Teaching with Magic Paper

I know that the sentence “The year is 2022” is just a bland statement of fact, but it hits my ear like a voice-over in a trailer for a bad science fiction movie made in the 1900s. Blame Walter Cronkite; I grew up watching his TV series The Twenty-First Century (1967-1969) and came to indelibly associate the 2000’s with The Future. Now that I actually live in The Future, surrounded by many of its predicted marvels, my degree of enthrallment varies from marvel to marvel, but I never tire of the wonders of magic paper. You know the stuff I mean: you write something in one place and the paper makes copies of itself elsewhere so that people in those other places can read the words you just wrote. I’m sure you’ve all used it. I’m using it now.

Magic paper helps me with some problems that have long bedeviled classroom teachers like myself: How do you find out what’s going on inside your students’ heads in the midst of a lesson without derailing it? How do you get all your students to actively participate without having the class descend into chaos? How do you communicate with a large group of students without the conversation devolving into what math educator Henri Picciotto calls a “pseudo-interactive lecture” dominated by the teacher and the two or three most vocal students?

Back in the 1980s, educator David R. Johnson tackled these problems using what he called the “paper and pencil method” of getting real-time feedback on how students are doing. Under this model, the teacher asks a question, the students write down their answers, and the teacher sees what the students wrote. This was back before magic paper, so the teacher would have to physically move around to look at the students’ responses. To make the moving-and-looking more expeditious, Johnson suggested that teachers seat students in a U-shape arrangement with the teacher stationed at the center. For more on Johnson’s ideas, see his book Every Minute Counts (1982, Dale Seymour Publications).

Around that same time, educator William F. Johntz pioneered an initiative called Project SEED that I had the good fortune to be exposed to while in graduate school; see the essay about it on Henri’s Picciotto’s math education blog.1 SEED had an innovative and charming way of opening up an underused communication channel from the class to the teacher. In SEED classes, hand signals played a big role: hand signals for numbers, for operations, for equality and inequality; hand signals for agreement, disagreement, partial agreement, and confusion. A teacher could ask a class a question (plant a seed, if you will) and quickly reap a rich visual harvest of information, a panoramic representation of her students’ states of mind. This provided SEED teachers with even snappier feedback than Johnson’s paper-and-pencil method, though with some limitations (there are after all only so many hand signals you can teach a class2).

In the 1990s I turned my efforts wholeheartedly towards mathematical research, with teaching as a side activity that I tried to perform competently and compassionately but which didn’t arouse my highest passions. I read what people like Sheila Tobias and Alan Schoenfeld and Uri Treisman and Liping Ma were writing, and some of their ideas affected my teaching, but mostly I taught my students in the same ways that I had been taught. In particular, when I asked a question, I waited until a reasonable number of hands were raised (or until I gave up on waiting for more hands to go up; I never felt comfortable cold-calling students). I would pick someone whose hand was raised (trying to pick whichever of the hand-raisers had spoken up the least so far that day), and then respond to that person as if The Class had just spoken to me through its Chosen Representative. But of course the students who spoke up weren’t representative of the class as a whole.

Fast forward a few decades to the Covid-19 pandemic. Suddenly I was teaching over Zoom with very little relevant experience. As my time permitted, I took some online classes in how to do online teaching, and one of the tricks I learned was Chat-Storming. I quickly grew enamored of it. In Chat-Storming, I ask a question and none of the students answer right away, because I’ve told them not to. Instead, students compose answers in the Chat field of their Zoom portal but don’t press Enter/Return until I say “Okay, submit your answers.”

Then a flood of feedback drops down on my head as all the students answer at once. If it were an auditory overlay of student responses, it would just be a roar of white noise, but it’s visible, searchable, interpretable. I can’t tell which students were quick and which were slow, but I can see at a glance what the students as a whole think, and also look at individual answers in as much detail as I wish. It’s as easy to visually scan the magic paper (in this case, Zoom’s chat log) as it is to scan the hand signals in a Project SEED classroom, and the responses have a higher information content. Chat Storms use the visual information channel William Johntz championed, but with more bandwidth. As a bonus for the teacher, students aren’t able to peek at each other to try to assess whether their instincts are right or wrong based on whether the apparently “top” students agree with them; they’re on their own, and must make up their own minds.

