Does the Opportunity–Propensity Framework predict the early mathematics skills of low-income pre-kindergarten children?

Prior studies have shown that the variables described in the Opportunity–Propensity (O–P) Framework have successfully accounted for the mathematics and science achievement of students in grades 1–3 and 8–12. The two goals of the present study were to (1) determine whether the O–P Framework could also account for individual differences in the early mathematics skills of low-income, pre-kindergarten children and (2) determine whether latent variables constructed from measured variables would account for performance in the manner specified in the O–P model. The O–P Framework assumes that high achievement in mathematics is a function of three categories of factors: (a) antecedent factors, variables that operate early in a child’s life and explain the emergence of opportunities and propensities, (b) opportunity factors, variables that measure a child’s opportunity to learn mathematics content at home and school, and (c) propensity factors, variables that capture a child’s propensity for learning in terms of self-regulation, motivation, and prior cognitive skills. To test the fit of this model for low-income children during the year before they attend kindergarten, the authors conducted a secondary analysis of achievement and background data from the Early Childhood Longitudinal Study-Birth (ECLS-B) Cohort data set. Structural equation modeling indicated significant associations between the antecedent factor, opportunity factor, and propensity factor, and between the opportunity factor and pre-kindergarten mathematics achievement. The results confirmed the fit of the model and identified the kinds of learning experiences that could promote the acquisition of mathematics skills in low-income children and improve their readiness to learn in first grade and beyond.     

An assessment of a social–cognitive model of academic performance in mathematics in Argentinean middle school students

This study tested a set of hypotheses derived from the model of academic achievement in mathematics of the Social Cognitive Career Theory in a sample of Argentinean middle school students. To this aim, 277 students (male and female; age: 13–15 years) were assessed using the following instruments: logical–mathematical self-efficacy scale, mathematics outcome expectations, mathematics performance goals, and mathematics ability test. All of these instruments had been adapted for use in Argentinean students. Academic achievement in mathematics (i.e., grades obtained on regular school exams) was the variable to be modeled through the path analysis technique. The analysis allowed identification of interrelations among the variables and identification of direct and indirect effects. Academic achievement in mathematics was partially explained by the model. Overall, the results support the theoretical postulates of Social Cognitive Career Theory.     

Fostering children’s relationship with nature: exploring the potential of Forest School

Forest School offers children opportunities to explore the natural environment regularly over an extended period of time. It is based on six key principles, one of which states Forest School fosters a relationship with nature and develops long-term pro-environment attitudes through regular personal experiences in nature. We consider to what extent this aim is achieved by exploring the published evidence. We identify that a relationship with nature could be demonstrated under six themes although there is little robust evidence to support this. We suggest further research is carried out directly linked to this aim of Forest School.     

How do the spellings of children with dyslexia compare with those of nondyslexic children?

Children with dyslexia are believed to have very poor phonological skills for which they compensate, to some extent, through relatively well-developed knowledge of letter patterns. We tested this view in Study 1 by comparing 25 dyslexic children and 25 younger normal children, chosen so that both groups performed, on average, at a second-grade spelling level. Phonological skill was assessed using phoneme counting and nonword spelling tasks. Knowledge of legal and illegal letter patterns was tested using a spelling choice task. The dyslexic children and the younger nondyslexic children performed similarly on all the tasks, and they had difficulty, for the most part, with the same linguistic structures. Supporting the idea that older dyslexics spellings are quite similar to those of typical beginners, we found in Study 2 that experienced teachers could not differentiate between the two groups based on their spellings.     

