Making Your Own Sense

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

Tag: study skills

  • I did it my way

    It’s Second Semester Orientation at Adelaide University right now, and last week I helped out at a session called Starting Strong for new students. One part of that session got the students to discuss procrastination, and I told the students that one thing that causes procrastination for me is being told to do a task via a process that doesn’t work for me. I spend days and sometimes months trying to bring myself to do it their way and only when I say, “Stuff it I’m doing it my way,” does the task actually get done.

    For example, when I wrote the Maths Foundations course, the project managers wanted me to tell them from the outset what the topic titles were and not to worry about the more finely-grained ideas and individual pages until later. And I agonised for ages trying to figure it out. But then I realised that until I decided what the ideas and pages were, I wouldn’t know how to organise them into topics or to give titles to those topics. So in the end that’s what I did. I brainstormed everything and then organised it. And it worked out better because then I had done the second part of what they wanted too.

    For another example, people sometimes give me pre-prepared slides for presentations with pre-prepared sections and I struggle for a long time to try to think about my presentation in terms of the section titles they give. Not until I completely ignore the slides and attempt to speak out the talk without slides do I have a chance of making slides. Or indeed making the decision of whether I want slides at all.

    On that same theme, I remember in high school my teachers made a huge deal of making note cards for presentations and even asked us to hand them in, and it actually prevented me from preparing effectively at all. I find cards very difficult to use because I can’t see what’s coming in the presentation and how it’s related to what came before. Knowing I had to make cards stressed me out and stopped me from planning the talk because I was worried about them. (Now I write notes on an A4 sheet of paper and refer to the sheet during the presentation, because I can see the whole presentation at once and can tell spatially where I am up to.)

    Another example: Many resources about problem-solving or assignment planning include advice to formulate a plan or break a problem into smaller steps. I have seen many students paralysed by this advice, because they can’t see a path all the way to the end and so can’t break the path into smaller steps. Because they can’t see how to do the entire thing, they do nothing at all. Much more productive advice is to find any smaller part and do that, and then decide what to do again after you’ve done that. (Yes I know there are many problems that really do have to be planned out from the start, but there are many many that don’t and even if you find yours does have to be planned out that way, just doing anything will usually help you realise more planning is needed.)

    I’m sure there are lots more examples. There are two things from opposite perspectives that I hope people take away from this:

    1. If you find yourself avoiding a task, consider whether one reason you’re avoiding it is because you were told to approach it in a certain way, and consider approaching it in a different way.
    2. If you give tasks to people, try your hardest not to enforce a specific order to do it or a specific way to organise things, even if you think it’s helpful to do so. It may be the opposite of helpful.

  • Three hours in the MLC Drop-In Centre

    Last week, I had one of those days in the MLC Drop-In Centre where I was hyper-aware of what I was doing as I was talking with students and by the end I was overwhelmed by the sheer volume of things I had thought about. I decided that today I might attempt to process (or at least list) some of it for posterity.

    A Real Analysis student was worried about a specific part of his proof where he wanted to show that a function was increasing. The function was a kind of step function with more and more frequent steps as it approached x=1. It was blindingly obvious from our sketch of the graph that it was increasing, but every time we tried to come up with a rigorous argument we just ended up saying something equivalent to “it is because it is”. I remember thinking about whether it was just obvious enough to just say it without proof, but shared the student’s desire to make the nice clean argument. I don’t remember now if he ended up with a neat argument or not, but I do remember finishing with the moral that obvious things are often the hardest things to prove.

    Before we even got to this frustrating little bit of the proof, I was reading through what he had done already, and I noticed that he had written “x is < 1” and commented that this wasn’t a grammatical sentence because the “<” contains an “is” already – it’s “is less than”, which makes his sentence “x is is less than”. This led to quite a discussion about the grammar of “<“, which is complicated by the fact that it can be read aloud in multiple ways depending on context. We investigated some other sentences containing < or > he had seen written before to see how it was pronounced there. In my head I was reminding myself to pronounce written maths aloud more often when I’m with students so they can learn the correspondence between speech and writing in maths.

