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

        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 mathematical knowledge out of the experimental instrumentation of algorithm construction (Heugl, Klinger, & Lechner, 1996). In secondary schools, students, however, may not be allowed to follow all of these cycles given limits on time and effort.

In a similar vein, Beeson (1998) laid out design principles for building a CAS for educational use in his paper. His principles consider student-oriented mathematics activities that seem to be essential for learning: glass box, cognitive fidelity, curriculum integration and level of mastery. Given a problem, the intermediate steps toward a solution are explicitly generated as a glass box. Moreover, Beeson's principle emphasizes the cognitive fidelity of a program that combines the textbook-based curriculum according to the levels of students. His program, Mathpert as CAS-based tutor program, was realized according to such a principle.

With graphing calculators, students can explore their own mathematical ideas and tasks with the black box/white box principle. Martinez-Cruz & Contreras (2002) showed that students could experience "what if" situations with TI-92 to explore, conjecture, examine, test and prove open-ended problems. Even though the participants of their study were pre-service secondary teachers, this model could be applied to secondary school students. By performing calculations, they tried to seek meaningful patterns and make conjectures that could be proven. Due to time limits, the activities to be proven were scarcely completed while intensive algebraic computing, as well as graphing for conjecturing, was performed. Later, the instructor tried to engage the participants in modifying the conditions of the original problem as part of posing the problem.

Two Types of Manuals

The structure of TI-92 paper manuals consists of three parts: input description, input keystrokes and a screen shot. The benefits of the electronic version of a CAS manual are twofold inasmuch as it can serve as a worksheet and provide online help. It is noted, however, that graphing calculators provide neither separate worksheets that are usually displayed in the output screen nor an online help function. Two types of manual construction are reviewed in the form of electronic media.

Manual as Worksheet

Worksheets are commonly used as complementary materials to textbooks during classroom teaching. Worksheet materials contain a set of tasks and questions that elicit guesses or conjectures that will help in proving a problem. Students are encouraged to work on and explore each task in sequence with worksheets. Using worksheets, students produce archival materials that they must then manage for easy access and quick reference.

When mathematics teachers instruct students using a CAS, they prepare lesson plans in advance and design instructional strategies that will take advantage of the CAS. In reality, it is not quite clear how to integrate the use of a CAS with mathematics teaching due to the strict content organization embedded in the existing mathematics curriculum (Peschek & Schneider, 2001; Schneider, 2000). To overcome such stumbling blocks in classroom teaching, students are often asked to control their own learning sequences by referring to the worksheets provided by the teacher.

When embedded into a Mathematica notebook, worksheets play the role of either a workspace or CAS-based calculator. Guided by the instruction provided in each worksheet, students can concentrate on mathematical concepts without being interrupted by heavy computational duties that can be simply dispatched to the CAS. For example, Calculus Wiz (http://www.wolfram.com/products/student/calcwiz/) belongs to this category of Mathematica development. Another aspect of designing a worksheet reduces a teacher's responsibilities for assessing student work. A Mathematica notebook as a worksheet can be sent directly to the teacher for immediate feedback either in a LAN or in an Internet environment. The necessary file transfers make teacher-student interactions more productive by reducing the number of technical chores to be performed.

Manual as Online Help

An online help system first appeared in the Macsyma system (Wester, 1999). Developed using Smalltalk language, Macsyma had an object-oriented help system. Eventually, Mathematica adopted the capability that is now widely used and which provides direct access to the CAS in the notebook format (Wolfram, 1996). The distinct feature of this online help system is that users can easily find contextual information by giving a keyword input. Beginning students seem to prefer following step-by-step instructions to pursue solutions even though middle or advanced students are encouraged to start with step-by-step algorithm building according to the glass box or white box principle (Beeson, 1998).

The recent version of the online help system in Mathematica can perform step-by-step illustrations that correspond to the white box phase of a CAS thanks to a feature that can directly manipulate mathematical expressions inside the help system. In other words, students can work on intermediate steps of problem solving in the pre-fabricated worksheet inside the help system. Students can also write down what they explore with black-box algorithms that are immediately executed by CAS. Work done in the worksheet can be stored in the online help system for future reference.

