The end is near....

          After the last 12 exciting weeks, our math class has come to an end. Throughout this course, I have learned a great deal of new and innovative methods that I would be able to use in the classroom. 

          At first, I was very nervous about how to teach the various concepts in mathematics. Math was never my favourite subject. Therefore, the requirement to teach math during my placement made me feel a bit uneasy at first. However, through each week, I was able to gain a better knowledge of different resources and activities that a teacher could use in order for students to succeed in their learning.

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          Throughout these past weeks, I have gained a great foundation of resources I will be able to use during my teaching placements. My two favourite activities that I have enjoyed learning about are the "Measurement Olympics" and the "Area Dice" game. I found these games to great tools to use in the classroom as ways to keep students engaged and interested in what they are learning. One of the main goals to keep in mind in a classroom is to maintain momentum and motivation of your students. The way in which to do this is to create a positive learning experience for students. By creating activities that are interactive for students, each type of learner will be able to connect and retain information based on their learning style. 

          Through the completion of this course, I have gained a great foundational knowledge of different resources I could use to help myself teach various math concepts. Despite my uneasiness at the beginning of this journey, I believe this has changed to a more confident attitude. I will be able to take my newly grasped knowledge and apply it to any future placements I have; ensuring that all students have a positive learning experience. 

The Importance of Assessment

          Over the course of the last 10 weeks, teachers candidates have been learning a variety of different activities and methods to help engage students in the classroom. Through these activities, teacher candidates are able to learn and apply these new findings during their time at their placement.

          Through these tasks, there is an important question that is taken into consideration: What is the importance of students completing these task?

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As educators, an important role we take is to evaluate the learning capabilities of students. Over the course of a year, students will be given ample amount of chances to demonstrate their learning to their teachers. It is of grave importance that teachers give students enough chances to demonstrate their knowledge of concepts in all subjects. 

          Due to the great importance of assessment, teachers must take their assessing role seriously. Therefore, it would be of great assistance to teachers to have their forms of assessment saved in an organized manner. One way in which to maintain organization is to keep a binder of assessments. During our session, our instructor displayed her use of math assessments in her various binders. Each binder contained various assessment tests, with calendars to help the instructor plan out units; and the consequently assessments that follow in order to ensure proper retention of information. 

         There are some primary forms of assessment that are common in all school settings. They include: tests, quizzes, written and oral communication evaluations. Through these four primary types of assessment, teachers were able to make note of the differences among their students; in terms of their learning styles and what levels they are at.Through the results of these assessments, teachers are able to modify their teaching techniques in order to help a student's learning process.

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          A great resource for teachers to use is the Growing Success document. This ministry document provides teachers with the necessary resources they need in order to create useful forms of assessment. Through the use of this document, teachers are able to have a guideline to help them succeed in evaluating their students throughout the year; and in turn, this will lead to the academic success of their students. 

          In the long run, assessment provides teachers an insight into the learning styles of their students. The better the teacher learns to get to know their students, the better the outcome will be of academic success in the students. One of the major goals of teaching is to provide the best possible educational and learning experience for students. Through proper assessment, teachers will be able to provide a great learning environment for al types of learners in their classroom.  

Data Management and Probability!!!

          When someone mentions the concept of data management and probability, the first aspect that comes to mind is taking chances; not knowing what the outcome will be. Generally, I am not a big risk taker. However, I was always interested in completing exercises in this math concept since we were able to apply this concept to real-life situations.

Pagliaro, T. © 2016.

         Just as usual, our sessions began with an innovative hypothetical lesson given by one of our teacher candidates. This lesson was a great interactive way to help students get familiar with using the proper terminology for this math unit. Before the activity began, the candidates reviewed the terms of probability and data management in order to re-familiarize ourselves with the terms. This is exactly what an instructor would do with their class in order to gain a foundation of what each terms means. Affirming the terms was then followed by the activity portion. Each table was given spinners; with different sized sections of different colours. Each group was given the task to complete 20 spins and determine the probability of spinning on certain colours. All members will have a chance to spin the spinner and record the data in a tally chart. In a classroom, the next step would be for students to find the mean, median, mode and range for this set of data; followed by recording the data in a graph. The spinner activity would be a great way for instructors to get students to actively participate in their learning process for this math concept.

