Nearing the end

As I reach the end of my first year teaching high school, I am thinking about what was good and what wasn't.

Calculators and decimals. In geometry, many of the students have a hard time finding an exact solution: tan36 becomes a decimal, sqrt 2 becomes a decimal, pi becomes 3.141. At my school, it is considered "okay" to do this until we get to volume, surface and circles then the students are overwhelmed. Questions I have heard, "is r=(1/5)^(1/3) an answer?", "can I take the square root of pi?". Through the year I have told some students they weren't allowed to give decimal answers. These students had no problem with the switch. I told the other students to give the exact solution time permitting. Side note: my class had the highest average on the entrance exam into algebra 2 so I wonder if forcing them to give exact solutions from the beginning helped.

I have taken to writing lecture notes in latex. I include exercises after each topic. As a class we go over a new concept, do an example problem or two and I set the class loose on in class exercises. Some students enjoy this and do well. Some are able to read the examples, do the exercises, check with me and then start their homework. I have several students in my regular pre calculus class that need to see a "similar" problem worked out first (the problems can only differ by constants). Generally this has meant my notes are organized by solution method. How do you find an explicit formula for an arithmetic progression given a_1 and d. How do you find an explicit formula if you are given a_n and a_m. I have avoided this somewhat in my honors class by introducing an idea and then including exercises which lead the student to discover what they need to know. Example: an arithmetic progression has a common difference between terms. Have them write out 5 terms using the recursive definition. Suggest that a_2 can be written as a_1 plus something. a_3 then can be written as a_2 plus something and then a_2 can be written in terms of a_1 and so on. They figure out the nth term formula. Some students find it frustrating. Other don't bother and just go to the book and look up the formula. Some enjoy figuring it out. I have found that it's difficult to engage all of my 32 students in my class.

Review days. These have been pretty good. I post a set of in class problems the class before the test. Everyone works on them. I go around asking probing questions and directing people towards the answer. Generally it has shown me what I have taught well and what I haven't. It also gives me an opportunity to introduce new methods of solving problems. Example: we had been computing partial sums of arithmetic progressions where the first term is not a_1. I had originally demonstrated how that is equal to S_n - S_m where m is the index of the first term in the partial series. In the review I had the students compute the same partial sum but each problem had a different explicit formula for the terms and a different starting and ending index. This lead to a discussion about how the same number of terms were being added whether we were adding terms 2-5, 9-12 or 25-28 and that a_2=a_9=a_25 so the a_1 in the partial sum formula could stand for "the first term you were adding" as opposed to "the first term in the sequence." This has been good because the class cycles back to topics we had discussed and the students view them in a new light but it also means that some students don't figure those things out until right before a test. This seems to mostly be a pacing issue. The school syllabus usually calls for one day per section in the book. I find it difficult to discuss the many ways a problem can be approached in just one day and add in an extra day. The most "actual learning" seems to occur on the review days.

There are no unanswerable questions in Euclidean Geometry

I guess we can delight or confuse our students by telling them that every question has an answer in Geometry (as they study it) but possibly not in their other math classes. From Wikipedia:

Tarski designed his system to facilitate its analysis via the tools of mathematical logic, i.e., to facilitate deriving its metamathematical properties. Tarski's system has the unusual property that all sentences can be written in universal-existential form, a special case of the prenex normal form. This form has all universal quantifiers preceding any existential quantifiers, so that all sentences can be recast in the form . This fact allowed Tarski to prove that Euclidean geometry is decidable: there exists an algorithm which can determine the truth or falsity of any sentence. Tarski's axiomatization is also complete. This does not contradict Gödel's first incompleteness theorem, because Tarski's theory lacks the expressive power needed to interpret Robinson arithmetic (Franzén 2005, pp. 25–26).

It would be sort of interesting to talk a little about geometry as dealing with thr intuitive notions of points, lines and space versus a purely logical deductive geometrical model as Tarski provides.

Preps...?

What are preps? All the teacher blogs talk about them. My interviewer asked me about them. What are preps? It makes no sense.

Why go in a straight line? Triangles.

I've read Lockhart's Lament so I've been thinking about problems to pose to students that will get them to own their definitions, theorems, etc.

Proposition 22 of Book 1 has me thinking. It is a construction proof to build a triangle given 3 line segments such that the sum of any two segments is larger than the other (third) segment.

I'm pretty sure more or less everyone feels that "the shortest path is a line" so we ask "Let's say you see a sandwhich somewhere in the room. Why do you walk in a straight line to it (assuming there is a clear path between you and said sandwhich)?" The answer will probably be because that's the quickest/shortest/whatever way to get there. Okay, let's put a small wall between you and the sandwhich.

      |
You   |   Sandwhich
      |

How do you get to the sandwhich now? Hopefully the students say walk in a straight line to the edge of the wall and then in another line to the sandwhich. Good. It might be interesting to ask if the path they took it shortest since it would bring up a lot of itteresting points about "shortest" beign subject to constraints. Now we make it interesting. Put students somewhere and a sandwhich somewhere else. Measure the distance between and then say "I want you to show me a path to take from where you are to the sandwhich that is (1) A feet long and (2) is made of two line segments like in the wall example". My guess is that they will come up with the isoceles triangle whose base contains the sandwhich and start point. Make it more interesting. If the path has to be A feet long what are all the different paths you can take? Maybe to push them in the right direction give them string and a measuring tape and split up the groups with half at the start point and the other at the sandwhich.

