## And the weeks fly by

**Oct. 5th, 2012 | 12:17 pm**

**location:** US, Wisconsin, Madison, Dane, N Paterson St, 323

I'm honestly not sure how things are going teaching-wise. I know that there are some students who're enjoying the class, and being challenged by it, but there are also some who don't seem very engaged.

One thing I've been doing is setting a short online quiz on each section, which has to be taken before the class in which we cover the material. This means the students are doing the required reading, which is great, but they don't like it much! I also notice a difference in the depth of questions on the days when they've had to do a quiz before class - and the quiz questions that stumped them are a good focus for discussion. I've been a bit stunned about how difficult they find reading a few pages in the textbook and answering questions on it. Don't they teach comprehension in schools anymore?

I bought a short handbook on teaching geometry, which has an example of a complex instruction-type task: drawing a bunch of quadrilaterals, then joining the midpoint of each side, and working out a conjecture about the shape of the resulting quadrilateral. We started this on Monday, and the reports are due in today - I'm interested to see what the come up with.

*Posted via LiveJournal app for iPad.*

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## Teaching geometry, round 2.

**Sep. 19th, 2012 | 03:48 pm**

**mood:** tired

I've been wanting to incorporate my reading about complex instruction, but so far have not got there. However, when working in groups I've asked the students to be aware of the group dynamics - is there someone who is monopolising? someone who's not saying anything? I also asked them to try to see that neither of these situations occurs to much. I've noticed some good cases of a student drawing another student out. Maybe next week I'll formalise the group work a little more - if I can find the time and energy!

Monday was the first quiz. I decided to make a bit more of a formal exercise of the grading, so once the students were in groups I explained that they need to put together a model solution for their question, with a grading rubric, then grade the questions. I also asked them to categorize each error as arithmetic (or careless) or conceptual, and if conceptual, what concept was misunderstood, and how. Each group will hand in a report summarizing their findings, as well as noting any overall conceptual error they noticed across the class, and any particularly good solutions (very clear, or solving the problem in a novel way). They seemed to take to it all with gusto, and the groups were busily discussing their grading when I came into the classroom today.

Last week we spent one of the classes in the computer lab, working through a basic geogebra tutorial. There were quite a few comments of 'This is fun!', which was great. The quiz questions which they handed in on Friday were generally of a high standard, with a few really outstanding, interesting problems. I overheard a few discussions of possible quiz questions today - it seems like there's a bit of competition going for who can come up with the best one!

Today I talked a little about the van Hiele model of geometric understanding - it's the pedagogical model on which the Common Core standards are based, and provides a useful framework for teaching geometry. The basic idea is the geometric concepts and abilities are developed sequentially, and it's necessary to master each level before moving to the next. The levels are, roughly,

Level 0: Visual. Recognizing basic shapes by appearance, without attention to parts, attributes or properties (so might identify any figure with 3 vertices as a triangle, even if the sides aren't straight, but might not identify a triangle as such if it is in a funny orientation, or is extremely long and thin).

Level 1: Analysis. Recognizing and naming the properties of geometric figures, but not understanding relationships between different types of shapes (so will not identify a square as a type of rectangle.)

Level 2: Abstract: Able to logically order properties, create meaningful definitions and make informal arguments (so can deduce that a square is a parallelogram, and also a rectangle).

Level 3: Students can construct proofs, understand the roles of axioms and derinitions, and know the meaning of necessary and sufficient conditions. High school geometry.

Level 4: Rigor. Students understand the formal aspects of deduction, can compare mathematical systems, understand non-Euclidean systems, and use indirect proof and proof by contradiction.

As I mentioned each level, we looked at the corresponding questions on the van Hiele geometry test, which give 5 questions for assessing understanding at each level. I think maybe I didn't give the students enough time to read the questions, although we were able to see some practical examples of what is expected at each level.

Having said all that, I used scroobius's awesome elfling as an example of someone who seems to have been born knowing Level 1 concepts.