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          One of the great reasons why I always enjoyed data management and probability because of the real-life application of the unit. Using real applications during day to day life will help students retain and fully understand the concepts of this unit. My favourite demonstration of this real-life application during this week's session is using Tim Horton's "Roll Up the Rim to Win." Each student in the class is able to purchase a beverage with a Roll Up the Rim cup. The students then hypothesize what they think the probability of winning would be for each person in the class. Once each student rolls up their cups, they can document the data in a tally chart; followed by finding the mean, median, mode and range for the set of data. Using this trip/activity is a great way to engage students and keep their interest in the concepts. In addition, it will also help students to gain further knowledge and understanding of these concepts. 

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          This week has proven to be very resourceful in applying data management and probability concepts in various forms. In my opinion, using real-life applications of these concepts will improve a student's retention of the information. In addition, using online resources and manipulatives, such as spinners, will allow students to gain some hands-on experience in defining the various terms of this unit. Through this week's activities, I have learned that a great way for students to maintain interest in a subject is to incorporate real-life aspects into the lessons. 

Time for Measurement!!!

          On your mark, get set, and time to measure...It's the Session 9 Measurement Olympics!!!

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This week by far has been my favourite week of activities. Over the years, one of my favourite units in math is measurement; partially because I had little difficulty understanding the concepts of the unit. This week, the teacher candidates had the pleasure of exploring various hands-on activities involving the unit of measurement. As you read further on in this post, I will further discuss the engaging and interactive activities that educators can use to help students understand measurement concepts. 

          One way in which to get students excited about a new topic or concept is through the use of media. Whether it be an infographic or video, students will become engaged with visual; and prepared to learn more about the concept. Therefore, this is exactly how we started our lesson. Our instructor chose to display a kid-friendly video on measurement using a TV show "Sesame Street." The video allows students to be introduced to measurement terms and concepts before the commencement of the unit. 

          Firstly, the class got to explore an interesting activity named "Area Dice." This is an interactive game where students are able to practice plotting points on a cartesian plane; as well as practicing the concept of area. Students work in pairs and are given one sheet of paper with boxes; a piece of graph paper if the teacher chooses. The students are responsible of rolling the die twice in order to attain the measurements for the length and width for the area. Once this is completed, the student will shade in the appropriate area. The goal of the game is to shade in as much area as possible on the sheet. This activity is a great way for students to interactively practice getting familiar with area. Shading in the area will help students to visualize what consists of length and area. In sum, the "Area Dice" game provides students with an outlet to practice and develop their skills in the concept. 

Pagliaro, T. © 2016.

          After this intriguing activity, we played, what I think, is the most interactive activity yet. This activity involved incorporating the aspect of the olympic games and the math concept of measurement. Each student would be put into a group with at least 4 other members. Each group receives a paper indicating the 5+ stations, or "games," that each member of the group will be able to participate in. The students as a group work together to make an estimate of the measurement, followed by checking the actual length of the measurement. This game promotes the development of collaborative working skills; while helping to improve the retention of measurement concepts. I really enjoyed this activity as an engaging lesson for students. Working collaboratively will help students learn measurement concepts while applying them to real situations. Below is an image of the worksheet of the various stations that can be given to students:

Pagliaro, T. © 2016.

          In my opinion, I felt that this week was the most intriguing week yet. Using the concept of olympic games would make learning the unit of measurement engaging to students. In future classes, I can see myself using this activity since it is a great way to create an active learning environment for students.

  

Geometry and Spatial Sense

          This week's sessions focused on exploring problems dealing with geometry. Throughout my math classes over the years, geometry was a unit I was on the fence about. I would retain some concepts of the unit and struggle with others. But as the years progressed, I was able to gain a grasp of the geometric concepts. 

