Mathematics Portfolio - Tanya Paladino

Final Reflection on Teaching Mathematics (Week 15 Reflection)

This course has provided me with the opportunity to develop my approach to teaching mathematics and expand my way of thinking regarding how to accommodate many types of learners.  The three key ideas that I will take away from this course are the importance of differentiated instruction, incorporating student-based learning through activities in the classroom, and involving technology in classroom activities. It will be my goal as a teacher to incorporate these concepts into my mathematics classroom because I believe it will greatly benefit the students, allow them to be engaged in the lesson, and create a positive learning environment; which makes for an effective teacher.

Differentiated instruction is an essential part of classroom planning because it is evident that not all students learn in the same way. It is important to differentiate instruction in order to continuously engage the students, grasp their attention, and hopefully inspire a will to learn. Differentiated instruction can occur in many different ways, such as differentiating content, process, product, and the learning environment. It is important that we are making accommodations for students when necessary, incorporate their interests, involve the students through many different types of activities: both teacher, and student centered, teach a concept multiple ways, use manipulatives, and incorporate technology, to name a few. We have discussed both instrumental and relational understanding in class and how important it is for students to learn mathematics catering to both types of learning. As teachers we need to ensure that students are developing both types of understanding, which can be done based on how we differentiate in our classrooms. It is important that we do not follow the same routines and methods each day while teaching, but rather try many different teaching styles, which will also allow us to grow as educators. An exceptional teacher will provide various options to students so that they can expand their thinking processes as well as develop a love and joy for learning.

Student-based learning through activities in the classroom has become an important concept frequently discussed in class. I have learned many effective ways to implement this notion through the student-centered activities we created and participated in throughout the course. Student-centered activities not only get students out of their desks and involved in their learning, but it also creates a positive and inclusive learning environment where students engage in peer-tutoring and grow as a class. It is essential that teachers are not just lecturing and providing the students with information while teaching, but rather allowing them to participate and get involved. Student centered learning allows students to work as a team with their peers and it allows them to develop other skills such as: confidence, patients, and the ability to voice their opinions. Student centered learning enables students to learn from their peers when sharing thoughts and ideas.

Using technology in the classroom

Technology in the classroom is becoming a major necessity in schools, and it is important that we are able to utilize technology to reach students on another level. New technological advancements are constantly occurring, and it is important that teachers are keeping up with the pace, since students are constantly immersed in a technological world. It is important to actively involve the students in the use of technology opposed to just discussing the applications of them, because hands on learning is critical when dealing with technology. Using technology in the classroom will engage the students and provide them with the opportunity to understand a mathematical concept in a different way!  Technology is an asset that all teachers should understand and incorporate into their classrooms.

Overall, this course has greatly benefiting my growth as an aspiring teacher, and it has allowed me to understand a different perspective and approach to teaching that will enable me to become a better educator.

Mathematics Activities in the Classroom (Week 14 Reflection)

Sharing with the Class

This week’s class focused on two learning activities that were intended for a grade 12 College level mathematics class. The first activity involved Trigonometric Functions, specifically SOH CAH TOA and the Sine and Cosine Laws. The students were divided into partners by randomly selecting names out of a hat, and each pair was given a worksheet that involved solving for angles and sides of right triangles. This promoted group work and peer tutoring because both parties in the group had to properly write out the solution on their own sheet in order to obtain the next activity worksheet. The next worksheet involved Oblique triangles using the Sine and Cosine Law, and it encouraged creativity and critical thinking skills, because one partner had to create the question for the other partner to solve. This was a great activity because in order to come up with a question that is solvable, the students have to really understand the material. This is a great indicator for teachers to assess if the students are learning the necessary skills to solve oblique triangles. After solving each question written by their partner, the students were to write their solutions on chart paper and present their question and answer to the class. This was beneficial because it allowed the students to share different types of questions and solutions to the class, which students can copy into their notebooks for future studies in a regular classroom setting, and furthermore, this implements Reinforced Learning.

