Volume XVII , issue 2 ( 2014 ) | back |

Structuring the subject matter of arithmetic, I | 51$-$75 |

**Abstract**

In the case of construction of the block of numbers up to 100 (block $N_{100}$), all processes that lead from observation to the creation of abstract concepts are traced and didactically shaped. Sums with summands in the block $N_{20}$ having the value exceeding 20 are used to extend this block to the block $N_{100}$. The addition and subtraction of two-digit numbers is treated and, for the sake of understanding, all intermediate steps are expressed in words and symbols. But when these operations are performed automatically these steps are suppressed and the expressing in words is reduced to its inner speech contraction. The block $N_{100}$ is a natural frame within which multiplication is introduced and where the multiplication table is built up. In the school practice the meaning of multiplication is esatblished through examples of situations having the structure of a finite family of finite equipotent sets which we call multiplicative scheme. Some suitable models of multiplicative scheme (as, for example, boxes with marbles) are used to esatblish main properties of multiplication. Let us also add that we build multiplication table grouping its entries according to the ways how the corresponding products are calculated and we find that these ways of calculation should be learnt, instead of learning this table by rote.

**Keywords:** Block of numbers up to 100; addition and subtraction of two-digit numbers;
multiplicative scheme; main properties of multiplication.

**MSC Subject Classification:** 97H20

**MathEduc Subject Classification:** H22

The relationship between the change of variable theorem and the fundamental theorem of calculus for the Lebesgue integral | 76$-$83 |

**Abstract**

We discuss an interplay between five versions of the Change of Variable Theorem and the Fundamental Theorem of Calculus for the Lebesgue integral. We show that, under certain assumptions, they imply one another.

**Keywords:** Change of variable theorem; fundamental theorem of calculus;
Lebesgue integral; absolutely continuous.

**MSC Subject Classification:** 97I40, 26A42

**MathEduc Subject Classification:** I55

Geometrical versus analytical approach in problem solving---an exploratory study | 84$-$95 |

**Abstract**

In this study we analyse the geometrical visualization as a part of the process of solution. In total 263 students in the first year of study at three different universities in three different countries (Poland, Slovakia and Spain) were asked to solve four mathematical problems. The analysis of the results of all students showed that geometrical visualization for problems where there is a possibility to choose different ways of solving is not as frequent as one would expect even if the problem is hardly solvable without it. Especially Spanish students prefer to solve the problem analytically. Visual reasoning is mostly regarded as an intuitive, preliminary stage in the reasoning processes and nothing needs to be done to develop it. On the basis of our results we consider it necessary to regain the use of geometry in the classroom and encourage visualization, the use of the figure, the physical and spatial intuition.

**Keywords:** Geometrical arguments; analytical arguments; visualization; problem solving.

**MSC Subject Classification:** 97C30, 97G99

**MathEduc Subject Classification:** C30