### Drawing parallels

05:05 pm
(Section 3.3)

Really, the textbook makes me cross sometimes! So we've covered parallelograms and trapezoids, and we know that the interior angles of a parallelogram or trapezoid sum to 180 degrees. So you'd think that the textbook would use this to introduce a bunch of facts about parallel lines, since they basically all follow from this. But no! So I went off road today and showed them how to get the parallel line facts from stuff we already know, rather than the way the silly textbook does it. So there.

Then is was time for another worksheet, and the 5 students who didn't present solutions yesterday got to do their solutions on the blackboards.

I also gave everyone an index card, and asked them to write down at least one thing about the way I was running the course that wasn't working for them, and one thing that was. I do this every semester I teach, and so far it hasn't gotten easier; it's still nerve wracking when they hand the cards back.

The feedback overall was helpful (it always is) and nobody hated everything, so that's ok. There are several students who need to get to work soon after class, and one of them wrote that he gets stressed if I go even a little over time, since he only has 20 minutes to get to teach his first graders. I'm going to see if people would mind starting/finishing 10 minute earlier; that should make things easier for all concerned. A couple of students mentioned that they weren't comfortable solving problems by themselves in front of the class yet, and would prefer to keep doing it in pairs/groups for a bit. I knew I was pushing them this week, so I'll ease off a bit. At least they've all done it once, though, so hopefully next time won't be so scary. There were a couple of comments about the length of the homework assignments (too long!). I'm not going to change that - I think Monday will be the day for my 'to get a strong maths brain, you have to do plenty of reps, just like if you want strong biceps, you have to do plenty of reps' talk. One student said that I could ask more questions during the 'lecture' portion. Will work on that - I do try, but at the same time I'm trying to get through the material quickly, so we can get on to interactive stuff. Which now that I write it down is just silly. Doh! After I'd graded the first homework set, I chose the best solution to each problem and put together an answer key from those; one student said she wasn't happy having other students able to see some of her homework. Since I'm spending too much time on the course already, I'd decided not to do that again anyway - it's quicker for me to just scan my solution key. So that's ok! Some students would like more time for asking questions about the homework; I need to factor that in.

The wiki, which has solutions to homework problems, the daily review, a gallery of blackboard solutions, and a list of definitions and facts learned to date, seems universally popular. Next time I teach this, I think I'll make the students responsible for keeping it up to date; to me it still feels a bit too much like my website, rather than the course wiki. They also like writing quiz questions and grading, and most people like that the focus in class is on doing problems, rather than just going over the required material.

So overall, I think it's going well. I'll address the problems that I can, and make sure to keep doing what is working. They write their first midterm next week, so we'll see how things are really going then. Eeep.

### The truth about proofs

04:59 pm
(Section 3.1)
Today I put a geometry problem on the board, and wrote down the textbook's suggested 'Teacher's solution'. Then we talked abou the anatomy of a good solution, and using correct abbreviations for reasons, and some other stuff. Then I gave them a worksheet with 10 problems, and had them work on those while I copied the problems onto the blackboards. When they were done, I got a volunteer for each of the first 5 problems, and they wrote up their solutions and talked the class through what they'd done. I particularly didn't want them to talk about the problems together, so that they might not have the solution correct. They were somewhat nervous, but I think I kept the atmosphere light and friendly, and let the other students ask questions and point out errors, or ways that the proof could be clearer. At the end, there were 5 beautiful solutions on the board, and the presenters looked pretty happy. I took photos, and posted them on the wiki in our Gallery. There wasn't time for the other 5 students to do their problems; we'll do that tomorrow. I know that I was pushing the students a bit today, since they haven't had to work individually in front of the class today; I hope it was ok.

### Teaching the controversy

08:43 pm
(Sections 2.4 & 2.5)

Hmm. It turns out that despite never having taught this material before, I have firm view about what is right, and what is wrong, and I will tell the students when I think that the textbook is wrong. Where did this come from? Today we talked about quadrilaterals, and the 'hierarchy' - a square is a special kind of parallelogram etc. For some reason, the textbook declares that a trapezoid (that's trapezium to us UK English speakers) is a quadrilateral with one and only one pair of opposite sides parallel - so a parallelogram is not a trapezoid. This makes me cross. Before class I asked The Google, and it turns out that there are different views on this, so I did pointed that out, and explained why I felt it makes more sense to define a trapezoid as a quadrilateral with at least one pair of opposite sides parallel. But I'm still a bit surprised by vehemence of my reaction.

