Introspective Writing: Teachers, the Good and the Bad

    I'll start with an account of my least favourite math teacher. They struggled to convey information, they demeaned the students, they failed to interact meaningfully with the class or the students' learning. A particularly egregious error was that they provided no resources outside of class, this mandated that every student show up to class where the students must find what little knowledge was dispersed to allow them to succeed on the tests. This taught me that students will mitigate the harm of your bad teaching when given resources to do so and that more resources in the students' hands tends to lead to the student adapting their learning to your teaching. Furthermore, tremendously late marking, on the order of a month or more, lead to an interesting case where students were interacting with material further in the course with no guarantee that their fundamental understanding of the early material was accurate. 

    My favourite math teacher would have to be an IB math teacher in high school. They made classes engaging but more importantly comfortable. In a quickly paced course they made sure to take time for concepts to sink in. They made sure to mention their own learning, as they were doing a masters at the time, emphasizing how important it is to continue learning and how no one knows everything. After some negative experiences in the IB program they were instrumental in rekindling my love of learning, something that has been invaluable to me in the years following my time in their class. 

Math Puzzle: Locker Problem

    Like many math problems a good first step is to reduce the problem to a workable size. Let us for this example imagine a scenario with 4 students and 4 lockers. The first student closes each locker, the second student opens the second and fourth lockers, the third student opens the third locker and lastly the fourth student closes the fourth locker. There are a number of patterns we could gather from this example, but I would like to point out that after the nth student the nth locker's state is set and remains unchanged for the remainder of the process. Furthermore, we can observe that the state of each locker is dependent on how many students had interacted with that locker prior and that lockers are interacted with by the students who are their factors. We notice that lockers 2 and 3 are open with 2 factors and lockers 1 and 4 are closed with 1 and 3 factors respectively. From this we can extend a hypothesis that any locker with an even number of factors is closed and any locker with an odd number of factors is open after the process. The set of open lockers describes the set of perfect squares. This is due to the fact that factors come in pairs of 2 and numbers which have a pair of "duplicate" factors will have an odd number of total unique factors.

Entrance Slip Sep 14: Richard Skemp

    I'd like to first point out the rather interesting mention of I.Q. when discussing the intelligence of a child in the text. For me this emphasized just how long ago this text was written. As, to my knowledge, I.Q. tests as a significant/holistic representation of intelligence has not been a common stance of educators since before I was born. Despite this, it is somewhat depressing that much of the instrumental teaching that is mentioned in this text I experienced in my own education. 

     While, I tend to agree with Richard Skemp on the value of relational learning in mathematics, the devils advocate section certainly caught my eye. I think the point about instrumental learning being the fastest way to get correct answers on a page is interesting when brought into the adjacent field of computer science. In particular, Computer Science is a vast and fragmented field of study even more so than math in my experience. When dealing with a particular implementation of a user interface, or a sorting algorithm, or a file storage system there is value both in understanding it at a surface level as well as knowing every part of it. Furthermore, when there are many different implementations of every tool, you have to make decisions on what is valuable to learn and what isn't. 

    The last point which grabbed my attention was the point that skills may be needed in other classes before it is feasible to teach students to a full relational understanding of the topic. I think it is a fascinating challenge to try and flatten the interconnected topics of math into a linear learning process. I know that in my experience there were at least two occasions where topics didn't "sink in" until months or years later when other pieces of information completed my understanding of the topic.

 

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