My Math Experience and Teaching Grade 8

When I first started my Bachelor of Education degree at BrockUniversity, I had no idea of what classrooms I would be teaching in my placement. I did not not know the school or the grade level I would be teaching. 

My placement is at St. Matthew Elementary School and I am teaching a grade 8 classroom math. I'm not going to lie, it was very intimidating at first. It is extremely important to build rapport with the students while ensuring that lessons are covered and students are having fun. I was extremely humbled when my teaching associate asked if I would like to teach the next unit of math, which was geometry, when I started my internship. Seeing that my teaching associate trusts me and has helped me get on track shows me the importance of building connections within the teaching environment.

Games are always a great way to get students engaged with material. My first lesson was 45 minutes and consisted of a brief overview of concepts that grade 8 students should be familiar with at this point in the year. I gave them a chart to fill out as we progressed through the lesson and decided to surprise them with a Kahoot quiz. I wasn't sure if they knew what Kahoot was or if they would enjoy the activity, but was relieved at the excitement they expressed. One of the central lessons I learned in math class was that the best learning takes place when students are having fun. I got to experience many new activities and games this year that I had never heard of. Some of the approaches to teaching were informing as well and I plan on implementing some of the strategies set forth in my math class.

Collaboration is a big part of learning in a math class. Allowing students to work with each other can be extremely beneficial as they are communicating ideas, resources, and thoughts with one another and can potentially build off of each others knowledge. Math seems to have a bad reputation in the school system as it is always the subject that gets labelled as "boring" or "too hard". This is not the case in my classroom. My teaching associate has made great efforts to engage students and uses technology as a platform to captivate students. From the various activities that he creates in the classroom to the online communication, my teaching associate has made sure that every student has had an equal and equitable opportunity to learn.

Watching every student teach a small lesson in math class has been very enlightening to me. While I was able to teach a lesson on the surface area of a circle, many of the other presentations showed me techniques and strategies that can be used across grade levels. Every lesson had an activity or game, which I consider to be the action part of the lesson. These activities and games varied between students, but collaboration seemed to be at the heart of almost every lesson taught. I don't think this was coincidental. I don't think math is a one-way street and that students need to talk with one another and build an understanding of there own. I hope that the knowledge and skills I developed in this math class will transcend to my teaching practice and show students how fun math can be.

Data Management and Probability

This week we got to look at the familiar concept of data management and probability. As usual, we went through some fun activities and got our minds thinking. The growth mindset is still at the core of ever lesson we have had up until this point and should be the centre of every lesson taught. Kids need to know that their brain is a muscle and that every time it challenged or faced with a struggle, it has the opportunity to grow stronger. Perseverance is a a skill that all students need to develop in order to succeed. Our friend Mojo shows us how some students view academics and the school setting. You're either born smart or you're not? I don't think so!

Mean, Median, Mode, and Range
One of the fundamental concepts rooted in data management and probability is the use of mean, median, mode, and data range. The mean is the number that people are usually referring to when they talk about "the average". It can be calculated by adding up all the data values and dividing by the number of values. The median is the middle value when the data is in order. This value can be a whole number or half way between two numbers. The mode is the most frequent value in a data set. A data set can have one mode, multiple modes, or no mode at all. The range is the difference between the least value and the greatest value and shows the spread of data. Understanding these terms and their role within a set of data is necessary for developing an understanding of data management and probability.

Bar Graphs
Data can be visually represented in a multitude of ways, such as pictographs, line graphs, and circle graphs. One of the visual representations that we focused on in class is the bar graph. Bar graphs are displays that use the lengths or heights of bars to represent quantities. While there are many variations of bar graphs, our class specifically looked at is the Stem-and-Leaf Plot. The stem-and-leaf plot is a way of both organizing and displaying numerical data. In this graph, numbers are grouped together by their place value. The "stem" is determined by the digits greater place value while the "leaf" is determined by the digits lesser place value. As shown in picture to the right, the "stem" of 1 has two leaves, 6 and 7. This means that within this data set, there are the numbers 16 and 17.



