Making Your Own Sense

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

Tag: teaching

  • Showing how to be wrong

    After writing the previous blog post (Finding errors by asking how your answer is wrong) and rereading one I wrote three years ago (Who tells you if you’re correct?), I got to thinking about how students are supposed to learn how to check if they are right.

    It occurred to me that, at least at university, we almost always show students how to be right, but almost never show them how to be wrong. We give them highly polished examples in lectures that proceed smoothly from the original information to the final answer, and then we move on. We very very rarely check our answers to see if they are correct, and even if we do check them, they are correct.

    So the students never see any examples of how to deal with the situation where they are wrong. Is it any wonder, then, that they don’t know to find and fix their own errors?

    If we’re going to give them examples in front of the class, I think we could spend a bit more time showing them how to check their work, and at least sometimes we should actually find errors that need fixing and fix them. Then they might actually see some strategies they can learn, rather than simply being lost when they’re wrong.

  • 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.

  • 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?

  • A constant multiplied on will stay there

    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.

    You can read the rest of this blog post in PDF form here. 

  • But I don’t like cricket

    When I was in primary school, one of my teachers once tried to teach us averages using cricket, and it is one of my strongest memories of being thoroughly confused in maths class.

    I’m pretty sure my teacher thought that using cricket to teach averages was a great idea, but (for me at least) it was a very bad idea, for three main reasons. First, I didn’t actually know the all rules of how cricket was scored. I had played cricket before, but this amounted to hitting when I was supposed to hit, running when I was supposed to run, and trying to catch when I was supposed to catch. I had never actually scored anything or been told how this was done. So all his discussion of average scores was basically meaningless to me. Second, there’s this technical detail in cricket batting averages that has to include “not out” somehow, which makes it not like normal averages. He spent most of his lesson discussing this detail and I ended up not knowing what a traditional average was, letalone a cricket average. Third, and most importantly, I didn’t like cricket. As an exercise-induced asthmatic, the running wasn’t pleasant. As someone with low coordination, I tended to be out pretty quickly as a batter, and so spend a lot of time just sitting on the bench. And as a fielder, well, the chance of actually interacting with the game as a fielder in primary-level cricket is quite low. So the mere mention of cricket turned me off. If cricket is what averages are for, then I really didn’t want to know about averages.

    And this story embodies the dangers of using “real life applications” to teach maths:

    • Students don’t know the context: If students aren’t familiar with the context of the application, the discussion will be meaningless to them, which often leaves you teaching the context itself rather than the maths.
    • The context is too complex: Most contexts are more complex than the thing you are trying to teach, and to deal with this complexity, you often cloud whatever it was you were trying to teach (or end up changing the context so much it doesn’t make sense any more).
    • Students might be turned off by the context: The application itself has a high chance of simply not being interesting to the students at hand, and they will transfer this disinterest to the maths.

    All three dangers are real and present in every classroom, especially the third one. Yet I have lost count of the number of people who have responded to the question of “how do I motivate my students to learn topic X” with “just tell them about application Y”. No-one seems to recognise the possibility of disengaging students by telling them about application Y.

    I’m not entirely sure what to do about it, unfortunately. If you have a group of students at university who are all studying the same degree (say Mechanical Engineering), then you have a good chance of picking an application they will be interested in, but even then almost always you have the second danger of complexity getting in the way. You could conceivably get the students themselves to seek out applications of the concept to things they personally are interested in, but some maths concepts simply aren’t used in varied enough places. And you could just show them a huge number of different applications so that they are sure to be interested in at least one of them (a linear algebra lecturer recently did this with eigenvalues). But of course, you yourself would have to know all these applications.

    In the end, I think we need be aware of the dangers so we can keep an eye out for students disengaging. Also, I think we need to make sure that the students are comfortable with the maths itself, and we need to be excited about the maths itself, whether we use a real-life application or not. Then the students who don’t like cricket might be able to be interested in just the maths.

