Making Your Own Sense

Reflections on maths, learning and the Maths Learning Centre, by David K Butler

Tag: questions

  • Questions with a morally wrong answer

    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.

  • Struggling students are exploring too

    I firmly believe that all students deserve to play with mathematical ideas, and that extension is not just for the fast or “gifted” students. I also believe that you don’t necessarily need specially designed extension activities to do exploration – a simple “what if” question can easily launch a standard textbook exercise into an exploration.

    This is lovely, but one problem is those students who on the face of it don’t want to play. The majority of students I work with in the MLC are not studying maths for maths’s sake, but because it is a required part of their wider degree. That is, I am helping students who are studying maths for engineering, or calculations for nursing or statistics for psychology. A lot of these students just want to be told what to do and get the maths over and done and don’t like “wasting” time playing with the ideas.

    Or so I thought. I have realised recently that actually they do like playing with the ideas. I just couldn’t see that this was what they were asking for.

    One of the questions I like to use to play with maths is “what if?” To ask it requires the asker to notice a feature they think might be different, and wonder what would happen if it was, and so investigate how this feature interacts with the other features of the situation. This investigation of how ideas interact is exactly what mathematics is, to me. I am very very used to doing it at extracurricular activities like One Hundred Factorial, and it’s easy to do with students who are doing very well with their maths and have the breathing space to wonder about this stuff because they’ve finished their work. For students working on an assignment they are really struggling with that they wonder about the usefulness of for their degree and which is due in the next 24 hours, it’s not so easy.

    Only maybe it’s a little easier than I thought, because I started to notice that the students were already asking questions about the connections between things.

    A very common question students ask around exam time is “What would you do if the question was like this …?” as they suggest a change in a small detail that makes it more similar to the past exam question they have at hand. I used to get really annoyed at this sort of question, but then I realised that in order to ask this question they had already noticed that the two problems were similar in most respects and different in this one. Sure, they may be motivated by a belief that success in maths is about a big list of slightly different problems and remembering ways to deal with each, but on the other hand they have noticed a relationship and are trying to exploit it, This is a mathematical kind of thought, to look for similarities and differences and relationships, and we can hang on tight to it and actually learn something!

    Another kind of question students ask is “Why is this here in this course?”. I used to get annoyed at the whinging tone of voice here, but then I realised that a student is begging for a connection between this and the rest of what they are studying, and connections is precisely what understanding maths is all about. I can respond to it by saying that yes I am also confused about the curriculum writer’s logic for including it, but then we can search out connections together to see if we can find them.

    A third kind of question is the one where the student looks at the lecture notes or solutions and asks “How did they know to do this?”. Sure, it’s usually motivated by wanting to be able to successfully do it in their own exam in restricted time, but on the other hand it does recognise that there ought to be reasoning involved in that decision, as opposed to just guessing. This expectation of reasoning is the beginning of believing that they themselves could reason it out too.

    My second-last kind of question is “Why is this wrong?” as the student points to the big red cross on their MapleTA problem on the screen. Sure they just want to make it all better so they can submit the damn thing. But they are also recognising that there must be a reason why it’s wrong, and so are looking for meaning. These students are often ripe for the experience of looking at the information and how it’s related to what they’ve entered, thus doing exploration. They are also usually ready to try various different ways of entering the result, or different strategies of getting a right answer to see where the edges of the idea are.

    The final kind of question I want to mention isn’t even a question, it’s an exclamation of “It doesn’t make sense!” Even this is telling me that the student thinks it ought to make sense. They are crying out to do something to make sense of it. Often they describe being almost there and needing something to push them over into understanding. This student is ready to explore the edges of the understanding they do have to see where the nonsensical stuff can fit, or where their ideas need some tweaking to fit together better.

    It is very helpful to me as a one-on-one or small group teacher, or even a lecturer with a big room of students in question time, to be able to see the questions struggling students are asking as cries for sense-making exploration. It doesn’t matter that they are struggling or don’t understand things. Indeed, being in a situation of not knowing is exactly what we are doing for the fast students when we give them extension activities, so why is the everyday maths the slower students are not understanding any different? In my experience students like being treated as if they are behaving like mathematicians when they have these struggling sorts of questions. And it was so much easier to treat them that way after I realised they are behaving like mathematicians when they have these kinds of questions.

  • Who is worthy to ask stupid and smart questions?