          Initially, a couple of the teacher candidates gave a demonstration on a activity a teacher could do in constructing geometric figures with a Grade 5 classroom. This activity was called "Marshmallow Geometry." Firstly, a review was done of different shapes; focusing on the number of faces, edges and vertices each shape has. After this quick recap, each table was given toothpicks and small marshmallows. Each person at your table had to chose a shape and make a 3-D representation of the shape; whether it be a prism or pyramid. I thought this activity would be very interactive for students. Allowing the students to be able to make the shapes themselves will help students to visualize the shapes when it comes time to drawing them on paper. Although the marshmallows may pose an issue, depending on the maturity of the class, a teacher could easily change the marshmallows to a non-edible item. Overall, this is an engaging activity that I could see using in my future classroom. Here are some pictures displaying this fun activity. 

Pagliaro, T. © 2016.

          Once this demonstration was complete, we continued to explore various other activities that could be done in the classroom in regards to geometry. Each candidate was given a sheet of paper divided into boxes. Each box contained a number of shapes with a total number of sides. Students would then have to fill in each box of the different shapes that would satisfy the two requirements. For example: 2 shapes, 8 edges -> possible answers could be 2 squares or 2 rectangles.This activity could also be done in collaborative setting, in pairs or the class as a whole. The teacher could use paper bags labelled with the number of shapes and total edges. The class could then guess as to what shapes are in the bag to satisfy the two requirements. In short, this activity would be an engaging way to help students develop their confidence in recognizing the different shapes, how many vertices and how many edges each one has. Here is an image of the practice worksheet. 

Pagliaro, T. © 2016.

          Another fun demonstration was playing "Guess Who: Geometry" as an activity. Instead of using people, and asking yes or no questions about the characteristics of a person, the students use geometric shapes to play. Students play in pairs and ask each other questions about various characteristics of geometric shapes in order to figure out each other's shape. 

Pagliaro, T. © 2016.

          The final demonstration focused on the Spatial Sense portion of the unit. Our instructor used the game Battleship to get students engaged about using a cartesian plane. This game allows students to practice plotting points on a white board cartesian plane. This is a great activity to use with students in order to help them develop their skills on plotting points. Here are some images of the fun activity. 

dPagliaro, T. © 2016.

          In short, we explored many innovative activities to make learning geometry and spatial sense fun. I personally enjoyed the Battleship gam the most to use with future students. However, all these activities provided engaging environments for students to develop their skills in these math concepts. 

Patterning and Algebra

          This week's session focused around the concept of patterning and algebra; and the various resources educators can use to help students improve their skills in this concept. Throughout my years learning math, I've always enjoyed learning the basic fundamentals of patterning and algebra. By high school, when the concepts became more difficult, the positive feeling started to dissipate. However, in general, I believe I have a strong basis of this concept.

          Firstly, one of the teacher candidates gave a sample lesson for students. This activity involved practicing to identify various patterns based on images and a table. The handout provided a great way for students to develop a basis of identifying the pattern; what is the operation and number by which the pattern is increasing or decreasing? Overall, a great way for students to develop a foundation of the basic skills needed for the concept of patterning.

          As a class, we continued this development of skills by practicing various problems about pattering and developing an algebraic expression. This was modelled by output # = input # x __. This algebraic expression will help students to predict what the pattern will look like for a higher term, such as 100. A sample problem is found below.

Pagliaro, T. © 2016.

           The next step our class took was to demonstrate these patterns with the use of manipulatives; instead of just a regular table. Each table was given a bag with various patterns indicated  Each member of the table chose a pattern and had to construct this pattern with square coloured chips (math manipulatives). After each table was done making their patterns come to life, the class did a gallery walk to see the different ways in which people represented the pattern. As a class, we noticed that most individuals represented the pattern flat on the surface of the table. However, there were some individuals who chose to build the pattern by expending upwards; stacking the math chips on top of one another. I enjoyed this method a great deal since it would allow students to visually represent each of these patterns and hopefully make more sense of this math concept. Below are some images of this activity; the pattern with the rule beneath it.

                                             Pagliaro, T. © 2016.                                                                                       Pagliaro, T. © 2016.

