All entries for Saturday 06 November 2010
November 06, 2010
A Mathematician's Lament
A long time ago I read something that really summed up the dire state of Maths education. Happily, I came across the exact same thing quite recently. Please, read this:
It really explains so well what has gone wrong. I still, although I'm quite used to it by now, get shocked by how few people think they can "do" maths, and how "difficult" maths is perceived to me. If I had to sum up the process of "doing maths" I would say it is
1. Pattern spotting
3. Proving/disproving the conjecture.
Humans are naturally great at 1, just watch an episode of "Deal or no deal?" to see the ridiculous and elaborate patterns people have spotted and are basing their decisions on.
2 is also very easy. All you have to do is make up a rule. Doesn't have to be true and you don't have to give any evidence whatsoever towards it being true (that's part 3). 3 gets nasty, yes. It requires a great deal of creativity and perseverance. For research mathematicians, they don't even know if their conjecture is true or false, so may spend ages attempting a proof that is doomed to fail simply because their conjecture is false. And you can't tell from the conjecture how easy the proof will be either. If you're given something to prove that is at the right level for you, then proving stuff is great. You have to be so creative, much of the time you end up bringing in areas of maths that seem totally unrelated. It gets really frustrating ("character building!") when you can't do it though, which is why I think part 3 is something to be thankful that someone has done for you.
This is why maths shouldn't be seen as "difficult". Maths is reliable, it always works. If some conjecture has been proven then you can use it, build all sorts of things on it, and it won't fail you. In science, gradually new theories replaces an old ones (the earth is flat, anyone?) so what you're learning isn't concretely true like maths is. Plus if you're unlucky, something will be changed while you're still around (Pluto isn'ta planet!?). Maths doesn't get outdated, like media or technology or ICT might be. Much of the maths we learn is very very old. It gets built on and added to, rather than be replaced. In short, the fundamentals of what makes maths maths are awesome and everyone should have a natural affinity towards it.
But what we learn, and teach, in school rarely fits into one of those 3 catergories. It's what could be put as
4. Applying the theorem.
(Conjectures get an upgrade to theorem when they are proven to be true.) Applying a theorem is what much of school maths seems to be. Here is Pythagoras' theorem. Now apply it to all these very boring little triangles. If I'm really trying I'll make you find a variety of sides, vary the notation, and perhaps even draw some triangles at funny angles. I'm going to try really hard with my teaching to do more of parts 1-3 (especially 2, who can'tmake up any old wild claim?).
Reflections on Week 5
It's been a long and tiring week. Monday was INSET. It was on assessment (hello, Master's topic!) and focused on the importance of formative assessment. Much was made of Black and Wiliams findings and suggestions. I didn't learn much about assessment itself, having spent the previous week studying this stuff, but that didn't bother me at all. I completely buy into formative assessment so I was far more interested in seeing the school's take on it and see if the Maths Department supported it. Everyone was enthusiastic about it - my Professional Mentor even has a stash of traffic light cups and lolly sticks to borrow if I want to try them out.
After the whole staff sessions we went into Departments to discuss how to implement it. Some of them have been doing peer-assessment already as part of their previous CPD. Lots of them liked the names-on-sticks idea but will make laminated versions to save money, or use a random name generator on their laptop instead. I'm going to do names on sticks, probably with my Year 7's. I also love mini-whiteboards so I'll be using those where I can. I'm debating using the traffic light cups with my year 7's since they behave well (there will be a temptation to mess around/break them) but I'm afraid of accidentally not noticing someone sat with theirs on red and them getting upset with the lack of help.
I've been given the task of taking past papers for the final GCSE exams and splitting the questions up into categories. It's a big o'task but right up my street - I used to do it myself for my own exams.
On Tuesday and Wednesday I was back in uni. More of the same really. Some more peer assessment. There was what I can only describe as a sales pitch by some calculator company. I'm quite morally opposed to that sort of thing, though, given what it was, it was done very well. It was by a current maths teacher who uses the stuff himself (it's some sort of fancy calculator which can draw graphs, do dynamic geometry, spreadsheets etc. It could be wirelessly hooked up to the teachers computer which was nice - so the teacher could take screenshots of everyone calculator to monitor progress). The teacher running it was excellent and gave me some excellent ideas for teaching.
The main idea he used was the idea of much of maths being special cases. For example, Pythagoras' theorem is a special case for squares attached to a triangle, when there is a 90 degree angle. So he'd give them some dynamic geometry and ask them to find out when the sum of two of the areas (which it calculated for you) equals the other. Then get them to discover it's when there is a 90 degree angle. I really liked it, it gives far more meaning to the theorem as just a string of letters doing stuff. It also hopefully avoids the misconceptions that Pythagoras works for any triangle, and emphasises which sides it is which are added. (That came up on the INSET day actually, we had a question "For which of these triangles does a^2 + b^2 = c^2 hold?" with a load of right angled triangles with the a, b, and c labelled in various ways. Lots of staff said they all do because pythagoras holds for right angled triangles, overlooking the fact that c must be the longest side.)
