The Prime Factorisation of Me

I had a student ask me the other week, “Teachers are always writing reports about us. When do we get to write a report about the teachers?” Well, I thought, why not?

Actually that’s a lie. I’d already decided to gather feedback from my classes weeks ago, but the fact that one student asked this question was a convenient way to start this post :). But seriously, isn’t the whole point of writing student reports so they know what they need to do to improve (well, that’s the theory at least). Though they really aren’t the experts they think they are on what teachers do, students are more than willing to let us know what they think of how we do our jobs. If we want to improve as teachers, it might be a good idea to at least listen to them from time to time.

I decided to use the “Keep, Change, Start, Stop” format and set up a Google docs form to collect responses from students. I thought I might share some of the feedback I got and maybe even respond to it. In each section, I’ve tried to order it from most common to least common feedback. The first class I got responses from was Year 9.

Keep

New Things Thursday. YES! YES! YES! This was overwhelmingly the most popular response. If you missed it, I introduced New Things in this post. I will absolutely continue it in Term 3.

Monday Science Pracs. Something I haven’t explained yet on this blog: our school has a slightly different Year 9 program, the result of which is that I actually teach a combined maths and science subject we call “Exploration” with another teacher. I mostly focus on the maths, but we have one lesson a week with both teachers in the room on Monday, which is when we tend to do science pracs.

Pythagoras’ Therom or however you spell it. Good. I wasn’t planning on getting rid of it. Not that I’d have a choice in the matter anyway. But more significantly, I’ve been trying a few different things in this unit (including New Things), which students seem to be more enthusiastic about.

Your beard. I grew a beard over the last school holidays, which has split student opinion between “It’s awesome” and “It’s scary”. I’m sorry to the jealous boys in my class, I think it’s only going to be a one term thing.

Change

Less book work. I’m pretty sure you meant “less textbook questions” by this. I agree – I’ve been guilty of falling into the too-many-textbook-exercises trap on a few occassions. I’ve been trying to get away from using it as much as possible lately – I promise I’ll keep doing that.

Let us leave class to fill up our water bottles. No. Just no. You know it’s a school rule, and you know you have recess and lunch to fill it up. Don’t waste my time just because you can’t mangage yours properly. (Can you tell I’ve had this conversation a lot over the semester?)

Make questions easier. If you mean “make questions clearer and more approachable”, then yes, I will try to do that. If you mean “let me not have to think in your class”, then no.

Make maths more fun. I’ll keep trying to do this. But my idea of this might be a little different than yours. I believe that maths is fun (because I am a crazy person), and my job is to help you understand it better so you can see how awesome it is. (But if you mean “teach us with different maths activities”, then sure, I’ll make more of an effort to do this).

Start

Most of the feedback here was repeats of the first two questions, but there were some new points.

Short quizzes at the start of the lesson where you time us. I’m pretty sure I did something like that two years ago when I had your class in Year 7, and you guys hated it! But OK, I’ll think about it.

Free time on Friday. No.

Give us more feedback on our work. I agree, this is something I need to work on.

Yes. Ummm, what? (This really was one of the responses. The same student wrote “No” for Stop).

Stop

Stop counting. I assume by this you mean the thing I do to get your attention when you’re all too noisy? (I quietly count by fives, which lets the class know I’m waiting for them. I go up by five because I used to say “five seconds, ten seconds, …” but they kept correctly pointing out that I usually only take three seconds for each count). There’s a simple solution to this – if you’re all quiet when I ask you to be, I won’t have to do it 🙂

Nothing, I guess. Well, that’s good to hear! But I’m under no illusions. Even if I get to the point where my class thinks I’m perfect, I know there will be improvements I can make.

I think that’s a good start for thinking about where I am with my teaching at the moment, but there’s a lot more reflection to do over the two week break. (And anyway, this is only one of my classes). If you’ve never gathered feedback from your students before, it could be a good idea. It’s a little scary giving students permission to examine and comment on your teaching practice, but they know what you do in the classroom better than anyone else. But it can’t stop here. The only feedback of value is that which leads to improved pedagogy – and I think that’ll take harder work than just letting students write a few things about you.

