may send questions at all hours of the day. They may be working on their homework at 2am. You want to be careful to set boundaries, say giving yourself 24 hours to respond, especially because it’s a chat platform where usually we expect responses right away.

You can hold office hours on Zoom using a personal meeting room. Dianna Torres and others said that they have times that students know that they will be on. Dianna’s students share their computer screens with her to ask questions about homework. Some have students hold their work up to the camera.

Yvonne has held office hours by appointment on Skype, trying to schedule at least two students at a time so they can have each other to talk. Something she has found useful is to have a movable webcam. She uses this to point at paper that she writes on, and also to give “privacy”. For instance, if she wanted to give a student time to work out an idea, she has found it helpful to ask: “Do you want some time to think about this by yourself?” Usually, students say that they do. She then asks, “Would it be helpful for me to face away?” Students usually say that it would be helpful. She then says, “I will leave the audio on to help me follow up, but otherwise I will give you this time.” She then turns the webcam to point to the keyboard and listens for a lull in conversation and also for points that she wants to probe.

Briefly, and to over simplify, FERPA means that you cannot share students’ identities, assessments, or grades with anyone but the student. When teaching online, you may be recording sessions that include student talk. Sharing these sessions may violate FERPA.

According to Zoom’s website, “Zoom enables FERPA/HIPAA compliance and provides end-to-end 256-bit encryption.” To share Zoom videos, use course management software (e.g., Canvas, Blackboard). Check with your institution about FERPA requirements, ADA, and resources available to you and to students. Some universities have online tutoring available, for example.

- Assessment at scale: How do we minimize cheating on exams, especially for large courses?
- Math resource centers: How do we convert tutoring centers to online platforms?
- Group work: How do we do group work and keep students engaged?
- FERPA: What are all the implications? Legal ramifications of incidental disclosure of information are still unfolding. Instructors should probably contact their University legal department for their institution’s guidelines.

A few words on assessment. This was perhaps one of the most sobering moments of discussion. Several people asked, “Are we setting up students for failure if we allow cheating?”

We realized people are ready to have thoughtful conversations about why we test, who is harmed by issues with academic integrity, and where things should be on the scale of “let it go” to “strict” in this unusual time.

In some large lecture courses, some instructors have seen students ace online homework while failing proctored exams. While there are many reasons for this, including test-induced anxiety and impostor syndrome, there is unfortunately also the reason that students have a friend next to them who answers online homework questions for them.

On the other hand, Yvonne has had the experience of homework grades predicting both midterm and final exam grades, even when homework allowed collaboration, the midterms were proctored, and the final exam was take-home.

We hope you find at least some of this helpful. We also hope the act of gathering, sharing ideas and concerns, and struggling together will be constructive. May you find the community and support you need as we work through these next months.

]]>“I am sad this class is going to be over,” said one student. “What am I going to do with myself?” asked another during the last week of an Intermediate Algebra class that I taught last summer at the Lincoln Correctional Center (LCC) with Meggan Hass, then a University of Nebraska graduate student.

Meggan and I were sad, too. It’s not often that we hear these types of comments from students, but as I have learned, the unexpected can happen when one teaches in prison.

Here is my story.

It started with a visit by Bryan Stevenson, author of *Just Mercy*, to Nebraska Wesleyan University, where I work. Stevenson is a lawyer who defends incarcerated individuals, many of whom are on death row. In his talk, he urged us to:

“Get proximate. Get uncomfortable. Change the narrative. Have hope.”

I was sold. In the coming months, I put in a request to adjust my Spring 2018 sabbatical.

When Stevenson called for the audience to “get proximate”, he was encouraging us to be closer to those who are suffering or excluded. In response, I arranged to teach classes at the Nebraska State Penitentiary and at LCC during my sabbatical.

The classes were non-credit bearing, so I had liberties in the content. I wanted to pick a topic that wouldn’t require a strong math background but that would be interesting. The 8-week course I developed was “Combinatorics and Probability.”

Not knowing anything about the prison system, I asked basic questions of corrections staff: Will paper and pencils be provided? (yes) Can I move around freely in the class? (yes) Are calculators available? (yes) Can they work in groups? (yes) What can I bring into the facility? (notes, book, pencils, calculator) Can I bring dice and playing cards? (no) How can we determine who is accepted into the class? (basic math proficiency, no recent misconduct)

I was scheduled to meet my students for two hours each week, but the prison staff would regularly let the classes run long. For my students, the longer, the better. The chance to think about something different and be away from the drudgery of the units was welcomed. My students never missed a chance to tell me how much they loved to learn and how education was what they needed more than anything.

Although I had carefully planned the course before it began, I made big adjustments once it started. My students’ backgrounds were quite disparate, creating some unease for me. But I adjusted the material by giving them more time to “count” things using brute force, with the idea that it would facilitate the process of making conjectures. I also asked students with stronger backgrounds to help those with weaker ones. That worked beautifully.

Towards the end of the 8 weeks, one student asked me what surprised me most about teaching them. He wondered if I was scared. No, I wasn’t scared. However, I had a preconceived notion that asking students to participate would make them feel vulnerable; that made me uncomfortable as I walked into class on the first day. The big surprise for me was how willing they were to ask and answer questions. In fact, my main time management issue was trying to address all of their questions. Most questions were about the specific content, but some extended into vocabulary and notation: “Did the question mark first show up in mathematics or in other writing?” (I always researched those questions I could not answer and reported back to them. Incidentally, the history of the question mark is fascinating, and I encourage you to research it.) To cover the main concepts adequately, I occasionally omitted a few examples.

The centerpiece of Bryan Stevenson’s directive to “change the narrative” is to talk honestly about the historical roots of racism and poverty. For me, changing the narrative was more personal. It was about refocusing where I put my energies; I wanted to continue teaching in prisons.

The following academic year, I taught a course in Algebra 1 at LCC. At the prompting of the assistant warden there, I subsequently co-taught, with Meggan Hass, an Intermediate Algebra course in Summer 2019, through a program at the University of Nebraska-Lincoln (UNL), supported by private funding.

Most of the students who had taken Algebra 1 with me applied to take the Intermediate Algebra course. All applicants needed to have a GED or high school diploma. Beyond that, UNL gave me full discretion in choosing participants, so I created a simple application form and held interviews.

Among other things, I asked about the hardest thing that they had ever done. The responses gave a sense of how varied their backgrounds and perspectives were: “Working at Long John Silver’s,” “Your Algebra 1 class,” “Fitting pipes for an oil rig.” But each talked about these hard things with the kind of conviction that conveyed their seriousness about this educational opportunity.

As Meggan and I began preparing the course, a question we had to answer was how much “discovering math” we wanted to put into our course. For example, we needed to decide if we wanted our students to “discover” the rules of exponents working with others, or if it would be better to offer more guidance. The challenge I had had when letting the students in Algebra 1 spend time discovering math was that some students got distracted or overwhelmed. So, we made worksheets for each section and worked through these as a class, asking students many questions along the way. As an example, to show that 3^{4}3^{5}=3^{(4+5)}, we asked our students to write all of the factors of 3 on the left side of the equation to see why it would be the same number of factors of 3 on the right.

I have a fascination with words and so for each class period, we had a “word of the day.” We delighted in vocabulary such as *vociferously*, *chagrined*, *vexillology*, *verisimilitude*, *polyglot*, *polymath*, *apoplectic*, *belie*, *ebullient*, *capacious*, and *pacific*. Riveting discussions ensued with some of the students reaching for a dictionary to uncover the word origins.

On the last day of Combinatorics and Probability, we held a certificate ceremony. I prepared certificates with one of my favorite quotes:

“We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills”

– President John Fitzgerald Kennedy

I chose this quote because it connected to their eagerness to learn and their willingness to challenge themselves.

The students asked if they could speak. They talked about how important education was to them. One shared that few people are willing to go into the prisons to volunteer and that it was significant when people did. They asked me to tell people about what I was doing so that it would encourage more to do the same. It was emotional.

These thoughts were echoed at the end of the Intermediate Algebra class, too. In this class, there were 26 homework assignments, 3 quizzes, 3 exams, and one final exam. Of these, only one homework assignment was not turned in by one student. In fact, this student had completed it, but accidentally left it in his unit. None of the homework turned in was incomplete and it was done with exceptional care. Attendance was almost perfect; we had one excused absence and no unexcused absences during the entire course.

