8P29 in review

Math: Growth Through the Year

My first thought of math was similar to many of my peers on the first day of classes. After all it was only a few days prior that I dragged myself through completing the math review tests that was required through the course. I thought to myself if I could barely finish the refresher how will I get through a full course on math? Then I walked into a classroom filled with twenty-five other students that were as nervous as me, and a teacher who wanted to change our opinions of math.  Together we were able to survive the math class, learn multiple methods of teaching the subject, felt comfortable with the topics and even had a little fun on the way!

The Key Points

Open Questions

Through the entirety of the math course Pat, our teacher, wanted us to create open ended problems and encouraged us to do so weekly. At first I did not understand the importance of the open-ended questions as I felt that math could only be done in certain ways to get the right answer. Quickly my mind was blown at how wrong I was. Every lesson started off with a question that we were encouraged to talk to our peers about and solve our own unique way. After everyone had a chance to solve the problems we shared our answers. The answers from each group varied yet we could all end up at the correct answer. Having these open-ended questions allowed for the students to work the way in which they felt was best suited for the question and allowed all students to participate. Open-ended questions will definitely be an idea used in my classrooms and something that I took away from this math course.

Children’s Literature

Story time! When I first thought of reading I immediately associate it with literacy.  Who knew there would be so many books out there that are used to teach math? The books were brought in multiple times throughout the year to reinforce the lesson that was being taught that day. Whether it was fractions and the use of the Hershey Chocolate bar or geometry and The Greedy Triangle children’s literature is a great way to grab the attention of the students. The books will even take some of the teaching off your hands, as it will explain to students the properties of a triangle or aspects of fractions. Children’s literature can be used for much more than story time.

A Student Focused Classroom

The classroom is a place for students to learn and grow. The teachers should encourage the growth of the students and we must put more resources toward this.  Students should feel welcomed to share their answers in math and not be afraid to make mistakes. Classroom congresses, group work, elbow partners all promote growth for the students. The communication can allow the students to talk to each other and solve their problems rather than struggle through them alone. As a teacher candidate I learned that collaboration in math is something that should be embraced and not frowned upon.

Growth Mindset

The biggest takeaway I had from this course is that of a growth mindset. Being open to new teaching methods, new ways of completing the task or even just trying something different are

all examples of having that mindset. An idea of positivity goes a long way in any subject, especially in math were mistakes can be built upon. Students should be encouraged to grow and have a positive outlook on math. Teachers should also have this growth mindset and not be afraid to try new things such as incorporating a game or having more group work. We can always improve and having the growth mindset allows us to do so!

Overall, this math course has enlightened me of so much. While I find the four points to be the most important there are countless other resources that I was able to gain by completing the course. With the completion of this course as well as my others I’m excited to go into my placements and put in to practice my newfound knowledge.

Data Management

Explaining the why in math

This past week in math we looked at data management and probability. When first hearing this unit I immediately think of estimation and using the phrases almost always or very unlikely. While this is a part of data management and probability I never really understood why would use estimation in real life. It was not until this class that it was actually explained to me. Using estimation can help us in the world because we do not always want the exact answer and we can get a good idea of how many or how likely something is to occur by estimation. Having this information presented to me made me realize that though I was completing the questions in math class I never fully knew why. The reinforcement of ensuring students know why they are completing the question and how it can be used in life was once again apparent. Being able to relate the questions to real life can gain the students interest and allow them to remember the information easier.

Data Collection

Data and how its collected and used is a major theme throughout the chapter in Making Math Meaningful was. One of the main points that was also echoed in class was ensuring that students are creating appropriate questionnaires and planning of how to gather the data. Students should not only be conducting a survey of 5 people and then displaying that information through a graph. The survey and coinciding graph should have a purpose for the student. Gathering the results and conducting a survey is only half of the expectations. Students then need to display the data in forms that are purposeful. The graphs that the students choose should help represent the information that they are trying to convey. Students should be able to look at the graph and make inferences or comparisons. The collection and display of data are a main concept in data and more time should be spent to ensure students have the proper skills and knowledge to do so.

