The purpose of the present study was to examine: 1) the nature of pre-service elementary and middle school teachers' knowledge of isometries; 2) the relationship between visualization, analytic strategies, and knowledge of the prototypical defining properties of isometries; and 3) how the use of the isometry modules in the iDGi dynamic geometry environment can improve these pre-service teachers' understanding of isometries and use of spatial visualization. The participants for the study consisted of four volunteer pre-service elementary and middle childhood teachers from the Department of Teacher Education working towards a Bachelor of Science in Education degree at a public research university in the midwestern United States. Each pre-service teacher participated in one-on-one, video-recorded teaching experiments implemented with instructional modules on isometries in the Individualized Dynamic Geometry Instruction (iDGi) environment. iDGi is a computer/iPad-implemented, learning-progression-based, dynamic geometry learning system for intermediate/middle school that can be used by students independently or in classrooms with teacher guidance. Individual case studies for each participant detail the development of knowledge of isometries exhibited by participants while completing iDGi isometry modules and how their visualization and analytic, property-based reasoning interacted. Overall, the findings of the study indicate that the iDGi environment enhanced participants' understanding of isometries. In general, students moved away from Level 1 visual-holistic reasoning toward Level 2.3 formal property-based reasoning, with the largest shifts in reasoning occurring for rotations and reflections (although some students continued to struggle with reflections about obliques lines). Even though participants used various configurations of spatial visualization, analytic strategies, and knowledge of properties when reasoning about isometries, the iDGi isometry m (open full item for complete abstract)

The relationship between spatial reasoning and success in mathematics has been well established. However, the ways in which spatial reasoning is used in mathematics learning is not clearly understood. Many current studies investigating spatial reasoning in mathematics quantitatively correlate aspects of students' mathematical proficiency with standardized measures of spatial reasoning. To extend the research that connects spatial ability and the learning of mathematics, there is a need for studies that elaborate the ways in which students use spatial reasoning while learning specific mathematics content. The present study examines the relationship between spatial reasoning and geometric reasoning in the context of 2D isometries, specifically reflections and rotations. Rotations and reflections are often included in spatial reasoning assessments. Moreover, isometries are important formal mathematical concepts, which have properties, specific parameters, and lend themselves to analytic reasoning. Thus, isometries are a foundational topic for studying the link between spatial reasoning and the learning of mathematics. The present study investigated student learning of reflections and rotations using a special dynamic geometry environment. Drawing on a constructivist view of learning, the environment integrates dynamic motion with a grid in which students can learn the geometric properties of isometries while using visualization and analytic reasoning. The methodology of the study is one-on-one teaching experiments. The research questions motivating the study are the following: (1) What types of spatial reasoning do students use while learning about reflections and rotations in the special dynamic computer environment? (2) How do students' analytic and mental imagery strategies interact while learning about reflections and rotations in the environment? (3) In the context of learning about reflections and rotations in the environment, how is property- (open full item for complete abstract)

Many researchers have argued that proving is essential to doing and knowing mathematics because it is the basis of mathematical understanding (Cirillo & Herbst, 2012; Hanna & Jahnke, 1996). However, research studies conducted on students' performance with constructing proofs has found that the majority of high school geometry students are unable to construct valid proofs (McCrone & Martin, 2004; Senk, 1985). The present study contributes to this literature and focuses on the formal proofs that use triangle congruence postulates, which students construct in high school geometry. This qualitative study utilizes a psychological constructivist perspective to investigate how students construct and reason about geometric proofs. The following research questions are answered. What are the different ways that students think and reason while attempting formal geometry proofs that use triangle congruence postulates? How can the information gained from answering the first research question be used to start developing a first draft of a learning progression for geometry proofs? The study presents detailed descriptions of the cognitive processes that students use to construct and reason about geometric proofs as well as describes some of the errors and difficulties students exhibit when doing proofs. The data was collected by administering a series of one-on-one semi-structured task-based interviews to six high school geometry students who were asked to complete a series of proof problems. Students were interviewed for four to five one-hour sessions in which they "thought aloud" as they worked on twelve proof problems. All interviews were video recorded and later transcribed. Data analysis methods implemented were the constant comparative method and retrospective analysis. Findings suggest that most of the proofs that students wrote were not formally correct, but that many students wrote proofs that were not reflective of the often sound proof reasoning they stat (open full item for complete abstract)

There are usually two concerns for die casting designers, thermal characteristics and fill pattern because they are closely related to casting quality and die life. The traditional way to obtain the results is numerical simulation. However, due to the high computational cost, numerical simulation is not a perfect tool during the early stages of product development. In this study, a quick algorithm to compute the equilibrium temperature of the die and ejection temperature of the part is presented. The equilibrium temperature is defined as the time average temperature over a cycle after the process reaches the quasi steady state. This can help the cycle and die cooling/heating design. A few models to compute the heat released from part are tested and the combined asymptotic and surrogate model is applied. Special attention is paid to heat transfer calculation at the part-die interface and computational efficiency improvement. The algorithm also addresses the modeling of cooling/heat line, spray effects and techniques for die splitting at the parting line. The algorithm has been implemented in the software CastView based on the finite difference method. The previous algorithm used in CastView for fill pattern analysis based on geometric reasoning is redesigned. In this qualitative method, the flow behavior is calculated using the cavity geometric information. Many shortcomings in the old algorithm were fixed and improved. The new algorithm includes considerations which affect the flow behavior, such as flow resistance, more flow angle search and influence within neighborhood. Special attention is paid to computational efficiency improvement. The fill pattern algorithm for die casting process is adapted for slow fill processes including gravity casting and squeeze casting. The dominant term for flow behavior for different process is defined from dimensionless Navier-Stokes equations. Based on this analysis, the fill pattern algorithm for die casting is modified for slow (open full item for complete abstract)