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?