    I was called over to help a Quantitative Methods in Education student with her research assignment because she had mentioned she was struggling with statistics (and I’m the default stats support person). Eventually I helped her figure out that her problem wasn’t statistics at all, but rather that she didn’t understand the wider context of the research, or what the data actually represented, or what the actual goal was. Mainly this came about by me not understanding the context, data and goal and kept asking her questions about those things in order to try to figure out where the statistics fit into what she was trying to do. We talked about how she might go about gaining the understanding she needed in order to come up with the specific-enough questions about variables that statistics would be able to help her to answer. I could see her heart sinking, so I reassured her I would still be here next week to talk about the statistics should she need that support later, but also from our discussion that she would probably be okay with that part when it came.

    I returned to the table I came from to discover even more statistics, this time a statistics theory course in second year of the maths degree. This student was struggling to come up with a linear regression proof which was pitched in the assignment using linear algebra. For this small part of the proof, he had to show that the columns of some matrix were linearly independent. I did what I always do with linear algebra proofs and helped him remember various definitions and facts relating to the vocabulary words in the problem, then we chose a representation of linearly independent from the list we had made. He commented that I go about proofs in a very different way to him, and I said I was merely following the problem-solving advice I had already put up in that big poster on the wall, see? In the end we finished with the moral that coming up with connections between things is a great way to solve problems and to study, even if it means ignoring the goal for a moment.

    I saw a student on the other side of the room that I had talked to months ago and I went to see how she was doing. When I saw her last, she was questioning her decision to study Statistical Practice I and indeed her whole decision to come to university at all, while simultaneously worrying about letting down her son who was the one who had urged her to come to university. Now she revealed she was systematically working her way through the course content so far, making sure she understood everything before classes returned and the last set of new content arrived. She asked if that seemed like a good plan and I said I was so pleased to see her making plans for her study and persevering with her course. I suggested she do some work soon to connect together the ideas in the different topics, since it’s the connections between ideas that will create understanding. I recommended a mind-mapping idea I had seen on Twitter recently (thanks Lisa).

    At the next table I came to, some students and one of my staff were having a discussion about function notation, in order to help a student in our bridging course. They were talking about how unfortunate it is that something like f(x) could be interpreted as multiplication in some contexts and function action in another. I agreed that it was unfortunate, but it was just the way maths language has come to work and it was too late to change it now. I related this to how the word “tear” is interpreted differently depending on context, which seemed to help everyone accept the fact, but not necessarily be happy about it! My staff member brought up how it’s made more complicated by the fact that we then write “y=f(x)” or even “y(x)=…”. I flippantly said that this was making a connection between graphs and functions, which is a whole separate discussion. Of course everyone wanted to hear more about this and suddenly I was giving a little seminar on the fly about function notation and graphs and the connection between them. I was keenly aware of everyone watching me, and of the choices I had to make with my words, especially when I made some mistakes along the way. Here’s more-or-less what I said [with some of my thoughts in square brackets]:

    There are many ways to think about functions, but one of them is that a function takes numbers and for each one it produces another number. [I thought about saying it doesn’t have to be numbers, but decided that was just going to distract from the discussion today.] It could be a formula that tells you how to get this other number, or there could just be a big list that says which ones produce which ones. For example, there’s a function which takes 1 and gives you 4, takes 2 and gives you 5, takes 3 and gives you 7, takes 5 and gives you 8. It’s the one that adds 3 to every number. The brackets notation is supposed to highlight that there’s a starting number and the function is a thing that acts on it to produce a result. So we could write something like “addthree(2)=5”. [I’m not sure what motivated me to write this. I’m sure I’d seen someone talk about this on Twitter somewhere, but anyway it seemed right at the time.] Sometimes we want to talk about the function as a thing in its own right so we give it a letter-name like “f” (for function) so that we can talk about it more easily. Or we only have some information about it but don’t know what it really is, so we can give it a letter name until we know its proper name. [I was reminded at this point about a Rudyard Kipling story featuring the origin of armadillos but chose not to mention it. I also thought here that I could have done another example earlier of a function acting like lookupthelist(2), but we’d moved further than that and I didn’t want to go back.]

    And what about the connection to the graph? Well you could draw a nice number line and write down what result each number produces. [I started drawing pictures at this point.] You could visualise this by drawing a line of a length to match the result attached to each number on the line. [I accidentally drew the lines representing the input numbers when I drew my picture, and had to go back and change them later. No-one seemed to mind that much.] You could have a ruler to measure how long each of these is when you needed it. Or you could attach the ruler to the edge of the page and line up the drawn lines with it. And now suddenly you’re locating the end of each drawn line by numbers on two axes. This is coincidentally just like the coordinate grid which is something else we’ve learned about before but not directly connected to functions. In the coordinate grid, the y refers to this second number and the x refers to this first number and you can describe shapes by how the x and y are related to each other. It doesn’t have to be y = formula in x, but that is a useful way to do it if you created your shape from a function.

