“What is a Circle?”: Hands-On Problem Solving in Geometry

Theorems are the foundation of geometry. Being able to identify what theorem fits a particular geometric situation is the most challenging part of this subject. It’s not just about memorization; it’s about applying ideas to a particular problem.

Sometimes students can draw on their own prior knowledge, but some of the more complicated theorems require students to explore and be creative. As teacher Jaime Hall reminded her Accelerated Geometry class recently, “Math was created by philosophers. When you get to high level math you really start talking about philosophy. You enter into theoretical conversations.”

Grappling with the unknown in math is relatively new for the bright students in this accelerated course. It can be a struggle. Ms. Hall explains that when she asks students, “Set up what you know is true, now what conclusions can you draw,” it can be tough: “When I introduce a ‘what if’ or a ‘now what’ and compel students to move beyond one set idea or image, it gets really challenging. It’s a big leap in how they learn math.”

Ms. Hall and I teamed up earlier this month to find a way to help students make that leap. She chose to focus on the Circle unit, since students have struggled with this in the past. Ms. Hall realized that she needed to make learning more tangible and hands-on.

Together, we also took a leap by introducing her students to the “Investigating Angles in Circles” activity, which was inspired by a lesson shared via Mr. Reddy’s Maths Blog (“Circle Theorems and Hoola Hoops”). This hands-on, kinetic learning activity had students work in teams to build solutions to theorems out of circles and cords purchased from a craft store. Each team member had a role to play (Author, Photographer, Labeler, Measurer — and some students invented the role of “Analyzer” as well) as they moved from station to station creating a visual representations of the hypothesis of each theorem using the tools provided. Student teams photographed the labeled circle they created at each station to “prove” their solution, and later shared their data with their teacher via Google Dive.

While some students revelled in this student-driven learning environment, others struggled. Here are some of the conversations I observed:

  • S to JH: “Is this right?”
  • JH: “It’s not about ‘right.’ You should make mistakes. Look at the theorem. Tell me what you have here. What do you see?”
  • JH: “Is this quadrilateral inscribed?
  • JH: “Can you show me what you mean by that using the tools?”
  • S to S: “Do Your Job!”
  • S to JH: “We are waiting for you to validate our answer!”
  • JH: “I’m not going to validate your answer. This is about taking risks, using what you know to solve the theorems — even if it’s not perfect.”
  • S to S: “Wait, what is true about it? It’s asking ‘what is true’?”
  • S to S: “Look at these two triangles. They are proportional to each other: That’s it!”

To be sure, this “leap” was not easy for all learners. Here are comments from two students:

  • “I didn’t really like that we had to figure out the formulas for ourselves.”
  • “It was frustrating at times.”

Yet 85.5% of the students confirmed via an anonymous survey that they appreciated learning in a “different way:”

  • “It was fun because it was hands on and we got to move around the room. We also got to work in groups and share our ideas.”
  • “It was fun because we got to build/construct different things, which is nice because we get to visualize it.”
  • “I got to have fun while doing math.”
  • “It was more interactive and fun.”
  • “We do not usually physically create the problem, then derive the theorems from the thing that we created.”
  • “I like problems that I can see visually rather than on a piece of paper.”
  • “I appreciated the hands-on experience and ability to get up and move around. I also enjoyed the environment of the activity. I felt less pressure to understand right away since it was about discovering theorems as opposed to mastering them.”

What was the “Best Part” this activity for students?

  • “Walking around.”
  • “I liked building circles hands on.”
  • “It was fun collaborating with others to learn the new material.”
  • “The increased understanding.”
  • “Not having to sit for a solid hour.”
  • “Trying to figure out how to place the angels and measurements.”
  • “I think that the best part was seeing the different theorems instead of just having them described to us.”
  • “The selfies:).”
  • “All of it!”

In terms of the learning goals, Ms. Hall and her students would agree that the results were mixed. As she explains, “Coming out of this assignment, did they understand the theorems? I’m not so sure, but they do understand the vocabulary. That is stronger because of this activity.” Student comments reinforce this: “There is a lot more than can initially be applied to circles than I originally thought. Before, Circles seemed so abstract, but now they make sense.”

Along with learning about circles, students also took some time to reflect upon themselves as learners. Here’s what they had to say:

  • “I learned that I am able to form my own ideas without a teacher teaching them to me.”
  • “I am a strong visual learner who can reach conclusions through visuals.”
  • “I learned that I like being taught information instead of figuring it out for myself.”

Armed with the knowledge gained from this learning experiment, Ms. Hall and her students (well, most of them) are eager to tackle more hands-on activities as they continue to make the leap to more independent problem-solving in math.

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