Making Your Own Sense

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

Category: Reading

Reflections on books and articles I have read.

  • Book Reading: The Classroom Chef

    Over the weekend, I read “The Classroom Chef” by John Stevens and Matt Vaudrey. This is a post about my reaction to the book.

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

    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.

    This comment was left on the original blog post:

    John Stevens 20 November 2016:

    David,

    Thank you for this. Rather than dropping a big reply here, I blogged about it:
    http://www.fishing4tech.com/fishin-solo-blog/shoulders 

  • 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?

  • 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.

  • Research reading can of worms

    Today’s blog post is about my experience attempting to become better read in the area of education research, and I’m sorry to say I’m not going to be glowingly positive about it. As the title suggests, it just seems to get out of hand so quickly.

    Let me explain.

    The MLC’s job is to support all students in learning and using the maths they need or meet in their coursework. An important part of this job is to support the people teaching the coursework itself to do their teaching in ways that will most help students to learn.

    While I have many good ideas, I wouldn’t be doing my job properly or in a scholarly way if I didn’t check out what people already say about teaching. Moreover, there’s nothing like an academic to not take good advice unless it is backed up by peer-reviewed research!

    So I try to read education research literature about the courses and concepts the students I help are learning.

    And there is the first cause of the can of worms: the students I help come from all sorts of different disciplines and even within the one discipline they are learning all sorts of different concepts. Every day there is at least one new concept that I have to wonder about how it could be taught better. And so I have an ever-increasing list of things to look up in the education literature.

    Then, when I come to look up the education literature online, there are any number of papers which may or may not actually be about the concept I am interested in today. If they are related, then they usually introduce at least a few new terminologies or refer to other people’s work which I then need to look up. Alternatively, they aren’t related, but they are usually related to some other thing I am also interested in. So the list of papers I am interested in reading gets longer again.

    And then the final problem is that education research is not nice and neat but never fully or adequately answers the question, and usually leaves you with more questions than answers. (As I have discussed before: Frayed Research.) So the can of worms is fully open now and they are wriggling all over the place.

    I’m not sure how to deal with this problem. I may need to figure out a specific area of interest and just ignore everything else. (This is more-or-less what I did during my maths PhD.) It’ll be hard though, because I really am interested in a lot of different things, and I feel like I am letting people down by not looking into things as carefully as I would like.

    For now I’ll try to wrestle with the worms as they come. You’ll see a new category of post called “Education Research Reading”, where I talk about a paper or few that I have read and what I think about it. It may not be systematic or thematic, but I hope you’ll come along for the ride.

    (Don’t worry, though. You’ll still see the standard fare of object lessons, metaphors, teaching ideas and musings about the coolness of maths.)

    These comments were left on the original blog post:

    Sophie Karanicolas 5 December 2014:
    Dear David, don’t despair, there is some good stuff out there, they are just hard to find. We will find a good one for you to read! Have a great weekend.

    David Butler 5 December 2014:
    Thanks Sophie — but that’s part of the point. In some areas there is too much good stuff out there! Better than no good stuff I suppose…

    Maureen Coffey 16 December 2014:
    “… education research is not nice and neat but never fully or adequately answers the question …” Indeed, this is because pedagogy or more specifically didactics still lacks the underpinning of a scientific framework anyone can agree on. If nurtritionists wre split about the idea of whether intestines played a role and if chewing was truly necessary for digestion they’d probably be fired from faculty and rather be treated in mental homes. But if educators fail to accept research on how the brain functions and instead expound lofty theories they seem to still be admired … How would different subjects require different teaching if different foods do not require different “stomachs”?