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.

A retrospective-longitudinal examination of the relationship between apportionment of seat time in community-college algebra courses and student academic performance

During the past decade, there has been a dramatic increase by postsecondary institutions in providing academic programs and course offerings in a multitude of formats and venues (Biemiller, 2009; Kucsera & Zimmaro, 2010; Lang, 2009; Mangan, 2008). Strategies pertaining to reapportionment of course-delivery seat time have been a major facet of these institutional initiatives; most notably, within many open-door 2-year colleges. Often, these enrollment-management decisions are driven by the desire to increase market-share, optimize the usage of finite facility capacity, and contain costs, especially during these economically turbulent times. So, while enrollments have surged to the point where nearly one in three 18-to-24 year-old U.S. undergraduates are community college students (Pew Research Center, 2009), graduation rates, on average, still remain distressingly low (Complete College America, 2011). Among the learning-theory constructs related to seat-time reapportionment efforts is the cognitive phenomenon commonly referred to as the spacing effect, the degree to which learning is enhanced by a series of shorter, separated sessions as opposed to fewer, more massed episodes. This ex post facto study explored whether seat time in a postsecondary developmental-level algebra course is significantly related to: course success; course-enrollment persistence; and, longitudinally, the time to successfully complete a general-education-level mathematics course. Hierarchical logistic regression and discrete-time survival analysis were used to perform a multi-level, multivariable analysis of a student cohort (N = 3,284) enrolled at a large, multi-campus, urban community college. The subjects were retrospectively tracked over a 2-year longitudinal period. The study found that students in long seat-time classes tended to withdraw earlier and more often than did their peers in short seat-time classes (p < .05). Additionally, a model comprised of nine statistically significant covariates (all with p-values less than .01) was constructed. However, no longitudinal seat-time group differences were detected nor was there sufficient statistical evidence to conclude that seat time was predictive of developmental-level course success. A principal aim of this study was to demonstrate—to educational leaders, researchers, and institutional-research/business-intelligence professionals—the advantages and computational practicability of survival analysis, an underused but more powerful way to investigate changes in students over time.

A study of indicators discriminating between the recipients and non -recipients of the International Baccalaureate Diploma

This study examined relationships among variables in the Pre-International Baccalaureate (Pre-IB) Program admissions criteria and the Pre-IB Program course grades to discriminate between recipient and non recipient groups of the International Baccalaureate (IB) Diploma award. The study involved a multiracial sample of 142 IB Diploma graduates between the years 1992 and 1996 from one IB magnet school. The IB school is located within an urban high school of a predominantly Black student enrollment. A discriminant function analysis found that the highest correlations between predictors and the discriminant function were 9th- and 10th-grade mathematics and 10th-grade science course grades. Ninth-grade course grades of science, 9th-grade and 10th-grade course grades of English, foreign language, and social studies, and 7th-grade Iowa Tests of Basic Skills (ITBS) Reading Comprehension scores were also highly correlated to the discriminant function. The ITBS Battery and subscores of Vocabulary, Total Language, Total Work-Study, and Total Mathematics subscores in seventh grade and a grade point average from language arts, social studies, science, and mathematics in seventh grade were not highly correlated to the discriminant function. Recommendations were presented in the areas of curriculum and instruction, guidance services, student mentoring, and decision-making processes which would parallel the IB examination procedure and thereby enhance the alignment of the IB Program enabling more students to become recipients of the IB Diploma award. ^

Self -regulated learning: The role of a journal in the learning process for students and teachers