          We continued to build on this manipulative method by expanding on the original pattern. The algebraic expression that was used no longer just contained a number that is multiplied by the term number, but rather adding an operation; such as output # = 2 + 4. The representation of the constant number added a bit of an extra challenge. The same process occurred for these questions. Each member picked a pattern from the bag, then displayed this pattern on the table using the math chips. Once all tables were complete, the class completed another gallery walk. Here are some images of the representations of these patterns.

                           Pagliaro, T. © 2016.                                                                                         Pagliaro, T. © 2016.

            Lastly, the class was able to explore the innovative app called "Dragon Box." Our instructor described this game as an opportunity for students to practice algebra without viewing algebra in a traditional form. She also disclosed that she uses this app all the time with her students. Therefore, I thought this app must be a great source for students. At first, I found the game was a bit confusing. However, after playing around and figuring out the elements of the game, I could see how this would be a great game to help students develop their algebraic skills. Instead of solving for "x", students must try to isolate for the dragon. Through trial and error, students will become familiar with the process of eliminating various boxes to get the dragon by itself. As students progress, the levels will get harder and harder; which will help them develop their skills for the algebra concept. Despite having to pay for this app,  I believe this would be a great app to use with students in the classroom.

          To sum up, this week has proven to be very informative for the concept of patterning and algebra. Through the use of manipulatives and various technological resources, such as the app "Dragon Box," I believe I would use any of these resources in my classroom.

Proportional Thinking

          This week's activities focused around the concept of proportional thinking. At first, I was unsure of what this concept entailed. I knew that I had probably learned this concept at some point in my education. However, at this point, I was not sure of what this was. After some guessing, proportional thinking  was defined as "the ability to compare two things using multiplicative thinking, then applying this to new situations." Through this definition, the class was then able to apply their proportional thinking skills to solve various problem solving skills. 

          The first question was called "The Giant's Hand" Problem. Without being given any information, other than the print out of the hand size of the "giant", we were asked to figure out the approximate height of the giant. The image of the giant's hand is found below:

Pagliaro, T. © 2016.

I found this problem a bit challenging at first glance. Through my table group's step by step process, we were able to figure out a rough estimate of the giant's height; which is about 10 ft. We used a ruler to measure the length of the whole hand; which measured about 12 inches (a foot). Then, we each measured our own hands to compare to the giants hands, as well as comparing our own heights. Our next step was to make a ratio equation to help us solve for x. The equation could have looked like 12 inches/ x = 7 inches/ 5.8 ft. I used 5.8 ft to sub in my own height and 7 inches for the length of my own hand. These two values can be substituted for any individual who us trying to complete this question. Through this equation, we found the answer to be 9.9 ft (rounded to 10 ft). Overall, a great question to help students learn how to set up a proportional equation and solve for the unknown variable. 

           The second source we used was a video from Daniel Meyers. This video was focused on the method of 3-Act Math. I found this video very interesting since it promoted people to think about various questions one would ask themselves if they watched this video on building a pyramid out of pennies. The main question was "How many pennies are needed to make this pyramid?" The process is divided into three steps: 1) Watch the video and ask yourself a series of questions, 2) what information would one need to solve the problem, and 3) solve the problem. Here is a link displaying the three steps:

http://threeacts.mrmeyer.com/pyramidofpennies/

This method really allows individuals to problem solve and think of innovative ways to complete the problem. I thought this process is very engaging for students, and individuals in general, to think outside the box and use their pre-existing knowledge to come up with an answer to the problem while promoting inquiry skills. Below is the full video of the experience:

          Lastly for this week, the resource I chose to explore is called "Dirt Bike Proportions." This is a great tool for students to use in the classroom as an extra practicing resource. Through this engaging tool, students are able practice and develop their skills for the concept of proportions. Despite the game being fast paced, students will be able to improve their mental math skills and get faster at solving for "x". In addition, I enjoyed the fact that students have the choice to work independently or work in small groups collaboratively to solve the problems.

          Overall, this week has been very informative. I have learned about new tools and resources to  use in their classroom to help future students become more comfortable in various math concepts. 

Fun with Integers

          For this week, the session focused on learning fun an innovative ways for students to explore and improve their integer skills. 

