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. ^

A comparison of content-based and context-based teaching on ninth-grade mathematics achievement

The purpose of this study was to determine if an experimental context-based delivery format for mathematics would be more effective than a traditional model for increasing the performance in mathematics of at-risk students in a public high school of choice, as evidenced by significant gains in achievement on the standards-based Mathematics subtest of the FCAT and final academic grades in Algebra I. The guiding rationale for this approach is captured in the Secretary's Commission on Achieving Necessary Skills (SCANS) report of 1992 that resulted in school-to-work initiatives (United States Department of Labor). Also, the charge for educational reform has been codified at the state level as Educational Accountability Act of 1971 (Florida Statutes, 1995) and at the national level as embodied in the No Child Left Behind Act of 2001. A particular focus of educational reform is low performing, at-risk students. ^ This dissertation explored the effects of a context-based curricular reform designed to enhance the content of Algebra I content utilizing a research design consisting of two delivery models: a traditional content-based course; and, a thematically structured, content-based course. In this case, the thematic element was business education as there are many advocates in career education who assert that this format engages students who are often otherwise disinterested in mathematics in a relevant, SCANS skills setting. The subjects in each supplementary course were ninth grade students who were both low performers in eighth grade mathematics and who had not passed the eighth grade administration of the standards-based FCAT Mathematics subtest. The sample size was limited to two groups of 25 students and two teachers. The site for this study was a public charter school. Student-generated performance data were analyzed using descriptive statistics. ^ Results indicated that contrary to the beliefs held by many, contextual presentation of content did not cause significant gains in either academic performance or test performance for those in the experimental treatment group. Further, results indicated that there was no meaningful difference in performance between the two groups. ^

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. ^

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.

The mathematics anxiety of bilingual community college students

Math anxiety levels and performance outcomes were compared for bilingual and monolingual community college Intermediate Algebra students attending a culturally diverse urban commuter college. Participants (N = 618, 250 men, 368 women; 361 monolingual, 257 bilingual) completed the Abbreviated Math Anxiety Scale (AMAS) and a demographics instrument. Bilingual and monolingual students reported comparable mean AMAS scores (20.6 and 20.7, respectively) and comparable proportions of math anxious individuals (50% and 48%, respectively). Factor analysis of AMAS scores, using principal component analysis by varimax rotation, yielded similar two-factor structures for both populations -- assessment and learning content -- accounting for 65.6% of the trace for bilingual AMAS scores. Statistically significant predictor variables for levels of math anxiety for the bilingual participants included (a) preparatory course enrollment (β = .236, p = .041) with those enrolled in prior preparatory courses scoring higher, (b) education major (β = .285, p = .018) with education majors scoring higher, and (c) business major (β = .252, p = .032) with business majors scoring higher. One statistically significant predictor variable emerged for monolingual students, gender (β = -.085, p = .001) with females ranking higher. Age, income, race, ethnicity, U.S. origin, science or health science majors did not emerge as statistically significant predictor variables for either group.^ Similarities between monolingual and bilingual participants included statistically significant negative linear correlations between AMAS scores and course grades for both bilingual (r = -.178, p = .017) and monolingual participants (r = -.203, p = .001). Differences included a statistically significant linear correlation between AMAS scores and final exam grades for monolingual participants only (r = -.253, p < .0009) despite no statistically significant difference in the strength the linear relationship of the AMAS scores and the final exam scores between groups, z = 1.35, p = .1756.^ The findings show that bilingual and monolingual students report math anxiety similarly and that math anxiety has similar associations with performance measures, despite differences between predictor variables. One of the first studies on the math anxiety of bilingual community college students, the results suggest recommendations for researchers and practitioners.^