Now I’m back in the physical classroom again, but I still use Chat Storms because they’re the best way I know to create in-class engagement that also can be used for assessment of participation3 and gives me realtime feedback on what students understand and what they don’t.4

Does anyone know who came up with the Chat Storm? If you have any leads, please share them in the Comments!

Among its other virtues, the Chat Storm taps into the seldom-utilized positive power of boredom. In the past, I would sometimes force a class to speak by using the brutal tactic of not saying anything. If a teacher pauses for long enough, someone will break the awkward silence, once the students realize that you’re willing to wait as long as it takes. Chat Storms do something similar. When a Chat Storm is going on, the classroom is a really boring place. Nothing is going on except a lot of people thinking and writing on their magic paper. If you’re a student in such a classroom, you’ll quickly realize that nothing interesting is going to happen; you might as well join in the thinking and writing.

Well, maybe that’s not entirely true. If you’re stuck in a silent classroom with nothing but a smartphone, there are approximately infinity things5 you can do on your phone besides participating in a Chat Storm. And indeed some students who are prone to being distracted by their phones (such as students with ADHD) have told me they prefer a more traditional style of teacher-student interaction. But the majority of students like what I’m doing; they find that Chat Storms are enjoyable, keep them engaged, and provide feedback on how they’re doing.

“Excuse me, professor: could you give an example of a Chat Storm?” (I hear a reader of this essay ask).

Excellent question! I’m so glad you asked that!

When I assigned my discrete mathematics class the task of forming the logical negation of “Everybody’s a critic” using the Chat Storm format, I got a few expected wrong answers of the form “Nobody’s a critic” but also a couple of instances of a wrong answer I hadn’t expected: “At least one person is a critic.” This led to an unplanned discussion of the meaning of negation that I hadn’t realized the students needed, and I enunciated a criterion I hadn’t thought of before: if you can imagine a universe in which p and q are both false, or a universe in which p and q are both true, then p and q are not negations of each other. The students found that criterion helpful; I think I’ll teach it on purpose in the future.

Here’s another example from my recent teaching: when explaining the basics of set theory I arranged a Chat Storm in which I solicited mnemonics for keeping ∪ (union) and ∩ (intersection) straight. In the past I have pointed out to students that ∪ looks like the U in the word Union, and that by a process of elimination the other symbol ∩ can be deduced to mean intersection (the “other thing”). Some of the students had the same mnemonic I’d come up with. But one suggested a mnemonic I hadn’t seen before, pointing out that ∩ resembles the lower-case “n” in “intersection”. I think that’s a keeper too!

I happen to use Zoom and Chat but I know of people who use other kinds of magic paper: Miroboards, Mentimeter, Mattermost, and others. There’s also polling software, but polls often need to be prepared in advance, and many polling systems only allow multiple choice. One thing I like about Chat is its spontaneity (I can whip up a Chat Storm on the whim of a moment) and its open-endedness (if a student wants to include a joke or comment in their answer the system will permit this expression of their individuality).

But going back to what I wrote in the first sentence of this essay, the fact is, the year 2022 is bad science fiction, just as 2020 and 2021 were; if I’d prophetically written an acurate account of the pandemic back in the 1970s and tried to sell it as a novel, no publisher would have touched it. (“Dear Sir: This blatant Andromeda Strain ripoff somehow manages to be scientifically over-detailed, politically implausible, mind-numbingly boring, and deeply depressing all at the same time.”) Yet the ongoing viral storm has had some silver linings. The best one is the advent of mRNA vaccines, which are likely to have marvelous applications to improving people’s health in years to come. But somewhere down my private list of silver linings I’d put this new way of engaging my students.

