Christmas in Turkey

We just got back from a week-long trip to beautiful Turkey! Even though we had not much time to look around, from what we could see, it was truly an amazing mix of East and West, of modern and ancient cultures. Geoff and I both agreed that it is incredible that you can hear the songs calling Muslims to prayer 6 times a day in the backdrop of Istanbul, yet the city is vibrant with modernly (and risquely) dressed women. The city, at least, seems to embrace people's choice of lifestyles, more so perhaps than most parts of the West. (I think that in the States, as much as we claim to embrace liberal ideas, if you choose to pray 6 times a day towards Mecca, people at your work place would definitely look at you funny.)

Istanbul itself is also an amazing mix of cultures -- there are bar-lined streets galore, such as in the Taksim Square area, terraces overlooking the hilly city, trams that run from tourist point to tourist point, and yet the guitar and drums music seeping out from the bars are traditional-sounding, somewhere between Arabic and Indian.

The Turkish food is incredible; there are elegant restaurants to match the best of Europe. Three restaurants in our hostel area that we can immediately recommend are De L'Artiste, Morro, and Solera. Of these, Solera was my favorite, because they serve up local Turkish wine, coupled with delicious cold appetizers that are of a local variety, but beautifully done with elements of savory surprise. The city also has countless bazaars -- the only one we visited was the Grand Bazaar, but the price and the quality of the goods there were fantastic. Between Geoff and me, we bought: a silver necklace, an exquisite mirror for my sister, a lamp set, a rug, a leather-made silhouette puppet, and a beautifully woven pillow case. :)

While in Turkey, we did the typical touristy thing. We flew into Istanbul, spent a few days there, took a flight out to Izmir, spent a couple of days there on an all-inclusive tour, and then flew back to Istanbul for a few more days. Istanbul was amazing because it was a party spot, but backdropped by the ancient buildings. It's incredible to think about the unique culture exported from Istanbul to the rest of the world over the centuries. It is of little wonder that it prides itself as the Cultural Capital of the World.




This is me fake-crying because of the weather. :) It was snowing our first day in Istanbul! Colder than in Berlin!!!


While we were in Izmir, we got to visit Ephesus, the third largest ancient city. A good amount of the stuff has been rebuilt from the excavated material, and the excavation is still on-going, but it was still impressive to see the ruins left after thousands of years.





Besides Ephesus, we also got to visit Pammukkale, which is a beautiful calcium bicarbonate deposit formed by centuries of active hot springs. Some of the springs were cooled greatly during the winter, but others were still bath-water warm! You can only walk through the labyrinth of springs with bare feet, but we braved the cold anyway....





As an added bonus of going to Turkey during the off-season, we got to stay at a five-star hotel for a night as part of our all-inclusive trip to Izmir. The food was delicious and our room had bubble jet stream bath, and a view of the ocean. It had been a long time since we had fallen asleep to the sound of ocean waves outside of our window, so it was a real treat. (Especially because we had anticipated staying at a hostel.)

Anyway, here were some other random things we did:

Geoff was fed some stuffed clams by a cute-looking Einstein man! Clams stuffed with rice is apparently a local specialty.


We visited a local (Geoff's favorite) shisha spot. Surrounded by colorful Turkish lamps and lots of locals, it was the perfect spot to enjoy some apple tea and some shisha.



Here is Geoff inside the Circumcision Room at the Topkapi Palace.


We couldn't take any pictures at the Whirling Dervish religious ceremony, but it was pretty cool and inside a fixed up bath house.


Speaking of bath houses, we had our first experience with a Turkish bath house. For the equivalent of about 30 to 35 Euros, you can strip down naked, lie inside a sauna, and then have a person scrub you clean. (It's not coed. Geoff and I were in separate parts of the bath house; he had an overweight man leaning over him asking him, "Is this GOOD? IS THIS VERY GOOD?" while scrubbing him down. I had an old lady and she also asked me if it was very good while I lay completely naked and she scrubbed me down. It was very unique -- definitely an experience I would recommend.) The bath house we went to was a traditional one -- it had been running since the 1400s.