Assessing kindergarteners’ mathematics problem solving: The development of a cognitive diagnostic test

This study aimed to develop an instrument for assessing kindergarteners’ mathematics problem solving (MPS) by using cognitive diagnostic assessment (CDA). A total of 747 children were recruited to examine the psychometric properties of the cognitive diagnostic test. The results showed that the classification accuracy of 11 cognitive attributes ranged from .68 to .99, with the average being .84. Both the cognitive diagnostic test score and the average mastery probabilities of the 11 cognitive attributes had moderate correlations with the Applied Problem subtest and the Calculation subtest of the Woodcock–Johnson IV Tests of Achievement. Moreover, the correlation between the cognitive diagnostic test and the Applied Problems subtest was higher than that between the cognitive diagnostic test and the Calculation subtest. The results indicated that the formal cognitive diagnostic test was a reliable instrument for assessing kindergarteners’ MPS in the domain of number and operations.     

An examination of the preparation and practice of grades 7–12 mathematics teachers from the Shandong Province in China

This paper presents the findings of a study that examined the preparation and teaching practice of ten teachers of grades 7–12 from the Shandong province in China. This study revealed that a multi-dimensional training system has been developed to help the teachers gradually build up their knowledge base for teaching. The findings of this study indicated that the prospective teacher education emphasized a deep understanding of advanced mathematics, while the teacher professional development provided meaningful and effective ongoing activities for the enhancement of teaching skills. The teacher professional development appears to be an essential supplement to the prospective teacher education for improving teachers’ performance and developing teaching expertise.     

Changes in children’s cognitive development at the start of school in England 2001–2008

Since 1997, England has seen massive changes in the Early Years including the introduction of an early childhood curriculum, free pre‐school education for three‐year‐olds and local programmes for disadvantaged communities. Many of these initiatives took time to introduce and become established. Beginning in 2001, and each year thereafter until 2008, the authors collected consistent data from thousands of children when they started school at the age of four on a range of variables, chosen because they are good predictors of later success. These included vocabulary, early reading and early mathematics. Children from the same set of 472 state primary schools in England were assessed each year. This paper contributes to the existing studies of educational trends over time by examining the extent to which children’s scores on these measures changed over that period; in general, they were found to have remained stable.     

To teach or not to teach the conceptual structure of mathematics? Teachers undervalue the potential of Principle-Oriented explanations

Principle-oriented explanations have demonstrated to foster students’ mathematical understanding, as they integrate conceptual and procedural information to make the solution process tangible to novice students. Teachers, however, often omit conceptual information when explaining procedures. In two experimental studies, we tested the hypothesis that teachers’ tendency to omit conceptual information may have occurred, as teachers generally devalued the potential of principle-oriented explanations. In Study 1, we randomly provided two cohorts of secondary students (N = 129) with principle-oriented versus procedure-oriented explanations on four mathematical topics. Afterwards, students answered a knowledge test. We replicated previous findings that students with principle-oriented explanations outperformed students with procedure-oriented explanations on the knowledge test (application test, transfer test). In Study 2, we gave mathematics teachers (N = 69) these explanations as judgment materials. Teachers randomly rated a balanced set of four explanations of Study 1 which varied in their procedure- versus principle-orientation. We found no significant differences between teachers’ judgments of principle- versus procedure-oriented explanations. Content analyses of the justifications revealed that teachers were more concerned about students’ overload when judging principle-oriented explanations than procedure-oriented explanations. Our findings replicated the beneficial effects of principle-oriented explanations for novice students’ understanding. Furthermore, they highlight the need to sensitize teachers for potential effects of providing principle-oriented explanations.     

The role of cognitive flexibility in young children’s potential for learning under dynamic testing conditions

The aim of the current study was to investigate to what extent children’s potential for learning is related to their level of cognitive flexibility. Potential for learning was measured through a dynamic testing procedure that aimed to measure how much a child can profit from a training procedure integrated into the testing process, including the amount and type of feedback the children required during this training procedure. The study followed a pre-test–training–post-test control group design. Participants were 153 6–7-year-old children. Half of this group of children were provided with a standardised graduated prompts procedure. The other half of the participants performed a non-inductive cognitive task. Children’s cognitive flexibility was measured through a card sorting test and a test of verbal fluency. Results show that cognitive flexibility was positively related to children’s performance, but only for children in the practice-only condition who received no training. These outcomes suggest that dynamic testing, and more in particular, the graduated prompting procedure, supports children’s cognitive flexibility, thereby giving children with weaker flexibility the opportunity to show more of their cognitive potential as measured through inductive reasoning.