So if you walk by my classroom and see my students glued to their phones while I’m standing by silently, looking around at students who aren’t looking at me, don’t assume that nothing is happening; a Storm is probably brewing.

Thanks to Sandi Gubin, Henri Picciotto, and my discrete mathematics students.


#1: Project SEED was a wonderful embodiment of the idea that you can set students up to discover deep mathematical ideas for themselves through artfully constructed activities. A teacher might start a SEED class by asking “What do you get when you add an odd number of odd numbers?” and then guiding the ensuing discussion. I could say a lot more about the Project SEED approach to teaching and about the many aspects of it that resonate with me, but I wouldn’t do a better job than Henri Picciotto has already done in his essay.

#2: I wonder how subjects are taught at Gallaudet University. Since everyone there is proficient in American Sign Language, there’d be opportunities for a whole class to respond to a teacher simultaneously in ASL; that might work really well not just for math but for other subjects too.

#3: I’ve written C programs and shell scripts that allow me to quickly determine, using the .txt files Zoom creates, how often each student wrote something in the chat at that particular class meeting. I’m happy to share the programs with anyone who’s interested.

#4: It’s fun to structure a sequence of Chat Storms that allows students to advance their understanding in manageable increments, and deeply satisfying to watch things sink into their brains, as evidenced by rising performance over the course of ten minutes. Of course, sometimes hardly anyone gets the right answer to a question I’ve asked. That means I didn’t design that part of the class well, and that’s never an enjoyable discovery. But the discovery enables me to learn what’s not working and fix it.

#5: There’s a wonderful exhibit at the Boston Museum of Science illustrating just how versatile smartphones are. It’s a replica of a room full of dozens of appliances, every single one of which has been rendered more or less obsolete by the smartphone. Even the stapler? Well, there are apps that have the specific purpose of combining multiple pieces of magic paper into a single magic document. Some of my students use those apps when they submit their homework on Blackboard. So yes, even the stapler.

5 thoughts on “Teaching with Magic Paper

  1. David Jacobi

    I think your approach is valuable for many reasons. One of them is that it deemphasizes the perceived link between mathematical aptitude and the speed with which one answers questions. I find to learn math well, I must grindingly reproduce each step on my own, rather than memorize a formula.

    Liked by 1 person

  2. Joshua Greene

    Apologies for some questions about basic details, rather than something deep: do you set up a zoom video call for each class and then just use the chat function with everyone in the same space?

    Do you ask structured questions (yes/no, True/False, or other multiple choice) or are they more open-ended?

    Do the students know LaTeX or have some other consistent way to enter mathematical expressions?

    Art of Problem Solving online courses use a similar structure for student answers. The whole lesson is text chat-based. When the instructor prompts students for responses, the responses are buffered so that only the instructor and moderators can see them. Instructors/moderators can reply back to individual students and/or approve the comments for publishing to the whole class. The system accepts LaTeX as input and then shows the interpreted output. Not all students enter LaTeX, but the fraction is high in the more advanced courses.

    I don’t know how much the rest of the instruction in an AoPS lesson is customized to respond to student misconceptions. Also, publishing the student comments returns a bit of speed/competitive feeling. Students I know who have used it refer to getting their comment published as “getting on the podium” like a medal winner.

    Liked by 1 person

    1. jamespropp Post author

      No apology needed; basic details are important! Yes, I set up a video call but don’t use the video or audio — only the chat. Seems silly but it works. Sometimes I give open-ended questions (such as “What do you think of chat storms?”, “What’s a good mnemonic for union and intersection?”), and other times it’s a calculation (“What is 4 + 4 in the group Z_5?”). Most of my students don’t know LaTeX, so they’ll just use “x” or “*” or “times” for multiplication, etc.; as long as it’s clear what they have in mind I don’t care what symbols (or words) they use. I wasn’t aware that AoPS uses something similar, but I’m not surprised; some pretty smart people run it.



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