A fundamental goal of education is to equip students with self-regulatory capabilities that enable them to educate themselves. Self directedness not only contributes to success in formal instruction but also promotes lifelong learning (Bandura, 1997). The area of research on self-regulated learning is well grounded within the framework of psychological literature attributed to motivation, metacognition, strategy use and learning. This study explored past research and established the purpose of teaching students to self-regulate their learning and highlighted the fact that teachers are expected to assume a major role in the learning process. A student reflective writing journal activity was sustained for a period of two semesters in two fourth-grade mathematics classrooms. The reflective writing journal was analyzed in search of identifying strategies reported by students. Research questions were analyzed using descriptive statistics, frequency counts, cross-tabs and chi-square analyses. ^ Results based on student-use of the journals and teacher interviews indicated that the use of a reflective writing journal does promote self-regulated learning strategies to the extent which the student is engaged in the journaling process. Those students identified as highly self-regulated learners on the basis of their strategy use, were shown to consistently claim to learn math “as well or better than planned” on a weekly basis. Furthermore, good self-regulators were able to recognize specific strategies that helped them do well and change their strategies across time based on the planned learning objectives. The perspectives of the participating teachers were examined in order to establish the context in which the students were working. The effect of “planned change” and/or the resistance to change as established in previous research, from the teachers point of view, was also explored. The analysis of the journal data did establish a significant difference between students who utilized homework as a strategy. ^ Based on the journals and interviews, this study finds that the systematic use of metacognitive, motivational and/or learning strategies can have a positive effect on student's responsiveness to their learning environment. Furthermore, it reflects that teaching students “how to learn” can be a vital part of the effectiveness of any curriculum. ^

Effects of worked examples and algebra problem -solving skill on error and cognitive load

This study determined the levels of algebra problem solving skill at which worked examples promoted learning of further problem solving skill and reduction of cognitive load in college developmental algebra students. Problem solving skill was objectively measured as error production; cognitive load was subjectively measured as perceived mental effort. ^ Sixty-three Ss were pretested, received homework of worked examples or mass problem solving, and posttested. Univarate ANCOVA (covariate = previous grade) were performed on the practice and posttest data. The factors used in the analysis were practice strategy (worked examples vs. mass problem solving) and algebra problem solving skill (low vs. moderate vs. high). Students in the practice phase who studied worked examples exhibited (a) fewer errors and reduced cognitive load, at moderate skill; (b) neither fewer errors nor reduced cognitive load, at low skill; and (c) only reduced cognitive load, at high skill. In the posttest, only cognitive load was reduced. ^ The results suggested that worked examples be emphasized for developmental students with moderate problem solving skill. Areas for further research were discussed. ^

Modeling instruction: Positive attitudinal shifts in introductory physics measured with CLASS

Among the most surprising findings in Physics Education Research is the lack of positive results on attitudinal measures, such as Colorado Learning Attitudes about Science Survey (CLASS) and Maryland Physics Expectations Survey (MPEX). The uniformity with which physics teaching manages to negatively shift attitudes toward physics learning is striking. Strategies which have been shown to improve conceptual learning, such as interactive engagement and studio-format classes, provide more authentic science experiences for students; yet do not seem to be sufficient to produce positive attitudinal results. Florida International University’s Physics Education Research Group has implemented Modeling Instruction in University Physics classes as part of an overall effort toward building a research and learning community. Modeling Instruction is explicitly designed to engage students in scientific practices that include model building, validation, and revision. Results from a preinstruction/postinstruction CLASS measurement show attitudinal improvements through both semesters of an introductory physics sequence, as well as over the entire two-course sequence. In this Brief Report, we report positive shifts from the CLASS in one section of a modeling-based introductory physics sequence, for both mechanics (N=22) and electricity and magnetism (N=23). Using the CLASS results and follow up interviews, we examine how these results reflect on modeling instruction and the unique student community and population at FIU.

An Assessment of the Performances of Several Univariate Tests of Normality

The importance of checking the normality assumption in most statistical procedures especially parametric tests cannot be over emphasized as the validity of the inferences drawn from such procedures usually depend on the validity of this assumption. Numerous methods have been proposed by different authors over the years, some popular and frequently used, others, not so much. This study addresses the performance of eighteen of the available tests for different sample sizes, significance levels, and for a number of symmetric and asymmetric distributions by conducting a Monte-Carlo simulation. The results showed that considerable power is not achieved for symmetric distributions when sample size is less than one hundred and for such distributions, the kurtosis test is most powerful provided the distribution is leptokurtic or platykurtic. The Shapiro-Wilk test remains the most powerful test for asymmetric distributions. We conclude that different tests are suitable under different characteristics of alternative distributions.