The role of *language in early childhood mathematics

Math literacy is imperative to succeed in society. Experience is key for acquiring math literacy. A preschooler's world is full of mathematical experiences. Children are continually counting, sorting and comparing as they play. As children are engaged in these activities they are using language as a tool to express their mathematical thinking. If teachers are aware of these teachable moments and help children bridge their daily experiences to mathematical concepts, math literacy may be enhanced. This study described the interactions between teachers and preschoolers, determining the extent to which teachers scaffold children's everyday language into expressions of mathematical concepts. Of primary concern were the teachers' responsive interactions to children's expressions of an implicit mathematical utterance made while engaged in block play. The parallel mixed methods research design consisted of two strands. Strand 1 of the study focused on preschoolers' use of everyday language and the teachers' responses after a child made a mathematical utterance. Twelve teachers and 60 students were observed and videotaped while engaged in block play. Each teacher worked with five children for 20 minutes, yielding 240 minutes of observation. Interaction analysis was used to deductively analyze the recorded observations and field notes. Using a priori codes for the five mathematical concepts, it was found children produced 2,831 mathematical utterances. Teachers ignored 60% of these utterances and responded to, but did not mediate 30% of them. Only 10% of the mathematical utterances were mediated to a mathematical concept. Strand 2 focused on the teacher's view of the role of language in early childhood mathematics. The 12 teachers who had been observed as part of the first strand of the study were interviewed. Based on a thematic analysis of these interviews three themes emerged: (a) the importance of a child's environment, (b) the importance of an education in society, and (c) the role of math in early childhood. Finally, based on a meta-inference of both strands, three themes emerged: (a) teacher conception of math, (b) teacher practice, and (c) teacher sensitivity. Implications based on the findings involve policy, curriculum, and professional development.

The relationship between the secondary mathematics curriculum, college persistence, and success at an urban community college

According to Venezia, Kirst, and Antonio (2003) and Barth’s 2002 Thinking K16 Ticket to Nowhere report, the disconnect between K-12 and postsecondary education was a contributing factor to high attrition rates. Since mathematics emerged as a primary concern for college readiness, Barth (2002) called for improving student transitions from K-12 to postsecondary institutions through the use of state or local data. The purpose of the present study was to analyze mathematics course-taking patterns of secondary students in a local context and to evaluate high school characteristics in order to explore their relationships with Associate degree attainment or continuous enrollment at an urban community college. Also, this study extended a national study conducted by Clifford Adelman (The Toolbox Revisited, 2006) as it specifically focused on community college students that were not included his study. Furthermore, this study used the theoretical framework that human capital, social capital, and cultural capital influence habitus—an individual’s or a group’s learned inclination to behave within the parameters of the imposed prevailing culture and norms. Specifically, the school embedded culture as it relates to tracking worked as a reproduction tool of ultimate benefit for the privileged group (Oakes, 1994). ^ Using multilevel analysis, this ex post facto study examined non-causal relationships between math course-taking patterns and college persistence of public high school graduates who enrolled at the local community college for up to 6 years. One school-level variable (percent of racial/ethnic minorities) and 7 student-level variables (community college math proportion, remedial math attempts, race, gender, first-year credits earned, socioeconomic status, and summer credits earned) emerged as predictors for college persistence. Study results indicated that students who enter higher education at the community college may have had lower opportunities to learn and therefore needed higher levels of remediation, which was shown to detract students from degree completion. Community college leaders are called to partner with local high schools with high percentages of racial/ethnic minorities to design academic programs aimed at improving the academic preparation of high school students in mathematics and promote student engagement during the first year and summers of college. ^

The efficacy of an interactive computer system for teaching developmental mathematics to college students

