Undergraduate students study the topic of binomial series expansion as part of their Calculus course. The purpose of this study was to explore the mental constructions of binomial series expansion of a class of 159 students. Data were collected through a written assessment task by each member of the class. A convenient sample of eleven students…

Both scientists and children make important structural discoveries, yet their computational underpinnings are not well understood. Structure discovery has previously been formalized as probabilistic inference about the right structural form--where form could be a tree, ring, chain, grid, etc. (Kemp & Tenenbaum, 2008). Although this approach…

Mental representations of numerical magnitude are commonly thought to undergo discontinuous change over development in the form of a "representational shift." This idea stems from an apparent categorical shift from logarithmic to linear patterns of numerical estimation on tasks that involve translating between numerical magnitudes and…

Openness has a long genealogy in education. Whether through the use of post, radio, television and digital technologies, extending learning opportunities to more and a wider range of people has been a significant aspect of educational history. Transcending barriers to learning has been promoted as the means of opening educational opportunities in…

Multi-voxel pattern recognition techniques combined with Hidden Markov models can be used to discover the mental states that people go through in performing a task. The combined method identifies both the mental states and how their durations vary with experimental conditions. We apply this method to a task where participants solve novel…

In the Basic Local Independence Model (BLIM) of Doignon and Falmagne ("Knowledge Spaces," Springer, Berlin, 1999), the probabilistic relationship between the latent knowledge states and the observable response patterns is established by the introduction of a pair of parameters for each of the problems: a lucky guess probability and a careless…

Although prior research has shown that experts tend to overestimate or underestimate what laypersons actually know, little is known about the specific consequences of biased estimations for communication. To investigate the impact of biased estimations of a layperson's knowledge on the effectiveness of experts' explanations, we conducted a…

Although estimations typically possess correct answers, these answers may be difficult to demonstrate to others. However, providing external information may increase their demonstrability. In this experiment, individuals (N = 60) and 6-person groups (N = 360) generated estimations with or without frames of reference. We hypothesized that…

This study examined adult students' informal knowledge of percent and its relationship to their computational skills. Sixty adults studying in urban and suburban adult education programs were interviewed to ascertain: (1) their ideas of the meanings of benchmark percents, 100%, 50%, and 25%, as they appear in advertising and media contexts; (2)…

To examine how conceptual knowledge about fraction magnitudes changes as students' learning progresses, 5th and 7th-grade students were asked to solve fraction magnitude problems that entailed finding a fraction between two given fractions and then to evaluate solutions for similar problems that were modeled for them. When the given fractions…

Studied were the misconceptions that preservice elementary teachers have about multiplication and division. Results indicated that they are influenced by the same primitive models as students; the most common errors made by both groups are quite similar. (MNS)

In the study, a part of longitudinal research focused on the emergence of mathematical knowledge structures in a learner's mind is presented. It concentrates on the analysis of introspective data gained from the author's study of a non-standard arithmetic structure in terms of the model of abstraction in context (Hershkowitz, Schwarz and Dreyfus).…

An understanding of fraction addition can be thought to involve two quantitative ideas: (1) the understanding that adding to an original quantity increases its size, and (2) a sense of how much increase occurs. Both of these ideas should underlie or inform a child's approach to problems involving fraction addition and thereby constrain the class…

Kindergarten (n=30) and grade 1 (n=15) students of contrasting socioeconomic backgrounds were interviewed to assess their knowledge of number word sequences and spatial patterns and their ability to recognize numerals and use counting to establish numerosity of visible and screened collections. Findings indicate greater variation in abilities…

Examined is the situation in which pupils invent mathematics on their own and teachers' reactions to this situation. The assimilation of students' original ideas into correct mathematical concepts is discussed. (CW)