Cape Town to Kasane

My mom’s comment on my last blog post was “uh.”  I assume she meant reading the mathyness felt like listening to adults in Charlie Brown talk, so I’m following it up with one full of photos of time spent with Courtney, Ali and friends.  Last month, I was extremely lucky because two friends traveled all the way across the ocean to visit for a week.  We made two amazing trips – one to Cape Town, South Africa and another to Kasane, Botswana.  

The trip started at a beach house with Drew and Rowan, two new friends I met though Courtney.  They live in Johannesburg and are both musical directors who often travel to Cape Town for work.  Rowan needed to be in Cape Town for a two-month run of Sweeney Todd and is staying at the beach house of a friend.  All I knew from Courtney was that we were staying at a “bungalow” with “bunk beds” and that it was small. I was shocked to arrive to a gorgeous house.  There were no bunk beds.  Courtney arrived the day after Rowan, Drew and I.  We kept asking if we should tell her how nice it was.  We decided instead to send her texts like “Well, at least there are towels.” 

These are some shots from the balcony and inside the house.  There are also some other random scenes from walking around town. 

I learned a little bit about the history of Cape Town.  On Saturday night, we went to see a musical about District 6, an area in the center of town that was established in 1867.  By the turn of the century it was a thriving, exciting, mixed neighborhood.  As the play described it, the harmony of the different working class people living in District 6 was an affront to the apartheid government.  In 1966 it was declared “whites only.”  In the following decade, 70,000 people were forced out of their homes and the people were racially separated.  This even included sending members of the same family to different places.  The play ended with the announcement that everyone had to move and did not get into what followed.  It was more a celebration of the utopia of the time period when District 6 was thriving.  Much of it was in Afrikaans, so Courtney and I didn’t understand all of the references and jokes, but for the most part we could follow along and loved it. 

We also got a chance to visit Robben Island, which is the site of the prison where Nelson Mandela served 18 of his 27 years.  The former prison now serves as a museum.  The guides are all former political prisoners and it was their idea to turn the site into to a museum. 

I did not hike up Table Mountain.  Instead, we took a funicular that magically transported us to the top at lightening speed.  It was fantastic. We got up there moments before sunset, took some photos, bought some wine and cheese at the store on the top and chilled out.  Literally.  I had no idea how much colder Cape Town is than Gaborone.  Sorry, Chicago, Cape Town is the real Windy City.  As a side note, a tour guide in Chicago once told us that “windy” actually refers to political gossip, not weather.  I’m not sure if that is true or not.  Anyway, the wind in Cape Town is no joke.  There are trees along the coast that are growing parallel to, okay maybe more like 45­⁰ angles from, the ground.  We did go on a beautiful hike at the end of the peninsula out to a shipwreck. 

Our friend Toni told us that kids call those little blue specks on the ground “blue bottles.”  They are washed up sea-creatures that make a popping sound when you step on them.  We also went to Courtney’s all-time favorite spot in Cape Town, a beach full of penguins.  I wonder if you can figure out from the photos which one she named “Judgmental Penguin.”  The rest of them had much more affectionate names, but I don’t know which is which from the photos. 

We spent the second half of the week in Kasane, which is in the north-east corner of Botswana. 

There is a place where the four countries of Botswana, Zimbabwe, Zambia and Namibia all meet up in one spot.  We didn’t find this exact place. Kasane was absolutely gorgeous.  We stayed at The Old House and loved it not only for its beauty and convenient location, but because we also booked our activities directly with them.  For a slightly (and I think barely) cheaper price, we could have made arrangements with a separate tour company.  I saw some of those vehicles out on the game drives and they were packed with people.  On the other hand, we felt like part of a family for four days.  Albert, who we saw every day, picked us up at the airport, drove the boat on the Chobe cruise and took us on the all day drive.  We loved it all and his company so much that we booked a short morning drive on our last day there.  We found out on our last day, while hanging out the airport as we were leaving and he was meeting another guest, that when he told us he just started three months ago, he actually started only three weeks ago.  Jaw drop. We also went on a day trip to Mosi-ao-Tunya, also known as Victoria Falls.  Here are some photos, including some zip-lining.  And, no, we are not the fools sitting in the water at the top of the falls.  They were on the Zambian side, we were on the Zimbabwean. 

UB and GSS

March?  March?!?  What happened to February?  How is time going so fast?  How have I failed to write anything here for over a month?  Why am I being so dramatic about this? 

Classes at UB

In addition to the Mathematics and Society course, I’m taking the Land, Labour and Liberation history course.  In both of them, we are preparing for group presentations.  For M&S, the class was split into three groups – Egypt, ancient Babylon and Greece.  Each group will present on the society and what mathematics emerged from that society.  I’m in the Egypt group and learned that they used 256/81 (which is about 3.16) to approximate Pi.  At first I was surprised and thinking about how much closer 22/7 is to Pi, but we are talking about 4,000 years ago.  Also, the area of a circle with diameter 9 units is extremely close to a square with side lengths 8 units.  You’re welcome. 

