Husserl's two notions of completeness: husserl and hilbert on completeness and imaginary elements in mathematics

In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures he gave in Göttingen in 1901 and other related texts of the same period, a problem that had occupied Husserl since the beginning of 1890, when he was planning a never published sequel to Philosophie der Arithmetik (1891). In order to solve the problem of imaginary entities Husserl introduced, independently of Hilbert, two notions of completeness (definiteness in Husserl's terminology) for a formal axiomatic system. I present and discuss these notions here, establishing also parallels between Husserl's and Hilbert's notions of completeness. © 2000 Kluwer Academic Publishers.

Creating Meaningful Homework and Implementing Homework Presentations to Aid in Mathematics Instruction

In this action research study of sixth grade mathematics, I investigated the use of meaningful homework and the implementation of presentations and its effect on students’ comprehension of mathematical concepts. I collected data to determine whether the creating of meaningful homework and the implementation of homework presentations would have a positive impact on the students’ understanding of the concepts being taught in class and the reasoning behind assigning homework. The homework was based on the lesson taught during class time. It was grade-level appropriate and contained problems similar to those students completed in class. A pre-research and post-research survey based on homework perceptions and my teaching practices was given, student interviews were conducted throughout the research period, weekly teacher journals were kept that pertained to my teaching practices and the involvement of the students that particular week, and homework assignments were collected to gauge the students’ understanding of the mathematics lessons. Most students’ perceptions on homework were positive and most understood the reasoning for homework assignments.

Journal Writing in Mathematics: Exploring the Connections between Math Journals and the Completion of Homework Assignments

In this action research study of sixth grade mathematics, I investigate how the use of written journals facilitates the learning of mathematics for my students. I explore furthermore whether or not these writing journals support students to complete their homework. My analysis reveals that while students do not access their journals daily, when students have the opportunity to write more about one specific problem--such as finding the relationship between the area of two different sized rectangles – they, are nevertheless, more likely to explain their thoughts in-depth and go beyond the traditional basic steps to arrive at a solution. This suggests the value of integrating journal writing in a math curriculum as it can facilitate classroom discussion from the students’ written work.

Meaningful Independent Practice in Mathematics

In this action research of my seventh grade mathematics classroom, I investigated how students’ explanations of math homework would improve their learning in math. I discovered these explanations can be very beneficial in helping students to improve their understanding of current skills although it did not affect all students. As a result of this study, I plan to incorporate these student explanations in my instruction next year but not as a daily expectation.

Oral Communication and Presentations in Mathematics

In this action research study of my classroom of eighth grade mathematics, I investigated the attitudes of students toward mathematics along with their achievement levels with the use of oral presentations in my Algebra class. During the second semester the class was divided into groups of two for each presentation, changing partners each time. Every other week each group was given a math problem that required more work than a normal homework type problem. On the last day of that week the students gave a short presentation on their problem. I discovered that while there was no significant evidence that student achievement increased, the students did enjoy the different aspect of presentations in a math class. I plan to implement presentations in my classroom more often with the intent to increase student enjoyment.

Generating Interest in Mathematics Using Discussion in the Middle School Classroom

In this action research study of my classroom of 8th grade algebra, I investigated students’ discussion of mathematics and how it relates to interest in the subject. Discussion is a powerful tool in the classroom. By relying too heavily on drill and practice, a teacher may lose any individual student insight into the learning process. However, in order for the discussion to be effective, students must be provided with structure and purpose. It is unrealistic to expect middle school age students to provide their own structure and purpose; a packet was constructed that would allow the students to both show their thoughts and work as a small group toward a common goal. The students showed more interest in the subject in question as they related to the algebra topics being studied. The students appreciated the packets as a way to facilitate discussion rather than as a vehicle for practicing concepts. Students still had a need for practice problems as part of their homework. As a result of this research, it is clear that discussion packets are very useful as a part of daily instruction. While there are modifications that must be made to the original packets to more clearly express the expectations in question, discussion packets will continue to be an effective tool in the classroom.