On the last day of class, the students presented Meggan and me with homemade pop-up cards. They each wrote a personal note of gratitude. We were equally grateful to have been their teachers. In their course evaluations, they talked about the class teaching them much more than math – one said he finally could believe in himself. They shared that the class made them feel like human beings.

It gave them hope.

There are many reminders that you are in a prison when you teach there. But, just like in any class I teach on my campus, each student comes to class with a different history, each student learns differently and each challenges me as a teacher. In my prison classes, I was stretched as a teacher in consequential ways; I had to adjust quickly to different backgrounds, find different ways to explain material and rally around my students to help them build their confidence. I came away seeing the immense value there is in offering incarcerated individuals the opportunity to learn.

* * *

*Note from the author*: If you think you might be interested in such an opportunity, and you would like an experience you might fit into a summer, I am happy to share my course materials from Combinatorics and Probability (class worksheets and homework sets). Email me at kpfabe@nebrwesleyan.edu. They are imperfect but can serve as a starting place.

In Marilynne Robinson’s *Gilead*, Reverend Ames testifies that each person in his flock has “a kind of incandescence in them… quick, and avid, and resourceful. To see this aspect of life is a privilege of the ministry which is seldom mentioned.” I think this is a privilege of the educator too. But our flock’s life cycle, the fourteen weeks from the cradle of the first day syllabus to the grave of the final grade, has a different rhythm to that of a liturgical calendar.

In a semester where I have over 100 students, witnessing each student’s luminosity is challenging, and by the time I get to know even half of them the semester is likely to be halfway through. So I ask my students to help me to see them. I ask what their values are, to tell me about the work they’ve done and are about to do, and to share their hopes and aspirations as people. I ask them about the concrete challenges that they might face as the semester progresses and what they wish previous mathematics teachers understood about them.

I do this because understanding my students holistically aids in my support of each student individually and can cultivate active learning communities. I do this because if the semester gets rough on my end — it invariably does — I have a wealth of prior empathy and insight to fall back on so that my students succeed in their goals. I do this because knowing only their placement scores and/or GPAs is not enough! There’s also a body of research supporting that when students’ values are affirmed in STEM [i], they are more likely to succeed in that course and continue as a STEM major.

I ask for their help on the first day because it is most expedient and it is the most important day of the semester [ii]: Consulting my students on that first day establishes a framework for the class, that we are starting a dance between the content of our discipline and the context from which each person comes; and in this dance both partners matter equally and each will have their turns to lead. The first day is also the liminal space between their previous experiences in mathematics and what this new course promises to be, so it provides an opportune time to reflect and to project, to celebrate and to prepare.

The questions I will discuss below are intended for junior/senior math majors but can be adapted for other populations of students [iii].

*What was your path to becoming a math major? Do you identify as a ‘math major’? How is this choice perceived by others and how do you respond to these perceptions? *

These questions are general yet typically elicit responses about formative experiences, far beyond what I originally anticipated when I first put this questionnaire together. Students tell me if their confidence is low, if they have a passion for physics, if their vocation is in teaching. They let me know if college was not the expected path for their family members, or if it was absolutely expected, each of which exerts its own distinct type of pressure. Students also tell me about their perceptions of local and global mathematics culture and how they situate themselves within it; in doing so they invariably tell me about their values.

Students also share personal issues outside of the major. Some will say they are holding down a full-time job and then I know that they might need some flexibility on deadlines [iv]. Others might be returning from some time away from college necessitated by mental health, financial, or family concerns: I tend to pair these students with partners who are more collaborative than competitive. For example, my typical Quantitative Finance or Engineering students — not inherently, but culturally, competitive — tend to be a poor match for someone who needed a break from school for a period of time.

The responses allow me to tailor my interactions with my students but also to customize some of the content of the course. For example, when I am assigning projects in the second week of Abstract Algebra, I can assign a reading project on matrix groups to the student who is very interested in physics. Similarly, if the class has future teachers then I will assign a project on transformation groups relating to isometries and how that relates to the state’s high-school mathematics curriculum.

The next questions ask the student to reflect on their time at university. It asks for a self-assessment of what courses went well, why they are here for this course, and the extent to which they have learned from positive and negative past experiences in collegiate mathematics. These questions are implicitly asking the students to transition from values to responsibilities, to a sense of ownership of their education. The questions also give me a very practical insight into their mathematical motivation and the gaps in their existing knowledge we might need to address.

*What’s your favorite area of mathematics? Your favorite ideas/results? What course do you wish you could have avoided? What course do you wish you could take again? What would you tell yourself if you could travel back in time and talk to your freshman self? *

My final questions concern mentorship. Knowing my student’s values and motivations, how can I support my student’s future plans? The responses can also be used to calibrate the course’s emphases to where they can have the most impact.

*What are your future plans in terms of vocation? Do you hope to enter into a job in industry? Are you preparing to be a high school teacher? To go to graduate school? To run for the hills, join a commune and live off the land? What non-math skill do you wish you were better at?” *

I distribute these questionnaires on the first day of class and ask the students to return on the second day of class with their responses. At no point do I make these responses public. The questions demand vulnerability on the student’s part so I am compelled to reciprocate by trusting my new students with my own answers to these questions. I do so by responding verbally to these same questions on the first day of class. I answer as I think my twenty-year-old self would have and also how my middle-aged self understands these responses today. I worry that I’ll look like a fool but it never works out that way.

For an “active learning”[v] context, where vulnerability and play are partners of discovery and rigor — a joining of many dialectics in the hopes of achieving a synthesis — the affirmation of students’ values may be yoked to their success in the class.

Miyake et al. provided evidence that when women in an introductory physics class “wrote about their most important values” twice in the semester, the discrepancy between male-female outcomes reduced substantially. Further, that “the benefits were strongest for women who tended to endorse the stereotype that men do better than women in physics.” In a different STEM context, Jordt et al. examined the discrepancy between underrepresented minority students and white students in introductory biology classes and concluded that the relative “underperformance of URM students could be mitigated” by a similar values intervention. The biology education lab of Sarah Brownell has more investigations along these lines.

I would suggest — as an axiom rather than an empirical finding – that the more a person feels affirmed in class the more likely they are to wholeheartedly participate in groups with others: the more likely they are to trust, the more likely they are to take risks. Indeed, group work without intentional individual affirmation is more likely to promote a tepid conformity — the “just get it done” treatment of the worksheet that defers authority to the student perceived as the group’s strongest or, failing that, the instructor — rather than the form of radical consensus that we as instructors wish for. The type of consensus I mean is one where each participant seeks to hear the other and where students’ collective achievements surpass their individual struggles with the material. It’s my suspicion that if we sincerely engage in an active learning practice without paying attention to individual affirmation, we risk cementing real and perceived hierarchies among students and, in doing so, exacerbate the very types of privilege and status quo that we might be trying to supplant — not just in our class but, more broadly, in our discipline’s culture. This is why affirmation matters.

Returning to *Gilead*, the epistolary work referenced in the introduction, Ames writes with the urgency of a dying father anticipating who his young child will be and what he will need from the world. He professes a nuanced and practical faith with decency, introspection, and a transparent and messy imperfection. For the teacher, the stakes are lower than those in Ames’s letter but the exercise, imperfectly anticipating the world as a young person might experience it, requires Ames’s same sense of optimism leavened with a dusting of realism and the acceptance of frequent futility.

To help our students navigate the outer world and the universe of our discipline, we can ask them to share what they think of as their true north, their stars within and beyond our discipline, the terrains they’ve passed through. We can ask which mountains they want to climb and let the contours of our courses, our programs, and our discipline’s culture be influenced accordingly. But if we want to know the answers to these questions, we have to ask our students. And honor the trust they place in us when they answer. In doing so we give our class a greater chance to become a community in which everyone is affirmed as a person, where collective risks are more likely to be taken, and where opportunities are increased for every student to join our discipline fully.

[i] Setting aside my deep reservations about the use of the term “STEM”; I’d prefer if we called it “MASH,” dumping engineering and technology for the arts and humanities, cf. https://sententiaeantiquae.com/2018/11/28/science-and-humanity/

[ii]The importance of the first day is in Chapter 2 of *Geeky Pedagogy: A Guide for Intellectuals, Introverts, and Nerds Who Want to Be Effective Teachers, by *Jessamyn Neuhaus. West Virginia University Press, Morgantown, West Virginia, 2019.

