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”.
Tag: drop-in centre
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SQWIGLES: a guide for action and reflection in one-on-one teaching
It’s university holidays again (aka “non-lecture time”), which means I’m back on the blog trying to process everything that’s happened this term. Mostly this has been me spending time with students in the Drop-In Centre, since I made a commitment to do more of what I love, which is spending time with students in the Drop-In Centre. When trying to decide what to write about first, I realised that a lot of what I wanted to say wouldn’t make much sense without talking about SQWIGLES first. So that’s what this post is about.
SQWIGLES is an acronym that we use to help our staff (and ourselves) when teaching in the MLC Drop-In Centre. It is a list of eight actions we can do to help make sure our interaction has a better outcome and make it more likely students will learn to be more independent.
SQWIGLES
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.
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Brackets
I had a meeting with an international student in the MLC on Friday who has having a whole lot of language issues in her maths class.
She was from the USA.
Yes, the USA. Her problem wasn’t the everyday English; it was with the different terminologies for mathematical things here compared to her experience where she comes from. Her only experience with vectors was in physics where a vector is a quantity with magnitude and direction, whereas in maths class here, a vector is usually just a list of coordinates. She knew how to find a derivative, but had never heard the word “differentiate” used for that action. She had only ever used the word “anti-derivative” for what we call an “integral” most of the time.
I was talking through the transpose of a matrix (something else new to her) and how it interacts with other operations on matrices, and how since it’s normally written as a power it takes precedence over nearby multiplications like powers do. She asked me if, here in Australia, we still use the same order for what is supposed to be done before other things. I said, yes, we do, and told her that most local high-school students use the acronym BEDMAS to describe that order.
She wrote underneath the acronym familiar to her: PEMDAS. First she focussed on the fact that one had DM and the other MD, but reconciled that quickly saying, “Well I suppose they go in the order they come and so it doesn’t matter which way around they are.” But she had no idea what to do with the B.
I told her the B stood for brackets, and I drew what brackets look like: ( ), [ ], { }. And then she freaked out. To her, those things are called parentheses ( ), brackets [ ] and braces { }. I said, yes, those are the official titles of those things, but here in South Australia they’re thought of as different kinds of brackets. If we want to distinguish between them we’ll call them round brackets ( ), square brackets [ ] and curly brackets { }.
And suddenly a light came on for her and a whole lot of stuff people had said this week made sense. She also suddenly understood the very odd look her class tutor gave her when she mentioned the word parentheses. “Yes,” I responded. “Most maths students and tutors here would never have heard the word parentheses.”
And what happened next? Well we’ll have to wait and see. I think we made some excellent progress, and we agreed to keep meeting across the semester to help deal with anything more that might come up.
For me, I’m so glad I knew a little about the differences between Australian and American mathematical-English. (Thanks MTBoS!) And perhaps if anyone is reading this, then you will know too.
Australia USA ( ) round brackets parentheses [ ] square brackets brackets { } curly brackets braces PS: I find it interesting how the Australian acronym BEDMAS references a general term “brackets” which covers all shapes of bracket, whereas the American acronym PEMDAS references a specific term “parentheses” which only covers one shape of bracket.
These comments were left on the original blog post:
Claire 31 July 2016
This is very interesting. I teach high school in Southern California. When I teach the Order of Operations, my students have seen it before from middle school. They were either taught PEMDAS or Please Excuse My Dear Aunt Sally to learn the order. I always use these acronyms as a way to highlight the limitations of some of the “tricks” they use to memorize math concepts.
It’s interesting that the parenthesis, brackets, and braces are all called “brackets” where you are. Having 1 word for that is super helpful. I tend to say that they are all “grouping symbols” and if it were up to me, the acronym would be GEMDAS. Because then, for example, absolute value symbols are grouping symbols.
We also discuss why MD doesn’t matter the order… That they are inverse operations and part of a mathematical family. Same with AS.