Many students are entering colleges and universities in the United States underprepared in mathematics. National statistics indicate that only approximately one-third of students in developmental mathematics courses pass. When underprepared students repeatedly enroll in courses that do not count toward their degree, it costs them money and delays graduation. This study investigated a possible solution to this problem: Whether using a particular computer assisted learning strategy combined with using mastery learning techniques improved the overall performance of students in a developmental mathematics course. Participants received one of three teaching strategies: (a) group A was taught using traditional instruction with mastery learning supplemented with computer assisted instruction, (b) group B was taught using traditional instruction supplemented with computer assisted instruction in the absence of mastery learning and, (c) group C was taught using traditional instruction without mastery learning or computer assisted instruction. Participants were students in MAT1033, a developmental mathematics course at a large public 4-year college. An analysis of covariance using participants' pretest scores as the covariate tested the null hypothesis that there was no significant difference in the adjusted mean final examination scores among the three groups. Group A participants had significantly higher adjusted mean posttest score than did group C participants. A chi-square test tested the null hypothesis that there were no significant differences in the proportions of students who passed MAT1033 among the treatment groups. It was found that there was a significant difference in the proportion of students who passed among all three groups, with those in group A having the highest pass rate and those in group C the lowest. A discriminant factor analysis revealed that time on task correctly predicted the passing status of 89% of the participants. ^ It was concluded that the most efficacious strategy for teaching developmental mathematics was through the use of mastery learning supplemented by computer-assisted instruction. In addition, it was noted that time on task was a strong predictor of academic success over and above the predictive ability of a measure of previous knowledge of mathematics.^

Analysis of Students’ Misconceptions and Error Patterns in Mathematics: The Case of Fractions

This study analyzed three fifth grade students’ misconceptions and error patterns when working with equivalence, addition and subtraction of fractions. The findings revealed that students used both conceptual and procedural knowledge to solve the problems. They used pictures, gave examples, and made connections to other mathematical concepts and to daily life topics. Error patterns found include using addition and subtraction of numerators and denominators, and finding the greatest common factor.

Estratégias de trabalho com blogs no ensino de geometria em turmas de 5° ano do ensino fundamental

This paper presents a survey conducted through collaborative work, which took place in a suburb school in the city of Uberlandia-MG. The research is characterized as case study and has a qualitative approach. Had the objective to look for different strategies of teaching and learning through the use of technology in pedagogical practice. Regarding the methodology in this research, we analyzed the work with the support of blogs, whose pages were used for student records and discussions directed to the geometry content. The students who were attending the fifth (5th) year of elementary school were invited to participate in this project. However, the research subjects were only those students who accepted the invitation to participate in the study through statement signed by parents. The project was developed with 30 students in the second half of 2014 and another 30 in the first half of 2015. The physical space at school, where most of the project activities were done was at the computer lab. In the process of compiling the data, at school, the following instruments were used: field notes produced by the entire project team, photographs and footage of the activities produced in the computer lab and in classroom (recorded by the research team) questionnaires, interviews, virtual space records: the blogs. The results of this research mainly focused on the analysis of the fifth year student‟s productions records in blogs. Regarding the conclusion, the research has shown that blogs, software and differentiated dynamic studies attracted the student‟s attention, leaving them mostly instigated by the unknown. Gradually, students built their own knowledge from their mistakes and successes. The entire work process enabled the computer lab to be an environment that is used not just to solving computerized and tedious drills. The blogs production work in groups, developed in students the reading and writing of both the mother language as symbols and mathematical nomenclature. The interaction between students became noticeable throughout the project, since it provided the student‟s personal growth, respect, tolerance and mutual cooperation. In this sense, we concluded that the project greatly contributed to the students' literacy process in the mother language, mathematics and computer literacy.

GeoGebraTUTOR : développement d’un système tutoriel autonome pour l’accompagnement d’élèves en situation de résolution de problèmes de démonstration en géométrie plane et genèse d’un espace de travail géométrique idoine

Travaux d'études doctorales réalisées conjointement avec les travaux de recherches doctorales de Nicolas Leduc, étudiant au doctorat en génie informatique à l'École Polytechnique de Montréal.