Literacy Practices in and out of School in Karagwe : the Case of Primary School Literacy in Rural Tanzania

This study has investigated the question of relation between literacy practices in and out of school in rural Tanzania. By using the perspective of linguistic anthropology, literacy practices in five villages in Karagwe district in the northwest of Tanzania have been analysed. The outcome may be used as a basis for educational planning and literacy programs. The analysis has revealed an intimate relation between language, literacy and power. In Karagwe, traditional élites have drawn on literacy to construct and reconstruct their authority, while new élites, such as individual women and some young people have been able to use literacy as one tool to get access to power. The study has also revealed a high level of bilingualism and a high emphasis on education in the area, which prove a potential for future education in the area. At the same time discontinuity in language use, mainly caused by stigmatisation of what is perceived as local and traditional, such as the mother-tongue of the majority of the children, and the high status accrued to all that is perceived as Western, has turned out to constitute a great obstacle for pupils’ learning. The use of ethnographic perspectives has enabled comparisons between interactional patterns in schools and outside school. This has revealed communicative patterns in school that hinder pupils’ learning, while the same patterns in other discourses reinforce learning. By using ethnography, relations between explicit and implicit language ideologies and their impact in educational contexts may be revealed. This knowledge may then be used to make educational plans and literacy programmes more relevant and efficient, not only in poor post-colonial settings such as Tanzania, but also elsewhere, such as in Western settings.

How teachers think about the role of digital technologies in student assessment in mathematics

This study concerns teachers’ use of digital technologies in student assessment, and how the learning that is developed through the use of technology in mathematics can be evaluated. Nowadays math teachers use digital technologies in their teaching, but not in student assessment. The activities carried out with technology are seen as ‘extra-curricular’ (by both teachers and students), thus students do not learn what they can do in mathematics with digital technologies. I was interested in knowing the reasons teachers do not use digital technology to assess students’ competencies, and what they would need to be able to design innovative and appropriate tasks to assess students’ learning through digital technology. This dissertation is built on two main components: teachers and task design. I analyze teachers’ practices involving digital technologies with Ruthven’s Structuring Features of Classroom Practice, and what relation these practices have to the types of assessment they use. I study the kinds of assessment tasks teachers design with a DGE (Dynamic Geometry Environment), using Laborde’s categorization of DGE tasks. I consider the competencies teachers aim to assess with these tasks, and how their goals relate to the learning outcomes of the curriculum. This study also develops new directions in finding how to design suitable tasks for student mathematical assessment in a DGE, and it is driven by the desire to know what kinds of questions teachers might be more interested in using. I investigate the kinds of technology-based assessment tasks teachers value, and the type of feedback they give to students. Finally, I point out that the curriculum should include a range of mathematical and technological competencies that involve the use of digital technologies in mathematics, and I evaluate the possibility to take advantage of technology feedback to allow students to continue learning while they are taking a test.

Children's performance estimation in mathematics and science tests over a school year: A pilot study

The metacognitve ability to accurately estimate ones performance in a test, is assumed to be of central importance for initializing task-oriented effort. In addition activating adequate problem-solving strategies, and engaging in efficient error detection and correction. Although school children's' ability to estimate their own performance has been widely investigated, this was mostly done under highly-controlled, experimental set-ups including only one single test occasion. Method: The aim of this study was to investigate this metacognitive ability in the context of real achievement tests in mathematics. Developed and applied by a teacher of a 5th grade class over the course of a school year these tests allowed the exploration of the variability of performance estimation accuracy as a function of test difficulty. Results: Mean performance estimations were generally close to actual performance with somewhat less variability compared to test performance. When grouping the children into three achievement levels, results revealed higher accuracy of performance estimations in the high achievers compared to the low and average achievers. In order to explore the generalization of these findings, analyses were also conducted for the same children's tests in their science classes revealing a very similar pattern of results compared to the domain of mathematics. Discussion and Conclusion: By and large, the present study, in a natural environment, confirmed previous laboratory findings but also offered additional insights into the generalisation and the test dependency of students' performances estimations.