Levels of Graph Comprehension
Looking at a graph requires numerous levels of comprehension. This is necessary so that information is not misinterpreted. Many graphs display information that can be looked at in different ways. An example of this is the graph to the right that focuses on tax rates. The bar on the left is much higher than the bar on the right, approximately 5 times larger. What you don't see, however, is the rest of the bar as you are focused in on the top part. The bar on the left represents a tax-rate of 35% while the bar on the left represents 39.6%. When we teach students about graphs, we need to be mindful of the three levels of comprehension. These include reading the data, reading between the data, and reading beyond the data.

Early Probability
Another focus of this weeks class was probability. Probability is a measure of likelihood. It can be expressed qualitatively or quantitatively as a fraction or decimal between 0 and 1 or an equivalent percent. When young students are asked to describe events of probability, they develop a probability language. Words like always and never as well as likely and unlikely are established early in this vocabulary and help students understand how probability plays a role in their daily lives.

One of the activities we engaged with in this class was understanding probability through the use of dice. We were asked to roll every number on the dice and record how many rolls it took us to roll each number. While the odds of a rolling a number are 1 in 6, it took most students many more rolls to completely roll every number. An even easier activity that can be used to teach young students about probability is flipping a coin. This activity would be much simpler than rolling a dice because there are only two sides on a coin, therefore the probability of every outcome is much larger than any outcome of rolling a dice.

Probability and Data Management in Our Daily Lives

Probability and data management play an important role in our lives in many different ways. Many of our actions throughout the day are based on data management and trends that we have seen throughout our lives. During this time of the year we usually have colder weather. With a higher probability of cold weather as compared to summer weather, we are more likely to wear a coat. Every December for the past 10 years, I have worn a coat at least once. This set of data helps inform me of what is to come in December as far as weather and helps determine a probability of wearing a coat.

Students already have an idea of data management and probability, they just don't know it yet. They know that they are in school from Monday to Friday during the school year and the likelihood of a snow-day is usually small. They know that they will most likely be eating pizza on Friday because they have had pizza every Friday for two months. This understanding of data management and probability is basic, but can be developed in more depth.

The Purpose of Assessment

Clapping Activity: How Assessment Works
This week in math class we started off with a fun activity. Several volunteers went to the front of the room and were divided into to groups. One group were the judges and three other volunteers were the contestants. The first contestant was asked to clap, which she did. The judges then each gave her a grade between 1 and 5. The second contestant was also asked to clap, but was given a slight advantage. The second contestant was given insight to the activity; she knew that she was supposed to clap and saw the contestant before her clap, giving her an idea of what she needs to do. By the time the third contestant was asked to clap, she had seen the other two contestants perform the activity and was given a success criteria that was created by the judges.

The purpose of assessment is to improve student learning. It is an ongoing process and takes place in numerous forms. When we ask students to clap on the spot without giving them information, we are setting them up to fail. When we provide them with examples and a success criteria, however, we are giving them the opportunity to succeed and learn. This activity showed me that putting a student on the spot and asking them to perform an activity, it makes them feel uncomfortable and unconfident. When you explain to a student what they are being asked to do, they can have fun and enjoy learning while being assessed.

Collaborative Activities
We also continued this week with more collaborative activities. In groups of six, we were asked to go around the classroom and solve the various puzzles presented to us. Each member in the group received a clue that they must orally share. This means that each member in the group had to vocally engage in the activity; you couldn't have another student look at your clue. These puzzles varied in difficulty and setup. There were a few activities where we were asked to create a shape using coloured building blocks. There was another activity that asked us to create shapes using toothpicks. There were also games that asked us to find a specific number from 1-100 using the clues we were given. In each game it was not only crucial that each member read out their clue, but that every member was actively participating to make sure that their clue was seen in the activity.

Collaborative activities are a fun way to get students out of their seats and focused on learning, which was specifically math in this case. We have talked about the use of manipulatives throughout this course and continue to see the value in using them within a classroom. I really liked these activities as they took a problem that could be solved by an individual and made the problem solvable only through collaboration. These types of activities help students develop their learning skills and work habits while developing positive peer relationships.

