Final Reflection: A New Perspective on Teaching Math

Exciting. Thought-provoking. Challenging. Insightful. Encouraging. These are just some of the words that come to mind when reflecting on my experiences in this math course. Over the past twelve weeks, my understanding of how to effectively teach math has been completely transformed for the better. Before this course, I was apprehensive about teaching math. Now, I feel energized and excited to take what I have learned in this course and to apply it in the classroom by creating a positive and engaging learning environment where my students will have the opportunity to develop a meaningful understanding of math!

The Importance of a Growth Mindset 

One of the biggest takeaways from this course is the importance of a growth mindset. Before this math course, I held the common misconception that some people weren't good at math. This is simply not true! Research shows that every child has the ability to excel in math. As a teacher, it is critical that I do not foster a belief in my students that they are not good at math but rather that I create a positive learning experience where every student in my class truly believes they have the ability to succeed. I need to establish a learning environment which promotes a growth mindset where my students understand the power of yet. I want my students to understand that mistakes and struggles are an important and valued part of learning which can be used as stepping stones leading to growth and understanding. Mistakes should not be stigmatized but rather should be embraced as a powerful learning opportunity! Rather than focusing on speed and efficiency, I need to reward hard work. As Carol Dweck discusses in her TED Talk entitled "The power of believing that you can improve", it is important for me to to praise the process the students are engaging in: their efforts, their strategies, their focus, their perseverance, and their improvement.

TED [2014, December 17] "The power of believing you can improve."
Retrieved from https://www.youtube.com/watch?v=_X0mgOOSpLU

Making Math Meaningful 

The Importance of Relational Understanding

This course has also taught me the need for students to develop a relational understanding of math. After completing this course, I now have a much better understanding and awareness of how intimately connected mathematical concepts and ideas are. As a teacher, I need to ensure that I develop lessons which help students to explore and discover how various mathematical concepts and ideas are connected. Rather than compartmentalizing different mathematical ideas and learning how to do specific tasks quickly using given steps or formulas, students need to explore ideas and learn about processes so that they are able to  develop an overall understanding and build a conceptual structure where mathematical ideas are linked. By exploring and making connections, students will develop a deeper, more meaningful understanding of math and will be able to adapt and apply their knowledge and understanding to new and diverse tasks. 

The Value of Manipulatives 

This course has also taught me the value of manipulatives in helping students to develop a more meaningful understanding of mathematical concepts. Before this course, I held the misconception that manipulatives were only helpful for students who struggled with math and thus would be kept at the side of the classroom as an option for students who need it. I now realize that manipulatives are valuable for all students as they make abstract concepts concrete, allow students to actively explore math concepts, and encourage students to prove their knowledge and understanding in a meaningful, concrete, and visual way.

Using chocolate bar pieces to explore fractions.
Olij, B. © 2016

Marian Small's discussion of the Van Hiele Taxonomy of Geometric Thought in Chapter 17 of Making Math Meaningful really drove home the value of manipulatives. In order for students to develop their geometric thinking, they need to have geometric spatial experience. For students to develop their spatial awareness and their understanding of the qualities and properties of shapes, they need to have opportunities where they can explore and discover these concepts through physical interaction with concrete materials. As a math teacher, facilitating opportunities where students work with a variety of manipulatives to explore ideas and demonstrate their understanding will be an integral part of my lesson plans.

Using tangrams to explore shapes.
Olij, B. © 2016

The Creation of Effective, Open-Ended Math Problems 

Another important takeaway from this math course is how to create effective, open-ended math problems which help my students to develop a more meaningful understanding of math. Effective, open-ended math problems help students to see math as sensible, useful, and doable! These problems should be grounded in engaging and relevant scenarios which help students to understand how math is useful and applicable to their own lives. Effective, open-ended math problems should also have a wide base and a high ceiling. This means that students at all levels of understanding should be able to engage with the problem. Students who are not as comfortable with the mathematical concept involved in the problem should still see the problem as challenging but doable and should be able to get started while students who are performing at a higher level of math should have opportunities to extend the problem in order to further challenge themselves. Effective, open-ended math problems should also involve more than one possible answer and more than one method of solving the problem. As a math teacher, I need to respect the diversity of thinking that occurs in my classroom. It is not fair or right for me to expect my students to all solve a problem using the same algorithm. Rather, I should create problems where students have the opportunity to use different algorithms to find the solution so that they understand that there is no single correct answer or single correct method of solving math problems. 

