Author: Shaun Carter

Spring holidays continue to roll on. When you live in a town as small as mine you don’t get many opportunities to get shopping done, so today involved driving an hour and a bit away to do that.

I’ve already mentioned that some of my shopping habits can be a bit weird. I’ll buy anything as long as I can justify it by thinking “there might be a maths lesson with this somewhere.” Odd purchases today included lots of colours of play-dough, bathroom scales, shower curtain, pool noodles (I don’t have a pool) and party streamers. Hopefully you’ll eventually read blog posts about the lessons these half-baked ideas turn into – otherwise I’ve just been wasting my money.

I did get some actual useful stationery stuff too. Coloured whiteboard markers (good ones, too) on clearance! Woo!

And I did get Mum’s birthday present too. She sometimes reads this blog, so I’ll say no more about that.

Most expensive purchase: new lawnmower. Saying I don’t enjoy gardening would be the biggest understatement of the year, but it’s still got to be done. I already owned a lawnmower, but it’s not very good. Seriously, the only way you can adjust the height is by removing the wheels and attaching them to a different hole. You could say that it just isn’t going to cut it anymore (I am so, so sorry).

Anyway, who cares about that. Most exciting purchase: new laminator! I’ll be honest, my school has a pretty good laminator, so I really didn’t need to buy one myself. But I like the idea of being able to get stuff done at home. Yes, I really am so lazy that I’ll avoid the two minute drive it takes to get to school if I can.

And really, it was there in the shop and no-one was there to tell me I shouldn’t buy it. So I did.

A new toy means testing it out first chance you get. So I finally got around to making my Continuous Probability Distributions posters. These are a follow up to the Discrete Probability posters I blogged about previously.

Downloads

• continuous probability posters.pdf
• continuous probability posters.docx

If you want to edit the document, you’ll want the font Matiz.

I put together this geogebra file to make the graphs. I didn’t really put much effort into tidying up the file, but someone might make use of it.

“Index Laws” is one of the topics that students at my school seem to struggle with. Okay, sure, they can use the laws, and when given an expression they can usually simplify it. But I’m convinced that they actually don’t understand indices as well as they think. There seems to be a lot of “pattern matching” going on, without much thought to what that little number above the other number means.

One lesson, I wrote 3⁴ on the board and asked different groups of students what it meant. The first answer was always “three to the power of four”. When I asked them to explain what that meant, they confidently told me it meant “4 is the number of times you times the 3”. Which sounds right, sort of.

But when I asked them to explain what number 3⁴ is equal to, I got some very concerning answers. Some told me it was 12. Some reasoned that 3² is 9, so this must be double that: 18. Others knew it was 3×3×3×3, but reached for the calculator to work out what that was. Surely they can do 27×3 in their heads? It wasn’t that long ago we were using the distributive law with numbers.

My class knew what indices were. But they didn’t understand them.

So what do you do when you misunderstand something about numbers? Reach for the number line!

Now, my students are pretty resistive to using number lines. They seem to think they’re too simplistic and beneath them somehow. They think because they can imagine what they look like in their heads, they don’t need to have it physically in front of them. But they usually change their mind once we do get them out, because they work. Of course, they’ll forget that and complain about them again next lesson. (Then there’s the fact they keep calling them “time lines”…)

So, multiplication is repeated addition. That’s obvious … or is it? I think that’s something we take for granted that kids know. But I wonder if years of memorising “times tables” helps kids hide that they don’t really get what “timesing” (as my students insist on calling multiplication) is.

So I had students write the integers -3 to 7 along the top of a number line. I asked the class, “If we choose a value on the line, and move on space forwards, what happens to the number?” The immediate response was “you go up by 1”. I’m trying to get my students to be more precise with their language, so after discussing a little bit, this was changed to “adding 1”.

“What if we chose to add something other than 1?” I asked. So, underneath the line, we wrote numbers so that each step was adding 3. A few students noticed that the numbers along the top and bottom were related – you can just multiply by 3. Which a bit more discussion, we realised you could move between the two sets of numbers by multiplying or dividing by 3 as appropriate.

“Does this work for other numbers?” one student asked. (Hooray!) “I don’t know,” I said, “maybe we should try and find out?” (By now my students are well aware that when I say “I don’t know”, I really mean “I do know because I planned the lesson this way, but you’re not going to get me to do your thinking for you.”) So, students chose their own numbers to add.

Using 1/3 was only my idea after the lesson. I wish I had gotten at least some of my students to try it with a fraction.

So I asked another question: “What if when we make a step, we multiply instead of add?” After a discussion about needing to start at 1 instead of 0, we did a few number lines using multiplying.

