MATHEMATICS FOR TEACHING PORTFOLIO

Portfolio Activity #1: Reflecting on Reflecting… My First Blog Post to Now

Since I have the opportunity to implement a mathematics portfolio that reflects on various things that we have learned this year, I would like to start by sharing how about how much I have grown both personally and professionally from my first blog post back in September, and how utilizing reflective practices with Blogger has made me a more critical mathematics educator. In my first blog post, I set a variety of goals that I wanted to accomplish throughout the EDBE 8F83 course. This was one of the paragraphs I wrote in my initial blog post:

“The main goal for this course is to gain valuable tools and resources that will help me teach mathematics at the I/S level. I am truly passionate about the topic of mathematics, and I want to find ways to share that passion with other students. I want to find ways that will help students in finding that connection between the theoretical mathematics they learn in the classroom and bring those math concepts into practical contexts. Also, as technology is emerging and becoming more prevalent in the classroom, another goal of mine is to ensure that I can apply technology into my own mathematics classroom and make it meaningful for all of the students involved in the learning process”.

Throughout my blog posts, I realized that the first thing we discussed in EDBE 8F83 was the difference between growth and fixed mindset, and between conceptual and procedural understanding of mathematics. From the beginning of my blogging to the end, I reflect on how I used to learn as a secondary student and how I was fixed mindset, and procedural understanding was my method of learning mathematics. However, I realized that for success in post-secondary mathematics, I needed to understand the conceptualizations behind the theoretical perspectives of mathematics as opposed to simply memorizing the ‘steps’ of each math problem. I also realized how important it is as a future educator to have a growth mindset, meaning that learning is fluid and thinking about the ‘high-ceiling’ that many of the problems that we did in class were composed of.

I realized throughout reading my blogs that doing many of those problems using Japanese Bansho methods and using various problem-solving strategies are what can aid students in learning conceptual understanding, and ultimately reach their maximal potential in the mathematics discipline. Below are some photos of various activities we did in class that required problem-solving skills that I/S students could acquire throughout their educational careers:

Another thing I wish I would have thought of at the beginning of my blogging journey is the importance of the mathematical process expectations that are outlined in the Ontario curriculum. I realized how important these process expectations also relate to the learning skills and work habits that we assess; such as collaboration, initiative and self-regulation; which are skills that are imperative to be an effective mathematics student. The process expectations are as follows:

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Throughout our first semester we used our collaborative skills to attempt various questions using our problem-solving skills. One question I remember we did was maximizing the size of a box. Here are some solutions that were completed in class:

At the beginning of my blogging, I did not realize how many tools you could use and processing strategies for a similar type of question. This is where I realized that a lot of my blog posts had something in common… the discussion of inquiry! Inquiry-based learning evokes collaborative strategies coupled with investigational skills where the teacher behaves more as a constructivist, meaning that they are more of a facilitator as opposed to the ‘expert’. This is where I realized that students can make meaning of their own solutions, and that not all questions have one concrete solution. This is where I had an ‘aha’ moment and noticed that this is where students can connect the theory of mathematics into practical contexts that they can use in their everyday lives, which was one of the goals that I mentioned in my very first blog post. So, I am glad that I did these inquiry-based learning activities as it made me understand how I can connect each student to their own practical goals.

Lastly, the last goal I had in my first blog was the incorporation of technology within my classroom. The main foci of my blogs in the second semester were related to the learning leading activities that our peers did within the EDBE 8F83 classroom that would be useful to teach for I/S students. Many of the resources that our peers presented incorporating a wide range of technological applications, such as Desmos, a multi-purpose application that helps with graphing, algebra and geometry. Pixton, an application that combines the usage of media arts and mathematics to create scripts that can be math-based in nature. And Kahoot, a website where students use their cellular devices to answer multiple-choice questions that can help for reviewing specific concepts and test preparation. Overall, being exposed to these technologies and working with my peers to discuss these technological resources in the classroom have given me comfort throughout both the reflective process of blogging about technology, and actually using the interface presented in the applications.

Portfolio Activity #2: Digital Word Problem: St Catharine’s Transit – Is it Worth it?

In September, we were asked to get into small groups and complete five ‘digital word problem’s’. That is, we were asked to make real-life problems that are interwoven in various Ontario curriculum expectations for I/S mathematics. The purpose of this assignment was to use an online forum to critically analyze the problem and discuss the contents of the word problem and its relationship to theory and practice. I really enjoyed the digital word problem activities as it was a great opportunity for us as prospective teachers to collaborate and create questions that may be useful in our own classrooms in the future. There were five roles that each of our group members cycled through for each digital world problem: resource manager, facilitator, curriculum connector, implications detective and the recorder. Each role made us reflect through a different lens for each digital word problem, making us better reflectors, and ultimately better educators. I chose on the digital word problem I created when I was the resource manager, who is the sole creator of a digital word problem with sample questions that could be asked to an I/S mathematics classroom. My question was as follows:

Public transit is a necessity in many larger cities. However, there are different scenarios where you can save money, whether it’s using a short-term bus pass or an annual pass. Some potential questions that could be asked are as follows:

1. How many times would you have to ride the bus in a month (using the attached fares) to equal the cost of the 31-day pass?