This family idea is helpful to explain where roots and logs might go in the order of operations.
This post gave me a lot to think about.
Geoff Coates 1 August 2016
I helped an African student once who learned his high school maths in French. They used commas where we use decimal points and full stops where we use commas in large numbers. He was very confused for a while …
David Butler 2 August 2016
This came up in the Chemistry labs last semester, when all the pipettes had their volume listed as “0,25mL”.
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Do you get tired of the same topics?
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!
Short answer:
Hell no.
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Finding errors by asking how your answer is wrong
One of the most common situations we face in the MLC is when a student says, “I’m wrong, but I don’t know why”. They’ve done a fairly long calculation and put their answer into MapleTA, only to get the dreaded red cross, and they have no idea why it’s wrong and how to fix it. One of the major problems is that many students can’t tell if it’s because they’ve entered the syntax wrong, or done something wrong in their algebra, or completely misinterpreted the question, or if MapleTA itself has a bug and isn’t accepting the correct answer.
The other day, I was helping an Engineering Maths IIA student in exactly this situation. He was solving a differential equation and his answer was wrong, but he didn’t know why. As usual in this situation, I encouraged him to think of a way they could check his answer for himself (I commented on this a few years ago, actually Who tells you if you’re correct?). In this case, subbing the solution back into the original equation is a useful approach.
When he subbed his solution into the left-hand part of the equation, he got a result of -3/16 cos(1/4 t). Unfortunately, the right-hand part of the equation was -3 cos(1/4 t). So yes, his solution really was wrong. This left us with the much more difficult question of how to fix the error.
In a sudden flash of inspiration, I realised that the way that his solution was wrong might tell us something about the kind of error he had made. How could he have gotten -3/16 cos(1/4 t) instead of -3 cos(1/4 t) when he subbed into the equation? Perhaps because his solution was 1/16 of what it should be. I went looking for a 1/16 but couldn’t find one. Ok then, how could you produce a 1/16 in a less direct way? Perhaps you could divide by 4 twice. So this time I went back through his working looking for 4s. Like a moth I was attracted to this line in his working: “A/4 + B/4 = -3 ⇒ A + B = -3/4”. Of course! Dividing by 4 instead of multiplying by 4 would have the same effect as dividing by 4 twice, which could totally have produced that 1/16.
I was floored by the amazing effectiveness of this approach, and I wondered that I had never thought to do it before. It seems like such an obvious way to come up with something specific to look for. Admittedly it might not always yield useful results, but the evidence from this episode suggests that it might, which is certainly better than no strategy at all!
The student himself was suitably impressed and you could see him consciously committing the idea to memory for future reference. So now at least two people have a new strategy to find errors: when you sub your answer in to the original and it doesn’t work, investigate the way that your answer is wrong – it might help you find something specific to look for to find your error.
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Pretending not to know
Yesterday the Maths 1M students handed in an assignment question that asked them to prove a property of triangles using a vector-based argument. It’s not my job to help students do their assignment questions per se, but it is my job to help them learn skills to solve any future problem. This kind of problem, I find, is really hard to learn generalisable skills from.
For most students you really need to be there every step of the way as you try to solve the problem together, so that at the end you can look over what happened and figure out the sorts of things that made it possible to come up with a proof today. Students need to be hear the sorts of general self-questions you ask to help progress your thinking, even if they don’t lead anywhere straight away, and they need to see the dead-end paths you went down only to come back and go a different way.
The big problem is that if you’ve already seen the proof of this 15 times this week, it’s very very easy to guide students down a particular path that they could never possibly think of by themselves first go. It’s very easy to ask specific leading questions rather than general questions that might not lead anywhere. It’s very easy to push them away from the dead-end paths towards something that will give a result more quickly. You want to avoid doing that as much as possible, and the only way I know to do that is to pretend you haven’t seen the solution.
