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• Illustrated analysis of Rule of Four using Maple

  383-404

  György Maróti

  Views:

  128

  Rule of Four, as a basic didactic principle, was formulated among the NCTM 2000 standards (see [14]) and since then it is quoted by numerous books and publications (see [4], [9], [12]). Practically we can say it is accepted by the community of didactic experts. The usage of the Rule of Four, however, has been realized mainly in the field of calculus, in fact certain authors restrict the wording of the principle to the calculus itself (e.g. [3]).
  Calculus is a pleasant field, indeed. A sequence of values of a function provides us with example for numeric representation, while the formula and the graph of the function illustrate symbolic and graphical representations, respectively. In the end by wording the basic features of the function on natural language we gain textual representation.
  This idyllic scene, however, becomes more complex when we leave the frame of calculus. In this paper we investigate the consequences of the usage of Rule of Four outside calculus. We discuss the different types of representations and show several examples which make the multiple features of representation evident. The examples are from different fields of mathematics and are created by the computer algebra system Maple, which turns out to be an excellent tool for illustration and visualization of the maim features of mathematical objects.
  Next we introduce the concept of basic representation and rational representation, which is considered as the mathematical notion of "didactic usable" or "didactic rational" representation. In the end we generalize the notion of numeric representation, which leads us a more widely usable didactic principle which can be considered as a generalization of Rule of Four.

• The investigation of students' skills in the process of function concept creation

  249-266

  Gyöngyi Szanyi

  Views:

  183

  Function is a basic concept of mathematics, in particular, mathematical analysis. After an analysis of the function concept development process, I propose a model of rule following and rule recognition skills development that combines features of the van Hiele levels and the levels of language about function [11]. Using this model I investigate students' rule following and rule recognition skills from the viewpoint of the preparation for the function concept of sixth grade students (12-13 years old) in the Ukrainian and Hungarian education system.

• Rational errors in learning fractions among 5th grade students

  347-358

  Tímea Karika

  Views:

  195

  Our paper focuses on empirical research in which we map out the errors in learning fractions. Errors are often logically consistent and rule-based rather than being random. When people face solving an unfamiliar problem, they usually construct rules or strategies in order to solve it (Van Lehn, 1983). These strategies tend to be systematic, often make ‘sense’ to the people who created them but often lead to incorrect solutions (Ben-Zeev, 1996). These mistakes were named rational errors by Ben-Zeev (1996). The research aims to show that when learning fractions, students produce such errors, identified in the literature, and that students who make these kinds of mistakes achieve low results in mathematics tests. The research was done among 5th-grade students.

  Subject Classification: 97C10, 97C30, 97C70, 97D60, 97D70, 97F50

• Trigonometric identities via combinatorics

  73-91

  Beáta Bényi

  Views:

  234

  In this paper we consider the combinatorial approach of the multi-angle formulas sin nΘ and cos nΘ. We describe a simple "drawing rule" for deriving the formulas immediately. We recall some theoretical background, historical remarks, and show some topics that is connected to this problem, as Chebyshev polynomials, matching polynomials, Lucas polynomial sequences.

  Subject Classification: 05A19