The Role of Facilitating 

This course has also taught me that direct instruction is not a very effective method of teaching mathematics. As a math teacher, my role is not to teach students through direct instruction what the equations or steps are for solving math problems. This will not create a positive learning environment or help my students to be engaged learners who develop a meaningful understanding of math. Rather, my role is to act as a facilitator in an environment where my students are actively involved in their learning by exploring, questioning, taking risks, and discovering as they build and share their understanding of mathematical concepts and ideas. This will create a much more positive and engaging learning environment where my students feel confident, empowered, and excited to learn about math! Rather than me directly instructing my students, my students need to have the opportunity to take ownership of their own learning! My role is to guide and support my students by asking effective questions which promote inquiry and thinking and encourage students to build new understandings and connections and to communicate their thoughts and ideas. The math congress which Marian Small discusses in Chapter 4 of Making Math Meaningful and which we experienced in class is a great example of how teachers can facilitate discussions and empower students to take ownership of their learning as students explain their work, ask questions, and share feedback. 

Overall, this math course has been an incredibly insightful experience which has taught me to see math and the instruction of math in a new, much more positive light. Over the past twelve weeks, my fear of teaching math has been transformed into eagerness and excitement as I now feel like I have a much better understanding of how to effectively teach math and how to create a positive experience for my students when they learn math. I have learned to embrace math with an open mind. I have learned to let go of the one-method, one-answer mentality of solving math problems and to instead embrace alternative algorithms and open-ended math problems. I have learned to let go of worksheets that involve rote questions and to instead embrace engaging, relevant math problems and creative math activities that encourage students to explore, take risks, and discover as they actively build their knowledge and understanding. I am eager to share this newfound enthusiasm and open-mindedness with my students and to continue to develop and build on my understanding of how to effectively teach math in an engaging and meaningful way! 

Exploring Data & Probability

Have you ever thought about how often you make estimations or determine the probability of an event in your everyday life? Whether it's estimating how long it usually takes to drive to work in the morning or determining the likelihood of our favourite sports team beating their rival in an upcoming game, we are constantly involved with measurements and estimations of central tendencies and probability. Given the fact that data and probability are so relevant to our everyday lives, it is critical that we as educators create a learning environment where our students have the opportunity to explore and develop a more meaningful understanding of these topics.

Making the Measures of Central Tendency Meaningful

In her discussion of the measures of central tendency in Chapter 21 of Making Math Meaningful, Marian Small not only describes what mean, median, and more are but also when each of these measures are useful. When teaching my own students, I would definitely take Small's approach of teaching not only the meaning of the three measures of central tendency but also their usefulness with regards to different scenarios and different sets of data. 

In order to understand what the mean or average of a data set is, my students and I could talk about what it would mean to make everything fair or to level the bars. In class, we were given word problems where students received an unequal number of items. We were challenged to use Cube-A-Links to show how we could split these items evenly among the group of students who sharing them. 

Olij, B. © 2016

Olij, B. © 2016

Marian Small discusses how the mean is useful when the numbers in the data set are fairly close together but that the mean may not be the best indicator of the "average" group size when there are one or two extremely high or extremely low values. In order to demonstrate this concept to students, I could ask them to level bars where there was a considerable difference in the number of Cube-A-Links each bar contained. As students leveled the bars, they would be able to visually see and physically experience how the size of the leveled bars (the average) did not accurately represent the initial sizes of the various bars in the data set. 

I could then have my students explore the concept of a median in order to understand how the median is a valuable measurement when the data contains one or two extremely high or low values. Students could interact with several different data sets and compare the median and mean for each of these sets in order to see how the median can sometimes provide a more accurate reflection of the data. 

When discussing when the mode is a useful measure of central tendency, we could use the example of a glove factory. In order to know what size glove the factory should focus on manufacturing, the glove designers could look at which size glove is most frequently bought in order to determine the most common glove size. In order for our students to understand the measures of central tendency and when it is appropriate to use them, it is important to explore and discuss examples of these measures in real-life, engaging scenarios. 

Exploring Probability Using TinkerPlots

During class, we also had the opportunity to explore TinkerPlots which is a data analysis software where students can manipulate a set of data in order to create graphs and other representations which compare different properties of the data. In Chapter 22 of Making Math Meaningful, Marian Small discusses the importance of using a large set of data in order to more accurately determine the trends or the probability of an outcome or event. TinkerPlots is a particularly valuable resource for the classroom as it offers data cards which contain large sets of data for students to interact with. TinkerPlots is also very beneficial for visual and tactile learners as it is very easy for students to sort and manipulate the colour-coded data by clicking and dragging different points on the plot to explore different connections between various properties and to create colourful graphs. At the start of the activity, students could develop a number of "I wonder if..." statements related to the data set. Students could then explore whether their hypotheses were true by sorting and analyzing the data in order to determine what the trends and connections are among the various properties of the data set. TinkerPlots is a fantastic resource for encouraging students to develop a more meaningful understanding of data and probability as they make sense of real data and recognize trends in an interactive, visual, and tactile way. 

