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1
Hart, Hilary. "Mathematics Vocabulary and English Learners: A Study of Students' Mathematical Thinking." BYU ScholarsArchive, 2010. https://scholarsarchive.byu.edu/etd/2573.
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This study examined the mathematical thinking of English learners as they were taught mathematics vocabulary through research-based methods. Four English learners served as focus students. After administering a pre-performance assessment, I taught a 10-lesson unit on fractions. I taught mathematics vocabulary through the use of a mathematics word wall, think-pair-shares, graphic organizers, journal entries, and picture dictionaries. The four focus students were audio recorded to capture their spoken discourse. Student work was collected to capture written discourse. Over the course of the unit, the four focus students used the mathematics vocabulary words that were taught explicitly. The focus students gained both procedural and conceptual knowledge of fractions during this unit. Students also expressed elevated confidence in their mathematics abilities.
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Hannula, Markku. "Affect in mathematical thinking and learning /." Turku : University of Turku, 2004. http://kirjasto2.utu.fi/julkaisupalvelut/b/annaalit/B273.html.
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Monteleone, Chrissoula. "Critical mathematical thinking in young students." Phd thesis, Australian Catholic University, 2021. https://acuresearchbank.acu.edu.au/download/cb06753760247f43b88bfde14ea04bc78463c1734aa47d3ca60129d4d5e7c8ec/2879980/Monteleone_2021_Critical_mathematical_thinking_in_young_students.pdf.
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The aims of the study were to investigate critical mathematical thinking in young students, and teaching actions/questions that help these young students exhibit their critical mathematical thinking. A key finding was the conceptualisation of a Critical Mathematical Thinking Framework for Young Students. This framework (a) articulates the key characteristics young students exhibit as they engage in critical mathematical thinking, and (b) can be used by teachers to help them identify critical mathematical thinking within the classroom context. Additionally, specific teacher questions that support young students to exhibit critical mathematical thinking were delineated. The study determined that teachers play a pivotal role in supporting young students to exhibit their critical mathematical thinking.
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Argyle, Sean Francis. "Mathematical thinking: From cacophony to consensus." Kent State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=kent1337696397.
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Lane, Catherine Pullin. "Mathematical Thinking and the Process of Specializing." University of Cincinnati / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1307441324.
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Stillman, Gloria Ann. "Assessing higher order mathematical thinking through applications /." St. Lucia, Qld, 2001. http://www.library.uq.edu.au/pdfserve.php?image=thesisabs/absthe16747.pdf.
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Cardella, Monica E. "Engineering mathematics : an investigation of students' mathematical thinking from a cognitive engineering perspective /." Thesis, Connect to this title online; UW restricted, 2006. http://hdl.handle.net/1773/10692.
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Tanner, H. F. R. "Using and applying mathematics : developing mathematical thinking through practical problem solving and modelling." Thesis, Swansea University, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.639162.
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Using and Applying Mathematics in the National Curriculum focuses on the development of reasoning skills through problem solving, and links the processes involved in learning new mathematics with the application of existing knowledge to new situations. This thesis begins by examining how mathematical knowledge is constructed and how children make sense of it. The nature of practical problem solving and modelling is considered and the literature on teaching and learning problem solving and modelling is reviewed with particular emphasis on metacognition and social practices. The research reports a quasi-experiment in which 314 pupils aged between 11 and 13 followed a mathematical thinking skills course and were compared with matched control pupils using pre-tests, post-tests, delayed texts and structured interviews. Assessment instruments were devised to assess pupils' mathematical cognitive development, their metacognitive skills and their metacognitive self knowledge. Statistical data were supported by participant observations. On average, experimental pupils performed slightly better than control pupils in metacognitive skill and mathematical development in the post and delayed-tests. The content of the mathematical development test had not been taught directly by the course and far transfer is claimed. The teachers were divided into four groups according to teaching approach, based on analysis of the qualitative data. The most successful teachers used a flexible form of scaffolding and encouraged reflection. Their classes demonstrated a substantial advantage over their controls in metacognitive skill, metacognitive self knowledge and mathematical development. Recommendations are made about teaching approaches.
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Reyes-Santander, Pamela, David Aceituno, and Pablo Cáceres. "Mathematical Thinking Styles of Students with Academic Talent." Pontificia Universidad Católica del Perú, 2017. http://repositorio.pucp.edu.pe/index/handle/123456789/123827.
