Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition

Subtraction can be understood by two basic models—taking away (ta) and determining the difference (dd)—and by its inverse relation to addition. Epistemological analyses and empirical examples show that the two models are not relevant only in single-digit arithmetic. As curricula should be developed in a longitudinal perspective on mathematics learning processes, the article highlights some exemplary steps in which the inverse relation is discussed in light of the two models, namely mental subtraction, the standard algorithms for subtraction, negative numbers and manipulations for solving algebraic equations. For each step, the article presents educational considerations for fostering a flexible use of the two models as well as of the inverse relation between subtraction and addition. In each section, a mathe-didactical analysis is conducted, empirical results from literature as well as from our own case studies are presented and consequences for teaching are sketched.     

Social-Emotional Learning Profiles of Preschoolers' Early School Success: A Person-Centered Approach

Examined how aspects of social-emotional learning (SEL)-specifically, emotion knowledge, emotional and social behaviors, social problem-solving, and self-regulation-clustered to typify groups of children who differ in terms of their motivation to learn, participation in the classroom, and other indices of early school adjustment and academic success. 275 four-year-old children from private day schools and Head Start were directly assessed and observed in these areas, and preschool and kindergarten teachers provided information on social and academic aspects of their school success. Three groups of children were identified: SEL Risk, SEL Competent-Social/Expressive, and SEL Competent-Restrained. Group members differed on demographic dimensions of gender and center type, and groups differed in meaningful ways on school success indices, pointing to needed prevention/intervention programming. In particular, the SEL Risk group could benefit from emotion-focused programming, and the long-term developmental trajectory of the SEL Competent-Restrained group requires study.     

Students’ Conceptions About ‘Radiation’: Results from an Explorative Interview Study of 9th Grade Students

One basis of good teaching is to know about your students?? preconceptions. Studies about typical ideas that students bring to the science classroom have been and continue to be a major field in science education research. This study aims to explore associations and ideas that students have regarding ??radiation??, a term widely used in various fields and necessary to understand fundamental ideas in science. In an explorative study, the perceptions of 50 high school students were examined using semi-structured interviews. The students were 14?C16?years old and were chosen from 7 different high schools in an urban area in Austria. Following an interview guideline, students were asked about their general associations with the term ??radiation?? as well as about their general understanding of different types of radiation. A qualitative analysis of these interviews following the method of Flick (2009) revealed that the students?? associations were, to a great extent, very different from the scientific use of the term. Several conceptions that could inhibit students?? learning processes could be identified. Consequences for the teaching of the topic ??radiation?? in science lessons, which are based on these preconceptions, are presented in the conclusion.     

The Crucible of Classroom Practice: Alchemy and Early Professional Learning in Secondary English Teaching

This paper explores the metaphor of the classroom as a ‘crucible’ for early professional learning where beginning teachers forge professional identities in complex, unpredictable, paradoxical, affectively and physically potent contexts of practice. It works into the dissonances and contradictions of the micro-narratives embedded in the accounts that three high-achieving early career English teacher graduates give of their early professional learning in schools. It traces the conflicts they experience in the gap between ‘economies of performance’ and ‘ecologies of practice’ within which they improvise new professional identities as teachers. Their effective ‘transmutations’ into teachers are contingent on a bewildering range of factors, many of them out of their control. As the medieval transmutation of lead into gold was analogous to personal transformation, purification and perfection, so new teachers aspire to ideals of practice that may be impossible to achieve. The paper moves beyond simplistic conceptualisations of theory–practice binaries in new teachers’ work and evaluates the impact of new pressures from globalised discourses of standardisation and professionalism that erase singularity, particularity and relationality. Rather, it posits professional learning as complex identity work that is variously constrained and enabled by local institutional factors and broader policy contexts that affect the capacity and desire of beginning teachers to introduce and sustain innovative pedagogies in their English classes.     

How to develop mathematics-for-teaching and for understanding: the case of meanings of the equal sign

What kind of mathematical knowledge do prospective teachers need for teaching and for understanding student thinking? And how can its construction be enhanced? This article contributes to the ongoing discussion on mathematics-for-teaching by investigating the case of understanding students’ perspectives on equations and equalities and on meanings of the equal sign. It is shown that diagnostic competence comprises didactically sensitive mathematical knowledge, especially about different meanings of mathematical objects. The theoretical claims are substantiated by a report on a teacher education course, which draws on the analysis of student thinking as an opportunity to construct didactically sensitive mathematical knowledge for teaching for pre-service middle-school mathematics teachers.     

Kindergartenbesuch und späterer Bildungserfolg

This paper analyses the impact of attending kindergarten, the most common early childhood care and education program in West Germany, on children’s placement at secondary school level. The analysis is based on an education production function that controls for parental background and regional variation in the educational system. The analysis confirms the hypothesis, that kindergarten attendance can have a long-lasting positive effect on educational chances. In four of six specifications a longer attendance of early childhood care and education increases the probability of attending a higher level secondary school. However, there are diminishing returns to early childhood education.     

Behavioral and relational correlates of low self-perceived competence in young children

It is developmentally normative for preschoolers (aged four to five years) to have self-perceptions that are typically positive and even over-inflated. However, because not all children have positive self-perceptions in early childhood it is important to understand factors that influence the development of self-perceptions, especially low self-perceptions, in early childhood. Therefore, this study examined the associations between self-perceptions and teacher ratings of social behaviors, subtypes of withdrawal, and peer ratings of acceptance/rejection in preschoolers. Participants included 199 children (107 males, 92 females) attending two early childhood programs in a midsized community in the western United States. The major findings were that children in early childhood who have lower self-perceptions of their abilities than their peers tended to engage in more reticence, solitary-passive withdrawal, and solitary-active behavior.     

Anthropomorphically speaking: on communication between teachers and children in early childhood biology education

In this study a particular kind of figurative language, so‐called anthropomorphic speech, is analysed in the context of science activities in a preschool setting. Anthropomorphism means speaking about something non‐human in human terms. Can any systematic pattern be seen with regard to when such speech is used? Do children and/or teachers introduce this kind of talking and how is it responded to by the interlocutor(s)? Of 128 instances of anthropomorphism found, 24 were made by the children and 104 by the teachers. Children sometimes respond in line with the introduction of such speech but they also at times reject this way of speaking. Anthropomorphic speech is discussed as a strategy for the teachers in handling the dilemma of how to connect with children’s experiences and terms, on the one hand, and developing children’s understanding, on the other hand.