Making Your Own Sense

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

Category: Reading

Reflections on books and articles I have read.

  • Book Reading: You’re Not Listening

    This blog post is about the book You’re Not Listening by Kate Murphy, and in particular my reactions to it from a teacher’s perspective.

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

    First, I want to apologise to Chelsea Avard for borrowing the book from her little student leadership library and holding onto it for a whole year while I got round to reading it and then got round to writing about it. Thanks for your patience and thanks in advance for forgiving me for the slightly battered state the book will be when I return it.

    Second, I want to say what an excellent book it is. It has a lot to say about what listening is and is not and how it makes a difference to us in everyday life. It’s full of vivid stories to illustrate and lovely turns of phrase, and it is very clear that Murphy clearly researched it extensively. I would recommend anyone to read it.

    Ok, now on to the actual purpose of the blog post. The book is not a teaching book, and it doesn’t even mention teachers as far as I recall, but I am a teacher and I can’t help reading it from that perspective. As a teacher who mostly works one-on-one with students, what Murphy had to tell me felt particularly relevant. And yet I’ve had trouble organising it into a cohesive blog post. I have decided to give up and just list some things I learned. Most of them are about teaching. Some of them are just life lessons. I may or may not distinguish them.

    Thoughts from the book

    Your brain thinks faster than people speak. Use it to listen better.

    In Chapter 6, Murphy clearly describes exactly how conversations feel to me, which is that the person is talking, but my mind is buzzing all around thinking about all sorts of things. Almost all of those things are related to what the other person is saying – memories their stories brought up, how I feel about the way they’re talking, what I am going to say next – but only a small amount is actually about the other person’s meaning.

    Murphy asserts this is totally normal because your brain thinks at least twice as fast as the other person speaks. Reading that, I was so glad that I wasn’t some sort of freak. However, she recognised that the problem with this normal state is that you can get wrapped up in your thoughts and miss what the other person is saying. Also, she points out that a lot of what people say isn’t contained in the words they speak, but in their silences and their body language and the things they look at.

    What she suggests is to use all that extra mental bandwidth to listen better. Instead of letting your mind wander or worrying about what to say next, direct your attention to all those extra things that aren’t audible that might tell you more about what they mean.

    This spoke to me powerfully as a teacher, particularly the bit about missing important things when you are planning what to say. In the MLC with a student, my mind can be very much occupied with what explanation or example I’m going to give the student in response to what they’re saying. But in worrying about this while they’re talking, I am missing important things. I’m missing what they’re telling me about how they feel about their maths. I’m missing the pauses that tell me when they’re figuring things out or struggling to articulate their thoughts. I could be a much more responsive teacher if I used my mental powers to think about their meaning while they are talking rather than think about mine.

    You don’t have to respond right away.

    The big problem in my line of work with the advice above to use your brain’s bandwidth to focus even more on listening and watching is that you really do feel like it’s your responsibility to respond immediately as soon as the student stops talking with useful explanations or advice. The desire to plan your explanation while they are still talking isn’t just your mind wandering, it’s motivated by a fear that you need to be immediately helpful and clever.

    However, Murphy argues that actually silences are really important in conversation. Space between speakers allows everyone time to process. And most importantly, it’s a sign of respect for the speaker that the listener takes time to process. She relates stories of many different cultures around the world where they have an expectation of silence in conversation as a sign of respect. I know students have said to me they appreciate me telling them “just let me think about that for a minute” because then they know I’m working on something good.

    I’ve heard this advice about silence before, for example in Making Number Talks Matter, and also in a recent UX Research Methods training. But not until reading Murphy’s book did I connect it to the idea of giving yourself the space to really listen in the first place. Knowing that it’s ok and actually respectful to think quietly before responding to students means that you are free from the expectation to have something ready and you can actually focus on listening interpretively. It has taken a huge worry off my shoulders as a teacher that I wasn’t even aware I was worrying about before.

    Assumptions stop you listening.

    Murphy has a whole chapter titled “I know what you’re going to say: assumptions as earplugs”, and that title really does sum it up extremely well. It just shows her skill as a writer that she can pack so much meaning into just a title.

    The idea is that when you assume things about people, it stops you listening to them clearly, and sometimes stops you listening at all. If you literally think you know what someone will say, then you think you have no need to hear it.

    The most shocking thing Murphy does is explain how finishing someone else’s sentences is proof that you’re not actually listening. Yes two people being able to finish each other’s sentences is often used as proof that two people are in sync, but she related stories from marriage counsellors who noted that couples who are really familiar with each other just assume the other person doesn’t need to be told things or doesn’t need to tell them things and so people never find out important stuff.

    This idea that assumptions stop you listening really gave me pause, because I have seen it first-hand in my own work. A student asks me a question about Question 3 on their assignment, and I assume they are struggling in the same place I would struggle, so I tell them how to deal with that part, even though it might not be the place they need help at all. A student is studying engineering, so I assume they don’t want to know the theory behind the maths and I only tell them procedures, even though they might actually want to make sense of it. A student asks a question about the topic at hand and I just respond to the key words and the question I assume they probably have, but their actual question is about quite a different thing. A student  is slow to answer my question, so I assume they don’t know and I answer for them, but actually they are just taking time to figure out how to say what they want to say.

    There are so many times when I assume what a student wants or needs rather than listening to their actual words and demeanour or seeking more information. I really need to turn off my assumptions and find out more about the actual situation before responding.

    People won’t tell you things if they don’t think you want to hear them.

    There were a couple of quotes close together that really spoke deeply to me.

    Researchers at the University of Utah found that when talking to inattentive listeners, speakers remembered less information and were less articulate in the information they conveyed. Conversely, they found that attentive listeners elicited more information, relevant detail, and elaboration from speakers, even when the listeners didn’t ask any questions. So if you’re barely listening to someone because you think that person is boring or not worth your time, you will actually make it so.

    Think of how you, yourself, might tell different people different things. It doesn’t necessarily have to do with the type of relationship you have with them or the degree of closeness. You might have once told a stranger something you hadn’t told anyone else. What you tell, and how much you tell, depends on how you perceive the listener at that moment. And if someone is listening superficially, listening to find fault, or only listening  to jump in with an opinion, then you’re unlikely to make any kind of meaningful disclosure and vice versa.

    The things that spoke most deeply to me were the things that can derail the students I work with saying anything useful:

    • Not worth my time
      Phew this one is hard. All those second- and third-year students who I have said are second and third priority after first-year students, what does knowing they’re not my priority do for them sharing things that are important? But they still have deeply important things about studying and learning maths to discuss, even if I don’t know their content.
    • Listening superficially
      I can do this so easily. I can wait until they say a key word and then launch into an explanation in response, and totally ignore the things they’re saying about how they feel about maths and their experiences studying it, or just ignore how what they say tells me about how much they already understand.
    • Listening to find fault
      How many students’ experience of maths in the past is that someone is waiting for them to be wrong? Knowing this, how might they shut themselves up if the first experience with me is telling them something they did wrong?
    • Listening to jump in with an opinion
      Phew this is so easy to do as a teacher. They’ve come for help, so I feel  the need to provide it as soon as possible, so I wait until they say something I could comment on and make the comment. But they may not want a comment on that thing and they are unlikely to give me what they really want because they know I’m not really listening.

    I believe that  to be the best teacher and support worker I can be, I need to know the students’ context and know what they already know, but if I really do believe that, then I need to get out of the way so that they will actually tell me what I need to know in order to help them.

    Other people’s thoughts really are more interesting than anything you have pre-prepared to say.