You’re going to have to pretend that you really don’t know how to do it and you really are just figuring it out with them today, and pretend to be surprised that something turned out nicely, and pretend to be frustrated when things don’t. It’s a real art and it takes a lot of practice and a lot of energy to pull it off.
I was very pleased the other day when I did pull it off. I was helping some students with this proof, and I said and did all the right things, including the dead-ends and everything.
After these students were happy with what we’d achieved and had a nice moral about problem-solving to take away, I turned to my other side to help the student who had been sitting there patiently. He had a whole different kind of proof to work on (mathematical induction), and I started as I often do by looking up the definition and writing that down, then saying “Now I’m not sure if this is going to help yet”. He responded to this by saying, “I don’t think I’ll ever believe you again when you say that.”
You see, I had helped him with the geometry proof only a couple of days before, and he had patiently sat there listening to the deja vu of me go through all the same things I went through with him. I looked him in the eye at that point and he said, “That was very impressive.” And he meant it. It’s nice when someone appreciates your craft.
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My cat’s bottom
Did you know that cats have scent glands just inside their bottoms that are constantly being filled with liquid and are squeezed as their poos come out, and if their poos are too skinny the glands are not squeezed enough and get over-full making them very painful and inflamed? Neither did I, until my cat Tabitha Brown started bleeding out of her bottom.
The vet literally reached into Tabitha’s bottom and squeezed the glands, making scent-liquid squirt all over his face (I have very great respect for vets, especially this one!). Then he told us some very useful things about how to look after our cat properly, such as the fact that many cats need Metamucil (yes, human Metamucil) every so often.
The point of this story is that we would never have known about how to look after our cat properly if she had never been in this pain. Because of her bleeding bottom, we now know for her, and for any future cats we might have in our home, to keep an eye on the size of their poos and give them Metamucil if the poos are consistently a bit skinny. All because of her bottom.
As always, my life makes me think about my teaching…
Students often say that I am a better teacher than their own lecturer, and it has become clear to me that the main reason this might be true is because I spend so much time with students. I am there when they have problems, and so I am able to see the problems students might have, and thus I am better equipped to help future students. There are so many things about the way people learn or don’t learn that I would never know if not for me being there when the students were in pain. Not that I wish anyone pain of course! I’m just saying their pain helps me prevent unnecessary pain for future students.
So thank you to all the students who were in pain and let me be there when they were. You make me more able to help others.
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Let’s
One of my friends and a past MLC staffer graduated from her PhD yesterday (congratulations Jo!). One of my strongest memories of Jo is when she told me something about my teaching that I never knew I was doing, but that she saw as an essential part of what I was trying to achieve at the MLC. That’s what I want to share with you today.
Jo noticed that there was a particular word I used rather often when working with students in the MLC. It was only a small word, but it was able to make a difference to the way students felt about our interactions.
The word was “Let’s”.
Very often, we tell students to do things that will help them learn, but usually those instructions come out as commands: “Find the definition in your notes”, “Read the original assignment instructions”, “Have a go at this first part of the question”, “Look at this handout”.
Many of the students who come to the MLC are in a fairly heightened state of stress; some of them really feel embarrassed that they had to ask for help; at least a few are just waiting for an excuse to pack up and run if they feel threatened. These students need reassurance that we really are committed to helping them today – that we’re willing to stick with them until they get it.
It turns out you can let them know you want to stick with them by simply adding a “let’s” to the front of your commands: “Let’s find the definition in your notes”, “Let’s read the original assignment instructions”, “Let’s have a go at this first part of the question”, “Let’s look at this handout”.
It’s amazing how one simple word (and the attitude that goes with it) can make such a huge difference to the experience a student can have of learning today.
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When will I ever use this?
“When will I ever use this?” is possibly a maths teacher’s most feared student question. It conjures up all sorts of unpleasant feelings: anger that students don’t see the wonder of the maths itself, sadness that they’ve come to expect maths is only worthwhile if it’s usable for something, fear that if we don’t respond right the students will lose faith in us, shame that we don’t actually know any applications of the maths, but mostly just a rising anxiety that we have to come up with a response to it right now.