One of the VCE Maths Methods topics I don’t think I’ve covered particularly well in the past has been the estimation of definite integrals using rectangles. By the time I get up to it, we’ve usually had so many interuptions during the first half of the year (this year is no exception) that I’m trying to catch up and often rush through the material with some short board notes and a few textbook questions.

This year I made a conscious decision to improve the way I introduced it. As much as sketched diagrams help a little, I think they still leave the concepts pretty abstract. I wanted to make my examples more concrete. I wanted to set the stage for explaining the definite integral as the limit of a sum. Hopefully this will also later help draw attention to the significance of the Fundamental Theorem of Calculus.

My solution (which I’m starting to realise is the solution to pretty much everything!) was cutting and pasting. Initially my students weren’t terribly keen on it, and thought it was a waste of time when they could just draw it, but I think they came round to the idea – kind of. One of them asked an off-topic question, which they started by saying, “Since we’re not really doing maths today…”. That’s OK, I’ve still got a semester to convince them that cutting and pasting absolutely is maths. (Actually, the question was a really interesting one, but I’ll wait until another post to write about it.)

I gave the students a sheet with the same graph of y = -x² + 4 repeated four times, on grids where 1 cm = 0.5 units. I also gave them coloured paper to cut into strips 2 cm and 1 cm, to use as the rectangles. The result is below:

With more time, I would have liked to have also done left and right endpoint rectangles, as well as midpoint rectangles and trapezia (I stuck to the boring smartboard for these). Personally I prefer upper and lower rectangles because you can talk about how they, besides being an estimation, form upper and lower bounds on the actual value, and you can demonstrate that the bounds get closer as the number of rectangles increases. But that could just be my pure maths background – I’m sure I’d prefer a faster method for calculating if I had an applied background.

Thinking ahead to next year, this could be followed up with some sort of technology to quickly demonstrate with even more rectangles – maybe using Excel or Geogebra, or maybe even the CAS has some way of doing that. I’ll need to investigate those options. That way we could experimentally find the limit, and hopefully comfirm it matches the definite integral we find using calculus.

EDIT: I meant to include a link to the worksheets here, I’ll add them once I’m near a computer.

EDIT (23-6-2015): A year later (to the day, apparently), I finally got around to adding those worksheets!

Downloads:

• Upper and Lower Rectangles.docx
• Rectangle strips.docx

I taught my last lesson for the term last Friday. Reports are written. My year 12s have their holiday study homework. My desk is tidy (well, tidier than it was). My list of paperwork to get through is significantly smaller than it was. There’s only one more thing to get done before I can go on holidays:

Go to Canberra with years 7 & 8 for a week. Hooray?

I have to admit I’m not really as excited as I should be right now. We left school at 6:30 (and I never cope that well with early starts as it is), I just had to growl at a couple of kids who were getting overly excited over a game of Mario Kart, and my leg hurts after I failed to jump out of the way of a hockey ball on Saturday. Plus I know how little sleep I’m going to get over the next week, and I remember how stupidly cold our national capital was last time I went on this excursion – and it’s already been stupidly cold at home.

But I really am trying to focus on the positives. Above all, we put ourselves through things like this to give our kids experiences they can’t possibly get by just staying at school. One of the problems our kids have growing up in a small town it that they sometimes don’t really think about the existence of the wider world – hopefully visiting Canberra gives them a better sense of their place in the nation. But these are some things I’m personally excited about:

• As crazy as it might sound, I’m looking forward to spending a week with these students. I haven’t taught either class before (aside from a few replacement lessons) so I don’t really know them that well. This is my chance to get to know them before I probably teach them in the next few years.
• Even though maths is “my thing”, I do tend to nerd out on history a bit. I’ve been to the Australian War Memorial a couple of times, but I’m keen to explore there again. I’m also a bit of a politics nerd (I’m not that political, I just find the process fascinating) so I always get a bit of a kick out of Parliament House and the High Court. (Maybe I should have left this point as “I’m a bit of a nerd” and that would have summarised it all?)
• Questacon! (aka the National Science and Technology Centre.) Last time I was there, I just wandered around with the kids, which was awesome enough by itself. This time, hopefully I can get some lesson ideas out of it.
• I’m really looking forward to not going ice skating as we’ve done on previous trips. That’s the kind of stress we really don’t need.
• I’ve had a coffee since I started writing this, so I’m actually awake now.
• When I get home, I’m going to sleep for a week.