    The previous section was about how the way you (don’t) listen affects how much people tell you, and I made it all about how I need information to help people the best I can. Except the thing Is, I think people would prefer someone who actually was interested in what they have to say for its own sake, rather than just as a means to an end. I have been working on fostering the belief that all students have existing thoughts worth listening to, and this book confirmed how true and important this belief is.

    This quote says it clearly:

    The most valuable lesson I’ve learned as a journalist is that everybody is interesting if you ask the right questions. If someone is dull or uninteresting, it’s on you.

    Also it’s just a vibe running through the whole book: it’s not just that it’s good to listen to people, but that people are worth listening to. Listening can be hard work, but it is worth the work, because everyone has something interesting to say, something you can learn from.

    It’s not about you: shifting versus supporting.

    Murphy spends a lot of time talking about how much time we spend talking and not listening, even when we think we’re listening. She describes two main types of response a listener makes: the shift response and the support response. When you make a shift response, you shift the focus to yourself; when you make a support response, you keep the focus on the other person, supporting them to continue and to share what’s important to them. This is one of her examples:

    Sue: I watched this really good documentary about turtles last night.
    Bob: I’m not big on documentaries. I’m more of an action-film kind of guy
    (shift response)

    Sue: I watched this really good documentary about turtles last night.
    Bob: Turtles? How did you happen to see that? Are you into turtles?
    (support response)

    I have to say I am repeatedly guilty of the shift response. Just now I responded to someone telling me about something that happened to them with a story about my own experience, when I could have so easily asked them more about theirs. Not to say you should never talk about your own story in a conversation, since of course a conversation moves both ways, but you miss out on so much if you do it too early.

    Support responses tend to be questions, where you seek more information from the speaker, but there are lots of questions that are still shift responses. There are ones where you set up what you want them to say such as, “Wouldn’t you agree that …?”,  and the ones which really just describe your own thoughts or shift the conversation to an entirely new topic that was on their mind already. I found it appropriate that the example of this cited in the book was from an academic at a conference.

    In my work in the MLC, I am guilty of shift responses so often, especially when the conversation turns to study skills or experience in classes. I always end up just saying what my experience is, when I could learn so much more by finding out about the student’s. You could argue that the students need to know they’re not alone in their frustration, and you could argue that the might appreciate knowing how someone else dealt with similar situations. However, listening just a little longer will also let them know their feelings are valid, and will tell you whether they need advice at all or just want to vent. (Not to mention that my experience as a student is now 20 years old, and might not be as relevant as I feel it is.)

    Also I am suddenly reminded of the distinction between focusing and funnelling questions: a focusing question while a student is doing problem-solving helps a them student focus on the relevant details at hand so they can use it in their problem-solving, whereas a funnelling question pushes them towards a path you the teacher have in your head. This is very similar to the concept of the shift and the support response. Both shift responses and funnelling responses are about you, whereas support responses and focusing responses are about them.

    The listening itself is what helps people.

    The final thing that stood out to me is that people don’t usually want or need your advice. The very act of listening supportively helps people to achieve clarity and sort out what they want to do. Here’s a useful couple of quotes:

    Being aware of someone’s troubles does not mean you need to fix them. People usually aren’t looking for solutions from you anyway; they just want a sounding board. Moreover, you shut people down when you start telling them what they should do or how they should feel. … Your answer to someone else’s deepest difficulties merely reflects what you would do if you were that person, which you are not.

    The solutions to problems are often already within people, and just by listening, you help them access how best to handle things, now and also in the future. … If you jump in to fix, advise, correct, or distract, you are communicating that the other person doesn’t have the ability to handle the situation.

    This is a really hard thing to hear because we are so used to providing advice as the way to help people. It’s particularly hard to hear for someone in my line of work, where people are literally talking to me because they do actually want help. But if I really do believe that people have thoughts of their own and I really do believe that all people are capable of figuring stuff out, then the best way to show that belief is to listen to them. And if I’m honest, whenever I’ve let them, they really do surprise both me and themselves with what they figure out on their own.

    Conclusion

    This book was full of such interesting and compelling stuff. I’d recommend it to anyone to read. I think I have listening on my mind even more than before after reading it, both in my life and in my work in the MLC. To sum up, here are my titles from above again, to hold onto for the future:

    • Your brain thinks faster than people speak. Use it to listen better.
    • You don’t have to respond right away.
    • Assumptions stop you listening.
    • People won’t tell you things if they don’t think you want to hear them.
    • Other people’s thoughts really are more interesting than anything you have pre-prepared to say.
    • It’s not about you: shifting versus supporting.
    • The listening itself is what helps people.

    Seeing them all together really highlights to me how much they interact with each other, and how much they all hang off the idea that other people have a lot to say that is worth hearing. Thanks for reading.

  • Four levels of listening

    Listening is one of the most important aspects – no, scratch that – the most important aspect of my work in the Maths Learning Centre.

    It is not obvious to people starting out tutoring in the MLC that this should be the case. To a beginning tutor, it seems that it’s their job to explain things to the students, and to show them how to do stuff. But even if the actual goal was to explain, you can be much surer which explanation to give the student if you first listen to their current understanding. More importantly, you can never improve as a teacher unless at some point you listen to the students to see how well your explanation has gone.

    But how do you go about doing the business of listening? This blog post is about my interpretation of a framework that describes different levels of listening for the purposes of teaching, which I read about in two papers:

    1. Davis, B (1997) Listening for Differences: An Evolving Conception of Mathematics Teaching, Journal of Research in Mathematics Education, 28, 255-376
    2. Yackel, E, Stephan, M, Rasmussen C and Underwood, D (2003) Didactising: Continuing the work of Leen Streefland, Educational Studies in Mathematics, 54, 101-126

    I spoke at a conference about this framework some years ago, and I have been meaning to write about it ever since. I am finally actually writing about it now (and you are reading it). My thinking has evolved a little since then, so you get the updated and extended version.

    The papers

    Davis 1997

    In the  first paper, Davis tells us about how he and schoolteacher Wendy reflected on the types of listening Wendy did in her classroom, and how they were related to her beliefs about what mathematics is and what the teacher’s role is in helping students learn it. It is a truly fascinating and powerful paper and I recommend everyone read it.

    Davis notes from previous research that “the quality of student articulations seemed to be as closely related to teachers’ modes of attending as to their teaching styles”, which is a very deep observation. Before, I said that at the very least a teacher needs to listen in order to figure out what to do next, but this says even more that the way you listen may change the very things the students say. Davis goes on to give three vignettes from Wendy’s classroom to display three types of listening.

    1. Evaluative listening

    When a teacher is listening evaluatively, their reason for listening is to evaluate the correctness of what the student is saying. Ultimately, they are “listening for something in particular, rather than listening to the speaker”. The vignette describes a whole-class discussion where student responses are dismissed until the exact right one was finally accepted. Even right responses that were perceived to be in the wrong form were dismissed. This reminded me of so many times when I had been a frustrated student in such class discussions (and several times I had been a teacher leading one).

    I had never quite put my finger on why this felt frustrating until reading this quote from the paper: “No one is attending to the answer in a way that will make a difference to the course of subsequent events”. The teacher in such discussions is waiting for the right response in order to continue on their pre-planned course. As Davis says of Wendy’s vignette, it was “a teaching sequence that seemed impervious to student input”. It sounds harsh, but Davis was more forgiving than that. He noted that Wendy was indeed seeking information from the students. She could see that the students were or were not able to give the responses she hoped for, and she could see that her lesson was more or less successful based on how quickly students could use her explanations to produce the right answers. The listening was doing exactly what she wanted it to do: evaluating.

    2. Interpretive Listening

    When a teacher is listening interpretively, their reason for listening is to interpret what ideas are actually happening inside the student’s minds. They are still usually seeking to bring students to the understanding they perceive as the correct one, but now they do it through figuring out how to talk about ideas in shared ways that move students forwards in their thinking.