Learn Troop. 2014, December 27. "TinkerPlot Basics."
Retrieved from https://www.youtube.com/watch?v=wPFfIurEnUg

Improving Student Learning: Cooperation, Technology, and Assessment

This week we explored several different topics in math: cooperative games, the use of technology in student activities, and valid, reliable, and meaningful assessment. While these three topics may seem very different, they share one common goal: improving student learning.

Cooperative Games

We began the class by participating in some cooperative learning activities. There were three different types of activities: building stick figures using toothpicks, solving a number puzzle using a hundred chart, and building a structure using Cube-A-Links. At each of these stations, each group remember received one hint or clue that would help to solve the puzzle. Each person was responsible for reading their clue out loud as no one else was allowed to see or read it. As a team, we pieced our clues together in order to solve the puzzle at the station.

Olij, B. © 2016

I loved participating in these cooperative group activities! I particularly liked that each group member had their own clue which they were responsible for. I find that if group activities are not carefully planned, it is easy for one or two students to dominate the conversation while the more timid or less confident students stay silent and don't really participate. By giving everyone a clue and creating a requirement where each student is responsible for reading their clue, every student is contributing and every student's voice is being heard. Every member of the group feels needed and valued as the students work together as a team to solve the puzzle. Cooperative games are a great way to promote a positive and engaging learning environment! 

Student Activities Involving Technology

This week's Learning Activity Presentation's focused on how we can incorporate technology into our lesson plans in order to enhance student learning. I chose to use the Chocomatic Gizmo from Explore Learning to develop an activity where students created rectangles which shared a common area but had different lengths and widths. In order to put this problem-solving activity into context, I created a scenario where students were "Chief of Chocolate" at the Chocomatic Gizmo Company and thus they had the responsibility to develop a chocolate bar collection where each chocolate bar in the collection shared the same area but had different dimensions. In order to ensure the activity had a wide base and a high ceiling, I allowed students to choose the number of squares in their collection. Students who struggle with math could choose a friendly number that they were comfortable with while other students could challenge themselves by choosing a larger or more complex number.  

© Gizmos. Retrieved from www.explorelearning.com.

The Chocomatic Gizmo is a great resource for enhancing student learning as it encourages students to take risks, explore new ideas, and make connections. It allows students to represent new knowledge in a non-linguistic format, use manipulatives to explore new concepts and put them into practice, and generate and test hypotheses. This inquiry-based, student-centred approach is important for helping students to develop a deeper understanding of math. 

Assessment 

As teachers, we also need to ensure that our assessment serves to improve student learning. In Chapter 3 of Making Math Meaningful, Marian Small discusses the characteristics of good assessment. One of the characteristics that stood out to me was that our assessment should be "useful in assisting students to assess their own learning" (p. 38). It is important that our students have opportunities for self-assessment so that they can take initiative to reflect on their learning and develop strategies for how to improve in the future. When students have a clear understanding of the learning goals and success criteria and are active, engaged, and critical assessors, deep and meaningful learning happens! 

Exploring Measurement with a Growth Mindset

This week’s math course focused on measurement. As Marian Small discusses in Chapter 19 of Making Math Meaningful, measurement is something that children are naturally curious about. Children are interested in finding out how big or small, heavy or light, or hot or cold things are. As teachers, we need to tap into this curiosity and to develop fun and challenging activities that encourage our students to explore measurement and to develop a deeper, more meaningful understanding of the topic.

The Importance of a Growth Mindset

In class, we were assigned a word problem where we were challenged to come up with two rectangles which had the same perimeter but areas that differed by six units. I struggled to find the solution for this problem! While my partner and I were able to find rectangles that had the same perimeter but different areas, we did not find two areas that were different by 6 units. While it was frustrating not being able to find a solution on my own, at the end of the activity I still felt like it was a valuable learning experience. I was able to learn from my peers when they shared their solutions and I was also able to practice calculating the area of shapes and creating shapes that have the same perimeter but a different area.  

This experience was also a valuable reminder about the importance of a growth mindset. As teachers, we need to ensure that our students understand that struggling with math concepts and questions and working hard to solve problems will be rewarded. We need to ensure that we are not creating an environment where students think that those who finish solving problems or answering questions first are smarter or better than those who take more time. Our students need to understand that spending time investigating math concepts, working with manipulatives, and discussing math problems and ideas are all a critical part of developing a deeper, more meaningful understanding of math. I want to create a learning environment where my students truly believe that struggling with math and making mistakes are a natural and valuable part of the learning process.

© Big Change. Retrieved from http://big-change.org/growth-mindset/.

Exploring the Relationships between Different Shapes in Measurement

During class, we also spent a considerable amount of time working through a problem that involved estimating and measuring the circumference, radius, diameter, and surface area of circular objects and converting various metric units of area. This activity was contextualized in an engaging scenario where we were members of a design team whose task was to determine how many decorative tubes could be made from one large sheet of steel. In order to work through this activity, we used toilet paper rolls as a cardboard model of the tubes and string or tape measures to help us measure the various dimensions of the tube.

