Open Mathematical Education Notes

49--62

Authors: Daniel A. Romano

The goal of this paper is to present to the international academic community some specificity in mathematics education of the Republic of Srpska [RS] primary school teachers. The paper draws on some previously published texts on the mathematical and methodical literacy of the RS prospective primary school teachers. The observations on this segment of mathematical education in the RS are the final comments of the author within the project „Research of mathematical and pedagogical literacy of prospective primary school teachers” which is realized by the Scientific Society of Mathematicians Banja Luka. The author deeply believes that accepting these observations and assessments can be a very useful argument to our academic community in negotiating with the RS social-political community on rising the quality of mathematics education. With this commitment, the author has revealed several examples of failures in the education of prospective elementary school teachers in the following domains: (a) specific geometric knowledge necessary for teachers; (b) specific methodological-geometric knowledge necessary for teachers, and (c) understanding of the process of teaching and the process of student's learning of geometry. These omissions in mathematics education of elementary school teachers are the systemic failures and cannot be eliminated within the existing educational paradigm by stronger efforts of the teachers themselves according to the author’s deep belief.

9--28

Authors: Sergei Abramovich and Michael L. Connell

The content of this paper stems from an earlier inquiry into the use of computers in secondary mathematics teacher education. The advent of the modern day digital tools (such as Maple and Wolfram Alpha) capable of sophisticated symbolic computations calls for the revision of technology uses set forth in the early years of teaching mathematics through problem solving [26]. Nowadays, technology not only facilitates problem solving to the extent of making it just “easy”, but provides learning opportunities for deeper inquiries into seemingly sealed for non-professional mathematical investigations that require formal reasoning. The activities described in the article are connected to recent standards for teaching mathematics and recommendations for the preparation of teacher candidates published in the United States and elsewhere in the world. Through the suggested activities one can appreciate the integration of mathematical understanding, conceptual knowledge, procedural skills and technological competence.

65-104

Authors: Sergei Abramovich, Jonathan Burns, Scott Campbell and Arcadii Z. Grinshpan

The paper shows that mathematics as an applied tool – when used consistently at the elementary, secondary, and post-secondary educational levels – has true potential to improve the teaching of the subject matter, increase students’ interest in STEM disciplines, and bridge the gap between the formal and engineering approaches to mathematics education. The adoption of the action learning techniques that have successfully been used in academia in the contexts of business management, social sciences, and teacher development is proposed as signature pedagogy for K-20 mathematics education. The appropriate use of technology and hands-on activities as an enhancement of mathematical applications to real-life projects of different degrees of complexity is discussed. Many examples are given.

19-39

Authors: María Ángeles Navarro and Pedro Pérez Carreras

Starting with an intuitive understanding of the idea of one-to-one correspondence via counting collections, we develop the notion of functional relationship in discrete and continuous settings. This concept image build up enables us to pursue the study of growth and rate of growth in the context of sequences and simple functions given by analytic expressions as well as the study under which conditions growth and its rate coincide, triggering the appearance of the number e as an added bonus. Under the constructivist perspective which focuses in individual thinking and the continuing act of creating learning opportunities, we shall use a carefully planned semi-structured Socratic interview in conjunction with a computer-generated tool. The dialog tests the solidity of those ideas and arguments, which students bring and that are based in prior educative experiences, when challenged to dig out and verbalize their own beliefs and questioning techniques try to ease the difficulties most students have with concept translations to and from verbal representations (some more considerable than others) which are the key to understanding and communicating effectively. The tool, apart from easing the burden of calculations, promotes higher levels of engagement encouraging students‟ participation, provides them with numerical data and visual realizations which, paradoxically as it may sound, facilitate abstraction by linking the processes of discovery, understanding and conceptualization.

101-113

Authors: Mirela Mrdja, Daniel A. Romano and Marina Zubac

In this paper, the authors attempt to make the analysis of unacceptable and missed student responses when calculating a limit of functions. Based on the collected data and derived conclusions, relying on APOS theory and the SOLO taxonomy, we estimate that the concept of limit values and processes with that concept (but not procedurally using this concept) for the majority of the student population is acceptable with considerable difficulty. А reconstruction is offered of student‟s mental images using categorical terms RBC + C theory of abstraction and Sfard‟s theoretical model for the learning of mathematical concepts.