Mathematics for Teaching Exit Portfolio: My Journey

Introduction

          As I begin my mathematics teaching portfolio I would like to go back and reference my very first blog post where I said, as a future mathematics educator, questions to consider for my future blog posts are as followed:

“Why do we need to teach/learning mathematics?”
“How to differentiate student learning in the classroom?”

“How many ways are there to solve multiple math problems?”

“How can we help individuals think critically, solve problems mathematically while connecting the math problems to their everyday lives?”

“How can we make mathematics fun for everyone?”

“How can we teach math using technology?”

          Throughout the year in our mathematics teaching class, we have been engaged in a wide variety of mathematical tasks such as online problem solving tasks, digital forum discussion, leading learning activities, creating a curriculum unit plan, determining how to incorporate technology and looking at math from many different points of view. I have watched myself grow as a teacher through my experiences in practicum and learning how to teach my subject areas of mathematics and chemistry from colleagues and professionals.

Mathematics for Teaching Portfolio #1: The Three Part Lesson Plan

          One of the most significant strategies that I have learned through my mathematics for teaching is the Three Part Lesson Plan and how to structure problem solving within a lesson plan. One thing I struggled with throughout high school was problem solving. Every time I had to solve a series of problems I was lost, confused and frustrated and often gave up. The method of the Three Part Lesson Plan approach to solving word problems will help me in my future to structure my lessons. In our class we had the opportunity to design a student-centered three-part mathematics lesson plan that address some course expectations from the mathematics curriculum document. I wanted to take an opportunity to reflect more on the aspects of the Three Part Lesson Plan and how I would implement this into my future and additionally, reflect on the activity that I made for this assignment. 

          Before I get started, I wanted to refresh on the aspects of the Three Part Lesson Plan. Firstly, the key points of the three part lesson plan are to serve two main purposes: to explore, develop, and apply understanding of a mathematical concept (teaching through problem solving) and to guide students through the development of inquiry or problem-solving processes and strategies (teaching about problem solving). In general, as a future teacher, one of my goals of this course was to develop awareness of strategies for teaching problem solving and differentiated instruction opportunities. The Three Part Lesson Plan is structured as followed:

Before/Getting started: 10-15 minutes

         Get the students to be cognitively prepared for the lesson problem by having them think about ideas and strategies they have learned and used before. The teacher organizes a revisit concept, procedure or strategy related to the lesson’s learning goal. This portion of the three part lesson plan evokes prior knowledge, skill and strategies to then be used in the “working on it” portion of the lesson plan.

During/Working on it: 30-40 minutes

          The students are actively solving the problem. They work in small groups, in pairs, or individually to solve a problem and record the mathematical thinking they used to develop solutions. In this part of the three part lesson plan students are developing independence and confidence by choosing methods, strategies and concrete materials they will use, as well as different ways to record their solution either through words, pictures, symbols. Throughout this part of the lesson plan the teacher than acts as the mediator and circulates the classroom by making observations about the way students are interacting and taking note of the mathematical models of representation, methods and strategies and mathematical language students are using. This leads to an assessment as and for learning opportunity as the teacher can give immediate feedback or pose questions to provoke further thinking or have other students explain their plan and method of solving the problem.

After/ Consolidation and Practice: 10-15 minutes

          The teacher strategically co-ordinates student sharing of solutions to the lesson problem, using a mathematical instructional strategy (e.g. math congress, Bansho or a gallery walk) By using such a strategy, the teacher can facilitate a whole-class discussion whereby students explain the mathematics in their solution, methods and strategies and discern whether classmates use the same or different methods. This portion of the three part lesson plan is excellent as it is a way to learn how others in your class are thinking and determine what mathematical processes they are using. Additionally, through rich mathematics classroom discourse, students develop and consolidate their thinking and the learning goals of the particular lesson. This then leads to the teacher learning about their students and the discussion will guide the direction for future lessons and activities. 

          The Three Part Lesson Plan involves the three parts illustrated above. We used a “placemat” strategy which encouraged collaborative thinking amongst our colleagues.  We had to use the three part lesson plan and come up with how we would use the growing dots problem in our classroom. I found this method to be useful as it allowed us as future teachers come up with the best possible way to approach this problem in a real life setting. This can also be a useful tool in our future to use with our colleagues to have consistent teaching and lessons throughout the grade levels. This placemat strategy can also be used for students to work together to promote critical thinking and collaboration amongst classmates. It provides an opportunity for each student to have their individual idea and record their response then come together and combine everyone’s idea to make the best group decision to the posed problem.