  • Out-of-body teaching experience

    I have had a couple of new staff start in the MLC this semester. As part of the selection process they have to do a trial session in the Drop-In Centre, with me observing how they teach in order to give them feedback.

    Every time this happens, it has a very unusual effect on my own teaching in the Centre – I start having out-of-body experiences! I find myself watching myself as I’m teaching. I’ll be sitting there working with a student, and simultaneously watching and listening to what I’m doing. A constant undercurrent of questions is flowing beneath my words and actions: Are you really listening to what the students’ understanding is? Was that a good question to ask them? Why haven’t you gotten them to write this instead of you? Did you stop to check if they knew they learned something they can use on their own?

    In some ways, it’s disconcerting to have an experience like this – to feel so consciously aware of my teacher conscience as if it’s another person. But in other ways I like it. Most of the rest of the time, I only get to think about what I’m doing with students later when it’s too late to do anything about it (and I mentally kick myself), but when I have this self-awareness, I can change for the better while I’m still with the student.

    I wouldn’t wish it on anyone all the time, but I do wish I could more easily give others this sort of out-of-body experience sometimes, because it really is beneficial I think. Perhaps we should all spend more time observing other people teaching where we have the responsibility to give feedback on others’ words and actions. It might make us think about our own actions more.

    This comment was left on the original blog post:

    Lyron 4 September 2015:

    This happens to me quite rarely, but its invariably been a positive experience whenever it has — although I agree, I would not want it to happen all the time, but I would like if it happened more often. I have no idea how to induce such a state in myself though, it just… kinda happens, occasionally, for no apparent reason. Almost always only when I am in a good mood. 🙂

  • Obscuring the GST by making it simple

    I was helping out at Roseworthy Campus yesterday as the Vet Medicine students were learning about budgeting for a Vet Clinic as a business. One aspect of this was calculating the amount of the cost of goods and services that was GST (stands for “Goods and Services Tax” – in other countries it’s known as VAT or Sales Tax). The Excel sheet they were working in already had the formula worked in and it was this: GST = (Total Price)/11.

    You can read the rest of this blog post in PDF form here. 

  • Education research reading: effective feedback

    After warning months ago that there would be more posts about my research reading, but I didn’t follow through. Finally here is a “Research Reading” post. This one is about how feedback helps students learn. I’ll discuss several papers which list principles/challenges for providing effective feedback.

    Gibbs, G and Simpson, C (2004) The conditions under which assessment supports student learning, Learning and teaching in higher education, 1, 3-31

    In this paper, the authors put together 10 conditions under which assessment helps students to learn, as gleaned from the research literature at the time and their own experience with actual students. The point is that assessment does drive learning in the sense that many students won’t engage with a course unless there is some sort of assessment. However, assessment doesn’t always drive the sort of learning that you want, and sometimes actually prevents people from learning. The nature of the assessments themselves can affect the amount of study, the focus of the study and the quality of the study. Also, and more importantly, the nature of feedback on the assessments makes a huge difference to whether and what students learn. They collect together 10 conditions around these themes under which assessment helps students learn. (These are quoted verbatim from various pages across the paper, with my translations and paraphrases beneath):

