Biographies:

Drs. Behrouz Aghevli, World Bank, and Mark Spikell, George Mason University have been collaborators on the invention, creation and dissemination of physical manipulatives since 1990. They have published or have in press four different physical manipulative products for the teaching of mathematics. And, they have completed all of the research and pilot testing for two additional ones. Their work has attracted considerable attention in a segment of the publishing world know as supplementary materials publishers for school and home markets. Dr. Spikell, is a Professor in the Graduate School of Education and is Coordinator of Math and Science Leadership Programs at the Ph.D and M.Ed. Levels. He has published more than 8 books and 20 articles in this area and is known nationally for his work with physical manipulatives. Dr. Aghevli is an inventor, educator and author. He works at the World Bank as a systems analyst and is an affiliate professor at George Mason University where he works with math education doctoral and master degree students interested in developing school curriculum materials. He has extensive experience working with children and teachers in school settings.

Manipulators are hand held electronic devices which create dynamic images of two and three dimensional geometric shapes. The shapes can be manipulated by slides, flips, turns, and scaling to solve animated puzzles and play action games. In the home market, manipulators are captivating devices for fun and recreation. In the school market, manipulators are fascinating devices for teaching topics in problem solving, geometry, algebra, and pre-algebra mathematics.

In K-12 schools, it has long been advocated by many that traditional teaching in mathematics leads to the memorization of concepts and skills but not to understanding and mastery, except for a relatively small percentage of students.

Traditional teaching in this context is teacher centered instruction in which teachers do a lot of talking and telling while students do a lot of passive listening and memorizing. This type of teaching is best characterized by the phrase, the teacher is the sage on the stage. Teachers define terms, give directions, explain problems, answer questions, and otherwise present information to students.

In contrast, non-traditional teaching is student centered instruction in which teachers have a very different role, one best characterized by the phrase, the teacher is the guide on the side. In this method of instruction, teachers do very little talking and telling. Instead, they create an environment where students become active learners through hands-on activity with concrete objects, called manipulatives. In such an environment, students do lots of investigating, exploring, solving, discussing and explaining to their peers and to the teacher.

For those who believed in non-traditional teaching, the focus of learning has shifted from the role of the teacher to the role of the students. As a result, it became necessary to rethink how concepts and skills need to be presented to students. Lecturing and telling is out. Experimenting and discovering is in. That led to the now popular notion today that mathematical concepts and skills should be presented in a hands-on activity centered environment in which four sequenced stages are followed, if mastery understanding and learning is to occur, not just rote memorization. These stages are:

1. Concrete stage -- introduce concepts and skills through the manipulation of physical objects (called manipulatives) that embody the concepts and skills;
   
2. Pictorial stage -- reinforce concepts and skills by presenting visual representations of the manipulatives (and manipulation) that embody the concepts and skills;
   
3. Symbolic stage -- learn concepts and skills through reading, writing and expressing alpha-numeric representations of the concepts and skills;
   
4. Abstract stage -- internalize, master, understand, and apply the concepts and skills.
   

   Until now, most educators have viewed the pictorial stage as being the use of static pictures, drawings, images, etc. I have never supported this approach, but the limitations of doing other wise (providing dynamic images) were just too great. And, as I have suspected for years, this reliance on static images has proven to be inadequate and insufficient.

   Now, based on my work with Behrouz Aghevli, I am convinced that what has been needed in order for the non-traditional approach to instruction to be effective are not static visual representations of the manipulatives but, rather, dynamic ones. These dynamic visual representations (virtual manipulatives, if you will) are effectively computer generated visual versions of the actual concrete manipulatives. And, now that the technology exists to make manipulators possible, I see the way to develop and deliver the virtual manipulative and its supporting curriculum materials as companion products to accompany the concrete manipulative and its supporting curriculum materials.

   Behrouz and I are now at the stage where we are ready to build on the existing concrete manipulatives we have developed (and others too) and to proceed to the actual development of the virtual manipulatives in the form of devices we call manipulators.

   It is our goal to have teachers and students in schools use both our manipulatives and manipulators, thus providing the first two necessary stages for the eventual mastery of math concepts and skills. It is in these critical first two stages that teachers are less well prepared themselves and therefore, more ready and willing to accept help from the outside in the form of concrete manipulatives, technology, and supporting resource material.

   We believe the symbolic and abstract stages are the ones most teachers and text books are well equipped to provide. And, we believe these latter stages should be left in their capable hands, with minimal intrusion by or from the providers of manipulatives or technology.