Do you hand out papers in class, but don’t like all the time and running around?
Sometimes a tiny little low-tech tip can make a happy difference in your classroom. When I came up with this one, I wished I had known about it 20 years ago. Maybe I can save you some time!
You have a stack of papers to hand out. (If you no longer hand out graded homeworks, exams, or reflections, then this tip isn’t for you! I’ll include a picture my wife took of our dog at the end so this isn’t a total loss.) If you have a class of 20, the solution is simple: Hand them out as the students walk in, chatting a bit, all casual If you have a class of 200, the solution is simple: You aren’t going to do it. You have already figured out a different way to get those homeworks back.
But what about the in-betweens? I have a class of 60. I think there is something to be said for having that connection, I bring their paper to their seat, from my hand to theirs. Even the people who sit in the back get a personal “hello,” and eye-contact. But how do I make this go faster?
Oh my god, you are going to love me so much in a second.
When my students turn their papers in, they write three things in the upper right hand corner: Their name, my name, and “Front”, “Middle”, or “Back.” That’s it. Then after I grade the papers I sort them into three piles. Only three piles – takes less than a minute. And then I stack ‘em up and hand ‘em out the next day.
I never have to run from one end of the room to the other. I hang out in the back, then the middle, then the front, and Hey! I’m at the front of the room ready to teach.
Seriously – try it!
And this is a picture my wife took of our dog. I’ve titled it “Born to be Wild.”
Every exam I give has the same last question on it. The result has been wonderful, and I think you will agree if you try this technique.
Here’s the last question I give on almost every one of my exams:
Give me your initial impression as to the fairness of this test. Did it cover what you thought it would cover? Was it too long or too short? Was the difficulty what you expected?Feel free to tear this page off, and hand it in later (You can slip it under my door or give it to the Math Office with my name written on it) or turn it in with your test.
This feedback has, over the years, helped me write much better tests. More importantly, the feedback I get from Exam One has helped me to write a better Exam Two, particularly in a new class (like when I, a math professor by training, taught oral and written comp). I’m going to advocate that you try this on your exams, and want to point out why I think this question has worked so well.
1) Timing. The students see the question right after they’ve finished the test. Before they’ve talked about it to their friends. Well before they’ve seen their grade. I have a consistent baseline. (Almost all of the students who answer this question do it on the test itself, and then some come in a few hours later under my door.)
2) Scope. The scope of this question is narrow so I get good information instead of broad impressions. I get feedback about one specific test. I could broaden it to “What do you think about the course? What do you think about my teaching in general? What’s your opinion about this University?” and I would get the broad generalizations we get in student evaluations. But because this is a focused question, I get precise answers.
3) Structure. It starts with the open question about fairness. Students who have a strong point negative point of view will let me know what they think about that! But I also want to hear from the students who aren’t in the “Strongly Agree!” “Strongly Disagree!” camp. By following it up with three sample questions: “Did it cover what you thought it would cover? Was it too long or too short? Was the difficulty what you expected?” they are given ideas what they can think about. And some will just give one-word answers to those three questions – that’s fine. I’m glad to get that information.
4) Coverage. There are some important things I want to get right. I want to get expectations right. I don’t want the students to be surprised by either the content or the difficulty of a test. Sometimes a test is hard or easy because the topic is hard or easy. I don’t believe in artificially making a test on easy material harder, or making a test on hard material easier. But the students should be prepared going in. Similarly, sometimes content will be on a test that the students would prefer never to see again – I have to make that choice. But the students should know what they are in for. And length – I want to get that right. At this point in my career I usually do, earlier in my career I often didn’t. But the students are the ones taking it. Their opinions on that point are vital to me. And all of that is part of the umbrella of fairness. Students tend to be angriest not when they get a bad grade, but when they believe their bad grade was unfair. And they are right to be angry in that case. I am sad when someone with power over me makes a decision I don’t like. I’m infuriated if I think it is an unfair one. The questions I ask were deliberately chosen to hit on these main aspects of my exams.