In conclusion, as you can examine from the photos posted of my classmates work below, we all had a slightly different way of viewing the problem by using various visual representations. I believe this activity allowed us to expand our viewpoints and got us as future educators to step out of our shell by accepting that other answers are correct and add value to our learning. This activity facilitated rich observation and discussion of mathematical thinking. (Fosnot & Dolk, 2001, p. 1) This process can be easily implemented in the classroom as the “Action” part of the lesson as it allows the students to defend and explain their thinking while the teacher can facilitate a group discussion. Math congress provides an opportunity for students to exchange their ideas and discover new strategies to solve mathematical problems through dialogue, pictures and symbols. This strategy can be extremely useful in my future career as a mathematical educator and it involves critical thinking, problem solving and collaboration. Using a Math Congress at the end of a lesson as a summarizing activity will be useful as it encourages group work, and can demonstrate the students thinking, as well as sharing ones thought. It is useful from a teacher’s perspective because it provides a platform for the teacher to guide students’ mathematical thinking towards mathematical concepts and processes (Kotsopoulos & Lee).

          In addition to this, we needed to than reflect on what we learned for the Three Part Lesson Plan and implement this activity into something we could use in our future classroom. An assignment from earlier on named “Three Part Lesson Plan: On the Frozen Pond” was an opportunity for myself to use what we learned in class and place it into future practice. The problem I focused on was the one below, and the following teacher guide was also made to be used in the classroom to assist with this problem solving question. 

The problem will cover the following curriculum expectations and mathematical processes.

Overall:
àDetermine, through investigation, the optimal values of various measurements
à Solve problems involving the measurements of two-dimensional shapes

Specific:

àDetermine the maximum area of a rectangle with a given perimeter by constructing a variety of rectangles, using a variety of tools, and by examining various values of the area as side lengths change and the perimeter remains constant. 

Determine the maximum area of a rectangle with a given perimeter by constructing a variety of rectangles, using a variety of tools, and by examining various values of the area as side lengths change and the perimeter remains constant.

Reasoning and Proving: develop and apply reasoning skills (e.g., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments

Representing: the students will be asked to use visual representations, explanation with words, and symbols/equations. Create a variety of representations of mathematical ideas, connect and compare them, and select and apply the appropriate representations to solve problems

Communicating: communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.

          In this lesson, the teacher will be observing the students talking amongst their peers and discussing the minds on question which were posed at the beginning to develop a platform for the rest of the lesson. The teacher will be using this minds on activity as a diagnostic tool. They will be looking to determine how well the students are grasping the concepts of perimeter, area and maximization before getting started with the action portion of the lesson. If the students seem to be struggling with the ideas posed, the teacher can choose to show the students more examples and explanations surrounding the concept. Throughout the action portion of the lesson, as the students are working in their groups, the teacher will be circulating the classroom and making notes on how the students are understanding the problem. If the teacher wishes they can use a checklist, ranking system or anecdotal records. These tools will record the students’ achievement of the success criteria and the mathematical processes used. The teacher will also need to record the students’ collaboration and self-regulation skills. They will be specifically looking at:

        1. How the student responds to the ideas, opinions, values, and traditions of others

2. How they work to resolve conflicts and build a consensus to achieve the group goals

3. How they share information, resources, and expertise and promote critical thinking to  solve problems and make decisions

4. The student’s ability to seek clarification or assistance when needed

5. How the student perseveres and makes an effort when responding to challenges

When the students post their solutions around the classroom, the teacher will need to assess the groups’ understanding of the problem solving process and maximization of the area of rectangles. The teacher will need to check if the students include various elements of the problem solving process. For example, did they include a visual representation of the problem, did they use equations or symbols, and did they include any words or explanations. The teacher can gauge the students understanding of the problem solving process by checking the number of problem solving elements they have included in their solutions. To gauge the students’ understanding of maximization of area, you will need to assess their specific solutions. For example, did they find that the maximum area of a rectangle with a given perimeter, had the dimensions of a square? You can find this through examination of their solutions, but you can also find this from in the form of conversations and discussion with the students. In terms of assessment this activity focuses on giving the students feedback and engaging in mathematical discussions through dialogue. 