I found this activity particularly valuable as it encourages students to understand how different shapes are related in terms of their measurements. When we cut our cylindrical tube and flattened it out to make a 2-D shape, we discovered that it was a rectangle and we were able to see how the length of the rectangle’s base is the same as the circumference of the circle and that the rectangle’s height is the same as the height of the circle. As we worked through this activity, I thought about how empowering this activity would be for students! If a teacher were to simply explain to students the relationship between a rectangle and a cylinder through direct instruction, many students would likely feel disengaged or confused. In sharp contrast, this activity encourages students to take more ownership of their learning as they actively investigate and explore the relationships between shapes using various manipulatives and discussing their ideas with their peers. What a great experience for our students!

Olij, B. © 2016

Another relationship that students could investigate and explore with regards to measurement is the connection between the area of a circle and the area of a parallelogram. As Marian Small discusses in Chapter 19 of Making Math Meaningful, and as Christian mentioned in his Learning Activity Presentation, the sectors of a circle can be arranged so that they form an “almost” parallelogram:

Retrieved from Making Math Meaningful to Canadian Students, K-8: Third Edition, p. 501.

I was amazed when I read this section of the chapter as I had never seen this explanation before! This visual deconstruction helped me to better understand why we use the radius when calculating the area of a circle. As a teacher, I would love to develop a word problem or activity that created an opportunity for my students to work with fraction circles to transform a circle into a parallelogram. By actively exploring these kinds of connections, students can develop a better, more meaningful understanding of why we use the formulas we do when calculating the measurements of various shapes.

Encouraging Hands-On Learning and Asking Questions

For this week's class, we had the opportunity to explore geometry and spatial sense and to further reflect on how to create an effective, engaging learning environment for our students.

Encouraging Hands-On Learning

One of the big takeaways from this week was the importance of incorporating hands-on learning in math class, especially for geometry. In Chapter 17 of Making Math Meaningful, Marian Small discusses the Van Hiele Taxonomy of Geometric Thought which contends that a child's spatial experience is critical in developing their geometric thinking. In order for students to develop spatial abilities and a strong understanding of shapes and their properties, they need opportunities to physically interact with shapes. I can certainly relate to this research. When I can touch the shape's faces, edges, or vertices or rotate the shape in my hands to see it from different angles, I have a much better understanding of the shape's properties. 

One way to provide students with these important spatial experiences is through the use of tangram squares. Students can combine different tangram squares to create a variety of shapes such as triangles, squares, trapezoids, parallelograms, and pentagons. This activity of dissecting and combing shapes can help students to gain a better understanding of the properties of shapes. For example, students might discover that a parallelogram can dissect into two congruent triangles. This might be helpful in the future when they need to calculate the area of a parallelogram. 

Olij, B. © 2016

During the learning activity presentation, Lianne introduced another way to interact with shapes. This time, we created 3-D shapes by using jujubes and toothpicks. This hands-on activity is particularly helpful for students to gain a better understanding of what a shape's edges (the toothpicks) and vertexes are (the jujubes). 

Olij, B. © 2016

Another example of how to provide students with spatial experiences came from Marian Small's Making Math Meaningful. Small suggests using pattern blocks to allow students to sort shapes based on their common properties. This can help students to understand how the different shapes are related to one another. 

Retrieved from Making Math Meaningful to Canadian Students, K-8: Third Edition, p. 399.

Asking Effective Questions 

This week I was also reminded of the importance of asking my students questions. I had an "ah-ha" moment during class when Pat discussed how asking students questions is a way of showing our students that we have faith in them. I had never thought about the importance of questioning in that way before, but it is so true! Students feel confident and empowered when they take responsibility for their learning. I found the Capability Building Series document entitled "Asking Effective Questions" to be very insightful. As the document discusses, asking questions encourages students to actively create their knowledge as they build new understandings and connections. Not only do teachers need to ask questions, but they need to ask effective questions that promote inquiry and thinking. 

While all eight of the tips that the document offered for asking effective questions were insightful, there were three tips that really stood out to me. The first is to "pose questions that actually need to be answered." While this may seems obvious, I know that I can fall into the habit of asking students rhetorical questions. This is not very helpful as it simply provides students with the answer and doesn't allow them to engage in their own reasoning. Another helpful tip is to "keep questions neutral" by avoiding qualifiers such as easy or hard as this can intimidate or discourage students. As a teacher, I need to choose my words carefully. The last tip that really stood out to me is to "provide wait time." When time is short and it feels like there is a lot to get done, it can be easy to rush the students. By allowing a wait of even just three seconds, this will likely result in a better quality and quantity of responses. Many students need time to digest information and to formulate their thoughts or words; it is important that I give them time to clarify and articulate their thinking. 

