(1) The following is from one part of a chapter of the International Handbook of Mathematics Education that Christoph Selter [Paedagogische Hochschule Heidelberg, Heidelberg, Germany] and I wrote - it is concerned with calculators and computers at the elementary school level and is section 2.4 of that paper.
[International Handbook of Mathematics Education, Kluwer Academic Publishers, Dordrecht, The Netherlands, Edited by Alan Bishop and others -- two volumes: Can be ordered from Kluwer Academic Publishers, Order Department, P.O. Box 358, Accord Station, Hingham, Massachusetts, 02018-0358 (firstname.lastname@example.org; fax at (617) 871-6528; phone: (617) 871-6600)]
2.4 Calculators and Computers [Becker and Selter]
Of the most recent developments that have important implications for mathematics education, probably none are more important, nor more urgent or controversial, than technology; in particular, calculators and computers (Fey 1989; Kaput 1992; Burkhardt & Fraser 1992; Bright, Waxman & Williams 1994; NCTM 1987, 1991a; National Research Council 1990). Dealing with technology is
* urgent<italic>,</italic> because computers and calculators are so ubiquitous in business and everyday life - and growingly more in education - and will become even more so as greater numbers of children and young people use them and as the prices go down;
*important<italic>,</italic> because the technology is available, in use, and has potential to effect major changes in the teaching and learning of mathematics (Cornu 1992), the curriculum (Ernest 1989), the cognitive development of learners (Wachsmuth & Becker 1986) and thus, the education of teachers; and
*controversial, because on the one hand, some people have the fear that learning basic mathematics will be undermined (Shumway 1989, p. 15), whereas on the other hand some people think that dealing with the standard algorithms is a waste of time.
Let us deal with the calculator first. It offers many possibilities (Clarke & Kelly 1989; Shuard et al. 1994, p. 192; Spiegel 1988). The calculatorÂ´s full potential, however, cannot be realised without a change in both teaching styles and teachers' conceptions about their use (Groves & Cheeseman 1993). Apart from just checking answers, we see their most important use in
* helping children to decrease computation time and effort in order to concentrate on the core of a problem and to enable performance with large and messy numbers,
* facilitating the exploration of numbers and operations with numbers,
* encouraging inquisitivesness and creativity through experimentation (problem solving),
* developing estimation and mental computation skills by checking reasonableness, and
exploring features of the calculator itself and the (dis)advantages of its use in different situations.
Some elementary teachers as well as parents still seem skeptical about the use of calculators because it appears to them that it may threaten computational proficiency. But research findings point in a different direction. Already in the late seventies, Bell et al. (1978) identified a variety of elementary school teaching materials, activities and children's responses that support the conclusion that calculators can be used to enhance the learning of mathematical skills. Suydam (1983) synthesized the research reported by the Working Group on Calculators of the Second International Mathematics Study. While the evidence accumulated at that time varies across the countries in the study, "the overwhelming majority of the data - from both formal experiments and information exploration - has supported the conclusion that the use of calculators does not harm achievement scores" and that "the use of calculators can promote computational achievement, as well as the learning of other mathematical ideas" (p. 638). By the mid-eighties, Hembree & Dessart (1986) did a meta-analysis of 79 calculator studies and concluded that the use of calculators in instruction does not harm computational skills and enhances problem solving skills and development of concepts.
These findings are in harmony with the results of the Calculator-Aware Number (CAN) project in England and Wales (Shuard et al. 1991) in which children devised ingenious, nonstandard algorithms and knowledge of large numbers, decimals and negative numbers much earlier than in the traditional curriculum. The project was undertaken with six-year-olds and emphasized mental computation - teachers in the project did not teach the traditional algorithms. A test of understanding on a variety of mathematical ideas provided evidence that project children performed better than traditionally taught pupils with respect to understanding and mental computation, and were also more enthusiastic and persistent (Shuard 1992). Overall, the research findings advocate the use of calculators in elementary school, but whether this should be the case from the very beginning of formal schooling onwards is a question requiring further long-term research. Especially, the training of the basic arithmetical skills and abilities should not be replaced by a premature use of the calculator.
In contrast to calculators, the public and the educational community seem to be more open minded regarding computer use in schools (National Research Council 1990; Ernest 1989). Nevertheless, there are several constraints such as an insufficient number of high-performance computers, limited access to high-quality software, and inadequate preparation of teachers; in addition, most of the programs for elementary school are simple drill-and-practice programs that restore to life the ideas of behaviorism and programmed instruction in a new <bold>'</bold>disguise<bold>'</bold> (Kaput 1992). Although there is a place for such programs in the learning process, they should not be used before a solid basis of understanding is established. On the one hand, they may be more diversified and possibly more <bold>'</bold>motivating<bold>'</bold> for the practicing of skills than just doing pages of raw computation. On the other hand, we should also recognize the disadvantages in their use and use great care to evaluate software that might lead us to see education simply as 'entertainment,' as Postman (1985) once put it.