Improving Health Literacy in Connecticut: A Prescription for Healthy Lives and Communities

The Practicum Project is a supervised service-learning experience that integrates curriculum with hands-on experience in a public health setting. All 2nd year students are expected to work collaboratively in assessing the extent, causes and public health responses to a selected public health problem confronting citizens of Connecticut. The focal topic for the 2006 Project was Health Literacy in Connecticut. During this past spring, 17 students of our program, working alongside and in partnership with more than 75 community-based stakeholders across Connecticut, completed over 1800 hours of service-learning in pursuit of answers to 3 questions: Can the present and future burden of health literacy be estimated for Connecticut? What is the current capacity of Connecticut’s health and social service system to halt the crisis we confront today? Can new policy and regulatory strategies be put forth to reduce the severity and scope of the problem? This occasion and the accompanying report mark the completion of their project and acknowledge the considerable contributions that many have made to the success of this educational experience. Through those combined efforts, students gained experience and skill addressing one of the most significant public health issues of our time; also, they gained insight into the breadth and capacity of our public health system and established invaluable relationships with public health practitioners, agencies and institutions around the state. Their report documents a rich campus-community partnership to advance public health goals.

Motivation and Learning in Mathematics Pre-service Teachers

Based on a review of literature of conceptual and procedural knowledge in relation to intrinsic and extrinsic motivation, the purpose of this study was to test the relationship between conceptual and procedural knowledge and intrinsic and extrinsic motivation. Thirty-eight education students with a mathematics focus (elementary or secondary) in their junior, senior, or fifth year completed a survey with a Likert scale measuring their preference to learning (conceptual or procedural) and their motivation type (intrinsic or extrinsic). Findings showed that secondary mathematics focused students were more likely to prefer learning mathematics conceptually than elementary mathematics focused students. However, secondary and elementary mathematics focused students showed an equal preference for learning mathematics procedurally and sequentially. Elementary and secondary students reported similar intrinsic and extrinsic motivation. Extrinsically motivated students preferred procedural learning more than conceptual learning. While there was no statistically significant preference with intrinsically motivated students, there was a trend favoring preference of conceptual learning over procedural learning. These results tend to support the hypothesis that mathematics focused students who prefer conceptual learning are more intrinsically motivated, and mathematics focused students who prefer procedural learning are more extrinsically motivated.

Assessment of markers of medication literacy in a US-Mexico border community

Each year an estimated 180,000 people in the United States (U.S.) die as a result of medication errors, now considered a major public health problem. If a patient cannot correctly act on information related to medication use or "Medication Literacy" there is an increased potential for error. Medication use issues are unique on the US-Mexico border because they include high rates of herbal products and medication products from Mexico as well as issues related to the preferred language (English or Spanish) of the patient. To evaluate the medication literacy in a US-Mexico border community, this retrospective study evaluates 180 subjects representing four diverse economic segments of a metropolitan US-Mexico Border community who have taken a Medication Literacy Assessment. The assessment tool has been created to understand how patients interpret medication information for prescription, over-the-counter, herbal, and Mexican medication product use, and how they problem-solve medication questions. The Medication Literacy Assessment tool specifically assesses document literacy (e.g., prescription labels), prose literacy (e.g., patient leaflets), numeracy (e.g., calculations and measurements) as well as qualitative data related to medication use practices. The main hypothesis of this study is that the ability to interpret and use medications will vary based on education, language (Spanish or English), and recruitment sites (economically diverse communities). The results will provide information to better characterize medication use in a primarily Hispanic population on the US-Mexico border and may be used to influence policy decisions regarding prescription and over-the-counter product information.^