Exploring Patterns and Algebra

When you were a student in elementary school, did you think that patterns and algebra were two separate units in math that never really crossed paths? This is the misconception that I had as a student in elementary school. This week's class gave me the opportunity to delve into the topic of patterns and algebra and to discover some useful teaching strategies and resources to incorporate in my future math class.


Building Connections

We began our exploration of patterns and algebra with a matching exercise where several patterns were demonstrated in four different ways: a table of values, a graph, an equation, and a stage-by-stage block diagram. Our task was to group the four different illustrations which represented the same pattern. I found this task to be very helpful in demonstrating how patterns and algebra are so closely connected. As we collaborated to discuss how an equation, table of values, graph, and block diagram were linked together, I was able to understand how an algebraic equation forms from generalizing patterns to create a bigger picture of the relationship. This activity reminded me of the importance of encouraging my students to discover the connections between different areas of math. From my own experiences and observations, I think it is easy for students to get so caught up in the minute details that they lose sight of the big picture of how math concepts are connected. I find that when students are able to see the bigger picture, they are able to develop a deeper, more meaningful understanding of the math concept they are learning.

Olij, B. © 2016

Teachers as Facilitators

This week we were also able to gain a better understanding of the value of facilitation as one of our group members took on the role of facilitator during our matching activity. The second video of The Three Part Lesson in Mathematics describes how teachers can serve as facilitators in the classroom. The role of the facilitator is to ask questions that encourage students to make connections, make predictions, justify their answers, debate ideas, and explain their reasoning. The video provided some examples of open-ended questions that a facilitator might use such as "How did you do this?" or "How do you know...?" or "How else might you solve this?" Rather than give students the necessary information through direct instruction, the facilitator is there to guide and support students as they discover and explore the key concepts. This was another important reminder for me that as a teacher I need to create a learning environment where my students are active learners. It is not very beneficial for my students if I simply give students the information to memorize through rote learning. Rather, I want to facilitate a collaborative learning environment where my students take ownership of their learning as they explore, question, and share their ideas.


Gizmos 

Another helpful resource that allows students to explore patterns and algebra is the Function Machine found on Gizmos. In this app, students can create a table of values by dropping different numbers into a function machine. They can then look at the table of values to determine what the function or expression of the machine is. I would definitely encourage my students to use this app to become more familiar and comfortable with identifying patterns and forming equations as it allows students to experiment with patterns and equations in a fun, interactive, and engaging way.

Screenshot taken from the "Function Machines 1" app on Gizmos
https://www.explorelearning.com/ 

Math Congress: A Strategy to Promote Math Talk

This week we had the opportunity to participate in a math congress. This experience reminded me of the importance of encouraging student discourse in math class. I want to create a learning environment that encourages rich dialogue, sharing, and thinking so that my students can develop a deeper, more meaningful understanding of the concepts they are exploring.

Math Congress: What is it?

In Chapter 4 of Making Math Meaningful, Marian Small discusses the lesson strategy of a math congress. The goal of a math congress is to debrief the various strategies or big concepts that students use to solve a math problem. Before the actual congress takes place, students work in pairs or small groups to solve a math problem and write their solution on a chart-sized poster. Each group then holds a mini-congress with another group in order to share their work, check their answers, explain their thinking, and ask questions and give feedback about their peers’ strategies. For each group, one student is chosen as the leader. The leader’s responsibility is to facilitate the discussion and to ensure that every member of their group has a clear understanding of the strategies used. Once the mini-congresses have taken place, the whole class gathers for a large congress. In this session, the teacher strategically selects two or three groups to present their solutions and explain their thinking. The teacher chooses a variety of strategies that encourage students to see the key concepts, make mathematical generalizations, and understand the reasoning behind various mathematical strategies. Thus, the final congress session helps students to consolidate the information that has been presented throughout the various steps of the math congress.

Seeing Different Strategies of Problem-Solving

As we participated in the math congress, I was truly amazed to see all the different strategies or approaches to solving the math problem we were given about determining which store had the better deal on cat food. When we started the problem, I honestly thought there would only be two or three different ways to solve the problem. To my amazement, we saw at least six different ways of doing the math problem! One method involved creating fractions and finding a common denominator in order to compare the two fractions. Some groups did this by finding equivalent fractions that had a larger common denominator while other groups did this by reducing the fractions so that they had a smaller common denominator. Some students chose to found a common number of cans while other students chose to find a common dollar amount. Another method involved physically drawing the money out to determine how much money each can cost. Another method involved creating a ratio chart. It was an empowering and eye-opening experience that reminded me that I need to reinforce in my students the idea that there is never only one right method of solving a math problem.