[iii] The questions I ask for introductory classes like Precalculus can be found at my website: http://educ.jmu.edu/~osheaem/teaching/who-are-our-students.pdf

[iv] Art Duval advocates for flexibility too in “Kindness in the Mathematics Classroom,” (this blog, Feb 2018). Duval’s main question is in response to Francis Su’s essay on “grace” in the classroom, how does one practically implement Su’s vision for mathematics praxis?

[v] As expected, “active learning” is quite the broad church. My own style emphasizes the centrality of reading outside of class, some elements of flipped classrooms and IBL, and mastery-based assessment. See “What Does Active Learning Mean For Mathematicians?” by Benjamin Braun, Priscilla Bremser, Art M. Duval, Elise Lockwood, and Diana White, Notices of AMS, February 2017.

[vi] Thanks to Cynthia Bauerle for pointing me to some of the science education literature on affirmation.

]]>Executive Director, Math Circles of Chicago

The New York *Times* recently published an article entitled “The Right Answer? 8,186,699,633,530,061 (An Abacus Makes It Look Almost Easy)”. Its lead photograph features over 100 children seated at desks, facing forward, working individually. This is yet another in a long series of public relations disasters for mathematics. This depiction of mathematics is nothing new, and I most suspect readers experienced no cognitive dissonance in seeing mathematics represented this way.

*Traditional Forms of Math Enrichment — and the Problem with Contests*

Mathematicians collaborate to explore exciting open-ended questions. Unfortunately, this may be the world’s best kept secret. The problem? We have two major gateways to participation in the community of mathematics: the classroom and the contest. The classroom, of course, is universally familiar, and innumerable efforts have been made to improve the student experience in the classroom.

Here I focus on the second gateway, the contest. When I say ‘contest’ I refer to the kind that looks like that Abacus competition pictured in the newspaper. These are quite common. For the majority of children competing for the first time, contests have a number of common features:

- A large group of students gather in one location. They do not talk to each other.
- Students complete closed-ended problems with unique correct answers, which is necessary so that responses can be judged impartially.
- Contest problems are generally predictable. This allows students to memorize formulas and tricks to save time during the contest. Frequently students prepare for contests through practice problems that turn contest
*problems*into*exercises*—known results that can be completed efficiently to save time during the contest, preserving time for checking answers, since partial credit for thinking is not given. Speed is prized. - At the end, four classes of prizes are awarded: 1st place, 2nd place, 3rd place, and no place — with this last group the mode by a wide margin. One can argue that this is a feature of any competition, but the ramifications of competition in a subject as fundamental as mathematics are quite serious. Giving up on math is not same thing as giving up on chess or basketball.

Again, there are many exceptions to this general description of contests, but those exceptions are usually experienced by those who have previously been successful in close-ended contests. One can argue students can get exposed to beautiful problems that they themselves might extend. A colleague once told me that the best part of the contest was after papers were submitted and kids started to “sing the contest” — eagerly talking to their friends about the most interesting problems.

I’ve seen students benefit from contests. But I do think it’s time to reflect on some of the downsides of this kind of enrichment. As we seek to expand access to the world of mathematics, do we really want our main form of math enrichment to narrow the gateway?

*More Subtle Problems*

I’ve known many high school students who have done well in contests and were subsequently motivated to major in mathematics. In turn, I suspect there is a selection bias that supports math contests as the main form of math enrichment. If you didn’t do well in contests, you didn’t major in math, and therefore you haven’t stuck around later on to question the practice.

For others, contests have a negative impact on identity. As I mentioned, “no place” is the most common ranking for the large majority of contest participants. **Contests send and reinforce a fixed mindset message.** Children often compete as part of a team from their school, and schools with more resources tend to perform better. But when less-prepared children from underresourced schools compete, they may see other children achieving at a higher level, and may be led to believe that they just aren’t as good as others out in the wider world. This is particularly pernicious when it comes to underserved communities and communities of color.

Teachers are often inspired by contests, benefit from writing them, and gain personal connections with students that they might not otherwise have formed. But, as I mentioned, preparing for contests prioritizes efficiency over depth. Few contests reward alternative solutions or depth of thinking. Speed matters; hence, game theory dictates that we ought to teach procedures.

Finally, and perhaps most subtly, the world of math enrichment largely embodies a “Field of Dreams” approach: “If you build it, they will come.” My question: *who* comes? If you look at the participants in existing contests, you are more likely to get boys, children from more affluent schools, and people who have the social capital to know where to find the contests. And there is a ‘Matthew Effect’–minor advantages accrue early in life, so that by the time children are participating in contests, what seems like a fair assessment of talent is really just a piling on of advantage.

The math enrichment gateway needs to change, or the usual suspects will be the only ones making it through the gate. We will continue to lose underdeveloped talent—children with latent ability who will never reach their mathematical potential. Career choice, economic mobility, and civic engagement will continue to be unnecessarily limited for many.

*What Might We Do Instead? The Case for Math Circles*

In the last five years, the single greatest impact on my thinking about teaching has come from Alan Schoenfeld’s Teaching for Robust Understanding (TRU) Framework. TRU asserts that to create powerful math learning environments, we need to attend to five crucial dimensions: mathematical connections, cognitive demand, access, agency/identity, and formative assessment. The ongoing program of research and practice of Schoenfeld and his team is working to show that these five conditions are both necessary and sufficient for robust learning to occur, and to explore effective, efficient ways to make them happen.

TRU is being applied in classrooms and it applies equally well to math enrichment settings. Math circles, festivals, and summer camps can be designed to be equitably accessible in a way that contests simply cannot be. Students exploring open ended problems for a math symposium have an opportunity to experience agency at a level that neither the classroom and the contest rarely provide.

Let’s consider math circles. Math circles usually occur outside of regular school hours, where interested children investigate novel mathematics in sessions led by an adult with a strong affinity for math.

Like math contests, math circles can be cognitively demanding. But the phrase ‘cognitive demand’ can be deceptive. Sometimes it’s read simply as ‘hard’. But — ironically — making mathematics hard is not difficult. TRU describes cognitive demand as: “The extent to which classroom interactions create and maintain an environment of productive intellectual challenge conducive to students’ mathematical development. There is a happy medium between spoon-feeding mathematics in bite-sized pieces and having the challenges so large that students are lost at sea.”

Contests rarely provide appropriate levels of cognitive demand to a broad range of students. A math circle, with a more classroom-like environment can be designed to provide that “happy medium” for individual students. Students can work at their own pace and have a personally rewarding experience.

The advantages of a math circle become fully clear in other dimensions of the TRU framework. Because they do not center around competition, math circle sessions are more welcoming spaces for a broader range of students. We can make our sessions accessible. We can improve the likelihood that more diverse students identify with the subject, consider it to be fun and worthy of long-term pursuit. Problems can be fine-tuned to become easier or harder based on the pace at which a given student digests the new material.

And, let’s not forget the mathematics itself. According to the TRU Framework, in order for a classroom to be mathematically powerful, “The mathematics discussed is focused and coherent, …[and] connections between procedures, concepts, and contexts… are addressed and explained.” Typical contests throw disparate problems at students, where mathematical connections between problems can be non-existent. Compare that to a math circle—based, perhaps, in one main rich activity, or a chain of problems that feature a strong connections that can reveal surprising and beautiful results.

*What’s next for math circles?*

While I see the potential of math circles, I also recognize that if we do not implement them thoughtfully, we can end up reproducing many of the shortcomings of contests. The field of dreams approach can still limit who attends, and we may end up serving the same children we served before with contests. Perhaps those children will have a better experience than in competition, but the audience might still be made of the usual suspects. We need to consider where (and when) math circle meetings are held; whether we can hold them free of charge; and how we can forge partnerships with community leaders to thoughtfully recruit children in underserved communities.

Moreover, as we consider serving a more diverse group of children, we need to consider how to serve those children effectively. A math circle leader needs to know how to teach well. They should also know how to build connections with and between students, while also having a deep understanding of the mathematical connections at hand. (I admit this is a significant topic unto itself, and that teaching well is an enormous challenge—but it’s absolutely essential that it be addressed for this enrichment based in student collaboration to be successful.)

Four years ago, I became the Executive Director of the Math Circles of Chicago (MC2). We aim to provide more equitable access to high-quality math enrichment. Since I began, we’ve quadrupled in size and now serve over 800 students.

We strive to make math enrichment accessible by reducing economic hurdles, fighting barriers introduced by geographic and school segregation, attending to student identity during sessions, and promoting interpersonal connections.

