While some researchers place negative numbers on a so-called extended mental number line to the left of positive numbers, others claim that negative numbers do not have mental representations but are processed through positive numbers combined with transformation rules. We measured spatial associations of negative numbers with a modified implicit…

Competence with rational numbers is essential for mathematics proficiency in secondary mathematics. However, many students struggle with rational number concepts, and students with mathematics difficulties struggle even more. The purpose of this study was to examine the effects of an intervention that incorporated the use of explicit instruction…

Number line models provide a visual aid for students to examine the relationship of integers with each other and facilitate learning of integers and integer operations. Such models are typically used when students are asked real-life problems. This study employs a qualitative case study design to perform an in-depth analysis of how middle grade…

This paper shares how fourth grade students solved a unique type of number line task. Instead of giving students the unit and asking them to locate a fraction on the number line, students were given two points on the number line and asked to either find 1, or another number on the number line. This type of task is called "reconstructing the…

Perceptual judgments result from a dynamic process, but little is known about the dynamics of number-line estimation. A recent study proposed a computational model that combined a model of trial-to-trial changes with a model for the internal scaling of discrete numbers. Here, we tested a surprising prediction of the model--a situation in which…

Perceptual judgments result from a dynamic process, but little is known about the dynamics of number-line estimation. A recent study proposed a computational model that combined a model of trial-to-trial changes with a model for the internal scaling of discrete numbers. Here, we tested a surprising prediction of the model--a situation in which…

This qualitative case study of emergent practices in a multi-paradigmatic field at one department in Norway explores how performance metrics in science intra-act with staff composition and resource allocation, potentially affecting paradigm trends. Inspired by agential realism, we discuss four key metrics as the material core of an assemblage…

A pivotal information about the structure of students' comprehension underlying from which the knowledge is gained can be identified through the mental model. This study aimed at describing the six levels of students' mental model in comprehending the concept of the integer. The subject of this research was 40 students consisting of 20 students…

In this article we aim to deepen our understanding of the content and nature of the early childhood teacher's knowledge, focusing on those aspects which might promote students' algebraic thinking. Approaching arithmetic from the viewpoint of algebra as an advanced perspective and considering the analytical model "Mathematics Teachers'…

The number line is a model that is used to measure, count, order and even operate, which requires a symbolic interpretation. Then, we investigate the conceptions of 72 in service secondary teachers, when they manage the model of the number line associated with order, spatial location and the relative position between numbers and marks. In a…

In mathematically literate societies, numerical information is represented in 3 distinct codes: a verbal code (i.e., number words); a digital, symbolic code (e.g., Arabic numerals); and an analogical code (i.e., quantities; Dehaene, 1992). To communicate effectively using these numerical codes, our understanding of number must involve an…

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories -- the…

Learning rational number concepts is acknowledged as an important task but many learners find it difficult to make sense of them. This paper reports on a case study of the learning in a short (nine-lesson) learning programme for Grade 8 learners in a Namibian school, which sought to use visual models (circle area, bar area and number line) to…

The rational number knowledge of student teachers, in particular the equivalence of fractions, decimals, and percentages, and their comparison and ordering, is the focus of this article. An instrument comprising multiple choice, short answer and constructed response formats was designed to test conceptual and procedural understanding. Application…

A central issue in the mathematics curriculum is that we want students to make connections. This issue has been analysed in a series of curricula and instruction design and analysis studies. Moving towards mathematics connections--and away from treating mathematics as a body of isolated concepts and procedures--is an important goal of mathematics…