Olij, B. © 2016

Olij, B. © 2016

Olij, B. © 2016

Olij, B. © 2016

Promoting Discourse and Deeper Understanding

I love the idea of incorporating a math congress into my math lessons! As a student in elementary school and high school, I had the tendency to stick to the strategy that worked for me without making a conscious effort to consider alternative methods. I also struggled with explaining my math reasoning; while I knew what I did, I did not always know why I did it. Students who participate in a math congress are not only presented with various strategies, they also have the opportunity to ask questions, seek clarifications, and provide feedback about these strategies. The sharing, reasoning, questioning, and explaining involved in a math congress promote an incredible amount of math talk! I love that the group leader has the responsibility to ensure that every member of their group has a clear understanding of the strategies used. This means that every student leaves the math congress having learned at least one other method of approaching a math problem. Every student leaves the congress feeling empowered as they understand not only every way a solution was found but also the reasoning and mathematical concepts behind these solutions. 

Keeping Fractions and Ratios Fun and Feasible

When you were a student, did you dread math problems that involved fractions? Did you find math operations with fractions frustrating or intimidating? If you did, you certainly were not alone! This week’s math class demonstrated how students should not need to fear fractions and how we as teachers can helps our students to see fractions and ratios as fun and doable!

Incorporating Stories and Manipulatives

In class, we heard a story about a man named Mr. Tan whose most treasured tile shattered into seven pieces. Mr. Tan was able to piece the tile back together, and in the process discovered many different interesting shapes. At the end of the story, each of us was given seven tangram pieces which represented Mr. Tan’s tile. We were given the opportunity to explore different shapes using the tiles and to see whether we could piece the square tile back together like Mr. Tan did. We were able to combine different tangram pieces and determine their relation to the whole tangram square.

Exploring different shapes.Olij, B. © 2016

Recreating Mr. Tan's tile.
Olij, B. ©2016

I love the idea of teaching math lessons through the use of stories and corresponding manipulatives! Often times students have the misconception that math is boring or distant from their own daily lives. Stories encourage students to use their imaginations and explore the world of math. As a student, my highlight of the day was when the teacher read me a story. In my placement, I see that my students have the same joy and excitement when listening to stories. Why not tap into this love of literature and integrate stories and math to help bring math to life for our students?

Introducing a “New” Way to Divide Fractions

The moment that stands out to me most from this week’s class was when we looked at an alternative way to divide fractions. The traditional method of dividing fractions, the method that I learned as a student in elementary school, is to multiply the first fraction by the reciprocal of the second fraction. While I was able to use this method as a student, I distinctly remember that I never understood why we had to invert the second fraction and change the division sign to a multiplication sign. I simply did what I was taught without questioning the logic behind it.

This week, I was truly amazed to learn that students do not have to use the algorithm of inverting and multiplying in order to divide fractions. A simpler, more logical method is to simply divide the first nominator by the second nominator and to divide the first denominator by the second denominator:

This method follows the typical rules for division and is therefore more natural and far less intimidating for students. I cannot wait to show this alternative method to my students! 

Using Relatable Examples

This week I was also reminded of the importance of using relevant examples that students will find engaging. For example, we started off the class working through a math problem that incorporated the game “Red Light Green Light.” The resource Paying Attention to Proportional Reasoning provides examples of various math problems that involve dogs, bicycles, and baseball cards. These examples demonstrate that, as a teacher, it is important that I tune into the interests and experiences of my students and incorporate them into the explanations and word problems that I develop. This will help my students to see math as fun, engaging, and applicable to their daily lives.  

The Value of Manipulatives

The week’s math class, which focused primarily on fractions, demonstrated the value of manipulatives in learning. Manipulatives allow students to think and reason in more meaningful ways as students view and physically interact with physical objects that represent abstract mathematical ideas. Chapter 1 of Making Math Meaningful to Canadian Students, K-8 (Third Edition) discusses how research since the mid-1960s has shown that “the use of manipulative materials – concrete representations of mathematical ideas – is powerful in developing mathematical understanding” (pg. 4). In this blog post, I would like to focus on three reasons why manipulatives are powerful learning tools.

Manipulatives Make Abstract Concepts Concrete

Manipulatives allow students to interact with physical objects that illustrate abstract mathematical concepts. By viewing and manipulating physical objects, students are able to connect abstract mathematical ideas to the real world. Manipulatives help students to understand how math relates to their personal lives. For example, in class we read The Hershey’s Milk Chocolate Bar Fractions Book by Jerry Polatta which helps students to discover and understand equivalent fractions.  As the teacher reads the book, students use their own chocolate bars  as a manipulative by breaking  the chocolate bar into pieces and organizing those pieces into various groups to create different fractions. Thus, students are able to visually see and physically create a fraction such as one-quarter or ten-twelfths in order to understand what these fractions really mean. Chocolate bars are also a fun (and delicious!) manipulative as they help students to see how fractions are found in their daily life when they divide and share food.