    Davis notes that teaching sequences with an interpretive listening stance need to have materials that “serve as a commonplace for learners to talk about ideas, enabling the process of re-presentation and revision”. For example, in the vignette, Wendy used two-coloured chips to help her students talk about adding and subtracting negative numbers. In my own teaching in the MLC, drawings or play dough often play this role.

    3. Hermeneutic Listening

    When a teacher is listening hermeneutically, they are listening not only to interpret what their students are thinking, but also to understand how their own thinking relates to that, and how the group as a whole understands. This is my description of it, anyway. Davis has several long paragraphs discussing philosophical and theoretical standpoints, which are a bit heavy (though his style makes it much lighter than I’m sure it could have been). The two main takeaways for me are that understanding isn’t only something that lives in one person, but lives in the shared communication of many, and that teachers listen not just to help students grow their understanding but also to change the teacher’s own understanding. A relevant quote: “Instead of seeking to prod learners toward particular predetermined understandings, Wendy seems to have engaged, along with her students, in the process of revising her own knowledge of mathematics.”

    Davis notes that this type of listening seems to go hand in hand with a teacher’s conception of what mathematics itself is. You have to be prepared to believe that mathematics concepts have multiple valid ways to understand and describe them, and that mathematics is at least in part a construction of a community, all of whom (including novices) have a part to play in the construction. Otherwise you won’t be ready to listen in this way.

    A final comment on the terminology… The word “hermeneutic”, no matter how often I look up definitions, still remains more-or-less meaningless to me. It seems to refer to a type of inquiry that in itself seems difficult to describe and has different meanings in different disciplines, so I can’t borrow meaning from whatever it means elsewhere to make it meaningful in this context, like Davis seems to have done for himself. This makes it hard for me to hold onto the framework.

    Yackel et al 2003

    In the second paper, the authors are thinking about how teachers structure and restructure their instruction, a process they call “didactising” after Leen Streefland. The reason the paper is here in a post about listening is that the way that you get information about what needs reworking in your instruction is to listen to the students.

    I have used the word instruction, as opposed to teaching, because it’s the word the authors used. And they really do seem to be thinking about instruction, in the sense of a sequence of explanations and activities you do with students. The main theme of the paper is about how listening to students helps design these instructional sequences, which I do not question the importance of. It’s just that the overall feeling I get is that students aren’t quite real people but sources of data, and that making good instructional sequences is a good in an of itself, as opposed to being something for the students. I’ve been a bit too dramatic there, and it’s not really as bad as I’ve made it sound, but still my feeling is that it dances a little too far from viewing the students as people.

    Anyway, the most useful thing in the paper for me was a new terminology for what Davis called hermeneutic listening; these authors call it “generative listening”.

    3. Generative listening

    These authors decide to use the word generative rather than hermeneutic because it’s easier to process for their purposes. They say, “Listening in this way can generate or transform one’s own mathematical understandings and it can generate a new space of instructional activities.”  While Davis was more focused on the way that hermeneutic listening changes the listener and the community’s understandings, these authors are more focused on the way generative listening generates new instructional activities. I’m happy to have both in my life. I think it’s important to recognise that teachers still have to decide what to do each day and that listening can help them make those decisions!

    I’ll finish off with three questions the authors list to help people focus on generative listening, which really do bring it back to the students as people at the last moment: “How does student thinking suggest alternative ways of thinking about particular mathematical ideas? How does student thinking suggest what mathematical ideas are experientially real for them? How can the instructional sequence be redesigned to capitalize on the fresh points of view that students offer?”

    Some thoughts

    From these two papers, we have a framework with three levels of listening: evaluative, interpretive, and generative. The authors of those papers focused a lot on the mindset of the teacher, and how this makes a difference to how you attend to what the students are doing and saying. Davis talks a bit about the kinds of questions people ask when they have those mindsets. But it occurred to me that even if you have a particular mindset, if you ask the wrong questions, you still won’t get the information you need. So yes the kind of question you ask is evidence for the kind of listening you hope to do, but also the kind of question you ask can also dictate the kind of listening you have to do, because you will only get certain kinds of responses.

    For example, if you ask a yes-or-no question (eg Is this a subspace or not?) or a direct question about factual information (eg What is the definition of subspace?), you are unlikely to get much information about what a student is thinking, even if that’s what you hoped for. You will have no choice but to simply evaluate their response.

    And there is one question that is famous for giving you no choice for what to listen to: “Does that make sense?” If you ask this of a whole class, students will usually give no response at all. If you ask it of a single student, they’ll say “yeah ok”. So basically it tells you nothing at all: to ask this question is to give no opportunity for you to listen. So actually there is another lower level of listening: not listening.

    And if we’re talking about not listening, then there is something worse than asking “Does that make sense?”, which at least shows you think things ought to make sense, and theoretically has a chance of a student saying “no” and so giving you some information to work with. What’s worse is asking no questions of any kind. It is amazing how often a maths teacher, even one-on-one, will speak continuously for half an hour with no opportunity for the student to say anything. I always feel such a sense of shock and shame when I realise I’ve done this and that I have absolutely no idea how the student is going.

    I think sometimes the impulse to talk continuously comes from a belief that it’s your main job to provide explanations, and sometimes it comes from believing in the power of an explanation you’ve worked hard to perfect. However, even if maths teaching were transmission, that process can’t possibly be perfect, and so you really do need to check in every so often! As Davis says, “Implicit in the act of questioning is a certain lack of faith in the transmission process.” I think everyone needs to have that certain lack of faith.

    So it’s good not to have total faith in the power of a single explanation. But what should you have faith in? I think you need to have faith that students actually are thinking. Implicit in the interpretive listening stance is the assumption that there is something to listen to. You have to believe that students have ideas if you seek to interpret them. If you don’t believe they do have ideas already, then of course you don’t seek to listen to them. For me, this is a huge part of working in the MLC that changes the whole approach. The next level above this is to believe that students have ideas that can change your own, which is where generative listening lives.

    My version of the framework

    So, finally, this is my interpretation of the listening framework of Davis (with the third level renamed by Yackel et al). There are a lot more aspects to this, such as the nature of the teacher’s role, but this version helps me think about what I am doing with students on the fly. You can download a handout PDF version of the framework  if you want.

    Level 0: Not listening

    Goal:

    • Tell what the teacher thinks is important
    • Give clear explanations

    Types of questions:

    • Not asking questions
    • “Does that make sense?”

    Beliefs:

    • Faith in the power of the teacher’s explanation
    • Students are waiting for your ideas

    Level 1: Evaluative listening

    Goal:

    • Judge student responses against a standard
    • Get a specific response so you can continue the plan

    Types of questions:

    • Yes/no questions
    • Direct questions about raw information
    • Results of calculations

    Beliefs:

    • The teacher’s explanation is not perfect
    • Students are waiting for your ideas

    Level 2: Interpretive listening

    Goal: 

    • Decipher the sense that students are making
    • Understand student thinking
    • Create a shared language to describe thinking

    Types of questions:

    • Open-ended questions about thinking or process

    Beliefs:

    • Students are reasoning
    • Student ideas are worth listening to

    Level 3: Generative listening

    Goal:

    • Jointly explore ideas
    • Discover new ways to think about or to learn concepts

    Types of questions:

    • Open-ended questions about thinking or process
    • What-if questions and I-wonder questions

    Beliefs:

    • Students are reasoning
    • Student ideas are worth listening to
    • Student ideas can change yours

    Final thoughts

    I have deliberately numbered the types of listening and called them levels, because I wanted to explicitly say to myself that some are higher than others. However, I don’t want to say that you should never seek to provide clear explanations and never listen evaluatively. Of course you should explain things when you need to, and of course there are times when you need to know students can do things in a standard way. And I also don’t want to say you should spend all your time listening generatively. That would be exhausting for everyone. It’s just that the types of listening definitely do progress in how student ideas shape what happens, and it is definitely a good thing for students to feel that what they think and do makes a difference to the outcome.

