It was O’Week a couple of weeks ago, when new students arrive on campus to find out how uni works and the services they have access to. Our tradition for the last several years is to play Numbers and Letters on a big whiteboard out in public as a way to engage with students. This year I discovered a way to help people engage: write something on the board that is not a solution.
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.
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.
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
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.
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 🙂!
The premise of the book is to use cooking in a restaurant as a metaphor for constructing teaching in a classroom. It’s a good metaphor, and executed well. Warm up routines are “appetisers”, being prepared is “setting the table”, creating curiosity before giving answers is “entree”, things you do to make life more fun in the classroom are “side dishes”, and assessment is “dessert”. The commitment to the metaphor is even more impressive than that, with the contents page called the “menu” and the references section called “secret ingredients”. Just looking at the menu was enough for me to want to read the book more closely.
The major messages the book seemed to be getting at were the following:
Don’t be afraid to really love maths in front of your students.
Give students the chance to show you how they understand in their own way. Posters and videos don’t feel fun for everyone when they are being assessed on it.
Set things up in your class activities so students are curious about something. It doesn’t have to be “real world” and it definitely doesn’t have to be serious, it just has to have a question that needs an answer. Some silliness and shock value will make it taste better, but the setup for curiosity is the really important bit.
It’s a risk to try something new with your teaching, but your students will appreciate it and you can’t learn without it.
It’s only a short book, so it didn’t go too deeply into any of those, though there’s probably more examples of the ideas in action on the companion website. But still I reckon classroom teachers would get something good out of it.
Unfortunately, for me, I had trouble as I read this book because early on John and Matt described their early teaching experiences and it brought back a whole lot of unpleasant memories for me. Their description of the days when they felt that perhaps teaching wasn’t for them actually made me cry with my own memory of feeling the same way. The worst part was that I knew Matt and John stuck with it and are now writing books about teaching, whereas I left. My first school I quit before the end of my first year there (there were a number of reasons for this), and the second school I stuck out the full year, but at the end of that I went back to uni to do my PhD in geometry.
Continuing to read the book, Matt and John talk a lot about being brave enough to take risks in the classroom. I am sorry to say that all this did was make me feel like my own reaction to these early stresses was chickening out. I felt like I had been a coward and let down the students I could have had by leaving teaching. Moreover, as they describe some of the fun things they did in their classrooms, I think back to some of the similar things I did and wonder if there was something wrong with me that they didn’t make a huge difference to how I felt about teaching.
Thinking about it more, I have found one possible factor that made my experience different to Matt and John’s: support. In the book, they both describe the support they received from their school leadership and from instructional coaches in their early years of teaching, sometimes without them having to ask for this support. I had neither of those things at my first school. At my first school, I was it for maths and science and my principal was a bully who repeatedly undermined me to the students when I was not in the room and attacked my relationship with my wife. At my second school, it was better since there were more other teachers to lean on, but still I was pushing against a curriculum leader who actually said aloud that maths was “a collection of problems and a procedure to solve each one”, and a school leadership who weren’t committed to helping me improve, only to telling me I needed to. Plus, this was before Twitter, so there was no #MTBoS.
Looking back, I think the critical lack of support was one of the major causes of me giving up on teaching high school. Reading the descriptions of support in this book made me weep for poor past David. Of course I know that it has all turned out for the best because it has meant that here I am at Uni doing the best job in the world, but I couldn’t say totally enjoyed the experience of reading it. Sorry Matt and John.
UPDATE: Check out John’s reply at his own blog – follow the link below.
Trig substitution is a fancy kind of substitution used to help find the integral of a particular family of fancy functions. These fancy functions involve things like a2 + x2 or a2 – x2 or x2 – a2 , usually under root signs or inside half-powers, and the purpose of trig substitution is to use the magic of trig identities to make the roots and half-powers go away, thus making the integral easier. One particular thing the students struggle with is choosing which trig substitution to do.
In Maths 1A here at the University of Adelaide, they learn that says that, given a function of x defined as the integral of an original function from a constant to x, when you differentiate it you get the original function back again. In short, differentiation undoes integration. And then they get questions on their assignments and they don’t know what to do. They always say something like “I would know what to do if that was an x, but it’s not just an x, so I don’t know what to do”.
It’s university holidays again (aka “non-lecture time”), which means I’m back on the blog trying to process everything that’s happened this term. Mostly this has been me spending time with students in the Drop-In Centre, since I made a commitment to do more of what I love, which is spending time with students in the Drop-In Centre. When trying to decide what to write about first, I realised that a lot of what I wanted to say wouldn’t make much sense without talking about SQWIGLES first. So that’s what this post is about.