In LLL, we have mostly been talking about Zimbabwe when it was Southern Rhodesia.  The professor, or lecturer as they say, broke us up into groups of four.  Each group chose one of these topics to present to the class.  Our group chose (8) and we are going to define colonial developments in terms of four categories – communication/transport, education, infrastructure and agriculture.  Each of us will take one of the categories, discuss some of the developments, try to name and quantify the colonial capital used, but ultimately argue that the work could not have been accomplished without the African labour.  (I guess I’ll just go ahead and add a “u” to labour and colour for the next few months.)

GSS

In the first week of February, my advisory brought me to GSS.  A senior secondary school is like 11^th and 12^th grades in the US.  Both the headmaster and the head of the mathematics department are former students of my advisor at UB and I’m lucky they are willing to help out their former teacher by hosting me.  The headmaster had read my proposal and saw that I was interested in working with a small group of students and pulling them out of class.  He suggested instead that I actually work with a whole class of students. 

From there the head of the department, brought me to the mathematics office.  There are 12 teachers and eleven of them are men.  They are welcoming and I am free to make arrangements with the teachers to visit their classes and come and go from the campus.  It is also just a short walk from UB.  In the past month, I’ve been visiting the school regularly and have made it to most of the teachers at least once.  The term began at the end of January, but only the Form 5 students (12^th graders) were there.  January is the beginning of the academic year.  It is summer here, so this is kind of like arriving back from summer break.  The Form 4 students do not arrive until after their Form 3 exams are marked.  Not all Form 3 students get to move on to Form 4, it depends on how they do on their end-of-year exams.  In January and February they wait to find out if they will continue with school.   So for the month of February, the campus was quiet because only half of the students were there.  Last week, the Form 4 students arrived but they spent the time in the auditorium having orientation.  Yesterday I observed a class where the teacher was seeing his Form 4 group for the first time.   They start with Algebra and it is likely that this will be the module I will use for my inquiry project, but more on that later. 

In February, the Form 5 students were studying transformations of objects in coordinate geometry – rotation, reflection, translation and enlargement (which we usually call dilation).  So that is mostly what I’ve been observing so far.  At first I thought enlargement seemed like a misleading word.  What if you wanted to shrink something?  Well, they use “enlargement” for making objects both bigger and smaller.  If they were making it smaller, they would just say it is an enlargement with a scale factor of ½, for example.  But here is what I really want to talk about…

The center of dilation

Why aren’t we making a bigger deal about it?  In all the problems I’ve seen back home, the center of dilation is the origin.  This seems to be such a norm that it is hardly worth noticing that there even is a center of dilation.  Maybe this is because it is just easier to multiply all the coordinates of the object by the scale factor to get the coordinates of the image.  Like in this example, the scale factor is 2. 

But until I was sitting in that class, I hadn’t thought about how we don’t have control over where the image lands if we always keep the center at the origin.  So there I was in class and the teacher drew a triangle on the board, marked a center and gave a scale factor.  The coordinates of the vertices were clear, but they were not labeled like (2, 1) on the graph.  In fact, in all the transformation lessons I observed, no one made a big deal about writing the coordinates like this at all.  I suspect it has something to do with more of an emphasis on the visualization of the transformation and less of an emphasis on a procedure such as (x, y) à (2x, 2y).  For example, during a rotation lesson, students had to rotate an object (about a point), and no direct method was given for how to do this.  The teacher did an example, and then gave a few more problems to work on.  I noticed many students holding up their hands like this and then rotating them, drawing a point then doing it again. 

But back to the enlargements.  The teacher was moving really fast and I was not following what he was doing to get the coordinates of the image.  He was counting from the coordinates of the object to the center and then using that information.  At the same time, he was also saying that if the scale factor is positive, the image will be on the same side of the center as object but if the scale factor is negative the object will be on the opposite side of the center as the image. 

           

He drew the image on a small board in the front a room with 43 students, it was extremely accurate and all the students were drawing it in their notebooks.  He asked for a volunteer to come to the board and do the next problem.  The room was quiet.  He waited.  I had been introduced in the beginning of the class, but wasn’t sure how much, if anything, I should say.  So, with all sincerity, I said it would really help me if someone could do another example because I have seen enlargement before, but not about a point.  The whole class started laughing.  My accent?  My question didn’t make sense?  There was something that had an unintended double-meaning?  No idea.  But a young man volunteered and explained with complete clarity how to count the vertical and horizontal distances from the object to the center and then how to manipulate those distances based on the given scale factor (including what to do if it is positive or negative) and find the image.  I totally got it.

I later asked the teacher if he knew why everyone laughed.  “Because a teacher said they didn’t know something.”

This has me thinking about dilation/enlargement questions U.S. geometry students encounter.  I took a look at the three most recent NYS Regents Common Core exam questions those that deal with dilation.  Then I looked at the enlargement section from the textbook they are using at GSS.  I’m not trying to make a comparison between the two countries and I’m definitely not trying to start a Common Core conversation.  I just thought it was really interesting to notice how much the visualization of the shape comes through in the questions they are using at GSS and how something seems to be missing in these Regents exam questions.  I mean, dilating a line?  What is that?