    1. Sufficient assessed tasks are provided for students to capture sufficient study time
      Since students often don’t study unless there are assessed tasks to do, there need to be enough assessed tasks to make them study enough. One big one at the end will usually not be enough since they’ll only study nearby to it.
    2. These tasks are engaged with by students, orienting them to allocate appropriate amounts of time and effort to the most important aspects of the course.
      Students will glean what is important to learn from your assignments, so make sure the assignments allow them to engage with the most important things in the course.
    3. Tackling the assessed task engages students in productive learning activity of an appropriate kind.
      Many assessed tasks encourage students to do activities that either aren’t productive (like endless searching online) or aren’t appropriate.
    4. Sufficient feedback is provided, both often enough and in enough detail.
      Students need feedback often so they can use it to learn and improve. A numerical grade only, or a comment like “check solutions” are not enough detail!
    5. The feedback focuses on students’ performance, on their learning and on actions under the students’ control, rather than on the students themselves and on their characteristics.
      Too often we tell students about whether they are smart or lazy, especially when we do it face to face.
    6. The feedback is timely in that it is received by students while it still matters to them and in time for them to pay attention to further learning or receive further assistance.
      Feedback on Topic 1 after you’ve already moved onto Topic 2 is effectively useless. Not receiving feedback on Assignment 1 before they do Assignment 2 defeats the whole point of feedback!
    7. Feedback is appropriate to the purpose of the assignment and to its criteria for success.
      Too often we give feedback on things not actually listed in the assignment criteria, or which will not actually improve student marks in future.
    8. Feedback is appropriate, in relation to students’ understanding of what they are supposed to be doing.
      Students often don’t know what the assignment is for or what your expectations are. To say “give reasons” is meaningless if they thought they did, or if they didn’t realise that was part of the purpose! So sometimes feedback needs to tell them what the purpose actually is.
    9. Feedback is received and attended to.
      How you do this is tricky, but there is evidence to suggest that students will be more likely to read their feedback if you don’t put a grade on it.
    10. Feedback is acted upon by the student.
      The best-case scenario is if you let them fix up their assignment or do a followup task so they can actually use the feedback straight away.

    One thing I particularly like about this paper is its grounding in the experience of the actual student. Feedback is seen in the light of how the student responds to it and whether this response is producing the learning you and they hope for. This is an important perspective to hold on to when you are planning any teaching! I particularly like the idea that your feedback might be completely invalidated by the student’s own beliefs about what the purpose of the task is, and that therefore sometimes what they need is to be given feedback about what the task is actually for.

    Nicol, DJ and Macfarlane-Dick, D (2006) Formative assessment and self-regulated learning: a model and seven principles of good feedback practice, Studies in Higher Education, 31, 199-218

    Just as the title so clearly states, the author put forward a model of how students use feedback, and then list seven principles of good feedback.

    The big idea is that students already have their own internal feedback process. All external information, including our feedback to them, is processed through their existing understanding, their goals, their motivations and their beliefs, and then produces internal feedback on how to act. The key idea is that our feedback to them is processed in exactly the same way as any other external information — it has to be processed and turned into internal feedback before it produces action. When you think about it, this is pretty obvious, but it still sounds revolutionary!

    Their list of seven principles of effective feedback is very similar to Gibbs and Simpson’s paper above, but it is all presented through the lens of students learning to self-regulate. I’ll quote the list verbatim and put my translations and comments in between.

    1. Good feedback practice helps clarify what good performance is (goals, criteria, expected standards);
      Students already have their own thoughts about this, and need to a more accurate picture in order to evaluate their own performance. Moreover, the expectations for a task are usually rich and nuanced and so can’t just be expressed in a rubric or handout. The feedback helps to work through those nuances.
    2. Good feedback practice facilitates the development of self-assessment (reflection) in learning;
      We need to explicitly provide ways for students to reflect on their work, so that they practice the art of assessing their own work.
    3. Good feedback practice delivers high quality information to students about their learning;
      Quality is defined as helping students to take action to close the gap between their current standard and the goal.
    4. Good feedback practice encourages teacher and peer dialogue around learning;
      Like under 1, the dialogue helps to sort out the nuances in the expectations. It can be whole-class dialogue if there are logistical issues with talking to every student.
    5. Good feedback practice encourages positive motivational beliefs and self-esteem;
      In particular, it focuses on the growth rather than fixed model of intelligence and ability, because a fixed model has been shown to demotivate people.
    6. Good feedback practice provides opportunities to close the gap between current and desired performance;
      Tying in with number 3, it’s best if there is actually an opportunity to act on the advice given. For example, resubmitting work or using it for subsequent work.
    7. Good feedback practice provides information to teachers that can be used to help shape teaching.
      It’s best if the opportunity of giving feedback allows staff to change their own practices and learn from the students, so feedback is actually asked of the students too!

