Inverted Class: MAT137Y Implementation

The MAT137Y structure

Students have videos to watch before class. They meet with you for class (what used to be lectures) for three weekly hours, and with a TA in a small tutorial for one weekly hour. In 2020-2021, while teaching is online, we are replacing tutorials with additional office hours. Students also have practice problems, problem sets to be submitted, and tests. They can get help in office hours, or at our piazza online forum. The course coordinator deals with most of the logistics of the course.

Resources for students

Course Website. This is the central place where all information, resources, and links about the course appear. It may be on Quercus or perhaps on a separate website. The course website is maintained by the course coordinator for all students in all sections of the course. Don't replicate this.

Course Outline. This document explains all the administrative aspects of the course, expectations, and assessment.

Videos. There is a library of short YouTube videos (opens in a new tab), created specifically for this course, that students are expected to watch before each class.

Practice Problems. There is a collection of practice problems for each unit, with short answers.

No Required Textbook. The videos plus the practice problems remove the need for a textbook. There is an optional textbook based on the lecture notes of a former MAT137Y instructor. Students may use this resource if they prefer a traditional book over videos, if they want to see more worked out examples, or if they want an even bigger collection of practice problems. However, this textbook is not required.

Piazza. There is an online forum for the course on Piazza (opens in a new tab) where students can ask and answer each other's questions. We, as instructors, can answer questions as well, but we encourage students to help each other.

Resources for you

Instructor Guides. There are two instructor guides:

• One about content, the “Goals of the course”. This explains what the course is about, who our students are, and what we want students to learn (learning goals).

• One about structure, the “Guide to an inverted class”. This is the document you are reading.

Annotated Class Activities. You also have a library of class activities/exercises that I have used for each unit (including .tex files), many of them with commentary on how they normally go or what the point of the activity is. You are welcome to just use my activities directly, or to modify them, or to create your own.

Your responsibilities

Prepare your class. This means choosing the activities/questions/exercises you are going to give to students. As stated above, you may want to just take them straight from the library we have, or edit them, or write new ones. I recommend that you create slides (you have latex files for all the activities in the library). If you teach in person, you will be in a classroom where you can use a screen and the board simultaneously. Then you can simply project the questions on the screen so that no time is wasted by you or the students writing them down. If you teach online, you will want students to download the questions before class starts so they can work on any of them right away.

Maintain your website. The course website is maintained by the course coordinator. However, every instructor has a small website for their own section, linked from the main website. In it, keep a calendar of the days, the topics for each day, the videos students have to watch before class, and the slides you use every day. You are welcome to copy the same calendar that the coordinator has for simplicity. For the slides, you may choose to have a pre-class and post-class version, in case you do not want to spoil some punchlines, in case your plan changes, or in case you have more than one plan. If you teach in person, you may prefer to post slides only after each class. You only need to post the questions, not the answers.

Teach your class. This is the main task, isn't it? More on this below.

Hold office hours. We keep a collective calendar of office hours for all instructors and TAs on the course website. We invite students to attend any office hours they want. They are not restricted to their instructor's.

Admin tasks. The course coordinator takes care of most of the logistics in the course, but they will ask you to help with some administrative task. This could include, for example, writing problem set solutions, handling regrade requests, or supervising TAs' grading.

Invigilation. We need every instructor and TA to invigilate each term test and the final exam. Please mark your calendars right away and give them priority.

Final exam grading. All instructors help grade the final exam. We normally use Crowdmark or Gradescope, so this is done electronically.

Piazza. You are very welcome to join Piazza and answer students' questions every now and then. Even if you never participate, you will find that merely checking occasionally what kinds of questions they are asking is quite helpful to understand where students are and what they find difficult.

So what do we do in class anyway?

1. Assume students have watched the videos you told them to for this day.

2. You project a question on the screen, you have students work on it, then you resolve it somehow. To make this more concrete I am including samples of class activities in the next section.

3. Depending on the types of questions, you will probably be asking students to spend some time thinking or working individually, some time discussing or working with their neighbours, some time listening to your explanations, to vote between options, or to offer you answers or explain to the whole class.

The coverage issue.

