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We made it home and I’m incredibly tired, but now it’s school holidays 🙂 After re-reading my post about going to Canberra with Year 7 & 8, I’ve realised how pessimistic I sounded. In fact, it was a really good time, and our students were very well behaved. There’s a few things I want to write about in detail, eventually, but for now I’ll just make a note of a few of the things that happened:

• I really enjoyed the National Portrait Gallery. I know! Not really what I expected either. A lot of this was from our tour guide, who didn’t just show us the artworks, but involved a number of different educational activites and got the students to think about what they meant within a theme of “Australian Identity”. I wouldn’t be at all surprised if she used to be a teacher.
• The War Memorial continues to be one of the most moving places I’ve been to. I really need to find an opportunity to visit Canberra again so I can spend a whole day there.
• Questacon is still one of my favourite places in the world. Of course this was always going to be my highlight, but I think it was for a lot of the kids as well. If you don’t know what Questacon is, it’s basically a gallery of a whole range of hands-on science demonstrations and activities (including a decent smattering of maths).
• Also from Questacon, I probably spent too much in the shop. I don’t get many opportunities to buy books about maths where I live, and my collection consists of mostly uni textbooks. So I picked up some holiday reading. (Admittedly, one of them is about physics).

• Also: pi mug!

• After I scored better than the kids at ten pin bowling, some of them have taken to calling me “Mr. Skills”. They starting talking about me similarly to how the internet talks about Chuck Norris. When we visited Parliament we met our local member – after they saw me chatting with him, these kids became convinced that I “know him from way back”. They started telling stories about me secretly being friends with the Prime Minister and actually being the one in charge of federal politics. I’m a little worried about where this might lead…
• Other than the strange rumors being started about me, it was great to get to know these kids and to have them get to know me. I don’t actually teach either of these classes, so a lot of them have previously just been names and faces to me. If I teach them in the future, it should make the start of the year a lot easier.
• I spent all week limping around Canberra after being hit by a hockey ball on my ankle last Saturday. But it’s OK now – it’s been balanced out by being hit on my other foot today. I think it really is time to have a rest.

This is just an idea I had tonight while on camp with Year 7 and 8. It doesn’t relate to anything thing I’m doing in class at the moment, but I wanted to get it written down somewhere before I forgot.

So we went ten pin bowling tonight, and I noticed that after each bowl the scoreboard showed a quick video of the pins getting knocked over (I wish I’d gotten a photo of it). Maybe this is pretty standard and you all know exactly what I’m talking about, but we don’t get to see much bowling in a town as small as ours.

Anyway, initially I couldn’t tell if the video was really recorded live with a camera I couldn’t see, or if it was faked by showing a prerecorded video with the appropriate pins being knocked over. (I’m pretty sure the videos were real, but that’s beside the point). It got me thinking – how many videos would need to be prerecorded in order to be able to show every combination of pins falling over?

The simple solution is this: there are ten pins, and each pin has two possible states, standing or knocked over. So the total number of combinations is 2¹⁰, or 1024 as any addict of “2048” would be able to tell you. Or if we imagine a more general sport called n-pin bowling, then the number of videos needed is 2ⁿ.

The thing is, this is not the first solution I tried. Instead, I tried to use combinatorics:

• There is C(10,0) = 1 combination with no pins knocked over.
• There are C(10,1) = 10 combinations with 1 pin knocked over.
• There are C(10,2) = 45 combinations with 2 pins knocked over.

etc. Then we add them together. This is much more work than the other method (particularly when trying to do it in you head while supervising a bunch of excited 12-14 year olds), but should give us our result. We can make this simpler by remember that combinations are contained in Pascal’s triangle, so we can just add the numbers in row 10 to get our result:

1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1 = 1024

But this should always work for our hypothetical game of n-pin as well, by adding the entries in row n. Because the number of videos should be the same regardless of which method we use, this leads to the following result:

The sum of the entries in row n of Pascal’s Triangle is 2ⁿ.

So that’s kind of cool. A more typical proof of that statement is to consider the binomial expansion of (a + b)ⁿ with both a and b set to 1. So now we can relate the problem to algebra as well!

I know this isn’t a lesson yet, but I wanted to get it down while it was still fresh in my mind. Hopefully I’ll remember to look up this post later in the year, when I actually have to teach this stuff! There’s also few possible extensions to this problem I thought of:

• Are there any combinations we can eliminate and not make videos for because they are impossible to occur?
• My 2048 reference was really just a joke. But thinking now, 2048 is all about powers of 2. Pascal’s triangle is (in a rather sneaky way) also about powers of 2. Is there some hidden connection between 2048 and Pascal’s Triangle?

For the record, I scored 133 and 126 in our two games of ten pin. Not great scores, but pretty good for me. I did start with two strikes, so now some of the kids think I’m some sort of bowling genius.

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.