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This study explores the predominant mathematical thinking style that students with academic talent used in solving mathematical problems. Thinking styles are preferences by subjects in the way of expressing mathematical skills against a task, in this case, visual, formal and integrated. We assessed 99 students from an academic support talent program, in a retrospective ex post facto study with only one group. We administered the questionnaire mathematical thinking styles of Borromeo-Ferri and determined that these students exhibited mostly an integrated style of thinking, which involves the use of symbols and verbal representations with visual expressions in solving mathematical exercises. They also show a strong orientation to address the problems of combined mode, which involves considering them as a whole at a time.<br>El presente estudio establece el estilo de pensamiento matemático predominante que utilizan los estudiantes con talento académico en la resolución de problemas matemáticos. Los estilos de pensamiento son preferencias por parte de los sujetos en la forma de expresar las habilidades frente a una tarea matemática, en este caso, visual, formal e integrado. En el marco de un estudio ex post facto retrospectivo de grupo único, se evaluó a un total de 99 estudiantes pertenecientes a un programa académico de apoyo al talento con el cuestionario Estilos de Pensamiento Matemático de Borromeo-Ferri. Los resultados indican que los estudiantes declararon orientarse hacia el estilo de pensamiento integrado, que supone el uso de simbología y representaciones verbales junto con expresiones visuales en la resolución de los ejercicios matemáticos, así como una significativa orientación a abordar los problemas de modo combinado, que supone considerar los problemas como un todo.<br>La présente étude établit le style de pensée mathématique prédominant utilisé par les étudiants ayant un talent académique dans la résolution de problèmes mathématiques. Les styles de pensée sont des préférences de la part des sujets sous la forme d’exprimer les capacités face à une tâche mathématique, dans ce cas, visuelle, formelle et intégrée. Dans une étude rétrospective sur un seul groupe ex post facto, un total de 99 étudiants appartenant à un programme de soutien aux talents universitaires ont été évalués, à qui le questionnaire Styles de Pensée mathématique de Borromeo-Ferri a été appliqué et déterminé que ce type de sujets déclare principalement un style de pensée intégré, ce qui implique l’utilisation de la symbologie et des représentations verbales ainsi que des expressions visuelles dans la résolution des exercices mathématiques. En outre, ils montrent une forte orientation pour aborder les problèmes de manière combinée, ce qui implique de les considérer dans leur ensemble dans le même temps.<br>Este estudo estabelece o estilo predominante do pensamento matemático usado por os alunos com talento acadêmico na resolução de problemas matemáticos. Os estilos de pensamento são as preferências dos indivíduos sobre a forma para expressar as capacidades em uma tarefa matemática, neste caso, visual, formal e integrada. Como parte de um estudo ex post facto retrospectivo de grupo único, foram avaliados um total de 99 estudantes de um programa de talento acadêmico. Foram aplicados nos alunos o questionário “Estilos de Pensamento Matemático de Borromeo-Ferri” e determinou-se que a maioria dos participantes declararam um estilo de pensamento integrado, que envolve o uso de símbolos e representações verbais com resolução de expressões visuais de exercícios matemáticos. Eles mostram também uma forte orientação para resolver os problemas de modo combinado, o qual envolve a considerá-los como um todo de uma vez.
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Coetzee, Carla. "Mathematical thinking skills needed by first year programming students." Diss., University of Pretoria, 2016. http://hdl.handle.net/2263/60991.
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The aim of this qualitative study is to explore and describe the mathematical thinking
skills that students require for a first level programming subject that forms part of the
National Diploma in Information Communication Technology (ICT) at a University of
Technology (UoT).
Mathematics is an entry requirement for many tertiary programmes, including ICT
courses, unfortunately the poor quality of schooling in South Africa limits learners'
access to higher education. From the literature it is evident that students lack fluency
in fundamental mathematical and problem-solving skills when they enter higher
education.
In this study, the concept of programming thinking skills is explored, described and
linked to mathematical thinking skills. An instrument (Mathematical and Programming
Thinking Skills Matrix for the Analysis of Programming Assessment) has been
developed and used to analyse examination papers of a first-year programming
subject (at TUT) in order to identify mathematical skills as these appear in
programming assessments. Semi-structures interviews were conducted with first-year
programming lecturers, examiners and moderators. The literature as well and the
results of the analysed data indicated and confirmed that mathematical thinking skills
are extremely important when learning to program. The results of the study indicate a
strong relationship between mathematical thinking skills and programming thinking
skills.
The outcome of this study is therefore a set of mathematical thinking skills that needs
to be developed when compiling a mathematics curriculum for first level programming
students studying towards a National Diploma in ICT.<br>Dissertation (MEd)--University of Pretoria, 2016.<br>Science, Mathematics and Technology Education<br>MEd<br>Unrestricted
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Groth, Randall E. Langrall Cynthia Willey Mooney Edward S. "Development of a high school statistical thinking framework." Normal, Ill. Illinois State University, 2003. http://wwwlib.umi.com/cr/ilstu/fullcit?p3087867.
