Transitions Project

Last modified 9.14.02.

Background on the
Transitions Approach

In transitions modules, a “technology” is any tool for doing or teaching mathematics, including hands-on manipulatives, pencil and paper, calculators, computers, and the Internet. The philosophy behind this definition is that every teacher should experience as many different ways of dealing with a mathematical problem as is feasible. Recent research supports this view. For example, Weiss, Matti, and Smith (1994) found only 45% of grades 1–4 teachers felt prepared to teach with calculators. In the transitions approach, however, teachers use not only manipulatives but also calculators and computers on topics traditionally treated with manipulatives only. The Internet is also part of the transitions approach. For example, some modules require students to download data from the Internet for modeling real-world phenomena, and others require students to download, test, and critique demonstration versions of selected mathematical software.

Project Goals and Objectives

The objectives are to

Develop and field test quality materials, possibly extending to commercially published textbooks, software, or lab manuals, for use in the mathematics content and methods courses of preservice teachers at KSU and other institutions.
Improve the ability, confidence, and willingness of K–12 preservice teachers to use manipulatives, calculators, computers, and the Internet in the teaching of mathematics.
Improve K–12 preservice teachers’ understanding of mathematical concepts.
Improve attitudes of K–12 preservice teachers towards both the learning and teaching of mathematics and towards mathematics as a discipline.
Improve attitudes of preservice teachers towards collaborative learning and student-to-student communication in mathematics.
Publish meaningful, unbiased analyses of the

developed materials’ effectiveness for Objectives 2–5.

Project Overview

Throughout K–12 mathematics education, teachers are expected to make appropriate use of technology (NCTM, 1997; Texas SSI, 1996; NRC, 1996). But how is a teacher to judge which technology is appropriate, and how does one use a given technology appropriately? In many programs, pre-service teachers use no more than one form of technology, if any, within the context of a single problem, yet as in-service teachers they are expected to make appropriate choices concerning different technologies. This produces teachers who insistently use only certain technologies to solve certain kinds of problems, much the way that mathematics students in general insistently use only previously encountered techniques. Since teachers teach as they were taught (NRC, 1989; Lortie, 1975), it is the critical responsibility of programs everywhere to provide pre-service teachers with experiences in making appropriate technology choices.

Definition. Because there is no general agreement as to what constitutes a “module,” it seems appropriate to at least define the term for this proposal. Here, a module is a self-contained instructional unit focused on a single topic or a set of closely related topics. The scope of a module is about the same as that of a single section in a textbook. For example, a chapter on elementary number theory might be divided into sections on primes and composites, prime factorizations, greatest common divisors, the Euclidean Algorithm, and least common multiples. By our definition, each of these topics would be appropriate for a single module. The term module is commonly applied to materials designed as supplements to traditional texts. This usage applies to our project as well, but here the view is also taken that sets of related modules can be merged to form an entire course and therefore take a primary rather than supplementary role. Indeed, this is a distinct possibility for three of the targeted courses. Finally, all transitions modules contain certain key components: introductory and explanatory narratives, definitions of new terms, directions for performing tasks, and questions that are distributed throughout the module (as opposed to all of them coming at the end.

Each transitions module introduces a central concept or problem then leads the students through a series of investigations using a number of different technologies, some of which are more or less appropriate than others. Students will discuss orally and in writing the appropriateness of each technology for the given topic, the effects of the different technologies on their mathematical experiences, and the potential application of their experiences as learners towards their goals as teachers. They will, of course, also be asked questions about the mathematics content. In summary, students will experience the applicability of each technology, see how the different technologies can be used together on a single topic, and critically analyze technological choices in mathematics classrooms.

Another important aspect of the typical transitions module is that it forces students to reflect on their own mathematical learning. Essays that compare and contrast the advantages and disadvantages of each technology, with respect to mathematical learning, are a required part of every activity. (Here, an “activity” could last several class periods.) Students must discuss how their experiences can be used to improve themselves as teachers. Analytical activities such as these encourage them to be reflective educators. Sometimes they have to create a transitions activity of their own by designing an appropriate mathematical lesson that uses multiple technologies. This will require considerable learning in both the mathematics and the technologies chosen for the activity. Finally, even with all its emphasis on technology, the transitions approach maintains mathematical thought and language as its focus.

REFERENCES

Blum, W, and Niss,.M. (1991) Applied mathematical problem solving, modeling, applications, and links to toher subjects—State, trends, and issues in mathematics instruction. Educational Studies in Mathematics. 22, 37–68.

Dossey, J. (1992) The nature of mathematics. In D. Grouws (ed.), Handbook of research on mathematics teaching and learning (pp. 39–48). New York: MacMillan.

Gilley, J. Wade, Kenneth A. Fulmer, and Sally Reithlingshoefer.. (1986). Searching for Academic Excellence: Twenty Colleges and Universities on the Move and Their Leaders. American Council on Education: Macmillan, New York.

Lortie, D. (1975). Schoolteacher. Chicago, IL: The University of Chicago Press.

National Council of Teachers of Mathematics. (1989) Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics. (1997). 1997–98 Handbook: NCTM Goals, Leaders, and Position Statements. Reston, VA: NCTM.

National Research Council. (1989). Everybody Counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press.

National Research Council. (1996). The Preparation of teachers of mathematics: Considerations and challenges. Washington, DC: National Academy Press.

National Science Board. (1996). Science and Engineering Indicators—1996. Washington, DC: U.S. Government Printing Office.

Raymond, A. (1997) Inconsistency between a beginning elementary school teacher’s mathematics beliefs and teaching practice. Journal for Research in Mathematics Education. 28, 550–576.

Texas Statewide Systemic Initiative. (1996). Guidelines for the Mathematical Preparation of Prospective Elementary Teachers. Austin, TX: Texas SSI.

Weiss, I., Matti, M., and Smith, P. (1994). Report of the 1993 National Survey of Science and Mathematics Education. Chapel Hill, NC: Horizon Research, Inc.

Wenglinsky, H. (1998) “Does It Compute? The Relationship Between Educational Technology and Student Achievement in Mathematics.” Cited by Education Week on the Web at http://www.edweek.org/sreports/tc98.