    This post was going to be part of the Virtual Conference of Mathematical Flavours, which you can see all the keynote speakers and presentations here: https://samjshah.com/mathematical-flavors-convention-center/ . The prompt for all the blog posts that are part of this conference is this: “How does your class move the needle on what your kids think about the doing of math, or what counts as math, or what math feels like, or who can do math?” In the end, it didn’t end up being there, because my computer started dying painfully at the critical time, but I still want to highlight the Virtual Conference anyway because it was a great idea.

    There are many things I could have written about this, but I think I will choose one thing that is about my approach in the MLC to student questions. In the MLC everyone is worthy to ask both stupid and smart questions.

    My Maths Learning Centre is a place where any student doing coursework at the Uni of Adelaide can visit to talk about their maths learning with a tutor (often me). People come to talk about all aspects of their maths learning in all sorts of places where maths appears, from dividing whole numbers by hand to understanding proofs about continuity of functions between abstract metric spaces. My point here today is that people from both ends of that spectrum and everywhere in between are allowed to ask questions that are about basics and questions that are about deep connections.

    Imagine a student who has always been good at maths, who finds things easy and quickly grasps abstract definitions. It is natural for such a student to fold their goodness at maths into their identity, which often means they become extremely embarrassed to show any sign of struggling. They’re supposed to be the smart student and this simple stuff is supposed to be obvious for them. So if they have a question about the basics, they hide it and hope it will come clear eventually.

    The thing is, having a question about something simple doesn’t make you stupid, and it doesn’t even make you not smart. Having a question about how to get from line 3 to line 4 is at the very least a sign that you’re paying close enough attention to wonder about that step; having a question about the definition is a sign that you know definitions are important; and having a question about some random bit of algebra or notation you happen to have never seen just shows you want to learn. In my Maths Learning Centre, I try to make it a place where everyone can ask a “stupid” question. Where stupid questions are treated with respect and answered clearly, with encouragement to make sense of what is happening.

    Now imagine a student who has always struggled with maths, who just never seems to understand the explanation the teacher is giving the first time, and who struggles to get through the first few of the exercises. It is natural for such a student to fold their badness at maths into their identity, which often means they don’t even try to understand things and just look for some step-by-step instructions they can follow so it will be over with as quickly as possible.

    The irony is, they never finish their exercises, so they never get to be part of that part of a maths class where the early finishers ask the deep and involved questions about theory and beyond-curriculum interesting stuff – the very stuff that can make maths a lot more fun. I know for a fact that students who feel they are bad at maths are intelligent people capable of logical and creative thought, and they deserve to ask their deep questions. So in my Maths Learning Centre, I try to make it a place where everyone can ask a “smart” question. If a student who is struggling asks about infinity or quaternions or what my PhD was about, I will damn well discuss it with them. If they look at the work they’re doing and ask how it is connected to some other bit of maths, we’ll explore that together. That curiosity is a treasure to be prized and I will not squash it by saying we have to get on with the assignment now.

    And you know what, it turns out that many a basic question is actually a deep and clever question after all. Recently a student who was struggling asked why it was ok to add two equations together. Not one student in my ten years of working at the MLC has ever asked that question! There must be something really special about the person who asks this question, right? And it’s a really deep question about the nature of equality. I want my Maths Learning Centre to be a place were it is okay for everyone to ask a question that is simultaneously stupid and clever.

    That’s all I have to say. I believe everyone deserves the chance to ask stupid questions and to ask clever questions and to ask questions that are simultaneously both. They are worthy to have their questions taken seriously and the answers discussed with respect for the humanity and intelligence of the asker. I have to always remind myself to give students the chance to ask these questions when I’m with them, especially students who are struggling to articulate the questions for whatever reason. And maybe if they’re not asking, I’ll sometimes ask the questions for them and we’ll answer them together.

    How will you welcome all people in your learning spaces to ask all kinds of questions?

  • Book Reading: Which One Doesn’t Belong – Teacher Guide

    This is another post about a teaching book I’ve read recently. This one is about the Which One Doesn’t Belong Teacher Guide by Christopher Danielson.

    (You can read this blog post and all other Book Reading posts in PDF form here. )

    It goes with a beautiful little picture book called “Which One Doesn’t Belong?”, which is a shapes book different from any you’ve ever seen before. In this book, each page has four pictures, and asks the readers to say which one doesn’t belong, and why. The fabulous thing about the book is that there is at least one reason why each of the four pictures doesn’t belong, and talking through these with children (or indeed anyone) is a rich conversation about the properties that shapes have and don’t have.

    The Teacher Guide is all about these rich conversations: why it’s important to have them, what you and your students/children can learn through them, and how to facilitate them. Chris has a friendly and welcoming style which draws you easily into a new appreciation of the sophisticated thoughts of children as they make sense of geometry and the world.