Image retrieved from www.secondgrademathematics.weebly.com/books3.html.

Olij, B. ©2016

Manipulatives Allow Students to Actively Explore

It is important to recognize that children are active learners. Manipulatives allow students to explore math concepts by actively engaging with concrete objects. Manipulatives encourage students to take risks, ask questions, test their ideas, and discover new knowledge about math processes and procedures as they experiment with physical objects. From my own personal experience, I have noticed that I am much more willing to take risks and experiment with different ideas when I am working with manipulatives. In elementary school, I loved to work with manipulatives such as Cuisenaire rods, tangrams, and fraction circles as it was easy to move them around and try different things. I found this easier and far less intimidating than trying to test something on paper.

Manipulatives Encourage Math Talk

Manipulatives are also valuable as they encourage collaboration and discussion. During my experiences as a volunteer in a Grade 3 classroom, I noticed that many student struggled  to discuss the more abstract ideas of math as they felt pressured to use the precise terminology. They were, however, able to verbalize their thinking and discuss mathematical procedures and processes when they described what they did with their manipulatives. Similarly, the teacher was able to promote math talk when she asked students to prove a math idea by using their manipulatives. Given the fact that each student might experiment with different manipulatives, there can be valuable discussions as students share their ideas and findings. For example, one student might use the manipulative of paper clocks to add fractions while another student might use fraction strips. These students could compare and discuss their strategies and results.

This week, therefore, I was reminded of the importance of manipulatives. Before this course, I had the misconception that manipulatives were only helpful for students who struggled with math. I now see that manipulatives are a fantastic resource for all students in the classroom as they encourage students to connect math to their daily life, explore new ideas, and verbalize their thinking. 

Math: Sensible, Useful, and Doable!

“When am I ever going to use this in real life?” “This is too hard!” “This doesn’t make any sense… it’s just a jumble of numbers!” “I don’t know how to do this the right way.” These are typical phrases that math teachers might hear from frustrated and disengaged students. This week in class, we briefly discussed the notion that students need to see math as sensible, useful, and doable. In this blog post, I would like to further reflect on the meaning of these three adjectives.

Sensible

Simply put, math needs to make sense to students. In her video entitled “Brain Crossing,” Jo Boaler discusses how students learn math most powerfully when they both think about numbers as symbols and visualize and draw those numbers as this causes different pathways in the brain to cross and connect. As a teacher, it is important for me to understand how students learn. If I know that students find math to be more sensible when they can visualize the numbers, I need to ensure I am incorporating resources that help students to visualize math (e.g. diagrams, manipulatives, and pictures).

Boaler, Jo. [S Lamb] 2015, August 25. "Day 2 - Brain Crossing" 

Retrieved from https://youtu.be/qZBjub36Bvs

Useful

In order to be engaged and to understand why math matters, students need to see how math connects to their own lives in the real world. As a teacher, I need to ensure I provide students with the opportunity to work with relevant math problems that pertain to the real world. Rather than always give the students questions involving strings of numbers (e.g. “16 x 3 = ?”), I can develop a word problem that is relevant to my students’ lives. For example, I could give the following problem: “Thomas is at the toy store and he wants to buy 16 Pokémon card packs. Each card pack costs $3. How many dollars would Thomas need to have in order to buy 16 card packs?” Thus, as a teacher, it is important for me to create problems and lesson plans that show students how math is useful in their lives.

Doable

It is also critical that students feel that they are capable of doing math. As a teacher, I cannot teach my students one way of doing math and expect them to only ever follow that one method. That is not fair to my students. Rather, I need to respect the fact that there are several different ways of carrying out a math problem and that these alternative algorithms are just as acceptable as the algorithm I personally might choose. This is not something that I personally experienced as a student in elementary school or high school. If I did not know how to do a math problem the way it was taught in class, I felt helpless because I had the mindset that there was only one correct method. Therefore, I understand the need to instill in my students the confidence that there are many different ways of making math doable and that they are not limited to one particular method. One algorithm might make more sense to a particular student or one algorithm might be more helpful for a certain set of numbers. Thus, it is helpful for students to see a variety of methods so they can find the methods that work for them. For example, here are just three examples of different methods we could use to solve an addition equation:

Olij, B. ©2016

Thus, this week’s class has inspired me to facilitate a learning environment where students see math as sensible, useful, and doable!

Embracing Open-Mindedness in Math

The first two classes of Junior/Intermediate Mathematics have come and gone and a common theme that has resonated with me is the importance of being open-minded and flexible. I find that often it is easy to stick to what you know or to what you believe has worked for you in the past.  It is easy to become complacent or set in your ways. It is critical, however, that teachers be open-minded to new ideas and strategies even if these new approaches push them outside of their comfort zone. Today, math is moving away from the rigid, one-way and one-answer approach that I experienced in elementary school and high school to a much more flexible and engaging approach that encourages inquiry, discovery, and creativity. While this new approach may seem intimidating at first as it is not what many of us are used to, it is also incredibly exciting to see such a positive change!

