My Weekly Report and Reflection 9 (Week 16)

The content for this week was very stimulating and it sparked a lot of engagement for the entire class! Functions is one of my favourite strands of mathematics. I love everything about this topic, from solving equations using algebraic methods to putting these equations on paper or graphing their them on technological software. This is also an important and applicable skill to develop, since it can be used in the real world in different ways. For example, equations of functions can be plotted with each other to show their relationship with each other and their point of intersection. One of Catherine’s questions, for instance, had us compare the prices to buy a pizza for two restaurants. The point of intersection of each function (based on the starting price, and the price of each topping) is something that people could use weekly, whenever ordering pizza! There are numerous other reasons that Functions is a crucial mathematical topic that is used by professionals worldwide to create things for the public to use.

I really enjoyed taking part, as a student, in the lessons that my peers taught. I thought it was rather insightful to learn about different and new strategies that I can use in my own teaching in the future. I found it very creative that each of my classmates incorporated a different method of teaching and learning for each of their lessons. Catherine had us play “Battleships” using different functions and equations, which made her lesson both fun and educational! This is a great way to motivate students to learn, especially the many students who find mathematics tedious and complex to begin with. Applying a game-design to education makes it much more interesting and, therefore, benefits student learning of the topic. Another station that was taught used a geometric approach to complete the square. I believe that most people prefer a hands-on and visual approach to learning, and this activity allows for a more visual and interactive way to complete the square. This is especially beneficial for children and adolescents, since it gives them another perspective on the technique of completing the square based on how it may have originally been invented. When problem-solving, using your hands and doing things in real-life makes it much more enjoyable and easier to visualize. Children tend to learn through play, so it makes sense to incorporate hands-on learning in the classroom. Geometrically putting together a square and a rectangle, then trying to add another thing to it, allows us to make a perfect square in a way that is more authentic than just solving an equation on a piece of paper. Lastly, Maxim introduced me to an online program that allows teachers to present slides to their students virtually. This resource is called Desmos, and it is something that I will certainly use in my future teaching! I had heard of Desmos before, but have never actually explored it at all. I was very surprised to see just how useful and engaging Desmos was from the student’s perspective. For anyone who has not tried Desmos before, I highly advise that you do so!

I am excited for the weeks to come. Specifically, I cannot wait to participate in more lesson stations as a student to my classmates. I am also looking forward to teaching my own lesson for a Grade 12 Data Management College level class!

My Weekly Report and Reflection 8 (Week 15)

Class 15 was another great one! We worked on factoring which was a great refresher since I haven’t done this type of mathematics in a while. We also gained some resources as future mathematics teachers, which we can use in our classroom in our careers! In fact, I was proud to be the first student in the class who completed the "pop-up" card! Check out the picture below to see my staircase! One thing that I wanted to focus this blog on was the TED Talk that we watched at the beginning of the class.

I love watching TED Talks, and the video we watched this week lived up to my expectations! The beginning of the TED Talk was about how mathematics has to do with patterns, as the speaker defines mathematics as being “about finding patterns (connections, structure, etc.), then representing these patterns with mathematical language.” He also states that math is about doing cool stuff. I found this definition to be useful because it is so true, and it expresses mathematics in a way that makes it seem doable and enjoyable. I feel like many individuals find math to be a challenging and discouraging domain, which they will never truly grasp. Many students feel like they have been defeated in the subject of mathematics, which is a stigma that educators must work to diminish in the classroom.

The speaker’s main claim in the TED Talk is that changing one’s perspective is is a critical skill to have when solving mathematical problems. He explains that every equation has multiple perspectives, and everyone’s point of view when looking at an equation may be different than someone else’s. this is an interesting point because it proves that there is no single best solution when approaching mathematical problems. The examples that the speaker uses are extremely beneficial and the way he discusses these illustrations are conducive to the viewer’s understanding of the topic. This is a concept that our class has been exploring all year, and it is something that could encourage students when struggling trying to solve a problem. In other words, if an individual is “STUCK” and having trouble in their mathematical processing, they can be motivated again once they realize that reframing the question may lead them to the correct answer. Of course, if they still cannot solve the equation, they simply need to look at the question from a new perspective, and keep doing this until they can finally solve the problem in front of them. Often times, understanding is only possible when the problem-solver takes a step back and looks at the bigger picture. The speaker acknowledges the fact that this is true for every subject matter, not just mathematics and science. The essence of understanding, then, is being able to change one’s perspective and adjust our point of view in order to learn more and more about something. He calls this ability to change perspective “empathy”, claiming that this competence is crucial when trying to understand something. It makes your mind more flexible and, subsequently, allows you to understand more about the world. He also explains how metaphors and analogies are an essential strategy to include in teaching and learning. I agree with this because I think that people learn better when a field that they are not so comfortable with (in this case, mathematics) is related to everyday life. These comparisons can serve as symbols with which the learner(s) can relate to ideas that they are much more comfortable with. Therefore, it is imperative that educators in any domain – especially mathematics – encourage their students to always change their perspective when looking at a problem, since there is never one single process to come up with a solution. It is only once students and teachers realize the importance of changing their perspective that they can truly understand how to solve mathematical problems and enable this understanding for others.

Bringing different experiences to the learning environment is necessary for individuals to gain more knowledge and enhance their learning. This is something that I think all educators must be aware of, in any classroom. It is the teacher’s responsibility, then, to connect their own experiences, and the experiences of their pupils, into their instruction.

My Weekly Report and Reflection 7 (Week 14)

The content for Week 12 was very interesting and provoked some great discussion! We started the session with each group presenting one of our Digital Math Word problems to each other. This was a good presentation for two reasons. Firstly, it was beneficial for us to explain our own problem and justify our problem solving to the rest of the class.