Activity Booklet with different colour Activities

The second activity presented was for a grade 12 College level mathematics class and it involved exploring Sinusoidal Functions with the technological tool, Geogebra. In this activity, the students were divided into partners, and they were each given a booklet with a variety of worksheets of different vibrant colours. Each team was given a different colour worksheet to work on and complete using the Geogebra program. Each coloured worksheet investigated a different parameter of a Sinusoidal Function. Geogebra is a very effective graphing program, which easily allowed students to graph a function and view the changes in the graph if the parameters change. After recording specific features of the graph, drawing it, and recording the transformations on the provided worksheets, the students shared their answers collaboratively with the entire class. A member from each group read their answers aloud so that the rest of the students could record the solutions in their booklet. This is a great exercise, however it greatly depends on if the students in the class are the hard working type, and will review their homework and look over the other group’s solutions. In addition, the main concept from each activity worksheet was written on the board for the entire class to see.  This activity was very engaging for the students, and they appeared to be more willing to participate knowing that their group was only responsible for one coloured worksheet, not the entire package.

Overall both activities were very effective in engaging the class and providing productive exercises for each mathematically concept. I believe both activities will be successful in a grade 12 classroom, and they will instill a positive and enjoyable learning environment for all.

Engaging Activities in the Classroom (Week 13 Reflection)

In class this week, we participated in learning activities that were designed for a grade 11 University/College mathematics class. The first activity involved the Domain and Range of Quadratic functions. We were each given three cards with different domain and ranges on them and we were asked to stand at the station set up around the room that applied to our cards. The stations around the room were signs that read: Word, Number line, List of Numbers, Inequality Statement. The objective was to then find people from other stations who had the same domain and range represented in a different way. This was a great activity because it allowed the students to understand that the domain and range of a function can be represented in many different ways. Additionally, this activity provided the opportunity for all students to be involved and participate in a discussion with their peers to figure the activity out as a team. I really enjoyed this part of the activity because it got the students out of their seats, and actively involved in their learning through peer collaboration.  After each card was matched in the correct place, the students were provided a worksheet to complete that displayed graphs and prompted the students to state the domain and range in at least two different notations. This worksheet was beneficial because it promoted reinforced learning, and allowed the students to collaborate once again to ensure everyone was grasping the concepts. I would love to try this activity in my classroom because everyone in the class seemed to really enjoy it, the material was well covered, and it created a positive classroom-learning environment.

            The second activity presented was also for a grade 11 University/College mathematics class, involving an investigation of the Exponential Growth and Decay rate. The idea of “chain letters” via e-mail was used to demonstrate an exponential growth. A person in the class was picked to start the chain, and the chain letter required the person who received it to send it to two more people. Thus, the first person chose two other classmates, and they all came up to the front of the class. The chosen students each chose two more classmates. This process continued until all classmates were chosen, and we were able to record the data in chart form. Furthermore, we arranged ourselves in a way that represented the exponential growth based on how we were chosen by our peers sending the e-mails. This was a great activity to try to reach the students through a real world example that they could relate to, however I would have preferred to have some structure as to how the students were chosen to receive an e-mail by their peers. To ensure that a negative classroom environment does not form as a result of students worrying that they will be picked last, it may be wise to have the students call out two names of students down the rows starting from one end of the classroom, or have them pick two names out of a hat. If the students are able to pick whomever they want it may become a popularity contest, and cause an unpopular student to feel embarrassed in front of their peers. After this exercise, the students worked in pairs on a worksheet that involved the use of a technological tool called Desmos. This was a great technological tool because it easily allows the students to graph out the equations given and make connections and comparisons between them. Furthermore, Desmos showed a clear and precise graphical representation of functions. The worksheet provided asked guiding questions that were very beneficial since it allowed the students to really think about what they were viewing on the Desmos graph.

Desmos

Overall, this is definitely an activity that I would like to use in my classroom, and I especially liked the final question of the activity, which asked the students to come up with their own real life scenario of exponential growth and decay. This question will truly test the students to see if they understand the concepts demonstrated throughout the activity.

                              

Activities in the Classroom (Week 12 Reflection)

Coin Flipper Technological Tool

In class this week, we participated in three effective learning activities that were structured in such a way that they were very entertaining and enjoyable for the class.  The first activity was for a grade 12 University mathematics class, involving Empirical vs. Theoretical Probability. The technological tool used in the activity was a virtual coin flipper, and it allowed us to generate empirical data for a specific number of coin tosses.

After collecting all of the data, we compared our findings to the theoretical data in our small groups, and then all together as a class. Further, the discussion questions listed on the worksheet were discussed as a class, which promoted group learning and collaboration. This was a great activity for a probability lesson because often students feel more challenged by probability, and working in groups created a positive classroom environment that enabled students to feel more comfortable to try and work together with their peers.