    What I want is to always be open to the opportunity of finding out how students think and possibly having it change the way I think. I also know that while beliefs definitely guide actions, it also works the other way too. If I spend all my time talking, I may come to believe implicitly that the students have nothing to say. If I spend all my time evaluating against a standard, I may come to believe implicitly that the students have nothing wonderful to say. I need to actively work in opportunities to listen at the higher levels, so that I never go too long without them.

    In daily work, where I spend most of my time one-on-one with students, this is even more important. Because when you’re right there next to the student, what a waste it would be to never hear the wonderful things they have to share, or to never make something wonderful together.

  • 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!

  • 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.

  • Book Reading: Play – How It Shapes the Brain, Opens the Imagination and Invigorates the Soul

    Looking back at my blog over the past few months, I’ve done a lot of these “book reading” posts. I really did mean to do some more on other ideas, but I felt I had to get these thoughts out of the way first. So here’s another book reading post, this time about the book “Play: How It Shapes the Brain, Opens the Imagination and Invigorates the Soul” by Stuart Brown (with Christopher Vaughan).

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

    Kassia Wedekind gave a talk about play in maths where she used some ideas from the book (you can see the video here ), but I didn’t realise what the book was until later when she responded to my request for how people define play. I immediately looked it up in the library and placed a hold on it so I could borrow it as soon as I could. I finished reading it pretty quickly, but I’ve only had the chance to stop and write about it now.

    The basic message of the book is that humans are actually wired to play and that play is essential for our survival and wellbeing, both individually and as a species. He doesn’t set out to apply this to education and certainly not maths education, but I have to say I agree with Kassia that it does indeed apply.

    The first few chapters describe what play is, and how it seems our brains are built with a need for play, even as adults. The last several chapters talk about applying play to various aspects of our lives including parenting, work and relationships. I will talk about Part One a lot, and then about Part Two a little.

    What play is

    One of the most interesting and useful bits of the whole book was Brown’s description in Chapter 2 of what play is. He is loath to give a definition, but he does describe a couple of lists of noticeable features that play tends to have. These were really useful both to broaden the scope of what can be called play, and to narrow my focus to the key features.

    His first list of features are things you can notice about play and people engaged in it. You can tell someone is playing when you notice most of these things (p17):

    • Play is apparently purposeless: that is, it seems to be done for its own sake and not because it has practical value.
    • Play is voluntary: it’s not required by duty or forced upon you.
    • Play has inherent attraction: it makes you feel good so you want to do it.
    • Play has improvisational potential: there is scope to put things together in new ways, to do things differently and try things out.
    • People who play experience freedom from time: they lose a sense of time passing.
    • People who play experience diminished consciousness of self: they stop thinking about their thoughts or how they look to others or whether they’re making mistakes.
    • People who play have continuation desire: they want to keep going and find ways to make it keep going.

    The last three are the aspects that Kassia mentioned in her ShadowCon talk, and at the time they really spoke to me. I really could imagine the times when I had experienced all three of these and they really were playful activities – those activities when you were so engrossed that you missed lunch or turned around to notice ten people watching you that you didn’t know were there. Interestingly, when I have noticed people watching, my play usually stops, or at least turns into more a performance, which isn’t really the same thing.

    So those last three just put names to how play already felt to me. What the full list added for me was a description of the sorts of things that might possibly encourage play. It would seem that an activity with potential to choose whether to do it, where there’s no particular performance goal in mind, and where you have scope to try different options, would be the sort of activity where play is possible. On the other hand, an activity that is tightly constrained with a specific goal in mind seems much less likely to produce play without the people involved being brave enough to break the constraints.

    The second list presented is a number of stages a player will go through as they play, taken from Scott Eberle. He says that all players may not go through all stages and not necessarily in this order. Still I agree with Brown that it’s still useful.

    • Anticipation: Curiosity and sometimes a bit of anxiety as you think about what will happen when you engage in the play.
    • Surprise: Something new happens or you see something a new way.
    • Pleasure: Usually caused by the surprise.
    • Understanding: Incorporating new ideas into what you know.
    • Strength: Being empowered because you have done something new and succeeded.
    • Poise: Feeling contented and composed.

    I particularly like the idea of anticipation. I can feel it when I do a puzzle and I’m investigating the ideas connected to it, getting a growing feeling that something cool is about to appear. And then it happens and it’s a surprise but not a surprise. When I dig into it to understand I feel ready to face something new in the future. That final idea of poise is also a good one. I see it as that quiet feeling of contentment, different from the intense spark of pleasure caused by the surprise, that allows you to feel ready to leap into anticipation once more.

    At this point I was in a dangerous place. I could recognise these aspects in myself and my children for activities that I knew already were play, but I hadn’t fully realised the extent of what Brown was trying to achieve here. The thing is, as he says on page 60: “play is a state of mind, rather than an activity.” Chapter 3 really opened my mind to understand that there are many experiences and activities that can be playful – even things that I might consider work. The key is in the state of mind that goes with it.

    In this chapter, Brown sets out a list of several “play personalities”. He describes eight types of people based on the sorts of things they prefer to play with, for example, the “creator”, the “storyteller”, the “joker” and the “director”. I’m not sure I agree totally that there is such a thing as a play personality, but I do agree that people would have certain preferences. The key thing I realised reading this is that there was a much wider scope of things that could be called play than I had realised. For example, the “competitor” plays by aiming to win. Their improvisation potential comes from figuring out how to get the furthest or the most within the confines of the rules, and how far they can push themselves. I had never considered this as play at all, since competition usually turns me off completely and makes things feel like work. But reading this I understood where the playfulness was and why a person competing against me may not actually have any grudge against me at all; instead I’m just a part of their game. Another example is a “collector”, who plays by collecting things. Their improvisation potential stems from the choices they make of what is in or out, or how to classify and order the collection of things they have. Again this was a completely new way to look at play for me.

    Why play is so good

    Brown argues that play is essential to help learn, to be creative and to be happy. He notes how in animals (in particular in bears), play helps them learn social cues that mean they can function successfully in their communities. Without play they can’t test the boundaries of what is acceptable and what is not acceptable. In Chapter 4: “parenthood is child’s play”, he outlines many interesting things that children learn through play, not least of which are the physical laws of nature and their own personalities.

    The really profound thing I got from the book was the importance of adult play. I have read some things recently saying that play is only appropriate for young children and shouldn’t be encouraged in high school because adults have to work. Brown argues that actually, adults are only capable of surviving work because they are able to infuse play into it. Work is soul-destroying when there is no scope for improvisation, no part of it gives you a choice, and when it focuses entirely on key outcomes you must be assessed against. These are the opposite of the lists of the features of play. On the other hand, given a choice to voluntarily do something, with freedom to try out things and fail, you are uplifted. So in short, making work more playful makes it more fulfilling. Even if you can’t change the constraints of your work, Brown says that allowing play elsewhere in your life will make work better too.

    Basically his argument is that as humans we need the opportunity to try new things in a safe environment, and only play allows for this. We need to be able to continue to develop, and play is the catalyst for development. As he says rather harshly on page 73:

    When we stop playing, we stop developing, and when that happens, the laws of entropy take over – things fall apart. Ultimately, we share the fate of the sea squirt and become vegetative, staying in one spot, not fully interacting with the world, more plant than animal. When we stop playing, we start dying.