The interesting idea for me is that you can do all this with their intuitive notion that straight is shortest. The legs of the triangles have to add up to more because otherwise that notion would be wrong. Also, the discovery process for the harder question is similar enough to the proof construction that going through the formal proof will illustrate how in mathematics exploration and proof are complimentary pursuits.

Exeter and Park Curricula: HS Moore Method?

In my blogging binge, I have repeatedly come across posts about the Exeter and Park Curricula and many discussions about their differences, similarities and foundation.

My first impression is that they are similar to the Moore Method with a crucial distinction. For the unfamiliar, the Moore Method is a teaching style which emphasizes having students prove theorems from a given set of axioms and present and presenting their results in class. The Exeter and Park Curricula seem to differ in that they don't propose any axioms or framework for the students to operate in. Instead, they seem to "front load" the conceptual work by having the students propose a "framework" in which to work (with gentle prodding and guidance from the teacher?). My guess is this is meant to avoid issues students have with axioms, new concepts, etc.--justifying and understanding why they are using the axioms or concepts since the student has already done that by proposing them! That seems to be the idea anyway.

Moore Method:

The way the course is conducted varies from instructor to instructor, but the content of the course is usually presented in whole or in part by the students themselves. Instead of using a textbook, the students are given a list of definitions and theorems which they are to prove and present in class, leading them through the subject material.

More Moore Method:

I do not consider it important in using the Moore method whether one lectures a lot, a little or none at all. It is important that there be regular interaction with the students so that the instructor knows how well the students understand the material. It is also important that they be given challenging problems, be motivated to work on them and to want to report their progress.

My undergraduate topology course was more or less a Moore class. It was an interesting opportunity to learn about something mathematically useless while developing real mathematical skills proving theorems and formulating conjectures. The course developed a topology on sequences using a modified lexicographical order for containment of sets and truncation and concatenation for set operations. Lesson learned: while the content might not have been "mathematically" worthy the experience and process itself justified the approach.

Purported as Opposed to Actual Goals

I have spent at least the last 9 hours reading math teacher blogs. NO SLEEP! YEAH! I have added most of the blogs to my list, Blogs I Read. I am happy that their experiences confirm my idea that writing about mathematics, mathematicians and how they are meaningful seems to benefit their students. It makes sense because I often figured out the most after I tried to distil something I thought I had figured out into an article for my fellow grad students to read. But reading their blogs has lead me to consider two questions: (1) Is it correct to say that "everything" in Geometry is from the time of Euclid/Archimedes and (2) How can I take advantage of the strategy of getting students to learn B by proposing that they solve A which requires B which seems to come from Out of the Labyrinth: Setting Mathematics Free? I will return to (1) later but will gather my current thoughts on (2).

I will be teaching Geometry, a class I've never taught before, and Pre-Calculus, a class which I have taught and TAed at a University many times before. I've been thinking about their relation to each other. Until recently, I had been of the mind that there is an obvious paradigm shift from Geometry to Pre-Calc but I am reconsidering my position. First, my copy of the Elements came bundled with several works by Apollonius of Perga on conics. So while I was reading, I was sort of primed to notice that Proposition 7 of Book I in the Elements more or less lays the groundwork for the idea that ellipses are well defined. This has me thinking that Pre-Calc probably naturally evolved from a classical class on conic sections to a modern class on basic function theory (transformations, symmetry, etc.). Motivate the modern material, functions, by classical questions about more advanced geometry concepts, conic sections. But thinking back on my high school math classes, I definitely did not get this impression. I never even thought about it in this way until now. Thoughts?

I have also had the interesting idea that it sort of explains why matrices are covered at the end of Pre-Calc. Geometry can be rigorously developed with linear algebra. Solving simultaneous equations is basically Gaussian elimination which gives us information about linear independence which tells us about co-linearity of points, orthogonality, etc. I don't think it's developed in that light though.

High School Geometry!

I recently found out that I am going to be teaching high school geometry this up coming school year. This will be my first time. This will also be the first time I teach high school.

At first, the best preparation I could think of was reading my old high school geometry text book and Euclid's Elements then I realized how vastly different they were. Euclid rarely dealt with the length of lines since he held himself to using an unmarked straight edge. The concept of measurement/number doesn't really appear until Book IX/X or so. He didn't measure angles. In fact, the only angles specifically named are right angles. Those two differences are huge.

I enjoy reading The Elements and working out the proofs. They are amazing demonstrations of constructive proofs! I don't really enjoy reading the high school text book but I can appreciate why there is a modern text book. The proofs, which are pleasingly elegant and logically organized, are also esoteric. They are probably hard to follow and boring to someone who hasn't studied advanced math, like high schoolers. This thought hits home even harder when I realize that the most educated of Greeks probably finished with studying The Elements. They had also probably studied Biology, Physics, Philosophy, Logic and Rhetoric before even setting out to work through it.

If Euclid isn't going to be my model, I am going to have to rethink my approach. Here are several books that I am reading to figure out what I should cover and how: Geometry in a Modern Setting, Designing Learning Environments for Developing Understanding of Geometry, Perspectives on the Teaching of Geometry for the 21st Century, Learning and Teaching Geometry, K-12. 1987 Yearbook and Geometry in the Mathematics Curriculum.