    There are a few key things Chris talks about that really impacted my thoughts about teaching and learning maths. I’ve organised them by quotes from the book:

    Commonly in maths class, student responses are compared to a standard answer key – the measure of what’s right is what’s in the back of the book, or what the teacher has in mind. In a conversation about a well-designed Which one doesn’t belong? task, the measure of what’s right is what’s true. – page 3

    I read this quote first when someone else tweeted it out of the book and it struck me as awesome then. In my job at the Maths Learning Centre, students are always asking me if things are right, as if the measure of rightness is if I say it is. But in most places in maths, correctness is measured by truth. Your vectors will either be an orthonormal basis for the subspace or not. A number is either prime or it’s not. You can tell if you’re right by thinking about whether it’s true. I very much want to see opportunities to talk about the truth of things with students, to put the measure of rightness outside an authority figure.

    The van Heiles haven’t argued that it is difficult to go from level 1 thinking directly to secondary school geometry; they have argued that it is impossible. If students don’t have experience and instruction building informal geometry arguments, they will not learn to write proofs. – page 8.

    Chris is referring to the van Hiele model of “how childrens’ geometric thinking develops over time”. In this model, there is a build-up from noticing that shapes look like things they’ve seen (level 0), to noticing properties that shapes have and don’t have (level 1), to relating properties between properties of shapes (level 2), to logically supporting claims about these relationships (level 3).

    The thing in the above quote that really struck me is the idea that it’s impossible to learn to write proofs without experiencing informal arguments first. I see so many students at university every day who struggle with proofs, and it makes me wonder that they maybe need more experience with informal arguments. Indeed, it makes me wonder if they need more experience simply noticing properties, since that’s an even earlier level. This is essentially applying the van Hiele models to other types of maths, but certain aspects of the progression still feel right to me, especially for things vaguely geometrical like vectors or matrices or graphs of functions.

    I wonder if a student struggling with proofs might benefit from talking through a progression like this, and then helping them have experiences at the earlier levels before helping them with proofs.

    Of course being able to state new facts is an aspect of learning, but much more important to me is being able to ask new questions. – page 21

    I had never thought of this idea explicitly before, but immediately I saw that new questions were important to me as well. I was reminded of the time someone asked me if my students were understanding my statstics lectures. I said that I wasn’t completely sure, but certainly the students were asking very deep and complex questions. Instinctively I knew that a new type of question indicated learning.

    Also, in the Drop-In Centre, there’s a certain joy when a student asks new questions you’ve never thought of before. They are wondering about the connections between things, which means they are learning, because learning is all about connections.

    I am excited to listen out for new questions as a sign of learning, and to tell the students that it’s a sign of learning to have new questons!

    … I hope you will begin to see geometry through children’s eyes as well as through the eyes of a mathematician. Mostly, I hope you will come to understand that these two views of geometry are not nearly so distant as the school curriculum might lead us to believe. – page 37

    Now, I already believe that children’s investigations and ideas are actually very close to the way mathematicians work. You can’t be married to a very excellent early childhood educator without coming to some appreciation of this! It’s so nice to have someone publish a book telling teachers and parents the same.

    Even more, this whole section is all about noticing and naming things and their properties. It’s about whether properties need names at all, or whether the objects that share those properties need names. It’s about what properties are important to make a thing a special thing and what aren’t, and in what context. It’s about the relationships between things. All of these are the work of professional mathematicians both pure and applied. And they are the work of children sorting out how the world works.

    The geometry of children and the geometry of mathematicians are definitely not so far removed.

    I have come to understand that talking about this difference is more important than defining it away. – page 54

    Along with the rest of this chapter, this quote got me thinking about a whole new way to approach definitions in mathematics. As a pure mathematician, definitions are very important to me, and I always used to start with the definition. But I know those very definitions took years and even centuries to come to their current forms, and I also know that humans don’t learn through definition but through comparison of things that do and do not fit an idea. I think this is precisely what Chris is getting at here.

    By skipping straight to the definition, we’re robbing people of a key part of mathematical thought, and we’re skipping them through the van Hiele levels before they’re ready. You don’t need a definition until you have a need to distinguish a thing from the other things around it. You don’t need a definition until you’ve noticed the properties you can use to define something.

    The classic example in my own teaching is subspaces in linear algebra. The properties used to define a subspace aren’t even discussed until the definition is given. Little wonder, then, that the definition is meaningless to students!