The Mathematics Anxiety of Bilingual Community College Students

Math anxiety levels and performance outcomes were compared for bilingual and monolingual community college Intermediate Algebra students attending a culturally diverse urban commuter college. Participants (N = 618, 250 men, 368 women; 361 monolingual, 257 bilingual) completed the Abbreviated Math Anxiety Scale (AMAS) and a demographics instrument. Bilingual and monolingual students reported comparable mean AMAS scores (20.6 and 20.7, respectively) and comparable proportions of math anxious individuals (50% and 48%, respectively). Factor analysis of AMAS scores, using principal component analysis by varimax rotation, yielded similar two-factor structures for both populations -- assessment and learning content -- accounting for 65.6% of the trace for bilingual AMAS scores. Statistically significant predictor variables for levels of math anxiety for the bilingual participants included (a) preparatory course enrollment (β = .236, p = .041) with those enrolled in prior preparatory courses scoring higher, (b) education major (β = .285, p = .018) with education majors scoring higher, and (c) business major (β = .252, p = .032) with business majors scoring higher. One statistically significant predictor variable emerged for monolingual students, gender (β = -.085, p = .001) with females ranking higher. Age, income, race, ethnicity, U.S. origin, science or health science majors did not emerge as statistically significant predictor variables for either group. Similarities between monolingual and bilingual participants included statistically significant negative linear correlations between AMAS scores and course grades for both bilingual (r = -.178, p = .017) and monolingual participants (r = -.203, p = .001). Differences included a statistically significant linear correlation between AMAS scores and final exam grades for monolingual participants only (r = -.253, p < .0009) despite no statistically significant difference in the strength the linear relationship of the AMAS scores and the final exam scores between groups, z = 1.35, p = .1756. The findings show that bilingual and monolingual students report math anxiety similarly and that math anxiety has similar associations with performance measures, despite differences between predictor variables. One of the first studies on the math anxiety of bilingual community college students, the results suggest recommendations for researchers and practitioners.

THE EFFECT OF PURPOSEFUL MATHEMATICS DISCOURSE IN THE CLASSROOM ON STUDENTS’ MATHEMATICS LANGUAGE IN THE CONTEXT OF PROBLEM SOLVING

This study examines how one secondary school teacher’s use of purposeful oral mathematics language impacted her students’ language use and overall communication in written solutions while working with word problems in a grade nine academic mathematics class. Mathematics is often described as a distinct language. As with all languages, students must develop a sense for oral language before developing social practices such as listening, respecting others ideas, and writing. Effective writing is often seen by students that have strong oral language skills. Classroom observations, teacher and student interviews, and collected student work served as evidence to demonstrate the nature of both the teacher’s and the students’ use of oral mathematical language in the classroom, as well as the effect the discourse and language use had on students’ individual written solutions while working on word problems. Inductive coding for themes revealed that the teacher’s purposeful use of oral mathematical language had a positive impact on students’ written solutions. The teacher’s development of a mathematical discourse community created a space for the students to explore mathematical language and concepts that facilitated a deeper level of conceptual understanding of the learned material. The teacher’s oral language appeared to transfer into students written work albeit not with the same complexity of use of the teacher’s oral expression of the mathematical register. Students that learn mathematical language and concepts better appear to have a growth mindset, feel they have ownership over their learning, use reorganizational strategies, and help develop a discourse community.

Matematik Öğretmeni Adaylarının Dörtgenleri Sınıflandırma Becerilerinin İncelenmesi

This study aimed to explore prospective teachers’ performance on recognizing quadrilaterals with their special cases and constructing a hierarchical classification of them. The participants consisted of 44 freshmen studying at a public university’s elementary school mathematics education department. Data was collected with a question form containing two questions at the first day of the geometry course taught in the second term of the first year. For quantifying the data of the first question, while students who identify the prototypes of quadrilaterals and their special cases were given 1 and 2 points for each correct answer respectively, -1 point was given for each incorrect answer. The similarity index was employed to quantify students’ concept maps. We investigated that students could detect the prototypes of the quadrilaterals but not their special cases. Additionally, the similarity index between majority of freshmen’ concept maps and the referent map was found as low or moderate.