What Makes a Good Assessment Plan?
We also looked at effective assessment plans this week and the characteristics associated with a good assessment plan. A good assessment plan:

• balances the measurement of both mathematics content and processes
• is appropriate for its purpose
• includes a variety of assessment formats
• is aligned with student needs and expectations
• is fair to all students
• is useful in assisting students to assess their own learning
• measures growth over time
• sets high, yet realistic, expectations for students

We also looked at some of the different ways in which assessment data can be gathered:

• portfolios
• performance tasks
• projects
• journals
• observations
• interviews
• homework
• exit passes
• tests and quizzes

One of the main focuses I took from this week's class is the need to step away from old-fashion questions, such as "Does anybody know..." questions. Throughout this course we have been developing open-ended questions where students are free to approach a solution using a variety of strategies. The purpose of a question isn't to see if a student knows the answer, rather to see if a student has a means of getting to that answer. Formulas can be extremely useful in math, but only if a student understands what the formula means. Trying to explain the surface area of a circle to a student can seem difficult without using the

π

r

2 

formula, but it shouldn't. There is a reason why people use that formula and breaking it down is a necessary step to teach students about surface area.

As the video to the above explains, assessment used to be simple. We would give something a try and it was obvious when we did not succeed. It was also obvious when we did. With 21st century learners, assessment has changed greatly. 21st Century learners need to synthesize knowledge, communicate clearly with others, and create solutions to problems that we don't even know exist. In order for students to become learners, rather than just graduates, they need personalized, engaging, and useful feedback on meaningful work. 

I Have...Who Has?

I Have - Who Has?
This week in math class we got to play a fun game called "I have...Who Has?", which can be adapted to virtually any unit of math. As this weeks lesson focused on measurement, the game related to the various terms that are associated with measurements. This game requires a prior knowledge in which students draw upon to answer questions. This game would be most effective at the beginning or end of a lesson. The game start off with a "I have statement" followed by a "Who has" question. Each person is given a card that has an answer and a question. When you hear a question asked that you have the answer to, you stand up and say "I have" and give your answer. You then say "Who has" and pose another question so the game continues. One of my favourite parts about this game is that there is no race to answer quickly. Everyone talks to one another to see what they have as answers and help one another answer the questions. This isn't the first time we have seen this activity being used in our math class and I plan on taking this activity with me wherever I teach as I feel this game can be adapted to all age groups and learning levels.

Measuring Length
While measuring length may seem easy to most adults, it is a concept that young students may struggle with learning. While it is easy for someone to say that a table is 60 centimetres long, the idea of centimetres may not be a familiar one with young students. This weeks math class taught me that measuring length can be broken down. Using nonstandard units as a form of measurement is a great way to introduce the unit to new learners. The picture to the right shows the measurement of a table. If someone told me that the table was 60 centimetres long, I would know what that means. A young student who is just learning how to measure length, however, may have a better understanding of the length if it was described in pencils as opposed to centimetres. Being able to visualize length using everyday items not only helps kinesthetic learners, it helps introduce a foreign concept.

Teaching the Area of a Circle 

This week in math class I had the opportunity to teach my classmates a lesson and receive constructive feedback. The focus of my lesson was the area of a circle, which is a concept that I struggled with in elementary school. The problem was that I was taught to use a specific formula, you know the one, but I wasn't taught what the formula meant. I was familiar with the radius, diameter, and circumference of a circle, but I was told to plug in numbers to the formula in order to generate an answer. The focus of my lesson would be to avoid giving a formula and instead give students an understanding of how a circle could be thought of as a rectangle, as shown in the picture to the right. The use of manipulatives was central to my lesson as I am continuing to understand how important they are for student learning.

