The Area Under A Distance-Time Graph

Teaching Mechanics

I love teaching A level maths, but I must admit to feeling a bit disappointed when I found out I would be teaching the Mechanics module (M1 from MEI, fact fans) because... Should I really be saying this? OK, I'll just say it: I find it boring. No, that's not the whole truth, I'm making excuses now. I have to admit: I don't always get it.

I did Mechanics 1 at A-level (also using the MEI exam board) and that's the full extent of my knowledge. No Mechanics 2, none at Uni, I didn't do Physics A level or even separate Physics GCSE. I've never really been a "scientist", preferring to see myself as a creative type. I was always better at English and French than at science.

Anyway, this year (2012) I've been team-teaching M1, sort of. I've basically been watching someone else teach it. And suddenly I'm learning things I never really understood the first time round. I'm starting to get it!

The Area Under a Distance Time Graph

But then I went and spoiled it all by doing something stupid like Googling "What's the area under a distance-time graph?"

I would hope that those reading this are aware of the concepts illustrated below:

When you draw a velocity-time graph, for example, the area underneath the curve gives you the displacement, and the gradient of the curve gives you the acceleration.

The other teacher of the class posed the question to the class "What's the area underneath a displacement-time graph?". My mind was immediately blown. With a bit of jotting down (the diagram above), I could quite easily work out that the units had to be ms (metre seconds, a bit like kilowatt hours), so I knew it was something to do with distance multiplied by time.

I knew that what I was looking for would be the blank in this sentence: "displacement is the rate at which BLANK changes". What could fit? Nothing seemed intuitive.

Absement

That was where that much-relied on search engine came in. After a bit of research (Googling), I found that the word I was looking for was "absement" (oh of course, you all exclaim) and that I was by no means the only geek in the world wondering what it was.

Absement is a port-manteau of the words absent and displacement (knowing which makes it no easier to understand) and there are not many results on Google for it (the eighth result is an English to Urdu translation page. Erm, cheers for that).

I will attempt to explain absement using an example. For another example, visit Wearcam.

You live 2km from school. You walk to school in 30 minutes, stay there for six hours, then return home, also taking 30 minutes.

A displacement-time graph would look like this:

Now let's consider the area under this graph. It would be given by integrating the curve above. You can sketch this curve easily: in the first section, the gradient is constant and positive, so the corresponding bit of our new graph would be increasingly increasing (curving upwards). The middle bit has zero gradient so our new graph will have a constant positive gradient. The last bit is the opposite of the first bit, so our new graph's last section will have a decreasingly increasing bit.

My sketch:

The next thing to do is calculate the numbers on the vertical axis.

Your absement at any point is given by the (average) distance you are from home multiplied by the time that you're there for.

So when you've just arrived at school, your absement for that point is 1000m (your average distance from home) multiplied by 1800. So that's 1 800 000 ms. Up to that point, your absement has been increasingly increasing. When you're halfway to school, for example, your average displacement was 500m, your time is 900, so your absement is 450 000ms (note that this is not half of 1 800 00ms). At the end of your second part (when you've just finished school), your displacement has been constant at 2000m, your time has been 6 hours which is 21600 seconds, so your absement at that point is 43 200 000 ms. You can calculate your absement at many different points so that you can plot a nice, smooth curve. It looks a bit like a cumulative frequency curve. I'll leave this to you as an exercise.

What's the point of absement?

Well if you were on a spaceship and you had some kind of mobile communication device, you might imagine that the further from Earth you are, the more power the device uses. Therefore you might measure its battery usage in metre seconds.



Emma x x x

Praise and Rewards: Overrated?

How often do you praise your students? I bet most of you would answer "not enough", because we're constantly being told that we should be praising and critiquing in the ratio of 4:1 (which I'm sure is completely arbitrary. Where are the calculations?) and that we should be constantly reinforcing good behaviours and boosting confidence.

But really? Really? The hype over praise has reached such a ridiculous level that we're now told we have to seek out things that students are doing and praise them for it. It's called "catching them being good" but it should really be called "searching frantically for something they do which isn't completely moronic". 

I sound quite bitter don't I? I'll dial it back a bit. I'm no educational expert, and I'm by no means an experienced teacher, but I still feel quite strongly that the advice we have been given about praise, and more so about rewards, is bad.

I have a confession to make. My name is Emma and I'm an over-praiser. I will meet every child's contribution to the lesson with a "well done!" or an "excellent answer!" or, more often than not, "that's not correct, but I'm impressed by your creativity!" I actually found myself praising a student for remembering to draw his margin with a pencil today. And this was a middle ability set. I hate the fact that I do this. It's a habit I got into during my training year, before I was confident enough to challenge any of the advice I'd been given. I'd spend hours trying my hardest to think of three "stars" for little Johnny whilst trying to cram all of my hundreds of "wishes" into one line. Result: Johnny walks away thinking he's done a good piece of work. It doesn't matter to him that one of the stars was "You wrote the date! (Smiley face)"

My revelation came during my NQT year. I was talking to a particularly inspiring teacher, and he admitted to me that he never gave out "points" for good work, good behaviour etc. He said he had never even logged into the online points system. I felt really smug then, because I had painstakingly given out every single point in my account every week since the start of the year. My smugness didn't last long, however. He told me he didn't really agree with the points system. He said that students shouldn't be behaving well and doing good work so that they can earn points, they should be doing it for their own self-satisfaction. This really struck a chord with me.

Giving out points encourages extrinsic motivation, where people act to gain external rewards and avoid external punishments. Intrinsic motivation is where people act for their own satisfaction. Studies have shown that using extrinsic motivation to get people to do a certain activity can lead people to see that activity as "work", a job for which they are paid. I have heard that it is common advice to tell parents not to reward their children for reading, because if they do, they will stop reading just for the fun of it. 

We all know that Pavlov et al have shown that using praise and rewards allows us to control behaviour, but I personally would rather teach students who are in control of their own behaviour. There's something quite sad really about a dog who salivates when a bell sounds. Would you like to teach a bunch of robots who immediately start working the second you say "VIVO miles"? Are they the kind of people we want in society? There's no one around to reward adults for not dropping litter on the floor, not sticking chewing gum under cinema seats and wearing appropriate clothing in public (I'm talking to you, large blonde lady from the number 5 bus).

And when you throw praise and rewards around willy-nilly, you devalue it. I don't think any of my kids ever really experience the feeling of pride puffing up in your chest. I'm sure my comments wash over most of them, a lot of the time. I watched the teacher mentioned above teach an A level lesson the other day. He asked the class how they thought speed cameras worked. After some answers and some discussion, a boy contributed an answer that I thought was pretty good (embarrassingly, I didn't know the correct answer). The teacher, without saying anything to the boy, explained the boys answer to the rest of the class, rewording it a bit so they would all understand. He then finished by pausing for a second, and then simply saying "that's exactly how they work". He didn't add "well done", he didn't even smile. He just said that, in a slow, low voice. But even from where I was standing, the pride that boy was feeling was palpable.

I must admit, I stopped giving out my points a while ago, and since then I haven't noticed a single bit of difference in my classes' behaviour, effort or achievement. My next step is to stop being so generous with my praise. I'm thinking I might aim to give out one really good bit of praise per lesson. 

Think about yourself as a student. What bit of praise really affected you? What moment of pride do you still remember? For me, it's the time my year nine history teacher (also the head teacher) wrote that my end of year project is the best he had seen for as long as he could remember. And he gave me a pen (my schoolmates will remember how scarcely these pens were given out). Embarrassing admission of the week: I still have the pen. It's in my "special box". 

Am I on my own here or do you agree? Comment below!

Emma x x x