by Joanne Tuxbury
Joanne Tuxbury studied techniques of participant observation at an NSF funded project at Simmons College in Boston and has had a chance to undertake some of this work in her own school. Here are some of the results:
Manipulative materials for math are more and more common in the primary grades. But what happens when a child reaches the upper grades of 4th, 5th and 6th? Do many of the manipulative resources that are successful with six year olds also support learning among ten to twelve year olds? When do students begin to make abstractions? And do manipulatives help them to make the jump to abstract thinking?
I have had a chance to test some of my own impressions during the past three years as I made a significant change in my teaching assignment from grade 1 to grades 4 and 5. As a primary teacher, I had been part of a team that was involved in an intense exploration of multi-age teaching. When a new 4th grade position was created because of higher enrollment, I volunteered to teach at that level, knowing that I would be able to "loop" with that class, teaching them a second year as 5th graders.
My class consisted of eighteen students, ten of whom were former first grade students of mine. This meant that I had substantial data on that group of ten as six year olds, as well as new information about them as they progressed from 4th to 5th grade. I was very excited about this challenge, as it gave me an opportunity to be a teacher-researcher. I wanted to see how this age group learned best and what I could do to maximize their learning opportunities.
My early primary classroom had been well stocked with commercial hands-on materials and a myriad of collections of scrounged materials (boxes, containers, etc.). Students had used these materials for free play as well as for more structured learning. I was constantly amazed by the patterns and designs students made to demonstrate their learning or their creative moments. No student text was used in math. Instead, we used Math Their Way and curriculum ideas from Box It, Bag It. Students proceeded at their own pace and various assessment strategies were used to evaluate progress.
On switching to 4th grade, I was delighted and challenged to learn that there were not enough math texts for this additional classroom. The school was in the process of exploring new math curricula to see which one we would adopt, but that was at least a year away. In the meantime, I had the opportunity to use a variety of strategies and math manipulatives to teach concepts and skills for this grade level.
The scope and sequence of the text-based curriculum became my reference point and some of the skill sheets from that series were used as one part of our assessment. Teacher observations and student "real life" applications activities completed my assessment format. Many of these student activities were photographed.
Fourth grade student work
Students recorded daily thinking and problem solving tasks in their math journals. Often this reflected cooperative group work which became the norm for much of the students work in math. When groups were active, I moved about the classroom, listening to them verbalize their thinking. Frequently, I would offer suggestions or ask "stretch" questions when appropriate. [Such questions are intended to provoke further investigation and discussions, leading to the use or exploration of additional math concepts and tools.]
In addition to the manipulatives known to the students and to me from the primary grades, we added new resources. Chip Trading, Decimal Squares and Fraction Bars joined our classroom collection of manipulatives. Students created their own rules for Chip Trading as one way of demonstrating their learning. Building on some common activities in Chip Trading, such as "3:1 Country," students developed other ratios to test and to do chip trading around. One example was the development of a game of 7:1. This was a complex investigation of a base 7 numerical system, although we did not use the term, "base."
What became a delightful surprise to me was the use of materials that the students referred to as, "Our old friends." These were the manipulatives they recalled using in first grade. Pattern blocks, unifix cubes, junk boxes, tiles, geoboards, dice and dominoes were used during instruction time but also became the objects of choice during free time. Elaborate designs and patterns were scattered about the room. They were multi-dimensional and intricate.
The novel, Bridge to Terabithia, by Katherine Paterson, triggered the most elaborate designs of all. I had challenged students to create a fantasy world with rulers, subjects and laws. They had to write about this world as well as build a concrete representation of it. They could build it at home or at school. Those who chose to build at school used every "old friend" that was available. The boundaries and structures became so complex that we arranged the desks into a rectangle to protect the structures from destructive forces such as the janitors vacuum cleaner. Each was photographed to document the project.
Toward abstract thinking
This two year "looping" experience, teaching 4th and 5th grades, is one I hope to repeat again soon. Having students for two years is like having, "Old friends." I knew my students in much more complete ways. I marveled at their accomplishments. I had studied Piaget but never really had the opportunity to directly follow a group of students from the pre-operational stages of first graders to the beginnings of abstract thinking of some fifth graders.
It was clear to me that students made appropriate use of manipulatives at the grade and ability level that they had reached. They were comfortable with inventing new uses for materials that they had used in the past and they were eager to test manipulatives that they had not seen before. Because classroom manipulatives are not toys with a limited age span, they are adaptable to the needs and inventiveness of many ages. Manipulatives can be used to represent so many ideas and to give visual reinforcement to math concepts. It is a mistake to limit their use to relatively young children, thereby missing opportunities to give valuable, concrete experiences to upper elementary and middle school students. .
- Joanne Tuxbury, an educator in Sunapee, New Hampshire, has been teaching for over 25 years.