There is one more, practical benefit to this question. I feel petty putting it on this list, it wasn’t an effect I expected, and it isn’t why I include the question. But here we go:
5) Ass-covering. You give a test. A student then complains to you, after it is handed back, that this test was unfair. This no longer happens to me. Even in that required course that students take because they have to. Even with the major (which shall remain nameless) that is notorious for having grade-litigious students. Because they get back their paper, with their grade, and a page on the back where they wrote “This was a fair test.” It isn’t just a “gotcha.” Dr. Catherine DeSoto’s book, “Lies of Omission” cites studies that show that if a person (including you) adopts a belief, even casually, and writes it down, then that belief will become internalized, probably permanently. Assuming that you give fair tests, when a student answers this question, they are cementing this knowledge, and are less likely to change their belief to “unfair” when they see their grade. And if your students are calling your test “unfair” right after taking it – that is really good for you to know.
Seriously – Make a note and try this out! Let me know how it goes!
I’m glad that there are anti-bullying measures being taken in schools, but we leave out the most important lesson.
“If speaking up seems too hard or not safe, walk away and stay away. Don’t fight back. Find an adult to stop the bullying on the spot.” From stopbullying.gov.
I saw the look on his face
The generation before me learned that if you are being bullied, learn to fight back. Kick the hell out of the bully and he will leave you alone. My generation learned that if you are being bullied, ask him to stop politely and he will see your good example, and stop. The generation after me got different advice, and now my daughter’s generation is getting still different advice.
I saw the look on his face. It wasn’t directed at me.
I have a big problem with statements that begin “People bully because -” or “People are bullied because -” or “This is what you should do -” or “Violence never works because -” or “Fighting back works because -” Depends on the bully. Depends on the victim. Depends on the surroundings. But I’m glad that people are at least talking about it. But something is missing.
I saw the look on his face. It brought me back to elementary school. But now I was 50. It wasn’t directed at me.
I also have a big problem with society trying to shove people in boxes. When my daughter was in pre-kindergarten, the anti-bullying programs made two boxes. Bullies and Victims. Which One Are You? By the time she was in Junior High it became three: Bullies. Victims. Bystanders. Which One Are You? And a lot of the education and grownup-led peer pressure was about what she should do as a Bystander. I can’t hate that completely. I learned a lot from a blog post about what to do if you see a Muslim woman being screamed at by a stranger on a bus or a train. But I still hate the boxes. I’ll tell you now, I was a Victim. Even at my age (50 is receding into the past too quickly) I find it hard to say that publicly. But I also took my turn as a Bystander. And sometimes as a Bully. Not as often. Not a particularly good one. But yeah. And a crucial thing is still missing.
I saw the look on his face. That look that conveys to the target, “You are garbage. You are not even worth the contempt I have for you. You are nothing.” It brought me back to elementary school. I had received that look. More than once. But now I was 50. It wasn’t directed at me.
So, I’m glad that there are these programs, and I hope they get better. I am disturbed that “Just punch him in the nose” is considered archaic, ignorant, and terrible advice. Sometimes it works. Sometimes, not all the time, but sometimes, the best course of action for Victims is to punch the bully in the nose, or kick him in the crotch, as hard as they possibly can. And if there are consequences, take them. And I’m not going to change that statement if the bully is a her. Or a them.
And sometimes the best course of action is not to hit back. To Ignore the bully. Or to tell the bully to stop. Or to talk, to understand where the Bully is coming from. Or to befriend the bully. Or to tell the teacher. (Or maybe try the pacifism I was taught when I was a kid, and is still found on anti-bullying websites. I’ve never seen that strategy work in any context, but the psychologists must have been basing it on something, right?)
But there is something important – something we all need to learn about bullying, and most of us have. Something you learned in school, no matter in which of the three boxes you spent the most of your time. You learned this from being bullied, watching others being bullied, or doing the bullying.
I saw the look on his face. That look that conveys to the target, “You are garbage. You are not even worth the contempt I have for you. You are nothing.” It brought me back to elementary school. I had received that look. More than once. Often. I was afraid to go to school. But now I was 50. It wasn’t directed at me.
He was a close friend. Still is. One of the kindest, most magnetic, most thoughtful friends I’ve had. I would not have believed it if I didn’t see it. I still find it hard to believe my own memory. That look on my friend’s face. It wasn’t directed at me.
