My Weekly Report and Reflection 5 (Week 7)

Week 7’s content was very interesting and incited some terrific discussion amongst the class! The reading was specifically interesting, and I found it difficult to comprehend. Nonetheless, after meticulously reviewing the chapter as a class, I was able to better understand it and it aided my knowledge regarding to the types of understandings that occur in education. Furthermore, reading this segment allows me to facilitate these mathematical understandings for my future students.

The “Dot Problem” allowed us to practice our problem-solving skills in multiple ways. This was also the goal of our problem-solving assignment, which is due next week. We must be able to solve problems in many different ways as a mathematics teacher, since our students will all think differently and come up with different solutions. This also demonstrates, mainly, how even the simplest questions can have several methods for answering. When we grade our students’ answers, then, it is crucial that we consider the various mathematical processes and strategies that they may use. For instance, my two group members and I compared our answers afterward and noticed that we all had something different from one another. When Joyce came over to view my answer, she asked me to explain it further by showing my thinking, instead of just discussing it. Once I started to show it to her, she told me that she had something entirely different in mind. She then illustrated what she had been thinking. This was a great learning experience; Joyce as I acknowledged the fact that students have many different learning styles and strategies, and I will need to be aware of this in my own classroom.

Furthermore, I found it fascinating to see how much “cleaner” some solutions looked compared to others. My answer, in particular, was very messy with a lot of correcting and crossing out. My sheet would, thus, be difficult for many people to read and to understand my thinking. Conversely, Gabriela’s solution looked much “nicer”, since it had no apparent mistakes and was colour-coded with neat diagrams. Although both Gabriela and I came to the same conclusion, some teachers may grade my solution less due to the fact that it is harder to follow. This is an issue that I must address when I evaluate and assess children’s answer. In other words, I cannot be bias to those students who think like I do or to those who come up with “neater” solutions.

As we progressed through the lesson and reached the “gallery walk” segment, Joyce introduced us to new part of the “gallery walk”. Once we all had time to post our solutions and walk around and observe the others’, we grouped the answers based on similarities. We “stacked” answers which appeared to follow the same – or similar – processes. This is advantageous because it is helpful for individuals to see each other’s answers, without being overwhelmed and confused with duplicates.

One challenge that I perceive as a future mathematics teacher is that it is sometimes difficult to respect other people’s views, especially when those people may not be as advanced as you are. Educating youth is one of the most challenging fields of work, since you must consider all perspectives, not just your own. In objective subjects like mathematics, therefore, teachers are forced to think about all of the potential solutions that their students may come up with, rather than just solving a question with one single solution that the teacher came up with. This unbiased thinking and assessment is a difficult skill to develop, but it is one that is mandatory in the profession of teaching mathematics.

My Weekly Report and Reflection 4 (Week 6)

The content for this week was very stimulating and it sparked a lot of discussion amongst the class! Although there were no reading, phot word problems, or professional reading circle discussion this week, it was still very valuable pertaining to my knowledge regarding teaching mathematics. In particular, I enjoyed learning how to plan and create the “Structure Problem-Solving 3-part Lesson”. It was interesting to investigate our own experiences as students in math classes. I noticed that, although Brock University strictly enforces teacher candidates and graduates to create a meticulous 3-part lesson before every class, very few, if any, of my past teachers seemed to do this. I was surprised to see how much emphasis is supposed to be put on the final “Consolidation and Practice” and “Follow Up” stage. I do not recall either of these phases being areas of focus as a student. Similar to how the Entry and Reflect stages are often underplayed by students while solving a problem, the beginning (“Getting Started”) and final (“Consolidation and Practice”) stages seem to be ignored by teachers while planning the lesson. Therefore, I now know just how important each of these phases are when creating a lesson, which I will use to guide my future lesson planning as a mathematical educator.

Joyce also had us learn about, and personally practice, what is known as the “Gallery Walk” strategy. This is a method of mathematical problem-solving in which the students complete their work, then they can either post their solutions or leave them on the desk for viewing. The name suggests the next part of this strategy, during which students walk around the “gallery” and view and reflect each other’s work. We practiced this using the “Open Box Problem”, which we solved in groups of 3, then walked around and used the RUBRIC writing to reflect on our classmates’ problem-solving methods with the same question. Finally, we had a group wide discussion on what worked and what didn’t, as well as how we could facilitate an understanding of and implement this method in our own classrooms in the future. It was very insightful to listen to and observe my classmates’ reasoning behind their methods, and to explain my own techniques when solving the “Open Box Problem”. It was also useful to hear how my colleagues would plan and implement their own 3-part lesson using this problem. This not only supported my knowledge and understanding on the topic, but also provided me with a realistic way to apply this thinking into my professional practices. The small groups were also beneficial as they allowed every member to provide input into potential solutions and each one of us felt like a significant member. Our individual ideas and thoughts could be brought up to the group and considered in a collaborative effort with the others. Additionally, I found it especially interesting to see that every group thought of similar, but unique, solutions to the problem at hand. Not only did this allow us to see that there are multiple ways to figure out a solution, but also that every individual has different ways of thinking and learning. This is crucial knowledge for our careers, since we will encounter hundreds of students, no one having the same learning style. There are numerous different ways to solve mathematical problems, and there is no single “correct” strategy; so teachers must be weary of this and encourage diversity in the classroom by using differentiated instructional methods.

My Weekly Report and Reflection 3 (Week 5)

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.