I have had many people say to me over the years, “But algebra is easy: just tell them to do the same thing to both sides!” This is wrong in several ways, not least of which is the word “easy”. The particular way it’s wrong that I want to talk about today is the idea that doing the same thing to both sides is somehow the only move in algebra, because it’s not even the most important or the most common move.
I think asking students questions is an important part of my job of helping students succeed. Good questions can help me see where they are in their journey so I can choose how to guide them to the next step, or can help to make clear the skills they already have that will help them figure things out for themselves. But there is a class of questions that shuts all of this down immediately. Here are some examples:
“Did you go to the lecture?”
“Have you started yet?”
“How many of the exercises have you done?”
These questions all have answers that are morally Right or Wrong. The answers a student gives make the student out to be a Good Student or a Bad Student. And if a student has the Wrong Answer, they will feel ashamed.
I know many people who believe it is very important to send students the message that they should go to lectures, start assignments straight away, and do all the exercises. While these are all things students could do to help themselves, they’re not the most important thing to focus on when they are here seeking support from me. They can’t change any of those things right now, so all a question like those does is make them feel ashamed. And, as Turnaround for Children CEO Pam Canto says in this blog post , “shame is toxic to positive outcomes”.
Shame is the feeling that you are a bad person, that there is something wrong with you. Guilt is a bad feeling about your actions, which is unpleasant, but may make you want to change those actions in the future. Shame is the next level, where you feel you have been exposed as the horrible person you really are. A person who feels shame won’t try to change their actions, they’ll just try to avoid situations that expose them, which will just make the problem worse. I don’t want this to happen to my students, and I certainly don’t want them to think that seeking support from me will expose them to shame, or they will decide not to seek help.
Once upon a time, I realised that I was causing a student shame, and I decided that I would give myself a new principle.
Never ask a question that has a morally wrong answer.
This is one of the rules I use to evaluate if my question is useful and choose a better alternative.
For example, I could ask “Did you go to the lecture?”, but there is definitely an answer to this question that is morally wrong and having to give that answer will cause shame. Do I really want to know if they went to the lecture? How will that help? Maybe what I really want to know is what the lecturer has to say about the topic, since that might be useful. In that case, I could ask “What did the lecturer have to say about this?” The student doesn’t have to reveal their attendance status to answer this question, thus avoiding the shame. Even better would be to avoid the awkward moment where they have to reveal they don’t know, and say, “It would be useful to know what the lecturer says about this. Can you tell me what they said, or tell me where we might go looking for that?”
For my second shame-inducing question of “Have you started yet?”, the first simple fix is to remove the “yet”. That implies they should have started already. The second fix is to think about why I want to know this? Maybe I want to know what they’ve done already so we can build on it. In that case I could just ask “What have you done so far?”, since that’s directly asking for the information I want. But there is still an implication that they should have done something, so causing shame if they have to reveal they’ve done nothing. So instead I could ask “What are you thinking about this problem?” or maybe “How do you feel about this problem?”. These let me get into their head and heart and I can help them move on from there. I might be able to ask them about what they’ve done so far later, or it might not even be important because they’ll tell me what they need to help themselves.
This second example highlights another principle, which is to ask open ended questions, preferably about student thoughts and feelings. This makes it much easier to ask questions without morally wrong answers, because there are no specific predetermined answers in particular! (Asking open-ended questions is actually one of the factors in SQWIGLES, the guide for action I give to myself and my staff at the MLC.)
So, I urge you, think about whether the questions you ask have a morally wrong answer, and if so, try a more open-ended question that is less likely to cause the shame that is so toxic to success.
These comments were left on the original blog post:
Todd Feitelson 12 September 2020:
Thanks for this, David. I appreciate as always the attention you give to the details of your interactions with your students. It can be excruciating to have to consider every word you speak, but it’s really important. It’s a big source of stress and anxiety for me. But, it’s critical.
I do wonder about the difference between a direct question (“Were you at the lecture?”) and a less-direct question (“Can you tell me what the lecturer said?”) As you stated, you want to avoid that awkward moment when the student might have to admit not being there, but it feels way more awkward (almost like a trap) to ask the indirect question. For me, setting up the situation and relationship where the direct question can be seen as just a way to gather the information without judgement is the trick. My goal can be to get that all out on the table and move forward, with the student seeing me (I hope) as an ally.
