Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Tag: student wellbeing

  • 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.

  • The memory of His Royal Highness

    Five people stand talking in a room which has crocheted coral attached to the wall and floor. Three are women and two are men. The two men stand next to each other looking down and smiling. The one on the left is me, and the one on the right is his Royal Highness the Duke of Kent.

    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!
    https://x.com/ashleigh_bryar/status/1297305993487896577 

    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.

  • Twitter and how not to treat my students

    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.

  • The importance of names

    Three years ago, my university’s Student Engagement Community of Practice collectively wrote a series of blog posts about various aspects of student engagement. I thought I would reproduce my blog post here, since it is still as relevant today as then.

    There is a lot that staff can do to engage students in the university community and in their learning, and a lot of these things have to do with the staff being engaged with the students. One way that any staff member can show their own level of engagement with the students is to learn the students’ names.

    Names are important. Your name is a part of your identity, and not just because it is what you call yourself. Your name may tie you to the culture or the land of your ancestors, or it may speak of your special connection to those you love. You may prefer to be called by a different name than your official one because your chosen name is more meaningful to you. What all of these have in common is that your name is an important part of your identity.

    For myself, my name is David, and I don’t like to be called Dave. I grew up in a community with several Davids and other people were called Dave, so being David kept my identity separate to theirs. Yet many people give me no choice and call me Dave without asking for my permission, despite me introducing myself as David. I find it intensely rude that someone would choose to call me by a different name than the one I introduce myself. On top of this, I am a twin, which means as a child I was forever being called by the wrong name entirely. We are not identical twins, and yet this still happened, because we were introduced as PaulandDavid, without an attempt to give us a separate identity. The fact that I was called Paul, or “one of the twins”, meant that I had no identity of my own separate to my brother. Being called David means that I have an identity of my own and this is important to me.

    For many students, these and worse are their daily lives. Imagine a student who no-one at university knows their name. They have no identity at university, can feel very alone and can quickly disengage. Yet according to “The First Year Experience in Australian Universities” by Baik, Naylor and Arkoudis  , only 60% of first year students are confident that a member of staff knows their name.

    Not having your name known at all is one thing, but being called by the wrong name can be worse. An international student has to deal constantly with being different to other students, and in the community at large has to deal with a lot of everyday racism. To have your name declared “difficult to pronounce”, or to have it declared as not possible to remember, is just another one of these everyday racist events. The person doing so may not be meaning to be racist, but it adds up to the students’ feeling of not belonging, to their feeling that they themselves are not worth remembering. Similar to me and my twin brother (only worse), they may have the feeling that others believe all international students are the same, so why remember them separately. In “Teachers, please learn our names!: racial microagression and the K-12 classroom” by Rita Kholi and Daniel G Solórzano , there are many examples of the hurt that such treatment of student names can have.

    So what can we do to learn our students’ names? Members of the Community of Practice suggested several strategies.

    One idea is to spend time talking to them and ask them what they would like to be called. You can’t learn their names unless you find out what they are! Be visible in your effort to pronounce it correctly, be adamant that you want to call them by the name they ask to be called. If you get multiple chances to talk to them one-on-one, ask their name again if you can’t remember and try to use it as you talk to them.

    Another idea is to print out photos of your students and to practice remembering their names. If you don’t have access to their photos, then it should not be hard to find someone nearby who can. (Though of course it would be excellent if there were a simple system whereby anyone teaching a class — including sessional staff — could get photos of their students!) Even if you can’t get their photos, simply working your way down the roll and remembering how to pronounce those names, or what the students’ actual preferred names are, is good exercise. The students are likely to appreciate the effort you put in here, even if they can’t know how much time you did put in!

    You may have your own ideas on how we can make sure we know students’ names. I’d encourage you to share them in the comments, along with any stories of how it made a difference to student engagement.

    I would like to work in a university where 100% of the students are confident that someone knows their name. We have hundreds (possibly thousands) of staff in contact with students on a regular basis. If each of us only learns a tutorial-worth of names, then we can surely meet that goal easily!

  • 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?

