Research Proposal (CI450 Fall,2000)

Report of the Independent Study
Fall 2000
Akihiko Takahashi

I have completed the research project entitled “AN INVESTIGATION OF
COMPUTER INSTANTIATED PROBLEM SOLVING (CIPS)” last semester. The purpose of this research project was to learn whether the CIM provided by the Internet can be used as an alternative to hands-on activities using concrete manipulatives in open-ended problem-solving. From the obtained results, it appears that the Computer Instantiated Problem-Solving (CIPS) may provide students with learning opportunities equivalent to those occurring with concrete manipulatives do. In conclusion, CIPS may contribute to mathematical reform in teaching and learning.
Although the results of the research project confirm a potential use of CIPS, the following issues are necessary to explore in order to apply the CIPS to mathematical lessons.
First, the CIPS of this research project applied only to small group activities and the experimenter did not play the role of a teacher. The experimenter did not teach students any mathematical concepts like teachers do; he just helped children to organize their group activities. Since open-ended problem-solving has been developed as an instructional strategy for classroom instruction through students’ collaborative work, the CIPS is expected to be applied in a classroom lesson, including teacher’s instruction, and to be examined in a classroom situation.
Second, although the CIPS activity is intended to establish the opportunity for students in transition from level 1 to level 2 in order to learn basic concepts to understand congruence based on van Hiele’s theory, the students’ activities have not been evaluated from a perspective of mathematical content. In order to help students to develop a basic concept of congruence, a series of lessons including the CIPS of this research project would be needed. Without going through a series of lessons, it is very difficult to evaluate students’ development of a basic concept.
Concerning the above two issues, the further research project is expected to focus more on application of CIPS in a classroom situation. The curriculum unit would consist of well-balanced activities including ones using concrete manipulatives and ones using CIM. This kind of research may enable educators to better understand how to introduce this technology into their classrooms.
In order to precede this research project, I have developed several CIPS for classroom activities through the independent study on the page on the web.
http://www.students.uiuc.edu/~takahash/MathActivities/
I have asked several American and Japanese elementary school teachers to use these problems in their classrooms and give me their feedbacks.
Addition to the above, I have updated my literature review on technology and mathematics education.

Technology and Mathematics Education

The use of technology in mathematics education is a major aspect of the reform movement because students can learn mathematics deeply if teachers use technology appropriately (National Research Council, 1989; NCTM, 1989, 1998, 2000). Although various computer software has been developed and although U.S. schools have started to bring computers in their classrooms, only a limited amount of software is available for mathematics lessons in both sufficient quantity and quality for classroom uses. Most of the commercially available software has been of the computer-assisted instruction ( CAI) drill and practice genre. Moreover, most of the studies about using software have been based on “relatively trivial content games,” which are intended for students to learn the syntactic transformation of a particular notation system (Kaput, 1992).
Among these studies, there is a study that looks at computer representation as a new form of manipulatives (Lesh, Post, & Behr, 1987). This case study compared computer-based manipulative and physical manipulative ‘Dienes blocks’. The major difference between physical manipulatives, such as Dienes blocks, and computer-based manipulatives is the constraints-support structure (CS structure) (Kaput, 1992). According to Kaput, any notation system is defined by rule systems and these rule systems determine “its allowable objects, allowable actions on them, and relation among objects”(Kaput, 1992, p.527). The important attribute of physical manipulatives is the lack of inherent CS structure on actions. In other words, whenever students use physical manipulatives, the constraints on supports for students’ action must be provided externally. On the other hand, computer-based manipulatives can constrain actions with equivalent relations among objects. The Thompsons (1992) developed computer-based Dienes blocks with constraints on actions and compared students’ learning based on the physical Dienes blocks and on the computer-based Dienes blocks in the identical teaching. He examined students’ written work on calculation items to determine how the discussion and activities that students experienced throughout the series of instruction influenced children. As a result, he found the different impact of instruction between the students who used computer-based Dienes blocks and those who used physical Dienes blocks. He also analyzed the data from selected students’ interview. The interviews were focused on students’ reasoning as expressed on the posttest. Although the use of manipulatives does not guarantee a student’s understanding of decimal numeration and construction of the notational methods for determining the result of operations involving decimal numbers, this research project contributes the important suggestion in using manipulatives. Students must first be committed to making sense of their activities and be committed to expressing their sense in meaningful ways before they make productive use of concrete manipulatives. The way of using manipulatives is very important for students to understand mathematics. The Thompsons concluded that teaching with the computer-based-manipulatives led fourth-grade students to a stronger understanding of the number system structure and of algorithms. This case study demonstrates the potential of computer use in hands-on activity based mathematics teaching and learning. Considering above, Thompsons’ computer-based Dienes blocks should be clearly distinguished from other computer software such as the CAI drill and practice genre software. Since Thompsons’ study, various kinds of computer-based manipulatives have been developed. Clements & McMillen (1996) investigated various computer-based manipulatives and argued that these can be the alternatives to concrete manipulatives if these are used appropriately.
Recent technological advancements make it possible to have interactive web sites that provide computer-based manipulatives without any special software. For example, the National Library of Virtual Manipulatives for Interactive Mathematics (Cannon, L. O., Dorward, J. T., Heal, E. R., Edwards, L., & Wellman, R.,1999) provides various computer-based manipulatives on the Internet. One of the most important features of this research project is using the Internet as a tool to provide manipulatives. The Internet can be one of the most powerful tools for mathematics teaching and learning because the Internet can provide classrooms directly with interactive software. It could bring great benefit not only to students but also to teachers. One of the benefits is schools do not have to have special software because only free web-browse software is needed for Internet use. This is very important for schools because many schools do not have enough money for purchasing computer software. Also, the Internet does not require schools to update their software. When the contents of the web sites need to be updated, only the pages on the web server need to be updated. This could be one of the important advantages for schools and teachers because schools do not have to spend their limited budget for updating software and teachers do not have to spend their time to update software on the computers one by one.

References

Cannon, L. O., Dorward, J. T., Heal, E. R., Edwards, L., & Wellman, R. (1999). National Library of Virtual Manipulative for Interactive Mathematics. Retrieved Nov 23, 2000, from the World Wide Web: http://www.matti.usu.edu/.
Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2(5), 270-279.
Kaput, J. J. (1992). Technology and Mathematics Education. In Grouws D. A. (Ed.), Handbook of research on mathematics teaching and learning (pp.515-556). New York: Simon and Shuster.
Lesh, R., Post, T., & Behr, M. (1987). Dienes revisited; Multiple embodiments in computer environments. In I. Wirzup & R. Streit (Eds.), Developments in school mathematics around the world (pp. 647-680). Reston, VA; National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Office of Technology Assessment. (1988). Power on! New tools for teaching and learning (OTA-SET-379). U.S. Government Printing Office.
Schoenfeld, A. et al. (1997). Student assessment in calculus: A report of the NSF Working Group on Assessment in Calculus. Washington, DC: Mathematical Association of America.
Thompson, P. W. (1992). Notations, conventions and constraints: Contributions to effective use of concrete manipulatives in elementary mathematics. Journal for Research in Mathematics Education 1992, Vol.23, No.2, 123-147.