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Developing a TI-92 manual generator based on computer algebra systems.

Article Excerpt
The electronic medium suitable for mathematics learning and



teaching is often designed with a notebook interface provided in a computer algebra system. Such a notebook interface facilitates a workspace for mathematical activities along with an online help system. In this paper, the proposed feature is implemented in the Mathematica's notebook environment. This paper illustrates how to produce a notebook interface for TI-92 graphics calculator manuals that can be embedded in an online help system based on Mathematica and Theorema. The TI-92 manual generator produces input descriptions, a sequence of TI-92 keystrokes and a TI-92 screen shot. The final part of this paper shows how static manual creation can be converted into web documents using Mathematica's Java package, called J/Link.

A Computer Algebra Systems (CAS), such as Mathematica, Maple and Derive, is a suite of complicated software parts that implement mathematical algorithms numerically, graphically and symbolically (Kajler, 1998; Wester, 1999). Teachers and students engaged in exploratory mathematics teaching and learning have paid attention to the increasing power of CAS. This phenomenon, however, has not received consensus among mathematics educators arguing whether the use of CAS was necessary or not (Buchberger, 1989). Recently, a number of researchers have considered various aspects of the CAS for promoting effective mathematical learning (Drijvers & Van Herwaarden, 2001; Heid, Hollebrands, & Iseri, 2002).

Since graphing calculators were required on the AP calculus examination in 1995, the mathematics curriculum has been greatly impacted by the computer algebra systems available in graphing calculators (Kennedy, 2002). When TI-92, one of the graphing calculators, came along with Derive inside, many mathematics teachers regarded the use of symbolic computation as an important problem-solving tool. By that time, they realized that graphing calculators could be effective in exploring and communicating mathematical concepts. Heid et al. (2002) report empirical research results along these lines.

The practical use of graphing calculators is not as straightforward as is the case for using technology in classrooms. Teachers first need to consider some obstacles using a CAS such as TI-92: 1) what a CAS produces is different from what students expect as results, and 2) students are confused about variables and parameters (Drijvers, 2000). For example, the fact that TI-92 can display some outputs that puzzle students implies that students might have difficulty continuing their mathematical work with non-trivial keystrokes for mathematical input expressions. As a consequence, it turns out that manuals written with elaborate annotations and detailed sequences of input keystrokes are essential for students' learning.

The objective of this study is to help teachers to easily produce TI-92 manuals with keystrokes, annotations and the screen capture capability of TI-92. Compared to Derive's input mode, it is not sufficiently easy for TI graphics calculator users who are secondary school students to type commands and arguments even though several explicit function keys and command keys are available. This consideration motivates how to create both the static manuals and interactive documents that can be embedded in a computational apparatus that can be operated according to the level of students' mathematical thinking (Karian, 1992; Tall, 1991). Not only does this study show how to produce TI-92 manuals automatically, but it also illustrates how to convert these manuals into dynamic documents in web environments.

Manual generators are designed in the following way: 1) to increase accuracy for key inputs, 2) to check the corresponding screen results, and 3) to prepare annotations that guide students to easily manipulate function keys and alphanumeric keys from procedural point of view. This design simplifies the structure of manual automation, even though it somewhat resembles a traditional way of producing a TI-92 manual. It should be noted that the structure of manuals can be easily created with programming capabilities of any CAS. Rewriting technique is the most prominent method to simulate TI-92's algebra part (Derive) with Mathematica due to its enhanced interface and rewriting language features (Rich & Stoutemyer, 1994). In the sequel, the steps for how to develop a manual generator are introduced with the languages built into Mathematica and Theorema (Buchberger et al., 1997).

Design Issues for TI-92 Manual Generation

The design framework for manual production consists of the work of Buchberger (1989) and Beeson (1998). To what extent should the black box/white box principle be considered in TI-92 manual generation? Here black box refers to using a CAS to find a result for an input expression as opposed to knowing the intermediate steps (white box). This principle suggested by Buchberger (1989) assumes that students are encouraged to start with known (complied) algorithms to do some numerical experimentation, detect meaningful patterns and produce conjectures. Once those conjectures are verified, then these become applicable algorithms for repeated use in the future. This cycle continues by building...

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