This is not a world of Bullies, Victims, and Bystanders, with a rule book on how to deal with each. And that’s the lesson. That Bullies aren’t always the stocky dumb guy with giant arms in Calvin and Hobbs, or the wiry hyperactive ass with the horrible mocking laugh like in A Christmas Story, or the Queen of the pack of Mean Girls like Caroline in Sixteen Candles. Sometimes they are really good people, your good friend, your sister, your father, your beloved aunt, the police officer who lives next door and hangs out with you in the backyard some summer evenings.. Really good people, except when they aren’t.
And I don’t think this is a lesson that can be taught with a powerpoint slide. Because it is unbelievable until you see it. I remember getting that look in elementary school, pinned to the ground and being punched in the face, but the real lesson was after “recess”, when the same person raised his hand, gave a great answer, and joked around with his friends, most of the class.
That’s the lesson we all learned. When a bully isn’t bullying, they are just like you and me. (And you and me may very well be bullies too – when we’ve depersonalized and dehumanized our victim enough to be convinced its okay. No crime in being cruel to a libtard, or a Muslim, or a babykiller, or an anti-vax mom, or a TERF. Amiright?) I agree that we should do everything we can to prevent bullying in school. And I hope we continue to get better at it, offering more than generalizations. And when that happens – when nobody has to be afraid to go to school – where are children going to learn this lesson we all learned? Because its hard to believe it until we have seen it. We need to figure out how we are going to teach it.
Once upon a time, I was the faculty liaison for No Shame theater – a monthly late night venue created by college students where groups of (mostly) students could perform five minute pieces without censorship. Almost nobody memorized their lines; people would have scripts in their hands, acting their hearts out. I’d occasionally write a sketch, get some student actors to join me, and perform.
On your way to the classroom where No Shame took place, you passed “The Pit.” This was a recessed lounge area, where writers without actors would meet up with actors without scripts. I was often called to on my way past the Pit – when a writer needed someone to play “Dad” or “Grandpa” or “Loan Officer” or “Elderly Man.” All informal, all cool.
Once upon a time, I had a script, and my student actor cancelled on me, so it was time to go into the Pit. And I said, “I have a role for a woman who can do comedy.” “I can do it, Doug.”
She knew my name and who I was, but I had no memory of seeing her before. She was close to my six feet tall, with broad shoulders and dressed feminine without being girly. She seemed a bit older than the other Pit-dwellers – I figured she must have been a grad student. “I’m Ariel,” she said, just slightly condescendingly. I handed her the script, and said, “You’re Janet. Can we run through it?”
Most people from the Pit didn’t run through their sketches. They’d get the scripts, read through it in their seats before the show, and perform it for the first time on stage. I liked to practice a bit first. Even though No-Shame was a stupid, low-stakes show, it was still a show, and doing well was important to me. So Ariel and I found a place, and we read through the sketch. It was … fine. She would do.
We entered the theater, and she went off to sit with her friends in the front, and I took my spot alone in the back row. Eventually our act was called and we took our scripts with us on stage. And…
Fucking magic.
She was brilliant. Her comic timing was perfect, her acting was perfect, every facet of her performance was amazing. When I write sketches, they are often inspired by some deep sadness that I deal with by making it into comedy. Nobody notices. But she clearly did – she knew exactly what was happening under the sketch and brought it out. And WE were brilliant. Something in the way she performed my stupid sketch made ME a better actor. It was like a Jacob’s Ladder spark going between our souls.
Five minutes passed. The audience went nuts. We made eye contact. “Good job” said one of us, and she went to sit with her friends in the front, and I walked to my spot alone in the back.
The next month, I didn’t ask anyone I knew to do my sketch with me. I was hoping, not really consciously, but kind of in the soup, I was hoping that she would be there in the Pit. I walked into the Communications Arts Center, and she left the Pit and greeted me at the door. “Hi, Doug. What have you got for me?” Not “Do you have anything?” but “What do you have?” She knew what was going on. I handed her the script, we went to our spot, ran through it twice, mediocrely, and then got on stage and the magic happened again. Just like last time.
I’ve told you everything I know about her. Her name was Ariel, she was tall, and about grad-student age.