Here is a liberating fact: you do not have to cover anything. In a lecture-based course we are often constrained by the need to “cover all the syllabus” (Sadly, “cover” means “I did it in class” rather than “Students understood it”.), and we always have to rush and move on. In this inverted class the videos and the practice problems are taking care of “covering all the material” so you are free to do more useful things. Your goal is to have students engage with the material, play with it, and think deeply for one hour. That is all. If you manage this, they will have sharpened their skills, improved their understanding, and gained intuition. Moreover, we are teaching them good habits and letting them experience how mathematics is actually learned. You want them to be as active as possible during the full hour.

This bears repeating: you do not have to cover anything. Even when we know this rationally it is hard in practice to let go of the desire to cover one example of every kind or tell them every single fact you can think of. There is no need for it: the desire to do so is for us to feel accomplished, not for them to have learned. So if you have one activity that is working very well and you decide on the spot to extend it and use the full hour on it, that is perfectly fine.

Some advice for running the class

Assume students have watched the required videos for each day. As hard as it is, allow those who haven't to be lost and waste an hour. If you begin each class with a “quick lecture” for those who have not watched the videos, they will conclude that they do not have to watch them, and they won't.

Always set very clear expectations. If students are not sure what you want them to do, they will do nothing.

Be active from the very first day. Try to create good habits from the beginning. You want them to be active from the very first day; otherwise, you will never overcome their inertia to stay passive. Do not begin the class until you have their attention: if you begin talking over a murmur, you are giving them permission to keep going. If you want them to work individually first without talking to their neighbours, insist on it; otherwise you are giving them permission to ignore your instructions in the future. When you want them to transition from discussion to listening, insist on it; otherwise you will never be able to gain them back.

Get them to stop frantically taking notes. If you are teaching in person, at the beginning of the term, the moment you post a slide on the screen they furiously write down everything on it and do not listen to a word you say until they are done. Do not let this happen. Remind them that the slides are posted and that you want them to use the time to think about the question before hearing an explanation. If they spend the time copying things down, they have wasted an opportunity to learn. This will require repeating, but it is important to get it right early on.

Walk around the class. When student are working individually or in groups, walk around the class. At the very least this gives you the chance to look at their notebooks and get very valuable feedback on how they are doing, whether something is too hard, whether they did not understand the instructions, whether they are all making the same mistake... Moreover, some students are too shy to speak in public and they will ask you a question if you are standing next to them, but not if you they have to raise their hand. Ask random students to tell you what they are doing; you will learn a lot about their thought process and what confuses them.

Choose different student to explain their answers. When you take answers from students, do not choose always the same students. I normally ask them to raise their hand and I choose who answers. If not enough hands are up after a long enough pause, perhaps it means they need more time or they need to discuss with each other before they are ready (or the question is too hard, or the question is too easy and they are feeling patronized). If you take the first hand that raises every time, it is easy to delude yourself into thinking that everybody is getting it when in fact only a few of them are.

Resist the temptation to fully explain the answer to all questions. A dangerous way to lead the class is to pose a question, pause for a couple minutes while students pretend to solve it, and then you explain it fully on the board. That is not an inverted class: that is a lecture in disguise. Resist the temptation! If students are not active, we are defeating the purpose of the class.

Prepare more questions than needed. In a 50-minute class I normally use 2 to 5 activities. I often prepare a few more activities for each class just in case. Then I decide which question to use next depending on how the class is going or on how much time I have left. If posting slides beforehand, I may post just my main plan, but I may end up changing it.

Explain why we do things the way we do. Explaining to students why we ask them to do what we ask them to do increases participation and buy-in. For example, I have to use the following speech multiple times in a term:

“Most of you did not vote for any of the options. Let's talk for a second. All of these questions are trivial for me. I do not know what is difficult or what is confusing you. When I ask you to vote or to answer, I do it because I need your feedback. I need that information to realize how you are doing and to decide what I should do next. If you do not reply, you are trying to make it as hard as possible for me to teach a useful class! We need to work together. So I am going to ask you again to vote, and this time everybody votes. If you are confused, hesitant, or you believe you may be making a mistake, then I particularly need you to vote, because I need that information.”

Aim your class at the middle 60%. Your class is large and heterogenous. If you focus on the top 20%, you may convince yourself that everybody gets something when most of them are confused. If you focus on the bottom 20%, you may waste time and bore the vast majority of students who are doing their part. For example, if students have worked on a problem that I think is reasonably accessible, and the vast majority of them has gotten it, I will not explain the solution on the board, even if a students asks for it. I will, of course, be very polite and invite them to ask me during office hours, just like I would do with a keener student who asked about Galois theory.