With quadrilaterals out of the way it was time for geometric construction. Since I'd asked them specifically to read the section before class, I made them put their textbooks away, and head for the blackboards. I'd scoured the department for all the blackboard rulers and compasses I could find, so there were enough that they could work in pairs. I gave each pair a construction to draw (e.g given a line and a point on the line, using only straight edge and compass construct a perpendicular to the line passing through the point). They had a lot of fun, and there was a fair amount of head scratching, since they couldn't consult their notes. Hee! The guy who was a math major had taken a higher level geometric construction course, so he was able to show a few really elegant constructions that we hadn't seen before. I added a few extra challenging constructions on some empty blackboards, and quite a few of them stuck around after class playing.

### Breaking news! Students love grading!

08:42 pm
Mondays are 'review and quiz' day. We started out with each person who was notetaker last week giving a summary of what was covered, so that we had about 15 minutes of review of last week's material. Then they wrote a 20 minute quiz, on that same material. There were three questions, so after the quiz I got them to divide into three groups, an allocated one question to each group. While they put together teacher's solutions for their question, and decided on a grading scheme, I dashed upstairs and photocopied all the quizzes, and then anonymized them by cutting off the names. Each group got the 10 solutions to 'their' question, and had to grade them, as a group. Wow! They really loved this! They said they'd never thought about things from the teacher's perspective, and realised just how many different ways there were to answer a problem. Also, quite a few mentioned that they'd assumed that it would be obvious how to grade a maths question - but when there are a bunch of different ways, or someone makes a mistake half-way through, suddenly it's not clear at all. The group who was grading the 'find the angle' problem got into a pretty intense argument; it was great!

### Yes, we really do need to talk about proofs

01:24 pm
I spent a lot of time grading today, and what I saw affirmed my realisation that whatever they've been doing in the previous (compulsory) maths courses has not been teaching them to write good 'Teacher's Solutions'. So what exactly is a good 'Teacher's Solution'? Well, according to the textbook a Teacher's Solution includes:
• a labeled picture that describes the entire problem (generally a bar diagram for word problems, or a redrawing of the figure for a geometry problem) with the unknown clearly labeled
• computations explained using simple language
• a clearly stated answer
This works for me, and we're going to practice it next week, oh yes we are.

The quiz questions were really well done, though - they obviously put a lot of effort into thinking about what to ask, and how to ask it. I'm pretty sure they've never allocated points though; maybe next week we can talk about that.

Homework all graded, scanned and emailed. Going to study algebra for the rest of the weekend...

### We need to talk about proofs.

07:08 pm
(Section 2.2, 2.3)

Aha! I knew they weren't doing the reading! Little rats! So today we talked about triangles, and introduced 'sum of interior angles is 180 degrees' and 'exterior angle equals sum of interior opposite angles'. The textbook has some lovely hands on demonstrations for these - for example take a paper triangle, tear off each angle, and rearrange them with the vertices together - and voila, you see that they add up to a straight line. There are a couple of other cute tricks, too. So I gave everyone a piece of paper, and asked them to turn it into any old triangle. Then I picked random people to show me how to use the triangle to 'prove' one of the facts - and no-one could do it! On the up side, when I demonstrated, they all went 'Oooh, I wish we'd done that when I was at school!'. I mentioned that I was expecting them to have read the necessary sections before each class, and they looked a bit bashful; we'll see how things go next week.

Then they worked on a couple of standard 'find the angle' problems, and presented solutions in groups. We need to talk about how to write solutions! I'm going to set aside a day next week for that, because what I saw made me wince (inside. Outwardly I like to think I was my usual calm, collected self. Rofl).

I'd photocopied a stack of designs with different symmetries for them to play with but we didn't get that far. My time management still needs work.

First homework handed in today. My plan is to grade it and email scans of the graded stuff back to the students, so that they can use it to prepare for the quiz on Monday. But right now, I need a nap. First week of teaching done, and I'm knackered.

### The many angled ones

07:51 pm
(Section 1.4, 2.1)

Lots of definitions and facts to go through today - all about angles and circles. I've been thinking for a while about blackboard style, and how best to present material. Just writing down a running commentary of what I'm talking about is not, I think, very helpful. So instead, today, at the start of the class I labeled one blackboard 'Teaching', one 'Definitions & Notation' and one 'Facts & abbreviations', and as I worked my way through the material I added info under the appropriate headings. At the end of the class I asked the students if that way of working worked for them. They all said they really like it, because it gives a clear summary of what's been going on, but it was confusing to know how to take notes, since I was going backwards and forwards, and adding stuff to each blackboard. So tomorrow I'll tell them to have three pages, labeled appropriately, and add stuff as I do!