Month Three: I walked into the building, she walked right up to me and said the same thing, “Hi, Doug. What have you got for me?” I handed her her script. Then she said, “I have something for you this time.” Her sketch was in a completely different style. It was more like poetry than a Comedy Sketch – more savory than sweet. Artsy. Something you would think she’d want a 20-year-old to perform with her. Someone skinny artistic with a 3 cm ear plug and a goatee. But nope – she wanted Ol Doc Shaw (although she never used my surname) and we ran through both of them in our quiet place. And she went to sit with her friends, I sat in the back, until my sketch came up, and then later when hers did. And we were brilliant. When we did her sketch, I acted in a way I never have before, and I was perfect for the part. We made each other perfect for the part.
And then, in the next month, for the first time, I wrote something specifically for her. It wasn’t ABOUT her – it was actually about my 4 year old daughter, and my fears about her growing up. But on the surface it was a sketch about a Dad and his daughter who had been home for her Freshman Year Thanksgiving break and was packing up to go back to school when her Dad found a baggie of pot in her luggage. Hilarious. And perfect for my sketch soul mate. In my mind I heard her performing the lines, I wondered how she would bring the character to life. Where she would stand, how she would move.
And I walked into the Communication Arts Center and she wasn’t there. One of my good friends and a brilliant actress was in the Pit, and I did the sketch with her instead and it went really well.
I never saw Ariel again.
(If you happen to know Ariel, please don’t say anything. Nothing would be gained by me knowing anything more.)
Another way of thinking about multiple choice where calculations are involved
So last year I wrote a multiple choice math exam and had this weird idea. Play with me for a sec…
How to write a multiple choice question – You write the question and the right answer, then you add “distractors.” It’s like a game: teacher figures out what the most common mistakes students will make are, and then puts the corresponding wrong answer on the test. So if students are likely to forget to convert “hours” to “minutes”, you do the problem without converting to minutes and put that answer down.
Here’s my thought…
What if you just effing didn’t? What if you didn’t include that distractor? Now your student makes the silly error, goes through the work, and that answer is… not a choice. (Don’t bring up “none of the above” – keep playing with me) So the student thinks, “I made a mistake somewhere.” And maybe the student realizes “Oh! A half hour is thirty minutes! Let me try that!” and gets it right.
What’s happened? We let the student make a error, and … fix it. They didn’t get PUNISHED, they got guided. If the student really had no idea what they were doing, they would have to guess, like in any multiple choice test. But if a student DID have the knowlege, and was just careless, they would have LEARNED something while taking the test.
So multiple choice tests involving calculations. What if we said, “To hell with distractor traps” and put the right answer down, along with other, random answers? Not playing the “what mistakes will you make” game?
An example of synergy for people who get annoyed by the word “synergy.”
Hey, Gang! This is a version of an article that was originally published in AIN Quarterly. For that journal, it was suggested that I take all the “maths” out. For this post, there are some optional maths which you can skip. Here we go!
Introduction
This is a story about a single moment. A moment where I forgot my improv knowledge, and remembered it before it was too late. A wonderful experience, continuing through today, that would not have happened had I not remembered. We all have our moments forgetting our AI knowledge. Sometimes we remember in time, sometimes we don’t. This is a story about one of mine when I remembered.
The Math(s) for the curious
Feel free to skip this part if you don’t like mathematics.The specific content is not the point of the story.Scroll down to “The Tale” to skip the math(s).
Think of a graph as a collection of facebook accounts. If two people are facebook friends, draw a line between them to indicate such. Here are three examples of graphs:
Notice that Graph 3 has two different groups of people, with no friends between them. Those groups are called components, so Graph 3 has two components. Good? Good.
A graph is Eulerian if everyone has an even number of friends, and only one component. Graph 2 is Eulerian. Graph 1 is not, because there are people with odd numbers of friends, and Graph 3 is not, because it has two components.
A graph is Hamiltonian if you can start with one person, and make a happy cycle between all the accounts and wind up back where you started, without using any person more than once. (You may suspect that the little circles can stand for more than just facebook accounts – you would be right). Graph 2 is not Hamiltonian, because any such cycle would have to use the middle person more than once. Graph 3 is not Hamiltonian, because you can’t get from one component to the other.