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Thesis (Ph. D.)--Illinois State University, 2003.<br>Title from title page screen, viewed November 10, 2005. Dissertation Committee: Cynthia W. Langrall, Edward S. Mooney (co-chair), Beverly J. Hartter, Sharon S. McCrone. Includes bibliographical references (leaves 199-212) and abstract. Also available in print.
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Calder, Nigel Stuart. "Processing mathematical thinking through digital pedagogical media the spreadsheet /." The University of Waikato, 2008. http://hdl.handle.net/10289/2662.
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Abstract This study is concerned with the ways mathematical understanding emerges when mathematical phenomena are encountered through digital pedagogical media, the spreadsheet, in particular. Central to this, was an examination of the affordances digital technologies offer, and how the affordances associated with investigating mathematical tasks in the spreadsheet environment, shaped the learning trajectories of the participants. Two categories of participating students were involved, ten-year-old primary school pupils, and pre-service teachers. An eclectic approach to data collection, including qualitative and quantitative methods, was initially undertaken, but as my research perspective evolved, a moderate hermeneutic frame emerged as the most productive way in which to examine the research questions. A hermeneutic process transformed the research methodology, as well as the manner in which the data were interpreted. The initial analysis and evolving methodology not only informed this transition to a moderate hermeneutic lens, they were constitutive of the ongoing research perspectives and their associated interpretations. The data, and some that was subsequently collected, were then reconsidered from this modified position. The findings indicated that engaging mathematical tasks through the pedagogical medium of the spreadsheet, influenced the nature of the investigative process in particular ways. As a consequence, the interpretations of the interactions, and the understandings this evoked, also differed. The students created and made connections between alternative models of the situations, while the visual, tabular structuring of the environment, in conjunction with its propensity to instantly manage large amounts of output accurately, facilitated their observation of patterns. They frequently investigated the visual nature of these patterns, and used visual referents in their interpretations and explanations. It also allowed them to pose and test their informal conjectures and generalisations in non-threatening circumstances, to reset investigative sub-goals easily, hence fostering risk taking in their approach. At times, the learning trajectory evolved in unexpected ways, and the data illustrated various alternative ways in which unexpected, visual output stimulated discussion and extended the boundaries of, or reorganised, their interaction and mathematical thinking. An examination of the visual perturbations, and other elements of learning as hermeneutic processes also revealed alternative understandings and explanations. Viewing the data and the research process through hermeneutic filters enhanced the connectivity between the emergence of individual mathematical understanding, and the cultural formation of mathematics. It permitted consideration of the ways this process influences the evolution of mathematics education research. While interpretive approaches are inevitably imbued with the researcher perspective in the analysis of what gets noticed, the research gave fresh insights into the ways learning emerges through digital pedagogical media, and the potential of this engagement to change the nature of mathematics education.
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Ozdil, Utkun. "A Multilevel Structural Model Of Mathematical Thinking In Derivative Concept." Phd thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614000/index.pdf.
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The purpose of the study was threefold: (1) to determine the factor structure of mathematical thinking at the within-classroom and at the between-classroom level<br>(2) to investigate the extent of variation in the relationships among different mathematical thinking constructs at the within- and between-classroom levels<br>and (3) to examine the cross-level interactions among different types of mathematical thinking. Previous research was extended by investigating the factor structure of mathematical thinking in derivative at the within- and between-classroom levels, and further examining the direct, indirect, and cross-level relations among different types of mathematical thinking. Multilevel analyses of a cross-sectional dataset containing two independent samples of undergraduate students nested within classrooms showed that the within-structure of mathematical thinking includes enactive, iconic, algorithmic, algebraic, formal, and axiomatic thinking, whereas the between-structure contains formal-axiomatic, proceptual-symbolic, and conceptual-embodied thinking. Major findings from the two-level mathematical thinking model revealed that: (1) enactive, iconic, algebraic, and axiomatic thinking varied primarily as a function of formal and algorithmic thinking<br>(2) the strongest direct effect of formal-axiomatic thinking was on proceptual-symbolic thinking<br>(3) the nature of the relationships was cyclic at the between-classroom level<br>(4) the within-classroom mathematical thinking constructs significantly moderate the relationships among conceptual-embodied, proceptual-symbolic, and formal-axiomatic thinking<br>and (5) the between-classroom mathematical thinking constructs moderate the relationships among enactive, iconic, algorithmic, algebraic, formal, and axiomatic thinking. The challenges when using multilevel exploratory factor analysis, multilevel confirmatory factor analysis, and multilevel structural equation modeling with categorical variables are emphasized. Methodological and educational implications of findings are discussed.
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Tan, Li-hua, and 陳麗華. "Primary school students' thinking processes when posing mathematical word problems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2001. http://hub.hku.hk/bib/B31962592.
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Tan, Li-hua. "Primary school students' thinking processes when posing mathematical word problems." Hong Kong : University of Hong Kong, 2001. http://sunzi.lib.hku.hk:8888/cgi-bin/hkuto%5Ftoc%5Fpdf?B23425155.