Resources to Help Teach

Two concepts that were examined in class were stem and leaf plots and tinker plots. Both resources will help the student gather and organize data. Stem and Leaf plots are especially useful when trying to find the mean median and mode of an array of numbers. This plot orders the numbers with the first digit(s) being the stem and the last digit being the leaf. The organization of the array through this method will allow for easy estimation or the calculation of mean median and mode if necessary. The Stem and Leaf plot place all data into context. A tinker plot is useful for the organization of a large amount of data. The plot places the data on a graph and allows the students to find certain characteristics of the data easily. www.tinkerplots.com is a great resource to use to show students the different aspects that can be utilized from creating an online tinker plot. These resources will help the students collect that the data and be able to make connections with it more easily.

 

 

Collaboration

Collaborative Working

A key concept from this week’s math class was collaborative working. The concept taught me and the other teacher candidates a great resource to use in the classroom. The idea of allowing the students to group themselves together in a way they wanted was a different experience to me. While we as students were given popsicles sticks with colors and numbers on them we were then allowed to divide up in a way in which we wanted to. We could have got into groups based on the color of our popsicle stick or the number we had – all the number 1s together or all different numbers from 1-6. The concept of choice allows the students to feel involved and does not make it seem like they were forced to work with particular peers.

Collaborative work also ensured that everyone was able to take part in the math lesson. Each member of the group was given a clue that they were to read to the rest of the group. That clue was only for that person and they had to read it to the others causing them to be involved. The interaction made sure that all members of the group participated and that the group was not dominated by a student that excels in math. An example of what can come from the collaboration is shown below as my group was able to solve the clues to complete the task given to us.

Assessment

Assessment is an influential part of a student’s learning experience. In the past assessment has been focused on what the student is able to produce on paper. Recently the narrative of assessment has begun to change to better reflect the knowledge and understanding of the student. According to “Teaching and Learning Mathematics,” the use of more informal assessment methods such as discussion or observation allow teachers to provide a mark that better reflects the students understanding. A students understanding should not be based on one test of performance task but should be based on their overall understanding of the math concepts. This week’s class along with the readings taught me that as a teacher we should be making notes about the success of the students. These notes will allow teachers to understand what they need to teach but will also allow them to document the success and understanding of the students.

Technology in the classroom

This week my classmates learning activity presentations examined the use of technology in mathematics. Two of the presentations used the website https://www.explorelearning.com/, which has multiple math activities for students of all ability levels. The ‘Gizmos’ as the activities are called allow the students to interact with the information that they are learning. By being able to interact with the content the students are learning they will be able to better understand and be more interested in the math topics. The Gizmos can be extremely helpful as a reinforcement activity or introduction into new topics. The Gizmos are just another example of the concept of ensuring that students are engaged in mathematics. Just in the past classes, the use of a student’s interest to teach math was reinforced as important and influential tool. 

Measurement

Measurement

I have! Who Has?

A great way to review concepts, be the main activity of the class or even the consolidation this math game creates an engaged and entertained class. This game allows for all students to take part and helps reinforce definitions or specific aspects of the math topic for the unit. A description of the game and some example card decks can be found online at Math Wire. The answers for the game would have to be known before hand and because of this the game would not be a good use for an introduction. The answers should also be simple so that students do not take to long answer and become disengaged. I can see this game easily being used in my practicum. The game does not take long to set up and provides for a fun form of review. Students will be able to have fun while completing math activities and hopefully this will keep them interested in the topics.

Different Area Same Perimeter

In class this week we examined the measurements of perimeter and area in rectangles. The main concept that we looked at was that even if shapes can have the same perimeter the area inside can be vastly different. The area does not stay the same if you change the dimensions of the shape. If asking students to find examples of this try to ensure that they keep track of the dimensions that they have tried. By keeping track of their attempts they will be able to see what has or hasn’t worked and can work more effectively because of this.

