This extensively updated teaching resource provides over 100 engaging, full-color visuals with explanations of how they can be used to stimulate mathematics learning, to explain mathematical concepts, and to assess students' mathematical understanding in grades K-8. Readers are provided with a strong mathematical background, downloadable copies of…

Introducing the distributivity property of multiplication over addition is a well-known challenge in mathematics education, especially in primary school. As a contribution, this paper presents the results of a cycle of design research that focuses on the design, implementation and evaluation of a modelling activity in which 2nd-grade students are…

The integer is a basic concept in studying arithmetic and algebra. However, students still frequently experience misconceptions, especially in negative integer, count operations. Traditional games are activities that are often carried out by students in coastal areas so that they are relevant to be used as a tool to construct learning trajectories…

"Where Mathematics Comes From" (Lakoff & Núñez 2000) proposed that mathematical concepts such as arithmetic and counting are constructed cognitively from embodied metaphors of actions on physical objects, and four actions, or 'grounding metaphors' in particular: collecting, stepping, constructing and measuring. This article argues…

The modelling approach to teaching and learning mathematics emphasizes the usefulness of mathematics in the real-world. The aim of the current study is to examine whether engagement in modelling activities provides learners an opportunity to expand their knowledge of a specific concept -- the "average" concept. Our data include…

Mental models are representations of students' minds concepts to explain a situation or an on-going process. The purpose of this study is to describe students' mental model in solving mathematical patterns of generalization problem. Subjects in this study were the VII grade students of junior high school in Situbondo, East Java, Indonesia. This…

The transition from preformal and propaedeutic generalization-actions to a symbolically explicit use of the concept of variable has been a matter of significant attention in mathematics education, for example in the context of generalization processes on a preformal level and regarding the specific nature of algebraic concepts. This contribution…

"Improving Mathematics in the Early Years and Key Stage 1" reviews the best available evidence to offer five recommendations for developing the maths skills of 3-7-year olds. Recommendations include integrating maths into different activities throughout the day -- for example, at registration and snack time -- to familiarise children…

There are many continuous quantitative dimensions in the physical world. Philosophical, psychological and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time, mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, density) and number can be…

Fourteen third graders were given numerical computation and division-with-remainder (DWR) problems both before and after they were taught the division algorithm in classrooms. Their solutions were examined. The results show that students' initial acquisition of the division algorithm did improve their performance in numerical division computations…

Discusses the principle elements for understanding and using mathematics, particularly arithmetic. Three aspects of arithmetic are considered: Understanding the concepts underlying a mathematical process, techniques or algorithms of arithmetic, and applications of the techniques to real situations. (SH)

MIDDLE-CLASS, FIRST-GRADE STUDENTS (100) WERE TESTED INDIVIDUALLY ON 4 ITEMS OF CONCEPT OF ADDITION AND CONSERVATION OF NUMBER. THE TEST ITEMS WERE IDENTICAL EXCEPT FOR THE NUMBER OF OBJECTS INVOLVED. FOR EACH ITEM, TWO PILES OF CANDY WERE PLACED BEFORE EACH CHILD AND THEN MOVED TOGETHER. THE STUDY SHOWED NO MAJOR DIFFERENCE IN THE MEAN…

Used are test and interview data to extract evidence as to what level tenth graders have coordinated the worlds of arithmetic and algebra and can move freely between them. More dissociation is shown than was expected. (Author/MVL)

Learning activities and questioning skills that teachers in Tasmania can use to determine a child's mathematics understanding are presented. Covered topics include division, multiplication, subtraction, addition, sorting and classification, one-to-one correspondence, patterns, ordering, and counting. Included for each topic are the aim, needed…

A LEARNING MODEL TO IDENTIFY FACTORS CONTRIBUTING TO THE DIFFICULTY OF A PROBLEM ITEM WAS SUPPORTED EMPIRICALLY, AND INDICATED THAT THE NUMBER OF STEPS REQUIRED TO SOLVE A PROBLEM WAS THE MOST IMPORTANT VARIABLE IN PREDICTING BOTH ERROR PROBABILITY AND RESPONSE LATENCY. THE MODEL, IN ORDER TO ESTABLISH DIFFERENTIAL PREDICTIONS OF DIFFICULTY IN…