Thinking in three dimensions: Exploring students' geometric thinking and spatial ability with the Geometer's Sketchpad

Current reform initiatives recommend that geometry instruction include the study of three-dimensional geometric objects and provide students with opportunities to use spatial skills in problem-solving tasks. Geometer's Sketchpad (GSP) is a dynamic and interactive computer program that enables the user to investigate and explore geometric concepts and manipulate geometric structures. Research using GSP as an instructional tool has focused primarily on teaching and learning two-dimensional geometry. This study explored the effect of a GSP based instructional environment on students' geometric thinking and three-dimensional spatial ability as they used GSP to learn three-dimensional geometry. For 10 weeks, 18 tenth-grade students from an urban school district used GSP to construct and analyze dynamic, two-dimensional representations of three-dimensional objects in a classroom environment that encouraged exploration, discussion, conjecture, and verification. The data were collected primarily from participant observations and clinical interviews and analyzed using qualitative methods of analysis. In addition, pretest and posttest measures of three-dimensional spatial ability and van Hiele level of geometric thinking were obtained. Spatial ability measures were analyzed using standard t-test analysis. ^ The data from this study indicate that GSP is a viable tool to teach students about three-dimensional geometric objects. A comparison of students' pretest and posttest van Hiele levels showed an improvement in geometric thinking, especially for students on lower levels of the van Hiele theory. Evidence at the p < .05 level indicated that students' spatial ability improved significantly. Specifically, the GSP dynamic, visual environment supported students' visualization and reasoning processes as students attempted to solve challenging tasks about three-dimensional geometric objects. The GSP instructional activities also provided students with an experiential base and an intuitive understanding about three-dimensional objects from which more formal work in geometry could be pursued. This study demonstrates that by designing appropriate GSP based instructional environments, it is possible to help students improve their spatial skills, develop more coherent and accurate intuitions about three-dimensional geometric objects, and progress through the levels of geometric thinking proposed by van Hiele. ^

Connecting visual and analytic reasoning to improve students' spatial visualization abilities: A constructivist approach

Current reform initiatives recommend that school geometry teaching and learning include the study of three-dimensional geometric objects and provide students with opportunities to use spatial abilities in mathematical tasks. Two ways of using Geometer's Sketchpad (GSP), a dynamic and interactive computer program, in conjunction with manipulatives enable students to investigate and explore geometric concepts, especially when used in a constructivist setting. Research on spatial abilities has focused on visual reasoning to improve visualization skills. This dissertation investigated the hypothesis that connecting visual and analytic reasoning may better improve students' spatial visualization abilities as compared to instruction that makes little or no use of the connection of the two. Data were collected using the Purdue Spatial Visualization Tests (PSVT) administered as a pretest and posttest to a control and two experimental groups. Sixty-four 10th grade students in three geometry classrooms participated in the study during 6 weeks. Research questions were answered using statistical procedures. An analysis of covariance was used for a quantitative analysis, whereas a description of students' visual-analytic processing strategies was presented using qualitative methods. The quantitative results indicated that there were significant differences in gender, but not in the group factor. However, when analyzing a sub sample of 33 participants with pretest scores below the 50th percentile, males in one of the experimental groups significantly benefited from the treatment. A review of previous research also indicated that students with low visualization skills benefited more than those with higher visualization skills. The qualitative results showed that girls were more sophisticated in their visual-analytic processing strategies to solve three-dimensional tasks. It is recommended that the teaching and learning of spatial visualization start in the middle school, prior to students' more rigorous mathematics exposure in high school. A duration longer than 6 weeks for treatments in similar future research studies is also recommended.

A comparison of the effects of social constructivist and traditional approaches to teaching on students' attitude and achievement in high school chemistry

