Week 5’s content was very interesting and incited some terrific discussion amongst the class! The reading, in particular, was interesting and somewhat difficult to comprehend. Nonetheless, after thoroughly reviewing the chapter in more depth, I was able to better understand it and it aided my mathematical processing abilities. Furthermore, reading this segment allows me to facilitate these mathematical processing skills for my future students.
The authors did an excellent job of explaining the process of conjecturing, which is the “backbone for mathematical thinking” (Mason et al., 2010, p. 76). They not only define – in detail – what a conjecture is, but they also explain the cyclic process of conjecturing. This progression involves a series of conjectures and justifications that will eventually make up a resolution. Possessing knowledge like this is imperative for conjecturing which, in turn, is crucial in order to come up with a resolution.
The activities that were presented in the lecture were also very helpful in my understating of the concepts discussed by Mason et al. (2010). …
In the consecutive numbers problem, we got a practical sense of the conjecturing process. Each group had to incorporate and apply the processes described in the chapter while answering the question that was asked. We also had to apply concepts from previous chapters by specializing and generalizing to come up with each conjecture, and to justify them. This forced us to apply previous knowledge that we have from the course with the new information we have on conjecturing. The consecutive numbers problem provided us with a great opportunity to practice conjecturing because the problem involved a lot of it. Because an unlimited number of positive integers can be created by adding two or more consecutive numbers, there were a lot of conjectured involved to see exactly which integers fit this criterion. Thus, we had to consider several conjectures, and try to prove/disprove each one. The second activity, pertaining to the light switches, was very similar. Again, my classmates and I were required to think of conjectures in order to come up with a resolution to the problem. Although I found the light switch activity a bit simpler than the first one, I do not think it was, theoretically, any easier. Rather, I truly believe that the second question only seemed easier because I had already completed the activity prior. This is one of the reasons that I love math so much and main motive of mine for wanting to teach math. I love the fact that the more you practice – the more mathematical problem you solve – the better you get at it! It was amazing to see how much faster I was able to conjecture and come up with a resolution for the second activity, after already practicing with the first. These are types of experiences that I aim to facilitate for my own math students in the future, and I look forward to doing so with them!
One challenge that stood out for me while reading
Chapter 4 was the complexity of the examples. I had trouble understanding each
example and found it very difficult to solve each one. At times, this
interrupted my understanding of the text and made it tough to get a firm grasp
of the material within. Often, I had to reread the examples and it took me a
lot of time and effort to solve each one. Nevertheless, I can take a positive
perspective on this and look at these difficulties from an optimistic point of
view. In particular, I now have a better, realistic, and even empathetic,
understanding of the struggles that my future students will experience on a
daily basis is the classroom. In fact, after completing these examples and
focusing on the mathematical processes (including conjecturing), I can now
empathize with math students and the issues that they undergo while solving
mathematical problems, especially with regards to thinking of multiple
conjectures. I now know just how much time, thought, and patience that I will
need to provide for these students in my teaching as a mathematical educator.
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