I found Daniel Meyer’s TED Talk entitled "Math Class Needs a Makeover" (see embedded video below) that we watched during the first class quite powerful and inspiring. Meyer discussed how “What matters?” is the most underrated question in math. He emphasized the fact that rather than work with questions that simply feed them the exact information that they need, students need to work with math problems that cause them to ask questions and become truly engaged with the material. I wholeheartedly agree with this sentiment. As a math student, I do not remember ever feeling particularly engaged with math problems.  I would read over the question, pick out the key pieces of information, plug them into the formula to get the answer, and then move on to the next question. Looking back, it was almost a robotic process. I rarely asked the questions why or how. Open-ended problems, which typically have several correct answers and several ways to develop an answer, offer students a fantastic opportunity to more actively engage with math as they question, reason, and discover. Students are forced to ask themselves questions about what matters. For example, the open-ended question that we looked at in class was to provide students with an image of a room and ask them how many people can fit in the room. Students would need to ask questions such as: Are all the people the same size? Can people stand on one another’s shoulders? Can the furniture in the room be removed or re-arranged? Thus, students are no longer robotically plugging in numbers, but they are actively problem-solving. This use of logic and critical thinking is imperative for students to truly understand math and to understand why math matters.

Meyers, Daniel. [TED-Ed]. 2013, August 1. "Math Class Needs a Makover." 

Retrieved from https://www.youtube.com/watch?v=qocAoN4jNwc

As a teacher, I need to be open-minded not only to incorporating new types of math problems into my classroom, but also to adopting new methods of doing basic math functions. I was reminded of this when we looked at subtraction in class. When I learned subtraction in elementary school, I was taught that when a digit from the top number was not big enough, it would borrow from the column to its left. However, this concept of borrowing, and never giving the number back, is illogical, unnatural, and confusing for children. It is an algorithm that is meant for computers, not human beings. A far better method is to choose a number and add it to both the top and bottom number in order to make the top number a more friendly number for subtracting. Here is an example of this method:  

Olij, B. ©2016

As we learned this alternative method, I thought of the article “Toward a Practice-Based Theory of Mathematical Knowledge” by Ball and Bass. The articles talks about how math teachers need to have pedagogical content knowledge. In other words, they need to consider what mathematical representations and explanations children find logical, useful, and helpful. The algorithm of borrowing in subtraction is not very logical or child-friendly, and thus teachers must be willing to look at how a child understands and interacts with numbers in order to find new approaches that will help the child to truly understand math. Thus, this new subtraction method was an eye-opening moment for me. When Pat first introduced this method, my natural reaction was to say: “What’s wrong with the way I learned it? Why do we have to change?” Yet after seeing how unnatural and illogical the old method of subtraction was, I left the class asking myself: “When I was a student, why did I just accept that algorithm that didn’t really make any sense? Why didn’t I ask why we were borrowing digits that we never gave back?” I now look forward to using the new methods and approaches in my future classroom!

Thus, the past two classes have been a much-needed eye-opening experience for me. I have been challenged to let go of the old ways of learning math and to embrace with an open mind the engaging, creative, and more logical methods of teaching and learning math. I have been truly inspired to help my future students joyfully discover why math matters as they actively engage with math in meaningful ways.

Welcome!

Hello and welcome to my blog! My name is Belinda Olij. I am currently a Teacher Candidate in the Junior/Intermediate Consecutive Education Program at Brock University.

As you can see above, this blog is entitled Math Matters. I chose this title because I felt the play on words aptly represents what I hope to discuss in this blog. In one sense, I will be discussing different matters about math. I will be sharing different ideas I have learned in class relating to math and how to effectively teach math. In another sense, I also hope to discuss and demonstrate why math matters. So many math teachers hear unenthused students ask “When am I ever going to use this in real life?” or “Why does this even matter?”  While I personally did fairly well in math in school, I never had a concrete understanding of math and I struggled to really connect with what I was learning. I went through the motions of plugging numbers into formulas without understanding how it worked or why it mattered. My goal is to give my students a more positive math experience. I want my students to get excited about math as they discover how valuable math is and explore all the amazing things math has to offer. I want my students to see, experience, and understand why math does matter.

I am really looking forward to this math course! My goal is to learn a variety of strategies, methods, and activities that will help children of all learning abilities and styles to become engaged learners who enjoy developing their math skills and potential. Methods of teaching have changed considerably since I was a student in elementary school and I am excited to learn about the different resources and techniques that are being used today. I am excited to gain a better understanding of how children think about math and how we can help children to discover their potential and develop their skills and confidence.

I encourage you to keep visiting my page over the next few months as I explore why math matters!