In 2015 MC2 had three sites, two on Chicago’s north side (the wealthier and whiter part of the city) and one on the near south side. Four years later we have circles at eight sites that are geographically dispersed around the city. This fall we opened a new site meeting on Saturday mornings in Back of the Yards, one of Chicago’s poorest communities, with an enrollment of over 60 students.

All of these programs are free to families.

Our population of teachers has diversified as we have grown. Initially many of our leading teachers were in graduate programs at universities like UIC, UChicago, and DePaul. While we continue to add more such teachers, we now have many more classroom math teachers, both from middle schools and high schools around the city.

The inclusion of teachers, particularly those teaching middle school, means that as a community that we have much more institutional knowledge of working with younger children (we serve kids in 5th to 12th grades, and in practice more than 80% of those served are middle schoolers).

We also provide workshops for our teachers—from 2015-2018 Dolciani Math Enrichment grants funded the development and implementation of workshops like, “What is a Math Circle?”, “Math Circle Teaching Basics”, and “Intermediate Problem Solving”, along with workshops built around group observations of math circle sessions. These workshops build connections between our teachers and build an esprit de coeur.

MC2 is still very much a work in progress. We need to improve in the evaluation of the work we are doing, particularly to measure whether it’s effective in meeting the needs of the varied children we are trying to serve. It takes time to build relationships within a community, with teachers who can both lead sessions and who can help advertise this opportunity effectively.

Messaging matters, and over time we’ve found that expressing clear core values helps us find the right teachers, community members, and family members.

- Math should be fun and empowering.
- Every child can do and can enjoy rich mathematics.
- Every child deserves equitable access to rich mathematics.
- Students should be agents of their own learning.
- Math can and should be collaborative.

This stance has made us discourage the use of the word ‘gifted’. We certainly draw many well-prepared students (among others), but we are careful to signal that the math experiences we offer are for everyone.

I started by talking about the popular image of mathematics, but changing that image is not enough. The substance of the experiences we provide—Competitive or collaborative? Cooperative or individual? Mathematically connected or disconnected?—will determine whether we can attract a broader audience to the intellectual joy that we know mathematics can offer.

The mathematical community is powerful. High stakes tests make mathematics a pathway to many opportunities, in college, in careers, and in our ability to influence our communities. Many of us spend a significant amount of time writing, coaching, and judging contests. I think it’s time to question how much time is spent this way, and how, proportionally, we might spend more time involved in collaborative, cooperative, connected mathematical experiences that provide access to the many, and a deeper experience for all.

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Texas State University

“[Functions] are completely different, which is what makes this course so challenging.” – Abstract Algebra Student

Functions are hard for students, even students in abstract algebra courses. Even if students have seen the definition and worked with examples of real functions throughout high school and college, their understanding might be stretched to a breaking point when it comes to ideas like homomorphisms on groups or rings. Fundamentally, we might know students don’t understand functions, but the extent to which they don’t understand functions goes deeper than we might think. In this blogpost, we will share some insights from our projects on key places where students’ idea of function can be detrimental to learning concepts of abstract algebra, and what we as instructors might be able to do about this.

The theme of our work is fortifying students’ understanding of functions while also forging through an abstract algebra curriculum. The issue of filling in gaps in prerequisite knowledge while still staying faithful to the current course is a challenge. We take on this challenge and hope that this post spurs others to take on this challenge with us, whether you teach abstract algebra or calculus, or any other course that relies on functions.

We have organized the post by projects led by the authors. We give an overview of each project and then describe what we have learned from each.

Project 1: The Group Theory Concept Assessment (GTCA)

Melhuish began developing this assessment as part of her dissertation, when she realized that there was no instrument that looked at student understanding of group theory (Melhuish, 2019). This assessment’s results, including over 1,000 responses from abstract algebra students over the past 5 years, as well as interviews with students about questions on the assessment, got us thinking: how do abstract algebra students think about function? Here’s one example. (And if this gets you curious as well, there are more examples in Melhuish and Fagan (2018).)

Function Understanding Thought 1: Abstract Algebra Students May Need a “Process” Understanding of Functions

Question (condensed): What is the kernel of ϕ(n)=i^n, ϕ:(Z,+)→(C,*)?

In total, fewer than 50% of students answered this question correctly. Many students, to our surprise, answered this item by providing a singleton element such as “{4}.” Yet, when we interviewed students about their responses, they typically provided an accurate definition that even contained language like “elements that map to the identity.”

We hypothesize that many abstract algebra students may still concieve of functions as “actions” (Breidenbach, Dubinsky, Hawks, & Nichols, 1992), meaning that they are not looking at a function holistically but rather element-by-element. Although this can helpful for direct computations such as “What is ϕ(4)?”, it is less helpful for finding preimages, particularly when the preimage has multiple elements (as kernels often do). For this, students may need to understand functions as a “process,” meaning that the function can be appreciated beyond acting on single elements. Students need to develop a process understanding to master early results in abstract algebra, such as the quotient struture of the First Isomorphism Theorem.

How might we address this? We suggest getting students to get their hands dirty with homomorphisms that are not one-to-one. Although students can recite the fact that a pre-image can have multiple elements, they might not have readily avaliable examples. After exploring examples, instructors can ask questions like, “How do you find all the elements that map to a particular location?” Students then reflect out loud and explicitly on how to generalize what they have explored. We also find it helpful to use relation diagrams during these discussions, such as those shown in Figure 1. Although these kinds of questions and diagrams can be used at many levels, including high school, we suggest that they are still worth taking up in advanced courses.

Project 2: Function Coherence in Abstract Algebra

As a next step, we interviewed a group of students to better understand how they connected functions in abstract algebra with their previous knowledge of functions in calculus and algebra. Our goal was to get a better understanding of their struggles to understand functions in an abstract setting (Melhuish, Lew, Hicks, & Kandasamy, 2019).

Function Understanding Thought 2: Abstract Algebra Students May Not Have a Mathematically Complete Definition of Function

We asked students: What is the definition of a function?

Across the six students we interviewed, only one could provide a complete definition for function (Student B). Here are the definitions the students gave:

Student A: A function is a relationship that maps members of the domain to a member of the range.

Student B: A function is a relation from one set to another where all the elements in the domain should be mapped to at most one element in the co-domain.

Student C: A function is a relation that takes an input and assigns it to exactly one output.

Student D: [An] equation that will do the same kind of operation to an input to get an output.

Student E: A function or a mapping takes a domain to a codomain following set rules.

Student F: A relation between two sets; can be one-to-one, onto, both, and/or neither.

When asked about their definitions, these students often appealed to the vertical line test (or in some cases, incorrectly, a horizontal line test.) However, in a setting like abstract algebra, this graph-based test can cause problems. When we graph a function, we study it analytically rather than algebraically.

How might we address this? We suggest not only stating the definition of function, but also getting students to talk about how this definition relates to maps in abstract algebra. Instructors can draw diagrams of homomorphisms, as well as show tables of homomorphisms, and help students see that one output is assigned to each input. By going back and forth between these diagrams and the formal definition of function, students have a better chance of seeing that the definition applies to abstract algebra.

Function Understanding Thought 3: Abstract Algebra Students May Not See Functions in Abstract Algebra as Functions

We asked students: Are the “functions” in abstract algebra the same as the functions in your prior mathematical experience?

While three students did think functions in abstract algebra were like prior functions, but with domains no longer limited to numbers, the other three students did not. One said, “What I thought of about function is always something to do with graph”; and another said that functions in abstract algebra don’t have a particular “rule.” These second three students—who focused on how different functions seemed in abstract algebra—struggled with the homomorphism tasks. Some believed that non-functions could still be homomorphisms. The first three students had more success with homomorphism tasks.

How might we address this? Here is a task we have found helpful: Which of the diagrams in Figure 1 could be a homomorphism? Which could be an isomorphism? Many students state that a diagram like the top-right could be a homomorphism (even though it is not a function.) This exploration could lead to discussion that both emphasizes the definition of function and explains that homomorphisms must themselves meet this requirement.

Figure 1. Relation Diagrams. Each arrow points left to right, so these are diagrams of relations from G to H.

Project 3: Orchestrating Discussions Around Proof (Easier Said Than Done)

We are currently developing curriculum materials for abstract algebra instructors who want to facilitate deeper classroom discussions while still meeting coverage goals. You may have noticed similarities in our suggestions so far: talk about definitions and properties more explicitly, and use function diagrams. These suggestions show up throughout in our materials. In this section, we share some obstacles that emerged, and also say why these types of moves were ultimately productive.

To begin, we turn to students’ responses to a sample proof that isomorphisms preserve the property of being Abelian.