They worked on today's worksheet together; it was mainly to do with using a protractor, being able to copy a drawn angle etc. Then I got them to talk me through the answers, while I pretended to be a student using a protractor for the first time (so I got to play with the big blackboard protractor for the first time). Turns out that using a blackboard protractor is harder than you might think; there are lots of things that can go wrong...

EDIT: a student just emailed me her quiz question, due tomorrow. It is beautifully done using GeoGebra. She said she got completely wrapped up in creating it, and had a lot of fun. Since she was one of the students who said 'all of maths' for what she feels uncomfortable with, I feel very happy.

### Not want.

11:21 pm
It is 11pm, and 32 degrees in my lounge. England, would you please come and fetch your damn summer, it has wandered over the Atlantic and is breeding with ours.

### Pride cometh ...

05:00 pm

(Section 1.2, 1.3)

Started out by getting groups of students to present their answers to the worksheet problems that we didn’t finish yesterday. Groups of three or four, with one student in charge of writing it up on the board, one in charge of talking us through it, and the others there for moral support. Seemed to work well; encouraged the watching students to ask questions, and suggest alternative answers if they did things differently.

Then I talked way too much. It was a real lesson to see the difference in the students’ faces compared with when they were discussing things and talking amongst themselves about the worksheet problems. I’ve asked them to read the sections that will be covered each day, and make notes, so that it’s not all new and I can go through it relatively quickly. I asked questions as I went along, and did get good responses, but they still looked passive and bored. I'd been talking about measuring length - the concepts that go into teaching this at various grades, what students are expected to know when - and I was going to go on to talk about measuring weight and capacity - the next two sections - but realised that I'd completely lose the class. Instead I got half the class to put together a presentation on teaching weight, using the framework for teaching length that I'd put up on the board, which was in fact a general framework for teaching measurement (Introduce the concepts of units - with hands on experiments- and why they're useful. Introduce standard units, and how to choose the appropriate unit for a particular measurement. Express a measurement as a multiple of the chosen unit. Do problems.) Then other half of the class taught the measuring capacity bit. It worked much better - and there was much hilarity, since I'm completely metric, and they're completely customary. We swapped hints about how to estimate different weights, and worked together to come up with ways to estimate conversions.

Oh, and I started a new thing: each day someone will volunteer to be the chief note-taker, and will give me an at-most two page summary of what was covered the following day, which I’ll post on the wiki.

I also explained about the mysterious ‘quiz question’ question on the homework: every week, as part of their homework, they need to put together a potential quiz question for their classmates on some aspect of the week’s material, give a short description of what it’s testing and how, and write a ‘Teacher’s Solution’ with point allocation. They perked up and seemed enthusiastic, and said no-one’s every asked them to do that before, so yay! I wonder what I’ll get on Thursday with the first batch of homework. I told them a little about GeoGebra (which I have to say is just about the awesomest piece of free software I’ve ever found - go and check it out, math people. It rocks.), and suggested that they might want to play with it to make their quizzes. They looked skeptical. I’d like to run a tutorial on it in the computer lab because I think it could be a fun teaching tool, but I think I’ll leave that for another semester, after I’ve got the hang of things…

### First day!

03:34 pm

(Section 1.1)

First day! Explained how the course will run: Tuesday, Wednesday, Thursday will be new material days, Mondays will be for review and quiz of the previous week’s work. Gave an overview of the topics we’ll cover over the course of the 8 weeks. Got each student to introduce his/her self, and say one thing they’re comfortable with in maths, and one thing they’re not comfortable with. Quite a big range of confidence, even though only 10 students: one maths major, and an engineer, who’re very comfortable in a maths environment, and 4 girls who’ve chosen elementary ed so they don’t need to do much math because they feel they suck at it. Also an already qualified elementary school teacher who is taking this as a CPD course, and wants to learn the ‘best’ way to present the curriculum. Ack; no pressure!

Gave a reasonably short overview of the first section, then handed out a worksheet I’d prepared, and asked them to work on it in groups. When they were done, talked through the worksheet, getting answers from the class and making notes on the board. Lots of discussion about the number of possible configurations of 3 planes in space; I’d brought spare pieces of paper, so people could play with them and explain what they were thinking.

I think I can do this! They seem to be a really lovely bunch of students, and so far have been enthusiastic and responsive.