If you’re the type of person who reads a blog like this, you’ll know where this is going – the result is the powers of whichever number we choose.

Once you get started with this, a lot of the rules related to indices start falling into place. Negative indices? Well, what number do you multiply by 2 to get 1? What number do you multiply by 2 to get 1/2? Very quickly, students were able to tell me that 2^-10 is 1/1024, without me having to tell them that a^-m = 1/a^m.

How about a^m×aⁿ = a^m+n? It makes much more sense as, say, “multiplying 5 four times then two more times is the same as multiplying 5 six times”.

Sorry about the bright colours. Did you know they make “neon” Sharpies? Did you know I already owned some but had forgotten? I got a bit excited when I found them.

This is where we stopped, but pushing this idea further makes this lesson just as appropriate for higher levels. What numbers fall between the integer indices? Is there a way of moving from from the bottom row back to the top. Get this: I actually had a student ask that last question. I had a year 9 student ask me about logarithms! I wonder if the confusion my Year 12s have around logs is a result of being confused about indices? In the same way that tables can hide misunderstandings of multiplication, I wonder if the textbook presentation of index laws can hide misunderstandings of indices?

One of the most exciting things in maths is realising a connection you didn’t see earlier. Hopefully this experience is what we’re giving students all the time, but as teachers we get it too. And sometimes, the connection seems so obvious (to someone with a maths degree), that’s it’s a little embarassing when you realise you didn’t notice it before.

Did you know that the volume of a pyramid (or cone) is one third of the volume of the equivalent prism (same base and height)?

Yeah… I didn’t know this until recently. I knew the formulas for the volumes of pyramids, and even derived them using calculus, but I’d never made the connection between them and prisms before.

I wanted to give my students the opportunity to make the same discovery. My school has a set of clear plastic prisms and cones, each with coloured plastic inserts of the net of the shape:

It took me a while to find these. I knew they existed because I remember them being bought, but everyone I asked didn’t know where they were. Well, it turns out the first colleague I asked did know where, but I was terrible at describing what I was after. Oops.

They come in matching pairs of the same colour and base, and they all have the same height. The only difference in each pair is that one’s a prism and the other’s a pyramid. I split the class into groups, and gave each group a pair without saying anything about them.

I asked the class to describe what they noticed about them. “They’re the same colour,” was the first reply. Okay, ask an obvious question, get and obvious answer. But after that, we get a few more suggestions:

• Each group had a prism and pyramid (or cylinder and cone – that group was very quick to point out every time I forgot to clarify that).
• The prism and pyramid in each group have the same base.
• The net of the prism has one more face than the pyramid. Even though this wasn’t the objective of the lesson, I loved this! The group that suggested it even gave an explanation why. One student from another group then said that if the base has n sides, then the pyramid has n + 1 faces and the prism has n + 2. Hooray! 🙂
• The shapes have the same height.
• They’re made of plastic (okay, not all of them were that noteworthy).

We’d already covered the volumes of prisms at length, so I asked them to look at the pyramid and think about how that volume compares. I gave the groups time to discuss. Interestingly, they all came back with the same answer – half the volume. Most groups had a reason too. One group said they thought if the pyramid was placed inside the prism, the volume outside the prism would be the same as the inside. Another said that the pyramid is like a triangle, and a triangle is half a rectangle.

I’m increasingly seeing the value in “wrong” answers. It was great that, even though the volume is not half, they were still able to give reasons. But also, no-one was completely happy with their answer. They thought it made intuitive sense, but weren’t satisfied without proof. Awesome 🙂

So, it was time to test it. I forgot to mention I did this lesson in the Science room, so there were sinks conveniently close by. The class realised that if their conjecture was true, they should be able fill the prism with water by filling the pyramid twice. So, they got to work:

Hmmm, this doesn’t really look half filled.

A filled prism.

Putting the shapes back together. You can also see this student’s formula sheet and their part-finished “New Things” page from this lesson.

Surprise! They all found they could fit three pyramids inside the prism.

Next I had students write down a formula for the volume of a pyramid/cone. They already knew the volume of a prism, so they used that to work out the pyramid. The nice thing about this (as opposed to them copying a formula off the whiteboard) is that they each wrote down the formula how they wanted. Some used ÷3, some wrote it as a fraction, some wrote 1/3 in front. But they all wrote a formula they were comfortable with and understood.

Areas for Improvement

I was really happy with how this lesson went, but I wasn’t particularly pleased with the end. We found correctly that the volume was one third, but we never really justified why.

I mean, sure, for a scientist, the empirical evidence gathered was fantastic. But we’re mathematicians, darn it. Aren’t we suppose to hold ourselves to a higher standard?