    Helping myself and others play

    One question that niggles at me is this: if play is a state of mind, then how can I help anyone to play, or even myself? How can I change anyone’s state of mind? It’s a big question, and Brown goes some way to answer it in the final chapter, at least for helping yourself to play. He suggests a few strategies and I think they fall into three main categories what I want to synthesise here:

    • Move: Brown says “motion is perhaps the most basic form of play”. He says that basically most of your cognition and perception are actually encoded into your brain in the first place via movement. Therefore movement has a way of shortcutting all of your cognitive and emotional inhibitions to play. In short, if you want to relearn to play, then move.
    • Be near others who are playing: Brown says several times across the books that one of the best ways to learn to play is to be with a dog, or a toddler. When you see them playing, you often can’t help but join in. I’ve seen it myself at One Hundred Factorial — people relearn to play by being with others who are playing.
    • Find a safe environment: There is nothing more toxic to play (and indeed general wellbeing) than being in an environment where people judge you or where you are afraid to be yourself. In order to be free to play, you need to be free from fear, so you need to find a place where you don’t have to be afraid. Usually for us adults, it has to do with having the right people around us. As he says on page 216: “If people around you cannot learn to understand your need for play, find people who do.”

    As a teacher and a team leader, I think these three things tell me a lot about how I can help my students and my staff to play and so be happier and more productive people. I need to work super hard on creating that environment where it is safe. My classrooms need to be places where it’s ok to make mistakes, to try out new ideas, to suck when you do something for the first time. I myself might need to be the person nearby who is playing so that others can see it’s ok and learn how. And I can get my students to move, at the very least to move their hands, to forge that new connection to the fundamental state of play.

    All in all, I really enjoyed the book. At times it did slip into the style of a motivational speaker making grandiose claims with references to specific people’s stories. But I still think the points it made are valid points, and it has made me think differently about what play is and where it fits in mine and my students’ life.

    This comment was left on the original blog post:

    Kassia Wedekind 8 June 2017

    I love your observations adults and play.

    It particularly makes me think that teachers need opportunities for mathematical play too! Both because I think experiencing the feeing of play and math together is really important for Ts when they’re thinking about what they want for their students and because I think the play process really lends itself to thinking about reasoning and how we figure things out in math.

    Thanks for writing about this and thinking about it with me too.

  • Book Reading: Making Number Talks Matter

    Here is another post about a book I’ve read recently. This time, I’m writing about the book “Making Number Talks Matter” by Cathy Humphreys and Ruth Parker.

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

    In Cathy and Ruth’s words, number talks are “a brief daily practice where students mentally solve computation problems and talk about their strategies”. I had heard people talk about them before and how they are a powerful way to help students come to a better understanding of how numbers fit together and to develop their confidence. So I read this book in the hope of finding out more about what they are and how to implement them. My goal was to eventually use number talks to help make a difference to Science and Health Science students, especially those with little maths experience or only painful maths experiences.

    I have to admit to you now that I’ve been trying to write this post for a couple of weeks now and I’ve been having real trouble. I think it’s because I had mixed feelings about the book at the time, but that looking back several months after reading it I have different feelings now than I did back then. I think the easiest way to write the post is to talk about some of those feelings first.

    While reading the introductory chapters, I had such hope for the power of Number Talks. Cathy and Ruth talked about how much students need to talk about numbers and make sense of things, rather than follow algorithms without making sense, and inside I was saying “Yes!” and I was inspired to keep reading. When I got to the next chapter where they described the standard routine for Number Talks, I felt a bit let down. The directions said to get students to put away all paper and pens, to ask them not to talk and to put up thumbs to say when they’ve got an idea, then to share answers before asking for strategies. My knee-jerk reaction was to feel very restricted by these directions. Looking back later, I am drawn much more to the rationales about each step: that no paper helps to focus away from algorithms and towards sensemaking; that no talking helps students to form their own ideas; that answers before strategies helps to get answers out the way to focus more clearly on strategies later. Focussing on the rationales helped me imagine how I might decide to change some of these to match the needs of the students I might be working with.

    On that note, the next chapter on Guiding Principles for Number Talks was I think the most useful chapter in the whole book. I kept coming back to it while reading the rest of the book to ground myself again. Indeed, the later chapters on specific strategies for specific operations got me a bit bogged down and made me feel a bit like I’d lost my vision of what we were trying to do here. I needed the touchstone of the Guiding Principles to pull me out of that feeling of slogging through. I’m going to come back to this chapter and talk about it in more detail because I want to end with the best bit!

    The next several chapters talk about various operations and number types and the various strategies that we might hear students using or encourage them to use. I found this a bit heavy-going, partly because some of the strategies were not natural to me and so I couldn’t think to try to recommend them to anyone! In hindsight I think it’s really good that I read this before trying any number talks because I am pre-prepared in order to not be surprised too much when students do some interesting stuff. Also, as I flick through them now, I am somehow more able to see how each strategy might apply to my current students. I think maybe having all those strategies floating in my mind while I’ve spent a few months helping my students make sense of algebra and calculus has helped me see where these strategies for operations tie in with the later maths concepts. I do need to say that even upon first reading, a useful thing about these middle “operations” chapters were the many vignettes of number talks in action that slowly gave me a better idea of how the discussion part of the routine is implemented.

    My very favourite part of all of the middle chapters were the special number talks that appeared in the chapter on fractions, decimals and percentages. These ones had students not calculate an answer but decide which of two numbers was bigger, decide if a number was closer to 1/2 or 1, or to place a fraction on a number line. These really gave me a better idea of the possible ways of using number talks to promote sensemaking than any of the previous calculation number talk ideas. I suddenly felt free to consider more options and therefore free to give it a go.

    And then the book finished off with a chapter called “Managing Bumps in the Road”. This was another chapter that was really useful for helping me be brave to try it myself eventually. Based on the roadbumps mentioned here, I reckon one of the major dangers is losing sight of the important goals of number talks outlined in those guiding principles at the start. This chapter helped refocus my attention on what’s important and gave some ideas for how to refocus this attention on the fly too.

    Which brings me to the end of the book. It was in some ways not the easiest book to read, but I did learn a lot about sensemaking and strategies and managing discussions. And as I said, the guiding principles mentioned early on were a very excellent thing I was able to take away from the book. Most of them are applicable to most of my maths teaching and not just to the specific routine of number talks. As promised, here they are:

    Guiding Principles for Number Talks

    from “Making Number Talks Matter” by Cathy Humphreys and Ruth Parker

    1. All students have mathematical ideas worth listening to, and our job as teachers is to help students learn to develop and express these ideas clearly.
    2. Through our questions, we seek to understand students’ thinking.
    3. We encourage students to explain their thinking conceptually rather than procedurally.
    4. Mistakes provide opportunities to look at ideas that might not otherwise be considered.
    5. While efficiency is a goal, we recognise that whether or not a strategy is efficient lies in the thinking and understanding of each individual learner.
    6. We seek to create a learning environment where all students feel safe sharing their mathematical ideas.
    7. One of our most important goals is to help students develop social and mathematical agency.
    8. Mathematical understandings develop over time.
    9. Confusion and struggle are natural, necessary and even desirable parts of learning mathematics.
    10. We value and encourage diversity of ideas.