  • Mr Johnson’s Rainbow

    I love reading and writing, and the way that people use words to express ideas fascinates me. So it is no surprise that when I was in Year 12, I studied the highest level of English available. My English teacher was called Mr Johnson and I hated him. (It wasn’t really, Mr Johnson – I’ve changed his name to write this.) The reason I hated him is expressed in this poem I wrote at the time:

    MR JOHNSON’S RAINBOW

    The afternoon sky was fretted
    With cotton shades of blue
    And the rainbow came, inspiring us all
    And on some old scrap paper
    My thoughts and feelings grew
    Some lines of verse upon the page did fall

    And then I took the poem
    A work all of my own
    And to my English classmates did I show
    That my poem was Quite Good
    To me it was made known
    Because my fellow classmates told me so

    But the teacher, oh my teacher
    Said: I know it is Quite Good
    But it is not what I would call My Way
    Your verses on the rainbow
    Are not the way I would
    Ever say the things that I would say

    For I, yes I my student,
    Am like the poet Donne
    As you are like the other poet Keats
    You like to write, like he,
    On emotion by the ton
    Where I do so much higher mental feats

    He went on by relating
    All the things that he would write
    And prattle from his open mouth did flow
    He said: the rainbow is
    All the colours making light
    So there must always be thingy, you know

    And I was quite inspired
    By this brilliant oratory
    And thought:
    Why don’t you write your own bloody poem if mine isn’t good enough for you?

    The same theme appeared in all of his feedback about all of my creative writing: he disapproved of the content I chose to write about, often saying that it wasn’t clever enough. He never gave me feedback on my expression of those ideas – no discussion of flow or characterisation or word choice or metaphor – only ever that the ideas themselves were not to his taste. One notable example was when we were asked to write a short story about a Far Side comic involving butterflies from the wrong side of the meadow, and so I wrote about the flowers sending rogue butterflies to attack the flowers on the other side of the meadow. I was marked down because I didn’t instead do something more clever, like write about some completely other thing only tangentially related to the theme of the comic. As you can imagine, I did not choose to study creative writing at university, and to this day I still have quite a fear of sharing my writing.

    Thinking about how this applies to my maths teaching, I wonder how often we tell students in maths “but it is not what I would call my way”. For example, those times when a student does a perfectly wonderful and correct solution to a problem, but then we tell them it has to be done this way instead. Or those times when we discount their excitement of maths applying to something that interests them to tell them they should be interested in the beauty of the maths itself. Alternatively those times when we get annoyed at the student who wants to understand the ideas behind a method and tell them to just do it and not worry about that. When a student asks me to check their work, do I critique their execution or do I criticise their ideas? What about all those times when I ask the class for what they notice/wonder and then wait until I get the one I was really hoping for? I am worried about students choosing to stop studying maths because we always judge them on their ideas.

    As usual, I don’t know what to do about this other than just be aware of it. Just yesterday when this was on my mind I was careful to say to a student how awesome I thought their Quarter the Cross solution idea was, before talking to them about how they might be more precise in their execution so other people could also be sure it was a quarter. I only hope I can have it on my mind a bit more often as I work with students on the everyday stuff in the MLC.

  • A public health approach to improving teaching and learning

    Making a big difference to student learning is a tricky business. Here at my university, there are a certain number of (wonderful) teaching staff who are champions of innovation, always making big changes to the way they do things and jumping onto any innovation as soon as it comes around. Yet the students not in those classes don’t see much benefit from it. Indeed, those staff who are not champions of innovation may do nothing for fear of having to adopt all at once All The Things they see the champions doing. A student who seeks regular support for their learning may make spectacular gains, but there are literally thousands of other students who don’t seek such support on a regular basis, and thousands of students who don’t really need spectacular gains but just a little bit extra. I have started to think that perhaps the best way to make a big difference is to find some way of encouraging a large number of small differences.

    This is essentially the way Public Health works. In Public Health you are concerned with whole population health initiatives, which are often of necessity a large number of small differences. For example, you may not cure the flu, but you might encourage 20% more people to wash their hands and so prevent the spread of infection and stop so many people getting the flu in the first place.