          The first resource the class explored was through one of our fellow teacher candidates. He chose to use cards in order to help students learn how to subtract and add integers. Red cards signified positive numbers and black cards signified negative numbers. The students had the goal of attaining the answer of 25 through adding or subtracting. The students play two rounds; one round adding the integers to eventually get the answer of 25 and the other subtracting integers to get the answer of 25. I thought this card game was a great way to get students engaged about learning the concept of integers. It gets the students to think on their feet and improve their mental math skills. I believe I would definitely use this in the classroom with my future students. 

Math Hombre. Integer Games. Friday February 4, 2011. Cards [Online Image]. Retrieved from http://mathhombre.blogspot.ca/2011/02/integer-games.html

          The next fun activity we completed was playing "Integer Football." I thought this was a great game to use with students to help the improvement of their skills. Students used a sheet with a number line; which was set up like a football field. One side had a negative end-zone and the other with a positive end-zone. The students are given a coin, in this case we were given a penny, and a die. In order to see how many places he or she moves, the student rolls the die. In order to determine whether he or she moves to the positive or negative end-zone, the students flips the coin; heads for the positive end-zone and tales for the negative end-zone. I thought this was a great game to use with students. It is a fun way that one can practice their skills, but using a concept that is relatable to many students. The game keeps students engaged and interested, while developing their skills.

Pagliaro, T. © 2016.

          Lastly for this week's forum, we had a choice of two resources to explore. I chose to explore the game "Orbit Integers." This is a great game for students to practice their integer adding and subtracting skills either independently or in a collaborative work group. The game allows students to bring out their competitive side and work to develop mental math skills. The more answers the student gets right the more the team space ship moves closer to the finish line. For the future, I believe that I will definitely use this resource for students in my class. 

          Overall, I have learned some great resources to use to help teach students integers. Learning these new methods has put me at ease and feel a bit more comfortable teaching and helping students learn the basic concepts of mathematics. 

Fractions and Decimals

          During this week's session, we touched upon some innovative ways in which we as educators can help students understand the concepts of fractions and decimals. 

Mme. Deminion Weekly. (2015). Fraction and Decimal [Online Image]. Retrieved from http://mmedeminion.weebly.com/number-sense-and-numeration.html

          To begin the class, two groups, one of which being my own group, demonstrated a small activity that teachers would be able to use in the classroom. The first group made an activity focused around fractions 

          My partner and I went first and focused on equivalent fractions. This was geared for a grade 4 class first learning about the concept of equivalent fractions. We used a "Folding Paper" method in order to develop a foundation of the concept for the students. Students simply need two pieces of paper. The students fold one piece of paper into the appropriate parts based on looking at the denominator of the first fraction. The students then shade in the appropriate amount of parts based on looking at the numerator. The students then use the other piece of paper to do the same folding action for the other fraction given. Since the pieces of paper represent the same whole piece (a.k.a the same denominator), we shall look to see if the amount of parts shaded are the same. If they are, then the fractions are considered equivalent. Below is an image of our activity.

Pagliaro, T. © 2016.

          For the second demonstration, the group chose to focus their activity on decimals; more specifically place value for a grade 5 class. They used an interactive worksheet in order to make the concept of place value relatable to students. They made use of a piggy bank analogy and saving money in order to get students engaged in the activity. The group also made use of manipulatives, the counting blocks, to give a visual representation for the money being saved. Overall, I believe this was a great interactive activity that a teacher could use in their classroom for this concept. Below is an image of the worksheet the group used.

Pagliaro, T. © 2016.

          Another activity we learned focused around fractions once again. Our instructor demonstrated a great innovative way to make equivalent fractions relatable to students. 

          Instead of using the traditional terminology of finding the lowest common denominator, LCM, she used clocks and time in order to help students understand this concept. In this case, the LCM would be time; 60 to represent the 60 minutes in one hour. The various clocks were then divided into 3 different fractions: 1/4 = 15 minutes, 1/2 = 30 minutes, and 1/3 = 20 minutes. Students will then be able to use these clocks as a foundation to help them understand adding and subtracting using equivalent fractions. In addition, the class also received a different method for teaching the values of different fractions through the use of a folding number line. Every student is given a rectangular piece of paper and then folds the paper into the various parts based on the denominator given; whether it be 4 (into 4 parts), 3 (into 3 parts), etc. These two activities are then put in our interactive notebooks for future reference.  Here are some pictures to help visualize the activities. 