The notion of worksheets and online help are quite limited, though, due to screen size. However, paper manuals are easily accessable in a text format for most of the graphing calculators. With the computerization of paper manuals in mind, Table 1 illustrates the features of TI-92 in terms of worksheets and online help.

Implementation of TI-92 Manual Generator

Compared to TI-92 paper manuals, an automatic manual generator system is equipped with a TI-92 computational simulator for output generation. The simulator has a rewriting parser that can generate special fonts for keystrokes and has a screen capture facility with VTI and KeyRecorder software. Table 2 compares several features of the TI-92 paper manual and the electronic manual generator.

TI-92 Simulator with Mathematica and Theorema

Theorema is designed for teaching logic and mathematics with the collection of theorem provers (Buchberger, Jebelean, Kriftner, Marin, Tomuta, & Vasaru, 1997). These provers consist of general provers that are based on set thoery and predicte calculus, and special provers that deal with polynomial properties proved by groebner basis computation. Theorema can generate automated theorem proofs in natural language style contained in Mathematica notebook. Besides the capability of proving, solving and simplifying, Theorema's language constructs enable students to deal with mathematical expressions that reflect conventional mathematics (Buchberger, 1996). With Theorema's tuple notaions, it is relatively easy to manipulate algebraic expressions, logical deductions and theorem proving. For the current work, TI-92 simulation only covers the Derive part installed inside TI-92 hardware.

In Theorema, the user can define his own definitions with universal quantifiers in terms of pattern matching. The main part of the simulator, in this case "TI simulator," is written by a recursive procedure that feeds an input expression. For each input, the simulator invokes one of the rewriting rules for algebraic manipulation. With this new intermediate state, another input is fed for the next execution of symbol manipulation, and so on.

The output is written with Mathematica commands for making different box styles with colors that mimic TI-92 layout. The current version of the TI-92 simulator only simulates algebraic manipulation without the graphical part using the recursive structure of the main program. The proposed simulator follows from the written manuals developed by TI and Kutzler (1997). Due to Mathematica's expressions, some parts are not identical with some of the TI-92 expressions. The output layout is quite rudimentary since there is no screen layout handler at this point. Most of the Mathematica commands can emulate Derive's algebraic commands with some slight modification.

Figure 1 illustrates input and output modes for the simulator. Teachers can simply load the manual generator from Mathematica and type in mathematical expressions to get the generated manual ouput. For example, the user can type "solve (x^2-2x-8=0|x>0)" in the input box to get the result "x=4" after clicking the "ENTER" button. Figure 2 shows what a Mathematica-simulated command "stepSolve" returns to the user. This command is explained in the following section.

Derivations for Equation Solving 

Equation solving in TI-92 is usually carried out by rewriting terms without showing any intermediate steps as a black box. This feature can be modified into a white box by considering the cognitive fidelities of the students according to different achievement levels. In fact, beginning and intermediate-level students seemed to prefer going through intermediate solving steps. With TI-92, however, students can explicitly manipulate different ways of transforming algebraic rules on their own. For example, Kutzler (1997) introduced several ways to isolate a target variable to find a solution given a system of linear equations. Our simulator rather directly generates a sequence of algebraic transformations including an answer all at once (see "stepSolve"). Currently, the number of transformation rules is less than ten since we try to capture general patterns associated with rewriting rules as simply as possible (Kajler, 1998).

[FIGURE 1 OMITTED]

A newly designed command with the TI-simulator is "stepSolve," by which students can watch how a linear equation can be solved in a step-by-step fashion. This feature complements the black-box type of TI-92 outputs with a white box principle. Another distinct part of this simulator allows the user to deal with shortcut computation in solving a linear equation. For example, the particular equation, "-6+5x=15+2x," can be memorized with TI command "ans(1)." Quick problem-solving is possible in this case by directly manipulating the whole equation by rewriting the expression in one step, (ans(1)+6-2x)/3.

