Recognising GOOD Software

New Tools, New Questions

Impediments, Imperatives and Implications

Features of Good Tools

Software tools such as LOGO, Cabri Geometry, Theorist, SyMan, Derive and Calculus T/L II exemplify the most positive features of computer technology as a medium for learning:

They place the user firmly in control of the technology;
They encourage and reward exploration and enquiry;
They offer capabilities impossible without the use of technology, and
They are naturally mathematical: the user is immersed in mathematical concepts and actions, and takes away from the encounter deep and versatile mathematical understandings.
With regard to algebra software in particular, certain features are highly desirable and determine the extent to which the software may be considered to be pedagogically appropriate:

simplified entry of algebraic forms (at least implicit multiplication and preferably correct symbolic forms, such as exponents, radicals and constants such as ¼);
two dimensional display of both input and output;
broad functionality with easy access to the range of available functions;
a clear and intuitive interface;
support and close approximation of algebraic procedures;
flexible and intuitive manipulation of algebraic forms, graphs and tables of values;
a simple record of each step of the transaction.

Features of Mathematically Able Software

In a recent three-year study of individuals use of mathematical software, significant reluctance was observed regarding the use of computer algebra software (the primary tools for this study were Theorist and Derive). Although participants invariably spoke positively of their use of such tools and found them helpful in a wide variety of mathematical situations, the majority rarely, if ever, chose them spontaneously. When they did use them, this was most often to verify results which they had obtained by traditional means. While it is quite acceptable to "check your answer in the back of the book," it is a form of "cheating" if such use occurs earlier. The message was clear, from students, preservice teachers and even classroom teachers - algebra is a solitary activity which must be mastered through repetition and individual practice.
Interestingly, this conflict with perceptions of "acceptable mathematical practice" was observed only in relation to computer algebra software; representational tools (especially graph plotters) appear to fit comfortably alongside existing instructional patterns, while tools which support the manipulations of algebra directly confront them.
Particular advantages include:

Open-ended software tools encourage active and meaningful learning experiences on the part of students: although initial use tends to be for verifying results obtained by traditional means, it is not a great step from this minimal use of the technology to one which involves verifying conjectures, a far more powerful and appropriate activity.

Technology which supports the processes of mathematical activity offers great potential as a medium for learning in the junior years and among those who find traditional manipulative approaches too difficult. From equation solving and completing the square to the more difficult processes of senior mathematics, open-ended software which is consistent with good pedagogical practice offers significant scaffolding for improved learning of even such traditional manipulative skills - learning first with the computer as support and then moving to an independent state. Access to appropriate tools offers students a level playing field for the learning of algebraic concepts and skills.

Finally, computer tools make both mathematical thinking and processes public. This serves not only to encourage verbalisation and discussion among student learners, but provides teachers with unique and important information for evaluating student thinking and understanding and assessing the extent and nature of student learning.

Mathematical software tools have significant implications for the teaching and learning process. The influence of such tools upon the interactions between teachers and students may well be far-reaching and rewarding.
In the shorter term, the principal uses of computer tools for mathematical purposes are likely to be for representation (using graph plotter and, less often, table of values) and verification of results. Although well-suited to support open-ended investigation, such use is likely to remain rare under the influence of a culture of learning which rewards closure and identifies algebra with "finding an answer" using automated and predetermined action sequences. Computer algebra tools may best be introduced into the current mathematics curriculum in two ways:

As means of supporting students in the learning of sequential mathematical procedures (such as equation solving in the early years). Computer tools which both support and make explicit the process provide a useful aid in such areas.
As tools for supporting open-ended assessment tasks, and so encouraging and motivating mathematical enquiry.

What is "strategic software use"?

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