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What might a School CAS look like? Some Thoughts and Considerations
The use of Computer Algebra Software with senior level and high ability students offers some obvious benefits for learning, and some very real motivations for widespread adoption. It is the use of CAS in the earlier years - the years when students are actually learning algebra - that remains highly problematic. Three issues present strongly when considering potential roles for algebra manipulation technology at such levels: Interface, scaffolding and concrete manipulative.
INTERFACE
There appear to be two key interface considerations:
Input: How do we assist the novice user to correctly enter their mathematical objects?
Functionality: What range of commands/menus need to be available/readily accessible for daily use?
For high school use, a minimal template, consisting only of fraction, exponent and (perhaps) matrix/ordered pair is likely to be sufficient. If kept to this minimum, it would be advisable for these to be "permanently" visible so that students always have a clear option for entering 2-D notation as it occurs.
In terms of functionality, high school algebra is really built around 3 commands: Simplify, Substitute and Solve. These may be extended to include expand and factor (and, later, differentiate and integrate). This minimal subset forms the basis for what is termed the Polynomial Toolkit
SCAFFOLDING
Since CAS arises from the needs of "working mathematicians" rather than from pedagogy, the focus tends to be upon delivering a result in the most expedient way: input -> output.
Consideration needs to be given to supporting students through algebraic processes, usually in a step-by-step way. Such an approach can be either static or interactive. It would seem likely that an approach that involved the student as active participant rather than spectator will be far more effective.
Some bases for these ideas are discussed in more detail at http://www.compasstech.com.au/TNSINTRO/TI-NspireCD/mystuff/dynamic_algebra.html .
CONCRETE CONSIDERATIONS
If we are truly considering the development of a computer algebra system specifically aimed at high school classrooms and viable for the learning of algebra, then we should also consider the key danger associated with the use of symbolic manipulation in schools: the main problem with our teaching of algebra up to this point has been the focus upon the symbolic manipulation (the syntax) at the expense of the meaning behind the symbols (the semantics).
Technology offers some unique opportunities in this regard, particularly through dynamic linking between the symbolic representation and geometric counterparts (consider some of the early TI-Nspire activities Ð the Falling Ladder, the PaperFold, the Beach Race. These powerful learning devices drew strong connections between the geometric, the graphical and the symbolic forms and offered powerful tools for building, not just algebraic facility, but algebraic understanding).
Similarly powerful has been the growing body of work devoted to dynamic "algebra tiles" which explicitly support the development of symbolic understanding and manipulative skill within a concrete framework. These ideas are explored further at: http://www.compasstech.com.au/TNS_Authoring/algtiles.html.
If we are to consider a CAS that is ideally positioned for use across the high school years, then this MUST explicitly support the link between the symbolic representation and geometric and graphical forms. Some form of dynamic algebra tile facility would seem to be highly recommended for this context.
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