[FIGURE 2 OMITTED]

Rewriting for Input Keys

Manual generation consists of three parts: 1) input description, 2) a sequence of keystrokes, and 3) a TI-92 screen shot. Currently, the TI-92 simulator cannot emulate the same screenshot of TI-92 as most TI emulators do. As developers can test TI emulators, such as Virtual TI (VTI), the manual generator sends the request of keystrokes to VTI to get the emulated screen shot. This task is done via the KeyRecorder application. The rewriting techniques built into Mathematica's language enables developers to generate verbalizations and a sequence of key fonts. All the built-in fonts for TI-92 keys have square boxes around them while the missing fonts are surrounded by curly brackets in the generated manual (e.g., {+}). Those missing fonts were not provided by WRI, which publishes Mathematica.

The rewriting technique implemented here consists of several IF and THEN rules. Each rule compares its pattern in the IF part with a given argument so that once it is matched then the THEN part is immediately fired and stops at that point. The overall rewriting part is shown in Figure 3 with the following input command:

ManualOut[sin ([phi]/4) {Enter} {Ctrl} {Enter}]

It is noted that the key for the Ctrl+ENTER key combination in TI-92 computes the numerical value of the previous input expression. In this case, the result of sin ([phi]/4) is .707107.

[FIGURE 3 OMITTED]

TI-92 Manual Generator Embedded in Mathematica

Similar to Mathematica's online help system, the manual documents created by TI-92 simulator are embedded in Mathematica's online help system shown in the Figure 4. In this way, students refer the specific keystrokes with short verbal descriptions in an executable mode inside Mathematica's online help system. The following figure illustrates TI-92 manual for the section of integration. In this integrated environment, students can easily browse manual documents while concentrating on mathematical concepts by reducing technical burden.

[FIGURE 4 OMITTED]

In this ongoing research, the emphasis lies on how to store and organize on-line help texts retrieved by each student in a coherent manner. This ends up with pursuing how to manage individual learning activities combined with an online help system. As soon as the student needs to manually test TI-92 commands, TI-92 simulator can be assessed directly from this online help system. In other words, the structure of TI-92 manual documents inherits all the indexing and retrieving capabilities from Mathematica's online help system, resulting in easy access to electronic mathematical documents.

The Web-based TI-92 Manual Generator

Mathematical documents are becoming more and more interactive either in a stand-alone format or on a dynamic web page. This interactivity is feasible mainly because of CAS-driven computation. The current static web documents will be directly interact with CAS when MathML/OpenMath is standardized and implemented on web browsers (Le, 1999). For the time being, though, there are several approaches for creating web-based mathematical documents: 1) html conversion, 2) the use of webMathematica, and 3) Java Servlet with J/Link.

Using Mathematica's "Save As Special ..." command, a user can easily convert Mathematica-based TI manuals to static web documents. From the user's point of view, it is quick and easy to browse web documents for reference purposes without being disturbed by the need to wait for a document to load. Commercially available software, such as webMathematica, has Mathematica Server Page scripts that automatically execute Mathematica code on the web. Once webMathematica is installed with a Java server plug-in, it is not difficult to create a dynamic Mathematica document that is executed by remote Mathematica in terms of CGI execution. A similar approach was implemented for Maple with Java API, called JavaMath (Solomon, Struble, Cooper, & Linton, 2000).

In this paper, a third alternative is considered for implementing web-based TI-92 commands that can be executed by a TI-92 simulator: a Java Servlet using J/Link. Mathematica can be connected with Java via J/Link, which means that a Mathematica notebook can contain any Java applets that can be controlled by Mathematica. For the first case, Mathematica can be regarded as an extended CAS that covers Java-written components. The latter case indicates that Java applets are in charge of a user-centered interface while Mathematica deals with the core parts of computational execution. The whole mechanism for this process is depicted in Figure 5. This figure depicts how a Java Servlet provides TI-92 manuals with a Java applet interface linked with a remote Mathematica kernel.

