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.

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

Can improving teachers’ knowledge of mathematics lead to gains in learners’ attainment in mathematics?

It is wellknown that the majority of South African learners achieve extremely poorly in mathematics. Many claim that one of the causes of this poor attainment is teachers’ weak knowledge of mathematics, and propose that improving teachers’ mathematical knowledge would improve learner attainment. However, the evidence-base for this proposed solution is currently relatively weak. We report on a quasi-experimental study examining the learning gains of Grade 10 learners from five secondary schools in the Johannesburg area whose teachers participated in a year-long professional development course aimed at improving the teachers’ knowledge of mathematics for teaching. Statistical analyses of pre- and post-test results show that the intervention group of learners (N = 586) taught by teachers who had participated in the professional development (N = 14) outperformed a matched control group of learners (N = 217) taught by teachers in the same schools (N = 7). An effect size of =0.17 for the intervention group is equivalent to two months’ additional progress. While the learning gains are small, they are statistically significant. These findings provide empirical support for claims that attending to teachers’ mathematical knowledge can impact learners’ attainment. Suggestions are made regarding the form and substance of such professional development

Improving students’ understanding of algebra and multiplicative reasoning: did the ICCAMS intervention work?

In this paper we report on the intervention phase of an ESRC-funded project, Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS). The intervention was designed to enable teachers to use formative assessment in mathematics classrooms by evaluating what students already knew, then adapting their teaching to students’ learning needs. A key feature was the use of models and representations, such as the Cartesian graph, both to help students better understand mathematical ideas and to help teachers appreciate students’ difficulties. Twenty-two teachers and their Year 8 classes from 11 schools took part in the intervention during 2010/11. Pre! and post-tests in algebra, decimals and ratio were administered to the students of these classes, and compared to a control group of students matched from the ICCAMS national longitudinal survey (using propensity score matching). The students in the intervention group made greater progress than the matched control

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

Improving students’ understanding of algebra and multiplicative reasoning: did the ICCAMS intervention work?

In this paper we report on the intervention phase of an ESRC-funded project, Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS). The intervention was designed to enable teachers to use formative assessment in mathematics classrooms by evaluating what students already knew, then adapting their teaching to students’ learning needs. A key feature was the use of models and representations, such as the Cartesian graph, both to help students better understand mathematical ideas and to help teachers appreciate students’ difficulties. Twenty-two teachers and their Year 8 classes from 11 schools took part in the intervention during 2010/11. Pre! and post-tests in algebra, decimals and ratio were administered to the students of these classes, and compared to a control group of students matched from the ICCAMS national longitudinal survey (using propensity score matching). The students in the intervention group made greater progress than the matched control