I’m dealing with younger students, but there is also some value in helping students see that going to the lecture is a valuable tool in learning. It’s water under the bridge once they’re asking for help, but they are asking for help, so it seems important to help them learn for the next time. In a shame-free way — that’s the challenge.
(Somewhat irrelevantly, it reminds me of what my English teacher wife wants to say when kids come and ask her how to improve their vocabulary scores on standardized tests — “Have read a lot since you were ten.” Not so helpful, so she remains positive and forward-looking, and avoids the shaming.)
Thanks for writing!
David Butler 12 September 2020:
Very helpful thoughts Todd. A decent relationship where you can ask questions to keep them accountable is definitely something that is desirable. For me, this is usually the first time I’ve met them so I don’t have that luxury. I will say that when I have used “It would be good to know what the lecturer said…” and we’ve looked it up, then they often say later that going to the lecture or at least consulting the notes is the thing they learned today.
This blog post is about a game I invented in February 2020, the third in a suite of Battleships-style games. (The previous two are Which Number Where and Digit Disguises.)
Once upon a time, I met a His Royal Highness the Duke of Kent.
The story of how that happened was pretty cool from my perspective, but every so often I wonder about it from his perspective. The Duke is the patron of the Royal Institution of Australia, and was in Australia just as they were displaying their installation of the Institute for Figuring’s Hyperbolic Crochet Coral Reef. So they organised a special viewing for him, to which a few key people were invited. One of those people was me, since I was the mathematician involved in the project.
I had a short but pleasant conversation with an old man about geometry and what it was like to be a man crocheting in public. We’re laughing in the photo, so somebody must have said something funny, though I don’t remember what it was. And that was it.
I suppose His Royal Highness meets a lot of people, so I might not make much of a difference to him. But every so often I think about how when he thinks about his time in Adelaide, there I am, if even for a moment (perhaps made just a little more memorable due to the rarity of meeting people standing in a roomful of coral made of yarn). I am part of the life memory of the Queen’s cousin.
Which gets me thinking…
Many of the students who I meet through my work in the MLC, I only meet once. Yet what happens with me will be a part of their memory of their time at university, however small. My short time with them is entangled forever with the events of their lives during their study. Will that moment with me make their memories more or less pleasant? Will it encourage them or discourage them? Will I just confirm their worst fears about themselves, or will it be at all like this student, who one visit to a lab in one class made all the difference?
From Ashleigh Geiger on Twitter 23 Aug 2020
For me, at school, the focus on memorising rather than understanding is what fostered a fear of maths. Returning to uni later in life and needing to use maths (biochem and genetics) showed me there was nothing wrong with my abilities, just my perception of my abilities. And the way the maths was taught in my degree showed me how powerful and interesting it can be, and helped me learn to enjoy it again. I never ended up visiting the MLC but having your team wander around in prac classes, peer over my shoulder and tell me I’d nailed it did wonders for my confidence! 😊 So whilst I did have experiences early on that made me fear/hate maths, take heart that they can be turned around by positive and empowering experiences with the right teacher. Love your work!
Every one of these students is important to someone, much more than a Royal who lives on the other side of the world is important to me. And I am there in their memory for better or for worse. This inspires me regularly to be the best I can be with each and every one of the students, especially that very first time I meet them.
Because once upon a time, His Royal Highness the Duke of Kent met me.
One of my favourite puzzles is the Twelve matchsticks puzzle. It goes like this:
Twelve matchsticks can be laid on the table to produce a variety of shapes. If the length of a matchstick is 1 unit, then the area of each shape can be found in square units. For example, these shapes have areas 6, 9 and 5 square units.
Arrange twelve matchsticks into a single closed shape with area exactly 4 square units.
I will tell you soon why this is one of my favourite puzzles, but first I want to tell you where I first learned this puzzle.
