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/