Figure 2. One sample proof.

Function Understanding Thought 4: Explicitly Analyzing Proofs for Function Properties Can Be Powerful, but Students May Need to Disentangle the Function Definition (well-defined; everywhere defined) from Function Properties (1-1; onto)

We asked students, “Do we need a function to be one-to-one and onto in order for the Abelian property to be preserved? Where do we see these properties in the proof?” (Consider the sample proof in Figure 2.)

Initially, we wrote this question thinking that that students would recognize that, in this proof, the fact that an isomorphism is “onto” is critical, while being “one-to-one” is not. However, we found that students identified the need for onto, but also (incorrectly) argued that onto was also needed to show the existence of ϕ(g) of any g∈G. Later in the proof, the students identified claims like ab = ba implying ϕ(ab)=ϕ(ba) as a consequence of the function being one-to-one.

In conversations with the two groups of students who have tested this unit thus far, we have found that abstract algebra students have not disentangled everywhere defined and onto, and 1-1 and well-defined respectively. The properties of 1-1 and onto play substantial roles in a number of abstract algebra proofs. However, we rarely contrast them directly with function definition properties which may leave properties conflated.

How did we address this? We used the students’ claims about where 1-1 and onto were needed as a springboard to return to the definition of function, 1-1, and onto. This allowed students to discuss what well-defined (everywhere-defined) meant, and how well-defined differs from 1-1 (onto). Through these conversations, the students disentangled the properties and eventually arrived at the conclusion that the property of “onto” is needed, but 1-1 is not.

Function Understanding Thought 5: Function Diagrams Can Be Powerful, But Students May Need to Support to Understand and Leverage Them

We asked students: How can you represent the function in the proof (that isomorphism preserves the Abelian property) using a diagram?

In an attempt to help students sort through these functional attributes and properties, we asked students to use function diagrams to make the ideas more concrete. To our surprise, students struggled with a number of aspects of representing function as diagrams.

Several students seemed to lack attention to the domain (G) serving as the first puddle and the co-domain (H) as the second. Although they labeled the puddles G and H, in numerous instances, the students also labeled image elements in the domain, obscuring where the elements came from, as in Figure 3. The students’ diagrams also illustrated a disconnect between notation such as ϕ(b) representing the image of b. In Figure 4, the student mapped element b to ϕ(a) and not ϕ(b).

Figure 3. Student function diagram displaying difficulty with technical aspect

As the proof contains numerous references to the image of particular elements, the lack of alignment between students’ diagrams and function notation may suggest that students’ understanding of functions (and the image of elements) may obscure important aspects of proofs. At the same time, this lack of alignment gives an opportunity for the instructor to help students develop stronger and more coherent understandings of function.

Figure 4. Student function diagram displaying difficulty with conceptual aspect

How might we address this? Although we have identified some unexpected obstacles that instructors may run into, we found overall that engaging students with function diagrams did serve an important role. Our recommendation is to prompt students to connect proofs to such representations. Through this process, it may become clear that students have some technical and conceptual issues related to functions that may otherwise stay hidden. By having a concrete representation, instructors and students can work to modify the diagrams to have normative features- and ultimatley leverage them as tools to analyze proofs.

Final Thoughts

This post is not meant to provide a thorough analysis of students’ function understanding. Instead we aimed to highlight some of the many roles understanding (or lack thereof) of functions may play in abstract algebra. As instructors, we may overlook something as simple as students having an accurate definition for function or seeing the functions in abstract algebra as examples of functions. However, these oversights might hide the fundamental role that functions play in an abstract course. In a setting like abstract algebra, students are stripped of many of their tools for understanding functions, such as relying on using graphical representations. Moreover, students in abstract courses often have to move beyond an action conception of function to a process conception. Additionally the role of function definition properties are essential when constructing or understanding proofs. However, we often leave the connections to function properties implicit. This is a disservice to students who find themselves in a setting where tools like the vertical line test cannot substitute for a function definition.

We have found much potential for explicit discussion around the definition of function, properties of functions, and function diagrams support students as they engage with various functions in abstract algebra. The biggest recommendation we can make is to not let functions continue to hide behind the curtain. Functions are everywhere in abstract algebra, and students’ understanding of functions can position them for varying levels of success (or lack of success) when engaging with abstract algebra tasks.

References

Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational studies in mathematics, 23(3), 247-285.

Melhuish, K. (2019). The Group Theory Concept Assessment: a Tool for Measuring Conceptual Understanding in Introductory Group Theory. International Journal of Research in Undergraduate Mathematics Education, 1-35.

Melhuish, K., Lew, K., Hicks, M., & Kandasamy, S. (2019). Abstract algebra students’ function-related understanding and activity. In Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education (pp. 419-427).

Melhuish, K., & Fagan, J. (2018). Connecting the Group Theory Concept Assessment to Core Concepts at the Secondary Level. In Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers (pp. 19-45). Springer, Cham.

]]>**I.**

What do primary/secondary math educators think of the teaching that happens in colleges? And — the other way around — what do mathematics professors think of primary and secondary math teaching?

I’m nearing my tenth year as a primary and secondary classroom math teacher, and every once in a while I end up in a conversation with a graduate student or professor who suggests (politely, almost always!) that math education before college is fundamentally broken. A few weeks ago, a mathematician told me that PhDs are needed to help redeem secondary teaching from its “sins.” Once, at the summer camp where I teach, a young graduate student told me that there is simply no real math happening in American schools.

Well — I disagree! But how widespread is that view? And why does it exist?

The flipside phenomenon is also interesting. When a mathematician criticizes primary/secondary math education, primary/secondary educators sometimes lash back. Often we point our collective finger at pure lecture. Primary/secondary educators tend to think of pure lecture as uniquely ineffective. It gives the teacher no knowledge of whether students understand the material, and students no chance to practice new ideas in class. It is rarely used in primary/secondary math classes. Still, pure lecture was the main teaching mode in my own college classes, across subjects. We therefore bristle at mathematicians critiquing our work; “Let those without pedagogical sin throw the first stone!” I’ve even said this before, or something not far from it.

I have heard this “anti-lecture” critique expressed by some primary/secondary educators, but I wonder how widely held this view is. Is it held even by some math professors? And, in general, do primary and secondary educators tend to see flaws in the way math is taught in colleges?

In short, I wanted to better understand how mathematics professors and educators of younger students relate to each other’s teaching.

Needing a way to approach the question, I created a survey and shared it on social media. I asked people to share their job, the highest degree they have earned, and their views on primary, secondary, and post-secondary education. So far, thirty-four people working in math or math education shared their reactions to these three statements:

- “Primary (K-8) math teaching is generally effective.”
- “Secondary (9-12) math teaching is generally effective.”
- “College (or University) math teaching is generally effective.”

They rated their agreement/disagreement on a scale of 1 through 5 and explained their response at whatever length they chose.

Disclaimer: since the survey made no effort to be representative, themes and patterns that emerged from the responses are at most suggestive. Nonetheless, suggestive themes and patterns did emerge, some of which surprised me.

There were three questions I wanted to more clearly understand:

- What exactly
*are*the issues people have with math teaching at the elementary, secondary and post-secondary levels? - Do people tend to have a rosy view of their own setting while leveling harsh critiques against what goes on in other settings?
- Where do extreme views of different educational settings come from, and why are they sometimes so deeply held?

The background question, though, the one driving this whole project, is whether there is any chance of coming closer together. What are the best ways for us to learn from each other?

**II.**

One mathematics PhD who responded to the survey rated the effectiveness of primary math teaching as just 2 out of 5. This same person rated secondary and college math both at 4. Here is how they explained the ratings:

My experience with a couple of districts is that the primary teachers up through Grade 4 tend to feel uncomfortable with math, so they do their best to opt out of teaching it as much as they can. I have some horror stories, including concerned parents of students who brought their math workbook home at the end of the year untouched.

Here is a college mathematics professor voicing similar views:

I think many elementary teachers are math phobic or have math anxiety, and this impacts how much time they spend on math. Also weak skills and knowledge of teachers could lead to misconceptions for students.

This view — that primary teachers tend to be uncomfortable with (or ignorant of) math, and therefore avoid it — showed up again and again in the survey responses.

I was expecting this, because it is maybe the most prominent critique of math education at any level. It’s the sort of thing that, every so often, pops up in the New York Times. A prominent version of this complaint, for instance, comes from Hung-Hsi Wu:

My own observation is that among teachers, especially elementary teachers, their prolonged immersion in textbook school mathematics has often rendered them incapable of routinely asking why, much less looking for the answer.