Shaping Lives with Geometry

Geometry and the Kinesthetic Learner 
"It is important to note that children's ability to conceptualize shape develops through different stages, and that this development is fostered by each child's experience" (Making Math Meaningful, 395). When teaching students about geometry, there needs to be a physical interaction for students to learn. Students need to physically touch and look at objects to gain a deep understanding of how shapes are formed and notice different aspects of shapes.

Geometrical Terms
Throughout class, we discussed several words and what they mean. Defining these words helped us to identify shapes and classify them. Some of these terms include:
     Similar Shapes - same shapes, but may have a different colour or size
     Congruent Shapes - Shapes are the same and equal
     Symmetry - Two parallel sides are the same, the shape can be folded in half and bot halves mirror                           one another
We also went through the different types of quadrilaterals, some of which include:
     Parallelogram - a quadrilateral with 2 pairs of parallel sides
     Rhombus - A parallelogram with all sides equal in length
     Rectangle - A parallelogram with 4 right angles
     Square - A rectangle with all sides equal in length
   
By allowing the class to collectively define terms and come to an agreement, it allowed us to work together effectively on activities throughout the class. When a common knowledge was shared, we were able to build off of one another's thoughts and complete activities.


The Greedy Triangle
One of the most helpful ways to teach students about shapes is through telling a story. The Greedy Triangle is a story about a triangle who want to gain additional sides to become a different shape. With every additional side, the triangle is able to take on different roles within the shape world. Students are able to see how one side changes a shape and that certain shapes have specific aspects. This book not only goes through various shapes, it also teaches students about individuality as the Triangle goes back to being a triangle because that is what he wants to be.



What did I take away from this week's lesson?
The biggest piece of knowledge that I took away from this week's lesson is the importance of hands on activities and visual representations. Geometry is a very specific topic in math and some students need to see shapes get off the page and take form in their hands. Manipulatives have been a central focus in many of our classes, but I feel that this is a topic in math that absolutely needs them.

Struggle Leads To Growth

Math Can Create Stress

"People who feel math anxious are unable to prevent their stress and worry about doing math from interfering with their ability to perform. Their worry about math so occupies their thoughts, it is hard for them to actually think about math" (Marian Small, Making Math Meaningful). Math is a subject that students struggle with because they are presented with concepts that they might not understand right away. Math is a process where there is a right and wrong answer, requiring hard work and understanding to get the right answer. Students may find math stressful because they know that if they do not get the right answer, they get the wrong answer. This thought is always present while in school, but I would argue is most prominent in a math class. 

How Can I Reduce Stress in a Math Class?

https://goo.gl/rpa3Gx

One of the biggest ways to reduce stress in a math class is to focus on an understanding of math as opposed to following a rigorous set of rules and formulas. Memorization can be extremely problematic in a math class because there are exceptions to rules and not all problems can be solved in the same way. Another way to reduce stress is to remove as many time restrictions as possible. Providing enough time to write a quiz or test is essential to a student's success and allows them to think about questions more clearly and not in a rush. Mistakes are not problematic; they are beneficial. They identify the area of learning that a student needs to focus on and can help in teaching math. It shouldn't be expected of students to learn math concepts immediately. Creating questions where students can approach multiple answers in multiple ways isn't just encouraged, it is necessary for learning. Stress due to math can be minimized by teachers who can identify their students' needs and act upon opportunities that present themselves within the math class.

Patterns and Math

There are several types of patterns in math and are seen on a daily basis. Students start working with patterns in elementary school and will continue working with them as they work their way through high school. A pattern represents an identified regularity. Within a pattern, there is always some element of repetition. The three types of patterns we looked at in class are Repeating Patterns, Growing and Shrinking Patterns, and Recursive Patterns.

Repeating Patterns - In this type of pattern, the shortest part of the pattern is called the core and it repeats itself. Repeating patterns can take on numerous forms and can be used by students to help predict what will happen next.

Growing and Shrinking Patterns - In this type of pattern, growing means the numbers increase in size and shrinking means they decrease in size. Along with number patterns, there can be growing and shrinking shape patterns. 

