Inquiry pedagogy to promote emerging proportional reasoning in primary students

Proportional reasoning as the capacity to compare situations in relative (multiplicative) rather than absolute (additive) terms is an important outcome of primary school mathematics. Research suggests that students tend to see comparative situations in additive rather than multiplicative terms and this thinking can influence their capacity for proportional reasoning in later years. In this paper, excerpts from a classroom case study of a fourth-grade classroom (students aged 9) are presented as they address an inquiry problem that required proportional reasoning. As the inquiry unfolded, students' additive strategies were progressively seen to shift to proportional thinking to enable them to answer the question that guided their inquiry. In wrestling with the challenges they encountered, their emerging proportional reasoning was supported by the inquiry model used to provide a structure, a classroom culture of inquiry and argumentation, and the proportionality embedded in the problem context

Student interpretations of the terms in first-order ordinary differential equations in modelling contexts

A study of first-year undergraduate students′ interpretational difficulties with first-order ordinary differential equations (ODEs) in modelling contexts was conducted using a diagnostic quiz, exam questions and follow-up interviews. These investigations indicate that when thinking about such ODEs, many students muddle thinking about the function that gives the quantity to be determined and the equation for the quantity's rate of change, and at least some seem unaware of the need for unit consistency in the terms of an ODE. It appears that shifting from amount-type thinking to rates-of-change-type thinking is difficult for many students. Suggestions for pedagogical change based on our results are made

Inferentialism as an alternative to socioconstructivism in mathematics education

The purpose of this article is to draw the attention of mathematics education researchers to a relatively new semantic theory called inferentialism, as developed by the philosopher Robert Brandom. Inferentialism is a semantic theory which explains concept formation in terms of the inferences individuals make in the context of an intersubjective practice of acknowledging, attributing, and challenging one another’s commitments. The article argues that inferentialism can help to overcome certain problems that have plagued the various forms of constructivism, and socioconstructivism in particular. Despite the range of socioconstructivist positions on offer, there is reason to think that versions of these problems will continue to haunt socioconstructivism. The problems are that socioconstructivists (i) have not come to a satisfactory resolution of the social-individual dichotomy, (ii) are still threatened by relativism, and (iii) have been vague in their characterization of what construction is. We first present these problems; then we introduce inferentialism, and finally we show how inferentialism can help to overcome the problems. We argue that inferentialism (i) contains a powerful conception of norms that can overcome the social-individual dichotomy, (ii) draws attention to the reality that constrains our inferences, and (iii) develops a clearer conception of learning in terms of the mastering of webs of reasons. Inferentialism therefore represents a powerful alternative theoretical framework to socioconstructivism

Effects of cooperation on information disclosure in mock‐witness interviews

Purpose: Forensic interviewers often face witnesses who are unwilling to cooperate with the investigation. In this experimental study, we examined the extent to which cooperativeness instructions affect information disclosure in a witness investigative interview. Methods: One hundred and thirty-six participants watched a recorded mock-crime and were interviewed twice as mock-witnesses. They were randomly assigned to one of four conditions instructing different levels of cooperativeness: Control (no instructions), Cooperation, No Cooperation, and No Cooperation plus Cooperation. The cooperativeness instructions aimed to influence how participants’ perceived the costs and benefits of cooperation. We predicted that Cooperation and No Cooperation instructions would increase and decrease information disclosure and accuracy, respectively. Results: We found decreased information disclosure and, to a lesser extent, accuracy in the No Cooperation and No Cooperation plus Cooperation conditions. In a second interview, the shift of instructions from No Cooperation to Cooperation led to a limited increase of information disclosure at no cost of accuracy. Cooperativeness instructions partially influenced the communication strategies participants used to disclose or withhold information. Conclusions: Our results demonstrate the detrimental effects of uncooperativeness on information disclosure and, to a lesser extent, the accuracy of witness statements. We discuss the implications of a lack of witness cooperation and the importance of gaining witness cooperation to facilitate information disclosure in investigative interviews

Understanding and supporting block play: video observation research on preschoolers’ block play to identify features associated with the development of abstract thinking

This article reports on a study conducted to investigate the development of abstract thinking in preschool children (ages from 3 years to 4 years old) in a nursery school in England. Adopting a social influence approach, the researcher engaged in 'close listening' to document children's ideas expressed in various representations through video observation. The aim was to identify behaviours connected with features of the functional dependency relationship – a cognitive function that connects symbolic representations with abstract thinking. The article presents three episodes to demonstrate three dominating features, which are i) child/child sharing of thinking and adult and child sharing of thinking; ii) pause for reflection; and iii) satisfaction as a result of self-directed play. These features were identified as signs of learning, and were highlighted as phenomena that can help practitioners to understand the value of quality play and so provide adequate time and space for young children and plan for a meaningful learning environment. The study has also revealed the importance of block play in promoting abstract thinking. Keywords: abstract thinking; functional dependency relationship; social influence approach; block play; preschool; video observation; qualitative researc

Generalization Strategies in Finding the <i>n</i>th Term Rule for Simple Quadratic Sequences

In this study, we identify ways in which a sample of 18 graduates with mathematics-related first degrees found the nth term for quadratic sequences from the first values of a sequence of data, presented on a computer screen in various formats: tabular, scattered data pairs and sequential. Participants’ approaches to identifying the nth term were recorded with eye-tracking technology. Our aims are to identify their strategies and to explore whether and how format influences these strategies. Qualitative analysis of eye-tracking data offers several strategies: Sequence of Differences, Building a Relationship, Known Formula, Linear Recursive and Initial Conjecture. Sequence of Differences was the most common strategy, but Building a Relationship was more likely to lead to the right formula. Building from Square and Factor Search were the most successful methods of Building a Relationship. Findings about the influence of format on the range of strategies were inconclusive but analysis indicated sporadic evidence of possible influences

Teachers and didacticians: key stakeholders in the processes of developing mathematics teaching

This paper sets the scene for a special issue of ZDM-The International Journal on Mathematics Education-by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers' and didacticians' engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice. © 2014 FIZ Karlsruhe