 

Pagliaro, T. © 2016.

Pagliaro, T. © 2016.

           Overall, I believe this week's activities were very educational. I learned a variety of new innovative activities to use in the classroom. Learning fractions and decimals can sometimes be difficult. Therefore, I hope to use these activities in the classroom with future students in order to make these concepts engaging and interactive. 

Learning New Ways!!!

          In this week's session, we learned new methods of solving the four basic operations in math: addition, subtraction, multiplication and division. We were all able to see 4 different methods of completing questions of each mathematical concept. After completing the various sample questions, I discovered that I use traditional methods for completing questions in all four concepts. However by the end of this session, I was very intrigued by some new methods I had never seen before.

          The first method I found interesting was solving subtraction problems. This method is called the Singapore Method. Instead of using the traditional method of regrouping, this method allows students to subtract the tens place of a number first in order to get the bottom number that is being subtracted to zero. I believe this method will help students a great deal if they are unable to grasp the concept of regrouping. In the image below, the four types of methods are demonstrated; with the Singapore method being visualized in the bottom left corner.  Since the first number to being the subtraction is 0, an individual using the traditional method would borrow from the number on the left. However, with this method, one subtracts the bottom number, in this case 6, from both numbers. This will ensure the bottom number to end in a 0; making it easier for the student to complete the subtraction.

Pagliaro, Teressa-Marie. Instructor's Photo. Image of instructor's slide. Retrieved from Teresa-Marie Pagliaro's photos. 

          The second method that struck my interest was solving multiplication problems using the Multiplying Lines method. Instead of using the traditional multiplication methods, using one's basic times tables skills, students draw a series of lines to obtain the same answers. I found this method extremely interesting due to the fact that it is a new perspective of looking at multiplication. The following link will give a demonstration of how this method is done.

https://www.youtube.com/watch?v=_AJvshZmYPs

          Overall, I believe teachers should always keep an open mind to different teaching methods. Each student learns in a different way. Therefore, we should all as educators keep an open mind to new ideas in order to create a comfortable learning environment for students.

Welcome to Math!

Hello fellow classmates,

       My name is Teresa-Marie Pagliaro and I am a recent graduate of McMaster University with a Honours BA in French. One of my great admirations is learning new languages and helping students become acquainted with the French language. Due to this admiration, I also took many courses in linguistics. My studies in linguistics allowed me to gain a different perceptive of the general makeup of languages; from the basis of learning sounds to the use of words in forming well structured sentences for communication and writing purposes. This same sort of learning process can be applied to learning math as well. For math, we must learn the basic foundation operations, such as addition, subtraction, multiplication and division, before we go on into learning new and more complex problems.

       Since high school, I have not taken any math courses. However, I do believe that I have developed a great foundation of mathematical knowledge through my elementary and high school education in the subject. Math was never one of my favourite subjects, but I managed to obtain a general understanding of its concepts. One way I have tried to re-familiarize myself with J/I concepts is working as a tutor at the Hamilton Oxford Learning Centre.  Primarily, I help students, from grades 1 to 8, with their English or French homework; but I occasionally work with math students. Working at an Oxford has helped me enhance my pre-attained math knowledge and allow it to grow as I review the various math concepts with students.
    
      By the end of this course, I hope to gain a better understanding of techniques that I will be able to use in the classroom. Many students find math difficult; as did I. As an educator, I believe it is important to make the students feel less anxious about learning and completing math concepts. Math concepts should not be feared. In order to promote the retention of information, teachers should make the concepts relatable to the students on some level. Whether it is through using sports, food, movies, etc. teachers can try to use any material to help the students remember the material they have just learned  Therefore, I would like to learn different strategies that will help engage students and create a fun learning environment in order to build confident math skills.

Clipart. 2016. Math Symbols Clipart [Online Image]. Retrieved from http://cliparts.co/clipart/7428.

Looking forward to the rest of the semester and future posts,

Teresa-Marie Pagliaro