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. ^

Effects of academic integration on the persistence and achievement of under-prepared college students

The purpose of this dissertation was to examine the effect of academic integration, defined in terms of instructor/student contact, on the persistence of under-prepared college students and the achievement of those who persist. The overall design of this study compared instructor-initiated contact with conventional contact. The dependent variables were persistence, achievement, motivation and anxiety. Information was collected by administering the College-Level Mathematics Test (CLM), the Perceived Affective Contact Questionnaire (PAC), the expectancy and anxiety components of the Motivated Strategies for Learning Questionnaire (MSLQ), a departmental final examination, and by accessing university records. The sample consisted of 130 college algebra students at a large, public university with a Hispanic majority. The main analyses consisted of a 2 x 2 x 2 ANOVA (treatment by ethnicity by gender) to test for differences in achievement, t-tests to compare motivation, anxiety and perceived affective contact scores for the two groups, Chi-square tests to assess differences in persistence, and Pearson Product-Moment Correlation to determine the relationship between pretest scores and achievement variables. Results indicated that neither instructor-initiated contact, gender nor ethnicity is related to persistence, motivation or anxiety. A significant disordinal interaction of treatment and ethnicity was observed, with Hispanic experimental students scoring significantly higher on a test of algebra achievement than Hispanic control students. ^ Academic integration, defined in terms of instructor/student contact, has a positive influence on the achievement of Hispanic students. This may imply their positive responsiveness to the relational aspect of contact due to traditional cultural values of interdependence, acquiescence to authority and physical closeness. Such interactive feedback is a means by which students with these values are recognized as members of the university academic community, prompting increased academic effort. Training in contact initiation to promote academic integration is implied, by the results of this study, for instructors dealing with first year students, especially instructors at institutions accommodating instructional methods to the needs of diverse groups. ^

The relation of sports participation to academic performance of high school students

Over the past one hundred years, interscholastic athletic programs have evolved to a place of prominence in both public and private education across America. The National Federation of State High School Associations (NFHS) estimates that approximately 3.96 million males and 2.80 million females participated in organized high school athletic programs during the 2001–2002 school year at over 17 thousand public and private high schools. The popularity of interscholastic athletic programs has resulted in continuous investigations of the relationship between high school athletic programs and academic performance. ^ The present study extends earlier investigations by examining the relation of athletic participation to several indicators of academic performance for senior high school students. This research examined: (a) average daily attendance of varsity athletes and non-athletes; (b) final cumulative grade point average; and (c) test scores on the tenth grade Florida Comprehensive Assessment Test (FLAT) in both reading and in mathematics. ^ Data were collected on 2081 randomly selected male and female high school students identified as athletes or non-athletes at ten public senior high schools in the Miami-Dade County Public Schools district. The results of the overall analyses showed a positive and significant relationship between athletic participation and educational performance. On average, athletes were absent fewer days from school per year than non-athletes and athletes earned a significantly higher cumulative grade point average than their non-athlete peers. A significant statistical difference was also found in the tenth grade FCAT test scores in both reading and mathematics for athletes and non-athletes when eighth grade FCAT test scores in reading and mathematics were used as co-variates. Athletes earned significantly higher Grade 10 FCAT test scores in both reading and mathematics than non-athletes. ^ Although cause and effect cannot be inferred from this study, the findings do indicate the potentially beneficial value of athletic programs in public secondary education. The study concluded that Florida high school graduation requirements might seriously consider the role of interscholastic athletic programs as a valid and essential extra-curricular activity. ^

Causal attributions for success or failure by passing and failing students in College Algebra

Success in mathematics has been identified as a predictor of baccalaureate degree completion. Within the coursework of college mathematics, College Algebra has been identified as a high-risk course due to its low success rates. ^ Research in the field of attribution theory and academic achievement suggests a relationship between a student's attributional style and achievement. Theorists and researchers contend that attributions influence individual reactions to success and failure. They also report that individuals use attributions to explain and justify their performance. Studies in mathematics education identify attribution theory as the theoretical orientation most suited to explain academic performance in mathematics. This study focused on the relationship among a high risk course, low success rates, and attribution by examining the difference in the attributions passing and failing students gave for their performance in College Algebra. ^ The methods for the study included a pilot administration of the Causal Dimension Scale (CDSII) which was used to conduct reliability and principal component analyses. Then, students (n = 410) self-reported their performance on an in-class test and attributed their performance along the dimensions of locus of causality, stability, personal controllability, and external controllability. They also provided open-ended attribution statements to explain the cause of their performance. The quantitative data compared the passing and failing groups and their attributions for performance on a test using One-Way ANOVA and Pearson chi square procedures. The open-ended attribution statements were coded in relation to ability, effort, task difficulty, and luck and compared using a Pearson chi square procedure. ^ The results of the quantitative data comparing passing and failing groups and their attributions along the dimensions measured by the CDSII indicated statistical significance in locus of causality, stability, and personal controllability. The results comparing the open-ended attribution statements indicated statistical significance in the categories of effort and task difficulty. ^