How to Measure

Teaching measurement can be a difficult topic especially if the first way to introduce it is by measuring with centimeters. Instead measurement can be introduced through three different ways. As seen below it can be introduced by comparison, non-standard units and standard units.

 

Retrieved from: Making Math Meaningful 472

These different concepts of measurement allow the student to better grasp how to measure. With comparison students should be able to recognize what object is longer by comparing to another. The comparison aspect also allows the student to develop vocabulary such as longer than or shorter than.  Measuring with non-standard units can help the students relate to the object that they are measuring. By using objects that are familiar to them such as a new pencil or popsicle stick the students can form a mental image of how long that object might be. Lastly measuring in standard units should be taught to the students as something that everyone will understand as the same length. Explaining to students that measuring with your feet may be different between two people if they do not have the same size feet. Measuring with standard units allow all people to have the same measurement of objects. Measuring with standard units should be introduced to students so that they see the value in everyone understanding the measurement as the same.

Geometry

Geometry and why it Should Be Physical

Geometry is one aspect of mathematics that should be made as physical as possible for the students. Having physical manipulatives will allow students to become more comfortable with the 2D and 3D shapes and have a better understanding of key concepts such as symmetry, congruency, and similarity. One of the main fundamental aspects of geometry is visualization, a study conducted in 2004 states, “visual reasoning as ‘seeing to think’” (Whiteley 2004). Since visualization is such an important aspect of geometry allowing students to use physical manipulatives will help them improve on their own visual thinking. Below is an example of an activity that presents geometry to the students in a physical form.

Important Terms

When learning about geometry this week three key terms became apparent. These terms are essential to teaching geometry and if the student is unable to understand these terms then they will have a difficult time throughout the geometry unity.

Similar: A shape is described as similar if they have the same shape with sides in proportion to another. Similar shapes are an enlargement or reduction of the other

Congruent: shapes are said to be congruent if one can be transformed into the other through a series of flips slides and/or turns. Congruence can also be used to describe specific components of shapes. An example of this is if the angle or side length are the same then the shapes are said to have congruent sides or angles. Congruency can be explained as a shape having equal properties, which could be the whole shape is equal to another or just a side or angle.

Symmetry: There are two types of symmetry, reflective and rotational. Reflective is when a shape is divided by a line or plane and the opposite side are mirror images of each other. Reflective symmetry is easily explained as being the same on both sides of the shape. Reflective symmetry is how many times a shape can fit over itself when it is rotated.

These terms should be a starting point for teachers who are introducing geometry to their students. With these words students will be able to have a foundation for the geometry unit. For more definitions in the geometry strand of mathematics, Geometry definitions  is a great resource that students can utilize to help them remember some more important terms.

The Use of Children’s literature

The story that was used to introduce shapes to our class this week was ‘The Greedy Triangle.’ This book was not only an overview of the properties of shapes and what they are called but also gave real life scenarios for what the shapes are used for. The story was about a triangle who was not happy and always wanted another side which transformed it into another shape. The story then told of all the different jobs that this shape could do in the world. I felt that this story was a great way to present the information for students. The storyline allowed the students a refresher on what certain shapes are called and how many sides they have. Most importantly this story allowed students to visualize how shapes are used in real life. By being able to have concrete examples of the use of shapes students can better relate to the geometry strand in mathematics.

It was yet another great week in math class. I was able to learn valuable lessons to how to teach geometry and even gained another children’s literature resource that can be used within mathematics. 

Patterning and Algebra

Class Overview

            This week in math class we covered the topics patterning and algebra. As a student I thought patterning was just the repetition of images or the continuous increase or decrease of a number of items. I also thought that algebra was the use of letters in math which was truthfully a little frightening. While enduring my math education I always thought that these units were two separate entities that had no relation to each other. In the class this week I quickly found out that I was wrong and that the connection between patterning and algebra is very strong and easy to see.

