Mathematics Teaching Portfolio

I created a mathematics teaching portfolio to showcase 4 of the items I have created or worked on, creating a description and explaining their relevance to my teaching. I found this extremely fun to do and after completing it I found it very cool and a great item to have. It helped me reflect on the items I have worked on throughout the entire course. Below is a part of my mathematics teaching portfolio.

After creating this portfolio I thought about how I could use it in the future. I feel it is a great idea to allow students to create their own mathematics portfolio of items from the whole course. It can be divided by units with at least one item per unit.

A portfolio is a purposeful collection of student work that tells the story of a student’s efforts, progress, or achievement. It must include student participation in the selection of portfolio content, criteria for selection, criteria for judging merit, and evidence of student self-reflection. It is excellent to incorporate this into your classroom because it will engage students in learning content, help students learn the skills of reflection and self-evaluation, and understand what quality work is. It also creates an opportunity to document student learning in ways other than the traditional assessment.

This is why it is a great idea to try and incorporate this into your classroom so that students have an end of year they can carry along with them for the future.

 

Teaching Trigonometry

The article "Teaching Trigonometric Functions: Lessons Learned from Research" by Keith Weber talks about how students do not fully understand trigonometry concepts because they only understand how to solve trigonometric ratios. When asked to solve application questions or questions that do not involve ratios or cannot be solved by using ratios, they get stuck. This is because they are only using their procedural knowledge to solve problems.

As teachers, we should amend our way of teaching so we are explaining and analyzing the concepts more deeply. For trigonometry, students will learn better if they are taught how to estimate using the coordinate grid and the unit circle. Understanding that the unit circle has a radius that is 1 unit, will help students estimate. They can look at the x and y components of a given angle based on how close or far it is to 1. This can help students understand/realize that the maximum value of sine and cosine graphs must be 1 and the minimum value must be -1. All other values will be between 1 and -1, which are easily estimable using the unit circle. Below is an image of the unit circle (Figure 1) which students are taught in school but this is just another variation of the special triangles. It does not teach you how to estimate an angle such as sin of 70 degrees.

Usually students are told to use the calculator and asked to round to two decimal places when asked to solve an angle that cannot be solved using special triangles (ex. sin 70). When plugging values into their calculators, students do not truly understand what their answer even means, it is just a number to them. If we start teaching students how to estimate, it will strengthen their understanding. Using the second figure, can see that the angle is being estimated by looking at the lengths of the x and y components of the terminal arm. By first being able to draw the prinicpal angle, create the components, understand how to find sine, cosine, and tangent, we are able to estimate any angle.

Use of Exit Cards

Exit cards are an essential part of student learning which I feel are not being incorporated enough.

Exit cards are written student answers at the end of a class or lesson to questions the teachers ask. Such rapid, informal evaluations enable teachers to rapidly assess the comprehension of the material by students. Teachers should use these because they provide teachers with an informal measure of how well students have understood a topic or lesson. They also help students reflect on what they have learned, check their understanding, see what they know and do not know, and what they need to study more or improve on. They also allow students to express what or how they are thinking about new information and to think critically.

As a math teacher it is important to understand that when teaching a unit, all concepts build upon each other and are very closely related. If a student does not understand something, it should not be ignored because they will easily and quickly fall behind. This will lead you to talking to them about why they got a certain grade after they receive a bad mark on their unit test. At this point it is too late because the unit is over and it is time for the next unit. That is why it is best to provide these exit cards everyday to ensure students are understanding the concepts daily. It can be a simple exit card such as one that asks how you feel about the concept (bad, ok, good, great), or it could be a series of questions (true/false, matching, multiple choice, short answer).

They are very easy to create and only take a few minutes to complete at the end of class so implementing this is a great idea to enhance student learning. Below are examples of exit cards I have created and used.

1. The area is the amount outside. T / F
2. With a perimeter of 30cm you could have an area of 56 ㎠   T / F
3. With a perimeter of 30cm you could have an area of 54 ㎠   T / F
4. There are a maximum of 4 possible dimensions of length and width for a rectangle with a perimeter of 12cm  T / F

Application Problems

When teaching math, students find it very hard to stay engaged and often think the material they are learning is pointless. They find the content irrelevant to their lives which causes disinterest leading to disengage. This leads to distractive behaviours which prevents learning for themselves and others. 

I created a real-life problem called a "photo problem", shown at the bottom. To create one, you just have to take a picture of anything and find a way to create a problem around it or relate it back to a mathematical concept. I took a simple picture of my family drinking tea and chatting. It can really be anything, as simple as an everyday activity. I related it to math by creating an action, like shaking hands, and asking how many handshakes would there be if each person (6 people) shook everyones' hand only one?

Doing this will always create a great problem for students to work on because it relates directly to their lives. They will feel they are not even doing math but rather they are just solving a life problem. It is also very easy for them to imagine what is happening, instead of wondering what the question is asking or what it means. It also keeps them interested and engaged because it is relatable, which also makes it fun for them to participate in.  As a teacher, it is a great idea to present these types of problems as a minds-on activity at the start of class to allow students to think about what the answer may be on their own, with or without previous knowledge. I encourage you to implement these types of problems within your classroom to enhance student learning. 