    I particularly like the continued focus in all of these on students learning how to manage the feedback process for themselves, which was mentioned as a condition for effective feedback by Gibbs and Simpson.

    Jonsson, A. (2013) Facilitating productive use of feedback in higher education, Active Learning in Higher Education, 14, 63-76

    This article is a review of research since 1990 into how students at university use feedback provided by teachers. About 100 studies were reviewed, mostly concerning student response to teachers’ comments on essays. Across all of them, there are many factors that might influence student use of feedback, but the authors identify five major themes common to most of the studies. They pitch them as challenges. Again, I’ll quote them verbatim, but with comments in between.

    1. Feedback needs to be useful.
      Here, “useful” means “able to be used”, funnily enough. If students are going to get the chance to resubmit the task, then they prefer the feedback to be about how to make this task itself better. If the feedback is on the final version of the task, then they prefer it to be about skills they can apply to future assignments.
    2. Students prefer specific, detailed and individualised feedback.
    3. Authoritative feedback is not productive.
      These two challenges are challenges because they work against each other. Students say they want lots of detailed individualised feedback. However, if there is a lot of detailed feedback, the students will often follow the instructions blindly, only making surface changes to the work in order to get incrementally higher grades. Indeed, feedback attached to grades will usually encourage students to use the feedback to guess how the grading was done, rather than to seek to improve qualitatively.
    4. Students may lack strategies for productive use of feedback.
      Students have many non-productive ways to use feedback: they might use it to tell them about their progress but do nothing to improve, they might simply delete the erroneous bit of their assignment, they might be motivated to “work harder” with no strategy for improvement. Basically, they need explicit guidance on how to use feedback to improve.
    5. Students may lack understanding of academic terminology and jargon.
      Students often don’t understand the terminology used to describe assessment criteria, or indeed the subject matter, which renders feedback meaningless. The authors suggest providing model answers with descriptions of why they are good/bad, and providing more opportunities to talk with students.

    The authors make the comment that much of the published research seems contradictory, basically meaning that the specific students, the specific teaching situation, and the specific discipline make a big difference to how feedback is used. They also note that almost all of the studies investigated student perception of feedback rather than asking them how they used it or observing them using it.

    Sadler, R. D. (1989) Formative assessment and the design of instructional systems, Instructional Science, 18, 119-144

    I didn’t actually read this paper, but it too has a list of conditions for feedback to be useful, and it was mentioned in all three of the above papers, so it seems incomplete to leave it out. Sadler lists three things that need to happen for students to close the gap between their current performance and the goal or expectation (this is my paraphrase):

    1. The student must know what standards they are aiming for
    2. The student must be able to assess their current performance in relation to the standards
    3. The student must have strategies to modify their performance

    What I find interesting about this list is that the success of feedback rests squarely on the skills of the student, which means the traditional method of telling students where they went wrong only has a chance of affecting the second point, and even then doesn’t help the student learn how to self-assess!

    Summary

    So, we have lists of 3, 5, 7 and 10 conditions under which feedback is useful for learning, with any number of specific recommendations. What do we make of all of it? Well it seems there are two main ideas. The first is that the feedback needs to be practically useable – it has to refer to things they can achieve, in a way that they can act on, and with opportunities to act on it. The second is that students need support to use feedback – they don’t know what assessment is for, or what we are looking for when we give them assessment, so we need to help them learn that. Also, interpreting feedback and putting it into action are specific skills that actually need specific training.

  • Sleeping through Miss Marple

    My wife and I like to watch mystery shows together like Poirot, Midsomer Murders and Miss Marple. Unfortunately I have a slight problem: when watching television in a comfortable position, I tend to drift in and out of sleep, no matter how interesting the show might be. This can be quite disasterous for mystery shows, especially ones with major unexpected plot twists.

    Just yesterday we were watching an episode of Marple called The Pale Horse and I woke up from a doze at the scene where everyone was gathered in the dining room to reveal the killer. And I had not the slightest clue what was going on.