    Imagine the benefit that might happen, not if a few lecturers rub out their courses and start again with flipped learning, but just if every lecturer simply labelled everything in Canvas/Blackboard so the students could easily find stuff. Imagine the benefit, not if a few course coordinators completely changed their tutorials to be about group discussion, but if every classroom tutor asked one “what if” question in every tutorial. These are not big things to change, but if a lot more people did them, I think the overall effect would be far-reaching. And they might seem like something you could actually do, as opposed to the big changes that are the usual fare of innovation.

    Personally I am trying to do more Public Health approaches to student support too. Instead of just visiting lectures to tell the students how to seek one-on-one support, I’m visiting with a five-minute message about interpreting assignment questions, or choosing to put in more explanatory working, or what a standard deviation is. If I can reach even half of a lecture of 500 students with one of those little messages, then I have made a big difference by making a lot of small differences.

    Unfortunately, Public Health doesn’t make for spectacular stories. Giving one person brain surgery to save their life after a horrific traffic accident is a spectacular story. On the other hand, lowering the speed limit in urban areas in order to make horrific accidents less likely is not a spectacular story, but it can be argued that it saves a whole lot more lives. I only hope I can convince the Powers That Be that my Public Health approaches to learning and teaching improvement are worthwhile, if not spectacular.

  • The unexpected fear of statistics

    Statistics is the cause of a lot of fear. There are thousands of students studying psychology, sociology, economics, biology, medicine, animal science and education who thought they would be free of mathematics and suddenly discover they have to deal with statistics. In the case of psychology it is absolutely everywhere: both in whole courses about statistics, but also embedded in almost every other course they do. For most of these students, their fear of statistics carries over from their existing fear of mathematics, and so as sad as it is that they are afraid, it’s not wholly unexpected.

    What is unexpected is the the thousands of students who have done the highest levels of mathematics at school, and are doing the most mathematical disciplines like physics, engineering and mathematics itself, and yet somehow have a deep aversion to statistics when it appears in their degree. Indeed, my own staff at the Maths Learning Centre often express a fear at having to help people with statistics. As I often say “Mathematicians are afraid of statistics in a similar way to how other people are afraid of mathematics.”

    But why? What is it that causes this fear of statistics in those with lots of mathematical experience? I have some ideas…

    One of the biggest reasons is that statistics isn’t 100% maths. A large part of statistics is whatever discipline the statistics is being used for today. That discipline dictates the kinds of data that can be collected, how it is recorded, and most importantly how it is interpreted after the statistics is done. In a generalist statistics course this will change moment-to-moment as each new assignment question brings up a new context. In Question 1 it’s ecology, in Question 2 it’s quality control in food production, and in Question 3 it’s engineering. Each new question requires you to think about a new context and understand various subtleties about what the context means. For a professional statistician, this is often what they say is the most exciting thing about their job. They absolutely love that they get to “play in everyone else’s backyard”. They love that the same tools can be used for a large variety of different problems. However, for many a maths student, this annoys them at best and terrifies them at worst. They didn’t sign up to learn ecology/food science/engineering; they signed up for maths. They prefer to work with the numbers and word problems have made them worried from a young age. Think of how terrifying it would be to suddenly be forced to do a course where every single problem is a word problem! And spare a thought for how hard it is to do the statistics when you don’t have a clue what the context means. As I said, each new context has subtleties that impact a lot on how the statistics is applied and interpreted, and not everyone will have the general knowledge (or indeed language skills) to understand those subtleties without considerable effort. And now imagine having to go through that effort for every single assignment question. It’s exhausting!

    I’m not really sure what to do about this particular problem, other than making sure you give students space and time to talk about the contexts. Don’t treat them like idiots for not understanding contexts they have never experienced before, and definitely don’t think they don’t understand the statistics just because they don’t understand the context. Allow them the grace to have to ask what a manatee is and to ask why manatee deaths would be expected to have anything to do with powerboat registrations. I can imagine assignment questions having links to further information so they can find out more, or a quick whole-class discussion about contexts when you hand out assignments – possibly something like a numberless word problem. They might go a long way to alleviate context fatigue.