Current views of the nature of knowledge and of learning suggest that instructional approaches in science education pay closer attention to how students learn rather than on teaching. This study examined the use of approaches to teaching science based on two contrasting perspectives in learning, social constructivist and traditional, and the effects they have on students' attitudes and achievement. Four categories of attitudes were measured using the Upper Secondary Attitude Questionnaire: Attitude towards school, towards the importance of science, towards science as a career, and towards science as a subject in school. Achievement was measured by average class grades and also with a researcher/teacher constructed 30-item test that involved three sub-scales of items based on knowledge, and applications involving near-transfer and far-transfer of concepts. The sample consisted of 202 students in nine intact classrooms in chemistry at a large high school in Miami, Florida, and involved two teachers. Results were analyzed using a two-way analysis of covariance (ANCOVA) with a pretest in attitude as the covariate for attitudes and prior achievement as the covariate for achievement. A comparison of the adjusted mean scores was made between the two groups and between females and males. ^ With constructivist-based teaching, students showed more favorable attitude towards science as a subject, obtained significantly higher scores in class achievement, total achievement and achievement on the knowledge sub-scale of the knowledge and application test. Students in the traditional group showed more favorable attitude towards school. Females showed significantly more positive attitude towards the importance of science and obtained significantly higher scores in class achievement. No significant interaction effects were obtained for method of instruction by gender. ^ This study lends some support to the view that constructivist-based approaches to teaching science is a viable alternative to traditional modes of teaching. It is suggested that in science education, more consideration be given to those aspects of classroom teaching that foster closer coordination between social influences and individual learning. ^

An innovative approach to teaching structural induction for computer science

Proofs by induction are central to many computer science areas such as data structures, theory of computation, programming languages, program efficiency-time complexity, and program correctness. Proofs by induction can also improve students’ understanding and performance of computer science concepts such as programming languages, algorithm design, and recursion, as well as serve as a medium for teaching them. Even though students are exposed to proofs by induction in many courses of their curricula, they still have difficulties understanding and performing them. This impacts the whole course of their studies, since proofs by induction are omnipresent in computer science. Specifically, students do not gain conceptual understanding of induction early in the curriculum and as a result, they have difficulties applying it to more advanced areas later on in their studies. The goal of my dissertation is twofold: (1) identifying sources of computer science students’ difficulties with proofs by induction, and (2) developing a new approach to teaching proofs by induction by way of an interactive and multimodal electronic book (e-book). For the first goal, I undertook a study to identify possible sources of computer science students’ difficulties with proofs by induction. Its results suggest that there is a close correlation between students’ understanding of inductive definitions and their understanding and performance of proofs by induction. For designing and developing my e-book, I took into consideration the results of my study, as well as the drawbacks of the current methodologies of teaching proofs by induction for computer science. I designed my e-book to be used as a standalone and complete educational environment. I also conducted a study on the effectiveness of my e-book in the classroom. The results of my study suggest that, unlike the current methodologies of teaching proofs by induction for computer science, my e-book helped students overcome many of their difficulties and gain conceptual understanding of proofs induction.

What Teaching Strategies Can I Employ to Improve Student Achievement in the Addition of Fractions with Different Denominators?

The purpose of this action research was to determine what instructional strategies could be used to improve student achievement in fraction addition. An eighth grade intensive math class practiced multiplication facts and hands-on applications of fractions concepts for 2 months. Pretests/posttests were used to measure improvement in computation and understanding.

Gaining insight into teaching: A phenomenological exploration of the lived experience of Teachers of the Year

What qualities, skills, and knowledge produce quality teachers? Many stake-holders in education argue that teacher quality should be measured by student achievement. This qualitative study shows that good teachers are multi-dimensional; their effectiveness cannot be represented by students' test scores alone. The purpose of this phenomenological study was to gain a deeper understanding of quality in teaching by examining the lived experiences of 10 winners or finalists of the Teacher of the Year (ToY) Award. Phenomenology describes individuals' daily experiences of phenomena, examines how these experiences are structured, and focuses analysis on the perspectives of the persons having the experience (Moustakas, 1994). This inquiry asked two questions: (a) How is teaching experienced by recognized as outstanding Teachers of the Year? and (b) How do ToYs feelings and perceptions about being good teachers provide insight, if any, about concepts such as pedagogical tact, teacher selfhood, and professional dispositions? Ten participants formed the purposive sample; the major data collection tool was semi-structured interviews (Patton, 1990; Seidman, 2006). Sixty to 90-minute interviews were conducted with each participant. Data also included the participants' ToY application essays. Data analysis included a three-phase process: description, reduction, interpretation. Findings revealed that the ToYs are dedicated, hard-working individuals. They exhibit behaviors, such as working beyond the school day, engaging in lifelong learning, and assisting colleagues to improve their practice. Working as teachers is their life's compass, guiding and wrapping them into meaningful and purposeful lives. Pedagogical tact, teacher selfhood, and professional dispositions were shown to be relevant, offering important insights into good teaching. Results indicate that for these ToYs, good teaching is experienced by getting through to students using effective and moral means; they are emotionally open, have a sense of the sacred, and they operate from a sense of intentionality. The essence of the ToYs teaching experience was their being properly engaged in their craft, embodying logical, psychological, and moral realms. Findings challenge current teacher effectiveness process-product orthodoxy which makes a causal connection between effective teaching and student test scores, and which assumes that effective teaching arises solely from and because of the actions of the teacher.