And, as a last note (harhar, no pun intended), I wanted to share with you my excitement to see the Sigma notation on a Turkish bill. I am pretty sure that in Turkey, they put random accomplished people (not just politicians) on bills, so they also put down some visual representation of why that person's famous. You know I had to share this:

Categorizing Student Mistakes

I am experimenting with a new method of grading exams, in which I look through the problems to determine what type of mistake the student made, and instead of writing a lengthy explanation of the mistake (which is what I typically used to do), I just circle the place where they messed up and then write down the category of the mistake.

So far, these are the labels I've come up with:

* Wrong approach - ie. the student was completely not on the right track.

* Conceptual mistake - the student had an inkling of what to do, but they made some severe error in the initial setup of the problem.

* Procedural mistake - the student understands at a high level what the question is asking and what procedures are required, but made some fundamental mechanical error in the procedure.

* Arithmetic mistake - mistakes involving combining decimals, fractions, or integers.

* Careless mistake - mistakes involving miscarried signs or wrongly recorded results, when the student exhibits overall competence in the process.

* Incomplete operation mistake - the student failed to completely answer the question or completely simplify their answers. Or, they were off to a good start and then bailed halfway...

* Mistake in interpreting the instructions - the student did not carefully follow the written directions and therefore did irrelevant calculations.

My hopes are that in this way, I can help kids to focus first on the bigger conceptual issues, and next on the other types of issues. I'd say that if a high level kid sees that they're consistently making the same types of careless or incomplete-operation mistake, it is valuable feedback for them to keep in mind for the future. Versus if a lower-performing kid sees that they're at least not missing the major concepts, then that is a good feedback for them as well, so that they know they would just have to focus on the procedural issues.

Thoughts? I'm grading semester exams as we speak. (sigh.) It's the last road block between me and a real vacation...

Doodle #3 and Rotating Calipers

I got a sketchpad and am fooling around on it, practicing sketching motions and emotions. Motions are easy, emotions are hard. So far, this is my favorite from the motions sketches:



If you're looking for some mathy updates, perhaps you should consider reading Geoff's very technical blog. He recently put up an implementation of the Rotating Calipers algorithm for finding the minimum bounding rectangle around any polygon. (The algorithm finds the best rotated rectangle, not just the best right-side-up rectangle, which would have been too easy.) His tech blog is super dry like the stock market books that he reads in his leisure time, but he likes it that way. :) He says he just wants to dump information on the internet to facilitate other people who might come across the same issues, so reader-friendliness isn't one of his main concerns.

Anyway, if you read through his implementation, you'd see that there's a lot of vectors math in there. It's neat... He's a computer programmer who actually uses high-school math on a regular basis to solve problems!

Doodle #2

Here's a second doodle I did today (you can click on it to see the full-sized version). My pen ran out of ink near the end, so I think this'll be the last sketch in a while and I'll go back to reading during my spare time.


I threw out the previous one (of the dancer) after I took a photo, since I had made it on scrap paper and there were other things on the back side. I think this one I'll keep. I didn't do such a good job on their faces, but I still liked the overall feel of the piece.

Ah, winter break...

I miss drawing. I think this Christmas, I'm going to buy Geoff and myself some charcoal and drawing pad, so that we can be drawing hippies on Saturday mornings. I made this today because I was bored and the art store was closed. It's based loosely on this picture, but of course I messed up on the arms since I was drawing with an ink pen (one I normally use for grading) and I hadn't made anything in years.



Hello, winter break. :) It's only Day 2, and we have already: partied a good bit, finished hanging all kinds of things up in our apartment (Geoff measured/built three art frames from scratch and stapled the canvasses to them! It was awesome watching him sawing and banging things together while I laid back, ate chocolate, and watched TV... but I'm extra happy that finally all of our Salvadoran art is hung up), and we rode our bikes today to eat yummy breakfast in the park. In about 5 days we will be off to Turkey. I can't wait!!!

PS. Geoff's parents sent us chilled champagne in the mail. Two bottles. I managed to convince Geoff to immediately crack open one bottle upon receiving them, because hey -- how often in your life would you get chilled champagne in the mail?! That seems like as good an occasion as any to enjoy them. :)

PPS. Our Christmas tree/bush is coming along. It's crooked and small (that's what she said?), but it's filled with holiday spirit! :) I am so excited about our first jointly owned Christmas tree ever!! (Last two years we lived in the tropics, and before that we each lived separately in NYC.)