This lesson is a student-centered lesson, where the teacher is acting as only a facilitator. The lesson focuses on one specific maximization problem. The lesson will start by introducing the problem with a series of guiding questions, then the students will be given plenty of time to discuss and create a solution to the problem within their groups. The lesson will be concluded by consolidating their learning through a class discussion and using a gallery walk to examine the various solutions the students found. Using a gallery walk will allow the students to develop a greater understanding of how their peers solve problems and this strategy will help develop critical thinkers and increase students’ collaboration skills. This activity develops students relational skills as the teacher and student develop greater skill sets and application of mathematical concepts which is engaging and allows the student to know what to do and can explain how they solved the problem. This leads to a positive classroom environment. Compared to if the teacher taught conceptually the student memorizes a rule or procedure but doesn’t understand why it works a certain way.

Students will be given the background information needed in order to solve the problem (definitions of area, perimeter, rectangle, maximizing, etc.) However, they will not be taught how to solve the problem before it is given. This process ensures that the students will not be simply memorizing the steps to finding the maximum area of a rectangle with a given perimeter. By having the students solve the problem with little guidance, we are ensuring the students are understanding the full aspect of their solution and how it relates to their final answer. The big ideas that the students will take from this lesson is that they will be able to describe relationships between measured quantities, connect measurement problems with finding optimal solutions for rectangles and develop numeric facility in a measurement context. 

Donna Kotsopoulos & Joanne Lee (2012) An Analysis of Math Congress in an Eighth Grade Classroom, Mathematical Thinking and Learning, 14:3, 181-198, DOI: 10.1080/10986065.2012.682958

Fosnot, C.T., & Dolk, M. (2001). Young Mathematicians at Work Constructing Fractions, Decimals, and Percents. Portsmouth, NH: Heinemann.

Mathematics for Teaching Portfolio #2: Digital Word Problem

Trick or Treat: Solve the Problem

          At the beginning of the year our class got split into two groups to undergo 5 digital math word problems throughout the year. The purpose of this assignment was to use forums to share exploration experiences with an original digital photo with an original math problem. This activity allowed us as collaborators to come together and discuss real world application and problem solving techniques to implement in our future classroom. Throughout the forum, each of us had our own “role” as either a Resource Manager, Facilitator, Connector, Implications Detective or Recorder. This problem I chose to reflect on was when I was the Resource Manager and I wanted to take the time to reflect deeper into this problem and on my colleague’s responses. The question I posed to my group was the following

“It’s Halloween and Allison and Lauren need your help. They have collected some chocolate! Allison has collected 3 kit kat and 3 coffee crisp which weigh 83g, and Lauren has collected 6 kit kat and 4 coffee crisp which weigh 135g. Help them determine the weight of an individual kit kat and the weight of an individual coffee crisp”

The forum continued and the following questions were asked by the Facilitator of the discussion.

“Is there any cross curricular connection to any other subjects in this question? If not, is there a way that this question can be modified to create cross curricular while maintaining the core of the problem?”

“Can the wording of the problem be modified to create more practical real life application?”

