No description available.

Thesis (M.S.)--Central Connecticut State University, 1999. / Thesis advisor: Dr. Philip Halloran. " ... in partial fulfillment of the requirements for the degree of Master of Science [in Mathematics]." Includes bibliographical references (leaves 74-75).

N/A / Thesis / Master of Science (MS)

No description available.

The purpose of this paper is to study convergence of certain continued fractions.

There are two main aims of this thesis. The first is to further develop and demonstrate applications of the combinatorial interpretation of continued fractions introduced in [Benjamin and Quinn, 2003]. The second is to investigate the theory of negative continued fractions, a relatively unresearched topic. That is, discuss the ways in which they are similar to and different from the regular class, describe how to convert between the two forms, and show that the central theorems concerning regular continued fractions also apply to the negative ones.

Fractions are very hard to learn. As the joke goes, “Three out of two people have trouble with fractions”. Yet the invention of a notation for fractions is very ancient, dating back to Babylonians and Egyptians. Moreover, it is thought that ratio representation is innate. And obviously, fractions are part of our everyday life. We read them in recipes, we need them to estimate distances on maps or rebates in shops. In addition, fractions play a key role in science and mathematics, in probabilities, proportions and algebraic reasoning. Then why is it so hard for pupils to understand and use them? What is so special about fractions? As in other areas of numerical cognition, a fast-developing field in cognitive science, we tackled this paradox through a multi-pronged approach, investigating both adults and children. Based on some recent research questions and intense debates in the literature, a first behavioural study examined the mental representations of the magnitude of fractions in educated adults. Behavioural observations from adults can indeed provide a first clue to explain the paradox raised by fractions. Contrary perhaps to most educated adults’ intuition, finding the value of a given fraction is not an easy operation. Fractions are complex symbols, and there is an on-going debate in the literature about how their magnitude (i.e. value) is processed. In a first study, we asked adult volunteers to decide as quickly as possible whether two fractions represent the same magnitude or not. Equivalent fractions (e.g. 1/4 and 2/8) were identified as representing the same number only about half of the time. In another experiment, adults were also asked to decide which of two fractions was larger. This paradigm offered different results, suggesting that participants relied on both the global magnitude of the fraction and the magnitude of the components. Our results showed that fraction processing depends on experimental conditions. Adults appear to use the global magnitude only in restricted circumstances, mostly with easy and familiar fractions. In another study, we investigated the development of the mental representations of the magnitude of fractions. Previous studies in adults showed that fraction processing can be either based on the magnitude of the numerators and denominators or based on the global magnitude of fractions and the magnitude of their components. The type of processing depends on experimental conditions. In this experiment, 5th, 6th, 7th-graders, and adults were tested with two paradigms. First, they performed a same/different task. Second, they carried out a numerical comparison task in which they had to decide which of two fractions was larger. Results showed that 5th-graders do not rely on the representations of the global magnitude of fractions in the Numerical Comparison task, but those representations develop from grade 6 until grade 7. In the Same/Different task, participants only relied on componential strategies. From grade 6 on, pupils apply the same heuristics as adults in fraction magnitude comparison tasks. Moreover, we have shown that correlations between global distance effect and children’s general fraction achievement were significant. Fractions are well known to represent a stumbling block for primary school children. In a third study, we tried to identify the difficulties encountered by primary school pupils. We observed that most 4th and 5th-graders had only a very limited notion of the meaning of fractions, basically referring to pieces of cakes or pizzas. The fraction as a notation for numbers appeared particularly hard to grasp. Building upon these results, we designed an intervention programme. The intervention “From Pies to Numbers” aimed at improving children’s understanding of fractions as numbers. The intervention was based on various games in which children had to estimate, compare, and combine fractions represented either symbolically or as figures. 20 game sessions distributed over 3 months led to 15-20% improvement in tests assessing children's capacity to estimate and compare fractions; conversely, children in the control group who received traditional lessons improved more in procedural skills such as simplification of fractions and arithmetic operations with fractions. Thus, a short classroom intervention inducing children to play with fractions improved their conceptual understanding. The results are discussed in light of recent research on the mental representation of the magnitude of fractions and educational theories. The importance of multidisciplinary approaches in psychology and education was also discussed. In sum, by combining behavioural experiments in adults and children, and intervention studies, we hoped to have improved the understanding how the brain processes mathematical symbols, while helping teachers get a better grasp of pupils’ difficulties and develop classroom activities that suit the needs of learners.

Recognizing the need for U.S. students' mathematics learning to be built on a solid foundation of conceptual understanding, professional organizations such as the National Council of Teachers of Mathematics (2000) and the Conference Board of the Mathematical Sciences (2001) have called for an increased focus on building conceptual understanding in elementary mathematics in several domains. This study focuses on an exploration of two aspects of Hill, Schilling, and Ball's (2004) mathematics knowledge for teaching: specialized content knowledge (SCK) and knowledge of content and students (KCS) related to fractions concepts, an area that is particularly challenging at the elementary level and builds the foundation for understanding more complex rational number concepts in the middle grades. Eight grades early childhood through four preservice teachers enrolled in a mathematics methods course were asked to create concept maps to describe their knowledge of fractions and interpret student work with fractions. Results showed the preservice teachers to be most familiar with the part-whole representation of fractions. Study participants were least familiar with other fraction representations, including fractions as a ratio, as an operator, as a point on a number line, and as a form of division. The ratio interpretation of a fraction presented the greatest difficulty for study participants when asked to describe student misconceptions and create instructional representations to change students' thinking.

Thesis (M.Ed.)--University of Central Florida, 2009. / Adviser: Juli Dixon. Includes bibliographical references (p. 65-68).

Thesis (Ph. D.)--University of Iowa, 1928. / Thesis note on label mounted on t.p. Bibliography: leaf 29.