SQWIGLES is an acronym that we use to help our staff (and ourselves) when teaching in the MLC Drop-In Centre. It is a list of eight actions we can do to help make sure our interaction has a better outcome and make it more likely students will learn to be more independent.
SQWIGLES
The story
It was originally Nicholas’ idea to have something like this. He wanted something to help the staff choose what to do in the moment, and also to help them reflect on their actions and choose ways to improve. We noticed that our staff (and ourselves) needed something focused on actions rather than philosophies, because then it could be used on the fly to choose what to do. Telling staff they need to be “encouraging” or “socratic” is not all that helpful when they don’t know how to put it into action. Yet this is what many documents giving advice to tutors do. So we decided to focus on the actions instead.
Back in 2013, I was away for the weekend and thought hard about what actions we could focus on and scribbled a lot in my ideas book until I came up with an acronym I could work with, and SQWIGLES was born. Over the past three years it has been remarkably good at helping our staff make improvements in their one-on-one teaching quickly, because it does focus on actions.
(Do note that we also have regular discussions about philosophy too, and one overarching goal to draw everything together: “In every interaction the student learns something they can use independently in the future”. But these are not as useful without a guide for action.)
Here is what the acronym means, with further details below:
(UPDATE 2018: You can see video of me talking about SQWIGLES at Twitter Math Camp 2017 here: https://youtu.be/u3v_7G5D5xI )
S: SPEAK your thinking
Every time we read or write in front of the student, we must speak our thinking. In order to learn how to do this stuff themselves, they need to learn about how to think, and they can’t do that without us speaking your thinking. I have yet to meet a telepath!
So, when reading student working, we can tell them what we are looking for and what we see. When reading a question to figure out how to do it, we can read it aloud and then say what our first thoughts are, including the ideas for attacking it that we are rejecting. When writing a solution to a maths problem, we can speak the decisions we are making.
This may seem such a basic thing it hardly seems mentioning, but believe me it does need mentioning. Our very new staff often need reminding of this – they will do a maths problem in front of a student without speaking at all. Or they’ll sit silently staring at a new problem, clearly thinking deeply, but not letting the student know what they are thinking. Or they’ll just present an existing solution without explaining the thinking that created it. We all need reminding to speak our thinking.
Q: Ask open-ended QUESTIONS
This action has two parts. First, we should ask the students questions, rather than just tell them stuff. Them responding means them thinking, which is much better than just watching. Second, we should ask the students open-ended questions if we can. There’s a much better chance of thinking then.
For example, instead of saying, “The definition of continuity is…”, we could ask a question like “What is the definition of continuity?”, or even better ask an open-ended question like, “What do you know about continuity?” Instead of saying, “Here’s how to start this problem…”, we could ask a question like, “How could you start this problem?”, or even better ask a more open-ended question like, “What do you think about this problem?”
Open-ended questions allow the students to tell us what they are thinking, whether it is what we were expecting or not, and so we can change our teaching accordingly. They may even surprise us by solving their problems by themselves if we give them the freedom to respond that way!
Some great all-purpose open-ended questions are “What do you think about that?”, “How do you feel about that?” and “Can you tell me more?”
W: Encourage them to WRITE
Writing is a great way to record thinking, and a great way to help structure thinking. So it’s a great thing to encourage the students to write. Us writing is not going to have nearly as much of an impact on their learning as them writing!
Moreover, if the goal is to learn how to solve problems, then the students need to solve problems. Us writing things for them is not going to give them this experience. Plus, everything looks easy when someone else is doing it!
So encourage them to write. This is the one I have the greatest struggle with. I have to remind myself of this one constantly and use every opportunity to hand them the pencil.
I: Help them to find INFORMATION
Students need to learn how to find the information they need for themselves. One of the most important skills they can learn at university is how to find information, rather than the information itself per se. So, we can help them learn how to find information. Moreover, if we just tell them, all they might learn is that the quickest way to get information is to ask us!
For example, instead of telling students the definition, we can go and find the definition with them. If they don’t have lecture notes to look it up in, there’s always the textbook or the internet – learning how to search the internet effectively is an important skill to learn! If they want examples to inspire them, we can help them find a source of these examples. If they need instructions on the expectations for their assignment, we can help them find these instructions.
Another worthwhile thing to note is that even if the student has a problem that it’s not our job to solve, we can always help them find information. At the very least we can help them find out who is the person whose job it is to help!
