Marilyn Burns one of the nation's most eminent math educators discusses in articles and lessons how to teach mathematics from early childhood to high school. Burns is the author of mathematics books, articles, and related materials.
I remember the first time I introduced a manipulative to my class. At a workshop, I had learned how students could use geoboards to explore the areas of shapes, a standard textbook topic. I was excited and ready, but nervous.
I gave a geoboard to each student, distributed a cup of rubber bands to each pair, and gave time for exploration. Within a minute, chaos reigned. The cups were empty and every geoboard was full. Some students slouched in their chairs waiting for instructions. A few strummed the rubber bands as if the geoboard were a guitar. Several students, attempting to remove rubber bands from the geoboards, instead sent them flying. Others disappeared under their desks to retrieve lost rubber bands. This wasn't what I had envisioned.
Since then, I've changed how I set the stage for learning with materials. Here are some ways I've learned to introduce geoboards.
For Intermediate and Upper Grades
I drew a model on the chalkboard of a geoboard's 5-by-5 array and distributed one board to each student (no rubber bands yet). I asked, "How many pegs are there?" When I did this recently with fifth graders, I gave students time to think and then called on Sarah.
"Twenty-five," she answered.
I responded, "You're right. How did you figure that out?"
She replied, "I counted five in a row and there are five rows, so I did 5, 10,15, 20, 25."
"Did anyone figure it out a different way?" I asked.
Kyle answered, "I did 5 times 5."
I continued until all students had a chance to report their methods.
I made a shape on my geoboard using one rubber band and drew it on the chalkboard geoboard. This helped introduce students to the geoboard dot paper.
"Does anyone know this shape's name?" I asked. Several handsshot up, and Mike said, "Quadrilateral."
I then said, "Listen to my question and raise your hand when you figure out the answer: How many pegs does my rubber band touch? Count not only the corner pegs, but any peg that touches the rubber band." I waited until practically everyone's hand was raised.
"Let's answer together," I said.
"Five," students responded.
"And how many pegs are inside my shape, not touching any side?" Again they said the answer in unison. I repeated this for several shapes.
I gave students directions to come up with nine shapes with rubber bands touching different numbers of pegs and with different numbers of pegs inside. I also gave pairs of students geoboards, rubber bands, and geoboard dot paper. I said, "First make any shapes on the geoboard. Then construct shapes to match my descriptions. On the dot paper, record one shape that matches each description."
The atmosphere was much different than my first experience. Some children still strummed the rubber bands, but the overall feeling was one of productivity.
The next day, I asked pairs to create as many shapes as they could that touch four pegs with two pegs inside. Students recorded their shapes on the geoboard dot paper. Later, they exchanged papers and checked each other's shapes. Finally, I asked students to write the names of the shapes.
For Younger Children:
I give children time to explore with the geoboard. They enjoy this, and some need the practice putting the rubber bands around pegs. Then I say, "Remove the rubber bands on your geoboard and start again, this time making a shape of something that can fly."
When everyone has made at least one shape that flies, I show them how to transfer their shape to geoboard dot paper. I use dot paper with only two geoboards to a page so their drawings are large enough for display.
Finally, I have students post their recorded shapes — kites, rockets, butterflies — on a huge graph with a different column for each object. We discuss how many of each shape we have, how many more of one shape we have than another, and how shapes are alike and different.