Comparison of Linear Functions in Middle Grades Textbooks from Singapore and the United States

Many U.S. students do not perform well on mathematics assessments with respect to algebra topics such as linear functions, a building-block for other functions. Poor achievement of U.S. middle school students in this topic is a problem. U.S. eighth graders have had average mathematics scores on international comparison tests such as Third International Mathematics Science Study, later known as Trends in Mathematics and Science Study, (TIMSS)-1995, -99, -03, while Singapore students have had highest average scores. U.S. eighth grade average mathematics scores improved on TIMMS-2007 and held steady onTIMMS-2011. Results from national assessments, PISA 2009 and 2012 and National Assessment of Educational Progress of 2007, 2009, and 2013, showed a lack of proficiency in algebra. Results of curriculum studies involving nations in TIMSS suggest that elementary textbooks in high-scoring countries were different than elementary textbooks and middle grades texts were different with respect to general features in the U.S. The purpose of this study was to compare treatments of linear functions in Singapore and U.S. middle grades mathematics textbooks. Results revealed features currently in textbooks. Findings should be valuable to constituencies who wish to improve U.S. mathematics achievement. Portions of eight Singapore and nine U.S. middle school student texts pertaining to linear functions were compared with respect to 22 features in three categories: (a) background features, (b) general features of problems, and (c) specific characterizations of problem practices, problem-solving competency types, and transfer of representation. Features were coded using a codebook developed by the researcher. Tallies and percentages were reported. Welch's t-tests and chi-square tests were used, respectively, to determine whether texts differed significantly for the features and if codes were independent of country. U.S. and Singapore textbooks differed in page appearance and number of pages, problems, and images. Texts were similar in problem appearance. Differences in problems related to assessment of conceptual learning. U.S. texts contained more problems requiring (a) use of definitions, (b) single computation, (c) interpreting, and (d) multiple responses. These differences may stem from cultural differences seen in attitudes toward education. Future studies should focus on density of page, spiral approach, and multiple response problems.

Investigating the Effects of Cognitive Apprenticeship-Based Instructional Coaching on Science Teaching Efficacy Beliefs

The overall purpose of this collected papers dissertation was to examine the utility of a cognitive apprenticeship-based instructional coaching (CAIC) model for improving the science teaching efficacy beliefs (STEB) of preservice and inservice elementary teachers. Many of these teachers perceive science as a difficult subject and feel inadequately prepared to teach it. However, teacher efficacy beliefs have been noted as the strongest indicator of teacher quality, the variable most highly correlated with student achievement outcomes. The literature is scarce on strong, evidence-based theoretical models for improving STEB. This dissertation is comprised of two studies. STUDY #1 was a sequential explanatory mixed-methods study investigating the impact of a reformed CAIC elementary science methods course on the STEB of 26 preservice teachers. Data were collected using the Science Teaching Efficacy Belief Instrument (STEBI-B) and from six post-course interviews. A statistically significant increase in STEB was observed in the quantitative strand. The qualitative data suggested that the preservice teachers perceived all of the CAIC methods as influential, but the significance of each method depended on their unique needs and abilities. STUDY #2 was a participatory action research case study exploring the utility of a CAIC professional development program for improving the STEB of five Bahamian inservice teachers and their competency in implementing an inquiry-based curriculum. Data were collected from pre- and post-interviews and two focus group interviews. Overall, the inservice teachers perceived the intervention as highly effective. The scaffolding and coaching were the CAIC methods portrayed as most influential in developing their STEB, highlighting the importance of interpersonal relationship aspects in successful instructional coaching programs. The teachers also described the CAIC approach as integral in supporting their learning to implement the new inquiry-based curriculum. The overall findings hold important implications for science education reform, including its potential to influence how preservice teacher training and inservice teacher professional development in science are perceived and implemented. Additionally, given the noteworthy results obtained over the relatively short durations, CAIC interventions may also provide an effective means of achieving improvements in preservice and inservice teachers’ STEB more expeditiously than traditional approaches.