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*(no subject)*

**Jul. 31st, 2012 | 12:10 pm**

**mood:** excited

From the little reading I've done on complex instruction, I decided to give the class more challenging group problems to work on ('group-worthy' problems). I figured that I only have two weeks left of teaching, and no energy, so sorting out the group dynamics probably isn't going to happen, but I could relatively easily find more challenging group work. So today I picked for of the most difficult area calculation problems, and assigned one to each group. I've also been reading that students on the whole prefer it if the teacher doesn't give hints, just lets them figure it out for themselves. So I sat myself down at the front of the class, and studiously ignored the plaintive meepings. An interesting thing happened! The problems were really tricky - certainly not the kind of thing that could be solved by inspection. Initially, I heard the only the 'strong' students in each group talking. Then they trailed off, and there was silence for a while. Then other people in each group started putting forward ideas, and suddenly everyone was chattering away, and gradually I heard cries of "Oh! That's it! We got it!". I asked each group how they got to the solution, and it was a definite team effort, with useful insights coming from multiple people.

Thought provoking.

After class I was chatting with one of the students who's doing his degree at Oshkosh (just taking this course in Madison because Oshkosh doesn't offer it during the summer). He was telling me about the previous course he did at Oshkosh, which was entirely group based; it sounded really excellent. He's going to bring me the textbook if he can find it, so I can see how it is structured. I really feel like I'm onto something here...

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## Why doesn't lj have an 'irascible' mood icon?

**Jul. 30th, 2012 | 11:07 pm**

**mood:** Hot 'n bothered

Link hopping from this article posted by

**schedule5**, I've started reading about 'complex instruction', which is structured around the notion that "there are two essential ideas to effective group work: 1) having a group-worthy task, and 2) recognizing and taking conscious steps to alleviate status issues that arise in any group activity, no matter how well-designed." I'm waiting for the main reference book on the subject, "Designing groupwork: strategies for heterogeneous classrooms" by E. Cohen, to be delivered to the maths library. So far, I'm taking comfort from the fact that it's a well known problem that status issues arise, so it's not just my classroom skills that suck.

I don't think that it helps that I'm dog tired at the moment; teaching 4 days a week, and studying flat out for my algebra qual, which is in THREE WEEKS, all in this crazy summer heat that just doesn't let up, has frankly just been exhausting. I'm finding it very difficult not to just give a straight lecture on each day's topic and answer all questions with the right answer, rather planning exciting hands-on activities and getting the class to develop a consensus answer to each question. Bah.

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## Week 6.5

**Jul. 25th, 2012 | 06:39 pm**

**mood:** artistic

Yesterday in class we were talking about similar triangles and indirect measurement, so I brought several tapemeasures to class, and got groups to work through the process of measuring a distance by measuring the apparent length of a meter stick placed at the end of the distance to be measured. I took the activity out of the activity manual that's one of the prescribed texts for the course, and modified it so that they did the calculation in 3 different ways. I was mainly curious to see what they thought of it - I didn't think it was a particularly well thoughtout activity. But the response was quite positive, with lots of discussion around getting the correct sketch of the problem, and how to solve with each method. Also a good discussion afterwards about how to make it a better posed activity, and ways to make it appropriate for different age groups.

Today we talked about the link between geometry and algebra. What I found interesting was how absorbed they became when I was telling the story of the ancient Greeks, and how they felt about mathematics - that for them mathematics was a part of their religion and philosophy, and ideas about 'ideal shapes' influenced how they thought about and did mathematics. I mentioned the platonic realm, where 'ideal shapes' live, and one student suggested that that's probably where good mathematicians go when they die :-)

There's a very different look on students' faces when they're absorbed as they were today, and when they're concentrating but not really engaged in a class. I always try to make every class a story, with exciting bits, and unexpected twists, and an overall story arc. I've been more concious of doing this since reading 'Mathematics and humour' several years ago. The guy who wrote it does an interesting analysis of why you'll often see mathematicians smiling or laughing at the end of hearing a proof; essentially, a good proof has the same structure as a good joke, with an unexpected or pleasing punchline. I've tried to build on this idea when planning my classes, so that there's a logical and compelling structure underlying the material that will pull the students in without them even realising it. Sometimes it even works! Today we'd got as far as building a number line with integers, only using geometric construction. I asked the class how we could put fractions on the number line. There were no ideas, but one of the students said excitedly 'Tell us, tell us! I can see you have a cunning plan! What is it?'. He really was excited to know! And when I showed them, there were a lot of ... giggles ... really, because it was a cute construction.

