Be more aware of the overall impact of using calculators in learning mathematics

Abstract

1. Procedural versus structural -- Students often experience enormous difficulties in their transition from learning arithmetic to learning algebra. These difficulties will trail some students to their college years. A typical problem of these students is to treat algebraic representations in procedural terms rather than objects that should be viewed in structural terms. Unfortunately, using calculators may reinforce this tendency and provide an "outlet" for those who do not quite understand how to operate on formal mathematical entities.
2. Conceptual understanding versus skill -- Because of the technology, complicated tasks can now be accomplished instantly. As a consequence, the embodied concepts are often wrapped up and become opaque to a user. Some students' performance exhibit that the skill and the efficiency of getting an answer from a calculator may cover a lack of conceptual understanding of subjects.
3. An epistemological issue -- To use and to respond to a calculator properly, one needs not only some technical knowledge about the device, but also a understanding of the limitation of it in representing mathematical objects. There are ample examples illustrating that calculators may give inaccurate or even incorrect information. As a learner, the danger of being misled is real.
4. Problem solving behavior -- A mathematics problem, such as solving a cubic equation, may pose very different cognitive challenges to a student depending on whether a calculator is allowed. It is found that students may overuse the technology in problem solving environment. Some students are indulged in the "trial and error -- looking for a pattern" heuristic when the technology is at their disposal and, thus, often either misinterpret a problem or fail to adopt an appropriate approach. Their problem solving abilities is getting confined instead of expanded.
5. Notation -- Mastering mathematical notations is never an easy task . Using calculator brings new difficulties in learning it. The physical limitation of a calculator, its self-contained and often weird rules, and its intention to simplify various procedures, all contribute to the problems. The documented examples show how some mistakes that many students committed in learning mathematics may be accepted by a calculator and added to their confusion. Several suggestions are made. It is stressed that improving rather than restricting the use of calculators is anticipated.