### The calm before the storm

06:32 pm

So. I'm going to be teaching 'Geometry and Measurement for Elementary School Teachers' during the summer session - 75 minutes a day, Monday-Thursday, for 8 weeks. I've been wanting to get involved with teaching potential teachers for a while, so I'm pretty excited. The course is designed to introduce students from the education department to the elementary school geometry syllabus. It covers learning to measure, geometric figures, finding unknown angles, unknown angle proofs, congruence and similarity, pythagoras, area and volume.

My aim going into the course, unsurprisingly, is to not only make sure that the students understand the material, but also that they feel confident teaching it. For a while now I've felt that a didactic style is not a great way to teach mathematics, so I'm looking at this course as a chance to try out some of my ideas. One of my main issues with lectures is that students see a 'grown up' write down a maths problem on the black board, and straight away solve it correctly. But maths doesn't work that way - or at least not all the time. My suspicion is that students go home, try a homework problem, get stuck, make mistakes, and conclude 'Well, I guess I suck at maths, otherwise I would be able to do this perfectly first time'. Which is not necessarily true at all. Whether my suspicion is true or not, immediately perfect solutions do lead to the other issue I see frequently: students don't learn how to confidently deal with things when they go wrong - which for me is a pretty essential part of mathematics (not to mention life).

While I was in Cape Town I was chatting about teaching with CF, who Knows Stuff about action learning, and later she forwarded me this link, which has made me think quite a lot about how to go about 'teaching' this course. I'm pretty sure that what I need to be is a catalyst, more than a traditional teacher - but how to do this? Given that I have been (and will continue to be) scathing about the 'education diploma' I did at my former cherished institution, which emphasized action learning, I do find it funny that I'm coming round to the idea, albeit somewhat differently from the 'let's spoonfeed the little darlings and make sure they're never, ever uncomfortable' view of the diploma teacher.

Given that I fully believe that the second best way to learn something is to teach it*, it seems to me that putting the students in the role of teacher as much as possible is a good way to go - especially since they're training to be teachers! CF also sent me this link, which includes as a principle of action learning 'not giving advice'. I've been pondering this; I think it might be quite profound.

*The best way is to invent it in the first place.

So. What do I hope my students will be able to do at the end of the course?

• Understand geometric definitions, how to build them intuitive notions, why they're important, and why a definition is formulated the way it is. Use examples to show what weird things can happen when bits of a definition are left out.
• Understand geometric 'facts', and how to use simple teaching tools to give an inductive or deductive 'proof'.
• Be able to confidently solve geometric problems, preferably in several different ways.
• Be able to pass on all of the above to the next generation in a way that will not cause the next generation to want to slit its throat, or refuse to do maths ever again.

### In a nutshell

11:43 pm
1. My dad's chemo has worked; he has been declared lymphoma-free. Super-extra-yay!
2a. I'll be in London from 22 May to 25 May. Yay!
2b. I'll be in Cape Town from 26 May to 10 June. Yay!
3. I've passed my first year of graduate school. Yay!
4. Over the summer I'll be teaching future elementary school teachers how to teach geometry to elementary school kids. Yay! Also, aieeee!

### I still ain'tn't dead,

11:00 pm
just working, mostly*. My Cape Town/London trip over Christmas was absolutely fabulous; I still feel warm and fuzzy thinking about all the happy time spent with friends and family. Lucky, lucky me! This semester feels a lot better than the last one: I'm much happier with my courses (algebra, differential geometry, geometric group theory and auditing analysis), and feel generally more settled. The main work this semester is preparing for the algebra qualifying exam in August: 6 hours, 5 questions, 1 bear of little brain. We have to pass one qualifying exam by the start of our fifth semester or we're out; the exams are offered twice a year, so I have two chances for this one. Eeep.

* And learning to swing dance. Is fun!

One thing I've realised this semester is that I have very definite ideas about teaching maths, which seem to have developed while I wasn't watching over the last twelve or so years of teaching. This has caused some friction between the lecturer I'm TAing for and me, which has been a little stressful, but has also forced me to figure out what I really think. I will almost certainly prattle about it all at some point, just not right now. Now, it's time for pictures:

### Hello world.

08:14 pm
I will be in London 22/23 Dec and 12-14 Jan, and CT 24 Dec-11 Jan. Hurrah. Now you know.

Still no snow here, but plenty maths.

As you were.

### (no subject)

10:03 pm
It turns out that the statement "No no, the lions in Cape Town don't pose any danger, they're fully urbanized", when uttered with suitably deadpan nonchalance, is completely believable. I don't think my logic professor will ever forgive me.

