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Article Excerpt
The mathematical problem solving ability of school children aged 12-13 years was assessed using the computer-based learning program ENCAL (Harrop, 2001, p. 97). The system helps children develop their concept of number and their skills with multiplication and addition with the help of a software calculator and some additional computer-based support. ENCAL is a novel method for improving the understanding of numerical structure in arithmetic through the use of computer-based multiple linked external representations. The system exploits three representations: iconic, calculator and datatree. Small-scale studies contributed to the design, and the results of a final evaluation study showed that the graphical constructs used in ENCAL are beneficial to understanding, suggesting that the approach can be usefully exploited in classroom mathematics education.
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Current computer software can create or solve arithmetic problems. However, a way needs to be found that will improve software used for the computation of arithmetic expressions that involve both order of operations and calculator use. More specifically, a computer-based learning environment needs to be established which will: (a) facilitate the conceptual and procedural understanding of arithmetic calculations involving order of operations; and (b) enable calculator calculations to be carried out whilst taking into account calculator behaviour. Arithmetic understanding (e.g., computational procedures) is a fundamental prerequisite for the evaluation of arithmetic expressions, particularly those that require the ordering of operations and the use of calculators.
Much computer-based software is directed at helping children plan and solve arithmetic word problems (e.g., SEMCALC, Schwarz, 1982; ICE, Kaput, 1989; TAPS, Derry, Hawkes, Diefenbach, & Kegelmann, 1993; ANIMATE, Nathan & Resnick, 1993; PLANNER, Schwarz, Nathan, & Resnick, 1996). However, in order to correctly compute an arithmetic function which results from a particular word problem, it is often first necessary to consider the order in which operations are to be carried out. The sequence in which operations are undertaken can be crucial when using a calculator, because simply keying in data from left to right could yield the wrong answer. Despite the fundamental importance of acquiring appropriate computational procedures as a basis for understanding the number system (as put forward by Ohlsson, 1987; Bell, Costello, & Kuchemann, 1983), and in particular with regard to order of operations using calculators (Ecker, 1989; Wiebe, 1989a), little or no computer-based research appears to address the issue of both conceptual and procedural understanding of multi-operator arithmetic computations with calculators using external representations once appropriate information has been elicited from the semantics of a problem.
Computation
An important aspect of problem solving is the fact that once information has been elicited from the semantics of a word problem, a child will need to compute the answer. During computation he or she will need to take into account: order of operations, parentheses, and calculator behaviour. These aspects of computation are referred to below.
Ecker (1989) highlights the computation problems of order of operations. For example, when considering 1 + 2 X 3, he states do you first add 1 and 2, and then multiply the result by 3, or do you multiply first, and then add? Technically, this is known as the issue of precedence: conventionally, multiplication precedes addition. This precedence (order of operations) hierarchy resolves ambiguity without the use of parentheses. Priority is given to operators in the following hierarchy: division/multiplication, addition/subtraction. Ecker claims that most confusion usually occurs when mixed operations are used which have different precedences, because the precedence has to be remembered and can be mistaken.
Where expressions use parentheses, then operations in the parentheses have top priority. When parentheses occur inside of parentheses, the innermost expressions are evaluated first.
Significant computation errors made by pupils occur through lack of understanding of parentheses (Demana & Osborne, 1988). For example, (3 + 5) X 4 is often thought to be equivalent to 3 + 5 X 4. Four-function (arithmetic) calculators contribute to such misunderstandings, and thus "fail to give correct values of mathematical expressions needed for appropriate prealgebra experiences" (Demana & Osborne, 1988, p. 3). Typical problems associated with calculator use are outlined below.
The Behaviour of Calculators: Why is it Obscure?
An important aspect of calculator use is that it structures arithmetic expressions in specific ways dependent on the logic system used, and consequently the behaviour of a calculator is often at variance with the computational procedures pupils have been taught. For example, in the conventional interpretation (7 X 10) + (6 X 7) = 70 + 42 = 112. However, if a four-function (arithmetic) calculator is used to compute this expression, then children who tend to have a strong left-to-right bias and who perhaps do not understand order of operations and the meaning of parentheses, could type in the data from left to right and get the answer of 532. Therefore, calculators could produce misinterpretations in children's understanding of arithmetic.
Wiebe (1989a) points out that most pupils need help when using calculators, "especially if they are using them with problems involving more than one operation..." (p. 36). He states that different calculators and computer software tools use different internal algorithms for computing answers and thus different answers may be given to the same problem. For example, the sequence 4 + 5 X 3 =, if entered into a four-function (arithmetic) calculator and an algebraic notation (scientific) calculator will give the answers 27, and 19 respectively. As Wiebe (1989b) points out, pupils are astonished that a calculator may produce incorrect answers--they usually assume that they have entered data incorrectly. The evaluation of expressions is a persistent problem. Even at college level, calculator users have trouble with premature commitment (Green & Petre, 1996), in that they tend to work from left to right rather than manipulating an expression (Mayer & Bayman, 1981).
Ruthven & Chaplin (1997) looked at the part played by calculators in children's numerical learning who were aged 10/11 years. One aspect of the study analysed how pupils approached number problems with and without the use of calculators. The research showed that regardless of current teaching policies, outcomes of calculator use could be much influenced by the manner of introducing and using calculators in the classroom. The successful solution of arithmetic expressions requires mapping, and the study also highlighted...
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