Matching Patterns

           We started out the week with a matching patterns exercise that had the students match a block pattern with a corresponding t-chart, an algebraic equation, and a graph. The exercise in completion is shown above. I was able to understand the relation between the t-chart and block pattern quickly as the t-chart just stated the stage and how many blocks were present in the pattern. From this chart I was also able to understand the graphing aspect, the left side of the t-chart was the x-axis number and the right side was the y-axis. Using the graphs and t-charts can help the student move from patterning to algebra easily. The organizational tools will help the student clearly see the pattern help them create an equation. Where I was confused was to how to write this relation as an algebraic equation. It was not until a later side that I was fully able to understand the relation. The slide that made it clear how patterns and algebra are related is shown below.

This picture easily explains how to relate a pattern to algebra. The ‘t’ stands for total number of blocks or output. The ‘n’ stands for the input or stage number. With a few input and output numbers given to the student they should be able to realize the pattern and quickly create an equation. The equation is simply put together by the pattern number or what the pattern increases multiplied by the input number and then adding an extra number to get the total output. This picture explains it clearly that that the number that increases the pattern will always be multiplied by the stage number.

Another Week, Another Teaching Tool

            This week showed me another tool that can be used in the classroom to enhance math education. The resource, Explore Learning, Gizmos, allows for a teacher to create a class and invite students in order to teach them more about a certain subject. The resource covers a large variety of topics and can help the students understand mathematics in a fun learning environment. The resource that I viewed in depth was the function machine as it went along with the topic of patterning and algebra. The function machine allows for the student to pick a number and drop it into the machine as it generates a pattern. The resource even has an assessment that the students can use to test their knowledge on the patterns and view how they are doing.

This week was a great one in math because it showed me the relation that patterning and algebra have. This week also provided me with another resource that I can introduce into the math curriculum in placement or when I am teaching in my own classroom.

Creating a Classroom that Revolves Around Students

Classroom Techniques           

        The week in math class was very influential for teaching techniques. Throughout the class session fractions was not the only lesson the teaching candidates and myself were being taught. The use of classroom techniques such as what makes for a good problem for students and classroom congress. These to teaching techniques provided the candidates with ways to ensure the involvement of all students.

What Makes a Good Problem?

            Every week in class we start out working on a problem within our table groups. The table groups are all able to start the problem immediately in a way in which they believe is best suited. The problems that we start with all have multiple aspects in common with each other. As a class we created the list below which describes the aspects that we feel create a good problem. As a teacher candidate I feel that creating these good problems are critical to having an inviting classroom. Questions that allow all students to begin and find answer allow for the students to feel a sense of accomplishment. As a teacher I will strive to make sure that in all subjects that the questions I assign to students will have the aspects of a good problem.

The Classroom Congress

            A classroom congress was a new idea that was presented to me in the previous week. A congress provides the ability for the students to have ownership of the work that they are completing. The congress focuses on student ideas who then explain to other students the way they created their solution. Leaders of the congress take on the role of trying to make sure that any questions have from other students are answered and all members of the congress understand the different solution methods. A congress is a creative way to provide choice and give freedom to the students. This choice that the students have will create better students as they take more interest in the work that they are in charge of. For more information and an example of a math congress go to Math Congress Video to better inform yourself on the use of a congress in the classroom. As a teacher I will hope to use this to provide my students a different way to present the information they have found. Students may feel more comfortable presenting to their peers and can gain a better understanding of the material if they explain it to their peers and feel free to ask questions.

Now for the Math Lesson … Integers

            Not much time was spent on integers but as a teacher I was still able to gain valuable information that will help me teach students. One of the largest problems with integers is applying rules to a problem where they do not apply. As a teacher we can hope to eliminate this mistake by teaching through example. Teaching through example will provide the students concrete material that they can refer to when trying to solve a similar problem. Instead of memorization of specific rules as a teacher I will strive to teach my students through the use of example to ensure that they are able understand what has been taught.