An Innovative Approach to Teaching Structural Induction for Computer Science

Proofs by induction are central to many computer science areas such as data structures, theory of computation, programming languages, program efficiency-time complexity, and program correctness. Proofs by induction can also improve students’ understanding of and performance with computer science concepts such as programming languages, algorithm design, and recursion, as well as serve as a medium for teaching them. Even though students are exposed to proofs by induction in many courses of their curricula, they still have difficulties understanding and performing them. This impacts the whole course of their studies, since proofs by induction are omnipresent in computer science. Specifically, students do not gain conceptual understanding of induction early in the curriculum and as a result, they have difficulties applying it to more advanced areas later on in their studies. The goal of my dissertation is twofold: 1. identifying sources of computer science students’ difficulties with proofs by induction, and 2. developing a new approach to teaching proofs by induction by way of an interactive and multimodal electronic book (e-book). For the first goal, I undertook a study to identify possible sources of computer science students’ difficulties with proofs by induction. Its results suggest that there is a close correlation between students’ understanding of inductive definitions and their understanding and performance of proofs by induction. For designing and developing my e-book, I took into consideration the results of my study, as well as the drawbacks of the current methodologies of teaching proofs by induction for computer science. I designed my e-book to be used as a standalone and complete educational environment. I also conducted a study on the effectiveness of my e-book in the classroom. The results of my study suggest that, unlike the current methodologies of teaching proofs by induction for computer science, my e-book helped students overcome many of their difficulties and gain conceptual understanding of proofs induction.

Gaining Insight into Teaching: A Phenomenological Exploration of the Lived Experience of the Teachers of the Year

What qualities, skills, and knowledge produce quality teachers? Many stake-holders in education argue that teacher quality should be measured by student achievement. This qualitative study shows that good teachers are multi-dimensional; their effectiveness cannot be represented by students’ test scores alone. The purpose of this phenomenological study was to gain a deeper understanding of quality in teaching by examining the lived experiences of 10 winners or finalists of the Teacher of the Year (ToY) Award. Phenomenology describes individuals’ daily experiences of phenomena, examines how these experiences are structured, and focuses analysis on the perspectives of the persons having the experience (Moustakas, 1994). This inquiry asked two questions: (a) How is teaching experienced by recognized as outstanding Teachers of the Year? and (b) How do ToYs feelings and perceptions about being good teachers provide insight, if any, about concepts such as pedagogical tact, teacher selfhood, and professional dispositions? Ten participants formed the purposive sample; the major data collection tool was semi-structured interviews (Patton, 1990; Seidman, 2006). Sixty to 90-minute interviews were conducted with each participant. Data also included the participants’ ToY application essays. Data analysis included a three-phase process: description, reduction, interpretation. Findings revealed that the ToYs are dedicated, hard-working individuals. They exhibit behaviors, such as working beyond the school day, engaging in lifelong learning, and assisting colleagues to improve their practice. Working as teachers is their life’s compass, guiding and wrapping them into meaningful and purposeful lives. Pedagogical tact, teacher selfhood, and professional dispositions were shown to be relevant, offering important insights into good teaching. Results indicate that for these ToYs, good teaching is experienced by getting through to students using effective and moral means; they are emotionally open, have a sense of the sacred, and they operate from a sense of intentionality. The essence of the ToYs teaching experience was their being properly engaged in their craft, embodying logical, psychological, and moral realms. Findings challenge current teacher effectiveness process-product orthodoxy which makes a causal connection between effective teaching and student test scores, and which assumes that effective teaching arises solely from and because of the actions of the teacher.