Volume XV , issue 2 ( 2012 ) | back |

An exploration of students' conceptual knowledge built in a first ordinary differential equations course (Part II) | 63$-$84 |

**Abstract**

This is the second part of a study published in a previous issue of this Journal whose main goal was to analyze and document the types of knowledge that university students exhibit to deal with fundamental issues that they had studied in a first ordinary differential equation course. Here, we focus on analyzing and discussing the following research questions: (ii)~How do students make sense of, interpret and deal with the concept of solution to an ODE? (iii)~What systems of representation do they use to represent and explore the information embedded in those questions in order to answer them? And, to what extent do the students privilege the use of certain type of representation?

**Keywords:** Learning of Mathematics; the concept of ordinary differential equation; representations.

An empirical study in convergence via iteration and its visualization | 85$-$112 |

**Abstract**

Our aim is to provide an educative experience for High School students leading to a precise verbal description of the notion of convergence of a sequence of numbers generated by an iterative process triggered by the visualization of an unending geometric progression. Iteration as a guiding idea is chosen because it is easy to grasp, opens the door to further mathematical topics of the curriculum and encourages the use of technology. The experience is structured as an interview where a computer-generated tool, providing data generated by iteration and their visual dynamic representations, is available as an unavoidable aid to our goal of getting insight before formality. It all ends up to a very simple observation: the step-by-step implementation of a manual/visual routine together with reporting verbally what you see and what you don't see at every step is the clue to the understanding of the Weierstrassian definition of convergence, showing that its reputation of unintelligibility is hardly deserved. A detailed discussion of the experience concludes the exposition.

**Keywords:** Iteration; symbol; visualization; convergence.

Monotonicity of certain Riemann-type sums | 113$-$120 |

**Abstract**

In this short note we prove with elementary techniques that the sequence $x_n=\sum_{k=1}^n\frac{n}{n^2+k^2}$ is increasing and its limit is $\frac{\pi}{4}$. Moreover, we give a sufficient condition for the monotonicity of some Riemann-type sums assigned to uniform subdivisions as a function of the number of the intervals from the subdivision. This mathematical content came up in a group discussion during an IBL centered teacher training activity and reflects a crucial problem is implementing IBL teaching attitudes in the framework of a highly scientific curricula (such as the Romanian mathematics curricula for upper secondary school).

**Keywords:** Monotone sequence; Riemann sums.

Maximally balanced connected partition problem in graphs: application in education | 121$-$132 |

**Abstract**

This paper presents the maximally balanced connected partition (MBCP) problem in graphs. MBCP is to partition a weighted connected graph into the two connected subgraphs with minimal misbalance, i.e., the sums of vertex weights in two subgraphs are as much equal as possible. The MBCP has many applications both in science and practice, including education. As an illustration of the application of MBCP, a concrete example of organizing the course Selected Topics of Number Theory is analyzed and one balanced partition is suggested. Several algorithms for solving this NP hard problem are also studied.

**Keywords:** Graph partitioning; computers in education; course organization.

On angles and angle measurements | 133$-$140 |

**Abstract**

In this note, we give a brief discussion on angles and angle measurements from the point of view of our teaching practice. Two things are clarified, namely definition of the notion of angle, and once that is done, various operations and measurements of angles.

**Keywords:** Angle; angle measurement.