     

Promoting equity in mathematics teacher preparation: a framework for advancing teacher learning of children’s multiple mathematics knowledge bases

Research repeatedly documents that teachers are underprepared to teach mathematics effectively in diverse classrooms. A critical aspect of learning to be an effective mathematics teacher for diverse learners is developing knowledge, dispositions, and practices that support building on children’s mathematical thinking, as well as their cultural, linguistic, and community-based knowledge. This article presents a conjectured learning trajectory for prospective teachers’ (PSTs’) development related to integrating children’s multiple mathematical knowledge bases (i.e., the understandings and experiences that have the potential to shape and support children’s mathematics learning—including children’s mathematical thinking, and children’s cultural, home, and community-based knowledge), in mathematics instruction. Data were collected from 200 PSTs enrolled in mathematics methods courses at six United States universities. Data sources included beginning and end-of-semester surveys, interviews, and PSTs’ written work. Our conjectured learning trajectory can serve as a tool for mathematics teacher educators and researchers as they focus on PSTs’ development of equitable mathematics instruction.     

Work–family relations among mothers of children with learning disorders

The study examined conflict and facilitation in work–family relations among working mothers of children with learning disorders (LD) or with typical development. The study also focused on three maternal personal resources (maternal anxious/avoidant attachment security, affect and sense of coherence) as antecedents of these work–family relations, and examined outcomes of such conflict and facilitation between the two life domains, as they affected patterns of close relationships within the family (child attachment and family cohesion). The sample included 96 mother–child dyads: 48 mothers and their children with LD; and 48 mothers and their typically developing children. Children’s ages were 8–12 years. All attended public elementary schools in urban areas of central Israel. Significant group differences emerged on mothers’ family to work conflict and on mothers’ work to family facilitation. Findings indicated that several maternal personal resources were antecedents of these relations and also contributed to differences in mother–child attachment relationships and family cohesion. Discussion focused on understanding work–family relations among these mothers of children with LD, as well as the influence of maternal personal resources on patterns of close relationships (i.e., child attachment, family cohesion).     

What helps children in a pupil referral unit (PRU)? An exploration into the potential protective factors of a PRU as identified by children and staff

Previous studies have focused on the views of excluded children or those finding themselves in Pupil Referral Units (PRUs). Few, however, have focused on exploring views from a resilience perspective. Studies linked to resilience have tended to focus on exploring factors through quantitative rather than qualitative measures. This piece of research aimed to explore the potential protective factors of one PRU through the valuable voices of children and staff. The views of six children aged between 9 and 13 years, all with varying forms of behavioural, emotional and social difficulties, and partway through a PRU placement, were explored through semi-structured interviews supported by the use of tools and techniques to facilitate discussion. The views of four staff members were also obtained. Findings reveal the powerful perspectives that children and young people can bring in relation to what might be helping them within a particular environment. A number of potential protective factors, both within the environment and within-child factors that may operate as mechanisms in fostering positive social and academic outcomes, are identified. Implications of these findings for professional practice, and for children finding themselves in PRU provision, are discussed, with a focus on reintegration and the potential impact of changing environments for these children.     

The art of teaching children the arts: music,dance and poetry with children aged 2–8 years old

In this article, the theoretical framework of developmental pedagogy is presented as a tool in studying and developing children’s knowing within the arts. The domains of art focused on are music, poetry and dance/aesthetic movement. Through empirical examples from a large‐scale research project, we illustrate the tools of developmental pedagogy and show how this perspective contributes to our understanding of children’s learning of music, dance and poetry. More specifically, we will analyse: (a) the important role of the teacher in children’s learning within the arts; (b) the importance of conversing when learning the arts; (c) what constitutes the knowledge, what we refer to as ‘learning objects’, to be appropriated within the three domains of art focused on; and (d) how to conceive of progression in children’s knowing within the arts.     