[FIGURE 5 OMITTED]

It is natural to develop the user interface of TI-92 with Java to mimic the actual layout of TI-92 calculators. By referring to TI-92 manuals, students can manipulate expressions directly with the interface while the real execution can be done with a remote TI-92 simulator via a J/Link connection. Figure 6 shows how the TI-92 manual can be generated in a web environment.

[FIGURE 6 OMITTED]

This developmental approach opens up a new way of creating and testing TI-92 manuals as a Web-based electronic medium. The result of students' work with manuals can be stored and classified according to a level of mastery. The natural consequence of this approach yields how to manage students' learning activities to ensure the students' steady progress. The comparison of the two different versions of manual generator is presented in the Table 3.

Discussion

This paper clearly deals with a technology-centered approach focusing on the development of the TI-92 manual generator. To aid teachers in using CAS for mathematics teaching, we presented the design framework for manual generation in three ways: 1) input descriptive annotation, 2) a sequence of TI-92 keystrokes, and 3) a TI-92 screen shot. The unique characteristics of the proposed simulator come from the automatic generation of TI-92 manuals that can be utilized either as stand-alone electronic documents or in a web-based medium.

The advantages of this study are that it allows math teachers to do a few technical things: 1) experiment with teaching scripts by finding out the most significantly meaningful keystrokes from the simulator before teaching, and 2) convert online manual documents to web-based tutorial materials. Math teachers who are also developers can create web-based worksheets based on the pre-generated outputs from the simulator. This type of research results has hardly ever been reported anywhere else.

From a pedagogical point of view, a highly relevant yet difficult question for the manual generator arises: How may one design TI manual generators to promote links between procedural and conceptual mathematical knowledge? Even though the work done in this paper only deals with the procedural part of algebraic manipulation in the TI-92, the manual generator is capable of generating partially conceptual annotations for procedural TI-92 keystrokes. With this limited capability, math teachers can explore their own ways of constructing teaching scripts in terms of conceptual-procedural-technical dimensions in order to resolve obstacles faced by students using CAS (Drijvers, 2000). A long sequence of computational keystrokes may correspond to a procedure, invoking several simple concepts in a composite way. In this way, math teachers can overcome difficulties in balancing the concept-technique seesaw that has always arisen from the use of a CAS.

Another important issue related to making Kutzler's TI manual interactive is using this computerization for something crucial to learning (procedural-conceptual links, for example) that may not be achieved by means of standard paper manuals. As Heid et al. (2002) remarked, this aspect of how students reason symbolically needs to be investigated further. The idea of 'encapsulation' as described by Dubinsky (1991) might be relevant to PC linkage given that encapsulation of processes into objects is an important step in reflective abstraction. He suggests that performing processes using a computer (possibly a CAS) may stimulate its encapsulation, which can also be examined by a TI-92 simulator.

On the other hand, it may also be important to examine how the applied manual computerization copes with the process of instrumentation relating to learners' acquaintance with and utilization of the tool whose built-in logic is to be gradually assimilated (Artigue, 2001). Along with the process of instrumentation, another practical question arises: how can teachers themselves produce manuals by using the examined approach? Math teachers are required to know how to handle Mathematica notebooks and online help systems. To do so, they don't need to focus on technical problems, but on the conceptual aspect of these problems. As is shown, the way to execute TI-92 commands is simply designed at this stage because we want teachers to not get stuck in the computer manipulations, but to focus on the connection between computer algebra methods and manual generations (Kutzler, 2000; Mahoney, 2002).

Conclusion and Further Work

Given a fixed set of commands, how is it possible to compose different commands to come up with a pedagogically meaningful output? This question concerns the technical aspects of manipulating TI-92 keystrokes to help students attain a level of understanding. As students gradually experiment with algebraic expressions and numbers with possible derivations in certain cases, they probably begin exploring some properties of the target mathematical expressions. The manual generator was designed with a TI-92 simulator to do this.

The manual preparation of TI graphing calculators requires not only document interface design but also content design. Rewriting techniques in the Mathematica language generates a concrete sequence of keystrokes, and the output of TI-92 execution is implemented in conjunction with the Theorema language. The main focus of this work lies in the automatic preparation of TI-92 manuals either in stand-alone or in web environments as an interactive medium. This approach ensures the flexible use of CAS notebook interfaces for manual browsing with the possible extension to worksheets and online help (Kajler, 1998), compared to paper manual production.