Effectiveness of the Teaching Enrichment Activities to Minorities (TEAM) program at increasing underrepresented students in gifted programs

This study examined the effectiveness of the TEAM (Teaching Enrichment Activities to Minorities) program in preparing and identifying underrepresented students for entrance into the gifted program. Miami-Dade County Public Schools (M-DCPS) developed the TEAM program as an intervention program aimed at developing student's thinking skills and critical thinking skills in all subject areas and prepare students for possible placement into the gifted program. ^ A systematic sampling strategy was utilized to select three TEAM schools from each of the six regions in M-DCPS for the sample, for a total of 18 schools. A pool of the students that participated in the TEAM program in 2003-2004 in the 18 schools selected were identified as the TEAM Sample students. A matching sample was created from 18 public schools in Miami-Dade County that did not implement the TEAM program in 2003-2004. The Matching Sample created a match for 806 students in the TEAM sample, for a total of 1612 subjects for the study. ^ This study used a logistical regression design to analyze the relationships of multiple independent variables, including: ethnicity, limited English proficiency, gender, free/reduced lunch status, grade level, reading achievement, mathematics achievement, and participation TEAM on the dependent variables of referral for the gifted program and eligibility into the gifted program. The first analysis found the variables of grade level, participation in TEAM, reading achievement, and mathematics achievement were all significant variables in determining if a student was referred for the gifted program. The second analysis found the variables of grade level, gender, free/reduced lunch status, reading achievement, and mathematics achievement were all significant variables in determining if a student was eligible for the gifted program. ^ Recommendations based on the results of this study include the expansion of the TEAM program in M-DCPS to include additional grade levels and schools. Additionally, adopting a broadened definition of giftedness and reviewing the screening and placement policies for potentially gifted students is recommended. Adopting multicultural and broader definitions of giftedness and constructing better tools and programs, such as TEAM, for assessing and identifying potential gifted students, represent small steps towards creating equitable education for all students. ^

An investigation of mathematics and science instruction in English and Spanish for English language learners

The contextual demands of language in content area are difficult for ELLS. Content in the native language furthers students' academic development and native language skills, while they are learning English. Content in English integrates pedagogical strategies for English acquisition with subject area instruction. The following models of curriculum content are provided in most Miami Dade County Public Schools: (a) mathematics instruction in the native language with science instruction in English or (b) science instruction in the native language with mathematics instruction in English. The purpose of this study was to investigate which model of instruction is more contextually supportive for mathematics and science achievement. ^ A pretest and posttest, nonequivalent group design was used with 94 fifth grade ELLs who received instruction in curriculum model (a) or (b). This allowed for statistical analysis that detected a difference in the means of .5 standard deviations with a power of .80 at the .05 level of significance. Pretreatment and post-treatment assessments of mathematics, reading, and science achievement were obtained through the administration of Aprenda-Segunda Edición and the Florida Comprehensive Achievement Test. ^ The results indicated that students receiving mathematics in English and Science in Spanish scored higher on achievement tests in both Mathematics and Science than the students who received Mathematics in Spanish and Science in English. In addition, the mean score of students on the FCAT mathematics examination was higher than their mean score on the FCAT science examination regardless of the language of instruction. ^

Effects of cooperative and individual integrated learning system on attitudes and achievement in mathematics

This study compares the effects of cooperative delivery (CD) and individual delivery (ID) of integrated learning system (ILS) instruction in mathematics on achievement, attitudes and behaviors in adult (16-21 yrs.) high school students (grades 9-13). The study was conducted in an urban adult high school in Miami-Dade County Public Schools using a pre-test/post-test design. Achievement was measured using the Test of Adult Basic Education (TABE) by CTB MC-Graw-Hill and Compass Learning. An attitudinal survey measured attitudes towards mathematics, the computer-related lessons, and attitudes toward group activities. Behavior was assessed using computer lab observations. ^ Two-way analyses of variance (ANOVA) were conducted on achievement (TABE and Compass) by group and time (pre and post). A one-way ANOVA was conducted on the overall attitude by group on the five components (i.e., content mathematics, delivery/computers, cooperative, partners, and self efficacy) and a one-way ANOVA was conducted on the on-task behavior by group. ^ The results of the study revealed that CD and ID students working on mathematics activities delivered by the ILS performed similarly on achievement tests of the TABE. The CD-ILS students had significantly better overall mathematics attitudes than the ID-ILS students and the ID-ILS group was on-task significantly more than the CD-ILS group. This study concludes that regularity and period of time over which the ILS is used may prove to be important variables although there were insufficient data to fully investigate the impact of models of use. Additionally, a minimum amount of time-on-system is necessary before gains can become apparent in innumeracy and increasing exposure to the system may have beneficial effects on learning. ^

The effect of instruction on the study behaviors and academic achievement of community college students

