After figuring out why many kids hate math, these longtime educators Bob and Ellen Kaplan created an after-school program in which children as young as 5 formulate equations of their own--and actually enjoy doing it. This paper discusses the experiences of the Kaplan couple, Bob and Ellen, on their Math Circle class during after-school sessions…

If one rolls a coin across a chessboard and it comes to rest on the board, what is the probability that it covers some corner of one of the grid squares? The online magazine "Plus" (2004) posed this problem for students to solve. It is a useful problem for several reasons: it introduces the idea of probability in a continuous sample space, it has…

Applications in engineering, science and technology within undergraduate programmes can be difficult for students to understand. In this paper, new results are presented which go some way to demonstrate and explain the problems faced by students in linking mathematical models to real-world applications. The study is based on student responses to…

In this study, the teaching experiment methodology is used to observe firsthand a gifted student's mathematical learning and reasoning. A series of teaching experiments was conducted with 1 gifted and 1 average 7th-grade student to investigate how the gifted student's mathematical concepts and operation constructions differed from those of the…

Here we give an example of a problem that could be beneficially investigated by AS/A level students. It is a geometry problem that they can profitably tackle by geometric (especially geometry software) and algebraic means. Such problems naturally lead students to the need for proof--an essential part of mathematics that is often lacking in current…

A inquiry-based approach called the "modelling discussion" is introduced for structuring beginning modelling activity, teaching new mathematics from examining its applications in contextual situations, and as a general classroom management technique when students are engaged in mathematical modelling. An example which illustrates the style and…

This study examined changes in 26 fourth-grade students' early conceptions of rational number representations as a function of receiving one of two curricular interventions. The first group of 12 students received a curriculum that emphasized constructing knowledge through extended problem solving with a single perspective of the rational number…

The objective of this research was to compare two types of models used in problem solving: those which give priority to intermediate qualitative representations, such as the episodic situation model, and those which are centred on the activity of the participant and are of a procedural nature. Two-step distributive problems were chosen that could…

This article relates to an empirical study based on the use of mathematical symbolism in problem solving, twenty-five pupils were interviewed individually at the end of grade one; each of them was asked to solve and symbolize 14 different problems. In their classical curriculum, these pupils have received a traditional education based on a…

The Chinese used the idea of generating equivalent expressions for solving problems where the problems from a historical Chinese text are studied to understand the ways in which the ideas can lead into algebraic calculations and help students to learn algebra. The texts unify algebraic problem solving through complex algebraic thought and afford…

Research suggests that student enrollment in developmental level courses at the postsecondary education level increases annually. This is occurring because underprepared and/or nontraditional students' reading, writing, or math skills are inadequate to meet the challenges of college-level work. This article focuses on self-explanations as an…

This multi-case qualitative study, conducted in two elementary schools, observed the self-efficacy experiences of 67 5th grade mathematics students in daily lessons. Four classrooms were observed a total of 40 times for 30 minutes each. Two classrooms each used traditional instructional materials and standards-based instructional materials.…

This paper describes a modification of a popular TV game show, "American Idol[R]," conducted in an undergraduate calculus course. The focal point of this game show is to get students to interact in competitive self-assessment. Student competition is an effective device to engage students in the learning process. Placing students in the role of…

A new computerized environment introducing a variety of metacognitive support mechanisms (MSMs) in different phases of the problem-solving process was designed to influence students' transfer from solving structured problems (near transfer) to solving open-ended problems (far transfer). Two hundred and thirty one students (aged 13-14 years) were…

There is evidence that spontaneous learning leads to relational understanding and high positive affect. To study spontaneous abstracting, a model was constructed by combining the RBC model of abstraction with Krutetskii's mental activities. Using video-stimulated interviews, the model was then used to analyze the behavior of two Year 8 students…