One of the students was a maths major, so he's pretty comfortable with the material. We've been doing some rather cute proofs with unexpected twists, and he has the endearing habit of giving a fencer's 'Sa-sa!' when we get to the punch line. I really love this; I'm not sure that the rest of the class gets appreciates it, but I love that there is one person who is seeing the elegance of the game.

After class one of the students stayed behind to tell me that she gets very frustrated when the maths major answers questions in class, because she can't understand what he's saying, so she has to ask me to answer the question again, which makes her feel stupid. It's only relatively recently that he's started answering questions in class, and he's obviously trying hard to be comprehensible, but still hasn't quite geared down to elementary math from university math. When he gives an aswer I try to guide him to be more comprehensible by asking appropriate questions, but obviously it's not helping enough. I don't want to discourage him from speaking - after all, he needs to learn how to answer question appropriately, so I'm not quite sure what to do to avoid frustrating the other students. This is a tricky course to handle: teaching the mathematical content, the methods of teaching the content and on top of that how to actually teach the content! I wish there was a course I could take on the maths education education!

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## Epic Teaching Fail

**Jul. 18th, 2012 | 09:55 pm**

**mood:** contemplative

Yes, well. The last week has been a bit of a write-off, as far as anything other than bare minimum teaching goes: the temperature has been in the high 30s-40s, without getting cooler at night, and I've just not been coping (and neither has my aircon). Even the aircon in the classroom has battled, and I noticed that everyone has been grumpy and lethargic (on Monday I had a couple of girls beg me to let them present sitting down, rather than standing in front of the board). Today when it was cooler, suddenly there was much more enthusiasm. I've kept meaning to update this blog, but it's been too hot to sit in my study, and now I'm just going to abandon the last week and move on.

So yesterday I made a fairly epic teaching fail. It was the day we talked about Pythagoras's Theorem, so beforehand I found 4 different proofs. One of them is similar to Euclid's original proof, but only uses areas of triangles, which we covered on Monday. I was very excited, because I thought is was a sweet proof, and so appropriate. So I wrote down The Theorem, and launched excitedly into the proof, and it was only after about 10 minutes of me talking, and them saying 'What?', that I realised I hadn't drawn The Diagram that was necessary to make the proof even vaguely comprehensible. Basically, I'd just written down a^2+b^2=c^2, without explaining that it's actually a theorem about the

__areas__of squares, even though it talks about side lengths. Wow, I felt stupid. I could feel that the class was grumpy and losing focus, so I cut my losses. I'd put together a worksheet with 3 other proofs, and I got them to work in groups and then present each proof; that was better, but there were still dirty looks coming my way at the end of the class.

I was chatting to some friends last night about the teaching fail, and one of them said 'Well, clearly that was an awesome teaching moment! You could've said "And now you know how not to teach this section!"'. I thought about that for a bit, and this morning I started the class by saying 'Ok, let's talk about my teaching fail yesterday. What did I do wrong? How could I have avoided it? How would you teach that section' etc. A pretty good discussion ensued, so at least it wasn't a complete waste. But I'm sad they didn't get to see the loveliness of the proof.

I think there was also something else going on yesterday that alienated the class. I'm usually pretty careful not to project 'math geek' when I teach; I make sure that I tell them about things I struggle with, and I work really hard to never just look at a problem and solve it immediately (even when I can). I alway talk through my thought process, so that they never have to deal with some annoying lecturer who just magically knows the answer. But I was so enthusiastic about the proof that I forgot to do that, and also I wasn't aware early enough that I'd lost them. So they got to see me in full 'math geek' mode, which I'm guessing is pretty alienating. Today I slowed right down, and even made the two math whizzes in the class talk through their thought processes in detail when they were answering questions, and the vibe overall was completely different. That in combination with the starting discussion made me realise that a good addition to this course could be getting the students to critique how I present stuff, and discuss how to make it better. Scary thought!