    The second reason is that statistics involves making decisions, the biggest of which is deciding what statistical procedure to do and which bits of your situation go with which parts of the procedure. With so many to choose from, and no consistent naming system for the various procedures even inside the one discipline, this is a hugely daunting task for the beginner. It all just seems like a big cloud of random stuff and the students often can’t see what it is that distinguishes between the procedures and what information is being used to decide one over the other. This is only compounded by the fact that part of the decision is made based on information that comes from the context the statistics is being used for today, which was already a problem. It’s further compounded by the fact that many who succeed in mathematics at school have done so by having a list of problem types and how to solve each one, and are not actually used to making decisions at all.

    I think a good dose of actually analysing that decision process and comparing situations that produced different decisions would go a long way to helping this, rather than leaving the decision to chance. Indeed, I’ve written before about how important it is to give students practice at the act of making the decision.

    The final reason I can think of right now is that doing statistics requires making a computer do what you want. This is a completely separate skill from understanding the context, understanding the maths and deciding what stats to do, and has a whole host of its own frustrations, not least of which is just getting access to the computer program itself or figuring out how to install it! And yet it is the gatekeeper of producing actual statistical results. Learning how to communicate with the statistics program is just one more language process that has to happen to succeed in statistics, on top of the decision-making and context-interpreting language processes I already mentioned! Added to this, for the mathematically experienced, they have spent a lot of time learning how to assert their independence from technology and rely on their own reasoning. To not be able to do something themselves and be forced to get a machine to do it for them leaves a bad taste in their mouth.

    Again I’m not too sure how to do something about this problem. Certainly you can make sure there is a lot of support available for getting the program to work, and for asking help specifically with the program. At university you could elevate it to the regular lecture time rather than leave it to practical classes that students may avoid. (Yes I know if they struggle with computer stuff they should go to computer classes, but humans are nothing if not illogical when emotions like guilt and fear are involved.)

    Now that I have written this all down, it occurs to me that these problems are a lot about language, and so this issue may be related to your high-maths-experience students avoiding language in much the same way that other students avoid maths. Perhaps the main thing we can do for them is help them process the fact that it will be about language and support them in their language, and perhaps help them realise that they have a lot more language skills than they thought they had. (Those of us who teach students at earlier stages in their lives might do well to help them realise that maths is all about language anyway!) On top of that, we can have some compassion on them because learning statistics actually is hard work.

    Megan 15 May 2018:

    I personally love statistics but I certainly know plenty of maths learners and educators who don’t. I think statistics is often mathematically simple (mostly plug and chug often with technology) but conceptually challenging. For example, even a simple p-test (covered in Year 12) is not entirely intuitive (Wouldn’t we want a large probability to reject H_0?…We reject for a large test statistic…). As a result, students who usually do well in maths are suddenly struggling not because they can’t do the maths but because they need more time to understand the context and interpret what the numbers actually mean as you say. Maybe more discussion of what is actually happening when we perform different tests could help.

    David Butler 25 May 2018:

    Thanks for the comment Megan. Yes I agree that people might struggle because it’s not the maths that’s the problem! I reckon more discussion of how the context relates to the maths might help them make the connection.

  • Book Reading: Choice Words

    This post is about my reaction to the book “Choice Words: How Our Language Affects Children’s Learning” by Peter H. Jonston.

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

    I was lent the book by Amie and I am very grateful to her because it really is a good book (though it was tough to read with the forest of sticky notes marking her favourite pages 😉).

    This thin little book is about how words have power to help children learn about reading, writing, learning, themselves and their place in the world. The majority of the book is a list of sentences spoken by teachers followed by an analysis of what those words mean for children’s learning. The focus is mostly on helping children learn to read and write successfully, but don’t let the “children” or the “read and write” fool you – I have so many thoughts swirling in my head about how this might possibly apply to my own teaching, and indeed my life.

    Unfortunately, “swirling” is the appropriate word for my thoughts right now. The fact that the book is structured around analysing specific utterances by teachers made it all very concrete, but on the other hand it is making really hard for me to process the information coherently. At the moment it’s just a big cloud of things to think more about, a lot of which overlaps. I’m finding it hard to tease things apart to find something I can apply first, or a way for me to consistently apply it so it’s useful for my students. I’ve decided the best thing to do is to write this post so I can attempt to process it all.