Recursive Patterns - In this type of pattern, each element in the pattern is defined based on the previous element or elements. The Growing Pattern shown in the picture to the right is an example of a recursive pattern as each element is one square greater then the element before it on the left. 

The Three-part Lesson in Mathematics

We also discussed the concept of a three-part lesson in class. A lesson can be broken up into three parts: Before, During, and After. An important aspect of this lesson is the reflection phase, which takes place between the During part and After part. This type of lesson works in conjunction with assessments and is an effective way to teach students.

Before - In this part of the lesson, students engage in work that draws upon their prior knowledge and what misconceptions they may have. Activating this knowledge helps teachers determine what students know and where to go with their lesson. Allowing students to activate this prior knowledge helps prepare them for the lesson and gets them thinking.

During - In this part of the lesson, students are presented with a problem to solve. Powerful problems allow for a range of solutions or strategies and provide students with choices. During this part of the lesson, students interact with the teacher and themselves to determine what is being asked in the question and how they can go about solving the question. Students can work in pairs or small groups to work collaboratively and reach joint-solutions. During this part, teachers should make sure that all students are working within their zone of proximal development.

Reflection on Student Solutions - The reflection process takes place between part two and three of the lesson. During this process, teachers reflect upon their students' learning process and look at the types of solutions that they presented to the problem. Teachers choose which solutions will be discussed in the After part of the lesson, determining how the solutions are linked and what mathematical language to use/focus on. 

After - This part is often called the consolidation and practice phase of the lesson. In this part of the lesson, the class consolidates through the summary/highlights of the lesson and students learn the big concepts within the lesson. Allowing students to present their solutions to the class helps them to understand their process and allows the class to learn. As a class, students have the opportunity to explain and respond to questions about the ideas found in their own solutions and listen to and question other solutions. Teachers should also allow for independent practice to ensure that students understand concepts outside of working collaboratively.

Growing Patterns: Group Activity

In this weeks class, we looked at an activity aimed towards understanding growing patterns. Working in small groups, we looked at a serious of graphs, formulas, charts, and visual patterns. There were four different patterns and each one aligned itself with one graph, formula, and chart. There was one graph, formula, chart, and pattern that was blank, however, and we were supposed to determine what the missing element looked like. 

Working collaboratively within the group allowed us to solve these blank cards and determine a relationship between the graph, formula, chart, and visual pattern. We also used building blocks to help visually determine what patterns were being used. 

This activity allowed every member of our group to jump in. Some members were more comfortable with the charts while others were comfortable using the formulas. Using each other's strengths allowed us as a group to understand the relationship between the cards and determine which patterns were at play. 

Gizmos

Here are some Gizmos to help teach students the concept of patterns. My favourite Gizmo from this list is Pattern Finder and I think it would be the most effective because of the visuals. Unlike the other two Gizmos, a pattern is not laid out in front of you. Instead, you must follow the frogs to see which lily pads they are jumping to and determine a pattern. 

 

Finding Patterns - Build a pattern to complete a sequence of patterns. Study a sequence of three patterns of squares in a grid and build the fourth pattern of the sequence in the grid.

Pattern Finder - Observe frogs jumping around on coloured lily pads. Find, test, and reason about patterns you see in their jumping

Pattern Flip- In the Pattern Flip carnival game, you are shown a pattern of cards. The first cards are face-up so you can see the pattern, and the rest are face-down. The object of the game is to determine which animals are on the face-down cards.

Creating Effective Math Problems

 What Makes a Good Math Problem?

This week in class we continued to talk about creating math problems and what characteristics contribute to an effective problem. Some of these characteristics we discussed include:

• The problem is relevant to the students
• There are multiple approaches and methods to solving the problem
• Students can use various manipulatives
• Wide base where everyone can join in
• The problem uses soft language 
• The problem creates discussion
• Students have the option to work together and collaborate
• There is a "high ceiling", which means that there is the potential to increase difficulty and challenge students