A discipline apart: the challenges of ‘Fostering a Community of Learners’ in a mathematics classroom

This is the fifth in a series that examines the challenges that teachers in different domains face as they attempt to implement the pedagogical reform ‘Fostering a Community of Learners’ (FCL). Here we focus on the relationship between FCL and the teaching of mathematics. We argue that it is possible to teach mathematics through the FCL pedagogy, but that doing so requires some rethinking of both mathematics instruction and FCL. In particular, we describe three shifts that aided a teacher's implementation of FCL pedagogy with mathematics: the teacher developed a new perspective on mathematics that emphasized the importance of having students learn both mathematical concepts and processes; the teacher developed a new understanding of the role of the teacher in mathematics‐education reform; and the teacher modified his understanding of FCL, coming to believe that a discourse community could be the basis for FCL pedagogy in a mathematics classroom.     

Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions

Twenty-four mathematics teachers were asked to think aloud when posing a word problem whose solution could be found by computing 4/5 divided by 2/3. The data consisted of verbal protocols along with the written notes made by the subjects. The qualitative analysis of the data was focused on identifying the structures of the problems produced and the associated ways of thinking involved in constructing the problems. The results suggest that success in doing the interview task was associated with perception the given fractions as operands for the division operation and, at the same time, the divisor 2/3 as an operator acting over 4/5. The lack of success was associated with perception of division of fractions as division of divisions of whole numbers and using the result of division of fractions as the only reference point. The study sheds new light on the teachers’ difficulties with conceptualization of fractions.     

Using culture as a resource in mathematics: the case of four Mexican–American prospective teachers in a bilingual after-school program

This paper explores Mexican–American prospective teachers’ use of culture—defined as social practices and shared experiences—as an instructional resource in mathematics. The setting is an after-school mathematics program for the children of Mexican heritage. Qualitative analysis of the prospective teachers’ and children’s interactions reveals that the nature of the mathematical activities affected how culture was used. When working on the “binder activities,” prospective teachers used culture only in non-mathematical contexts. When working on the “recipes project,” however, culture was used as a resource in mathematical contexts. Implications for the mathematics teacher preparation of Latinas/os are discussed.     

Promoting social inclusion in the early years of elementary school: a focus on children’s epistemic beliefs for moral reasoning

As classrooms continue to diversify, there is an increasing need to understand children’s inclusive behaviours and moral reasoning. Research shows that epistemic beliefs (beliefs about knowing and knowledge) can influence reasoning for adults, but we know little about this relationship in younger children or how classroom contexts relate to epistemic beliefs for moral reasoning. Thirty-one elementary school children (mean age 6.5 years) participated in epistemic beliefs and moral reasoning tasks in the first year of a three-year longitudinal study. Findings showed that while children described objectivist epistemic beliefs (right/wrong answers) about social inclusion, their justifications revealed an unexpected, more complex set of epistemic beliefs. Implications for moral pedagogies are discussed.     

Identification with early primary school mathematics: a home–school activity theory conceptualisation and methodology

This paper offers a new methodology for researching children’s identifications in the early primary phase that makes visible the ‘seeds’ of academic dis-/identities, which typically flower later in adolescence. It focuses on the ‘case’ of mathematical identifications (MI) to exemplify how children come to dis/identify from the academic curriculum more broadly. MI is defined as any act where cultural representations of being mathematical are reflected back on to the child by others or the self, as they engage in mathematical activity. Data are presented concerning Bilal and Nico to show identifications in a hybrid home–school mathematics activity that contradict or ‘align’ with MI elicited in school-mathematical activity. It is argued that contradictions between these MIs indicate ‘potential’ development towards disengagement (the case of Nico). This contrasts with home–school alignment involving Bilal’s ‘schoolified’ home activities, which (re)validate school-mathematical activity. It is further argued that this conceptual innovation offers potential for investigating identity in early childhood.