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.