Category: Reflections

Reflections on learning and teaching and research and life.

Really working together

Yesterday, I had one of those experiences in the MLC that makes me love my job.

The Maths 1B students were working on a linear algebra proof today, and as I came up to one of the tables, Fred (name changed) was explaining the beginning of his proof to the rest of the table. When I arrived at the table, he was leaning over two of the other students to point at a section of the lecture notes. He noticed I was standing there and said, “But David can probably do this better than me.” I responded, “Not necessarily. You keep going,” and I sat down in his chair.

Fred continued to explain, and I think he did a very good job. I was very pleased that he kept flicking through the lecture notes to point out different theorems, though I thought it was interesting that at no point did he write anything down.

Then one of the other students said, “so is that the end?” And Fred said that no, this was just the beginning, there was still more after that, and I could see in his eyes he was having a sinking feeling as he tried to think of how to move on to the next bit.

So I asked him if he could pass me a whiteboard marker. I stood up to the wall and said, “I just want to write down where we’re up to.” I asked the students he had been talking with to tell me what they’d done so far, and I transcribed it to the wall, asking them to explain why each line worked. And then we got up to the end of what they had already done.

“So what now?” I asked. There was a short silence, and then Fred piped up with a comment about what we needed to know next. I asked why that was important to know, and this started a discussion of what goal we were heading for.

And here is where the really great stuff happened. The students at the table offered suggestions of things to try, looked up definitions and theorems in their notes, helped each other refine their maths language, asked each other questions when they weren’t sure of things, welcomed new students into the discussion when they wandered over to listen, discussed how to make the proof their own when they wrote it to hand in, and basically really worked together to construct the proof. It was a pleasure to be a part of it.

It’s this sort of thing that makes my job such a joy – seeing students learning and supporting each other to succeed. On a day containing many other parts of my job that are much less joyful, it was something I really needed to see.

This comment was left on the original blog post:

Steven 8 February 2016:

Indeed, I also did enjoy reading your post regarding on how crucial and effective to have a group discussion i.e. working together. I’ll be sitting UMAT this year and I hope i can find someone/group as well to discuss on some UMAT questions and produce interesting results like the above.

5 February, 2016
Making sense of the effective population size formula

I was going to have a punchy title for this post, with a big moral to apply to the future, but I’ve decided I’m just going to describe to you what happened yesterday as I tried to learn some Genetics. You see what you can learn from my experience.

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

15 October, 2015
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.
25 September, 2015
The reorder of operations
The community of maths users the world over agrees that when evaluating an expression or calculation, some operations should be done before others. Mostly it’s to prevent us having to be needlessly specific about what order to do calculations in, mathematicians being very concerned with efficient communication.

My problem with the order of operations as usually stated is this: it’s wrong! It’s wrong because you almost always don’t do the operations in the order described. You don’t do all the multiplications before all the additions, and instead will often quite a few of the additions first because they are easier. And you don’t do the multiplications in the order they come, but rearrange them into some other easier order. And you often don’t do the brackets first, but instead choose expand them out in order to make the calculation easier.

You can read the rest of this blog post, and the other posts in the series across the years, in PDF form here.

The titles of the five blog posts are:
- The reorder of operations
- (Holding it together)
- The Operation Tower
- Replacing
- Sticky operations
11 September, 2015
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. 🙂

3 September, 2015
Pure play

The other day I did a workshop with students from Advanced Mathematical Economics III, which is more or less a pure maths course for economics students. It covers such things as mathematical logic, analysis and topology – all a bit intimidating for students who started out the degree with almost no mathematical background!

We had just spent an hour looking at relations: the definition as a subset of the set of pairs, the common usage as statements in natural language, various ways to visualise them, examples and non-examples of various properties relations might have, and proofs involving those properties. After all this point, one of the students said, with some surprise, “It really feels like it’s just playing around here.”

The student was right, of course, and I told him so! Pure maths is play. You have some ideas and you fiddle around with them to see how they fit together, how they’re similar or different, what other things are really the same things in disguise. Most of the time there’s no particular goal in mind, but even when there is, you often get distracted by something cool and end up somewhere totally different. This is exactly what play is. Watch any child in a sandpit and you’ll see the same behaviour.

It suddenly occurred to me that I have always seem maths as play, and have liked it the least when it looked like work. More than this, the maths courses I have had the most academic success in were the ones where I allowed myself to see it as play. These were the courses where I took the concepts I was learning and pulled them apart and put them together in new ways, where I tried to do things that may or may not be possible just to give it a go, and where I drew lots of pictures in vivid colour just because. And importantly, where I didn’t question what the point of any of it was but just ran headlong into whatever crazy idea the lecturer presented next to see what happened with it.

So perhaps the best way to approach a pure maths course is to see it as play! Surrender to “just playing around” and see where it goes. That’s the advice I gave to these students, and I’m hoping it helps them to have the freedom to learn.