There are 6 people in a room. Each person must shake everyone's hand only once. 

a) How many handshakes will there be?

b) How many handshakes would there be with 35 people? 

c) Can you think of a general formula in terms of n for the number of handshakes

Learning through Educational Games

Learning can be very boring for students if you stick to the same method every single day. As teachers we should be switching up the way we teach so that students are always engaged. A way to do that is through educational games. I created a game for a Grade 12 Advanced Functions class so that students can learn concepts involving the graphical and algebraical representations of rational expressions. I would like to share that game here so that you can include it in your class if you wish.

Purpose
Students will learn about the following features of rational expressions both graphically
and algebraically.

• Intervals (positive/negative, increasing/decreasing)
• Domain and Range
• Asymptotes (vertical/horizontal)
• Intercepts (x and y)

Students will choose a card from any category, they will read the question and answer the question
- if they get the answer right they will receive the amount of points on the card (1-7)
- if their answer is incorrect, another player will have the option to try to steal the point by
answering correctly
- if no one gets the answer, no one gets a point
The person with the most points at the end is the winner.

Another way to play the same game is as so:

Materials 

• Game Board
• Question Cards (4 Categories)
• Intervals, Domain/Range, Asymptotes, Intercepts
• 6 Playing Pieces
• Jumbo Category Die and Jumbo Movement Die

Number of Players

This game is designed for 2 - 6 players. For more than 6 players, players can create teams of 2. 

Objective of the Game 

Be the first player to reach “The End” and become the ultimate Smart Student 

Setup

1. Place each of the coloured deck of cards on the board in its designated corner
2. Each player chooses a playing piece and places it on the “Start” space

Playing the Game 

1. The player with the first birthday of the year (Player 1) is the Reader and asks the first question.
2. Player 1 rolls the Jumbo Category Die to determine the category of the first question.
1. Blue is Asymptotes, Green is Domain & Range, and Orange is Intercepts. 
3. Player 1 selects a question card from the category rolled. They read the question. Any player can yell out an answer at any time.
4. The player who answers the question correctly rolls the Jumbo Movement Die and moves their playing piece forward the number of spaces rolled.
5. If no player gets the correct answer, Player 1 wins the round, rolls the Jumbo Movement Die and moves forward the number of spaces rolled.
6. Play passes to the left and the next player becomes the Reader 
7. Spaces on the Board
1. Hard Question Space(Red Space): player must draw a card from the red pile “Intervals” and ask a question
2. Pink Space: player must move their playing piece back three spaces 
3. Yellow Space: player is silenced from answering or asking the next question
8. Tie Breakers: If two players answer the question at the same time the Reader decides which player wins.If the Reader cannot decide then they must choose a card from the red pile “Intervals”, whoever answers first out of the two will win the round. 

Winning the Game

The first player to land on “The End” is declared the winner and becomes the ultimate Smart Student.

Roles and Responsibilities of Teachers and Students

Both teachers and students have responsibilities they must fulfill in order to ensure student success. Teachers are sometimes held responsible for poor student grades because their test was "too hard" when in fact poor student grades falls under the lack of responsibility by the student.Students have roles inside and outside of a classroom which they sometimes don't realize. Teachers also have roles inside and outside of the classroom - different from the students' roles. 

Example: Student receives poor grade on test 

Student must:

• review concepts taught in class
• complete all assigned homework 
• attend class
• ask for clarification or assistance as required (seek extra help)
• put in effort to achieve high grades

Teachers must:

• teach concepts based on curriculum expectations 
• prepare lessons and practice for students (in class and at home) 
• help students as required
• answer questions students may have
• provide students with feedback on how to improve/what to work on
• provide materials needed 
• address different student needs 

Now, as you can see there are many factors that may have contributed to a student receiving a poor grade on their test. To say it is the teacher's fault or the student's fault is not fair. To find out more it, other factors need to be considered such as the marks of other students in the classroom. If they all did poorly it is likely the fault of the teacher but if it is just one student then it is likely his/her fault. 

Understanding the roles of both student and teacher are important because when combined together it is likely to result in students' academic success. Explaining these roles/responsibilities is essential at the beginning of the semester so that expectations are set and both student and teacher know what they must do. A great classroom arises from a mutual understanding between teacher and student.  

Teaching Strategies

Within a classroom it may be hard to think of a fun activity, get that activity started, manage your classroom, etc. I would like to share a list of strategies I have learned in my math class

Strategies when trying to organize students in groups:

1. Use a deck of cards 
2. Number students off 
3. Use an online group maker 
4. Allow them to choose their group 
5. Make your own card with 4 categories with a different item in each category (ex. Category: Animal, Item: Cat, Dog, Frog, Wolf, etc.
6. Have students pick lollipops out of a bag and group them based on the colour at the bottom of their lollipop
7. Have students draw out of a hat
8. visibly random groups   

Strategies when creating/during group activities:

1. Organize different activities by keeping them in large zip lock bags 
2. Use small white boards to compare different group responses 
3. Use laminated paper so it can be written on and reused
4. Use manipulatives to make the activities interesting and engaging
5. Have some "easy" activities not just "hard activities"
6. Desmos can be used as a great group activity especially when there are graphs involved
7. There are many fun and interactive games (What would you do?, parabola headbanz,etc)
8. Use vertical and horizontal permanent and non permanent surfaces.

Technologies to use in classroom: 

1. Ipads/laptops/phones/chromebooks/computers
2. Document camera 
3. Projector 
4. Smart board
5. Motion sensors 
6. Calculators/graphing calculator 
7. Online programs (desmos/excel)
8. Assistive Technology (hearing aids/microphone/speakers) 

Below are some pictures of examples