Entry: October 23rd Conference

I will be attending the BC TESOL 25th Annual Conference and AGM, on Friday, October 23rd, 2015. For more information, please see: http://bctesol.ca/conference/

Exit: What Influenced Me?

   Even though I was always the top math student at my elementary school, I don't remember any prominent, influential characters until I reached high school. One of my early high school math teachers was very helpful in cultivating my interest by bringing an enthusiasm and new outlook to the class. He would often start the class with tricky riddles or questions, give bonus problems to students who finished early, and kept his classroom open every lunchtime for anyone who played chess (and also offered an award of pizza to anyone who beat him). Through these gestures, as well as encouraging me to join any math contests, he was able to help me grow a stronger intrinsic motivation to learn math. In my final year of high school, I was going to transfer schools to take Pre-Calc 12, as my school didn't offer it. In response, he talked the school into running the class with only 6 students, which worked out amazingly as we were able to hold some of the classes on the school lawn. Even though I entered high school as one of the top math students, I still hold him responsible for my successes throughout my math education.


   In contrast, I had an awful experience with one math teacher in high school. In grade 9 or 10, I took an AP Math class with a new teacher (I believe it was a year-long substitute teacher). The first difficult came from the fact that he could barely speak above a whisper due to a medical problem. This made learning quite frustrating and difficult. Unfortunately, there was a greater problem. He was very distrustful and challenging towards the students. At the end of the semester he declared that every student (and this was an AP class) was failing. He then offered a bonus assignment to let the students make up some of the marks, in which he subsequently declared that every single student plagiarized their work and would receive a zero. The bad experience I had this year luckily did not discourage me from studying mathematics, but it did leave me more jaded towards the education system.

Entry: TPI Results

Here are my results:

  I was pleased to see that my nurturing attribute tested highly, and I'm quite certain that it was a result of my previous work experience. When I taught EFL in Taiwan, I was charged with teaching critical thinking and self-growth above all else, and the thought process was that their English would naturally strengthen due to the environment they were in.

   My transmission score was a bit low, but that was not too surprising. I have never made a lesson plan for secondary mathematics and had lots of trouble answering the survey questions on this topic, as they were hard to predict (due to the fact that I'm not actually teaching, but only thinking of how I would teach it). I believe that you need to have a strong grasp of the material in order to be able to teach it, but beyond that I am not sure. As a side note, I'm not sure why it says I have little intention to focus on transmission, yet I take strong action to use it.

   Social Reform is the one that I would have guessed would be the lowest, and I was not far off. When I decided I wanted to become a teacher it was because I liked to share my knowledge and help people understand things they struggle with, as well as to help people grow as an individual (ie. confidence, happiness, enthusiasm). I have never put much emphasis on social reform, because I felt that it would become a by-product of teaching self-growth and critical thinking. By challenging what they know about themselves and their ideas, they would use that to rethink their role in society.

   I am curious to see how these will change as I step into the classroom and take on the role as a real math teacher. I look forward to retaking the test and think it will be more accurate when I am not assuming what my actions would be.

Entry: How Many Squares in a Chessboard?

   If we want to find out how many squares on an 8-by-8 chessboard, I would approach it this way:

First, count how many 1-by-1 squares there are. This is easy as there are 8x8 small squares, or 64.

Next, I would count the 2-by-2 squares. This is tricky, as they will all overlap. I count 7x7, or 49, of these squares (since you count from right to left, but omit the last one, and the same from top to bottom).

(Using my counting method, there would be 6 2-by-2 squares, 

as denoted by the 6 colours, in this section of the board. 

I explain my method at the end.)

For 3-by-3  I get 6x6, or 36.
For 4-by-4 I get 5x5, or 25.
For 5-by-5 I get 4x4, or 16.
For 6-by-6 I get 3x3, or 9.
For 7-by-7 I get 2x2, or 4.
For 8-by-8 I get 1x1, or 1.

Here is my diagram for 7-by-7, so that you understand the method and can extrapolate for the other numbers:

Using this method, I feel that I am not missing any possible squares. Therefore, I feel that there should be 8^2 + 7^2 + 6^2 + ... + 1^2 squares. This is assuming that by square, we mean squares with a size that is divisible by the length of a single colored square (otherwise there would be an infinite amount). This means that there should be 8 possible square sizes. Using my thought process, I have concluded that there are 204 squares on an 8-by-8 chessboard.

As I was solving this question, my thoughts went from, "let's count how many squares there are," to, "there are too many squares to count, I need to find a pattern". From there I first decided to find how many different squares there were and then count each type. My thought process did not change to much other than the one shift, as my method worked and seemed effective enough for the small size of the problem. Afterwards I was trying to think of new methods and how other people might approach the question.

I think the best way to extend this problem would be to give a different shape (such as a triangle or circle) with a checkerboard pattern in it and to count how many complete squares are contained within it. I think this would be an interesting extension because you would need a new method for this approach (as opposed to giving them a larger square or rectangle). Another method would be to paint the smaller squares in a pattern of colours and ask them to find how many squares there are that don't contain a certain colour.

Entry: Instrumental and Relational Learning

   Our class discussion raised some interesting points regarding the different types of learning. I had been quite convinced that relational learning was the better method, especially after reading Skemp's article. I was quite surprised to see so many students arguing the opposite theory. I would now think that firstly: instrumental learning is better when you simply want to learn how to solve a problem and relational learning is better when you want to build a foundation for more learning. If I were to separate my math students into essential (the ones who simply want to learn the bare minimum) and advanced (the ones who plan to take higher level math) class, I think that I would emphasize relation learning to the latter group.

   I feel that to incorporate fluency of the material, it is essential to use relational learning. If I am teaching students how to find the area of a rectangle and I tell them to just memorize " A = a*b ", they will be able to solve that question, but that is all. If I give them graph paper and explain that area is the amount of space inside the rectangle, they will be able to build the formula themselves and, more importantly, they will have an idea of how to find the area of different shapes. However, if I wanted to teach them how to find the area of a circle, it is much easier to simply give them the formula, as they won't have the ability yet to have a relational understanding of the solution. Therefore, I feel that it is not simply a matter of which method to use, but a larger problem of which ideas of math to teach them first.

Entry: Skemp's Theory

     In response to Skemp's musings, I first paused at his mention of the young, intelligent boy who stumbled in his studies when he was forced to learn mathematics instrumentally. This stood out to me, likely contrary to the author's intent, and made me consider if some students would actually benefit more from an instrumental understanding of mathematics. Perhaps those who have a great difficulty with math can better understand this method, as I may not grasp the deeper meaning of a piece of art, but could benefit from studying the methodology.

     Next, I paused at his listing of how the methodologies affect the teachers (notably how instrumental understanding is easier to teach, grade, etc.). This made me curious about which ages and subjects would benefit from each method and how the progression would come about. Is it sometimes better to teach them the method and subsequently help them discover the reasoning? Or is it always best to build their knowledge from scratch? His analogy of the two methods in relation to a man mapping out the city was quite eye-opening. This is where I stopped reading, reprocessed my ideas of understanding, and became a stronger proponent of relational understanding.

     To elaborate, I always felt that to learn a subject such as math, you need to "understand" what you are doing. When I used that word, I had been unwittingly using Skemp's meaning of "relational understanding". Without it, you can simply solve the task at hand with an "instrumental understanding" but you will not have the ability to modify or create new methods when additional problems arise. I am of the opinion that, even though it is easier to teach and grade instrumental understanding, you will not be tapping into the potential of your students without giving them the tools to have a relational understanding.