2. How much money are you losing in a month if you purchased the 31-day pass and only went on the bus 10 times a month?

3. Assume you are a group of four people wanting to travel 5km to the nearest restaurant. Is it cheaper to take the bus, or would you be better off using a taxi service with a flat rate of $6 and costs an additional $0.75/km?

The reason why I posted this question is because it is great at the local level. This is something that all students in the Niagara region can relate to, whether or not they take public transit or not since we are also comparing the prices of other modes of transportation (i.e., driving your own vehicle). Transportation always seems to be a topic of discussion in many different practical contexts, whether it is the government or environmentalists. I also intentionally made this question by placing financial literacy within the hidden curriculum of this digital word problem, as it is important for students to be financial literate when progressing to further stages of life. Therefore, I felt that a transportation question is extremely necessary for I/S students to be exposed to within the mathematics classroom.

Jenna was the facilitator for this question, meaning she asked the group some questions related to the problem’s content after critical analysis. 

Jenna asked the following questions:

1. Since this question relates to both the academic and applied stream for grade nine math, how can we modify the questions to benefit the different learning strategies that occur is each stream?

2. St. Catharines is a smaller city and has limited public transportation options. If we expanded this problem to a larger city like Toronto, would we be able to create a question from it that relates to the curriculum expectations in higher grade level math courses?

After reflecting on these questions, it was clear that both of these questions do relate into both the academic and applied curriculum. The consensus in the group was that you could ask the academic class to extend their knowledge into maybe having multiple rates for a cab after a certain distance or communicate the idea of partial and direct variation in terms of the question and ask them to state similar examples of partial and direct variation that could be found locally in St. Catharines. The applied class could use a table of values or a graphic organizer to display their thoughts and answer the prompts accordingly. Jenna’s comment about potentially having higher-order thinking problems for a bigger city like Toronto was of particular interest for me to reflect on. Using multiple fares and rates could lead to a series of unknown variables that could lead to two variable systems of equations (i.e., found in MPM 2D and MFM 2P) and three variable systems of equations (i.e., found in MCV 4U). It was also interesting to realize how much this question could expand in terms of the question’s logic and thinking capacities.

 

Emily was the curriculum connector, and as we mentioned, the course fit the MPM 1D and MFM 1P course perfectly, with the following expectations as outlined in the curriculum.

Grade 9 Academic:

- Number Sense and Algebra
Overall: • manipulate numerical and polynomial expressions, and solve first-degree equations.
Specific: – solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods

- Linear Relations
Overall: • apply data-management techniques to investigate relationships between two variables;
• demonstrate an understanding of the characteristics of a linear relation;
Specific: – design and carry out an investigation or experiment involving relationships between two variables, including the collection and organization of data, using appropriate methods, equipment, and/or technology
– describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses
– construct tables of values, graphs, and equations, using a variety of tools
 

Grade 9 Applied:

- Number Sense and Algebra
Overall: • solve problems involving proportional reasoning;
• simplify numerical and polynomial expressions in one variable, and solve simple first-degree equations.
Specific: -make comparisons using unit rates
-solve problems involving ratios, rates, and directly proportional relationships in various context
-solve first-degree equations with nonfractional coefficients, using a variety of tools

- Linear Relations
Overall: • determine the characteristics of linear relations;
• connect various representations of a linear relation, and solve problems using the representations.
Specific: – identify, through investigation, some properties of linear relations
– determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation
– describe a situation that would explain the events illustrated by a given graph of a relationship between two variables
– determine other representations of a linear relation arising from a realistic situation, given one representation

The question also used these process expectations:

Problem Solving - Students are required to look at a real-world problem and find a logical and mathematical solution. 

Reasoning and Proving - Students will have to not only solve how much each method of transportation costs, but they also need to reason which method is the most cost effective and prove their answer using evidence.

Representing - Students will be representing a real-world problem using a variety of math forms, for example, numeric, graphical, algebraic, etc.
Communicating - Once the problem is solved, students will need to be able to communicate their solution effectively, as well as model their problem in a way that makes sense.

This clearly demonstrates the connection between the practicality of the digital word problem and the theoretical perspectives that the Ontario curriculum outlines for I/S students.