    I think it was at about this point that I asked the students how they felt about this. My staff member said he quite liked the “addthree(x)” thing, which he’d never seen before. The bridging course student said that really helped her too, as well as realising that there were two different things happening at once when you write y=f(x). I decided to move on at this point.

    The next student was studying Maths 1B and was doing a deceptively simple MapleTA problem which he was stuck on. It listed an open interval – I think his was (-4,5) – and asked him to give a quadratic function with no maximum on this interval, a quadratic function with neither a maximum nor minimum on this interval, and a cubic function with both a maximum and a minimum on this interval. He had answers entered for the first two questions, which MapleTA had marked correct, but he wanted to know why. He revealed a little later he had gotten these answers off a friend and that now he wanted to know how to get them himself. I’m always glad when students do this, because it’s the first step to them really understanding themselves. With the first question, we discussed what it meant to be a maximum, and how that definition played out on an open interval. Once we had done this, I decided that I should take my own advice and make the question more playful, more exploratory. So I asked him what he would need to put in in order to make the answer wrong. He was really intrigued and started thinking through what the function would have to look like in order to have a maximum. After a while, he hit upon the idea that the leading coefficient could be negative, and this is all it would take. This was followed by several attempts to test his theory by putting in bigger and bigger constant terms in order to see if they really didn’t make a difference. They didn’t, and he was most impressed with himself.

    In the second part, we talked about moving the function around until the minimum wasn’t part of the domain. I went to get a plastic sleeve and drew a parabola on it in whiteboard marker, then asked him where he could move it to get what he wanted. At first he rotated it. I congratulated him for thinking creatively but did point out that it’s not the graph of a function like it was before. After that he only moved it up and down. Finally I gently moved it a tiny bit to the left, and he caught the idea that he could move it sideways, which he did until the turning point was outside the domain. The issue then came with how to change the function to do this, and we discussed why his friend’s solution was able to achieve it. Again he went playful and started putting in really big numbers and numbers really close to the boundaries to confirm that yes really he could move it anywhere outside that domain.

    Now we moved on to the third part and we discussed how cubics look and tried to figure out how to make the two turning points higher or lower than the end points. He wanted to use derivatives, so I let him do that, and it was quite a long journey, though he learned a lot about derivatives and specifying function formulas along the way.

    At the end, he asked if he had done all of this the correct way. I replied that it was definitely a correct way. He took this to mean there was a better way, and I said there was possibly a faster way. This meant a further long discussion about specifying zeros of functions and talking about whether we needed them to be zeros or if the function could be higher up or lower down. When we had finished this he asked me to explain the steps of this method again so he could write them down. I said that he didn’t need to remember the method because he’s never going to see this question again. The point is not to remember methods of solving all questions, but to learn something, and look how much stuff you learned today!

    Even though it was very close to my home time, I decided to talk to one more student. This one was doing a course specifically designed for the “advanced” maths degree students. He had been set a problem in Bayesian statistical theory. This is not familiar to me, so he spent a lot of time explaining what was going on to me. There were a whole lot of things in his working that I thought could be solved by simply referring to facts he had learned in previous courses, and I asked if he had done those courses. He revealed that they were in fact prerequisites for being allowed to do this course, at which point I said it was okay to use results from prerequisite courses. The biggest problem at the end was that he had assumed he had to integrate the normal distribution density function by hand, because the lecturer had gone to the trouble of giving its formula in the assignment. I assured him that it was in fact impossible to find areas under that distribution by hand, and he was allowed to use a table or technology. Sometimes you just have to tell a student what’s impossible so they can do it another way.

    So that was my day. I had three hours in the MLC and talked through so many ideas about learning and studying and maths. I had to make so many decisions about my words and actions and teaching tools. And I was left with a lot to think about. I hope you’ve found it interesting too.

  • 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.

  • Past Exam Vision

    Students have just been told their exam results for Semester 1, and some of them are facing replacement exams. So we’ll be trotting out our standard suite of exam advice again, which will be all the more poignant now because these people tried to do it last time and failed!