          The overall responses to the problem allowed us as a group to dig deeper into the problem and try and relate the problem to other courses, which is a true challenge of mine to be cross-curricular. I believe that this problem can also relate to any science course as part of a lab activity if they are weighing certain items in a lab setting to find unknowns, as the question can be altered to weigh any object. Therefore, for this question to be more practical it can be changed to weighing any object that the student wants to weight or also develop two equations, two unknowns of a real life situation, for example “if a company sold 13 cupcakes and 8 coffees for 29 dollars on Saturday” and “if the company sold 11 cupcakes and 4 coffees for 19 dollars on Sunday” what is the price of each cupcake and coffee? Thus, relating this problem to a business perspective can get the students more interested than weighing chocolates. Also, the use of chocolates was to have a manipulative when learning how to solve a system of equations as I believed this was another topic of mind when in high school that I had difficulty visualizing system of equations. This was one of the main reasons why I wanted to share this problem as I found a way to visualize the process of substitution and elimination. I also had difficulty in grade 10 what the point of solving 2 equations and 2 unknowns therefore this problem has allowed me to unfold that thinking process to help my future students. Additionally, finding a way to visualize system of equations and having the collection of two or more equations with the same set of unknown can be hard to solve when first learning the concept. I believe adding a visual hands on learning experience to the classroom will benefit the student ability to solve the problem. A modification can be that the students can add candy to the problem to visually see subtracting the two equations. For example, if the students wanted to find a way to eliminate the x term, knowing that they can multiply Allison’s equation of 3x + 3y = 83 by 2 which would be 6x + 6y = 166 and then add the corresponding kit kat and coffee crisp to match the equation and then subtract the equation 6x + 6y = 166 minus 6x + 4y = 135, 6 kit kat minus 6 kit kat is zero and it can be a visual aid of solving for the unknown value. Therefore, this problem provides a more hands on experience in the math classroom.

In terms of what curriculum expectations this problem covers is as followed:

Grade & Level	Curriculum Strand	Overall expectations	Specific expectations
Grade 10 Academic	Analytic Geometry	Solve problems using analytic geometry involving properties of lines and line segments	Solve problems of two linear equations involving two variables, using the algebraic method of substitution or elimination Solve problems that arise from realistic situations described in word problems or representing by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method
Grade 10 Applied	Modelling Linear Relations	Solve systems of two linear equations, and solve related problems that arise from realistic situations	Solve systems of two linear equations involving two variables with integral coefficients, using the algebraic method of substitution or elimination Solve problems that arise from realistic situations described in word problems or representing by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method

Connections to the Mathematical Processes

Problem solving – students are required to solve a simple problem by working “backwards” to find the original weight of the chocolates.

Selecting tools and computational strategies – students must choose between the method of substitution and elimination to solve the problem, as well as they can choose graphical methods

Connecting – students can see how the skill of substitution and elimination can be used in real world settings.  When considering cross-curricular connections (such as to science), students can see even more clearly how this can help with real-life unknowns

Representing – the physical use of candies to create the equations helps students to see where these symbolic representations truly come from.  As an added bonus, giving students candy for the successful completion of a problem will definitely help their engagement

Communicating – students are to “translate” the word problem into a system of equations using the appropriate symbols and notations.  They are also required to “translate” their answer back to the original problem by answering the question: “What is the weight of an individual Kit Kat and the weight of an individual Coffee Crisp?”

          In my teaching I would use this as an assessment as and for learning tool as it will give the teacher an idea of where students are, and how students learn. The activity also provides an opportunity to understand how they like to learn themselves, are they visual or non-visual learners. To incorporate this question into a lesson the first thing I would have to do is ensure my students have the skills to pull the information needed from the question to create the linear equations. To make it more hands on within the lesson I would give the students the opportunity to recreate the problem by actually providing the scales and candies for them to use as props when thinking about the question. Applying the question to any context to fit time of year is easily done. This question could be implemented as a minds on activity to get students thinking about linear systems. It requires students to find the required values for the equation and place them in the correct places with the correct variables attached. It allows the teacher to see what parts of the equations students are comfortable with and can identify and indicates which areas of the system the students may need more practice with. Therefore, it will provide a benefit of feedback to the students also and help develop their understanding of the curriculum expectations mentioned.

Mathematics for Teaching Portfolio #3: Lesson Leading Activity

          The last activity I would like to reflect further on is the Lesson Leading activity and the importance of incorporating technology into the classroom and the addition of using the teacher resources available online at the OAME and other resources available to us as future teachers. In my lesson leading activity I used Desmos as a tool for an online learning experience for my colleagues.  

