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.