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Goggins, Lauren Lee. "Eliciting elementary preservice teachers' mathematical knowledge for teaching using instructional tasks that include children's mathematical thinking." Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file, 245 p, 2008. http://proquest.umi.com/pqdweb?did=1490070411&sid=13&Fmt=2&clientId=8331&RQT=309&VName=PQD.
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Stone, Jason C. "The Formation of Self-Constructed Identity as Advanced Mathematical Thinker Among Some Female PhD Holders in Mathematics and the Relationship to the "Three-Worlds" Cognitive Model of Advanced Mathematical Thinking." Kent State University / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=kent1436975429.
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Aizikovitsh, Einav, and Miriam Amit. "An innovative model for developing critical thinking skills through mathematical education." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79308.
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In a challenging and constantly changing world, students are required to develop advanced
thinking skills such as critical systematic thinking, decision making and problem solving. This
challenge requires developing critical thinking abilities which are essential in unfamiliar
situations. A central component in current reforms in mathematics and science studies worldwide
is the transition from the traditional dominant instruction which focuses on algorithmic cognitive
skills towards higher order cognitive skills. The transition includes, a component of scientific
inquiry, learning science from the student's personal, environmental and social contexts and the
integration of critical thinking. The planning and implementation of learning strategies that
encourage first order thinking among students is not a simple task. In an attempt to put the
importance of this transition in mathematical education to a test, we propose a new method for
mathematical instruction based on the infusion approach put forward by Swartz in 1992. In fact,
the model is derived from two additional theories., that of Ennis (1989) and of Libermann and
Tversky (2001). Union of the two latter is suggested by the infusion theory. The model consists of
a learning unit (30h hours) that focuses primarily on statistics every day life situations, and
implemented in an interactive and supportive environment. It was applied to mathematically
gifted youth of the Kidumatica project at Ben Gurion University. Among the instructed subjects
were bidimensional charts, Bayes law and conditional probability; Critical thinking skills such as
raising questions, seeking for alternatives and doubting were evaluated. We used Cornell tests
(Ennis 1985) to confirm that our students developed critical thinking skills.
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Spitzer, Sandy Margaret. "The role of graphing calculators in students' algebraic thinking." Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file, 135 p, 2008. http://proquest.umi.com/pqdweb?did=1601234511&sid=4&Fmt=2&clientId=8331&RQT=309&VName=PQD.
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Owens, Kay Dianne, and mikewood@deakin edu au. "Spatial thinking processes employed by primary school students engaged in mathematical problem solving." Deakin University, 1993. http://tux.lib.deakin.edu.au./adt-VDU/public/adt-VDU20050826.100440.
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This thesis describes changes in the spatial thinking of Year 2 and Year 4 students who participated in a six-week long spatio-mathematical program. The main investigation, which contained quantitative and qualitative components, was designed to answer questions which were identified in a comprehensive review of pertinent literatures dealing with (a) young children's development of spatial concepts and skills, (b) how students solve problems and learn in different types of classrooms, and (c) the special roles of visual imagery, equipment, and classroom discourse in spatial problem solving.
The quantitative investigation into the effects of a two-dimensional spatial program used a matched-group experimental design. Parallel forms of a specially developed spatio-mathematical group test were administered on three occasionsbefore, immediately after, and six to eight weeks after the spatial program. The test contained items requiring spatial thinking about two-dimensional space and other items requiring transfer to thinking about three-dimensional space. The results of the experimental group were compared with those of a control group who were involved in number problem-solving activities. The investigation took into account gender and year at school. In addition, the effects of different classroom organisations on spatial thinking were investigated~one group worked mainly individually and the other group in small cooperative groups.
The study found that improvements in scores on the delayed posttest of two-dimensional spatial thinking by students who were engaged in the spatial learning experiences were statistically significantly greater than those of the control group when pretest scores were used as covariates. Gender was the only variable to show an effect on the three-dimensional delayed posttest.
The study also attempted to explain how improvements in, spatial thinking occurred. The qualitative component of the study involved students in different contexts. Students were video-taped as they worked, and much observational and interview data were obtained and analysed to develop categories which were described and inter-related in a model of children's responsiveness to spatial problem-solving experiences. The model and the details of children's thinking were related to literatures on visual imagery, selective attention, representation, and concept construction.
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Qwillbard, Tony. "Less information, more thinking : How attentional behavior predicts learning in mathematics." Thesis, Umeå universitet, Institutionen för psykologi, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-100999.