Overall this week showed me that creating an inviting classroom and using examples can positively enhance the education of students.

Red Light! Green Light!

Making Math Fun
I would have never thought to teach fractions through the use of a playground game. This week in math class yet again I was astonished at learning just how fun math can be. To start the day of our class went back to our childhood and played the favourite children’s game, Red Light, Green Light. While we were all laughing and having a good time I was unable to see the connection to fractions until it was simply explained by Patricia. Fractions can be used as a way to describe a distance or far you have gone.  An example of this in the game was 7/16. This fraction symbolizes that the person is a little bit less than half of the way there. The language that one uses also will help students understand how much the fractions symbolizes. Using soft language such as, a little bit or almost all of it, can help students identify how big or small a fraction. Below is a picture of the game that we played in class and the distance that each student got to.

This quick example showed me the importance of two key concepts when teaching math. The first concept is the use of games to help students become engaged can increase their learning. The use of game was able to keep me interested in the subject and therefore made it easier for me to learn about fractions. The second concept was using language that students are able to understand. Using soft language will help the students better understand the concepts it makes it easier for them to complete the tasks.

Common Denominators in Division!

A second activity that we did in class this past week was dividing fractions. As a student in mathematics I was always taught that common denominators are only used for addition or subtraction. My mind was blown when I was told that common denominators can be used in division as well. This method even seemed easier to me and I was better able to understand the concept. Using common denominators could create a three tiered fraction if the top number was a fraction but the bottom tier of the fraction would be 1. We were also taught in class that anything over one is just the anything therefore taking away the three tiered fraction. Common denominators can allow the student to divide the fractions easier and not be worried the flip and multiply method. The following image is an example of how using common denominators to divide is possible.

This week in math class was outstanding! I was able to learn concepts that I can implement to keep my students engaged in the lesson. I was also able to better understand how to divide fractions and that the use of common denominators is helpful when doing so. Math class so far has been an outstanding experience as I am being taught math in way that I could have never imagined. 

Teaching Fractions with Chocolate

The best part of the week

            Teaching fractions is a lot easier when a story is involved. In class this week we were taught fractions through the use of the story called, “the Hershey’s Milk Chocolate Fraction Book.” The book teaches the students the fractions from 1/12 to 13/12 and also a variety of other unit fractions (1 as the numerator). The activity is an interactive story and got the whole class involved as many of us were eager to eat the chocolate that was associated with the activity. This activity made it clear to see the positive outcomes that can be found through the use of children’s literature, provided a resource for future teaching and also gave us a tasty snack.

Pallotta, Jerrry. "The Hershey's Milk Chocolate Fractions Book." 1999. Book cover. Retrieved from: http://www.scholastic.com/teachers/book/hersheys-milk-chocolate-bar-fractions-book-0#cart/cleanup

What I will take away

Memorization of steps of mathematical procedures are not important if students do not understand why they are doing them. This sentence from the previous week is something that I will bring into my future classrooms as a teacher. Countless formulas were taught to me as a student throughout my education. I would memorize the formulas and even the parts of the question from where to get the numbers to put in the equation to get the correct answer but I never fully understood what I was doing.  By forcing students to memorize a formula we are putting them at a disadvantage as we limit the ways that they are able to learn. If the students are able to understand what is asked of them, then they are more likely able to remember how to find the answer to questions such as what is the slope of the line? By allowing students to create their own way to arrive at the solution we are encouraging a growth mindset and helping them understand math better.