Manual generation will eventually be automated with a shell-like program that facilitates several slots, each of which sends a request to the user to fill in relevant information. The collected information can be reassembled to yield a structured and organized template in a specific order that will produce meaningful materials. This is quite similar to automated instructional design (Merrill, Li, & Jones, 1992). Another issue with manual generation comes from Theorema's capability of extracting equivalent classes given an input expression. This feature allows manual developers (teachers) to produce manual sections that are closely related to conceptualized modules according to equivalent key relations. It is expected that future versions will generate sequences of keystrokes for the execution of various commands, including commands in a graphical mode. The internal representation of Theorema as shown in the figure 1 can be applied to capture the keystroke sequences of a user's action in the particular learning mode such as exploratory learning (Buchberger, 1999). Much harder topic is Theorema's testability whethere there are any equivalent key relations among all possible keystroke sequences in terms of sequences lenghth or screen output.

This paper shows that it is feasible within a CAS to produce structured documents such as TI-92 manuals as an integrated part of an online help system. Aside from retrieving relevant information from the online help system, students can directly deal with tasks and problems using CAS-based manual worksheets. This powerful interface serves as an individual learning environment that can be further enhanced with an adaptive interface. More research into managing students' mathematical knowledge in several modes of learning is needed. Exploratory learning leads students to work on individual notebooks with a high degree of freedom, while manual-based receptive learning may result in repetitive practice with immediate feedback from the system. How to integrate manual generation with an adaptive interface and learning management system is left for future research (Henze & Nejdl, 1999).

Table 1

The comparison of worksheet and online help for TI-92

                        TI-92 worksheet    TI-92 online help

Learning theory         Black box/white    Learning by example,
                        box principle      hypertextual linking

Problematic issues for  White box is       Static reference,
teaching math           missing, possibly  online help is not
                        without thinking   available

Resolving the issues    Flexible           Mathematica's online
                        worksheet such as  help system with
                        Derive, provide    interactive
                        white box          worksheet

Feedback for errors     Not available      Not available

Logic commands          Boolean operation  Boolean operation

Table 2

The comparison of TI-92 paper manual and electronic manual generator

                   TI-92 paper manual          Electronic Manual
                                               Generator

Focus              Conceptual explanation      Automatic manual
                                               generation

Input              Typing into Word file       Mathematical expression
                                               in a generic form

Keystrokes         Typing special fonts        Special fonts are
                   individually                generated

Screen capture     Manually done with using    Automatically generated
                   TI-92 link cable            with VTI software

Output             Partial screen capture      TI-92 simulator
                   produced with TI-92         generates equivalent
                                               output

Software required  Word processing software    Mathematica, Theorema,
                   only (e.g., MS Word)        VTI, KeyRecorder

Table 3

Two types of manual generator with Mathematica

                      Mathematica online           Java servlet
                            help

      Content        Interactive TI-92      Interactive TI-92 manual;
                     manual; annotation,    annotation, keystroke
                     keystroke fonts,       fonts, screen capture
                     screen capture

     Structure       Keyword searching,     Web-based manual
                     example indexing,      generation, Java
                     hyperlinking, topic    connectivity, linking to
                     hierarchy              WBI

    Advantages       Highly structured,     Integrity with learning
                     interactive,           resources, platform
                     Mathematica notebook   independent
                     oriented

   Disadvantages     Mathematica            Mathematica, J/Link, Java
                     programming required   servlet programming
                                            required

Learning management  Teacher's extra work   Linking with database
                     with Mathematica       toward CMI
                     notebooks

Feedback for errors  Not available          Not available

Acknowledgements

This work was supported by the Korea Research Foundation Grant (KRF-2000-CA0109). The author is grateful to Dr. Bruno Buchberger (RISC, Austria) for his help and advice on this research.

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YOUNGCOOK JUN 

Sunchon National University

Korea

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