Once upon a time, about ten years ago, I met a lecturer at my university who was designing a course called Puzzle-Based Learning. It was an interesting course, whose premise was that puzzles ought to provide a useful tool for teaching general problem-solving, since they are (ostensibly) context free, so that problem-solving skills are not confounded with simultaneously learning content. Anyway, when he was first telling me about the course, he shared with me the Twelve Matchsticks puzzle, because it was one of his favourite puzzles. The reason it was his favourite appeared to be because the solution used a particular piece of maths trivia.
And the puzzle has become one of my favourite too, but for very different reasons! Indeed, I strongly disagree with my colleague’s reasons for liking the puzzle.
The first thing I disagree with is the use of the word “the” in “the solution”, because this puzzle has many solutions! Yet my colleague presented it to me as if it had just one.
The second thing I disagree with is that using a specific piece of trivia makes it likeable. I do recognise that knowing more stuff makes you a better problem-solver in general for real-world problems, but even so I always feel cheated when solving a puzzle requires one very specific piece of information and you either know it or you don’t. It seems to me it sends a message that you can’t succeed without knowing all the random trivia in the world, when of course it is perfectly possible to come up with acceptable solutions to all sorts of problems with a bit of thinking and investigating.
The main reason I love the Twelve Matchsticks puzzle is because it does have many solutions, and because this fact means that different people can have success coming up with their own way of doing it, and feel successful. Not only that, if I have several different people solving it in different ways, I can ask people to explain their different solutions and there is a wonderful opportunity for mathematical reasoning.
Another reason I like it is that people almost always come up with “cheat” solutions where there are two separate shapes or shapes with matchsticks on the inside. I always commend people on their creative thinking, but I also ask if they reckon it’s possible to do it with just one shape, where all the matchsticks are on the edge, which doesn’t cross itself. I get to talk about which sorts of solutions are more or less satisfying.
The final reason is that most of the approaches use a lovely general problem-solving technique, which is to create a wrong answer that only satisfies part of the constraints, and try to modify it until it’s a proper solution. This is a very useful idea that can be helpful in many situations. As opposed to “find a random piece of trivia in the puzzle-writer’s head”.
Which brings me to the second part of the title of this blog post. Let me take a moment to describe two different experiences of being helped to solve this puzzle.
When I was first shown this puzzle, my colleague thought there was only one solution, and his help was to push me towards it. He asked “Is the number 12 familiar to you? What numbers add up to 12? Is there a specific shape that you have seen before with these numbers as its edges?” He was trying to get me closer and closer to the picture that was in his head and was asking me questions that filled in what he saw as the blanks in my head. And there were also some moments where he just told me things because he couldn’t wait long enough for me to get there.
I did get to his solution in the end, and I felt rather flat. I was very polite and listened to his excitement about how it used the piece of trivia, but inside I was thinking it was just another random thing to never come back to again. It wasn’t until much later that I decided to try again, wondering if I could solve the problem without that specific piece of trivia.
So I just played around with different shapes with twelve matchsticks and found they never had the right area. And I wondered how to modify them to have the right area. And I played around with maybe using the wrong number of matchsticks but having the right area, and wondered how to change the number of matchsticks but keep the same area. This experience where I used a general problem-solving strategy of “do it wrong and modify” was much more fun and much more rewarding.
When I help people with the Twelve Matchsticks puzzle, this second one is the approach I take. I ask them to try making shapes with the matchsticks to see what areas they can make. I ask what all their shapes have in common other than having 12 matchsticks. I notice when they’ve made a square with lines inside it and wonder how they came up with that. Can they do something similar but not put the extra ones in the middle? In short, I use what they are already doing to give them something new to think about. I encourage them to do what they are doing but more so, rather than what I would do.
The approach where you have an idea in your head of how it should be done and you try to get the student to fill in the blanks is called funnelling. When you are funnelling, your questions are directing them in the direction you are thinking, and you will get them there whether they understand or not. Often your questions are asking for yes-or-no or one-word answers to a structure in your head which you refuse to reveal to the student in advance, so from their perspective there is no rhyme or reason to the questions you are asking.