What surprised me, though, was how frequently this concern (or ones like it) was also voiced by those working in primary and secondary education. In fact, the math PhD quoted at the top of this section is actually a high school teacher. Meanwhile, a 5th Grade math teacher rated primary teaching’s effectiveness at 2 out of 5, and explained that what primary teachers need most of all is the help of math pedagogy specialists. They can’t handle the math on their own, it seems. Likewise, a high school department chair blamed “lack of content knowledge” for a “strong focus on algorithms” at the elementary level.

The second half of this department chair’s worry — the “strong focus on algorithms” — appeared often on the survey as well, though expressed in slightly different ways. Here is a sample, along with the respondent’s professional role and their rating of primary education’s effectiveness:

Too often students are taught processes instead of concepts (8th Grade math teacher, 3 out of 5)

In the memorization and skills-based way mathematics is taught in most K-8 classrooms, I do not think teaching is effective. (High school math teacher/community college adjunct, 2 out of 5)

Are there shades of distinction between “algorithms,” “processes” and “memorization”? Maybe. But I think there is more in common than not in these complaints, and they are probably trying to say something like what Wu said above — there is no *thinking* going on in too many primary classrooms. The reason? Because teachers fear or misunderstand mathematics. I would say that this picture is the major critical narrative facing primary education.

To put my cards on the table, I think this narrative probably overstates the problem, though it gets at something real. I’ve heard something like this story told by many primary teacher educators and coaches — people who really would know. I do think this critical story misses two important things: (1) what looks like a mindless call for a procedure to an adult is often a thought-provoking and interesting problem for children, when presented appropriately and (2) a lot of good schools do a great job of helping math-phobic teachers teach math at a high level. They provide training, coaching and strong curricular materials that can help teachers overcome their fears and become more mathematically confident. My wife, for one, worked through a lot of her math phobia when she used the TERC Investigations curriculum to teach multiplication to her 4th Grade class.

So much for primary teaching. What about math teaching at colleges and universities?

My own college teaching relied almost entirely on pure lecture as a classroom teaching technique; often the teacher would not even pause for questions. I was fully expecting to hear this come up in the survey. I thought I would hear it from my primary/secondary colleagues, but I was curious to know whether any of those working in higher education would raise the “pure lecture” critique themselves.

The answer was no. The “pure lecture” critique certainly did come up, as expected. It was almost entirely raised by those working in primary/secondary teaching. The high school department chair (quoted above) mentioned “a preponderance of lecture as the instructional strategy” in college classes as an issue. An instructional coach at a middle school bemoaned “lecture-based math classes.” Another response, from someone who works with middle school students, critiqued “lectures that are too difficult to follow, or very hard to be engaged in.”

This “pure lecture” critique was not raised *at all* by professors or graduate students. It came only from primary/secondary educators in my survey.

However, college mathematics teachers did raise other issues. One college teacher (“assistant professor of mathematics at a small private liberal arts college”) described college math teaching as “highly variable, and depends strongly on how much the institution and the individual instructor value teaching as part of the academic job.” Another (“visiting Professor of Math at a 4 year college”) wrote that “the biggest issue for post-secondary mathematics is the mindset of *I know mathematics therefore I can teach it and I will have minimal pushback because I have a terminal degree in mathematics*.”

This narrative is different than the “lecture” complaint. It alleges that some professors either do not value teaching highly or that they are too confident in their ability to teach well. *Strong* content knowledge can perhaps present its own challenges for a teacher of mathematics; deep knowledge can make it difficult to understand the point of view of the struggling student. Curiously, this was only brought up by college teachers; primary/secondary teachers didn’t mention it.

To sum up the situation, in this informal survey, the knock against primary teaching was that its teachers avoid or misunderstand math. This results in students being presented with a distorted picture of mathematics as a subject devoid of thinking but full of procedures to follow. You could even hear this critique from primary/secondary educators themselves. On the other hand, primary/secondary educators were somewhat apt to critique college teachers as relying too heavily on pure lecture. College teachers themselves did not bring up lecture, but did mention the relationship of professors to the work of teaching as an area of concern.

One last thing: I haven’t mentioned what people said about secondary teachers! This is for two reasons: (a) there’s plenty to talk about with primary/university, but also (b) secondary teachers seemed sort of stuck in the middle on the survey. The critiques they (we!) received seem to me best understood as watered-down versions of the concerns leveled at primary teachers. Secondary teachers are taken to have stronger content knowledge, but in the survey we were said to be too procedural, too dry, too focused on memorization and not enough attuned to the needs of the discipline. There were more strong opinions about primary and college than secondary teachers, who seemed to get a slightly different, but weaker, version of the critiques aimed at primary teachers.

**III.**

I had assumed that the responses to my survey would be super-skewed, with everyone defending their own turf but taking issue with the work in other educational settings. But for the most part this was *not* true. Without getting statistically precise about this, people’s ratings stuck pretty close to whatever their own baseline was. People who thought math education was basically working *across the board *didn’t distinguish a lot between primary, secondary and post-secondary schooling. Likewise, there were many others who thought that math education was *across the board *not getting the job done, and they didn’t distinguish very much between settings.

(Getting statistically precise: people’s ratings did not deviate much from the *mean* of the three ratings they provided. The absolute deviation from the mean of their three ratings was 0.493, on average.)

I was surprised by this. I had expected mathematicians would have *much* harsher things to say about primary/secondary education, as compared to their own work. But, at least on my little survey, asymmetrical harshness turned out to be the exception rather than the rule. For instance, one respondent with a Master’s in mathematics said, “The early grades K-3 tend to explore more and most kids are not left behind.” That doesn’t sound damning at all!

Echoing this was a PhD and college mathematics instructor: “Especially with better training nowadays, I generally think that elementary teachers do a pretty good job.” That’s also pretty positive; this person gave primary and secondary teaching a 4 and college teaching a 3. People who were overall cheery about math education tended to be so across the board, even across educational settings.

The flipside also tended to be true. People who were critical of math education *overall *often made significant criticisms of their own educational setting. The source of many strident critiques against primary education, for example, came from those who oversee primary math education. One of my favorite lines on the survey was from a mathematician who explained their middling evaluation of primary education: “I read about it in NCTM literature.”

(NCTM is the National Council of Teachers of Mathematics, the largest professional organization for primary and secondary math educators in the United States. They have been trying to reform primary/secondary math education for decades and are often highly critical of the way mathematics is typically taught to younger children.)

This is interesting! It suggests that we are not exactly a profession divided against itself, as much as a profession that can’t seem to agree as to whether things are fundamentally broken or not. (I tend to side with the “not”s, for the record.)

In general, things were both less critical and less adversarial than I was expecting. People tended to judge the *entire *math education system, from primary through university, as a whole. One high school teacher said, “I’m certain that experiences differ widely (as with primary and secondary), but I had good university teachers and I learned a lot.” One mathematics teacher educator repeated the same comment for primary, secondary and college education: “Too much focuses on procedural knowledge and less on higher-order thinking.”

Both of those views makes a lot of sense to me; whether good or bad, we’re all in this together.

**IV.**

To summarize:

- There wasn’t a great deal of
*quantitative*polarization; people tended to be overall happy or overall unhappy with how things are going in math education. - There was, however, a great
*qualitative*difference in the issues critics recognized in primary, university and (to a lesser extent) secondary math teaching.

It’s worth dwelling on this, if only for a moment. Critics identify *entirely different flaws *in primary and college education. The “bad version” of primary teaching looks almost nothing like the “bad version” of college teaching. The stereotypes, to whatever extent they are believed, are completely unlike each other.

This is fertile ground for extreme views. Under these conditions, people can identify problems with other areas of math education and think to themselves, *nothing *like that is happening *where I teach* — and they would be largely correct. If you are a middle school teacher, none of your colleagues could ever be accused of talking for the whole period straight. And whatever problems exist in college teaching, nobody would ever accuse a professor of turning mathematics into nothing but mindless routines. Quite the opposite! Students are, absolutely, asked to think.

These differences, I feel confident in saying, result from deep differences between our teaching contexts. It is sometimes tempting to see the similarities between our educational settings — there are students, desks, whiteboards and textbooks — instead of uncovering the deeper structural differences. But the differences are vast! Here are just a few of the variables that differ in significant ways between primary/secondary and college classes: the number of students in a class; the frequency of the class’ meeting; whether we are accountable to a test or not; the ability of students to study the material independently; our control over the curriculum; etc. We could easily name more.