    One piece of advice we give is not to use past exams as your main study tool. So many students study for their exams by taking a stack of past exams and systematically working their way through each question and making sure they can do all the intimate details. This is a bad idea for several reasons. I’ll list some in dot point form:

    1. The course may have changed over time, so some of the questions will not be relevant to your course content anymore, while still other questions just won’t have appeared in past exams.
    2. The lecturer probably changed, so the style of the exam questions may be quite different to the exam you are about to do.
    3. No one exam can cover every concept in a whole course, and even several exams will miss something between them.
    4. Lecturers are not stupid, and so will generally always put something in that has not been done in an exam for the past several years, in much the same way that they don’t use yesterday’s questions today on a TV quiz show!
    5. You need to save at least a couple exams to do as proper timed exams in exam conditions or you won’t practice the skill of doing exams in exam conditions.

    ​​​But there is one more reason I myself had never really known fully until this last semester. It’s related to point number 3 above, but it’s even more pernicious:

    1. Questions in past exams are often cut-down versions of full problems designed specially to be dealt with in exams, and so will not necessarily help you actually understand the material.

    Let me explain how I fell into the trap of this peculiar kind of “Exam Tunnel Vision”.

    I never studied Differential Equations in a formal course as part of my degree. I managed to avoid all applied maths beyond first year by instead studying statistics, pure maths and Chinese. This means that pretty much everything I know about differential equations has been learned while helping students in the MLC. I have learned a remarkable amount, but there is a problem with my approach: I only see the parts of the course that students ask me about. And since students often study using past exams, the parts of the course I see do not necessarily represent the full picture. Now I do know full well that I should ask questions like “What would happen if it were this way instead?” and “Is there more stuff related to this?” and “Where does this fit in the bigger picture?” and indeed I do ask these things, but sometimes no matter how hard you ask, sometimes you can’t find this information without asking an expert.

    Case in point is the Frobenius method for solving differential equations. What happens is you are supposed to make an indicial equation, which for second-order equations will give you two solutions for r. Then for each value of r, you are supposed to do a process to find a solution. But here’s the catch: this final process is quite long, and so in exams and assignments the lecturer only ever asks students to do one of the solutions. Since my learning about differential equations was based entirely on helping students, I had never seen what you were supposed to do after this point! No-one ever asked, so I didn’t know.

    I had fallen into the very trap I warn students about: I had developed “Past Exam Vision” and couldn’t see beyond the exam to get the full understanding. In future I’ll be more careful, and now I have a good story to tell them to warn them about it. If I can fall into the trap, then anyone can!

  • Essay outlines, not plot summaries

    The Writing Centre put something on Facebook today about how to organise an essay and I’d like to quote something from the link they put up:

    Though there are no easy formulas for generating an outline, you can avoid one of the most common pitfalls in student papers by remembering this simple principle: the structure of an essay should not be determined by the structure of its source material. For example, an essay on an historical period should not necessarily follow the chronology of events from that period. Similarly, a well-constructed essay about a literary work does not usually progress in parallel with the plot. Your obligation is to advance your argument, not to reproduce the plot.

    From “Organizing an Essay “, by Jerry Plotnik, University College Writing Centre, University of Toronto

    While reading this, suddenly something about my own marks for essays in Year 12 made a whole lot more sense. How dearly I wish someone had said this to me when I was in high school!

    But of course, while you may sypmathise with my regrets, you may also be wondering why I am talking about essays when this blog is about the learning of maths… Well it occurred to me that the above quote applies to the writing about anything really, including your notes about your maths course.

    Some people when they study for a maths exam will start at the first lecture and proceed to write down everything they were told in the order they were told it. They make their official cheat sheet for the exam and it has headings “Lecture 1”, “Lecture 2” etc. These people invariably find that they don’t do so well in their exams, and I never found a decent way of explaining why it doesn’t work. It’s because they are basically writing a plot summary of their lectures!

    What they should be doing is writing an essay outline. In an essay, you are not just repeating what you saw, but synthesising it into new knowledge. Similarly, when you are studying a course, you should be reorganising the content into a a structure that makes logical sense, and where the connections between things are clear. This is not necessarily the same as the order it was taught in. (Indeed, sometimes you can’t teach things in this order because some things have to be learned before other things to fit with how your brain works, but the logical strucure works the other way around.) There’s no point simply repeating the things in the way you saw them – you’ve seen that already in the lectures themselves. Instead, you have to write something that shows your understanding of the content. You’re writing an essay outline, not a plot summary.