The approach where you riff off what the student is thinking and help them notice things that are already there is called focussing. When you are focussing, you are helping the student focus on what’s relevant, and focus on what information they might need to find out, and focus on their own progress, but you are willing to see where it might go.
Interestingly, when you are in a focussing mindset as a teacher, you often don’t mind just telling students relevant information every so often (such as a bit of maths trivia), because there are times it seems perfectly natural to the student that you need such a thing based on what is happening at that moment.
From my experience with the Twelve Matchsticks, it’s actually a rather unpleasant experience as a student to be funnelled by a teacher. You don’t know what the teacher is getting at, and often you feel like there is a key piece of information they are withholding from you, and when it comes, he punchline often feels rather flat. Being focussed by a teacher feels different. The things the teacher says are more obviously relevant because they are related to something you yourself did or said, or something that is already right there in front of you. You don’t have to try to imagine what’s in the teacher’s head.
So the last reason the Twelve Matchsticks is one of my favourite puzzles is that it reminds me to use focussing questions rather than funnelling questions with my own students, and I think my students and I have a better experience of problem-solving because of it.
I’ve just started teaching an online course, and one module is a very very introductory statistics module. There are a couple of moments when we ask the students to describe how they interpret some hypothesis tests and p-values, and a couple of the students have written very lengthy responses describing all the factors that weren’t controlled in the experiments outlined in the problem, and why that means that the confidence intervals/p-values are meaningless. When all we wanted was “we are 95% confident that the mean outcome in this situation is between here and here”.
It’s happened to me a lot before. Many students in various disciplines are extremely good at coming up with worries about experimental design or validity of measurement processes, and so they never get to the part where they deal with the statistics itself. They seem to treat every problem like the classic “rooster on the barn roof” problem, essentially declaring that “roosters don’t lay p-values” and choosing not to answer the question at all.
Don’t get me wrong, I really do want the students to be good worriers: they should be able to think about experimental design and validity and bias and all those things that impact on whether the statistics answers the question you think it does. But what they can’t do is use it to avoid talking about statistics at all! There are quite a few students who seem to be using those worries to discount all statistical calculations, and to sidestep the need to understand the calculation processes involved. “Your question is stupid, and I refuse to learn until it’s less stupid,” they implicitly say.
The weirdest part is that the assignment or discussion questions don’t usually discuss enough details for the students to actually conclude there is a problem. They say “the groupss were not kept in identical conditions”, but nowhere does it say they weren’t. I realise that in a published article if it doesn’t say they were then you might worry, but this is just an assignment question whose goal is to try to make sense of what a p-value means! Why not give the fictitious researchers the benefit of the doubt? And also, take some time to learn what a p-value means!
I do realise it’s a bit of a paradox. In one part of the intro stats course, we spend time getting them to think about bias and representativeness and control, and in another part, we get grumpy when they think about that at the expense of the detail we want them to focus on today. It must be a confusing message for quite a few students. But on the other hand, even when reading a real paper, you still do need to suspend all of that stuff temporarily to assess what claim the writer is at least trying to make. It’s a good skill to be able to do this, even if you plan to tear down that claim afterwards!
I am thinking one way to deal with this is to start asking questions the other way around. Instead of asking only for “what ways could this be wrong?”, ask “how would you set this up to be right?” And when I ask about interpreting a p-value, maybe I need to say “What things should the researcher have considered when they collected this data? Good. Now, suppose they did consider all those things, how would you interpret this p-value?” Then maybe I could honour their worries, but also get them to consider the things I need them to learn.
I have learned a lot from Twitter about how to treat my students, and most of it has been through being treated in ways I do not like. Recently I have been searching my own tweets to find things I’ve said before, and as I’ve dipped into old conversations, several unpleasant feelings have resurfaced when I read the way I’ve been treated. I don’t want to make my students feel that way, so I want to avoid doing those things to my students.
So, here are some ways I don’t want to treat my students, based on ways I have been treated on Twitter. To my shame, I have done most of these to others on Twitter too, and I am trying hard not to. I know most of the people who have done this to me will be mortified to know they have, so I am not going to call anyone out here. I just want to share what I have learned.
