When students sit on the couch in the office of Head of Lower School Lainie Schuster, their fingers are often drawn to the brightly colored acrylic shapes scattered on the coffee table. They flip, turn, and swap out the geometric pieces as they try to make them fit perfectly into the wooden form of an animal. Some students solve the challenge with surprising ease, while others struggle with the number of variables.
These pentomino puzzles, which have been created by Fay’s fourth grade math students, are more than just colorful games. They are an exploration of congruence and spatial relationships, two essential underpinnings of the fourth grade study of geometry.
Building pentomino puzzles is one of the rich mathematical tasks that help third and fourth graders explore, reason, problem-solve, communicate, and reflect upon the math concepts they are studying. These projects require no memorized rules and
are open-ended in answer and approach. “Some students who are number-crunchers can find these tasks difficult,” says Lainie, who also teaches fourth grade math. “On the other hand, students who are visual thinkers may find these tasks easy but struggle with multiplication and long division,” she says. “What I love about these projects is that kids begin to understand that this is mathematics, too!”
“In order for a child to understand something, he must construct it himself, he must re-invent it.” – Jean Piaget
Pentominoes, which are sets of twelve polygon pieces each composed of five equal-sized squares, have long been a way for Fay’s fourth grade math students to explore spatial reasoning. The pieces can be arranged in multiple ways to form a rectangle, an idea that is mind-boggling for students used to math problems that have one correct answer. Last year, Lainie worked with Design, Technology, and Innovation Teacher Allison Bishop to put a new spin on the pentomino unit by having each student create his or her own pentomino puzzle.
The first step in the project was for each student to create an animal shape of their choosing out of pentominoes. Next, each student transferred the physical gestalt of the animal onto graph paper and drew it, making sure that each of the 12 pentomino pieces fit within the sketch. Students then took their paper pentomino sketches to the Digital Literacy Lab and transferred their designs into 2D Design, a software program that enables students to design on screen and send their work to the laser cutter.
Transferring the design from paper to screen was the trickiest step for many students as they suddenly discovered that their pentomino pieces didn't fit, their giraffe’s neck was too long, or there were 57 squares in a design that should have 60! Allison explains that some students struggled with the concept that the pentomino structure, the paper sketch, and the digital design were all the same thing. “Eventually, students began to understand that they were just looking at the same information in three different ways,” she says.
The project also challenged students to develop a deeper understanding of some fundamental geometry concepts. Each pentomino animal, for example, has
an area of 60 squares, but the perimeter varies depending on the design. “The students gained a first-hand understanding of congruence, understanding that shapes are the same whether you flip them, turn them, or rotate them,” Lainie adds. “It’s one thing to do it physically, but it’s another to do it conceptually.”
“Learning is experience. Everything else is just information.” - Albert Einstein
In third grade, students use analog clocks to tell and write time to the nearest minute and to calculate elapsed time. They chunk time into 15 minute-increments and explore time conversions between second, minutes, and hours. Third grade math teacher Maura Oare admits that it’s not a unit that students find thrilling given the prevalence of digital clocks, but in addition to cementing a basic life skill, analog time-telling helps students to build number sense and practice their addition and subtraction strategies as they work on time problems.
As a hook to the unit, Maura and Allison work with each student to design and build their own working clock. Each student designs a clock face that holds particular meaning. This year, student designs included a football-shaped Superbowl clock, an asteroid hurtling toward earth, and even a clock shaped like a chicken! The only design constraints are that the clock can be no larger than 6”x6”, must have numbers and a space for hands in the center, and feature the student’s name somewhere on the clock face.
Using the 2D Design software, students created their clock designs and added numbers and tick marks around the edge for the minutes. The laser cutter cut out each shape and rendered the design features onto the clock face. The wooden clock faces were then brought back to the classroom, where students painted them, added hand-drawn designs, and set up an assembly line to fit each one with clock hands and motors.
In a forthcoming co-written article for the International Journal of Science and Mathematics Education entitled “Geometry and Design Thinking in the Elementary School,” Lainie and Allison explain that hands-on projects like the clocks and pentominoes offer an immersive math experience that challenges students’ problem solving skills and enriches their understanding of foundational concepts. Lainie and Allison note that the objects created in these projects “hold a story, and the story includes the design process, the pride of accomplishment, and the core subject learning objectives interwoven into the project.”
What is a meaningful mathematical task?
Activities like creating pentomino puzzles and clocks engage students in the study and application of mathematical topics in a way that enriches their understanding. According to the authors of Making Sense: Teaching and Learning Mathematics with Understanding (Hiebert et al. 1997) there are three features of meaningful mathematical tasks:
1. The tasks are problematic. Students view the problem as interesting. There is something to find out as a result of solving the problem and something to make sense of.
2. The tasks connect with where the students are. Students must be able to apply the knowledge and skill they already have to develop a method for completing the task or solving the problem.
3. The tasks engage students in thinking about important mathematics. Students must have the opportunity to reflect on important mathematical ideas and to take something of mathematical value away from the experience.