Wrapping up Joel's Cat Food Problem

We wrapped Joel's Cat Food problem this week with a Math Congress. In large groups, we were asked to discuss the different approaches we used to solve Joel's Cat Food problem. A few students were assigned leader roles where they explained how their group solved the problem and made sure that every student in the group understood the approach. After our group discussed three different approaches to solving the problem, we went on a Gallery Walk where we got to see how other groups solved the problem. Here are some pictures of the different approaches that students took. Our group discussed how some of the other approaches were more visually geared and how our approaches could seem difficult to understand for some students. One aspect of the solutions that all of us determined to be important was that the information given was shown. Knowing that Bob's store sells 12 cans for $15 and Maria's store sells 20 cans for $23 was shown on almost every piece of chart paper. Even though this information was provided on the slideshow, most students found it helpful to re-write this information on the chart paper.

Proportional Thinking

We continued the lesson by discussing the concept of proportional thinking and how we think about proportions on a daily basis. Proportional Thinking is the ability to compare quantities multiplicatively and ties into our previous lessons on Ratios and Fractions. Some of the ways we think proportionately can be as obvious as baking or as subtle as thinking about how much gas you need to get from Point A to Point B. Fractions and Ratios is not a foreign concept and plays an important role within life, so it is extremely important that students learn about proportions and how to compare them.

Integers, Gizmos, and CLIPS

We wrapped up the class this week by discussing integers. One of the biggest problems with integers is trying to apply "rules" that students have learned to a problem where the rule does not apply. There are many different ways to develop an understanding of integer size and operations and creating rules can limit a students understanding of how integers work. Gizmos is a website we discussed briefly discussed in class that can help students understand integers. Gizmos are interactive math and science simulations that help students understand concepts that they struggle with. There are several Gizmos that are aimed towards Integers that allow students to develop their own understanding as opposed to a definition created through rules. The term "integers" may be foreign to students, but the actual concept is more understood than a student might think. The Critical Learning Paths Supports website, or CLIPS, is an educational website that provides students with videos and activities to help understand math. One of the areas it focuses on is integers and how integers are embedded in daily tasks. Outside temperature will usually determine how someone dresses. When students are told this, they begin to understand how integers can be thought of as a whole numbers. This website allows students to explore the concept of integers and see how integers are present in their daily routine.

Fractions; The Other Half (part 2)

This week in class we continued our discussion on fractions. With last week's intro to fractions, we continued to develop multiple perspectives of how we see fractions.



Racing With Fractions!

One of the opening activities reinforced the idea that others may look at a fraction and see it differently.  Several volunteers were asked to go up to the front of the classroom and play a game of Red Light/ Green Light, which set the stage for the introductory activity. After a few minutes of students being sent back to the beginning of the race, we were asked to go back to our seats. A slide similar to this picture was shown to us.
Although the actual race results were not similar to the results we were shown, we began to think of the race in fractions. Henna finished one quarter of the race while Christian finished three quarters of the race. We were told to think of these fractions as the amount of race that the student finished. We were then asked to place them in order of race finished. With the understanding of fractions I gained from last week's class, I wasn't surprised to see the different ways that my peers had answered the question. I tried to visualize the fractions and place the racers along a line. I noticed other students were doing the same, while some were converting the fractions into percentages and listing the racers according to their percentage. This activity was fun and engaging as it got students out of their seats for some friendly competition and allowed us to approach the question using our own methods.

Dividing Fractions - Old vs. New


When I was learning the concept of dividing fractions, I was told to use a specific formula. I would start by flipping the second fraction. I would then change the division sign to a multiplication sign and multiply across the formula. The answer I got was right, but I completely avoided the concept of division and was confused as to why I got the right answer. The way I learned how to divide fractions in class is clear and makes more sense to me. Instead of flipping the second fraction and replacing the division sign with a multiplication sign, I just divided through. I get the same answer, which is the right answer, and I use less steps to get there. With this new understanding, I can divide fractions without getting lost. I will certainly not continue to teach the "cross-multiplication" approach and ensure that every student is exposed to this way of approaching division with fractions.