    It’s not just definitions either. I help a lot of students learn statistics, and one of the things that is never explicitly taught in your traditional statistics course is how to choose what is the most appropriate statistical procedure for the situation. I have been teaching this by focussing on some specific aspects of these procedures that statisticians use to distinguish things. Reading this chapter and this quote in particular helped me realise what I was doing was exactly “talking about this difference”. To distinguish between things you need to notice the properties that make them different, and to notice them, you need to compare things. I now have a much clearer idea of what I’m doing when teaching in the way I do.

    I want to spend more time putting students in situations where they notice the differences between things and have to talk about them, so that they can distinguish between things they need to, and so that the properties I use to define things make more sense.

    Thanks Chris for a most thought-provoking book.

  • SQWIGLES: a guide for action and reflection in one-on-one teaching

    It’s university holidays again (aka “non-lecture time”), which means I’m back on the blog trying to process everything that’s happened this term. Mostly this has been me spending time with students in the Drop-In Centre, since I made a commitment to do more of what I love, which is spending time with students in the Drop-In Centre. When trying to decide what to write about first, I realised that a lot of what I wanted to say wouldn’t make much sense without talking about SQWIGLES first. So that’s what this post is about.

    SQWIGLES is an acronym that we use to help our staff (and ourselves) when teaching in the MLC Drop-In Centre. It is a list of eight actions we can do to help make sure our interaction has a better outcome and make it more likely students will learn to be more independent.

    SQWIGLES

    An acronym SQWIGLES written down the page with the meaning of each letter written across. Speak your thinking. Ask open-ended Questions. Encourage them to Write. Help them to find Information. Guide their problem-solving. Listen to their thoughts and feelings. Explain another way. Help to Sum up the learning.

    The story

    It was originally Nicholas’ idea to have something like this. He wanted something to help the staff choose what to do in the moment, and also to help them reflect on their actions and choose ways to improve. We noticed that our staff (and ourselves) needed something focused on actions rather than philosophies, because then it could be used on the fly to choose what to do. Telling staff they need to be “encouraging” or “socratic” is not all that helpful when they don’t know how to put it into action. Yet this is what many documents giving advice to tutors do. So we decided to focus on the actions instead.

    Back in 2013, I was away for the weekend and thought hard about what actions we could focus on and scribbled a lot in my ideas book until I came up with an acronym I could work with, and SQWIGLES was born. Over the past three years it has been remarkably good at helping our staff make improvements in their one-on-one teaching quickly, because it does focus on actions.

    (Do note that we also have regular discussions about philosophy too, and one overarching goal to draw everything together: “In every interaction the student learns something they can use independently in the future”. But these are not as useful without a guide for action.)

    Here is what the acronym means, with further details below:

    (UPDATE 2018: You can see video of me talking about SQWIGLES at Twitter Math Camp 2017 here: https://youtu.be/u3v_7G5D5xI )

    S: SPEAK your thinking

    Every time we read or write in front of the student, we must speak our thinking. In order to learn how to do this stuff themselves, they need to learn about how to think, and they can’t do that without us speaking your thinking. I have yet to meet a telepath!

    So, when reading student working, we can tell them what we are looking for and what we see. When reading a question to figure out how to do it, we can read it aloud and then say what our first thoughts are, including the ideas for attacking it that we are rejecting. When writing a solution to a maths problem, we can speak the decisions we are making.

    This may seem such a basic thing it hardly seems mentioning, but believe me it does need mentioning. Our very new staff often need reminding of this – they will do a maths problem in front of a student without speaking at all. Or they’ll sit silently staring at a new problem, clearly thinking deeply, but not letting the student know what they are thinking. Or they’ll just present an existing solution without explaining the thinking that created it. We all need reminding to speak our thinking.

    Q: Ask open-ended QUESTIONS

    This action has two parts. First, we should ask the students questions, rather than just tell them stuff. Them responding means them thinking, which is much better than just watching. Second, we should ask the students open-ended questions if we can. There’s a much better chance of thinking then.

    For example, instead of saying, “The definition of continuity is…”, we could ask a question like “What is the definition of continuity?”, or even better ask an open-ended question like, “What do you know about continuity?” Instead of saying, “Here’s how to start this problem…”, we could ask a question like, “How could you start this problem?”, or even better ask a more open-ended question like, “What do you think about this problem?”

    Open-ended questions allow the students to tell us what they are thinking, whether it is what we were expecting or not, and so we can change our teaching accordingly. They may even surprise us by solving their problems by themselves if we give them the freedom to respond that way!

