Reflections on maths, learning and maths learning support, by David K Butler
Teaching resources I have made, and reflections on how they were created.
This blog post is about the Solving Problems poster that has been on the MLC wall for more than ten years in one form or another.
This blog post is about a metaphor I use when I think about the order of operations: the idea that the various operations are stickier than the others, holding the numbers around them together more or less strongly.
The titles of the five blog posts are:
I don’t like BODMAS/BEDMAS/PEMDAS/GEMS/GEMA and all of the variations on this theme. I much prefer to use something else, which I have this week decided to call “The Operation Tower”.
The titles of the five blog posts are:
Sometime in the past, I was approached by academics in the Faculty of Arts to discuss the numeracy skills of the students in their faculty. They wanted to discuss how they might include numeracy skills in some of their courses across all the degrees they teach. It was a lot bigger than the MLC could reasonably do, but I said I would certainly be able to do a small thing in a few courses, and certainly help their students in the MLC itself when they came to talk.
Then in January 2019, almost out of the blue, I was sitting down at a meeting with the Faculty of Arts Associate Dean Learning & Teaching, and the course coordinator for their core first year course called “The Enquiring Mind”. We were talking about how I might run a workshop for their students to introduce the importance to numerical skills for Arts students. We agreed on Week 4 of semester, and then I walked away into the Summer School exam period and O’Week and the crazy beginning-of-semester rush, with ideas percolating in the back of my mind for what I could possibly do in an hour.
You can read the rest of this blog post in PDF form here.
Also, I thought I would collect together the resources if you want a closer look at them.
Thanks for reading.
A lot of people introduce the derivative at a point as the slope of the tangent at that point, which to me is quite confusing. It seems to me that the reason we want the derivative is that it is a measure of the slope of our actual function at that point, not the slope of a completely different thing. To me, the thing is that the function itself is pretty much straight if we are close enough to it, so when we’re looking really close, saying it has a slope at this point is a meaningful thing to say.
For years I got at this idea by drawing a little magnifying glass on my function, and a big version nearby to show what you could see through the magnifying glass. But last year I decided to get technology to help me do this dynamically and I made a Desmos graph with a zooming view window that will show you the view on a tiny circle around a point on a curve. (Some of the bells and whistles here were provided with a little help from Andrew Knauft .)
Here’s what the graph looks like, and a link to the graph so you can play with it yourself: https://www.desmos.com/calculator/pa1cudpc07
I’ve used this to help students see that the curve itself really does look straight when you’re very close, and so treating it like it is straight there and saying it has a slope is a reasonable thing to do. Students are usually most impressed and love to play with it. It’s also interesting to put in a function with a kink in it like |x2-1| to show that the kink never goes away no matter how closely you view the function, so having a single slope there isn’t really reasonable.
Every time I’ve had to search my own twitter account to find the tweet where I shared it , and I couldn’t keep doing that, so here it is in a quick blog post for posterity.
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.
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 )
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.
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?”
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.
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!
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.
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!
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.
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.
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.
I have always loved the naming of quadrilaterals, right from when I first heard about it in high school. I’m not entirely sure why, but some of it has to do with the nested nature of the definitions – I like that a square is a kind of rectangle and a rectangle is a kind of parallelogram.
Most of the time you’ll see this classification organised in a sort of “family tree”. Like this:
(Warning: if you search for quadrilateral family tree in Google Images, you’ll get all sorts of things, many of which use American terminology, which is different to that here in Australia, and many others are just plain wrong.)
Of course, almost all people I meet are not aware of these nested definitions. Most students I work with are bamboozled when the answer to the problem of the largest area rectangular garden of fixed perimeter gives them a square. To them, a square is not a rectangle. Then again, there are others who aren’t aware that orientation is not one of the factors mathematicians use to name shapes. Just the other day someone asked me why a shape was called a diamond if it’s this way and a parallelogram if it’s this way. (That was a most interesting conversation about language and maths and maths language.)
Partly I think this is because the only examples we give of rectangles are non-squares — we never include squares in a collection of rectangles. Also, almost all examples we give of these things have their edges parallel to the top and bottom of the page. The casual viewer comes to believe that the orientation is part of the defining properties. A serious case of Beware of the Toast.
For a long time, I’ve wanted to make some posters showing collections of quadrilaterals including ones not parallel to the page edges and including the more special ones further down the tree. And yesterday I just decided that I would finally do it.
Here’s a picture of the majority of them. (You can find them in pdf form here .)
And finally, an upgraded family tree, showing heaps of examples of each type of quadrilateral. (You can find this in pdf form here .)
PS: If you want the original svg files in case you want to edit them (for example, to make them into local terminology), just ask.
This comment was left on the original blog post.
Anne Kelly 30 August 2016:
Hi David
Many thanks for putting together and sharing the Quadrilateral Family Tree. I plan to share it with the classes I teach.
Anne Kelly
Geographe Primary
WA
There are two terminologies in probability which many students are confused about: “independent” and “disjoint”. The other day I was working with a student listening to their thinking on this and I suddenly realised why.
You can read the rest of this blog post in PDF form here.
The above post has a special diagram to explain independence, which you can download in various formats:
At the end of last year, the MTBoS (Math(s) Twitter Blog-o-Sphere) introduced me to this very interesting task: you have a cross made of four equal squares, and you are supposed to colour in exactly 1/4 of the cross and justify why you know it’s a quarter. I call it “Quarter the Cross”.
You can read the rest of this blog post, and four other related posts, in PDF form here
The titles of the five posts in the series are:
Some resources linked from this post:
One of the most fundamental properties of the integral is that multiplying by a constant before doing the integral is the same as doing the integral and then multiplying by a constant. However, the way it’s presented here makes it look like a rule for algebraic manipulation – I can move a constant multiple in and out of the integral sign. I do actually use it this way when I want to do algebraic manipulation – it comes in handy when I’m creating a reduction formula, for example. But most of the time when I do an integral, I don’t use it that way at all.