I was responsible to cover any expectation in the grade 9 and 10 academic mathematics curriculum. For the purpose of this activity I chose to find technology that covers the curriculum expectations and mathematical processes as followed

Grade 10 Academic Mathematics, MPM2D

Overall Expectations:

v  Determine the basic properties of quadratic relations

v  Relate transformations of the graph y= x² to the algebraic representation y= a(x - h)² + k

Specific Expectations:

v  Identify, through investigation using technology, the effect on the graph y= x² of transformations (i,e., translations, reflections in the x-axis, vertical stretch or compressions) by considering separately each parameter a, h and k [i,e., investigate the effect on the graph y= x² , of a, h and k in y= x² + k, y= (x - h)² , and y= ax²];

v  Explain the roles of a, h, and k in y= a(x - h)² + k using the appropriate terminology to describe the transformation, and identify the vertex and the equation of the axis of symmetry

*The overall and specific expectations for this activity are found in The Ontario Curriculum Grades 9 and 10 Mathematics document, page 47

Mathematical Processes:

v  Reasoning and Proving: students will be applying reasoning skills by recognition of relationships between the parent function y= x² to the algebraic representation y= a(x - h)² + k with justification

v  Representing: students will be using the desmos platform to represent visually the roles of the values of a, h and k and explanation with words, and symbols/equations

v  Communicating – students will be communicating their exploration findings on the student handout, and with their peers throughout the activity using appropriate transformation terminology

The activity is easy to use for teachers and has many advantages, key indicators of students learning and special features that can be an extremely useful tool for teachers to use. The interface of desmos is easy to use and very user friendly. For example, on your teacher homepage for desmos it has a list of all the students names and what questions they have done, shown below, and the addition of each individual student response shown in the second picture. 

Desmos online platform is a way for teachers to track student progress and can be a beneficial tool to use at the end of a unit as review, or as a quick online graphing tool. The activity features a “pace” button where I can assign students either less or more of the activity based on their learning needs. Additionally, if students are gifted I can include additional application questions at the end to further their thinking. This feature is extremely useful for differentiated learning and a way to adjust the level of the learning activity. The activity features a “pause” button which can be used to pause all of the students at a certain point to have a discussion on what they have learned so far to ensure that everyone is understanding the mathematical concepts. (See photo below) This feature is useful for classroom management and adjusting the student’s use of the activity The activity features a slide called “sliders free-play” where students can explore on their own, in pairs or in groups the roles of a, h and k values in y=a(x-h)² + k by developing their own functions with the desmos activity and use appropriate terminology to describe the transformations. This feature is useful as it allows the students to discuss amongst their peers and communicate the mathematical concepts through engagement and exploration. In addition, there are many key indicators of student learning as the teacher has a summary tab and a list of all the students which include their answers to each slide, their pace at the activity and can identify where students are struggling throughout the activity. The teacher can keep a look out on their computer while circulating the classroom to see how everyone is involved in the activity. A handout that compliments the activity or the answers written online can be seen as a product that the teacher can use for assessment. Observations and conversations can occur while the teacher circulates the classroom

This particular activity can be extremely useful for student-centered learning and they are allowed to go at their own pace while teachers keeping track. The teacher can also see when students are struggling as they have that available platform to refer to while the students are working on the activity. With this activity I also made a package handout of all of the questions as through repetition students can re-write their typed answers on paper to get extra practice and prepare them for the final assessment task. 

On the final tab of this activity there is a section called “Sliders Free Play” which allows the students to collaborate and practice any type of transformation, this is an example of what it would look like for students. 

In conclusion, I believe utilizing technology into my future practice is something I will still need much more work on. In addition, desmos is an extremely helpful for visual learners and I also struggled with this exact topic and having that visual aid would have helped me in my previous math classes. Therefore, as a future teacher I would like to implement and use technology in my classroom to be used as a guide for students and an aid to help them understand the mathematical concepts being presented.

          In my previous experiences in secondary education (grade 9-12) there was a lack of use of technology and now that there are many options available, I am overwhelmed with the ability to incorporate technology into mathematics. Each and every colleague had an outstanding application of technology in their Lesson Leading Activity presentations which as expanded my technology toolkit and I can refer back to all of these activities in my future career and use the wonderful ideas that they have come up with. Without this activity, I would have not known about all of the various types of technology available to assist in teaching mathematics and can be used as a form of assessment instead of the typical “chalk and talk” style of traditional teaching. Therefore, as a future educator I strongly suggest incorporating technology into your classrooms to improve student learning and success in the classroom.

          Now that I come to an end of my mathematical journey I will always take a step back and look at all of the excellent strategies I have learned this semester that will truly help me succeed in my future career as a mathematics and science educator.