    Some great all-purpose open-ended questions are “What do you think about that?”, “How do you feel about that?” and “Can you tell me more?”

    W: Encourage them to WRITE

    Writing is a great way to record thinking, and a great way to help structure thinking. So it’s a great thing to encourage the students to write. Us writing is not going to have nearly as much of an impact on their learning as them writing!

    Moreover, if the goal is to learn how to solve problems, then the students need to solve problems. Us writing things for them is not going to give them this experience. Plus, everything looks easy when someone else is doing it!

    So encourage them to write. This is the one I have the greatest struggle with. I have to remind myself of this one constantly and use every opportunity to hand them the pencil.

    I: Help them to find INFORMATION

    Students need to learn how to find the information they need for themselves. One of the most important skills they can learn at university is how to find information, rather than the information itself per se. So, we can help them learn how to find information. Moreover, if we just tell them, all they might learn is that the quickest way to get information is to ask us!

    For example, instead of telling students the definition, we can go and find the definition with them. If they don’t have lecture notes to look it up in, there’s always the textbook or the internet – learning how to search the internet effectively is an important skill to learn! If they want examples to inspire them, we can help them find a source of these examples. If they need instructions on the expectations for their assignment, we can help them find these instructions.

    Another worthwhile thing to note is that even if the student has a problem that it’s not our job to solve, we can always help them find information. At the very least we can help them find out who is the person whose job it is to help!

    G: GUIDE their problem-solving

    This one is the most vague of the actions, but I really wanted to mention the problem-solving thing here! There are two ideas here: guiding is different from telling, and the problem-solving is theirs and not ours.

    We can tell students what to do, and we can do problem-solving for them, but it won’t achieve them learning something they can do on their own in the future. The best we can offer is guidance. We can give them signposts to point the way, like “What do all these words mean?” and “How is this thing related to that thing?” We can give encouragement. We can help them delineate smaller steps they need to follow.

    But we can’t do it for them. It’s their problem-solving and we can’t do it for them.

    L: LISTEN to their thoughts and feelings

    This is one of the most important things on the list. We need to listen to the students. We can’t figure out what they need without listening to them. We can’t figure out where they are so we can guide them further without listening to them. We figure out what information they need to find without listening to them. We can’t choose an alternative explanation without listening to them. And we can’t convince them we care without listening to them.

    It’s also important to note that we’re not just listening to hear if they are correct or not – we’re listening to their thoughts and feelings. We really want to and need to know what they think and how they feel. These are so very important to their experience of learning and our decisions of how to help them learn, especially the feelings. Never underestimate the impact how they feel has on their learning, whether positive or negative feelings!

    This is one of the reasons why open-ended questions are so very important – it gives us something to listen to!

    E: EXPLAIN another way

    At some point in our interactions with students, it is likely we will need to explain something. This letter is about not sticking to just one explanation or type of explanation. It is likely the student has heard at least one explanation and it hasn’t worked so far, so we need to be ready to try other ways to explain, especially if it was us who gave them the first explanation they didn’t understand!

    We can use a proof, or just an example. We can use an analogy, or a drawing, or play-dough, or movement. We can colour-code an existing example, or line several examples up next to each other. We can ask another student to have a go explaining it to them. There are so many options to try and the key is to keep trying, and build our repertoire of new ways to explain, and be brave in trying something new.

    There is also the issue of what exactly they need explaining, which might be different to what we are used to explaining. Sometimes they want the concept explained in all its mathematical detail, and sometimes they just want the general idea. Sometimes they need to feel philosophically comfortable with it, and sometimes they need to understand the mechanism of the procedure. Sometimes they need to know the acceptable ways to write it on paper. We can only know which thing they need an explanation of by asking questions and listening to the response.

    S: Help them to SUM UP the learning

    Most of the time, students come to us with assignment problems, and the only thing they will think about the interaction with us is that they got their problem solved. But this does not service the goal of them learning something they can do on their own. They probably did learn something, but if they don’t know they did, then how can they use it when they need to? So we can help them sum up what they learned today. At a conference once, they called this cognitive closure – a bit like grief closure, but for learning.

    Summing up what they’ve learned is good, but helping them to sum up what they learned is even better. Asking them what they’ve learned here, or what they can use in the future is an excellent open-ended question that will often give pleasantly surprising results.

    Conclusion

    So there is SQWIGLES: a list of actions we use when we work with students one-on-one. A list of things to reflect on when we think about our one-on-one teaching. Try using them in your own teaching and reflection.