Mr. Tan's Tangram


One activity that I was familiar with was Mr. Tan's Tangram. Each student was given a set of seven shapes and was told the story of Mr Tan. With the mixture of shapes, we were told that it was possible to make a square. I personally struggled to find a way to make a square, even though I had come really close on a few tries. After being shown the solution, I saw that I was really close to finding the answer. I liked this activity as I got to explore the way shapes fit together and come up with different layouts, even though I could not find the answer. This activity was brought up in the fractions lesson because it deals with ratios and fractions. The large triangles are equivalent to one quarter of the square each. Each of the smaller shapes can be placed within on of these triangles and be given a fraction corresponding to it value within the whole square. This activity is a great hands-on activity and allows students to learn by themselves not only how to make a square with the tangram shapes, but also grapple with the topic of fractions and how each of these shapes work within the whole square.

Fractions In Everyday Life

Students are taught about fractions within a short period of time. There is not a lot of focus within this topic of math, even though it is extremely important. I was taught to use a formula, as I discussed earlier, so I could quickly arrive at an answer without actually understanding the process. Fractions play an important role in the world and in many day to day activities. Whether you are baking, paying taxes, or building a house, fractions encompass our lives in many different forms. Without learning how fractions work, we miss out on an opportunity to learn and knowing how to perform day-to-day tasks.

Teaching Fractions

The way I will teach students fractions will be a lot different from the way I was taught. Much like the way I am learning about fractions now, I will use games and classroom activities to engage students with the material and concepts. Tug Team Fractions is a fun game that students can play to learn about comparing fractions. I would use this game along with the Race Activity we worked on at the beginning of class to teach students how fractions relate to one another and how you can compare them. As I continue this course, I am relearning essential math skills and learning how I could approach teaching these concepts through a fun and engaging way.

Introduction to Fractions!

This week's class covered the topic of fractions. We started the class by choosing any fraction that we wanted. I decided to use the fraction 3/4 as it is an easy fraction to work with. We were then asked what is important about our fraction and why is it a fraction? The fraction 3/4 is important to me because it is an easy fraction to work with and can help teach students about fractions. It is also easy to visualize as you can imagine a pizza cut into four and one slice is eaten. You can also visualize this fraction using quarters.

When we were asked why the numbers we chose were fractions, I tried to create a general definition that could be applied to every fraction. There is a numerator (the number on the top), a denominator (the number on the bottom), and a line between the two. This seemed to be the general definition for most students in the class.

However, most of the fractions created in the class were different. I started to notice that some students had created fractions that were a lot different than the one I created. The lesson then progressed into the types of fractions.


The first type of fraction that we looked at is called the Proper Faction. This fraction equals less than one. The fraction 3/4 is an example of a proper fraction.



Another type of fraction is the Mixed Numbers Fraction. This fraction equals more than 1. The fraction 1 and 2/4 is an example of a mixed numbers fraction.



The third type of fraction is called a Unit Fraction. This fraction is similar to the proper fraction as it equals less than 1. The difference is that the numerator is the number one.

One of the tools used to teach us about fractions is a book called "The Hersey's Milk Chocolate Fractions Book". This book uses a Hersey's milk chocolate bar as a manipulative to help students grapple with the understanding of fractions. Each table was given a Hersey's chocolate bar and was asked to break up the bar into twelve pieces. One of the reasons why I really liked this activity is because I think it will be effective with Junior/Intermediate students. Most students, if not all, know what a chocolate bar is and dividing it into twelve can help them understand how a portion of something (such as the chocolate bar) can be represented by a fraction. As the book was read to us, we were asked to group the chocolate in several ways that represented fractions. The book also uses pictures to help demonstrate how the chocolate pieces should be divided.


It was also beneficial to understanding fractions when we were asked to fill out a graphic organizer on fractions. This organizer was a chart we had to fill in that helped us to define what a fraction is and give examples and non-examples of fractions. This chart could be filled out by students as an activity either before they are introduced to fractions or after they are taught. It can be used to help the teacher understand what students think of when asked to define a fraction and guide students.