Cenário da produção acadêmica em história da matemática no ensino de matemática: uma análise reflexiva das teses e dissertações (1990-2010)

This research aims to make a reflective analysis about the academic production originated in the stricto sensu post graduation programs in the country, produced in the period of 1990 to 2010, in the field of History of Mathematics, especifically on works about the History of Mathematics in Mathematics education and that present pedagogical proposals that make use of the History of Mathematics in order to teach Mathematics. Defending the thesis that the researches on mathematics education with goals turned to the use of didactic proposals related to the history of mathematic that take in consideration the coherency between epistemological aspects inherent to mathematics history and anthological elements materialized on the conceptions of mathematics and mathematics history and of apprenticeship (implicitly or explicitly exposed) may originate significant contribution to the field of history of mathematics on education. Among these, nine were Master’s Degree dissertations and five PHD’s theses. The reflective analysis was accomplished from two matrixes; one from theoretical nature and the other, ontologic nature, elaborated from the pretexts of Sanches Gamboa, about the epistemological analysis from academic production in the field of Mathematics Education and the following theoretical perspectives in the field of History of Mathematics Education, that are: linear evolutionary theory, structural construtivist operative, evolutionary discontinuous, historical and socialcultural investigation and the use of activities estimulating the usage of verbal and nonverbal expressions. These perspectives were based on the works of Miguel and Miorim, Mendes and Radford. As results, we have detected some established dissonances between the categories related to theoretical and ontologic levels and the pedagogical proposal presented in these researches. On the other hand, we have discovered works that are able to establish consonances between the theoretical and ontological elements and the presented pedagogical proposal. These works carry significative contributions to the field of History of Mathematics applied to Mathematics pedagogical practice, inclusively presenting significative theoretical elements to the production of knowledge recognized as scientific in the Mathematics field

As compreensões do construtivismo de Ernst Von Glasersfeld e John Fossa: intermediando um diálogo em busca de novas significações

The present thesis, orientated by a letter sent by Ernst von Glasersfeld to John Fossa, is the product of a theoretical investigation of radical constructivism. In this letter, von Glasersfeld made three observations about Fossa’s understanding of radical constructivism. However, we limited our study to the second of these considerations since it de als with some of the core issues of constructivism. Consequently, we investigated what issues are raised by von Glasersfeld’s observation and whether these issues are relevant to a better understanding of constructivism and its implications for the mathema tics classroom . In order to realize the investigation, it was necessary to characterize von Glasersfeld’s epistemological approach to constructivism, to identify which questions about radical constructivism are raised by von Glasersfeld’s observation, to i nvestigate whether these issues are relevant to a better understanding of constructivism and to analyze the implications of these issues for the mathematics classroom. Upon making a hermeneutic study of radical constructivism, we found that what is central to it is its radicalism, in the sense that it breaks with tradition by its absence of an ontology. Thus, we defend the thesis that the absence of an ontology, although it has advantages for radical constructivism, incurs serious problems not only for the theory itself, but also for its implications for the mathematics classroom. The advantages that we were able to identify include a change from the usual philosophical paths to a very different rational view of the world, an overcoming of a naive way of thi nking, an understanding of the subject as active in the construction of his/her experiential reality, an interpretation of cognition as an instrument of adaptation, a new concept of knowledge and a vision of knowledge as fallible (or provisional). The prob lems are associated with the impossibility of radical constructivism to explain adequately why the reality that we build up is regular, stable, non - arbitrary and publicly shared. With regard to the educational implications of radical constructivism, the ab sence of an ontology brings to the mathematics classroom not only certain relevant aspects (or favorable points) that make teaching a process of researching student learning, empowering the student to learn and changing the classroom design, but also certa in weaknesses or limitations. These weaknesses or limitations of constructivism in the classroom are due to its conception of knowledge as being essentially subjective. This requires it to work with one - on - one situations and, likewise, makes the success of teaching dependent on the teacher’s individual skills. Perhaps the most important weakness or limitation, in this sense, is that it makes teaching orientated by constructivist principles unable to reach the goal of the formation of a community. We conclud e that issues raised by von Glasersfeld’s observation are absolutely relevant to the context of a better understanding of radical constructivism and its implications for education, especially for Mathematics Education.