Problem Solving Assignment

The problem solving assignment reinforced the notion that students should be allowed to find their own solutions to the questions that are asked. When completing the answers to the questions that I was given, I originally was only able to think of one way that I knew how to find the answer. I was able to remember the substitution method of algebra as that was the way I was taught to find the answer of what the variable represents in the equation. Upon first looking at the problem I was only able to solve the question in one way. It was not until a couple days later that I was able to go back to and understand the problem from a different perspective. By leaving the problem and answering the question in another method I was able to better understand the question. By finding the answer using a different method I was able to realize that there are multiple ways to arrive at the answer and to use the way that you understand the most.

Multiple Ways of Addition and Subtraction

What We Learned

This week in my math class we tackled the challenging issues of addition and subtraction. When I was taught math in elementary school the method of carrying numbers or borrowing was used to help the student with addition and subtraction. This way taught the students what to do, but not why or how they were doing the math. In class I was taught multiple strategies that teach the students addition and subtraction but also allow them to chose what method that they understand the best. The methods learned in class were: partial addition, compensation addition, constant addition, partial subtraction, compensation subtraction and constant subtraction. The students use these methods to understand the math but also remember what they are taught so that it will be of use to them in the future. The education on the addition and subtraction strategies are useful to me as it presents multiple ways in which to teach students. It also opened my mind to the idea that math can be done in a variety of ways while still arriving at the same final result. Having multiple ways to find an answer showed me that students will learn in a variety of ways and to be accepting of all of them.

The Best Part!

            My favourite part of math class this week was student interaction. I did not previously realize the importance that a good math problem can have for the students. Below is a snapshot that was taken in class that describes the different aspects of a good problem that gets students involved. The conversation around what makes a good problem made me realize that if students are not connected to the problem then they are less likely to be engaged with the math that is being taught. The use of good math problems and the interaction of students became more evident to myself when the learning activity presentations were ongoing. All three presenters during class time create good math problems for the class to solve and looking around the room at this time showed me how involved my peers and myself were with the question at hand. The interaction of the class proved to me how important it is to create student interaction in order to teach math.

Improvement

            Through this week I feel the biggest improvement I made was the open mind-set towards the new ways to teach math. When first entering math I believed that there was only one way for it to be taught and that it was uninteresting to me. Through this week as well as the previous I feel I have opened my mind to different approaches to math and the problems that are presented. By having this different mindset, I was able to enjoy the class and learn different techniques of teaching a fun and engaging math lesson. For future classes I hope to continue having an open mind to the new math lessons and learn valuable information that I will be able to take into my teaching career.

Welcome!

Welcome to my weekly math blog! My name is Christian Ferracuti and I am a teacher candidate at Brock University in the first year of the two-year consecutive education program. The title of my blog “J/I Math” represents the grade divisions that I hope to teach in the near future. In this blog you can expect updates on the different teaching techniques that can be used to teach the 21st century learner. Math education has changed rapidly over the past ten years and new teaching methods have been created. Over the next 3 months I will be posting weekly updates to the teaching concepts that will help math students have fun and be successful in math. Throughout this course I hope to learn how to teach math in an inviting way as well as learning math concepts that I will be able to retain. I also hope that this course will show me that math is a fun subject and not only certain students are able to be successful. Throughout my own math education, formulas were taught and expected to be memorized by the students in order to pass the course. With two weeks of the class completed it was very apparent to me that this is not the way math is taught anymore. The use of physical objects to help engage the students as well as help them create their own way of learning is a fascinating concept which I want to learn more of. Below there is a picture of the objects that we used in the previous math class. The use of physical objects/material was able to help me understand the math and will surely help the students. The main goal for this course is to become a strong math educator. The interest that a student has in the subject is related to the abilities of the teacher and the way that they teach. I hope that this course will enable me to become a great educator for math so that the future generation embraces the subject and is not afraid of math. In closing I invite you back to my blog for weekly updates. In the upcoming weeks I hope to document more of my experiences in the math class especially the use of materials to increase learning. I will also update the readers on how the materials were incorporated into the lesson and the other ways that the lesson can be taught. Hopefully through explaining the use of the materials will give myself or other teacher candidates more resources to use in the future when teaching math.