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It has been shown in experiments that a method of teaching where students are encouraged to create their own solution methods to mathematical problems (creative mathematically founded reasoning, CMR) results in better learning and proficiency than one where students are provided with solution methods for them to practice by repetition (algorithmic reasoning, AR). The present study investigated whether students in an AR practice condition pay less attention to information relevant for mathematical problem solving than students in a CMR condition. To test this, attentional behavior during practice was measured using eye-tracking equipment. These measurements were then associated with task proficiency in a follow-up test one week after the practice session. The findings support the theory and confirm previous studies in that CMR leads to better task performance in the follow-up test. The findings also suggest that students within the CMR condition whom focus less on extraneous information perform better.<br>Experiment har visat att en undervisningsmetod i vilken elever uppmuntras att själva komma på lösningsmetoder till matematiska problem (creative mathematically founded reasoning, CMR) resulterar i bättre inlärning och färdighet än en metod i vilken eleverna ges en färdig en lösningsmetod att öva på genom repetition (algorithmic reasoning, AR). Denna studie undersöker om elever under en AR-träningsbetingelse ägnar mindre uppmärksamhet åt information som är relevant för matematisk problemlösning än vad elever under en CMR-träningsbetingelse gör. För att testa detta mättes elevernas uppmärksamhetsbeteende under träning med hjälp av ögonrörelsekamera. Måtten ställdes sedan i relation till uppgiftsfärdighet i ett uppföljningstest en vecka efter träningssessionen. Resultaten stödjer teorin och bekräftar tidigare studier som visat att CMR leder till bättre prestation i uppföljningstestet. Resultaten tyder även på att de elever under CMR-betingelsen som fokuserar minst på ovidkommande information presterar bättre.
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Venter, Dalene. "Three-dimensional thinking in radiography." Thesis, Cape Peninsula University of Technology, 2008. http://hdl.handle.net/20.500.11838/1564.
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Dissertation submitted in fulfilment of the requirements for the degree Master of Technology: Radiography in the Faculty of Health and Wellness Sciences, 2008<br>Introduction
Research to date has not been able to agree whether spatial abilities can be
developed by practice. According to some researchers spatial ability is an inherited
cognitive ability, compared to spatial skills that are task specific and can be acquired
through formal training. It is commonly assumed that radiographers require general
cognitive spatial abilities to interpret complex radiographic images. This research
was conducted to investigate second year radiography students’ three-dimensional
thinking skills pertaining to film-viewing assessments.
Materials and methods
The experimental research strategy was mainly applied together with correlation
research. Two trials were run (in 2005 and 2006). The sample group consisted of
fifteen second year diagnostic radiography students in 2005 and twenty-three second
year diagnostic radiography students, of the same institution, in 2006. Each year
group was randomly divided into a control group and an intervention group. Two
instruments were used, that is a film-viewing assessment and a three-dimensional
test, Academic Aptitude Test (University) (AAT) nr. nine: Spatial Perception (3-D).
The whole class completed this basic spatial aptitude test, as well as a base-line film viewing
assessment, which focused on the evaluation of technique/anatomy of
second year specialised radiographic projections. The marks that the students
achieved in the fore-mentioned tests were compared, to determine if there was any
correlation between their performances in the different tests. A curricular
intervention, which was intended to improve applied three-dimensional skills, was
subsequently applied. The students executed certain modified radiographic
projections on parts of a human skeleton. For each radiographic projection, the
students had to draw the relation of the X-ray beam to the specific anatomical
structures, as well as the relation of these structures to the film. The related images
of these projections were also drawn. With each of the following sessions, films
including images of the previous session were discussed with each student. After the
intervention, the whole class wrote a second film-viewing assessment. The marks
achieved in this assessment were compared to the marks of the initial film-viewing
assessment to determine the influence of the intervention on the performance of the
intervention group. Following this assessment, for ethical reasons, the same
intervention took place with the control group. A third film-viewing assessment was
then written by all the diagnostic second year students to evaluate the overall impact
of the intervention on the applied three-dimensional skills of the class. The marks of
both the 2005 and 2006 classes (intervention classes) were compared to the marks
achieved by former classes from 2000 to 2004 (control classes), in film-viewing
assessments to evaluate the role of the curricular intervention over the years. The
students again completed the three-dimensional test, Spatial Perception (3-D) to
evaluate the impact of the intervention on students’ general three-dimensional
cognitive abilities. These marks were also compared to the marks of the third filmviewing
assessment, to determine if there was any correlation between the students’
performances in the different tests.
Results
The intervention groups did not perform significantly better in film-viewing
assessments after the intervention, compared to the control groups, but reasonable
differences, favouring the intervention group, were achieved. Statistical significance
was achieved in film-viewing assessments with both year groups after the whole
class had the intervention. The intervention year groups also performed significantly
better than the previous year groups (without the intervention) in film-viewing
assessments. The performance in general three-dimensional cognitive abilities of the
group of 2006 improved significantly after the intervention, but on the contrary, the
performance of the group of 2005 declined. There was a small intervention effect on
the performance of the group of 2006. Only a weak to moderate correlation between
the marks of the students achieved in the three-dimensional tests and the marks
achieved in the film-viewing assessments, was found.