There’s an interesting discussion in this pdf article [1] of the various responses that are commonly given to this question and their various drawbacks. The author is mainly concerned that we often inadvertently confirm the uselessness of maths by our very attempts to make it seem useful. While this is a legitimate concern, I have another one: in our attempts to justify the mathematics, we forget to listen to what the student actually needs.
In my experience, when a student asks this question, it’s a sign that they are starting to lose faith. They are having trouble motivating themselves and are seeking a reason to keep working at it. Being able to use it someday is the first thing they think of to motivate themselves, so they ask the question. But really most students will be happy with any reason that encourages them to stick at it today.
I had been thinking about this for a couple of days, after following a Twitter conversation and the comments on a post on Dan Meyer’s blog . Then one one of the students in the MLC actually asked the question, so I was all ready with my response. I said, “Actually, I’m not going to answer that question, but instead I have my own question to ask: how are you feeling about this topic right now?”
It is a testament to the trust I’ve built up with the students that he answered my question honestly! He said that he couldn’t see how the bits fit together or how they related to other things in the course. So I talked about how this topic fit in with the big ideas in maths, and how it connected with what they did last semester and last week. Then I helped him to organise some of the information in the topic so it was clearer how it was structured.
And you know what? After this discussion it didn’t matter so much that he might never use it. He had what he needed to have the courage to keep going, because I took the time to find out what was really bothering him.
[1] Otten, S. (2011) Cornered by the Real World: A Defense of Mathematics, Mathematics Teacher, 105-1, 20-25
Alexandre Borovik 27 April 2016:
It is like learning to swim: how many people actually have to use swimming for *practical* purposes?
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Really working together
Yesterday, I had one of those experiences in the MLC that makes me love my job.
The Maths 1B students were working on a linear algebra proof today, and as I came up to one of the tables, Fred (name changed) was explaining the beginning of his proof to the rest of the table. When I arrived at the table, he was leaning over two of the other students to point at a section of the lecture notes. He noticed I was standing there and said, “But David can probably do this better than me.” I responded, “Not necessarily. You keep going,” and I sat down in his chair.
Fred continued to explain, and I think he did a very good job. I was very pleased that he kept flicking through the lecture notes to point out different theorems, though I thought it was interesting that at no point did he write anything down.
Then one of the other students said, “so is that the end?” And Fred said that no, this was just the beginning, there was still more after that, and I could see in his eyes he was having a sinking feeling as he tried to think of how to move on to the next bit.
So I asked him if he could pass me a whiteboard marker. I stood up to the wall and said, “I just want to write down where we’re up to.” I asked the students he had been talking with to tell me what they’d done so far, and I transcribed it to the wall, asking them to explain why each line worked. And then we got up to the end of what they had already done.
“So what now?” I asked. There was a short silence, and then Fred piped up with a comment about what we needed to know next. I asked why that was important to know, and this started a discussion of what goal we were heading for.
And here is where the really great stuff happened. The students at the table offered suggestions of things to try, looked up definitions and theorems in their notes, helped each other refine their maths language, asked each other questions when they weren’t sure of things, welcomed new students into the discussion when they wandered over to listen, discussed how to make the proof their own when they wrote it to hand in, and basically really worked together to construct the proof. It was a pleasure to be a part of it.
It’s this sort of thing that makes my job such a joy – seeing students learning and supporting each other to succeed. On a day containing many other parts of my job that are much less joyful, it was something I really needed to see.
This comment was left on the original blog post:
Steven 8 February 2016:
Indeed, I also did enjoy reading your post regarding on how crucial and effective to have a group discussion i.e. working together. I’ll be sitting UMAT this year and I hope i can find someone/group as well to discuss on some UMAT questions and produce interesting results like the above.