Owen’s implications were related to the issues of potentially the question become outdated in the modern classroom due to the alternate transportations that have appeared within the last five years, such as Uber. Although this is true, there will always be citizens, especially in areas like Toronto and Ottawa who will rely on public transit for financial reasons. He also mentioned the lack of mentioning about saving money through transfers, which is something that could be used as an extension for the academic level or for senior level courses. Overall, this activity, and this problem in specific was deemed important by every member in our group due to the multiple areas of practicality that the question demands. As a future educator, I will ensure that I incorporate these types of inquiry-based activities for groups of students to solve and come up with structured problem-solving strategies for students to investigate each type of practical question. These problems are great for both assessment for, as and even of learning! Implementing this into the EDBE 8F83 course has made me a more critical, and socially aware educator in the field of mathematics.

Portfolio Activity #3: Can You Escape? A Creative Inquiry-Based Resource

I will reflect on one of the learning leading activities that really stood out for me in the course of MBF3C in the strand of data management. The activity was an escape room, which is a concept that has recently been utilized within the mathematics and science classrooms across North America within the last ten years. Although this was a college preparation course, the great thing about escape rooms is that they are flexible for all levels of I/S students (academic, applied, workplace, SSTW courses, etc.). Another good aspect to these courses is that assessment is flexible as well! You could use these escape rooms as a diagnostic (for learning), as a in-class activity (as learning), or potentially as an evaluative piece or test preparation (of learning). The escape room consisted of a box intertwined with a series of locks, and within the boxes were a series of clues where we would have to go and eventually find a key that was somewhere in the classroom in order to win a prize! In terms of the curriculum expectations that were covered in the activity we did, was the data management strand. Also, all of the process expectations are used within this escape room. This is because we are using logic that is actually outside of the strand to be able to solve some of these locks, which makes us more logical learners in general! Below are a series of pictures related to the escape room:

Although I found the whole resource interesting. There were three concepts that really stuck out in my experience. First, there was a directional lock which code is used by moving the cursor up, down, left or right. So, we were given a map that went up, down, left or right and we had to follow the correct answers to the problems in order to open the directional lock. We also had to use an ultraviolet light to solve problems related to a deck of cards (which were also in the box). The trick was there was no battery and had to open another lock to get the flashlight battery! Lastly, I loved the last part where we had to find a phrase that related to our box and go and solve a problem related to standard deviation. The answer that we obtained from there was supposed to lead us to a key in the room that opened our prize. Overall, the structure of the activity is embedded in tons of inquiry and opportunities for students to be engaged! Another thing I realized about this activity is how flexible this could be for differentiated instruction and how much escape rooms support the universal design of learning. You could make questions that are easier or harder depending on the box and then assign groups based on ability. This can also be used for fundamentally anything in the curriculum! There is really no restriction on how to use this resource in the classroom, which is why I will absolutely incorporate this escape room into my routine classroom planning!

In chapter four of Thinking Mathematically, there was much discussion about conjecturing and how it is applicable to our secondary school mathematics classroom. The main idea of conjecturing is focusing on reinforcing knowledge that was already previously acquired and learning from any errors that were made in the past and correcting them in the future. The goal of conjecturing is to get students to think outside of the box and ultimately acquire those fundamental critical thinking skills that mathematicians need. This is directly related to the conceptualizations of the escape room structure because the locks in itself give you feedback (i.e., if your code is correct or incorrect) and it reinforces potential mistakes that you have made in your calculations.

Perhaps one of the most important features of this resource is the fact that it is set-up in such a way that is assesses many learning skills and work habits that are imperative for our students to develop in order to be productive members of society. Collaboration is clearly a huge aspect of this project. Without teamwork, it would be difficult for the group to come to generic consensuses on correct codes and to analyze hints. Other characteristics that could be assessed are self-regulation, based off of the students’ ability to perform appropriately in an activity that requires a large amount of engagement. You could also assess initiative and see which students are actively participating within the group. This is important because the whole purpose of modern education is to make our students productive in the real-world. Without these important skills, they could have all of the knowledge from the curriculum but will struggle in the workplace setting.

This is such a fantastic, engaging tool and there is no doubt that every single mathematics educator in the world should look at this and modify it to their own classrooms. There are truly no major cons to this resource and the flexibility that the escape room structure provides is so imperative to meet all student’s need, which is something that is so difficult to do in some math classrooms.

This is the end of my mathematics portfolio. These three resources that were discussed throughout my portfolio have provided me with a multitude of learning strategies that will help me become a lifelong learner, and they have also given me teaching strategies that will allow me to reach each individual learner in my classroom. All of the reflections I have done in EDBE 8F83 have made me so critical in my reflective practices that it has turned me into an educator that is self-aware, and now I feel that I can be the best teacher I can be for my students.