Conclusion
The contrasting evidence between the data of the two groups (2005 and 2006) in the
three-dimensional tests and the small intervention effect on the performance of the
group of 2006, makes the intervention not applicable for the increase of general
spatial abilities. The results of this research show that the applied three-dimensional
skills of radiography students in interpreting specialised and modified projections
can be improved by intensive practice, independent of their inherited spatial
abilities.
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Rahman, Roselainy Abdul. "Changing my own and my students' attitudes to calculus through working on Mathematical Thinking." Thesis, Open University, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.518366.
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Over the years, I have seen my students struggle to move from elementary to advanced mathematical thinking. I believe there is a need for an effective mathematical pedagogy in the learning of advanced mathematics that explicitly promotes mathematical thinking. Researches in mathematics education have contributed much towards providing theoretical perspectives for understanding thinking, learning, and teaching. It has also provided description on aspects of cognition as well as evidence on the viability and consequences of various kinds of instruction. In this study, I will describe my experience in translating some of the theories into classroom practice appropriate to both my students' and my own circumstances. The purpose of my research was to bring about improvements to the teaching of Advanced Calculus (Engineering Mathematics) to engineering undergraduates with the aim that it would also bring about changes on how students think about the mathematics. A framework that guides the design of classroom instruction and activities to support and encourage students' use of their own thinking powers will also be highlighted. A model of teaching was finally adapted based on various approaches designed to invoke students' ability to use their own thinking powers, enhance their problem solving skills and promote soft (generic) skills that can contribute towards students' acquirement of necessary attributes as an engineer. The researcher will also be the teacher and thus the study was implemented using an action research and case study perspectives as various methods within these stances will ensure flexibility in responding to the dynamics of interaction between the teacher and her students. Every encounter with my students, within and beyond, the classroom, was considered and contributed to my reflection about my teaching, review of the strategies and consequently, changing the way I teach or interact with the students. The thesis will present a description of the challenges and obstacles to making changes from my own and my students' point of views. Analysis of the research findings will also add knowledge about the factors that influence students' learning. A model that depicts the process of change will be proposed.
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Dumitraşcu, Gabriela Georgeta. "Generalization: Developing Mathematical Practices in Elementary School." Diss., The University of Arizona, 2015. http://hdl.handle.net/10150/556959.
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The process of generalization in mathematics was identified by mathematics education and educational psychology research, out of many mental actions or operations, as a cognitive function fundamentally required in the thinking process. Moreover, the current changes in education in the United States bring forward the dual goal of mathematics teaching and learning: students should have strong and rigorous mathematical content knowledge and students should be involved in practices that define the status of doing mathematical work. This dual role is totally dependent on the process of generalization. This study uses theories and research findings from the field of algebraic thinking, teaching, and learning to understand how the third grade teacher’s edition textbooks of three mathematics curricula portray the process of generalization. I started my study with the development of a theoretical coding system obtained by combining Kaput’s theory about algebra (Kaput, 2008), Krutetskii’s two way of generalization (Krutetskii, 1976), and the five mathematical representations identified by Lesh, Post, and Behr (1987). Then, I used the coding system to identify tasks that have the potential to involve students in the process of generalization. The findings from my study provide evidence that following a well-structured theory, such as Kaput’s theory about algebra, allows us to identify tasks that support algebraic thinking and to create new ones with higher potential to involve children in the process of generalization. Such tasks may support the development of algebraic thinking as a continuous process that should start from early grades of elementary school.
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Lealdino, Pedro. "Didactic Situations for the Development of Creative Mathematical Thinking : A study on Functions and Algorithms." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSE1254/document.
