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.