Volume VIII , issue 2 ( 2005 ) | back |

Quadratic functions in several variables | 53$-$60 |

**Abstract**

In secondary schools students learn to investigate the behavior of the quadratic function in one variable, and to find the point where the function reaches its extremal value. The purpose of this article is to demonstrate how the idea which is applied to functions in one variable can be extended and applied to functions in several variables. We present the procedure to determine whether a quadratic function in several variables has a minimum or a maximum, and if it has, to find points in which the extremal value is reached. This procedure leads to several theoretical results.

**Keywords:** Quadratic functions in several variables; quadratic forms; extremal values; definitness, Sylvester Criterion.

Possibilities and limitations of Scientific Workplace in studying trigonometric functions | 61$-$72 |

**Abstract**

This paper presents the investigations regarding the role of computer in examining trigonometric functions, in particular in solving trigonometric equations and inequalities. The advantages and disadvantages of the applied procedure, are particularly emphasized.

**Keywords:** CAS; teaching-learning process; trigonometry.

Towards basic standards for research in mathematics education | 73$-$81 |

**Abstract**

Although there seems to be a consensus among mathematics educators that judgments of the quality of research should be based on a set of widely accepted criteria applied in a constructive, non-dogmatic way, such criteria are not yet available. Instead, because a common vocabulary has not been used, various sets of criteria for research quality have been proposed, and no one has attempted to uncover those latent issues concerning quality whose realization and a step-by-step utilization would be particularly beneficial for novice researchers. This paper proposes the grouping of criteria for the quality of research in mathematics education into three basic comprehensive standards. The appropriateness of the three standards is established through an analysis of a representative sample of proposed sets of criteria. The standards are examined in terms of suitable indicators and then applied to the realization of a study searching for the dimension of mathematics attitude that mainly influences mathematics achievement. Although the standards may appear to apply only to the evaluation of research reports, their use may also increase the quality of the design and management of research activities for both quantitative and qualitative research studies.

**Keywords:** Standards; research; mathematics education.

Problema vvedeniya fundamental'nyh matematiqeskih ponyatii v uqebnom processe [The problem of introducing basic mathematical concepts in the learning process] | 83$-$87 |

**Abstract**

Following the requirements elaborated by Academician S. M. Nikol'skii and Professor M. K. Potapov that fundamental mathematical concepts of school mathematics at any level have to be motivated by the practical use, the author treats a way in which variables enter school contents. Here, this way is condensed in the form of a series of definitions.

**Keywords:** Justification principle; defining basic mathematical concepts.

Division---a systematic search for true digits | 89$-$101 |

**Abstract**

For the sake of an easier understanding of the procedure of division, first this operation is conceived as continual subtraction. Taking into account place values of groups of digits of dividend, while proceeding with successive subtraction of divisor from such groups a slow, and in the real time feasible method of division is established. Based on this method, it is shown how a long division (quotients having two or more digits) is reduced to the short divisions (quotients are one-digit). In the case of short division, the first (second) guide number of divisor is defined which rounds down the corresponding dividend. Increasing by one the first (second) digit of divisor, it becomes rounded up and by dividing these guide numbers by the numbers that are increased by one, two methods of division are obtained, both having their advantages and disadvantages. In Section 6, a combined method is presented that should be at the top of practice in division. All three of these methods are clearly defined algorithms which are neat (free from trial and error correcting and erasing) and they successively produce true digits. This paper is intended for teachers, and it has a form containing all necessary details and examples that are within their reach.

**Keywords:** Long division; Short division; ``Increase-by-one'' methods.