    Number 2 and Number 5 in particular shone out to me at the time as my guiding lights for day-to-day teaching even outside of number talks. (Though looking through them now, 6, 7 and 8 are right up there too.) Two specific quotes from this chapter make these more real to me and are a good place to finish:

    While we may have a good idea about how students are thinking, we don’t really know until we ask. Authentic questions keep the mathematical focus where it belongs: on students’ reasoning – not ours.  (pp26)

    No strategy is efficient for a student who does not yet understand it.  (pp27)

    These comments were left on the original blog post:

    Mark Pettyjohn 25 May 2017

    Your post reminded me of some conflicted thoughts I’ve previously had. The nature of which were about:

    Number Talks (TM) vs. number talks

    Four or five years ago I became aware of Number Talks via Sherry Parrish’s book. I got an overview and then dove in with my class. The results were amazing, for me and for them. I think the principles of what I saw happening were highly aligned with the principles outlined by Humprheys and Parker.

    Then something peculiar happened. I wanted to share with others the good things happening, so I went back to my Number Talks book (again, Parrish not Humphreys), and it all looked so stilted. I watched the accompanying videos and they looked little like what was happening in my classroom. It was more a teacher driving strategies to students rather than principles 1-7, 9, and 10 outlined here.

    So as I was reading your reticence to write this post, I was feeling my own reticence back then about sharing Number Talks (TM) with colleagues because I didn’t feel like it captured in practice or in spirit what we were doing in my classroom. There’s enough confusion around terms in education that I was hesitant to add to it.

    That’s partially why I asked on Twitter if you had seen or done any yourself. I’ve found that a number talk is not always a Number Talk (TM) and I would imagine that extends to what Humphreys and Parker have here.

    But I really like the principles outlined in your post, and I think that if you can look back at a number talk (with your own kid or with other students) and see those principles reflected, then you done good.

    Susan Jones 25 May 2017

    I share the reaction to restrictions. I remind myself that it’s only for that chunk of time and…. I think I’d break it every once in a while for people like me who think wth their pencils. I see that obsession with algorithms on our 5-math-question survey at the beginning of our “transitions” course for students, which asks what 4 and a half x 2 is. In several years we’ve seen 1 or 2 students answer it correctly while half the students attempt an algorithm (many leave it blank).

    David Butler 26 May 2017

    Interestingly, in the book they do do some number talks where they suggest to let the students have pencil and paper. I think you really need to be looking at what message you are sending today and whether not having paper is going to help.

  • Book Reading: Becoming the Math(s) Teacher You Wish You’d Had

    This post is about Tracy Zager’s most excellent book, Becoming the Math Teacher You Wish You’d Had.

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

    I actually finished reading it back in January, and I live-tweeted my reading as I went.  The process culminated with this tweet:

    A tweet from David Butler @DavidKButlerUoA 25 Jan 2017. Text: I've just finished reading your #becomingmath book @TracyZager. This is the bit I liked. Video: Me flipping through all the pages in the entire book. https://twitter.com/DavidKButlerUoA/status/824183955012415495
    https://twitter.com/DavidKButlerUoA/status/824183955012415495 

    That’s what I thought about it at the time, but I haven’t sat down to organise my thoughts on it. Until now.

    I was first drawn to the book based entirely on its contents page. Check this out:

    • Chapter 1: Breaking the Cycle
    • Chapter 2: What Do Mathematicians Do?
    • Chapter 3: Mathematicians Take Risks
    • Chapter 4: Mathematicians Make Mistakes
    • Chapter 5: Mathematicians Are Precise
    • Chapter 6: Mathematicians Rise to a Challenge
    • Chapter 7: Mathematicians Ask Questions
    • Chapter 8: Mathematicians Connect Ideas
    • Chapter 9: Mathematicias Use Intuition
    • Chapter 10: Mathematicians Reason
    • Chapter 11: Mathematicians Prove
    • Chapter 12: Mathematicians Work Together and Alone
    • Chapter 13: “Favourable Conditions” for All Maths Students

    Is this not awesome? Here was a list articulating things about maths that I know are important and yet that I’ve struggled to articulate all my life as a mathematician and maths educator. Many of them cut straight to the heart of the difference between how I experience mathematics and how it usually is experienced in a classroom.

    • “Mathematicians use intuition” you say? Well, yes. Yes we do.
      But many a maths classroom is about following rules and avoiding the need for intuition.
    • “Mathematicians work together” you say? Well, yes. Yes we do.
    • But so many students think maths is only a solitary activity.
    • “Mathematicians make mistakes” you say? Well, yes. Yes we do.
      But mistakes are feared and avoided in most maths classes.
    • “Mathemaicians connect ideas” you say? Well, yes. Yes we do.
      But so many maths curriculums are just so many piles of disconnected procedures, even here at my own university.

    The contents page promised a book about the most important aspects of mathematical work and thinking, and a hope that it would give ways to bring these into the experiences of students in all maths classrooms.

    And the hope was made real.

    Each chapter starts out comparing how mathematicians talk about what they do and what students’ experience of it is. Then it moves on to detailed examples of the aspect of maths thinking in action in real classrooms, as well as strategies to encourage it both in your students and in yourself as a teacher.

    I didn’t expect to see this last point about encouraging these attitudes and thinking in yourself as a teacher. Yet it is the most compelling feature of the book for me. Indeed, I don’t think the book would have had nearly the impact it had on me (or the impact I see it having on others) without this constant message that to help your students experience maths differently, then you yourself need to experience it differently too. More than this, Tracy doesn’t just make this need clear, but actively and compassionately empowers us to seek out ways to fill it.

    Somewhere inside you is a child who used to play with numbers, patterns and shapes. Reconnecting with your inner mathematician will improve your teaching and benefit your students, and it will also benefit you.
    – Tracy Zager, Becoming the Math Teacher You Wish You’d Had, p39

    These comments were left on the original blog post:

    Tracy 10 May 2017:

    Misting up over here, David. You’ve cut right to the heart of it. Thank you so much. I feel incredibly lucky to learn with you.

    David Butler 10 May 2017:

    And I am lucky to have been able to sit and talk with you by reading your book. I really felt like you were there with me, encouraging me to be more.

    Susan Jones 10 May 2017:

    I’m still reading it. Your post makes me realize my privilege in not majoring in math 🙂 I never did lose the “play with the numbers” thing.

    David Butler 10 May 2017:

    Sadly, you don’t have to major in maths to lose the play-with-numbers thing. A good dose of standard high school maths teaching can safely banish that tendency, as Tracy described in the first chapter of the book!

    For me, my maths university degree is what actually freed me to play. I took Discrete Maths II in second year of university and it felt like all we were doing was playing with these ideas, and it was play encouraged by the lecturer.

  • 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.

  • Book Reading: 5 Practices for Orchestrating Productive Mathematical Discussions

    Writing about the teaching books I’ve read is fast becoming a series, because this is the third post in a row about a teaching book I’ve read.  The book I finished earlier this week is “5 Practices for Orchestrating Productive Mathematical Discussions” by Margaret S. Smith and Mary Kay Stein and coming out of the National Council of Teachers of Mathematics (NCTM) in the USA.

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

    I’ll get straight to the point: everyone in any sort of classroom where maths happens should read this book. It gives a simple and practical framework for using student work and class discussion to promote maths learning. The authors have a direct, clear style that make the nuances of the practices seem almost obvious, using careful studies of classroom scenarios to illustrate. Let me say again: read this book!

    In a nutshell, the idea of the book is that you can help students learn mathematical content by giving them tasks rich enough to be worth talking about and connected to the mathematical goals you have in mind, and then orchestrating class discussion of the methods students use and their connections. They give five practices, and a smattering of other strategies and ideas to guide this.

    I think this book should be required reading and/or the basis of training for staff who are teaching tutorials at university. University tutors are often given no training in teaching, and even then don’t get tools to help them choose what to do in their classrooms. In some schools here at the Uni of Adelaide, they are instructed to get students working in groups. This is great, but the part where the mathematical ideas of the week are brought out is not strong. I am hoping to take these practices to these schools, and to the ones where it’s more just another lecture, in the hope I can help to improve the learning happening in the tutes. I’ll be mentioning how I think it applies to tutorials as I go.

