Longitudinal study of low and high achievers in early mathematics

Background. Longitudinal studies allow us to identify, which specific maths skills are weak in young children, and whether there is a continuing weakness in these areas throughout their school years. Aims. This 2-year study investigated whether certain socio-demographic variables affect early mathematical competency in children aged 5–7 years. Sample. A randomly selected sample of 127 students (64 female; 63 male) participated. At the start of the study, the students were approximately 5 years old (M = 5.2; SD = 0.28; range = 4.5–5.8). Method. The students were assessed using the Early Numeracy Test and then allocated to a high (n = 26), middle (n = 76), or low (n = 25) achievers group. The same children were assessed again with the Early Numeracy Test at 6 and 7 years old, respectively. Eight socio-demographic characteristics were also evaluated: family model, education of the parent(s), job of the parent(s), number of family members, birth order, number of computers at home, frequency of teacher visits, and hours watching television. Results. Early Numeracy Test scores were more consistent for the high-achievers group than for the low-achievers group. Approximately 5.5% of low achievers obtained low scores throughout the study. A link between specific socio-demographic characteristics and early achievement in mathematics was only found for number of computers at home. Conclusions. The level of mathematical ability among students aged 5–7 years remains relatively stable regardless of the initial level of achievement. However, early screening for mathematics learning disabilities could be useful in helping low-achieving students overcome learning obstacles.This material is based on work supported by the Spanish Ministry of Science & Technology grant no. SEJ2007-62420/EDUC and Junta de Andalucia grant no. P09-HUM-4918

Fostering relational thinking while negotiating the meaning of the equal sign

In this article, we relate an experience in which we have used number sentences to begin to develop algebraic thinking. Working with third-graders during six sessions, we set out to explore the following questions: How do students’ conceptions of the equal sign evolve when considering and discussing varied True/False number sentences? Do students develop relational thinking while we negotiate the meaning of the equal sign? Do students retain the new interpretation of the equal sign over time? We successfully helped the students to broaden their conceptions through the different tasks but were only partially successful at initiating relational thinking. The particularities of both developments are here described

Constructing and Resisting Disability in Mathematics Classrooms: A Case Study Exploring the Impact of Different Pedagogies

This study demonstrates the importance of a critical lens on disability in mathematics educational research. This ethnographic and interview study investigated how ability and disability were constructed over 1 year in a middle school mathematics classroom. Children participated in two kinds of mathematical pedagogy that positioned children differently: procedural and discussion-based. These practices shifted over time, as the teacher increasingly focused on memorization of procedures to prepare for state testing. Two Latino/a children with learning disabilities, Ana and Luis, used multiple cultural practices as resources, mixing and remixing their engagement in and identifications with mathematics. Ana, though mastering the procedural performances necessary for success in the second half of the year, authored herself as separate from mathematics, creating distance between herself and those she considered “smarties.” Luis was identified as a creative mathematical problem-solver and was initially positioned as a “top” mathematics student. As the pedagogy shifted towards memorization, Luis resisted the pedagogy of procedures and continued to identify as a creative thinker in mathematics. Yet, his teachers saw him as increasingly disabled and eventually placed him in a group only for those in special education. This group, which Luis named the “unsmartest group,” was seen as least competent in mathematics by both teachers and students. The narratives of Luis and Ana highlight mathematics classrooms as relational and emotional and demonstrate different strategies of resistance to the construction of mathematical disability

Engaging with issues of emotionality in mathematics teacher education for social justice

This article focuses on the relationship between social justice, emotionality and mathematics teaching in the context of the education of prospective teachers of mathematics. A relational approach to social justice calls for giving attention to enacting socially-just relationships in mathematics classrooms. Emotionality and social justice in teaching mathematics variously intersect, interrelate or interweave. An intervention, usng creative action methods, with a cohort of prospective teachers addressing these issues is described to illustrate the connection between emotionality and social justice in the context of mathematics teacher education. Creative action methods involve a variety of dramatic, interactive and experiential tools that can promote personal and group engagement and embodied reflection. The intervention aimed to engage the prospective teachers with some key issues for social justice in mathematics education through dialogue about the emotionality of teaching and learning mathematics. Some of the possibilities and limits of using such methods are considered

Overcoming the Newtonian Paradigm: The Unfinished Project of Theoretical Biology from a Schellingian Perspective

Defending Robert Rosen’s claim that in every confrontation between physics and biology it is physics that has always had to give ground, it is shown that many of the most important advances in mathematics and physics over the last two centuries have followed from Schelling’s demand for a new physics that could make the emergence of life intelligible. Consequently, while reductionism prevails in biology, many biophysicists are resolutely anti-reductionist. This history is used to identify and defend a fragmented but progressive tradition of anti-reductionist biomathematics. It is shown that the mathematicoephysico echemical morphology research program, the biosemiotics movement, and the relational biology of Rosen, although they have developed independently of each other, are built on and advance this antireductionist tradition of thought. It is suggested that understanding this history and its relationship to the broader history of post-Newtonian science could provide guidance for and justify both the integration of these strands and radically new work in post-reductionist biomathematics

Developing Learning Trajectory For Enhancing Students’ Relational Thinking

Algebra is part of Mathematics learning in Indonesian curriculum. It takes one half of the teaching hours in senior high school, and one third in junior high school. However, the learning trajectory of Algebra needs to be improved because teachers teach computational thinking by applying paper-and-pencil strategy combining with the concepts-operations-example-drilling approach. Mathematics textbooks do not give enough guidance for teachers to conduct good activities in the classroom to promote algebraic thinking especially in the primary schools. To reach Indonesian Mathematics teaching goals, teachers should develop learning trajectories based on pedagogical and theoretical backgrounds. Teachers need to have knowledge of student’s developmental progressions and understanding of mathematics concepts and students’ thinking. Research shows that teachers’ knowledge of student’s mathematical development is related to their students’ achievement. In fostering a greater emphasis on algebraic thinking, teachers and textbooks need to work more closely together to develop a clearer learning trajectory. Having this algebraic thinking, students developed not only their own character of learning and thinking but also their attitude, attention and discipline. Key Words: Learning Trajectory, Relational Thinkin

Justifications-on-demand as a device to promote shifts of attention associated with relational thinking in elementary arithmetic

Student responses to arithmetical questions that can be solved by using arithmetical structure can serve to reveal the extent and nature of relational, as opposed to computational thinking. Here, student responses to probes which require them to justify-on-demand are analysed using a conceptual framework which highlights distinctions between different forms of attention. We analyse a number of actions observed in students in terms of forms of attention and shifts between them: in the short-term (in the moment), medium-term (over several tasks), and long-term (over a year). The main factors conditioning students´ attention and its movement are identified and some didactical consequences are proposed