    Maybe next time I see a student studying in an unhelpful way I will tell them they’re not supposed to write a plot summary, and I’ll give them an essay question instead: “Discuss the logical structure of the ideas in this course.”

  • Can I take a cheat sheet?

    The first maths exams for the year are tomorrow, so recently I’ve been talking to more and more students about exams. To be clear, I’m not complaining about this! It’s a really important part of the MLC’s role to give students advice about exams, since they have such a huge impact on the students’ experience of learning maths at uni. We can make a big difference to people by simply helping them cope with this stressful time.

    Anyway, one question that keeps cropping up is, “Can we take a cheat sheet into our exam?”, and the answer for the regular maths courses here at Uni of Adelaide is no. I’m not complaining about this either, because I believe that the process of memorising things strengthens connections in your brain that you will need for problem-solving. Moreover it’s easier to figure out how to do new things if you have a good stock of old things you are really familiar with.

    I’m not even complaining about the fact that students ask the question in the week before the exam. It’s perfectly reasonable to start looking around for ways to reduce your stress when you are in a stressful situation, even if your hope for relief is based on the slim chance that you just happened to miss a vital piece of exam administrative information.

    No, what I am complaining about is this: no-one seems to be teaching students skills to help them remember things while they’re teaching the maths! It seems obvious to me that if you expect students to remember things, you should support them in learning how to remember things. This is especially true if you know full well that they don’t have these skills already because the majority of them were allowed to take cheat sheets into all their exams in high school!

    Of course, it doesn’t change what my response is to the “Can I take a cheat sheet?” question. The response is to say no, and then give them some advice for how to remember things and talk to them about how it will help them do problem-solving if they do. I just wish more of that sort of thing was done at the moment they first learned the maths.

    Ok. My complaining is over. Now it’s off to the MLC Drop-In Room for the last day before the exam…

  • Bathelling in assignments

    The Deeper Meaning of Liff by Douglas Adams and John Lloyd  defines the word bathel like this:

    bathel (vb.) To pretend to have read the book under discussion when in fact you’ve only seen the TV series or movie.

    I do not like to bathel, and in fact it is one of my life’s ambitions to find and read the books on which the TV series and movies I have seen – especially those I saw as a child. This ambition has inspired me to read Tom’s midnight garden, The Children of Green Knowe, Anne of Green Gables, The Hundred and One Dalmations, Babe (aka The Sheep Pig), Archer’s Goon, Jumanji, Dot and the Kangaroo, Peter Pan, The Wizard of Oz, The Last Unicorn, Halfway Across the Galaxy and Turn Left, Finders Keepers and I’m sure several others I can’t think of right now.

    I was talking about the word bathel at the AUMS barbecue yesterday, and Nicholas called me to remind me that I had lost track of time and what I should be doing was helping students in the MLC Drop-In room. So off I went to help people with their t-tests, conics and subspaces. And it occurred to me while doing this that a small number of the students I was talking to were attempting to bathel about their coursework.

    These few students were attempting to use information they’d been told in their lectures to talk knowledgeably about a problem, without having tried to organise and connect the ideas first. They hadn’t sat down with their notes and some problems and tried to grapple with how these ideas can be applied. They had only seen the movie and not taken the time to read the book.

    (I should say at most students I talk to have a very positive attitude and do try to think through their course content deeply, using the MLC to help them learn to do this thinking!)

    A movie presents the ideas in a book most pertinent to the film-makers’ intepretation of the overall theme. And it does so in a small window of time without any pauses or breaks for thought. On the other hand, when you read a book, you can savour a particular page for quite some time, and flick back and forth as you read to check something you might have missed. And you can think about what the book means to you in the gaps between reading sessions.

    In the same way, the lecturer presents the ideas of a particular area of learning most pertinent to the overall themes of the course. And you don’t get the chance in the lecture to think through what it means to you and how these ideas are connected. To really understand you need to sit and savour it like you do when reading a book.

    I’d like to hope that I can encourage students to take the time to savour it, but if not, I’d at least like to teach them that bathelling is not the best way to go. Lecturers are pretty good at spotting people bathelling on assignments!