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## Detail oriented

**Jul. 10th, 2012 | 12:08 am**

**mood:** tired

So the class did really well at finding the mistakes in their midterms after I gave them the grades for each question, but before they saw their graded version! I was very impressed - there was only one student who didn't find the mistakes she'd made! I asked them how they felt about it, and it seems the general consensus was that it was intimidating and scary, but that they learned more than just looking through the graded version. A few students did say that having more of a hint for where the mistake occurred would be helpful; I'll think about that.

Today I did several proofs on the board, trying to work into the method the ideas I had yesterday about teaching geometry problem solving. Before diving into a proof, we figured out what specific thing we needed to prove, and then wrote a possible list of ways that thing could be shown. From there, thinking about facts we knew, we decided which routes looked viable. I worked each proof on two blackboards - one for rough work and thoughts, and one for 'final answer'. Got some good interaction going, but the faster students were a bit bored and frustrated. I need to make it more about 'how to teach the process', and less about 'solve this problem', I think. Oh - we also used a lot of colour on the figures - marking things that are given in one colour, things to be proved in another. Seemed helpful; will elicit more feedback tomorrow.

I'm still a section behind. Gah!

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## Stepping stones

**Jul. 9th, 2012 | 06:56 pm**

**mood:** contemplative

Much better for a weekend away! Today we talked about congruences, a lot. One of the things I found myself going on about, which isn't really covered in the book, is how learning about congruence (i.e. being able to tell that two figures are the same) is really all about orientation, and learning to mentally re-orient shapes. It's an important step for students to realise that the orientation of a figure is not really important.

I also suddenly realised why we did an entire section on geometry constructions: so that we can figure out what auxiliary lines can and can't be added to a figure, and also, given limited information about a triangle (for example) whether we can draw a unique triangle that satisfies the information, or not (which leads to the various tests for congruence).

It was a homework day and there were questions about one particular problem, so I put it on the board and got the class to help me complete it. After we were done, two of the students mentioned that they would not have know how to solve the problem, since it really required solving two equations simultaneously. I need to check when solving simultaneous equations comes into the syllabus - before 8th grade? And also, I think I need to put together a worksheet that helps the students figure out multistep proofs. I'm thinking maybe problems that include

- filling in the blanks in a given proof
- giving the correct order for a proof whose lines have been muddled up
- drawing a mind map of possible solutions, before actually solving the problem (is this something that I can make more structured?)
- writing a list of the facts that might be applicable to the problem

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## Not my finest.

**Jul. 5th, 2012 | 05:39 pm**

**mood:** cranky

Ugh. It was so hot that the classroom aircon just couldn't hack it; by the end of the class I was a limp, soggy rag of minimal math ability. I couldn't keep my focus or enthusiasm during the class, either, so was not very satisfactory, and ended up a section behind where I should be. I think my goal of two sections per class, which leaves me Mondays free for review, is overly optimistic, particularly as the material is getting harder. I'm very glad I'm going away for the weekend, as I'm just hot, tired and grumpy!

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## Testing, 123

**Jul. 3rd, 2012 | 09:48 pm**

**mood:** exanimate

I think overall it was still a bit long; I ended up giving them an added 15 minutes to the 60 originally intended. The general comment was that it wasn't very similar to the past exam papers available through the library, and that was disconcerting. I'd looked at those, and felt that it was similar, so I need to probe a little more and find out what seemed different.

I scanned all the exams before I graded them, and then once I'd graded them I emailed each student their ungraded exam, with their score for each question. For Thursday, they have to provide corrections for each question they didn't get full marks for, and the corrections will be graded. When I told them about this after the exam they were enthusiastic about this; we'll see how well they do at figuring out what they did wrong. Overall the grades were good - only one in the 50s, the rest in the 70s and above. I'm a bit exhausted from the grading marathon, so will write up more detailed observations about answers another time.

I'm very glad tomorrow is Independence Day. This teaching lark is not for sissies! Also, the temperature has been in the high 30s for several days in a row; today it reached 39. Since my apartment gets full afternoon sun, my little aircon has been battling. There are a couple of things I want to write about at some point: maths as narrative, and the psychology of teaching maths. But right now I think I'm going to have a cold shower. Again.