G: GUIDE their problem-solving
This one is the most vague of the actions, but I really wanted to mention the problem-solving thing here! There are two ideas here: guiding is different from telling, and the problem-solving is theirs and not ours.
We can tell students what to do, and we can do problem-solving for them, but it won’t achieve them learning something they can do on their own in the future. The best we can offer is guidance. We can give them signposts to point the way, like “What do all these words mean?” and “How is this thing related to that thing?” We can give encouragement. We can help them delineate smaller steps they need to follow.
But we can’t do it for them. It’s their problem-solving and we can’t do it for them.
L: LISTEN to their thoughts and feelings
This is one of the most important things on the list. We need to listen to the students. We can’t figure out what they need without listening to them. We can’t figure out where they are so we can guide them further without listening to them. We figure out what information they need to find without listening to them. We can’t choose an alternative explanation without listening to them. And we can’t convince them we care without listening to them.
It’s also important to note that we’re not just listening to hear if they are correct or not – we’re listening to their thoughts and feelings. We really want to and need to know what they think and how they feel. These are so very important to their experience of learning and our decisions of how to help them learn, especially the feelings. Never underestimate the impact how they feel has on their learning, whether positive or negative feelings!
This is one of the reasons why open-ended questions are so very important – it gives us something to listen to!
E: EXPLAIN another way
At some point in our interactions with students, it is likely we will need to explain something. This letter is about not sticking to just one explanation or type of explanation. It is likely the student has heard at least one explanation and it hasn’t worked so far, so we need to be ready to try other ways to explain, especially if it was us who gave them the first explanation they didn’t understand!
We can use a proof, or just an example. We can use an analogy, or a drawing, or play-dough, or movement. We can colour-code an existing example, or line several examples up next to each other. We can ask another student to have a go explaining it to them. There are so many options to try and the key is to keep trying, and build our repertoire of new ways to explain, and be brave in trying something new.
There is also the issue of what exactly they need explaining, which might be different to what we are used to explaining. Sometimes they want the concept explained in all its mathematical detail, and sometimes they just want the general idea. Sometimes they need to feel philosophically comfortable with it, and sometimes they need to understand the mechanism of the procedure. Sometimes they need to know the acceptable ways to write it on paper. We can only know which thing they need an explanation of by asking questions and listening to the response.
S: Help them to SUM UP the learning
Most of the time, students come to us with assignment problems, and the only thing they will think about the interaction with us is that they got their problem solved. But this does not service the goal of them learning something they can do on their own. They probably did learn something, but if they don’t know they did, then how can they use it when they need to? So we can help them sum up what they learned today. At a conference once, they called this cognitive closure – a bit like grief closure, but for learning.
Summing up what they’ve learned is good, but helping them to sum up what they learned is even better. Asking them what they’ve learned here, or what they can use in the future is an excellent open-ended question that will often give pleasantly surprising results.
Conclusion
So there is SQWIGLES: a list of actions we use when we work with students one-on-one. A list of things to reflect on when we think about our one-on-one teaching. Try using them in your own teaching and reflection.
When doing algebra and solving equations, there is this move we often make which is usually called “doing the same thing to both sides”. Quite recently this phrase of “both sides” has begun to bother me.
In the Drop-In Centre, the majority of students visit to ask for help learning in a very small number of courses, mostly the first-year ones with “mathematics” in the title. Of course, any student from anywhere in the uni can visit to ask about maths relating to any course, and we do see them from everywhere, but the courses called Maths 1X have between them a couple of thousand students per semester and that’s a lot of people who might need help to learn how to learn.
Anyway, the upshot of this is that I help people with the same topics semester after semester, year after year. Sometimes people ask me this question:
Question:
“Do you get tired of the same topics?”
Short answer:
No.
Long answer:
I actually really love the topics in first year maths. Row operations and the fact that they help to solve equations and decide independence and find inverses are fascinating. Nutting out how to do an integral is a fun game. Eigenvalues are the Best Thing Ever. And don’t get me started on conics and quadrics. To me, seeing them every semester is like watching the Muppet Christmas Carol every December. I get to be reminded of a story I love, and notice something little I had never noticed before every time.
Also, it’s not just the topics I get to see each semester, it’s the students learning the topics. So many of them have a perfectly appropriate and successful way of understanding it that never occurred to me and these make the topics fresh again. Who ever thought of checking vectors are parallel by making sure that cos of the angle between them is 1 or -1? Not me until yesterday when a student did it.
And then finally, I get to be there at the moment everything clunks into place and see the light in their eyes as they feel the buzz of understanding it for the first time. And that never gets old!