Now let’s talk about toughness. A graph is tough if it is hard to break. Specifically, it means the graph is connected – has only one component. If any one account is deleted by Meta, the company that owns Facebook, then the result will still be connected! If any twoaccounts are deleted, the result can have at most two components, no more. If threeaccounts are deleted, the result can have at most three components, etc. Graph 1 is tough. If you delete any n accounts, the result will not have more than n pieces left. Graph 2 is not tough – if you delete the middle vertex, you will have two components. Graph 3 is not tough – it is not connected.
That’s all the math(s) that is (are) in this story.
The Tale
I was teaching an international summer program for math-enthused students. We had spent a delightful few days talking about Eulerian graphs, Tough graphs, and Hamiltonian graphs. I came to the climax of this part of the course – I had the students get into groups and prove that IF a graph was Hamiltonian, then it was Tough.
As we were summing up our discussion, and I was mentally rehearsing the introduction to our next topic, one student, Harris Spungen, asked if the converse of my statement was true – if every Tough graph was a Hamiltonian graph.
I had anticipated that question, and smoothly put this graph on the board:
We verified that the Petersen graph was tough and that it was not Hamiltonian.
I was still on autopilot when Harris raised his hand, and told me he came up with a new conjecture:
I knew his conjecture had to be false, because if it were true, it would be in all the textbooks. So I said, “I don’t believe you are correct.” I thought for minute, trying to come up with a quick counter-example. If you’ve ever taught, you know how long a minute is when you are standing at the board, thinking. I couldn’t up with one, so I said “I’ll get back to you on this tomorrow,” and went back to my lecture plan.
I said “I’ll get back to you on this tomorrow,” and went back to my lecture plan.
I’d been teaching for close to 30 years, and won teaching awards at four different universities, and I said “I’ll get back to you on this tomorrow,” and went back to my lecture plan.
I’d been going around the country giving my successful “Improv for Educators” workshops, where one big lesson was how to roll with what is actually happening in front of you, and I said “I’ll get back to you on this tomorrow,” and went back to my lecture plan.
I was teaching an ungraded course where the syllabus was whatever I said it was, a course where I wanted the students to learn the thrill of mathematical discovery, a course in my favorite subject to think about, and I said “I’ll get back to you on this tomorrow,” and went back to my lecture plan.
Fortunately, I thought
“Wait! What am I doing?”
And then I said to my class, “Hey – you know what? Nothing else I had planned for today is as interesting as Harris’s conjecture. Let’s try to find a counter-example.”
The extroverted students were at the chalkboard immediately, drawing and arguing. I erased all my notes and pictures to give them room. The more introverted ones had notebooks out, sketching and experimenting, occasionally showing pictures to each other. Five minutes passed. Ten.
A few of the ones whose previous mathematical experiences all involved immediate success were getting frustrated and mentally checking out. And a couple of the ones who had confidence problems were also checking out.
But then one group of three boys at the board called out, “I think we got it.” I was in full on improv mode, and made myself look busy and distracted and asked the checked-out students for a favor – “Can you see if they are right?” And they were all too happy to be useful, running up to the picture, to see if it was a counterexample. It wasn’t – they showed it wasn’t. But by then someone else had a proposed counterexample, and off they went – the Harris Graph Verification Squad.
Fifteen minutes. Twenty. When the HGV squad were convinced that a counter example worked, a teaching assistant and I would take a look, dashing hopes to the ground. Everyone in the room was so excited. Had Harris come up with a new condition after all? Nobody knew, not even me. And that is another lesson from improvisation in education. Uncertainty is okay. Chaos is okay. There’s a joy in not knowing. Just like on stage, when neither the audience nor the actors know how our heroes will escape from the cave guarded by bears. The teacher didn’t know if Harris was right, the students didn’t know if Harris was right, and Harris was having the time of his life trying to prove himself wrong.
Thirty minutes. Ask a teacher friend how often you get a class of students completely involved in an activity for 30 minutes. Laughing, yelling, groups splitting and reforming. All looking for this holy grail, which in my mind I was calling a Harris Graph. While the HGV Squad ran from group to group, and the TAs were called in when they couldn’t break a potential graph, I was hanging back, because pedagogically I had decided – no, that is a lie – because I wanted to find a Harris graph, too. And when working alone wasn’t doing, it, I joined a group as an equal. Improvisation lesson: Think about status. Think about status reversals.