    Here’s a summary in my own words:

    “Practice 0”: Worthwhile tasks and mathematical goals

    You’re not going to be able to have a class discussion about a task which is routine procedure-following, because everyone will do it the same way. You need something that has some level of challenge and has decisions to make about how you do could do it – something actually worth discussing! Also, you need to have a goal in mind for what you want to achieve so that you have a chance of achieving something. This goal needs to be about the mathematical ideas involved. For example, about the connection between the different types of equations for lines, or about the distributive law, or about the relationship between squares and rectangles.

    This isn’t technically one of the five practices, since it happens “outside” the context of the discussion. Plus, you may not always have total control over the tasks that students have to do or the mathematical goals. (More likely a school teacher is in control of this, but a classroom tutor at university this will be less often true.) Even so, if you do have control, it’s very important, which is why the authors call this “Practice 0” a couple of times, because it’s needed before you even start.

    As I already said, in classroom tutorials, someone else often chooses the tasks. But you can add your own question to the end to make it more open to discussion. Maybe something like “What would happen if…” are good to extend learning. Someone else may set the goal, but it’s more likely the people coordinating your course won’t tell you what the learning goal is. So you’ll have to choose for yourself. It’s so important to choose the goal so that the tutorial doesn’t end up feeling like a whole lot of activity and discussion, but with nothing of substance to take away.

    Practice 1: Anticipate

    When you have a goal and a task, the first thing to do is anticipate how the students will respond to the task. At the very least, you need to do the task yourself, but even better, imagine as many correct and incorrect, helpful and unhelpful approaches as you can.

    One reason for this is so that you don’t have to make so many decisions on the fly during the class. You can figure out in advance some of the ways you will respond to these before you get there.

    I see another advantage and it is about putting yourself in the mindset of your students. We university teachers are often so blind to how our students think, and tutors are often very focused on their own way of doing things. By explicitly trying to think of multiple approaches, it can help to break down this egocentric focus we fall into.

    Practice 2: Monitor

    Once you’re in class and the students are working on their task, the role of the teacher is to monitor the students’ work and thinking. The anticipating you did earlier helps you to respond appropriately to them, and sets you up into a mindset where you’re focused on their thoughts, so even unexpected methods are easier to process. It’s while monitoring their work that you will make the final decision of how you want to run the discussion, and who will be involved. It’s also while monitoring their work that you’ll ask the students questions to help them learn in-the-moment.

    One thing I particularly like about this practice is how it gives us a focus while the students are working. Just the other day when talking to tutoring staff, they expressed a distaste for groupwork because it meant they, the teacher, weren’t “doing anything”. This practice says you’re not doing nothing – you’re monitoring.

    The authors recommend asking students two types of questions during the working (and hence monitoring) phase:

    • Ask questions about student thinking
      Help students while they are working to express their thinking about the problem and the maths. Actually ask them to tell you how they are thinking. This gets them ready for the discussion to follow, and also helps them with the problem-solving too.
    • Ask questions about maths meaning and relationships
      Help students to express what the maths ideas mean and what they mean to them. In particular draw out relationships between concepts. This is what your goal is ultimately, and it front-loads this discussion so students are ready for it.

    I see these two types of questions as really important for classroom tutors at university. Too often the questions we ask are about yes/no correct/incorrect answers, rather than about thinking and ideas. Encouraging tutors to focus on these types of questions makes thinking and meaning the focus of the learning activity.

    Practice 3: Select

    The last three practices are about making the discussion part of the class happen productively. They work together to help make sure that the discussion both uses student work, but also proceeds towards the mathematical goal. Also they prevent the random show-and-tell which often just ends up with students confused or with no particular idea of what they learned.

    First, you want to select what student work you want to discuss as a whole class, and whose work it will be. The authors list a few considerations here, not least of which is choosing students who up to now haven’t participated much in class. It’s worth noting that in their examples, even though students worked in groups, specific single students are asked to talk about their work, which means people can’t hide from participating! It also means that people can’t monopolise the participation either! We all know that one person who seems to think the tutorial is just there for them to show how clever they are. By preselecting students to show their work, you’re making it less likely for this person to take over.

    The thing I like most about the concept of selecting student work is that it has the potential to help students feel like their work is a valid and important contribution (which of course it is). By using student work and student generated ideas to forward the maths discussion, we can help them be more engaged in the learning and feel like we care about them. This is not a small thing to consider!

    I am particularly interested in applying this idea to classes where students are expected to do preparation for the tutorial in advance and hand it in (like they do in several courses here at Uni of Adelaide). At the moment, what usually happens in these classes is that students do the homework, hand it in, and then the tutor presents their own preprepared solutions. But think what might happen if the students handed in the homework, and the tutor used the homework itself as a tool for class discussion. I think it might help the students feel like their homework was actually worth all the effort!

    Practice 4: Sequence

    After choosing which student work to present, you need to choose what order it will be presented in so that you can progress towards the mathematical goal. The authors give a number of things you might consider with your sequencing. For example, you might want to choose to start with a solution method that a lot of people have so that everyone can get buy-in to the discussion (I did this when I did Quarter the Cross in my daughter’s classroom). You might want to start with a solution containing a misconception to get it out of the way. You might want to avoid a specific solution because it will just send everyone off on a tangent (though you might also want to talk to that student one-on-one separately). You might want to have two particular solutions in quick succession in order to be able to compare them.

    The important bit is to think about what order would be most helpful to get to where you hope to go. Importantly, the way you hope to make connections between ideas will dictate how you might sequence the students’ work.

    Practice 5: Connecting

    Now that you’ve chosen what student work to focus on in the discussion and in what order, it’s now time to actually have the discussion. It’s important here to remember there is a mathematical goal we’re working towards, which will often be about understanding a concept, and understanding is a sensation that happens when ideas are connected to other ideas. It’s our job to help students make these connections.

    The authors suggest five “moves” you can make during your discussion to make sure it stays focused on the connections you want to draw out.

    • Revoicing
      This is when you repeat what a student says to make sure you and everyone else heard it and understands it. Importantly it’s not about you making what they said more correct, simply making it heard. A good phrase to end with when you revoice a students’ words is “Is that right?” This lets them know that the point is to make their thought heard (not yours) and they get to decide if it’s been voiced right.
    • Asking students to restate someone else’s reasoning
      Instead of you revoicing a students’ words, you can ask another student to explain the reasoning. This includes even more students into the discussion in a more active way.
    • Asking students to apply their own reasoning to someone else’s reasoning
      This time you’re not just asking the other student to explain the first student’s reasoning, you’re asking them to explicitly explain the connection between two different types of reasoning (one of which is their own reasoning). For example, suppose you’re doing Quarter the Cross and John proved the house-shape was a quarter by cutting and overlaying, whereas Jane proved the L-shape was a quarter by folding, you could ask Jane to prove the house-shape works by folding.
    • Prompting students for further participation
      There are times where a student will close off with a quick answer, and it might be more productive if they stayed in the discussion a little longer. The questions listed above of asking them to explain their thinking or focus on the meaning and relationships are useful now as well. In the Maths Learning Centre, I find “Tell me more about that” to be a good all-purpose request to participate further.
    • Waiting
      This may seem paradoxical, but leaving some silence can help to promote discussion. The authors say that whenever anyone asks someone else to say something, it’s appropriate to give them plenty of time to respond. Giving them this time helps to actually make the point that their answer is important to you. You giving yourself time to form your response to their question helps to make the point that their question is important to you. Waiting a bit after an explanation to let it sink in before asking people for any questions helps to make the point that it does in fact take time to process information. These last couple were new thoughts for me (though obvious in hindsight).