I can see the various manipulatives used in class being extremely beneficial to teaching students about fractions. While the chocolate bar and book are extremely helpful, we were also given various shapes, egg cartons, and several informative sheets that broke down fractions. As I continue this class, one of the key elements that I am noticing is the use of several manipulatives. One type of activity may help some students to understand what a fraction is, but others may need a different manipulative to help them. Activities that engage students and promote collaboration is another important element to teaching. The Hersey's chocolate bar activity allowed us to work in groups and got us excited. The presentations that followed the lesson also promoted the use of manipulatives and collaboration amongst students. 

How to Subtract!

This week in class we learned about the importance of different algorithms and why it is important to promote several approaches to answering a question. With the EQAO assignment in mind, I have tried to not only see the importance of different algorithms but also attempted to answer math-based questions using different methods. We got to play around with some simple math expressions in class using different approaches. It was surprising to see that many students take different approaches that may have seemed confusing to me before this class, but now appear to be useful and easy ways to approach math. Understanding this will help me to become an effective teacher as I plan to implement several algorithms throughout my math courses.

387-146=241

Having developed my math skills over the years, it didn't take me long to figure out that
387-146=241. My approach to answering this question, however, is a way that can be extremely confusing to students who have a different learning process. I learned that it can be extremely effective to start with a number line, even if some students don't find it as useful as others.


By breaking up the equation into steps, students can understand the role of numbers and how they can be broken up into smaller numbers in order to make math easier. By starting on the right side with 387, students can move to the left (addition would move to the right) by blocks of numbers that they understand. In the picture to the right, 146 is broken up into three parts: 100, 40, and 6.  As students subtract the numbers and move along the line to the left, they arrive at 241. This method is known as the Partial Subtraction, which is subtracting from the greatest to the least place value.

Compensation Subtraction
Another method of approaching this problem is known as the Compensation Subtraction method. This method involves making a friendly number. By adding 4 to 146, students might have an easier way of solving the problem. 387-150 may appear less daunting than 387-150 and students may prefer this method. It is important to note that the 4 added to 146 needs to be subtracted at the end to balance the equation; when you borrow, you must return.

Constant Difference Subtraction
Much like the compensation method, the Constant Different Subtraction method involves making a friendly number. Unlike the compensation method, however, the equation isn't balance out at the end. If you add 4 to 146, you need to minus 4 from 387. The equation will read 383-150, but the answer will still be the same. Some students may prefer this over the compensation method, but both are equally acceptable.

During my placement I have been given the opportunity to watch students during their math class to see how they answer questions. While I helped students get to the right answer, I was surprised to see many using methods that I was never taught. The class I have been placed in is a grade eight elementary class and they are just beginning to learn the basics of algebra. Understanding the role of symbols within math can be challenging to many students. I was surprised to see that visual manipulatives formed the basis for many students' understanding. Breaking down the problem in front of the students is an extremely effective method and can be used at any grade. Another thing I noticed is that while my teaching associate promoted several approaches, he made it necessary for students to understand more than one method and use several on tests and quizzes.

Learning Mathematics all over again!

Welcome to my math blog! My name is Brandon and this blog is about learning how to teach Junior/ Intermediate level mathematics. In this blog I will be posting my experiences with learning mathematics and the approaches to math that I am currently learning. A lot of people say "I'm not a math person". I've caught myself saying this a few times and I'm starting to understand that this feeling about math derives from my experience with learning. Learning math was hard growing up, and I expect that re-learning math and how to teach math is going to take some time. I was always taught that there is one right answer and that there is a certain process or formula that needs to be followed, which is not the case anymore. I'm learning that even basic addition and subtraction can be learned through several teaching strategies. Understanding that there are different ways for students to understand math will hopefully allow me to teach students effectively. 

Here is a picture of some simple addition that caused some confusion. Everyone in the class knew that 272 + 272 = 544. However, our methods of arriving at this conclusion were challenged. Students may not understand one method, yet understand another. When teaching math, it is important to teach several strategies to ensure that students fully understand concepts and are not just regurgitating a formula or process.