By Thomas E. Hodges, Malisa Johnson, and Kyrsten Fandrich, Posted December 19, 2014 –   

In our previous blog post, we described the Cut-a-Card activity, which provides students with opportunities to visualize the effects of three-dimensional flips and rotations. Before giving students opportunities to create their own notecard puzzles, we engaged them in conversation. After students had time to re-create the initial design, we provided them with additional note cards to develop their own designs. Here are some of the puzzles our students created:

Each student’s design included the entire card and involved only straight cuts and rotations, a requirement of the puzzle design.

The Follow-up Conversation

After students had opportunities to re-create the initial puzzle and develop their own puzzle, we engaged them in a follow-up conversation about their strategy and design. The following partial dialogue shows how Malisa, the classroom teacher, chose to make students’ strategies public during the conversation:

Malisa: Let’s talk about re-creating the initial puzzle. How did you go about figuring out the design?

Ella: I started cutting and messing up, which used a lot of cards. So, I decided to stop and picture what was happening.

Malisa: So, you visualized in your mind what was happening. What did you think about?

Ella: Well, I tried to think about where the card was cut, and I tried to turn it back around in my mind so I could see what it looked like before it was turned.

Jamiel: Yeah, I tried to picture that it was all on the lined side of the card and what it would look like. And I looked at the ones that were messed up.

Malisa: And that helped?

Ella: Yeah. And it helped to slow down and think before keeping on cutting.

Malisa: So, taking your time, visualizing your design, and learning from the cards you already cut are some good strategies when figuring out the design.

We consider the ideas that students make sense of and take away at this point in the task to be important. Often students learn that getting correct answers and getting them fast are most important in mathematics. However, through tasks such as Cut-a-Card, students can see that the design process requires patience, learning from their mistakes, and mental anticipation for outcomes based on certain moves. Not only are these important conclusions for design but also for addressing the Standards for Mathematical Practice (SMPs)—namely, perseverance (SMP 1) and modeling (SMP 4). Students then went on to share their own notecard puzzles, discussing the moves they had made to create their designs.

By thoughtfully addressing task selection, we can simultaneously support students’ understanding of mathematics content while providing concrete images of what it means to learn and do mathematics. We have found that the Cut-a-Card task and related student designs deepen students’ understanding of three-dimensional models, symmetry, and rotations while promoting perseverance when solving challenging mathematics tasks. Following the Cut-a-Card task, teachers may pose subsequent tasks about orientation, reflections, and properties of geometric figures, including two- and three-dimensional shapes.

Your Turn

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	Thomas E. Hodges is an assistant professor of mathematics education at the University of South Carolina. He teaches field-based mathematics methods courses, capitalizing on opportunities for preservice teachers, teacher educators, classroom teachers, and elementary students to learn with and from one another. He published on the field-based design in NCTM’s 2014 Annual Perspectives in Mathematics Education and regularly contributes manuscripts to Teaching Children Mathematics, Mathematics Teaching in the Middle School, and Mathematics Teacher.
	Malisa Johnson teaches a self-contained fourth-grade class at Oak Pointe Elementary School in Irmo, South Carolina. In her thirteenth year of teaching, Johnson often hosts mathematics methods courses in her classroom and collaborates with university faculty and other classroom teachers on mathematics education publications. She is interested in productive discourse and students’ use of representations in mathematics classrooms.
	Kyrsten Fandrich is a Master of Arts in Teaching candidate at the University of South Carolina, completing her internship experience in Johnson’s classroom. She is interested in learning alongside her fourth graders through careful attention to students’ mathematical thinking.

 Archived Comments

I love the hands-on aspect of this task and the different visual representations of the transformations. Great activity.
Posted by: JaneW_27993 at 1/5/2015 7:27 AM