29 August, 2015
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.

14 August, 2015
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.

15 April, 2015
Inspiration, not instructions

We have a big problem-solving poster on the MLC wall that gives students advice for solving problems. One of those pieces of advice is that to decide what to do for your current problem, you could look at other problems for inspiration. Yesterday I saw the dangerous results of what happens if you look at other problems for instructions rather than inspiration.

Across the day I talked to several students about Statics, which is an engineering physics course concerning situations where objects are not moving (ie “static”). At least three of them had put their answers into the computer-marking system and found that they were wrong, and so they wanted help to figure out why. As always, I asked them to tell me more about the problem and how they solved it.

The students showed me the problem they were trying to solve, and then pulled out their lecture notes and showed me how they found a problem that was similar and followed the same procedure. Normally, this would make me extremely happy – students actually looking at their actual lecture notes independently? Bliss! But unfortunately, what they had done was notice how the example had the same letters in it as their problem and put the values of those letters from their problem in the formulas they saw in the example.

The big problem was that the forces in the book’s examples were perfectly horizontal, but the ones in their assignment were at an angle, so they couldn’t just put things in the formula. Oh dear.

These students had interpreted the example as a list of instructions for precisely what to do if they saw a similar picture with similar letters in it. They had interpreted the process of solving Statics problems as finding the right formula and putting things in the right places. Instead, Statics problems are more about making your own formula from the structure of the problem situation itself. When you read a Statics example, you are looking for inspiration for how to think about the problem, rather than for instructions for how to do it.

I am very glad now that when we chose the words for our poster, we used the word “inspiration”. In the future I will try to highlight that aspect of it a bit more, so that students can be looking for the right things when they perform the good practice of looking for examples in the notes.

This comment was left on the original blog post:

Terry Bennett 10 April 2015:

Great post/observation David,

I’ve come across the same thing in class – “.. are you trying to reuse the equations on pg44 of my notes? Did you draw a free body diagram of this particular problem?”.

Next step: the reflection “what can I do better next year to avoid/reduce this misconception?” (answers on a postcard please …). Thanks for the help in identifying this “recipe trap” I’ve inadvertently introduced.

Terry

10 April, 2015
Jack Frost’s Centre

On the weekend I watched the film “Rise of the Guardians” by Dreamworks Pictures, and it is a very enjoyable film. In it, Jack Frost is enlisted by the Man in the Moon to join the Guardians of Childhood—who already have Santa Claus, the Easter Bunny, the Sandman and the Tooth Fairy in their ranks—and together they fight the evil Pitch Black, who is the Bogeyman.

There is a scene in the film, where Santa Claus (called “North” by the other Guardians) gives some advice to the new recruit Jack. He tells Jack that he needs to find his centre – that part of him that guides how he will guard the children of the world against fear.

Spoiler alert! Jack’s centre is fun. His great epiphany is when he finds himself saying the phrase: “Don’t be afraid, we’re just going to have a bit of fun instead.” He realises that for hundreds of years he has been warding off the darkness of winter by helping children have fun in the snow. Because they are having fun, they forget to be afraid.

And then it was Monday. And not just any Monday either! It was the first day of O’Week – one of the busiest days in the year for the MLC. From 8am it was printing handouts and setting up the drop-in-centre and visiting preliminary lectures and playing Numbers and Letters in the Hub Central welcome centre.

This year we had PASS leaders helping us out in the welcome centre and I must say they were a great bunch of people who leaped into the role of seeking out students with cheerful fervour. Georgia in particular was super enthusiastic about getting people involved in the games, and indeed playing the games herself.

Interestingly, Georgia started the day saying she wasn’t all that good at calculations, and stood back letting others do it. But I persevered with her, encouraging her to have a go, asking for any ideas, writing down her slightly wrong solutions so we could work together to tweak them to make them work. By the end of the day she was our greatest champion for the numbers game, itching to rub it out and start a new one every time we drew the last stroke of the previous solution.

That night when thinking about this MLC success story, I suddently made the connection to “Rise of the Guardians”. Georgia had forgotten to be afraid because she was having fun. She had forgotten to be afraid of calculations, forgotten to be afraid of failing, forgotten to be afraid of looking stupid in public – all because what we were doing was playing a game.

Of course this is precisely why we have a mathematical art and play program at the MLC, but I had never so clearly realised the power that it actually had to change someone’s perspective.

And I never realised how much of the time that Jack Frost’s centre is mine too. Any number of times a day, I will do something to distract a student from their fear so that they can get on with learning the maths they need to learn: wear a maths-themed t-shirt, pull out the play dough, draw their diagram in texta or crayon, ask about their lecturer’s lovable quirks, or just simply yell out “That’s so COOL!”. In all of this I am saying to the student, “Don’t be afraid of the maths, let’s just have a bit of fun instead.”

26 February, 2015