    The chapter titles might be a good place to start. Here they are:

    1. The Language of Influence in Teaching
    2. Noticing and Naming
    3. Identity
    4. Agency and Becoming Strategic
    5. Flexibility and Transfer (or Generalizing)
    6. Knowing
    7. An Evolutionary, Democratic Learning Community
    8. Who Do You Think You’re Talking To?

    Even just listing those titles is helping me focus a bit more. While I was reading it, it might have helped me to keep a bookmark in the chapter heading so I could look back and remind myself what the big idea of the chapter was. Instead I found that I got a bit bogged down in some of the details as I went along and lost the focus. Now that I can look back from a higher vantage point, I reckon I might be able to pull out some bigger ideas…

    Chapter 1 is about how much our language has power to create reality, in particular the reality of the listener’s identity. If I were to hold on to just one thing from the whole book then maybe this message would be it: I can make the world different for another person by choosing the words I use.

    Chapter 2 is about how in order to learn and know what you have learned, you need to notice things. You need to notice how things are similar or different, how they are related or not. And then, things need to be named, so that it is possible to talk about them. This is remarkably similar to the Notice and Wonder idea from the Math Forum people, and to Chris Danielson’s way of getting to geometry ideas via Which One Doesn’t Belong. But here, Peter goes deeper than this. He suggests that you can notice and name not just content, but also your processes as you work as a group, your thoughts about yourself as a learner, the things you have learned so far, and your behaviour. It is a fascinating idea to me that you can apply the same noticing and naming to mental and social processes as you can to the properties of quadrilaterals. Something to hold onto from this chapter is that my words can draw attention to features worth noticing, and the act of noticing itself.

    Chapter 3 is specifically about identity. Peter talks about how we construct a narrative with ourselves as one of the characters and the words we use to tell this story shape the sort of person we see ourselves as. We as teachers can make a difference to identity by the words we choose. Something that struck me most strongly was using words that don’t give people a choice to opt out of the identity. For example, the question “What problems did you have?” assumes that there must have been problems, and asking someone what choices they made assumes they made a choice. This is what I want to hold onto from this chapter, that I can give someone courage to be a writer or mathematician by using words that put them into that character.

    Chapter 4 is about agency, and in a way is an extension of the previous chapter on identity. The identity in question here is that of a person who has power over their own choices. This chapter spoke to me most strongly as a maths teacher, since maths is a subject where so many students feel they have no choice and that choice isn’t even a thing that people ought to have (as evidenced by the constant request to tell them what to do). Peter advocates talking to students as if they did make a choice, and analysing the choices they could have made. This is one of the biggest ideas in the whole book to me, and I want most to hold onto this one as I go forwards.

    Chapter 5 is about transfer, that holy grail of teaching where students are able to apply what they learn in one area to another. Peter pulls together the agency and the noticing/naming from the earlier chapters as the main mechanism for this. More explicitly, the questions listed here focus on noticing explicit connections between things and also exploring the “what if” questions. He ends with a comment about the importance of play, which of course resonates strongly with me. The thing I want to hold from this chapter is the focus on connections, over and above answers.

    Chapter 6 is about knowing, and in particular about who holds knowledge and who decides when we know something. In many teacher-student interactions, the assumption is that it’s the teacher who knows and the teacher who decides what is true and when we are correct. Yet really one day when they leave our care, our learners will need to know how to be sure of things for themselves. The thing I want to hold onto here is that I can give my students the power over knowledge. This is especially important in maths, which is set up so that you actually can be sure of things through your own arguments, rather than having to rely on the authority of others.

    Chapter 7, while it has a very long title, is really about how our words can help people learn to work together. Peter has a lot of examples where teacher words encourage learners to consider the feelings and ideas of others, and to choose shared goals. He reuses the noticing and naming power of words to help learners notice their own group processes, and the identity-forming power of words to help learners put on the mantle of people who care about others. The thing I can hold onto from this chapter is that words can make group social and cognitive processes explicit in a way that makes them learnable.

    Chapter 8 is about the interplay between your beliefs and your words. As a teacher, if you believe your students are not capable of learning something, your words (and your silences) will reflect this. However – and this is the big thing I want to hold on to here – if you choose to change your words, then some of your beliefs might follow. I see this in using SQWIGLES with myself and my staff where choosing to ask open-ended questions changes the ways that students respond to you and therefore ways that you respond to them. Your beliefs about what students have to say can change through this change in your words.

