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.

My Weekly Report and Reflection 2 (Week 3)

The content for this week was very stimulating and it sparked a lot of discussion amongst the class! In particular, I enjoyed doing the handshake problem and the paper strip activitie in small groups. The small groups were nice because it allowed every student 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. Joyce let us walk around after and browse our classmates’ solutions, which was beneficial for learning as future teachers. I found it especially interesting to see that every group thought of a different procedure to solve the handshake problem. 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.

These activities also reflected the textbook reading of Chapter 1, which emphasizes the need for two mathematical processes when beginning to solve a question: specializing and generalizing (Mason et al., 2010). In the handshake and paper strip problems, each group had to incorporate and apply these processes while answering the question that was asked. We had to specialize by trying different examples of ways to think about the question. This made the question meaningful, and allowed us to begin to see any underlying patterns that could aid in our attempt to figure out a solution. During the specialization portion, we tried to find other ways of calculating how many handshakes the class completed, in total (78), and how many creases would be made in the paper. My partner and I, for the handshake problem specifically, tried three equations before coming to a confirmed “correct” answer. Upon pondering solutions, we came up with about four different ways that we thought we could solve the problem. We also tried using the RUBRIC format for recording our thoughts that was mentioned in the textbook. We were able to write “Stuck”, “Aha”, “Check”, and “Reflect” to help us reflect on our mathematical thinking and understanding of the question. Following this recording, we were able to narrow the formulas down to a “conjecture. In other words, we found one that we thought would work for the “handshake problem” with any number of people. We then moved on to the next stage – generalization. We tried to generalize our equation by ensuring that it worked for several variables (n = number of students). We tried to generalize our equation by setting the number of students (n) to 2, 3, 4, and the actual number, 13, and using our sensed pattern (formula) to articulate and justify the conjecture. Overall, it was fascinating to put these two crucial mathematical processes to practice and observing my classmates do the same, in various ways. Applying this reading in a real-life, mathematical setting with fellow classmates was a key learning moment for myself as both a mathematician, and as a future educator.

I only wonder, now, how I can better specialize and generalize, and facilitate this for others. I would like to come up with underlying patterns and possible conjectures more quickly in the future. Furthermore, I want to learn more about how to instruct students on how to use these processes when answering math questions. Some challenges that I foresee include:  not-as-advanced students; being misled into believing a pattern is right when it is too simple and only partly correct; students finding it difficult, unnecessary, or boring; or other problems. Therefore, I must ensure that I keep practising this and encourage my students to do the same by motivating them and providing feedback. I now know that children come to school at a young age with a “generalizing” power of mind (Mason et al., 1988), and it is my duty to use this existing schema and instruct all students on how to specialize and generalize. They must always test an observed pattern(s) with a variety of examples. It will also be critical to make sure that I build students’ self discipline so they persevere and continue to use the RUBRIC framework (in a natural way, not dogmatically), and explain why they should record their ideas, purposes, feelings, etc.

My Weekly Report and Reflection 1 (Week 2)

The content for Week 2 was very interesting and provoked some great discussion! In particular, the reading was interesting, and it was a good learning experience to complete the first Professional Reading Circle. It was nice to be able to work as a group with my classmates and to feel like I aided in their understanding of the reading. Furthermore, it was a good icebreaker to familiarize ourselves with each other and to get a feel for each other’s strengths, as well as becoming more comfortable with the others.

One activity that I could not exclude from this blog was the “Mystery Op” equation. Joyce arranged the class into three groups and handed out small cards with the same mystery operation, Δ, on them. Yet, each card showed a different equation involving this mystery operation. Our duty was to collaborate as a group and solve for an operation that would satisfy Δ for each equation. At first glance, I thought this was going to be a simple feat, considering my classmates’ and my own prior math knowledge and experience. However, after a lot of time pondering each other’s card, communicating with each, and (trying to) hold our frustration in, we still were left unable to solve for Δ! To our surprise, none of the three groups successfully completed the activity. This task, though frustrating and very difficult, was extremely fun and eye opening as a future educator. This is a potential activity that I could incorporate in my own classes to not only promote students’ problem-solving skills, but also to demonstrate the collaborative nature that can accompany solving math problems. Too often, I believe, people view math as an independent domain in which teamwork and cooperation with others is undermined. Students rarely see the benefits of working with others on mathematical dilemmas. Nonetheless, math is very much an interdependent field in which collaboration promotes extended learning. Activities such as Mystery Op support this claim that math classrooms benefit from different minds coming together.

In addition, it was advantageous to the learning of my peers and myself to take a look at the Ontario Mathematics I/S Curriculum documents and to read the first 28 pages. Not only did this introduce me to the curriculum and what it entails – which will help me in the future when reading this document more in depth – but it also gave me a better sense of why mathematics is such an important subject to learn. I can use this information in the future my referring back to it when I have reluctant students who may not always see the purpose in what I am teaching them. Furthermore, I now have a better understanding of the subject overview and the different courses that are offered at the secondary school level.

One question that I have for next class is how to be a stronger reflective practitioner in math. I have taken classes before that have emphasized reflecting on my own learning and that of others’. In fact, reflecting-on-practice seems to be a key theme in University courses and in the subsequent professions that follow. Thus, it would be helpful to go deeper and specify my reflective abilities to the domain of math and education. I feel that blogging about my experiences will be beneficial as it allows me to engage in such practices on (almost) a weekly basis.

Welcome Readers!

Hello all!

My name is Connor Sclater and I am currently in my fifth year at Brock University, taking Bachelor of Physical Education/Bachelor of Education (Intermediate/Senior), with Mathematics as a second teachable subject.

The current Blog has been created for my EDBE 8F83 - I/S Teaching Mathematics Part 1 course. It will be focused on my experiences and reactions based on the course content, my classmates, the professor (Dr. Joyce Mgombelo), and other matters that I might encounter in the next year. In other words, I will be blogging on my weekly experiences after I have completed all of the activities for lecture that week. For instance, my blogs may include things that surprised me during the session, any important points, emotions that I experienced, any actions I may want to pursue further, or anything else that comes to mind.

This blog will not only demonstrate notes that I have taken away from each lesson, but it will also serve as a tool of reflection for me throughout the course. It will address responses to questions, discussions about ideas, and information that has been presented (both online and face to face) for each week. I also expect it to facilitate similar reflections for my peers and for Joyce, since they will be able to view and consider what I have written. Therefore, I hope to reflect on moments, ideas, or thoughts that my classmates may have not understood, or missed entirely. Another expectation for my Teaching Math blog is to help anyone who may not be familiar with Mathematics and/or how to properly instruct students in the subject. I will attempt to use what I have learned to support the knowledge of other practitioners, parents, students, or other stakeholders involved. I anticipate that it will be a great opportunity to formulate my thoughts and ideas, and to connect new learning to past experiences.

My main goal for this course is to gain a basic understanding of how to teach students in the domain of Mathematics in an educational setting. Although I am already proficient at engaging in Mathematical processes myself, I lack much of the knowledge that is crucial for any professional educator - particularly, how to properly instruct and assess students' learning. In doing so, I look forward to familiarizing myself with the Ontario Mathematics Curriculum. Another goal that I have is to gain strategies and tools that pertain to teaching, learning, organization, assessment/evaluation, and lesson and unit planning. Finally, I will work towards building my foundation for Teaching Mathematics through authentic experiences and learning, while gaining teaching resources along the way.

I look forward to being a student in this course and to continue my journey of life-long learning! It's been a great ride here at Brock, and it's only getting better. I know that it will be an educational and fun experience, and I hope that you all enjoy experiencing it with me!