    Later I went back to see the bits I had missed and it turned out I had missed a total of about three minutes of viewing in small snippets, but these were precisely the key moments I needed to be able to follow that final revelation.

    This morning it occurs to me that some students I have helped in the MLC have been in a similar position with their maths courses. A particularly common example is in Maths 1A too with the word “span”. It is mentioned in passing in one of the early Algebra lectures, with the discussion lasting for a total of about a minute, and it doesn’t seem related to the content of the course at the time. But then later it becomes one of the most important ideas and is talked about as if they already know what it is. If they “slept through” the first mention, they’d be most confused! It happens in our own bridging course too, with the idea of a “unit vector”. It’s mentioned on precisely two pages in our course materials, and is very easy to miss. Students almost always completely ignore the word in their assignment and then struggle to get what we are asking them to do. (This is one of the reasons we hope to rewrite the resources in future.)

    As teachers, we need to remember that a maths course is not like a murder mystery. In a murder mystery it’s the fact that working out the case hinges on small details that makes it mysterious and fun. But maths courses don’t hinge upon small details, they hinge upon big ideas. We need to make sure that anything pivotal is mentioned several times and discussed deeply so that even if their attention wanders for a minute once, they can still pick it up again and still follow the story.

  • The Fear of Mollycoddling

    Recently I was a guest at a planning meeting for a certain school and ended up in a session where we discussed how we can better support students in terms of their wellbeing. We were shown a news report highlighting the fact that the suicide rate in professionals of this particular discipline is four times higher than the general population. One of the major factors mentioned in the news report was that professionals in this discipline are very unlikely to seek support from anyone when they are struggling, having been trained too well to be self-sufficient while they were students.

    Later, we discussed in small groups ways to support student wellbeing, especially with regard to helping them develop skills they can take forward into their professional lives, such as time-management. It was a heartwarming thing to see academics so concerned with the wellbeing of their students.

    However, a small number of people expressed concern that giving students support might be “mollycoddling” them. They worried that students wouldn’t learn the coping skills needed to deal with demanding professional lives if they were given support. I do agree that it is important for students to learn those coping skills, but I am not sure that it is entirely healthy to respond to the fear of mollycoddling by not giving support.

    Some lecturers I have met in other disciplines in the past seem to think that by refusing to give support they are doing the students good. They seem to think that if a student asks for support they are just lazy and expecting the lecturer to do it for them. For example, I talked to a student who asked a lecturer about one of the expectations for an assignment, but the lecturer refused to give the information and just said “it’s in the handbook”. Another student was confused by an assignment question and the lecturer said something like, “Well if you had done any preparation you would know what the definitions of those concepts were”.

    But most students are not lazy most of the time. The first student in the previous paragraph had showed me the handbook itself and the information in it was contradictory, and I encouraged them to seek help from the lecturer. The second student had been in the MLC every day for the previous week going over all the resources at their disposal before they asked the lecturer. They were anything but lazy.

    Admittedly there are a small number of students who actually are lazy, but you just can’t make the assumption that your student today is one of them. There is no telling whether a student is asking because they always hope someone will do it for them, or because they’ve tried everything they can themselves now they need your help.

    My greatest fear is not that I might be mollycoddling a lazy student. No, my greatest fear is that I might be teaching someone that it is wrong to seek support. This was one of the identified causes of the high suicide rate originally: a lack of ability to seek support. It seems to me that for some students, receiving support today could mean the difference between life and death.

    These comments were left on the original blog post:

    Fiona Brammy 17 December 2014:
    David, what a read! I am a little shocked that we have staff who don’t feel that support is fundamental.

    Peter Murdoch 17 December 2014:
    Sometimes I wonder if what appears as laziness is simply the fact that the right domino hasn’t been triggered by the right support so that comprehension falls into place. Surely all educational activities, in classes or not, ought to be designed to support students to find understanding and through this the confidence to support what matters in the world around them.

    David Butler 17 December 2014:
    I like that way of saying it Peter. They might actually be working very hard, just not in a way that makes it possible for them to understand or complete the task they have been given.