So, here is a list of things I don’t want to do to the students, because I don’t like it when they happen to me:
Offer solutions when they haven’t asked for any.
Interrupt their problem-solving process.
Ignore their feelings when they express them.
Tell them their feelings are wrong.
Respond with a story about me rather than seek more from them.
Completely ignore the main point of what they said and respond to just one word or phrase.
Respond to them telling me something they like by giving recommendations for new things.
Respond with a fire hose of even more technical terminology.
Tell them they are wrong to be confused.
Tell them the thing they like is wrong because I like something else.
Discount their success by pushing to the extension straight away.
Respond to everything with sarcasm.
Focus only on the bit I think is wrong.
My original plan was to elaborate on each of these, but I have kept coming back to this post for months and feeling overwhelmed with that task, so I think it’s time to just push send. And maybe it’s a good thing each time I read this to have to imagine what each of these things looks like. I’m hoping it’s useful to you to have to think about what these might mean too. But of course if you want me to explain a particular one of them more, do ask and I will do my best.
This blog post is about a piece of the MLC learning environment which is very special to me: the date blocks. It’s a set of nine blocks that can be arranged each day to spell out the day of the week, the day number, and the month. I love changing them when I set up the MLC in the morning, so much so that since the face-to-face MLC closed due to COVID-19, I brought them home and have been changing them each morning here in the dining room. The story of how this object came into the MLC is the reason it is so special to me.
The weekly puzzle session that I run at the University of Adelaide is called One Hundred Factorial. In the middle of the night, I suddenly realised that I have never written about why it is called One Hundred Factorial, and so here is the story.
The very beginning
Once upon a time I was a PhD student in the School of Mathematical Sciences at the University of Adelaide. Sometime during the third year of my PhD program (2007), I was asked to give a talk to the first year undergraduate students as part of an evening event where the goal was to hopefully convince them to keep studying maths at a higher level next year. I titled my talk “How to Tell If You Are a Mathematician”. I don’t remember any of the things I spoke about, except for one thing. Before I started talking, I put a puzzle up on the document camera. I did not mention the puzzle in any way or look at the screen at all. I just did my little talk as if it wasn’t there. But right at the end of my talk I said this:
The final and truest way to tell that you are a mathematician is that you haven’t been listening to any of what I just said, and instead have been trying to solve this puzzle.
Cue guilty looks and nervous laughter from all of the academic staff in the audience, which successfully proved my point. Anyway it worked. Several students came up to me to talk about the puzzle, and I was able to direct them to lecturers who could talk to them about their study options. Yay for puzzles, right?!
This was the puzzle I used so neatly to make my point about the mathematician’s mind:
The number 100! (pronounced “one hundred factorial”) is the number you get when you multiply all the whole numbers from 1 to 100. That is, 100! = 1×2×3×…×99×100. When this number is calculated and written out in full, how many zeros are on the end?
I don’t remember where I got the puzzle from, but it is a pretty famous one that’s been around for some time. I actually hadn’t even thought through a solution at the time either. I just knew that it mentioned a concept that had been in the first year lectures recently.
The puzzle sessions begin
The other thing that happened that night was that a group of students and staff stood at the blackboard in the School of Maths tea room to nut out a solution to the 100! puzzle. I can’t even remember if we finished it or not, but we did decide that we should get together regularly to solve puzzles together, and a weekly puzzle session was born. At the first session, we started with the 100! problem again, and an extension of it, which is to find out what the last digit is before all those zeros start. Then as the weeks went on, we would do puzzles that I would find and bring to the sessions.
When I finished my PhD in mid-2008 and took up the job in the Maths Learning Centre, I took my little puzzle session with me, and was able to invite more students to come along, and it slowly morphed into a student event more than a staff event, which really pleased me. In fact, a regular at these puzzle sessions for years was that first student who had come up to me after my talk at the first-year event, and he eventually became one of my tutors at the MLC.