Anyway, I hope your holiday spirits are bright. Setting up the Christmas tree made me all sorts of sentimental. It was the first time I had actually set one up without my sister around (even though it has been 7 or so years since we've spent Christmas together). Made me miss her extra much.

PPPS. You know that Australian claymation movie Mary and Max? Please watch it. It's phenomenal (although not really suitable for children) and made me both laugh and cry.

Holiday Geometry Activities

Today was the last full day before Christmas break, and another teacher and I had talked about gathering up both of our classes to do some fun holiday geometry. In the end, the other teacher was absent this week and then very busy when they returned, so they trusted me to just plan the session by myself.

This is how I structured it.

First, we did this snowflake prediction activity in partners. Everybody folded the papers together for the first snowflake, drew out their predictions, and cut it out. Then, I monitored that each pair of partners finished predicting the next two before I gave them each one piece of paper to have them cut out a snowflake to verify their predictions. (They split up what they would cut up, both to save time and save paper.) Then they proceeded to make more predictions, followed by more testing by cutting out snowflakes. This took about 40 minutes. Meanwhile, both the other teacher and I circulated to make sure that kids were understanding how to apply the idea of symmetry to making appropriate predictions.

Then, with the remaining 40 minutes, the kids got to choose between either doing a tetrahedron origami (mostly unassisted; the exercise was in reading and deciphering diagrammed instructions... the hardest part is reading the earlier instructions on how to create the regular hexagons out of a rectangular sheet of paper), or making a geometric sequence/recursive pattern (see below).


This second activity, by the way, was one that I learned at PCMI. :) You keep cutting each segment into smaller thirds (or any fixed fraction), and folding up the middle part. In the end, you end up with a very intricate design. I'll post a photo when I get a chance!

It was awesome! We wrapped up the class by talking a bit about the rotational symmetry of the tetrahedron and also about why we could fold hexagons up into tetrahedrons (same base shape, the equilateral triangle). It was a lovely way to inject some last-minute holiday cheers after all the heavy-duty algebra we had been doing.

Happy holidays!

Being the New Kid On the Block

I have to admit that my professional transition from El Salvador to Berlin has not been a warm-and-fuzzy one. At my old school, I had built a reputation with my students and their families, and their parents would regularly come up to tell me how much they appreciated their kids being in my class and how they wished that I could have stayed to teach their younger sons and daughters. At the end of my first year, an entire group of kids went to talk to the administration to request for me to move up with them to the next grade. Former students would come back to visit me, and even now they are sending me emails to let me know where they are off to next year for college.

When you move to a new school, for better or worse, you start off anew. You leave whatever reputation you have built up behind you -- the respect that you have gained from your colleagues and the administration, the affection from kids whom you've known over multiple years. Most of all, you leave behind the trust of your students and their families. When you move schools, you start again at the bottom of the totem pole and have to prove yourself every step of the way, to everyone who might be watching.

In my case, I took on the slower-paced classes in two grades this year, because 1. I didn't really mind, I enjoy teaching things at a manageable speed for the kids, 2. I wanted to make sure the kids at the bottom would get the extra attention/support that they needed. Well, in those two grades, I have had various resistance from a few kids who feel that they should belong to a faster-paced group. Those kids care not for fun learning or meaningful tasks; they just want to move along faster through the topics. How do you convince these kids that conceptual development is worth taking the time to get right?? I refuse to short-change their conceptual foundation in order to speed through the topics, and I don't believe that it is good for their mathematical growth in the long run, or good for their problem-solving abilities. (Case in point, one of those kids moved up to a faster-paced group on a trial basis, and went from getting 100% to getting 25% on assessments.)

ARGH! It is frustrating to feel like the new kid on the block. ...I know that being new doesn't mean I am less qualified to teach these kids, or that I'm making bad decisions for their learning. But it does mean that what I value carries a lot less currency around here, as far as my kids are concerned. sigh.