### (no subject)

02:07 pm
In the immortal words of Granny Weatherwax, 'I ait'nt dead'. Just studying. Here's a pic that sums up my life at the moment:

In summary:
1. tissues: M & I broke up
2. algebra textbook: wrote midterm last week
3. analysis textbooks: write midterm next week
4. coffee mugs: needed to pass 2, 3 above
5. 2 + 3 + 4 = Harry Potter when my brain overheats
6. empty plate: contrary to what our professor seems to think, mankind cannot live by analysis alone. Sometimes, one needs poptarts.

That is all. As you were.

### (no subject)

01:15 pm
I have, as I suspect do many bloggers, a detailed list of exciting* posts about my life, just waiting for the time and energy to write them. This is not one of those posts. This is an extremely geeky 'SQUEEEEEEEEEE!' post that goes thusly:
When I was working on my first** PhD, one of the signal processing techniques I investigated was Kalman filtering. I was somewhat miffed that it turned out not to work for what I was doing, since it was nifty, and I would have liked to learn more about it. Time has moved on, and my main interest now is sort of applied algebra - graph theory, group theory, that sort of thing. Anyway, today we got an email saying that Rudolf Kalman, he of the Kalman filter, is going to give a talk here next month. The topic? "A New Direction of Research in System Theory: Rebirth network synthesis via algebra and graphs."

* Ok. Possibly not exciting, since I just spent the last week alternating between unpacking boxes and studying analysis (which went back to being HORRIBLE. Dammit. But now we've moved onto linear functionals, which seem a lot better. Yay.)
**Rofl. Sorry. It just gives me the giggles to be able to say that. I'll stop soon. Probably.

### I've been chuckling all day

04:02 pm
This morning, in my early TA session, someone said the fateful words "But if x is zero, then isn't 1/x equal to zero?". There was a suitably appalled gasp from the rest of the class, and then a lone voice piped up from the back of the room "Oh. My. God. I think I just threw up a little.". Hee. What a great class.

I handed back a quiz I graded to the afternoon TA class. One student had written

when the correct answer was

She felt I should give her partial marks "because there's just a little bit of line missing". Bless.

I'd pretty much decided to follow the sage advice of my commentors, and drop analysis, but on Friday the topology lecture was absolutely appalling and incomprehensible, while the analysis lecture was actually compelling. I worked on analysis over the weekend, and have started to find it quite interesting. So it's bye-bye topology - or at least until next year, when there's a better lecturer. Yesterday was much more manageable only having classes

In other news, I have a bruise on the underside of my foot. I have no idea how it got there, but it's really painful (and purple). Weird.

### (no subject)

08:46 pm
A student just told me I am awesome to the 100th power. It's blatant sucking up, but it's still making me smile. Those little critters can be cute when they put their minds to it.

I TAed my first two calculus I discussion sessions yesterday. At the beginning of each session I got the students to each introduce themselves*, and to say something they struggle with in math(s), and something they find easy in math(s). I was surprised how well they responded to this - they really got into it, and by the time we started working through problems for the homework assignment, they were participating well, and asking questions and debating among themselves. I have to say, it is a real pleasure tutoring something I understand really well, after several years of lecturing difficult stuff. I had fun!

It's good that the TAing is fun and relatively easy, because graduate math(s) courses are TOUGH. At the moment I'm taking 4 - algebra, algebraic topology, real analysis and logic - but I'll drop one some time next week, when I've figured out which I like least. I love the algebra, and although the lecturer isn't too good I'm rather smitten by algebraic topology. At the moment I have no clue about logic - I'm still learning what all the words mean. However, the lecturer is very good**, and there's something compelling about it, even though I don't understand it yet. Real analysis isn't really floating my boat, and I'm tempted to drop it, except it's stuff I really aught to know if I'm ever going to call myself a mathematician. So I don't know which one should go. Perhaps logic? I can always take it next year, when I've got the analysis out the way. Sigh. So much math(s), so little time.

* The calc I course has around 1000 students, but TA sessions are limited to 24. With such a big pool of students, the TA sessions have students from just about every discipline requiring math(s) - engineering, physics, biology, pre-med, BA - so the students are unlikely to already know anyone else in the session.
** And today he was wearing this t-shirt. Is that not an awesome t-shirt? Not as awesome as the one and Robert gave me, obviously, but definitely up there...

### The smug, the sneeze and the spaghetti squash

08:37 am

Last Monday I moved into my apartment. Without doubt the highlight of the a very hot, dusty, tiresome day was finding a little note from the local post office saying I could collect this, a gift from and Robert. It makes me deeply happy, and apparently adds +10 attractiveness, since FIVE guys flirted with me when I wore it last Tuesday. Gosh.

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