The purpose of this study was twofold. It was designed to determine (a) the efficacy of an intervention designed to increase the frequencies of appropriate study behaviors on the part of community college students who were preparing for academic mathematics activities and (b) whether any increase in appropriate study behavior frequency was accompanied by increased academic achievement in college preparatory mathematics classes. A total of 126 Miami Dade College students participated in this study. Two developmental (remedial) mathematics classes were randomly assigned as the experimental group, and two others were assigned as the control group. All students also took a College Survival (SLS) class. The Study Behavior Inventory (SBI) and the Computerized Placement Test (CPT) were administered to the four classes. The SBI was used as a pre- and post-test. The SLS curriculum and classroom time were the same for both groups. However, students in the treatment groups received instruction designed to increase the frequency of effective study behaviors associated with the three factors identified in the SBI, while the students in the control group participated in activities that did not emphasize study behaviors. A series of analysis of covariance procedures were used to analyze four hypotheses. The first three hypotheses proposed that students who were instructed in the use of appropriate study behaviors would score higher on the three factors of the Study Behavior Inventory than those who were not. The fourth hypothesis proposed that a greater proportion of mathematics students who were instructed in the use of appropriate study behaviors would receive a passing grade in their course than students who were in the control group. The four hypotheses were tested at the .05 level of significance. This study revealed that students who received instruction in appropriate study behaviors scored higher in the use of appropriate study behaviors and in mathematics achievement than students who did not. Additional research is needed to investigate whether these effects would persist over time, or be found in subjects other than mathematics.

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.

Motivational Strategies to Increase Completion of Assignments in Mathematics Classes

The purpose of this paper is to share the results of an 8-week study that focused on the effects of incorporating real-life applications and rewards to measure their impact on student motivation. The goal was to reduce the number of students unmotivated to complete their mathematics assignments satisfactorily.

Improving Student Mathematics Achievement through Self Regulation and Goal Setting

As middle school students’ mathematic scores decline in comparison to other countries, researchers have found self-regulation to be a tool to improve students’ mathematics achievement. The following is an action research project conducted in a middle school where result showed an increase in students’ mathematics achievement.

Mathematics: Going Beyond the Academic Discipline

Mathematics is rigidly classified as an academic discipline. This determines curriculum content and teaching and evaluation methods. These methods can give rise to negative views of mathematics, resulting in increased math anxiety. Educators, therefore, need to look beyond the discipline to provide a classroom environment that meets students’ needs.

Effects of Cooperative and Individual Integrated Learning System on Attitudes and Achievement in Mathematics

The purpose of this study was to determine the effects of a computer-based Integrated Learning Systems (ILS) model used with adult high school students engaging mathematics activities. This study examined achievement, attitudinal and behavior differences between students completing ILS activities in a traditional, individualized format compared to cooperative learning groups.

The Predictive Power of Homework Assignments on Student Achievement in Mathematics

This study examined the relationship between homework performance (percent of homework completed and percent of homework correct), student characteristics (Stanford Achievement Test score, gender, ethnicity, and socio-economic status), perceptions, and challenges and academic achievement determined by the students’ average score on weekly tests and their score on the Florida Comprehensive Assessment Test (FCAT) Norm Reference Test (NRT) mathematics assessment.

A comparative study of high -achieving and low -achieving African -American students in an urban setting

The purpose of this study was to delineate which demographic and school variables were important for predicting the achievement of 10th grade African-American students. The sample population was divided into two groups: high-achievers, students with GPAs of 3.5 or higher, and low-achievers, students with GPAs of 1.5 or lower. Variables examined in the study included: gender; birth place; student's native language; exceptionality (ESE); history of English proficiency (LEP); SES (lunch status) in elementary and high school; the percentage of the Black student population in high school; and suspensions, absences, tardies, and the Stanford Achievement Test (SAT) scores in reading comprehension, mathematics computation, and mathematics applications in elementary and middle school. Two separate logistic regression analyses were conducted to determine which variables were influential in predicting achievement.^ Analysis 1 (N = 366), which included all the variables, except the SAT percentile scores, correctly classified 87% of the students as high-achievers or low-achievers. The results from Analysis 1 revealed that students who--were female; spoke a language other than English as their first language; did not apply for free or reduced lunch in elementary school; were in the gifted program; had no absences or tardies in elementary school; had no suspensions or tardies in middle school; and attended a high school with a lower percentage of Black students--were more likely to be high-achieving than low-achieving.^ Analysis 2 (N = 274) included all the variables and resulted in 94% of the students being correctly classified. It was found that students who--were female; were currently or previously classified as Limited English Proficient (LEP); did not apply for free or reduced lunch in elementary school; had no suspensions or tardies in middle school; and had higher percentile scores in reading comprehension and mathematics computation on the SAT in middle school--were more likely to high-achieving than low-achieving.^ The quantitative analyses were coupled with interviews from a purposeful sample of the population (N = 12) to gain additional insight about why some African-American students are succeeding in our schools and others are not. This study provides a viable means for assessing African-American students' achievement patterns in our schools. ^