Madison’s Memorial and Remonstrance, Jefferson’s Statute for Religious Liberty, and the Creation of the First Amendment

Jurists, scholars, and popular writers routinely assert that the men who framed and ratified the First Amendment were influenced by James Madison’s Memorial and Remonstrance (1785) and Thomas Jefferson’s Statute for Religious Liberty (1786). In this essay I demonstrate that there is little evidence to support these claims. Because these documents represent only one approach to church-state relations in the era, jurists and others who believe that the religion clauses should be interpreted in light of the founders’ views need to look well beyond these texts if they want to understand the First Amendment’s “generating history.

An exploratory study on mathematics teacher educators’ beliefs and understandings about computational thinking

This paper reports an investigation of mathematics teacher educators’ views and perceptions on computational thinking (CT) and its impact on mathematical learning. We conducted semi-structured interviews with experienced mathematics teacher educators, all of whom have extensive experience with the use of digital technologies for mathematical teaching and learning and report on data from two of them. Our aim is to offer insights into how CT is perceived and understood by them, to support them in self-assessing their possession of CT practices, and how to support mathematics teachers and students in gaining CT. We offer ideas regarding the promotion of CT and its integration in mathematics teaching and learning

Replicating a study with tasks associated with the equals sign in an online environment

This paper presents a case study of a conceptual replication study. We replicated the famous and widely cited task presented in Falkner et al. (1999), 8+4=__+5. In contrast to the original study, we administered the task with the same age group (Grade 6) in a different system (Denmark) and via a large-scale online learning environment (OLE), with a larger sample and two decades later. Our replication indicates that the Danish students performed very significantly better than the students in the original study. We discuss why this is the case and argue that OLEs such as the one we used provide an important opportunity to replicate, and thus better understand, similar results

Operationalising Vergnaud’s Notion of Scheme in Task Design in Online Learning Environments to Support the Implementation of Formative Assessment

This paper presents an implementation process model for designing and implementing tasks that provide formative feedback in the online learning environment of mathematics classrooms. Specifically, the model operationalises components of Vergnaud’s notion of scheme. The implementation process model features a task sequence guided by controlled variation and a ‘dual scheme idea’. Using such a sequence of tasks, this work illustrates how Vergnaud’s notion of scheme can be used to aid teachers in hypothesising about their learners’ understanding of problems involving linear equations, ultimately providing improved feedback for teachers and improved opportunities for student learning in online environments. In Denmark, the online environment matematikfessor.dk is used by approximately 80% of Danish K-9 students. The impact sheet to this article can be accessed at 10.6084/m9.figshare.19493846

A case study of an expert in computational thinking in the context of mathematics education research

We conducted semi-structured interviews with three experienced mathematics education researchers with great expertise in the design and use of digital technologies, including programming skills, to investigate their views and perceptions on computational thinking (CT) and its impact on mathematical learning. In this paper we report on our findings from one of them, Mark, and we suggest ways for adapting the very recent Mathematical Digital Competency (MDC) framework to encompass CT practices and dispositions. Our aim is to offer insights into how CT is perceived and understood by him, by prompting him to reflect on his own CT practices and competencies. We offer suggestions for an MDC framework for mathematics teacher educators that encompasses CT

Danish Students’ Understanding of Fractions: A Replication Study

In this paper, we report on a conceptual replication of a fractions test originally administered in the 1970s in England as part of the ‘Concepts in Secondary Mathematics and Science’ (CSMS) research programme documenting lower secondary students’ understanding of mathematics. We present data from a recent study conducted in Denmark with 336 students aged between 12 and 14 years, presenting descriptive results as in the original research. In addition, we use Rasch modelling — a technique that was not widely used in the 1970s — to further analyse the students’ understanding of fractions and to validate the test. Our findings indicate that Danish students’ fractional understanding in word or diagrammatic contexts is slightly stronger in comparison to that of the English students in the original study, whereas the case for computations involving fractions is the reverse. We discuss the possible reasons for these differences. Our study provides evidence that fractional knowledge involves a number of components and suggest ways in which the test might be improved

The use of alternative double number lines as models of ratio tasks and as models for ratio relations and scaling

In this paper we draw on ICCAMS project materials that used the double number line (DNL) to develop secondary school students’ understanding of multiplicative reasoning. In particular, we look at the use of a DNL, and its alternative version, as a model of ratio tasks, as a model for developing an understanding of ratio relations, and finally (but only briefly) as a model for developing the notion of multiplication as scaling

Defining a rhombus

Colin Foster, Jeremy Hodgen and Dietmar Kuchemann exlore how the use of examples and counter-examples can support students developing understandings of definitions

The use of alternative double number lines as models of ratio tasks and as models for ratio relations and scaling

In this paper we draw on ICCAMS project materials that used the double number line (DNL) to develop secondary school students’ understanding of multiplicative reasoning. In particular, we look at the use of a DNL, and its alternative version, as a model of ratio tasks, as a model for developing an understanding of ratio relations, and finally (but only briefly) as a model for developing the notion of multiplication as scaling