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• Some Pythagorean type equations concerning arithmetic functions

  157-179

  Ildikó Kézér

  Views:

  40

  We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x² + y² = z², we study equation f(x²) + f(y²) = f(z²) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.

  Subject Classification: 97F60, 11A25

• How do secondary school students from the Kurdistan Region of Iraq understand the concept of function?

  221-244

  Yusuf Fakhraddin Hussein

  Csaba Csíkos

  Views:

  101

  The study investigates secondary school students' understanding of the concept of function. The paper focuses on three main aspects: students' ability to define the concept of function; students' ability to recognize different representations of function; and students' ability to convert between different representations. A test was developed to assess the three main constructs of the study and administered to 342 students in secondary schools in the Kurdistan Region of Iraq. According to the results, students have diffculties in recognizing different representations of function and conversion between them. Connections between different parts of the test may provide hints on educational challenges of how to appropriately teach functions.

  Subject Classification: 26Bxx, 97D60

• Promoting a meaningful learning of double integrals through routes of digital tasks

  107-134

  Francesca Alessio

  Lucio Demeio

  Agnese Ilaria Telloni

  Views:

  108

  Within a wider project aimed at innovating the teaching of mathematics for freshmen, in this study we describe the design and the implementation of two routes of digital tasks aimed at fostering students' approach to double integrals. The tasks are built on a formative assessment frame and classical works on problem solving. They provide facilitative and response-specific feedback and the possibility to request different hints. In this way, students may be guided to the development of well-connected knowledge, operative and decision-making skills. We investigated the effects of the interaction with the digital tasks on the learning of engineering freshmen, by comparing the behaviours of students who worked with the digital tasks (experimental group, N=19) and students who did not (control group, N=19). We detected that students in the experimental group showed more exibility of thinking and obtained better results in the final exam than students in the control group. The results confirmed the effectiveness of the experimental educational path and offered us interesting indications for further studies.

  Subject Classification: 97D40, 97U70, 44A45

• Potential, actual and practical variations for teaching functions: cases study in China and France

  157-166

  Luxizi Zhang

  Views:

  35

  This contribution is based on two major hypotheses: task design is the core of teachers’ work, and variation is the core of task design. Taking into account variation in task design has a profound theoretical foundation in China and France. Developing my PhD with two co-supervisors, in China and France, I wish to seize this opportunity for constructing an analytic model of “teaching mathematics through variation” making profit of this theoretical diversity. This model distinguishes between potential variation and practical variation and is based on the process of transforming potential variation into actual variation, and of using practical variation for rethinking potential variation. The design of this model is based both on theoretical networking, and on case studies, in France and China. In this contribution, we will focus on a critical aspect in the two cases, from potential to practical variation.

  Subject Classification: 97-06