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La créativité est considérée comme une compétence cruciale pour le monde contemporain. La recherche décrite dans cette thèse a eu comme contexte principal le projet MC Squared. Réalisé entre octobre 2013 et septembre 2016. L'objectif du projet était de développer une plate-forme numérique pour le développement de C-books destinés à l'enseignement des mathématiques de manière à développer la pensée mathématique créative chez les étudiants et les auteurs. Cette thèse propose une analyse de la conception, du développement, de la mise en oeuvre et du test des activités numériques et non numériques dans le but d'améliorer et d'encourager la pensée mathématique créative ayant des fonctions et des algorithmes comme objets mathématiques à analyser. Les questions de recherche suivantes ont été soulevées à partir du problème: -Comment opérationnaliser et réviser les définitions existantes de la pensée mathématique créative? -Quels sont les composants nécessaires d'une situation ou d'un artefact permettant un processus de pensée mathématique créative? -Comment pouvons-nous évaluer l'avancement d'un processus impliquant la pensée mathématique créative?-Le modèle "Diamant de la créativité" est-il un outil d'analyse utile pour cartographier le cheminement du processus créatif? Pour répondre à ces questions, la recherche a suivi une méthodologie basée sur une recherche agile basée sur le design. Quatre activités ont été développées de manière cyclique. Le premier, appelé Function Hero, est un jeu numérique qui utilise les mouvements du corps du joueur pour évaluer la capacité de reconnaissance des fonctions. Trois autres activités appelées Binary Code, Fake Binary Code et Op'Art, visant au développement de la pensée computationnelle. Le modèle principal de cette thèse est le modèle "Diamond de créativité" pour cartographier le processus de résolution des problèmes rencontrés dans chaque activité, en évaluant le processus et les produits dérivés du travail des étudiants.Pour valider les hypothèses de recherche, nous avons collecté des données pour chaque activité et les avons analysées quantitativement et qualitativement. Les résultats montrent que les activités développées ont éveillé et engagé les étudiants dans la résolution de problèmes et que le modèle "Diamond of Creativity" peut aider à identifier et à identifier des points spécifiques du processus de création<br>Creativity is considered as a crucial skill for the contemporary world. The research described in this thesis had the Project MC Squared as the main context. Carried out between October 2013 and September 2016. The objective of the project was to develop a digital platform for the development of C-books for teaching mathematics in a way that develops Creative Mathematical Thinking both in the students and the authors. This thesis, entitled: Didactic Situations for the Development of Creative Mathematical Thinking proposes an analysis of the design, development, implementation, and testing of digital and non-digital activities with the aim of improving and fostering Creative Mathematical Thinking having Functions and Algorithms as mathematical objects to analyze. The following research questions raised from the problem: • How to operationalize and revise existing definitions of Creative Mathematical Thinking? • How can we assess the progress of a process involving Creative Mathematical Thinking? • How the "Diamond of Creativity" model is an useful analytic tool to map the Creative Process path? To answer such questions, the research followed a methodology based on an agile Design-Based Research. Four activities were cyclically developed. The first one, called: "Function Hero," is a digital game that uses body movements of the player to evaluate recognizability of functions. Three other activities called "Binary Code," "FakeBinary Code" and "Op’Art", aimed at the development of Computational Thinking. The main constructs of this thesis are: (a) the "Diamond of Creativity" model to map the process of solving problems found in each activity, evaluating the process and the products derived from the students’ work. (b) The digital game: "Function Hero". To validate the research hypotheses, we collected data from each activity and analyzed them quantitatively and qualitatively. The results show that developed activities have awakened and engaged students into problem-solving and that the "Diamond of Creativity" model can help in identifying and labeling specific points in the creative process
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Harris, Carolina. "Hur tänker elever? : Elevintervjuer som metod för att kartlägga elevers tankar kring matematikundervisning." Thesis, Södertörn University College, Lärarutbildningen, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:sh:diva-843.
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<p>During my time as a student of education I have learnt that it is my responsibility, as a teacher, to adjust the ways in which I teach to the needs, abilities, experiences, and thoughts of each individual child. What I have not yet gained much knowledge on is how to go about finding the children’s thoughts.</p><p>In this thesis I investigate the interview as a method of finding out how sixth graders think about their mathematics education. Four children were interviewed. In addition to these inter-views, as a means of giving a broader perspective to and a greater understanding of the chil-dren’s answers, one math lesson was filmed and the math teacher was interview on two sepa-rate occasions.</p><p>What I found was that a number of factors seemed to influence the children’s thoughts and answers, and that their answers were most likely not always a mirror of their thoughts. From this follows that we, as teachers, must be careful and not assume that we know about a child’s thoughts when, in fact, what we know is what the child chooses to communicate about his or her thoughts. I also found that the children seemed unaccustomed to speaking about mathe-matics in the way that I wanted them to. One reason for this seemed to be the way in which their teacher organized the lessons.</p>
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Lemon, Travis L. "Thinking on the Brink: Facilitating Student Teachers' Learning Through In-the-Moment Interjections." BYU ScholarsArchive, 2010. https://scholarsarchive.byu.edu/etd/2292.
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In order to investigate ways pre-service student teachers (PSTs) might learn to teach with high-level tasks and effectively incorporate student thinking into their lessons a teaching experiment was designed and carried out by the cooperating teacher/researcher (CT). The intervention was for the CT to interject into the lessons of the PSTs during moments of opportunity. By interjecting a small question or comment during the lesson the CT hoped to support the learning of both the students of mathematics in the class and the PSTs. This in-the-moment interjecting was meant to enhance and underscore the situated learning of the PSTs within the context of actual practice. Essentially the PSTs learned how to manage and improve the discourse of the classroom in the moment of the discourse. This study utilized both an ongoing analysis of the data during collection in order to inform the instruction provided by the CT and a retrospective analysis of the data in order to develop an understanding of the developmental sequence through which PSTs progressed. The results suggest the interjections provided to the PSTs served multiple roles within the domains of mathematical development for the students of mathematics and pedagogical development for the PSTs. A classification of the interjections that occurred and the stages of development through which PSTs passed will be discussed. Implications from this work include increased attention to the groundwork leading up to the student teaching experience as well as an adjustment to the role of cooperating teacher to be more that of a teacher educator.
