Jarmila Novotná

How Structure Sense for Algebraic Expressions or Equations is Related to Structure Sense for Abstract Algebra

Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense components, and that several components of university algebra structure sense are analogies of high school algebra structure sense components. We present a theoretical argument for these hypotheses, with some examples. We recommend emphasizing structure sense in high school algebra in the hope of easing students’ paths in university algebra.

Oborové didaktiky: vývoj – stav – perspektivy

Oborove didaktiky jsou vědnimi disciplinami zaměřenými na oborově specifickou dimenzi vyucovani a uceni ve skole i mimo ni. Jejich agenda saha od vymezovani a zdůvodňovani cilů oboroveho vyucovani a uceni přes výběr, legitimizaci a didaktickou transformaci obsahů až k metodicke strukturaci ucebnich procesů. Problemem ceskeho skolstvi je, že oborove didaktiky u nas nebyly po dlouhou dobu s naležitými naroky systematicky pěstovany. Přesto se rozvinuly do podoby relativně svebytných vědnich disciplin, jak o tom svědci tato kniha. Jejim cilem je podat pohled na vývoj a stav oborových didaktik v Ceske republice a naznacit perspektivy jejich dalsiho směřovani.

Cultural and linguistic problems in the use of authentic textbooks when teaching mathematics in a foreign language

This paper is a part of a longitudinal study focusing on qualitative aspects of learning in a foreign language in the development of cognitive processes in mathematics. The aim of the paper is to present a more complex analysis of textbook-based obstacles to communication. These obstacles originate in the process of vocabulary and grammar acquisition within a particular multicultural and sociocultural context. The study was carried out using mathematics textbooks from English-speaking countries which are used when teaching mathematics in English to Czech students.

Didactical Variability in Teacher Education

It can be often observed that some school students can use taught knowledge in new contexts, while others although familiar with the taught algorithms are not able to apply their knowledge in new contexts. The chapter starts from this observation. The main focus is consideration of these inter-personal differences as an effect of the teachers’ didactical variability (Sarrazy 2002) in the domain of word problems. Students of teachers with weak didactical variability in the domain of word problems are strong in standard problems, while those of teachers with advanced didactical variability may be weaker in them but better performing when solving non-standard, non-algorithmic problems (see e.g. Novotna, J., & Sarrazy, B. (2009). Teacher’s didactical variability and its role in mathematics education. In: Proceedings of CERME 6. Lyon, in print). The text deals with the crucial question how to increase teachers’ didactical variability.

Students and Their Teacher in a Didactical Situation

Giving students the space to actively participate in the introduction of new knowledge through their own independent discovery is one of the demands of pedagogical theory and curricular documents. For example, Czech official pedagogical documents demand that pupils develop their problem solving competence by “making use of the acquired knowledge to discover/identify various ways of a problem solution” (Framework Education Programme for Basic Education, 2007, p. 12).

Developing Mathematics Educators

This chapter addresses, from various perspectives, issues associated with teacher education and its development. Several categories of mathematics educators are characterized and their development and roles in the teaching/learning processes are summarized. Cooperation between teachers and researchers as well as the concept of teachers as researchers are discussed from different points of view. The crucial role that observations play at all levels is analyzed and illustrated by two different models of implementation of observations into teachers’ and researchers’ practice. Throughout the chapter the influence of the research of Guy Brousseau on mathematics education research is recognized.

Ornaments and Tessellations: Encouraging Creativity in the Mathematics Classroom

The presented paper focuses on encouraging creativity in mathematics lessons in heterogeneous mathematics classrooms. It is an extended version of a paper presented at ICME 2016 in Hamburg. It describes teaching experiments conducted within the frame of the project M3EaL: Multiculturalism, Migration, Mathematics Education and Language (526333-LLP-1-2012-1-IT-COMENIUS-CMP), a multilateral project whose aim was development of teaching units to support immigrant pupils and pupils from different socio-cultural backgrounds (m3eal.dm.unipi.it). The paper shows that if pre-service and in-service teachers face a situation in which they cannot rely on traditional textbooks and ways of doing mathematics—in this case a culturally heterogeneous classroom—they tend to be very creative in planning their lesson and at the same time encourage their pupils’ creativity. Thus cultural heterogeneity may be perceived as advantage as it may result in breaking down stereotypes in mathematics classrooms.

Traditional Versus Investigative Approaches to Teaching Algebra at the Lower Secondary Level

Van Ameron (2002, p. 3) stated that “Algebra is known to be a major stumbling block in school mathematics, both in the past and present. Historical studies on the developments of algebra education in the twentieth century show that the algebra studied in secondary school has not changed much over the years”. Sutherland (2000) presented a comparative study of algebra curricula in twelve countries all over the world.

Student’s Levels of Understanding Word Problems

In the paper I focus on the following aspects: grasping word problem assignments levels of understanding and students’ solution strategies and their relation to their understanding of the solved word problems. This is a part of broader research project aimed at the following questions: What is going on between the posing of a word problem and the students’ answer? What kinds of experience will help students to perform well on word problems? What sort of practice may help?