Oficina educomunicativa de produção de vídeos: construção de significados sobre problemas socioambientais

This research aims to analyze the meaning making by Elementary Education students from public schools of Uberlândia (Minas Gerais, Brazil) about the social and environmental problems of their surroundings as of audiovisual reading and expression skills. The theoretical framework is based on constructivism, through the study of contributions about: cognitive development; meaningful learning; cognitive processes and types of knowledge; principles of learning with technology; educommunication focused on critical media literacy; and critical environmental education. The study object is a video production workshop organized in nine meetings, from September to November 2015, attended by 15 students. The following data collection instruments were used: the materials produced by the participants, specifically guided critical media literacy, agenda, script and final video; researcher observations from his role as a mediator; and focal interviews. The analysis was divided into two axes: procedural knowledge – technical skills of critical media literacy and production; and conceptual and metacognitive knowledge – representation of social and environmental problems and metacognitive skills of critical media literacy. Data were coded in the form of a skill evaluation rubric and also in the form of graphs. Thus, despite the time constraints, it is inferred that the workshop helped students to deepen their understanding about the discussed content, which is reinforced by observing in the graphs how the constant progressive differentiation of more inclusive concepts occurred along the meetings. It is further considered that the workshop contributed to the students reflect on their way of learning through critical use of techniques of media literacy and production, which can be seen from the satisfactory learners performance in most elements evaluated by the rubrics, as well as from the success in identifying interlocutors, values and actions in the read and built texts, something revealed by the graphs.

O programa institucional de bolsa de iniciação à docência PIBID na formação inicial de professores de matemática

This research sought to understand the space training provided by Institutional Scholarship Program Initiation of Teaching to a group of students of Degree in Mathematics that had activities developed in the same public school. The goal is to qualify them for teaching practice for these basic institutions. We decided to conduct a qualitative study of type ethnographic case study. For a year and a half while we were at the meetings and activities of the Group, we did what we call as a participant observation. To obtain the data, we used different survey instruments: the researcher\'s field notes through his observation of everyday life of the group, photographs and filming of the activities, document analysis and database produced, physically and digitally, in addition to questionnaires and interviews with records written, which complemented each other and helped establish a triangulation of information collected. We analyze the trajectory of the group on three axes: on the first, we present and understand the paths taken by the Group in the process of setting up training spaces, and production of their professional training, in the second, we analyze how the space of PIBID is being integrated with others spaces of formations in the educational institution of the degree course in mathematics and, in the third axis, we understand the process of knowledge production of that group. The trajectory taken by the group was marked by a process of reflection and discussion systematic and collective, which favored the pursuit for be a better professional and also confirmed a possible path to be followed in initial teacher education.

Uma proposta de ensino de frações no 6º ano do ensino fundamental a partir da teoria histórico-cultural

This text presents developed in the Graduate Program in Science and Mathematics Education at the Federal University of Uberlândia, in which it was intended to answer the question: What are the pedagogical implications for the fractions concept learning for students of the 6th grade of elementary school that the teaching guide activities can provide? The objectives of this research were: a) analyze the possible pedagogical implications for the learning of the fraction's concept for students of the 6th grade of elementary school through guiding teaching activities; b) using the conceptual connections of the fraction to enable students to develop an abstract thought and c) investigate whether guiding teaching activities reflect on 'how to think' and 'how to do' of the student. Five teaching activities have been developed (MOURA, 2002) from the perspective of teaching guiding activity (TGA) and had as object of study the teaching of fractions for students in 6th year of elementary school. They have been prepared and proposed activities in which it was intended to investigate the use of history of mathematics as an aid in learning the conceptual fraction links (CARAÇA, 1951) by students. Such activities, for analysis, were organized into episodes and scenes (MOURA, 2004) and discussed how students deal with the measurement of whole quantity (all) and subunits (part); how they represent in verbal or written language. It is hoped that the research is set up as an important contribution to mathematics teaching area and may contribute to the initial and continuing training of mathematics teacher sand the formation of theoretical thinking of elementary school students.

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.