Volume VII , issue 2 ( 2004 ) | back |

A versatile tool to promote link between creative production and conceptual understanding | 61$-$70 |

**Abstract**

With respect to constructivist views on the nature of mathematical knowledge and the genesis of heuristic processes in the mind of the learner, the abundance of problems, richness of ideas and students' possibilities and power to develop intuition have to be redefined by utilizing versatile technological tools. This paper highlights how link between creative production and conceptual understanding may be promoted by use of a progressive pocket computer. It focuses on learning based upon the interplay of different representations of mathematical objects, the use of which would improve problem solving abilities as well as the understanding of underlying mathematical concepts. We describe this kind of learning and examine its empirical values by using a modification of a classical extreme value problem.

**Keywords:** Computer-based learning, conceptual knowledge, constructivism,
heuristics, mathematics education, problem solving, procedural knowledge,
representation, visualization.

A broader way through themas of elementary school mathematics, VII | 71$-$93 |

**Abstract**

An extension of the block of numbers 1--100 to the block of numbers 1--1000 is considered. The reason for singling out this block is a thorough elaboration of the procedures of decimalization. Represented as sums of hundreds and one- or two-digit numbers, addition and subtraction of three-digit numbers is reduced to easy and already known cases of performing these operations. Represented as sums of units, tens and hundreds, these two operations are also carried out on such summands. To the latter case, the shorthand forms of performing these operations on digits, are attached. Bound to the cases of one-digit multipliers or divisors, multiplication and division are treated similarly. By analogy, these acquired skills and understanding are expected to be carried over in cases of all other numbers, when operations are chiefly performed on digits. At the end, we summarize main points of our approach to the elaboration of arithmetic, being exposed in all seven parts of this paper that run under the same title.

**Keywords:** Block of numbers 1--1000, Decimalization.

What factors may influence collaborative problem solving performance? | 95$-$101 |

**Abstract**

By using a sample of 21 pairs of eleventh grade students (10 comprised mathematically-talented students), this study examined paired problem solving performance in terms of paired students' features concerning mathematical self-concept and cognitive empathy and found that collaborative problem solving performance was positively influenced by average mathematical self-concept for paired talented students. The talented pairs' bootstrapped data evidenced that this performance could be explained by a multiple liner regression model, where average mathematical self-concept for paired students and average cognitive empathy for paired students had zero or positive influence, whereas absolute mathematical self-concept distance for paired students and absolute cognitive empathy distance for paired students had zero or negative effect. Though not supported, the validity of that model was indicated by the average pairs' bootstrapped data.

**Keywords:** Paired problem solving, mathematical self-concept, cognitive empathy, bootstrapping, talented students,
mathematics education, upper secondary education.

Comparing two ways of elaboration of complex numbers | 103$-$106 |

**Abstract**

The aim of this article is to compare two well-known ways of introducing complex numbers. As a result, we find that the treatment of complex numbers as polynomials in one variable $i$, is the most acceptable for students. The concepts of ring and field play a hidden but crucial role in such an approach.

**Keywords:** Complex numbers, polynomials, ring, field.

Karamata's products of two complex numbers | 107$-$116 |

**Abstract**

In his paper Über die Anwendung der komplexen Zahlen in der Elementargeometrie}, Bilten na Društvoto na matematičarite i fizičarite od N.R. Makedonija, kn. I, Skopje, (1950), 55--81 (in Serbian, resume in German) and in the university textbook Complex Numbers, Belgrade, 1950, Jovan Karamata (1902--1967) maintains that, in planimetry, a complex number can assume the role of a vector, whereby in definition and solving of certain problems, there appears the product $\overline{a} b=A+Bi$ where $\overline{a}$ is the conjugated number of $a$, while $a$ and $b$ are complex numbers which correspond to free vectors $\vec{a}$ and $\vec{b}$. Using geometric interpretation of $a$ and $b$, Karamata expresses $A$ and $B$ as $A=|a||b|\cos \alpha$ and $B=|a||b|\sin \alpha$. In order to underline the geometric sense of these expressions, Karamata denotes them as $A=(a\perp b)$, resp. $B=(a\mid b)$ designating them ``orthogonal product'' and ``parallel product''. By means of these two symbols, considered as products, Karamata interprets those problems in planimetry which correspond to parallelity and orthogonality, and shows how they can be used in deriving of Papos-Pascal and Desargues propositions.

**Keywords:** Karamata, projective geometry, complex number,
vector, Papos-Pascal, Desargues.