Assessing the Impact of Computer Programming in Understanding Limits and Derivatives in a Secondary Mathematics Classroom

This study explored the development of student’s conceptual understanding of limit and derivative when specific computational tools were utilized. Fourteen students from a secondary Advanced Placement Calculus AB course explored the limit and derivative concepts from calculus using computational tools in the Maple computer algebra system. Students worked in pairs utilizing the pair-programming collaborative model. Four groups of student pairs constructed computational tools and used them to explore the limit and derivative concepts. The remaining four student pairs were provided similar tools and asked to perform identical explorations. A multiple embedded case design was utilized to explore ways students in two classes, a programming class P and a non-programming class N, constructed understandings focusing upon their interactions with each other and with the computational tools. The Action-Process-Object-Schema (APOS) conceptual model and Constructionist framework guided design and construction of the tools, outlined developmental goals and milestones, and provided interpretive context for analysis. Results provided insights into the effective design and use of computational tools in fostering conceptual understanding. The study found the additional burden of programming redirected students’ attention away from the intended conceptual understandings. The study additionally found, however, that pre-constructed tools effectively promote conceptual understanding of the limit concept when coupled with a mature conceptual model of development. Four themes influencing development of these understandings emerged: An instructional focus on skills over concepts, the instructional sequence, the willingness and ability of students to adopt and utilize computational tools, and the ways cognitive conflict was mediated

Misunderstanding Models in Environmental and Public Health Regulation

Computational models are fundamental to environmental regulation, yet their capabilities tend to be misunderstood by policymakers. Rather than rely on models to illuminate dynamic and uncertain relationships in natural settings, policymakers too often use models as “answer machines.” This fundamental misperception that models can generate decisive facts leads to a perverse negative feedback loop that begins with policymaking itself and radiates into the science of modeling and into regulatory deliberations where participants can exploit the misunderstanding in strategic ways. This paper documents the pervasive misperception of models as truth machines in U.S. regulation and the multi-layered problems that result from this misunderstanding. The paper concludes with a series of proposals for making better use of models in environmental policy analysis.The Kay Bailey Hutchison Center for Energy, Law, and Busines

The Impact of Data Sovereignty on American Indian Self-Determination: A Framework Proof of Concept Using Data Science

The Data Sovereignty Initiative is a collection of ideas that was designed to create SMART solutions for tribal communities. This concept was to develop a horizontal governance framework to create a strategic act of sovereignty using data science. The core concept of this idea was to present data sovereignty as a way for tribal communities to take ownership of data in order to affect policy and strategic decisions that are data driven in nature. The case studies in this manuscript were developed around statistical theories of spatial statistics, exploratory data analysis, and machine learning. And although these case studies are first, scientific in nature, the data sovereignty framework was designed around these concepts to leverage nation building, cultural capital, and citizen science for economic development and planning. The data sovereignty framework is a flexible way to create data domains, around developed key indicators to integrate appropriate cultural capital when working with Native nations. This design is intended to put scientific theory into practice to affect everyday outcomes using data driven decision making. This framework is a proof concept and represents both applied and theoretical metrics in design strength

Students’ Misconceptions of the Limit Concept in a First Calculus Course

Misconceptions of the limit concept were examined in 130 pre-engineering students in Dilla Universities. Questionnaire and Interview were deigned to explore students understanding of the idea of a limit of a function and to explore the cognitive schemes for the limit concept.The study employed a quantitative-descriptive or survey design. The empirical investigation was done in two phases. A questionnaire on the idea of a limit was given to 130 students during the first phase. During the second phase 14 interviews were conducted. Then, the results indicated that students in the study see a limit as unreachable, see a limit as an approximation, see a limit as a boundary, view a limit as a dynamic process and not as a static object, and are under the impression that a function will always have a limit at a point. Regarding the relationship between a continuous function and a limit were: Students think that a function has to be defined at a point to have a limit at that point. A function that is undefined at a certain point does not have a limit; Students think that when a function has a limit, then it has to be continuous at that point. Other misconceptions were: The limit is equal to the function value at a point, i.e. a limit can be found by a method of substitution, when one divides zero by zero, the answer is zero, Most of the students know that any other number divided by zero is undefined. The study concluded that many students’ knowledge and understanding rest largely on isolated facts, routine calculation, memorizing algorithm, procedures and that their conceptual understanding of limits, continuity and infinity is deficient. The outstanding observation was that students see a limit as unreachable. This could be due to the language used in many books to describe limits for example ‘tends to’ and ‘approaches’. Another view of a limit that the students have is that a limit is a boundary point. This could be because of their experience with speed limits, although that could always be exceeded. Lecturers ought to become aware of their students’ understanding and possible misconceptions. Diagnosing the nature of students’ conceptual problems enables lecturers to develop specific teaching strategies to address such problems and to enhance conceptual understanding.Finally, the study suggested that concepts such as limit, involves a construction process, students build on and modify their existing concept images. Lecturers, in teaching the topic of limit, could develop concepts first before embarking on techniques in problem solving. Students need to conceptualize first before applying the formula. Keywords: Limit Concept, Misconceptions; Limits of functions; Concept Image; Concept Definitio

Knowledge used for teaching counting: A case study of the treatment of counting by two Grade 3 teachers situated in schools serving working class communities in the Western Cape Province of South Africa

Knowing how to correctly count, is fundamental to the future mathematics success of young children. Earlier studies show that many South African primary school students underperform in mathematics even when evaluated with task below grade level. Reports suggest that this is a problem stemming from the poor pedagogic, and or content knowledge of classroom mathematics teachers. Shulman (1986; 1987) refers to this area of knowledge as Pedagogic Content Knowledge (PCK). In the field of mathematics teaching and learning, Ball, Thames and Phelps (2008) refer to it as Mathematics Knowledge for Teaching (MKfT). Teachers' mathematics PCK, comprises of three core knowledge domain: (i) Teacher's Knowledge of Content and Teaching (KCT); (ii) Teacher's Knowledge of Content and Student (KCS); and (iii) teacher's Knowledge of Content and Curriculum (KCC). Teachers' KCS was considered in this study as it concerns what teachers know about what learners know and how they learn. The general interest of this project was to study the construction of experience of mathematics (non-core domain knowledge) by genetic endowment on the basis of contextual data. More specifically, the particular interest of the study is on the construction of the experience of counting in the pedagogic situations of Grade 3 schooling. For that purpose, video records of mathematics teaching in two schools situated in working-class communities were analysed. The study adopted an Integrated Causal Model approach which drew on resources from different disciplines such as mathematics education, cognitive science, evolutionary psychology and mathematics. The study was partly framed by Bernstein's pedagogic device, particularly with respect to his notion of evaluation, as well as the inter-related constructs of PCK, MKfT and KCS. The theoretical resources used to describe computations were drawn largely from Davis (2001, 2010b, 2011a, 2012, 2013a, 2015, 2018) and related work on the use of morphisms as elaborated in Baker et al. (1971), Gallistel & King, (2010), Krause (1969) and Open University (1970). These resources were used to produce the analytic framework for the production of and analysis of data. The analysis describes the computational activities of teachers and learners during the recorded lessons, specifically the computational domains made available pedagogically. In so doing, I was able to provide more illumination on what is described as teacher's KCS for teaching counting at the Grade 3 level. From the generated data, the study finds that counting proper was restricted to the constitution and identification of very small ordered discrete aggregates which can be handled by human core domain object tracking system and approximate number system, and that an implicit reliance on numerical order derived from computations on aggregates was central to the teaching and learning of counting