These contextual differences should make us slower to come up with educational solutions for other people’s problems. If our educational settings are different enough that *bad *teaching looks different, it seems to me that *good *teaching in primary, secondary and college settings ought to look very different as well. This means that we can’t simply assume that the techniques that are useful for teaching math in one setting will also be useful in another. Which means that we should be cautious before offering advice to those who teach in other contexts based on our own teaching experiences.

It reminds me of something Neil Gaiman says about writing:

When people tell you something’s wrong or doesn’t work for them, they are almost always right. When they tell you exactly what they think is wrong and how to fix it, they are almost always wrong.

When those outside our professional setting tell us that our teaching is not as successful as it should be, we should listen. But as the criticisms and prescriptions get more specific, they almost always grow less useful, at least to me. The culprit is context; we rarely truly understand other people’s constraints.

I don’t intend to sound pessimistic. I think we really can learn from each other’s perspectives. But specific solutions and criticisms can only be supported by a deep understanding of the teaching context. That’s why people who move between these educational settings are capable of doing such important work. These are the PhD graduates who become high school department chairs, the primary teachers who attend mathematics lectures, the secondary teachers who pursue graduate work in mathematics. These are the people who can not only hear the criticisms, but can turn them into something that really works.

In various little ways I have benefited from others who have done this work. When I began teaching math to 3rd Graders at my school, I was committed to sharing “real” mathematics with these students. What surprised me, though, was just how real the mathematics of the curriculum is for these students. Just in the past few weeks I have heard some pretty amazing mathematical conversations about fairly straightforward mathematical questions; things such as 120 + __ = 210 and 3 x 8 = __. But, as mathematical critics of primary teaching have said, there is more to mathematics than arithmetic, and I wanted to expose my students to more. How?

Joel David Hamkins is a professor of logic. At some point I came across his blog, and found materials he posted. For several years, Hamkins had gone in to his daughter’s elementary classroom as a guest math teacher. Each year he put together a pamphlet of problems for the children, and was generous enough to share them online.

For the last several years, my students have loved doing his “Graph Theory for Kids” for a few days each year. They learn about circuits, planar and non-planar graphs, and chromatic numbers. They color maps and hear, for the first time, of the four-color theorem. I feel so grateful that Hamkins was able to really be there, in person, to teach his daughter and her elementary classmates. Do other professional mathematicians do classroom visits? If not, couldn’t they? What if our professional organizations were to organize such things, at some sort of scale?

Primary and secondary teachers have a unique sort of expertise in pedagogy, but I see no way for us to share it unless we tangle in-depth with the context of college math teaching. I have never heard of primary/secondary educators visiting college courses as guest speakers, but why not? Surely there are some who work in primary/secondary settings who could be invited to give lessons or talks in university math courses. We could try to adapt our methods to the university setting, and work together with professors to design different styles of lessons. Could this be a way to learn what primary/secondary pedagogy looks like in a different setting?

The truth is that I am optimistic that something like this cross-pollination is already occurring, though very slowly. For the past few years I have taught at a summer math camp in New York City. The students are all entering the 7th Grade; the faculty are about evenly split between middle/high school teachers, graduate students and college teachers. Every summer, I’m surprised by the sorts of pedagogical discussions we end up having. Assumptions are frequently challenged as we all look for a common language to describe our teaching. For six weeks, we talk daily about how we are structuring our lessons and helping our students. We’ve seen each other teach, we’ve seen the problems each has shared with their students, and we’ve shared our successes and struggles.

When camp is over we say goodbye to each other, taking whatever ideas we have learned over the past few weeks, and help students learn mathematics in all sorts of different classrooms, wherever they happen to be.

]]>After my day-to-day interactions with students, one of my favorite things about teaching is talking with other teachers. There is no shortage of amazing teachers who are working hard to make their classes better and improve student learning. Likewise, there are plenty of opportunities to find inspiration in our colleagues’ work, ranging from attending talks at conferences to simply getting coffee with coworkers to talk about how our classes are going.

A few years ago, I realized that the proportion of inspiring ideas that turned into measurable change in my classroom was essentially zero. As I thought more about this, I realized that *I* was the biggest hurdle to this change. There was a little voice in the back of my head with a constant and emphatic message: *No. I can’t do that, and here are fifteen reasons why. *

I know I’m not the only one who hears this voice. Of course, the reason we have these thoughts is that they are often true. No two people experience teaching in the same way. We have different personalities, different styles, and allow for organized chaos in different ways. As a community, it is easy for us to despair in the challenges we face in our teaching.

Joan Baez said, “Action is the antidote to despair.” At the end of the day we are all mathematicians and we have been trained in solving problems. To be apathetic in the face of the challenges put before us is antithetical to our training as problem solvers. And teaching, particularly teaching well, should be viewed as a problem that desperately needs to be solved. Like many real-world problems, the problem (“What does it mean to teach well?”) is not clearly defined. The data is messy. There is not one single correct answer.

In the rest of this post, I would like to discuss some methods for moving beyond the little voice that says “no” and changing your teaching without reinventing the proverbial wheel. And, as with many real-world problems, I will not answer the question at hand (“What does it mean to teach well?”) and instead I will address a different question – How do I teach better?

In 2012 I saw David Pengelley give a wonderful talk in which he outlined a rather intricate system of assigning and grading calculus problems of varying difficulties over several sections of the textbook, all of which were meant to be done by the students before class and accompanied by an email to the professor reporting on the section of the textbook they had just read. This happened for every class meeting. The mental yoga of keeping track of so much material at once made my head spin, and I did not (and still do not) have the confidence to implement something on such a grand scale. (A summary of David’s method can be found here: https://www.ams.org/publications/journals/notices/201708/rnoti-p903.pdf)

But the ideas in the talk were inspiring. I left the talk telling myself, “I should create opportunities for my students work on math more regularly.” I started giving mini-assignments in all of my classes on a daily basis. The mini-assignments consist of about five relatively simple problems related to the material covered in a given class, which are due at the beginning of the next class.

I can mark the papers and record grades in about 15 minutes per assignment (I do this by hand – if you use WebWork it takes no time at all). It helps make sure the students are engaging with the class material throughout the week instead of just the night before a bigger homework set is due. It also lets me focus my grading efforts on problems with more substance that better measure what the students are learning. This works well for my teaching philosophy and helps make the students stay on top of the course material.

Our teaching experiences are shaped by a large number of variables, including student preparation, class size, resources, research expectations, and other demands on our time. Because of this, the things that my colleagues who work at ivy league schools are able to do in the classroom often are not feasible for me, just as the things that have worked well for me in a class of 20 students might not scale well to a class of 300 students. However, rather than finding all the ways that a technique cannot be applied to your classes, look for ways to transform that solution into something that will work for you.

When I dissect the “no, because…” statements that persistently come to my mind, I find two main themes. The first main issue is one of practicality – what works for person X may not work for person Y – and we have already discussed this. A second main theme, whether we like to admit it or not, can be boiled down to an issue of discomfort or fear.

Change is hard. For most of us, our educational lives consisted largely of lectures to be observed and internalized. This is comfortable because it is what we know best. It is a known quantity in which we hold significant control. Letting go of some of that control is scary. What if it doesn’t work? What if the students don’t like it? What if I don’t do a good job? These are all legitimate concerns, and we can work through those concerns once we are honest with ourselves about their root causes.

Change does not happen overnight. Maybe you are interested in having a more active classroom or having a flipped classroom where students do more work outside of class, but you don’t want to give up complete control to the potential chaos that might come from this. Maybe you don’t have the time or resources to completely re-invent your differential equations course because you’re on sixteen committees and teach three classes per semester. I get it.

You don’t need to do it all at once, and you don’t need to do it all by yourself. Ask colleagues to lend you worksheets or materials. Find activities that help you move towards this goal and try them out a couple times over the course of a term. After a few years of doing this, you will have developed more resources and gained more confidence in this approach, homing in on techniques that work for your personality and style while also better serving your students.

New teaching methods aren’t going to be perfect the first time. You’ll probably mis-judge the difficulty of some tasks. Students might get frustrated. You might get frustrated. But it also won’t be a disaster. Students will still learn. So will you. It will be better next time.

Be prepared for the eventuality that a lesson won’t go as planned. It is hard to know how long an activity will take, but you can bet that it will usually take longer than you think. Don’t be afraid to change an activity in the middle of class if it is taking longer than you expected or the students aren’t getting it. You can always change your course schedule to adapt to this change. Besides, if you try five new lessons in a semester and two of them don’t go as well as you had hoped, that still accounts for a relatively small proportion of the overall class.

