Index
Index
USING THE TOOLS EFFECTIVELY
In observing the ways in which individuals chose to make use of available mathematical software tools, I learned much about both the tools and the users. The study revealed a strong preference for the graphical representation across all groups of participants. A much greater reluctance was observed regarding the use of computer algebra tools. 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. It appears that computer algebra tools were viewed as somehow "illegitimate." To use them to support an algebraic computation was seen as akin to "looking up the answer in the back of the textbook." This analogy is further supported by their use for verification purposes after completion of the process. 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 tools; representational tools (especially graph plotters) appear to fit comfortably alongside existing instructional patterns, while tools which support the manipulations of algebra directly confront them. In considering the beliefs and perceptions of the participants in this study regarding the nature of algebra, the ways in which it may best be learned and the role of computers in this process, consistent evidence was found to support the notion of a culture of mathematics learning: a shared set of beliefs and experiences which extended across all groups of participants.
This culture served largely as an impeding factor for the use of algebra software, characterised as it was by such features as:Index
- a view of mathematics as "answer-based," devaluing exploration and open-ended problem solving (those areas in which the software appears most effective);
- a view of algebra as primarily serving a symbolic representation purpose, with little usefulness beyond this role--algebra is not about modelling the real world, it is about "letters standing for numbers;"
- an emphasis upon individual efforts, devaluing both group approaches and the use of external aids (such as computer tools) coupled with a strong reliance upon individual as opposed to group aids--especially textbooks and hand calculators;
- a dependence upon the teacher as source of knowledge and direction in mathematics learning;
- a limited representational repertoire, dominated by symbolic and graphical forms;
- a lack of reliance upon individual judgement and confidence with regard to their mathematical processes--students appeared quite happy to conclude an answer while expressing little confidence in their result.
IndexFactors such as these militate against both the use and the perceived need for open-ended software tools which support and extend mathematical learning.
The use of the tools itself took on several quite distinct forms:
- Level 0: Non-Use: Although the software is available and appropriate, and the user has sufficient skill, no use is made.
- Level 1: Passive: The user is content for the tools to be operated
by another, but takes no personal initiative.
- Level 2: Random: Use is not goal-directed and bears no relation to the instructional context.
- Level 3: Reflexive: The user makes superficial and automatic use of appropriate tools.
- Level 4: Strategic: Use of the tools is deliberate. goal-directed and flexible, frequently involving multiple strategies for both exploration and verification.
IndexClearly there are some dangers associated with poor use of technology. There is also a goal for all who would use technology to encourage, as some have called it, an "intelligent partnership" between user and tool, characterised here by "strategic software use."
How then may we best achieve such a goal? The evidence of my study suggests that teachers may encourage strategic software use through the creation of a learning environment within which:
- students are comfortable with the available software tools. The interface should support ease of entry of mathematical forms and make the range of mathematical functions clearly available.
- mathematical tasks lie within the zone of proximal development of the students. Students must perceive the task as potentially achievable, although beyond their present capabilities unaided.
- students must be able to elicit from the task a mathematical object which is capable of signalling appropriate action strategies involving the integration of mathematical and computer-based actions.
- open-ended investigation is perceived as a valid means of achieving a solution, which may be only one of several appropriate responses to the task.
- The use of multiple strategies for verification must be perceived as a necessary component of mathematical enquiry.
- students must be motivated: persistence and some measure of personal commitment to the solution process must be evident.
The strategic use of mathematical software tools is indicative, not only of a high level of computer-based competence, but of insightful and strongly connected mathematical thinking. Conditions under which such use may be encouraged should be a feature common to all mathematics learning situations. Use of the technology is not an end in itself; rather, the technology may serve as a catalyst, encouraging changes in the current culture of mathematics learning towards the creation of a learning environment which is active on the part of the learner, which rewards exploration and understanding rather than recall, and within which mathematics assumes a vitality and significance which has been missing for so many learners in the past.
Index
IndexREFERENCES
Arnold, S. M. (1996) Learning to use new tools: A study of mathematical software use for the learning of algebra. Unpublished doctoral dissertation, UNSW.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.