The first activity was a “4 corners” type of activity that had students choose which corner of the classroom to stand in based on our resolution the selected problem. In this case, Joyce asked us what cylinder could be created by folding paper in order to maximize volume: a hamburger-type shape or a taller, hotdog-type shape? At first, the entire class went to the hamburger corner because we are all university mathematics students who are highly proficient at measurement and geometry. Then, Joyce asked us to think like a student would, and we dispersed into separate corners. It was advantageous for us, as future educators, to consider what students might be thinking. Teachers have the responsibility of considering their students perspectives and educating them on why they may be correct, or where they may have gone wrong. This “4 corners” activity is one that I can, and will, use in my own classroom in the future. Not only does it force students to take multiple perspectives and brainstorm different solutions and justifications, but it also creates small groups where all students are encouraged to discuss their thinking. Small groups are much more inclusive and more efficient than larger groups or the entire class. I think that using popcorn for the activity would be especially useful because it motivates children to complete the problem at-hand so they can eat it after.

Measurement and geometry is a mathematical unit that I have not done in a school environment in quite some time. I find that University courses are very advanced and comprise of mathematical strands such as Calculus, Statistics, Algebra, etc. More hands-on mathematical problems, therefore, usually do not occur in University due to an emphasis on solving equations and algorithms mentally or using technology/software. However, solving problems by manipulating diagrams/shapes, measuring, rearranging pieces, or other hands-on strategies, is still a huge part of mathematics at the school-age level and in many practical instances in real-life. Thus, it is nice to have classes like this one that focus on more rudimentary mathematical processes, which are both useful in real life and are what we will be teaching in our future careers.

Joyce divided us into three groups and lead each group through three separate stations, one at a time. She created handouts for us that could be given to Intermediate/Senior Mathematics classes in public school. These activities provided us with a hands-on, interactive way to solve measurement and geometry problems, including parameters such as area, perimeter, and volume. We had to manipulate the objects that we were given to create certain shapes, which extended our knowledge on how shapes translate to another form, either having the same parameters or different ones. I was proud to be part of the only group that solved all eight squares that could be made on the geoboard (see geoboard photo below). These activities could also be adjusted and implemented in any I/S grades and for any level of students’ abilities! 

I look forward to the weeks ahead and am especially excited for the “Teaching a Learning Activity” assignment in which I must teach my classmates a lesson designed for Grade 12 College level students in the unit of Data Management and Probability. I have never been part of a College level class, nor have I observed any lessons in such a class, so this will be a very interesting and education experience for myself as a future Mathematics Teacher!

My Weekly Report and Reflection 6 (Week 13)

For Week 13, the main purpose of the lesson was to reflect on our placements. Since physical education is my major, I sometimes have anxiety when thinking about teaching mathematics. Although I am extremely confident in my mathematical ability and have tutored many students in math in the past, I sometimes feel as though I will not have enough experience as a math educator once my career begins. In relative terms, I feel like I will be less prepared than my classmates because they have placements in Mathematics, whereas my current placement is in Health and Physical Education subjects. I will not have experience actually teaching math – as a professional – until I am in my final teaching block. This makes me apprehensive about whether or not I will have the experience necessary to be a professional Mathematics teacher in a secondary school setting. Physical education is a far different domain than mathematics, so I am nervous about having to teach both, when I have only focused mainly on the former domain.

One advantage to teaching both Mathematics and Health and Physical Education is that I think it will help be integrate both curriculums. Teaching an integrated curriculum is a major task in 21^st century education and is extremely beneficial for learners today. Math and physical activity may seem to be very distinct from each other, but I see many similarities between the two subject matters. For instance, pathways and directions (special awareness) in the gymnasium relates to coordinates and shapes in math. Furthermore, sport and physical activity rely heavily upon statistics, which is a mathematical course. Keeping records and quantifying physical achievements is how athletes progress in sport, which is dependent on math and statistics. Therefore, I can see statistics – among other math courses – being incorporated in physical education courses, and vice-versa.

It was really satisfying to be able to interview Alyssa and have her do the same for me. Talking about my placement was much needed, and I really enjoyed discussing it with Alyssa and the rest of the class. My placement did not go as well as I had anticipated, and my associate teacher had much more negative feedback than I expected. Instead of criticizing me during the teaching block and allowing me to work on my areas that need improvement each day, she gave me all the feedback at the end on the final day. She presented me with the feedback sheet, which had one line of what I did well – my lessons and activities. The rest of the page was entirely things that I need to improve on, and my associate teacher even wrote sideways to fit it all on the page. This was very surprising, and even somewhat overwhelming, to me since she had not discussed any of it with me beforehand. This not only made me feel overwhelmed with advice, but also was a bit of a shot to my confidence in my teaching abilities. Mainly, the teacher said that I need to work on my classroom management skills. I found her critique to be a bit redundant and unnecessary, as I think classroom management is something that I need to work on and believe it will come with experience. Therefore, I think my associate teacher could have just reminded me to work on my classroom management skills, rather than drowning me in negative comments about how poorly I am doing at the moment. I think it would have been much more beneficial if she had been a little more positive in her review, given the fact that it is only my first placement.

Nonetheless, the rest of my classmates and Joyce were extremely helpful and supporting. They reassured me that some teachers can be enormously nit-picky and expect perfection. Joyce also mentioned that if the problem escalated to the point that I could not withstand it, I could request a transfer. I am not considering this at the moment, but it is good to know for the future, in the case that I change my mind. Mainly, it is really great to know that I have an amazing group of colleagues and a terrific, caring professor in my mathematics class that I know I can count on to make me feel better! I hope to encounter teaching partners like this in the future as a mathematical educator!

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.