Should I be worried that the last two points were about how tired I am?

Weirdly, I’m also looking forward to the hours spent on our coach today and at the end of the week. I really haven’t had any spare time for a while now, so I can finally sit down and write a few blog posts that I’ve been thinking about. Stay tuned to either read my reflections of the first semester, or to watch my sanity slowly drain away. Could be either, could be both.

On Monday I had my first Year 9 lesson since introducing New Things Thursday last week. While I’d had a few positive reactions, I was still a little nervous about how the class felt about the whole thing. As it turned out, they were asking if we could do New Things, and were disappointed that they’re going to be away on Thursday. I was also disappointed, so I’d already decided to New Things on Monday anyway 🙂

As I put my New Things slide up on the IWB, my students were immediately excited about being able to share their new things – well, most of them were. It was only then that I realised two of my students were away last Thursday and had absolutely no idea what was going on. Oops. I told them to just go with it and found a gap in the conversation to explain it all to them.

Proving Pythagoras’ Theorem

Last week explained Pythagoras’ Theorem, and while I mentioned that theorem’s should be proven, I’d only hinted that the proof was still to come. This was Monday’s task. Like last time, we’d do this with a “New Things Page”.

I wanted to make the proof as simple to follow as possible. I also decided to use cutting and pasting of triangles and squares rather than just drawing diagrams – I hoped to help the kinaesthetic learners as well as the visual ones, and make it obvious that all the triangles are the same. The end result (well, my version of it) is below:

(I tacked on the algebra proof after the class – it’s my personal favourite proof, but we haven’t covered enough algebra yet for me to teach it. Also, I think my arty skills have improved since last time!)

The key point that makes the proof work is realising that the areas not covered by the triangles are the same on both large squares. When I asked the class which area was bigger, they debated with each other for a few moments before declaring it a stupid question because they were the same. (Hooray!)

I’m pretty keen to make New Things Thursday a permanent feature, but I’d like to see if the class stays enthusiastic about it once it’s, well, not a new thing. I asked a few students if they thought it was a good idea or if they thought I’d gone crazy. “Both,” was the answer. I guess that’s good?

One of the features of my Maths Methods classes over the last few years has been the scoreboard, for a game that no-one is entirely sure of the rules for. Well, there are some rules:

• If I make a mistake on the board and they spot it before I correct it myself, the class gets a point.
• If they claim I made a mistake when I didn’t, I get a point.
• I’m the only one who’s allowed to update the scoreboard (otherwise I get a point).
• Last year they were allowed to claim one ‘cake-point’ per week (that’s exactly what it sounds like: they get a point if they bring a cake to class). Honestly, I don’t mind it when they score this way 🙂

Plus there’s a whole heap of silly rules that tend to get made up on the spot, by both them and myself.

It’s a silly little thing we do which keeps them paying attention and keeps me on my toes. I’d like to say I planned it that way, but the game sort of just developed by itself. At the end of each year the outgoing class seems to tell the next year’s class about the game, so they claim points from me before I’ve even mentioned it.

The first year we did it, I won comprehensively. It was a tie last year (though I may have bent the rules a fair bit to catch up). This year, I think I’m in trouble.

As Unit 3 ended on Friday and Unit 4 starts this week. So it’s half time at the moment. The scoring’s been pretty slow this year – they’re fairly conservative with their challenges and aren’t willing to give up anything easily. Unfortunately for me, that strategy seems to be working:

Oh well. Everyone loves a good comeback.