At the end of a course where you’re trying new things, reflect honestly about the successes and struggles you faced. Were there common themes in the struggles you faced? How can you fix them next time? Make a list of three things you’d do differently next time. Don’t be too hard on yourself.

There’s a rule in improv comedy called the “yes, and…” principle. If you’re in the middle of an improv comedy skit, you can only react to the material you are given from your collaborators. You may have been ready to tell a very good joke about Care Bears, but now someone has decided that there’s a grizzly bear running around the stage and you need to act on that instead. You can’t stop the skit and ask for a retake, so instead you have to accept the reality of the grizzly bear and add your own brand of humor to it.

The same principle applies to teaching. We can all become better teachers by finding inspiration in others. This takes work, and it can be scary to take a risk and try something new in the classroom. In many cases, we fail to apply the lessons our peers have learned because we feel their experiences do not directly translate to our own. Next time you go to a talk about teaching, I challenge you to move beyond the naysaying gremlins in your head. View the reasons to say no as equations that bound the parameter space of your problem. Say yes to new ideas and apply them in your classroom in a way that works for you. Over time, these small changes can add up to more effective teaching.

]]>Elena Galaktionova sent us this article shortly before she passed away earlier this year.

Elena Galaktionova received her first introduction to mathematics from her favorite middle school teacher in Minsk, Belarus, her hometown. After she had finished her education at the Belarusian State University she went on to receive a Ph.D. from the University of Massachusetts in Amherst. Her area of research was representation theory. She taught mathematics for many years at the University of South Alabama, after some earlier stints at the Alabama School of Math and Science and the School of Computing at USA. In Mobile, Alabama, she was one of the organizers and teachers of the Mobile Mathematics Circle. The Circle has been going strong for 20 years. Later she recruited local teachers and a middle school principal to participate as a team at an AIM workshop on Math Teacher Circles. Upon return to Mobile she founded the Mobile Math Teachers’ Circle. Twice she gave presentations at the Circle on the Road conferences. Her work with local middle schools and her interests in home schooling were motivated by her love for mathematics. She cared deeply about math education. Sadly, Elena passed away earlier this year after a long battle with cancer.

In all my classes I try to teach reasoning, writing and problem-solving skills. I noticed that if a class is heavy on computations and dense in content, such as Calculus, the result of this effort is barely noticeable if at all. I recall a memorable moment in a multi-variable calculus class. The topic was optimization. My students knew just fine how to use the Lagrange multiplier method given a function and a constraint, thank you much. But it turned out they were helpless in the face of even the simplest application problems. Some of these students were studying Calculus with me for almost three semesters and their grades were good and I tried so hard to teach them what matters in mathematics the most. I remember a chilling realization at the moment, that we — the students and I — wasted three semesters.

A very different experience comes from another course. At our university it is called “Foundations of mathematics”. Unlike other math classes it does not have a lot of content. A bit of logic, set theory, relations, maybe some number theory. It is the first class where students are learning to write proofs. This is a writing-intense class. There are essentially no calculations. I collect the homework every class period and grade the same way one would grade an essay. My first requirement is writing in grammatically correct meaningful English sentences. This is not an easy task for most. A lot of students by the time they start this class learned to perceive math as number and symbol manipulations. At the beginning of the course I often see in students’ work words that are strung together in rather random fashion. We go together over some of the responses asking questions like : “Is this an English sentence? What is the meaning? Are all the terms defined? How could it be misinterpreted?”

By the end of the semester I observe a turn-around: there is a palpable effort from even the weakest students to put their ideas into words. The change is most noticeable in weak students. The struggle for finding the right words and writing in grammatically correct sentences may be still there. While they did not suddenly became great at math, their mental activity and learning efforts are much more productive, since they are consciously directed towards comprehension and expressing their ideas verbally with a degree of precision.

I wondered if my students noticed this change themselves; that was until I was approached at the end of the semester by two of my “Foundations” students who emphatically told me how this course entirely changed the way they view and approach math. This is reflected in their grades in other math classes as well. For example, one of my Calculus II students was taking “Foundations” concurrently. Her grade in Calculus II changed from a D at the start of the semester to a B towards the middle. Most notably, she enthusiastically confirmed and told other students how much taking “Foundations” helps with Calculus II, despite having no content in common.

What is most interesting to me is the quantum character of this change and that it was especially noticeable in weaker students.

Young children come to school as a blank slate. Yet they have the innate ability for reasoning, they have curiosity, they are eager to play and explore. Over the years their teachers influence their perception of what math is about. Two of the possibilities are:

- math is a manipulation of numbers and symbols according to a predetermined set of rules;
- math is communicated through meaningful statements.

The students who do not do well in mathematics typically view math as a manipulation of symbols. The “making sense” switch changes this so the students begin to read and communicate mathematics as meaningful, logically connected statements.

To summarize, here is what I observed:

- Both exclusively formal processing of math tasks and making sense of math tasks are learned, eventually habitual, behaviors. Either one becomes a mental process which is practiced and reinforced in every math class.
- Effective learning of mathematics does not happen until mathematical communication is perceived as meaningful statements.
- Students who view math as a formal manipulation of numbers or symbols will habitually direct their effort and mental energy toward this in a math class, unless they are given problems which naturally invite reasoning and stay away from using formulae and rules. In a class with a computational component, such as pre-calculus or calculus, even if a teacher tries to teach reasoning and making sense, it has relatively little consequence: under stress, such as homework due the next day or a test, such students revert to their habits. Some of them spend a fair amount of time studying and reinforcing these habits, often getting frustrated because of the little return for their efforts.
- A dedicated computation-free and writing-intensive class which stays away from problems that may suggest formal manipulation can turn on the “making sense” switch. Students start to perceive mathematics as meaningful statements. They look for logical connections between the statements. Their verbal skills are productively challenged.

The important qualities for such class, assuming the main purpose is to turn on the “making sense” switch:

- The class should be writing intensive.
- The tasks are such that students can rely on their existing reasoning skills, common sense, intuition. They should not be too abstract. For example, it is easier to find appropriate problems in logic, set theory, elementary geometry or combinatorics than in abstract algebra. This allows students to scrutinize math statements using their own “sensometer” and keep working with them until they can make sense of them.
- The course should be light on content and big on thought, allowing sufficient time to think and write about problems.
- The class should not include tasks which could tempt students into formal manipulation.
- Feedback on writing is continuously provided by the teacher: students’ attention is brought to details of their writing, the meaning of what is written, and how the writing could be improved.
- The time required for the “switch” to turn on in such class is less than a semester. This is my experience with undergraduates.
- This works even if this writing intensive class is taken in parallel with other, computationally intensive math classes.
- Once the switch is turned on it stays on in other classes, even in those with computational components, as long as teachers pay attention to making sense and reasoning.
- A practical aside on grading: a class where non-routine and sometimes difficult problems are part of homework presents certain challenges for grading. I told my students that if they could not solve the problem, they should write down their attempts, for example, how they used problem-solving strategies discussed in class, such as looking at related simpler problems or generating examples and trying to find a pattern and showing why it did not work. Adequate effort and quality writing would earn almost full credit. Of course, it is important to also include easier problems which are within reach for nearly everyone. If I did not have sufficient time to grade the full homework, I selectively graded 2 or 3 problems.

Unfortunately, the “Foundations” class is a sophomore level for math and math education majors. In fact, no prerequisites are needed for it. So we started to encourage students to take this class as early as possible, when noticing that it helps them in other math classes. There is no reason why a class with similar characteristics and goals is not taught to seventh graders. It would improve their learning of mathematics for years to come. As an example, in Russia, the class that perfectly fits the bill is Geometry class. Systematic study of Geometry starts in 7th grade and continues through 11th grade. The Geometry class meets 2 or 3 times per week. All statements and theorems are proven based on what is already known. Thus, Geometry is presented as a unified theory and not a random collection of facts. The students are expected to state definitions and prove theorems and they solve problems involving proofs and geometric constructions. Of course, there are Russian students who struggle with writing proofs and deriving formulae. But they are used to the concept of intrinsic reasoning and they know that they are expected to articulate it. In the U.S., ask a class of either seventh graders or freshman Calculus students why a particular fact or formula is true and the answer invariably will be “Because our teacher told us so ” or “Because it says so in the book”.

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