Does anyone know a decent way to explain why it’s a third, without relying on integration – this is a Year 9 class after all. Using calculus is such a nice proof, and also explains why changing from 2D to 3D changes from half the area to one third the volume. But I’m struggling to think of another explanation.

The class really enjoyed this lesson, as did I. But this one loose end is still bugging me.

This one isn’t going to be mathsy. Sorry. Just thought I’d share a little bit of what’s been happening with me lately. I really won’t be offended if you tune out now 🙂

My twitter description currently reads: Secondary maths teacher. Church musician. Hockey coach. Suspected human being. Obviously, this blog focusses on the first one, but the middle two combined last weekend to create some of the busiest few days of my life. (I don’t really have any evidence of the last one…)

So despite the fact I’m really not athletic at all, I coach our town’s junior hockey team (that’s field hockey for any international readers). Saturday was our Grand Final, which our club hosted, and we were playing in it. After finishing 1-1 at full time, we clinched the winning goal in extra time! Third premiership in a row!

The team immediately celebrated by spraying their coach (and maths teacher) with their water bottles:

One of the mothers decided I should be holding streamers for the photos. I’m not sure why. I must’ve been pretty happy, because I don’t usually grin like that when I’m soaking wet 🙂

The water thing seems to be their standard celebration. This is the same moment from a year ago:

See? I was getting water dumped on my head before it was cool.

Saturday evening was spent helping host our hockey association’s Grand Final Dinner, and Sunday lunchtime was our club’s presentations. So a lot of hockey for one weekend. As much as I enjoy the game, I’m happy the season’s over for now and I get my Saturdays back again.

For the record, I also play hockey, but my team has a proud tradition of choking in the first few weeks of the finals.

Sunday afternoon was spent preparing for my church’s monthly night service, which we very creatively call “Night Church”. I head up the music team, for which I play piano and guitar, and pretend that I can sing. My friend who was singing with me last night is a keen photographer, so she got this pic of me practicing:

Unfortunately I look pretty ridiculous with the guitar pick hanging out of my mouth.

So there you go, a little taste of what I do when I’m not teaching.

In theory last night’s #OzMathsChat was about Inquiry Based Learning, but we got sidetracked pretty quickly. Oh well, I’m sure we’ll come back to it sometime 🙂

Someone mentioned the difficulty in using inquiry in Year 12 classes because of the pressure to get through everything before the exams. The conversation moved to the way our senior secondary certificates are structured, and the possible negative effects that has on mathematics teaching in the classroom. I was fairly quiet last night because others had basically the same ideas as me, but I’d like to take some time to reflect on it a bit more.

It’s undeniable that I teach my Year 12s differently to my Year 9s. With lower years, I feel I have the time to play around with ideas, to set kids to discover mathematical concepts themselves and to ‘waste’ lessons on arguing whether a square should be allowed to be called a rectangle (yeah, this did actually happen recently).

But the constant pressure to “cover everything before the exam” makes it all too easy to fall into the old pattern of lecture-then-questions-then-exam-practice. And that’s not just me, the kids come into class expecting lessons to be like that. Having discussions is like pulling teeth. Whenever I try to include more investigation type activities, the looks on the kids faces basically say “Come on Mr. Carter, just tell us the answer so we can get back to the textbook exercises.” Not that they want to do the exercises, but anything that doesn’t look like exam questions feels like a waste of time.

Which is a problem. Because doing exam questions doesn’t feel like doing mathematics to me.

Then there’s the way the calculator impacts the class. The calculator requirement is so explicit in Victoria it’s even in the subject name: Mathematical Methods (CAS). TI really do have a great deal going. I’d love the freedom to use whatever tools I want, such as Desmos, Geogebra or just coloured paper for that matter. But no, the exam requires the calculator so we need to practice using the calculator.

I really do admire the US teachers who have to deal with high-stakes testing at every grade. The very existance of the exam changes the approach teachers and students take to maths, whether they like it or not. Last night, primary teachers also mentioned similar pressure to ensure kids are “ready for high school”, something I’d never really thought of before (that could be because I’m in a P-12 school).

Then there’s NAPLAN. I think it’s probably best if I don’t say anything about that.

So, enough ranting. What do we do? There needs to be a culture change, so that learning is the primary driver in the senior secondary classroom, not assessment. We can’t deny the importance of that assessment – the kids do depend on it for university entry after all.

But good teaching and learning is good teaching and learning – the existance of the exam doesn’t change that. The culture change needs to begin with us, the teachers who believe in it. The exams are not going away any time soon. But surely if we produce students who actually understand mathematical thinking and discovery, they can do just as well, if not better, than the kids who only know how to reproduce rules and examples.

Yes, the exams push hard. But we can push back. And in my classroom, that begins with me.