Nevertheless, computers have much more to offer than drill and practice; in fact, they can be used in conjunction with all parts of the constructive learning process, when embedded in a classroom culture where there is communication and cooperation (Hoyles, Healy & Pozzi 1992). There are several ways in which computers can be used; for example in the practicing of skills in a way that incorporates understanding or in simulations that enhance concept building (Klep & Gilissen 1989; Krauthausen 1995; van Galen 1995). At the moment, there is no clear evidence about the role of Intelligent Tutoring Systems (ITS) that aim at simulating an expert teacher who is able to perform a sensitive and constantly retuned diagnosis of a pupil's thinking, and on that basis give the right stimulations that may optimize the learning process (De Corte, Greer & Verschaffel, in press, pp. 69-70).
Many computer based systems can provide children with new tools for learning in an exploratory environment. The most popular is LOGO that is widely used in some countries, but often limited to turtlegraphics (Resnick 1988). Several studies have reported positive effects in the use of LOGO (Watson 1993) that can provide a rich environment in which pupils engage in collaborative mathematical activity (Hoyles & Sutherland 1990). At the same time, it should be noted that Papert's (1980) strongest claims for LOGO have not been confirmed (Hoyles & Noss 1992), the first and foremost of which was the 'cognitive-effects hypothesis' that stated that experience with LOGO would achieve major changes in general problem-solving skills (De Corte, Greer & Verschaffel, in press, p. 71).
We know that children learn with computers and that they learn about computers. Thus, learning how to make intelligent use of the computer and its software seems to be an important goal in the coming years, and this should begin in elementary school. In this respect, there is a need to question how much emphasis should be placed on learning how to program computers, since nowadays software is widely available and used for many purposes
Overall, the potential of computers for enhancing learning in mathematics has been demonstrated. Nevertheless, it remains a crucial task for mathematics educators - in cooperation with experts from several domains such as informatics, psychology, and pedagogy - to develop significant software that is in harmony with constructivist perspectives on learning and teaching. The standards of mathematics education should be the guideline here, not the technical limitations and possibilities of hard- and software.
(2) Some references:
Bell, A., Burkhardt, H., McIntosh, A. & Moore, G.: 1978, A Calculator Experiment in a
Primary School, The Shell Centre for Mathematical Education, Nottingham.
Bright, G., Waxman, H. & Williams, S.: 1994, Impact of Calculators on Mathematics
Instruction, University Press of America, New York.
Clarke, O. & Kelly, B.: 1989, 'Calculators in the Primary School - The Time Has Come'
in B. Doig (ed.) Everyone Counts. Proceedings of the 27th Annual Conference of the
Mathematical Association of Victoria, MAV, Melbourne, 27-30.
Fey, J.: 1989, Technology and mathematics education: A survey of recent developments and important problems. Educational Studies in Mathematics, 20, 237-272.
Groves, S. & Cheeseman, J.: 1993, 'Young ChildrenÂ´s Number Concepts - The Effect of Calculator Use on Teacher Expectations.' in B. Atweh et al. (eds.) Contexts in Mathematics Education. Proceedings of the 16th Annual Conference of the Mathematics Eduaction Research Group of Australasia, MERGA, Brisbane, 327-333.
Hembree, R. & Dessart, D.: 1986, 'Effects of Hand-held Calculators in Precollege Mathematics Education: A Meta-analysis.' Journal for Research in Mathematics Education 17 (2), 83-99. (**)
Kaput, J. J.: 1992, 'Technology and Mathematics Education' in D. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, NCTM, Reston, VA, 515-556.
Shuard, H.: 1992, CAN: 'Calculator Use in the Primary Grades in England and Wales''in J. Fey (ed.), Calculators in Mathematics Education, NCTM, Reston, VA, 33-45.
Shuard, H. et al.: 1991, Calculators, Children, and Mathematics, Simon & Schuster, Hemel Hempstead.
Shumway, R. J.: 1989, 'Technology, Mathematics and the International Congress' in T. Cooney (ed.), American Perspectives on the Sixth International Congress on Mathematical Education, NCTM, Reston, VA, 15-20.
Suydam, M. N.: 1983, 'Calculators in the Pre-Secondary School' in M. Zweng, T. Green, J. Kilpatrick, H. Pollak & M. Suydam (eds.), Proceedings of the Fourth International Congress on Mathematical Education, BirkhÃ¤user, Boston, 637-640.
(**)Especially useful - it is a meta-analysis of many different studies that have been done in the U.S. on calculator use.
(3) You should contact Gary Martin who heads the Research Section at the NCTM
headquarters for further information - I am sure he can help you ....
email@example.com or (703) 620-9840 X2191.
(4) You can get NCTM's position statements on calculators (#251) and technology (#264)
by 'Fax on Demand' -- call (800) 220-8483 and follow the instructions; alternately,
you can simply call NCTM at (703) 620-9840 and request a hardcopy - it will be