Finally, after about 45 minutes, one graph passed the HGV inspection, the TAs inspection, and my inspection. So I asked the entire class to help confirm. All 23 people in the room gathered around, students, teaching assistants, myself. Painstakingly checking Toughness by mentally deleting accounts and counting how many pieces of the graph remained. Looking for Hamiltonicity by tracing cycles in the air with our fingers.
I said to the class, “This graph is called a Harris Graph. We have proven they exist.”
Quick Math(s) Diversion Because I Must:
This is the formal definition of a Harris Graph:
A Harris graph is a tough, Eulerian, non-Hamiltonian graph.
The Rest of the Tale
Every summer that I teach this course, I tell the students the tale of Harris, and we look for Harris graphs. Some groups find one, some don’t find any. One motivated student wanted to curate a Harris Zoo on the internet, and she worked on it for a while, and then another motivated student (years later) picked it up, and made a website. There will be a link to it below. Students are eager to come up with new Harris Graphs, knowing they will get their discoveries in the Zoo.
I wrote a math education paper about Harris Graphs, and so now they are officially a Thing. I sent a copy to Harris, who was quite pleased.
A student was enthused enough to write code to verify whether a proposed Harris Graph was valid or not. It was too hard for me to use, but another student learned how to use it, and checked the graphs I used as examples in my paper. Ummm… most of them were correct.
A few years later a different student heard me say that I personally couldn’t use the code, and so he wrote different code and a user interface so simple that even Doctor Shaw could use it. Last summer, a student used that code in a program he wrote to find all the Harris graphs with 10 accounts or fewer.
Another student asked me to mentor him on a Harris Graph research project. He told me later that the project helped him get into the university of his choice.
One of my students, Shubhra, became one of my TAs, and she and I started working together on gathering all the results students came up with over the years, including my own results, and writing up a formal paper. We are in that process now. And we’ve added two former students as co-authors, including one who figured out how to take two Harris graphs, and combine them to make a new one! I would strongly recommend collaborating with someone on the other side of the world, by the way – I wake up to see the work she’s done while I’ve slept, and vice-versa. And we can zoom during my morning/her evening. Dozens of students have emailed me, asking me to help them understand mathematics research papers they were reading, on their own, as they researched graph theory results they hoped would help them with their independent Harris graph research.
We are thinking we may have to split this into two math papers.
Conclusion
Harris graphs have inspired students to embrace their love of mathematics, and sometimes to add the subject as a major or minor in college. And now they are a mathematical Thing, and when Shubhra and my graph paper is published, mathematicians who couldn’t care less about the Tale may still be motivated to take them up as a research topic. Over 500 students have looked for Harris graphs. I cannot picture teaching Graph Theory without having a Harris Graph day. But what made it possible? The skills I learned as an improvisor.
Let’s go back to the basic concepts that come up when I teach improv, that came up when I learned it. Listening. Yes-Anding. Status. Inclusivity. Honesty. Vulnerability. Persistence. Spontaneity. Think of your individual pet improv concept – I’ll bet you can find it in the soup above. Mathematics is a large subject, Mathematics Education is a large subject, and this tale was an example of how Improv training affected both of them.
Links! Links! Links!
If you have attempted to find your own Harris Graph, you may want to visit the Harris Graph Zoo! You can find it here: https://tinyurl.com/harriszoo
How do we begin? Let’s start light. Next time we will do a really long piece about math education and improv training.
DOUG: What kind of rabbits can vegetarians eat? BASHO: Dust bunnies!
DOUG: Why does Dracula hate pasteurization? BASHO: It makes his blood boil.
DOUG: What do you call it when someone lives among Vulcans long enough to start identifying with them? BASHO: Spockholm syndrome.
DOUG: I only know BASIC and JAVA – can you help me adjust to C+? BASHO: I can give you a few pointers.
DOUG: I went to a fancy seminar on pilfering BASHO: Oh? What were your takeaways? DOUG: The silverware
DOUG: How are we going to increase our container-making factory’s profits? BASHO: We’re just going to have to start thinking outside of the- DOUG: Don’t say it!