    It’s this last practice that we often don’t do in tutorial discussions. I was talking to some tutors from the Faculty of Arts recently, whose tutorials are traditionally only discussion. They talked about how often the discussion just goes for a while and then stops at the end of the class, without coming to any conclusion the students can take away about the concepts or the process of learning them. They recognised a need to explicitly make connections during the discussion. Over in maths tutorials, I think we assume the connections are obvious, but I can attest that they are not, if all the students complaining that the tute doesn’t teach them anything are anything to go by.

    Conclusion

    It may seem that I’ve given you the content of the whole book, and indeed my aim was to present the ideas clearly, mostly for my own future reference! But I would still encourage teachers and tutors to actually read the book. The vignettes of actual classroom use are vitally important to come to an understanding of what the practices look like and where they are useful, plus there’s whole chapters about how to seek support for teaching and how to include it in formal lesson planning that I haven’t even mentioned (until just now).

    I am excited to take the ideas here and use them to help support classroom tutors here at University. I think this book could really be a tool that people might actually get behind. Here’s hoping.

    To wrap up, here’s the headings in dot point form for future reference:

    • Practice 0: Worthwhile tasks and mathematical goals
    • Practice 1: Anticipate
    • Practice 2: Monitor
      • Ask questions about student thinking
      • Ask questions about meaning and relationship 
    • Practice 3: Select
    • Practice 4: Sequence
    • Practice 5: Connect
      • Revoicing
      • Asking students to restate someone else’s reasoning
      • Asking students to apply their own reasoning to someone else’s reasoning
      • Prompting students for further participation
      • Waiting

  • Book Reading: Math on the Move

    Over the last week or so, I have been reading the book “Math on the Move” by Malke Rosenfeld (subtitled  “Engaging Students in Whole Body Learning”). Ever since connecting with Malke on Twitter back in June or July, I’ve wanted to read her book, and I finally just bought it and read it. Now that I’ve finished, it’s time to write about my thoughts.

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

    The book is all about whole-body learning as it relates to maths and dance, mostly focussing on pre-school to Year 6. Some of you may be wondering why I, a university lecturer with a doctorate in pure maths, would be so very interested in something to do with dance-and-maths at the primary level. My first response to that is that you clearly need to get to know me a bit better! Perhaps start by checking out the following past blog posts: Kindy is awesome and The Pied Mathematician of Hamelin.

    My second response is that seeing things from a new perspective is one of the best ways to understand them better and to understand how you understand them in the first place. I was fascinated by this new medium of a moving body for thinking about maths and I wanted to get the benefit of reading the thoughts of someone who has already considered it deeply. And Malke Rosenfeld is just that person, because reading the book you can tell immediately that she has thought very deeply about it.

    The book has two main parts. The first part is about the concept of movement-scale activities and the body as a thinking tool in mathematics. The second part is about the Math in Your Feet program, which is also about the body and its movement as a thinking tool, but even more than that, that dance itself is a mathematical thing worth thinking about.

    The first part had me thinking from the moment I started reading. Malke argues that scaling up a mathematical idea to the scale where your whole body can interact with it or be it can give insights and understandings not available in any other way. Malke gives examples of number lines and hundreds-charts of a scale you can walk on, and building polygons out of knotted rope that has to be held by multiple team-members. My head was whirring with the possibilities. Immediately I imagined what it would be like to stand on a surface defined by a two-variable function and questions about directional derivatives occurred to me that never had before. Imagine what would have happened if I could actually stand on the surface itself!

    Malke makes the very important point that meaningful moving-scale mathematics learning is not about using your body to memorise things, or to copy what is on the page. It is about using your body to make movements that are intrinsically related to the thing you are trying to understand. Stretching your arms to copy the drawn shape of a linear graph while saying its formula is not really meaningful. Perhaps more meaningful would be walking on a graph drawn on a basketball-court-sized coordinate grid and explicitly discussing how you move relative to the x and y axes. (And just now writing this, I suddenly have this cool idea to really understand discontinuities as places in the graph where the mover has to literally jump to get to the next point.) The discussing I mentioned is important too – meaning happens when the ideas are discussed and compared.

    The second part of the book, as I said earlier, described the Math in Your Feet program. Children are given a two-foot by two-foot square to dance in and a number of possible ways to move. They create steps within this framework and work with partners to make dance steps the same and different, to combine patterns of steps into longer patterns, and to transform dance movements through rotation and reflection. There’s detailed information about how the program moves forward, and the ways to facilitate work and play and thinking and discussion, as well as lots of linked videos to really see the action. You could be forgiven as a high school or university maths teacher for thinking this part of the book doesn’t really apply to you as much as the movement-scale exploration of existing maths ideas. I say you could be forgiven, but you’d still be wrong.

    Firstly, there is a whole heap of very deep discussion on what it means to give the students the power over their own learning. Malke discusses the importance of clear simple boundaries, of precise language, of encouraging language, of reflection, of getting students to share, and of ways to help children to focus. All of this is vividly displayed throughout the Math in Your Feet chapters of the book, and what you can learn here would translate to all sorts of other teaching situations. It is worth watching all the videos jut to revel in Malke’s skill of never praising product but always excitedly praising participation and practice.

    Secondly, it is this part of the book that is the most mathematical, from my perspective as a pure mathematician. The dance moves within the tiny square space are an abstract mathematical idea that is explored in a mathematical way. We ask how the steps are the same or different from each other, identifying various properties that distinguish them. We investigate how these new objects can be combined and ordered and transformed. We try out terminology and notation to make our investigations more precise and to communicate both current state and how we got there. These are all the things we pure mathematicians do with all our functions, graphs, groups, spaces, rings and categories. The similarity of this to pure mathematical investigation in striking.

    I have been changed by reading this in ways that I am not capable of processing completely at the moment. Not until I have more chances to try out movement-scale investigation of maths, and mathematical investigation of movement, will I feel I have a handle on it. But it’s a pleasant sort of feeling all the same.

    One final warning: If you read this book, don’t attempt to do it in an armchair, or on the train, or while walking. It won’t work. In order to read this book effectively, you need to sit with access to a computer to watch the video clips, and with access to a 2 foot by 2 foot square on the floor to try the dance steps in. Also if you’re like me, you’ll need somewhere to write down quotes which speak deeply to you. Quotes like this:

    Using the moving body in math class is about more than getting kids out of their seats to get the wiggles out or to memorize math facts. Instead, we need to treat the movement as a partner in the learning process, not a break from it.  pp 1

    Using tangible, moveable objects (including the moving body) can be useful in math learning as long as attention is paid to the math ideas as well as what you do with the object. pp 13

    Using language in context to label, describe, and analyze this work is one of the most powerful ways to help learners create meaning and understanding. pp 112

    Grading or judging a child on his or her ability compared with others’ is harmful in this creative environment. This is a place where the focus should be firmly on the ideas expressed, not on the facility or ease of that expression. pp 146

    We want math to make sense to our students, and the moving body is a wonderful partner toward that goal. pp xvii

    Thank you Malke.

    These comments were left on the original blog post:

    Joy 5 December 2016

    What about people with disabilities? How can teachers and students teach/learn if they are disabled?

    David Butler 6 December 2016

    Malke has a section of her book specifically about how to include children with special needs. She also encourages teachers to use the children to show examples of dance, so a teacher with a movement disability I think would be very successful with the Math in Your Feet program.

    Jeremy 15 January 2017

    Very interesting David !! I am very glad to have found this blog,

    I’ll admit that the life size maths visualisation technique seems to me to be much more helpful than the maths in your feet program. I does sound like an interesting book 🙂!