Exponential Growth Activity Worksheet

            The next activity presented was also for a grade 12 University mathematics class, involving Exponential Growth. This activity was set up as a game where a virus broke loose and the “infected” students could infect other students by answering a posed question correctly before the other student.  If the “non-infected student wins, then they do not contract the virus, however the “infected” person must move on to another opponent until they infect someone. As the game progresses students are to be recording the number of infected people per day. At the end of the activity, after everyone was infected the students answered a series of questions on their worksheet based on the data collected. This game was very exciting for the students and everyone was engaged and motivated to participate. This is a great way to get the students involved in the lesson, and it situates them directly in the problem presented, by participating in the activity. Being involved in the game allows the students to have a better understanding of the material, since they were able to see the exponential growth from a hands-on perspective. In addition, the students were out of their seats and moving around the classroom, instead of remaining seated for the entire class. Overall, this is definitely an activity that I would like to use in my classroom, however my only concern is that it has potential to disrupt the classroom environment, if a student becomes embarrassed in front of their peers in an instance where they cannot infect another person due to continuously losing in the competitions.

            The last activity presented this week was for a grade 12 University Advanced Functions class, involving Polynomial Functions. This activity was an individual task for the students and it involved matching the presented polynomials to their respective graphs that were displayed on the handout. The polynomials were presented in a variety of ways and the students needed so use different methods in order to be able to match them to their appropriate graphs. Each graph was matched with different letters that were to be written in the space provided on the worksheet. Each worksheet had a different word puzzle on it, where they were to write the boxed letters from their worksheets on the board. At the end of the activity, everyone worked together to unscramble the letters written on the board from each worksheet. This activity was very enjoyable and all students were on track because they were interested in figuring out the word scramble, and they didn't want to let their fellow classmates down.

Trigonometry Group learning Activity (Week 11 Reflection)

In class this week, I lead a learning activity for a grade 11 University level mathematics class, for a unit in Trigonometry that involved Trigonometric ratios, and the Sine and Cosine Laws. The students were arranged into groups of 2-3 since we have a smaller class size, however with a regular class size, it would be best to have larger groups of 3-4 students. The instructions of the activity were explained to the class verbally, and the written instructions were displayed on the board to accommodate all students. In this activity, the students were given a worksheet and were asked to work together to solve the 4 questions provided. This activity also involved a word scramble, where the students were able to unlock some of the letters to the mystery word after answering each question on their worksheet correctly. In addition, each group was given a HELP card, where they had the option to call on another member from a different group for assistance if they were struggling with a question. The HELP card could only be used once, and it was added into the activity to limit teacher intervention and encourage the students to work together. The last question on the worksheet was a word problem that was to be written on chart paper and presented to the class. Each group received a different word problem on their handout, and their presentation needed to include a diagram, calculations, and their mathematical thought process. In a real class setting, after completing the activity, the students can be given the opportunity to walk around to view each groups work on the word problems, copy them down into their notes, and make sure they have a good understand of how to solve each problem.

First page of the worksheet

Some of the key points in this activity are that it is important to strategically place students into groups, ensuring that each group consists of students with a range of abilities. This will create a positive learning environment, and encourage peer tutoring and collaboration. This activity caters to both instrumental and relational understanding and it is best utilized as a review, after the students have learned these concepts in class. Thus, the teacher should refrain from quickly assisting the students, and wait until they have used their HELP card (assistance from another group) before stepping in to assist struggling students. The teacher should also be circulating the room to ensure the students are on track, working together, and that everyone is participating in the activity. Additionally, it would be beneficial for the teacher to keep anecdotal records or to use a checklist to track student progress. Another important note is that the teacher should ensure that the students have enough time to work through the worksheet before presenting their problem to the class. Lastly, the word scramble portion of the activity promotes literacy, and it was a great way to get the students interested and keep them on track, since they wanted to keep unlocking letters to solve the final puzzle.

This activity can also be structured as a competition, however the teacher should make that decision based on the students maturity level. It is important to ensure that the students aren’t rushing to win, and that they are all participating and learning through the activity. Another option would be to give each group a prize after everyone has answered all four activity questions and completed the word scramble. In this way, everyone will be able to enjoy a reward!

                                           

                                     Above is the second page of the worksheet for each group