At my school, classes get to spend one lesson a week in the maths computer room so I think later in the term I'll write some special lessons to do in there with these sort of ideas. I'm not interested in used the calculators (we can borrow them from the company if we want to) since I could spent that time getting better at the software I have access too - I doubt I'll ever see those calculators again so I'm not going to waste that time getting used to using them.
Thursday and Friday were back in school. It's quite daunting to know I'll be there until Christmas now with no breaks, also quite saddening that I'll be leaving there in such a short space of time. I've got more involved with the classes now, I haven't be actually given anything to do with them yet but managed to sneak a bit in anyway. I went to a Year 11 class for the first time, their teacher wasn't in. By lucky coincidence I met the (maths-phobic) cover teacher on the INSET day and she happily let me take a bigger role in the lesson. They were doing past papers for their imminent exam. I marked them as and when they finished (don't know if I was supposed to, oh well) which was very useful for me to see what they are asked to do in this exam, and how good they are. Some of them finished in half the time, and got full marks. Some of them only missed a few marks. I wrote comments on them like "ask xxxx to explain how she did this" (xxx being someone I'd marked and they'd gotten that question right) next to the empty question boxes, but the girls were just looking at the front page and saying "12/15 is good enough"! Not impressed! They'd also been allowed to work together which meant I marked the same incorrect answers time and time again. One of them didn't even make sense which makes it look like they just copied it down without even thinking. They also finished early and were told to get on with homework. I felt bad about this but could hardly contradict what the cover teacher had said, in any case my ideas (more practice papers; go through the worst done questions on the board) wouldn't had benefited all those who got full marks, which is most of those with nothing to do. One girl saw me specially to ask for more exam practice so I got her another paper. I felt really bad for her being in that situation. I'm used to having to try to chivvy pupils into the mood to learn, to have someone ready and willing to learn but prevented from doing do was sad to see.
I saw my Year 12's again and I feel more comfortable with them now. I've found with all new classes, as a teacher or as support, there is a period of time where I have to prove myself before they will accept me as good enough. I know I did it as a pupil! It was frustrating enough to be stuck enough on something to put my hand up for help, let alone to have the student teacher wander over, not understand what I'm saying as quickly as the teacher would, then sometimes not even be able to help so I'd have to wait for the teacher anyway. What a waste of time! I got more involved this time providing help and it's made me very thankful for my maths degree and the confidence it gave me with maths as "easy" as C1. I don't like to read through a pupils' answers at that level since the answer is long and spotting a mistake in a long piece of maths is hard. Instead I work through their method next to them without looking at their working. I get them to follow what I say out loud, confirming they have the same answer as me at various "check points" in the method. At some point, there is a discrepancy and we can then look back through the lines to see what happened. In this lesson, there was a tangent used instead of a normal, a x value substituted in for a y value, and someone taking something= -2/17, multiplying -17 across and leaving -2. Worryingly, in the last one, she didn't think that was a mistake (in the previous two there was a "oh no, I've made a stupid mistake" moment). As long as she's sat in the same place (learning names and faces is really hard!) I'll check on that idea again.
Next week is a busy week and hence this weekend will be a busy one. The more I do now, the earlier I can go to bed in the week! I'm doing a starter on Monday with my Year 7's. I'm running after school revision classes on Wednesday and Friday for the GCSE exam (I'm well excited about this). I have to make a poster to advertise said classes (less excited about this - computer designed posters are not my forte. I might give up and make one by hand with my paints and scan it in). I need to plan my Year 12 lessons to roughly divide the material between the lessons I have. I have two 2-hour lessons with them to do C1 sequences and series. It's less time than I was expecting but it's not that hard a topic, so as long as I present it in a way that's not scary (mental blocks against a bit of maths is far more effective at preventing learning than lack of ability) all should be ok. I'm really lucky that my mentor is happy to accept late planning. The way I like to work is to have a vague plan of concepts that need to be taught over a certain time frame. I put them in order and divide it up roughly. But I can only do a full lesson plan after the lesson previously, in order to match to what they need next. I also like to mentally note misconceptions and mistakes which are being made and check up on them next time (like the -2/17 girl in my C1 class). This way I can pose a similar question to the class in a "make sure you can do this type of thing" way rather than a "someone here made this mistake, all tell her she's wrong" sort of way. My mentor is happy for me to give her one lesson plan at a time, which makes for some tight turn arounds when I have the same class the following day! I'm so very lucky for this to be the case, one trainee has to have in plans 2 weeks in advance!