The name of the event
Over the years the puzzle session has had many names. We started out calling ourselves “People with Problems”, and then simply “Puzzle Club”. For a while it was called “The Hmm… Sessions” after the sound we made very often while thinking about puzzles. Indeed, there is a reference to the Hmm Sessions inside this very blog. But in 2012 after the website where I was hosting our online discussion was decommissioned, I decided it was time to change the name. I was also starting to think about moving the sessions out of the MLC itself and into a public space, and to match with this move I wanted a new name. I thought long and hard, and decided to name it after the first puzzle we ever did, the puzzle that first inspired staff and students to talk and think about maths together, the puzzle that helped students decide they really were mathematicians after all.
The legacy
So the regular puzzle session of the MLC became One Hundred Factorial at the end of 2012, and here we are in 2020 still going, so that now it’s been One Hundred Factorial longer than it’s been any other name. It’s been my testing-ground for new puzzles and games and teaching ideas, a place where I have made friends and welcomed people from around the country and the world. And it has become a glowing island of mathematical play in the middle of the stressful university life, and indeed the middle of a stressful life generally. In recent weeks it is a glowing island of community in a world of pandemic-induced isolation.
One Hundred Factorial reminds us that there is always something joyful to think about if you are looking for it, and that it’s okay to pause and ignore your responsibilities for a while to think about it, and that doing this with people is a source of shared joy. I hope the puzzle and the event can keep reminding us of that for a long time yet.
Context fatigue is a particular kind of mental exhaustion that happens after having to make sense of multiple different contexts that maths/statistics is embedded in. I feel it regularly, but I feel it most strongly when I have spent a day helping medical students critically analyse the statistics presented in published journal articles.
The problem with maths in context is that the contexts themselves require understanding of their own in order for the maths to make sense. This is nowhere more true than in statistics, where you have to use your understanding of whether you expect the relationship to exist, what direction you expect it to be, and whether you think this is a good or bad thing. The classic one in my head is an old first year statistics assignment where they used linear regression to investigate the relationship between manatee deaths and powerboat registrations in each month in some southern coastal American city. You have to know what a manatee is, what it means to register a powerboat, and why those things might possibly be connected in order for the statistical analysis you’re asked to do to make sense, not least because at least one part of the question will ask you to interpret what it means. When helping students read published articles recently, I’ve had to find out what’s been done to the participants in the study, how things have been measures, what kind of measurements those are, why they’ve been measured that way, and all sorts of little details to decide how to interpret the numbers and graphs that are presented.
Even ordinary everyday word problems are a minefield. Across two recent assignments, some financial maths students had to cope with album sales for AC/DC, flooding of the land a factory is built on including insurance, bull and bear markets, machines in a mining operation, committees with various named positions, road testing electric cars, contraband being smuggled in shipping containers. This is a lot of context that has to be made sense of before you can get a handle on the maths, and there is nothing in the question itself to tell you what any of this context means if you don’t already know. Even if you are already familiar with the context, you actually have to suspend some of your understanding in order to do the maths problem, because it’s much simpler than the actual situation any of the questions are talking about.
All of this interpreting is exhausting stuff! It just tires you out if you have to even a moderate amount all at once. You just feel like you don’t have any more energy to deal with any more today. That feeling there is context fatigue. Yesterday the first year maths students were doing related rates and every question was a new context with little nuances created by the context that had to be dealt with. Those poor students were exhausted after just one problem, letalone three or four.
As teachers, we need to realise that as the people writing the assignment questions, or at least people who have dealt with them before, we are much more aware of the details and nuances of the context than the students are, so we don’t have to work so hard to make sense of them. Not only that but we’re usually simply more experienced in both life and language than most of our students so it’s easier for us. Imagine the context fatigue you would get reading ten research papers in an unfamiliar area in one day (I feel this in real life regularly). That’s the sort of context fatigue your students have just from your assignment questions. Cut them a little slack, and make sure there is adequate time to process the context with appropriate rest time between context-interpretation. Also it wouldn’t be the worst thing to explicitly teach them strategies for making sense of context, such as ignoring the goal, and finding out about what some of the words mean. Strategies can make the work less intimidating, especially in the face of knowing how tiring it is already!
PS: If you’re in charge of tutors in a drop-in support centre, especially one that deals with statistics, please be kind. Context fatigue is real and tends to wear us down some days!