    I think I’ve achieved my goal in writing about this book, in that I have a much clearer idea about how I want to respond to it in my work. I have clarified how much of an impact my words can have on learners’ realities, which I knew, but not to the level of specific detail I did before. In particular I think I want to hold on most strongly to the idea that I can help learners to see themselves as having choice and capable of making that choice, changing both their view of mathematics and of their place in it.

    These comments were left on the original blog post:

    Simon 8 August 2017

    Well, that makes me want to read it – thanks for taking the time to review it David.

    Nic Petty 8 August 2017

    Thanks for this David. I am really interested in how this fits with the ideas on helping people to think more positively about mathematics and to think of themselves as mathematicians. Language empowers or disempowers people, and we really want the former.
    I have just bought the book. (Thanks to one-click on Amazon, I can have instant gratification.)

    David Butler 10 August 2017

    Thanks Nic. I’d love to hear your journey as you read it too.

    Mark Pettyjohn 8 August 2017

    It sounds like you are at the beginning of a tremendous journey. My book probably looks a lot like Annie’s. This thin book spawned so many thoughts.

    I see Nic commented above about identity. It looks like he teaches stats, which probably means high school students. That puts you two in a basket teaching students whose identities in mathematics have already been forged for a good portion of their academic careers. So while the end goals may be the same as someone like me working down elementary, I think you two have a much tougher go of it.

    Not that you shouldn’t! Last year I worked with a professor who was teaching a university course of mathematics for college students studying to be elementary teachers was enlightening. There were some serious challenges to developing a positive mathematical mindset, but I also saw some good progress.

    I know you will be keeping us updated, and I am looking forward to following along in the coming months and years.

    David Butler 9 August 2017

    Thanks for the comments Mark. It was motivating to think that my words might be enough to help people *change* their identities, as opposed to *form* them. I felt a lot of these things already, but the book helped me know them.

  • When will I ever use this?

    “When will I ever use this?” is possibly a maths teacher’s most feared student question. It conjures up all sorts of unpleasant feelings: anger that students don’t see the wonder of the maths itself, sadness that they’ve come to expect maths is only worthwhile if it’s usable for something, fear that if we don’t respond right the students will lose faith in us, shame that we don’t actually know any applications of the maths, but mostly just a rising anxiety that we have to come up with a response to it right now.

    There’s an interesting discussion in this pdf article   [1] of the various responses that are commonly given to this question and their various drawbacks. The author is mainly concerned that we often inadvertently confirm the uselessness of maths by our very attempts to make it seem useful. While this is a legitimate concern, I have another one: in our attempts to justify the mathematics, we forget to listen to what the student actually needs.

    In my experience, when a student asks this question, it’s a sign that they are starting to lose faith. They are having trouble motivating themselves and are seeking a reason to keep working at it. Being able to use it someday is the first thing they think of to motivate themselves, so they ask the question. But really most students will be happy with any reason that encourages them to stick at it today.

    I had been thinking about this for a couple of days, after following a Twitter conversation and the comments on a post on Dan Meyer’s blog . Then one one of the students in the MLC actually asked the question, so I was all ready with my response. I said, “Actually, I’m not going to answer that question, but instead I have my own question to ask: how are you feeling about this topic right now?”

    It is a testament to the trust I’ve built up with the students that he answered my question honestly! He said that he couldn’t see how the bits fit together or how they related to other things in the course. So I talked about how this topic fit in with the big ideas in maths, and how it connected with what they did last semester and last week. Then I helped him to organise some of the information in the topic so it was clearer how it was structured.

    And you know what? After this discussion it didn’t matter so much that he might never use it. He had what he needed to have the courage to keep going, because I took the time to find out what was really bothering him.

    [1] Otten, S. (2011) Cornered by the Real World: A Defense of Mathematics, Mathematics Teacher, 105-1, 20-25 

    Alexandre Borovik 27 April 2016:

    It is like learning to swim: how many people actually have to use swimming for *practical* purposes?