Christmas break cannot come soon enough. The last week has been fairly demoralizing, and I don't have a lot of umph left in me before the holidays. :(

Updates from Deutschland

I thought I'd take a minute and do some life updates. Time is flying by!

Things are going smoothly over in Berlin. I had put in a lot of work (ie. over 10-hour work days daily) from August to November to learn the new curricula and to earn the trust of my colleagues and student parents, and finally I was ready to re-focus on what I needed for myself. So, in a non-trivial gesture, I'm back doing yoga on a weekly basis and am LOVING it more than ever. (My new yoga teacher is really tough, but I love him!) I've also arranged for a private German teacher to work with me starting in the new year, since I feel like I cannot commit to 6 hours of classes like Geoff does during the work week. Socially, things are good and we've met a lot of fun people. :) Overall, despite the weather getting colder and the days getting darker, we are enjoying our first winter in Germany!

Recently, we got some free tickets to check out a Christmas market outside of Berlin, so we took the local train there last Sunday. It turned out to be a fairly small market, but the town was very charming!

The town had a lot of old people on bikes. I did not see a single young person riding a bike that day. I told Geoff that some day, we're going to retire to towns with old people on bikes. :) The town also had statues that looked like they were from old Grimmes' tales.


There were gondolas ferrying people back and forth between the two sides of the Christmas market. The ride was long and a bit chilly, but we had blankets and people were drinking hot mulled wine ("gluhwein"), which is common at Christmas markets and really all around Germany at this time of year.



We also saw a for-rent sign next to this cute little barrel of a room. It's even smaller than our apartments in NYC!!


Geoff took a photo of some locals moving a tractor via two gondolas.


All in all, it was a lovely day away from the city, and a much needed break from all of the stress I had been feeling from nearing the big semester exams. :)

I am looking forward to checking out some of Berlin's very own Christmas markets this weekend. Gluhwein, here we come!

Math in Psychology

Since my Kindle got fixed mid last week, I've been reading a rather delightful book called Thinking, Fast and Slow. I've read books like this one before, on the clever psychological experiments that people have managed to design over the years and what they reveal about the human mind. But this book, in particular, pulls together a lot of interesting bits that I've either read or heard over the years and organizes them into one cohesive and surprisingly elegant theory. (I won't spoil the book for you here, but it's worth looking into. The theory is elegant and seemingly simple, but the details are very interesting and not always so predictable or obvious.)

I am also surprised by how mathematical the author is and how easily he ties abstract math ideas into concrete experiments. For this, I highly recommend the book to math teachers. For example, the author talks about how if you get one person to look at a jar of pennies to guess at how much money is inside, that person might over- or under- estimate by a lot. But then, if you repeat the experiment with a large number of people, their average errors will actually approach zero (in the absence of a systematic bias), and therefore if you average all of their guesses, that average is actually going to be quite accurate. This is a logical idea that kids can grasp, and it's a nice extension of the absolute-value error concept. By the same token, he ties this in general to public opinions. If you survey a large enough sample population on a certain issue, in the absence of a systematic bias, the average of their answers will represent the truth.

Another issue that the author addresses is basic numeracy when reading current event reports or statistics in the media. He illustrates with a simple picking-colored-balls-out-of-a-box example why, with small sample groups, we end up with more extreme values more often. And then he extends this to why when you poll different counties for health information, it's easy to see rural counties with more extreme health statistics. Again, it's not impressive math, but the ease with which he ties math to something real is delightful.

And, as an aside, try to answer this question:

"How many animals of each type did Moses bring into the ark?"*

If you're like me and (the author so says) most people, you let your mental image of the ark prevent you from noticing that Moses is the wrong biblical character here in this context. Our mind has the tendency to smooth over the little inconsistencies using preconstructed expectations, in order to make its job easier. And that's both advantageous and troublesome, depending on the context.

Anyway, so far, I thoroughly enjoy the book! :)

Addendum 12/07/11: I've reached a part of the book where the author talks about how we let our stereotypes affect our judgment of the likelihood of certain combined events. For example, after being exposed to a description to a liberal woman, people -- even those who are mathematically inclined -- would rank the probability of her being a "feminist banker" to be more likely than her being a banker, even though any added details should diminish the overall probability! --What an interesting intersection of math and psychology!!