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Espinoza, Eduardo. "Elevers utveckling i den matematiska tänkande : Exempel från en fristående skolan profilerad i matematik." Thesis, Södertörn University College, Lärarutbildningen, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:sh:diva-848.
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<p>The primordial purpose of our studies has been carrying out a detailed research to describe methods or work procedures in the teaching and application of the mathematics, at a school based or alignment on the mathematics instruction. To be able to study the pupils in their development of the mathematical thinking.</p><p>We have carried out a detailed investigation, in the previously mentioned school using the ethnography observation methods directly in the place of the facts. Where it was possible to verify that the mathematics lessons were a consequences of the methods or work procedures which made us deduce that this school did every possible effort to stimulate all the pupils to be better and particularly talented pupils individually to develop one’s talent by means of the following results:</p><p>· Develop the logical thinking</p><p>· The self-critical ability</p><p>· The attitude of the teacher/communication</p><p>· A positive work atmosphere</p><p>· Organization of the school and the class</p><p>· Formation of the theoretical knowledge</p><p>Keywords</p><p>Mathematical thinking.</p><p>Pupils</p><p>Independent school</p>
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Basaran, Seren. "An Exploration Of Affective And Demographic Factors That Are Related To Mathematical Thinking And Reasoning Of University Students." Phd thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613422/index.pdf.
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There are four major aims of this study: Firstly, factors regarding university students&rsquo<br>approaches to studying, self-efficacy in mathematics, problem solving strategies, demographic profile, mathematical thinking and reasoning competencies were identified through the adopted survey and the competency test which was designed by the researcher. These scales were administered to 431 undergraduate students of mathematics, elementary and secondary mathematics education in Ankara and in Northern Cyprus and to prospective teachers of classroom teacher and early childhood education of teacher training academy in Northern Cyprus. Secondly, three structural models were proposed to explore the interrelationships among idenitified factors. Thirdly, among three models, the model yielding best fit to data was selected to evaluate the equality of the factor structure across Ankara and Northern Cyprus regions. Lastly, differences regarding pre-identified factors with respect to gender, region and grade level separately and dual, triple interaction effects were investigated through two two-way MANOVA and a three-way ANOVA analyses.
Exploratory and confirmatory factor analyses were employed to determine the factors<br>meaning orientation, mathematics self-efficacy, motivation, disorganized study methods and surface approach for the survey and &lsquo<br>expressing, extracting and computing mathematically&rsquo<br>(fundamental skills) and &lsquo<br>logical inferencing and evaluating conditional statements in real life situations&rsquo<br>(elaborate skills) for the test.
The three models commonly revealed that while mathematics self-efficacy has a significant positive effect on both fundamental and elaborate skills, motivation which is a combination of intrinsic, extrinsic and achievement motivational items was found to have a negative direct impact on fundamental skills and has a negative indirect contribution upon elaborate skills.
The results generally support the invariance of the tested factor structure across two regions with some evidence of differences. Ankara region sample yielded similar factor structure to that of the entire sample&rsquo<br>s results whereas<br>no significant relationships were observed for Northern Cyprus region sample.
Results of gender, grade level and region related differences in the factors of the survey and the test and on the total test indicated that, females are more meaning oriented than males. &lsquo<br>Fourth and fifth (senior)&rsquo<br>and third year university students use disorganized study methods more often than second year undergraduate students. In addition, senior students are more competent than second and third year undergraduate students in terms of both skills. Freshmen students outscored sophomore students in the elaborate skills. Students from Ankara region are more competent in terms of both skills than students from Northern Cyprus region. This last inference is also valid on the total test score for both regions. Males performed better on the total test than females.
Moreover, there exist region and grade level interaction effect upon both skills. Additionally, significant interaction effects of &lsquo<br>region and gender&rsquo<br>, &lsquo<br>region and grade level&rsquo<br>, &lsquo<br>gender and grade level&rsquo<br>and &lsquo<br>region and gender and grade level&rsquo<br>were detected upon the total test score.
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Munlo, Isaac. "Critical systems thinking, theory and practice : a case study of an intervention in two British local authorities." Thesis, University of Hull, 1997. http://hydra.hull.ac.uk/resources/hull:5718.
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This thesis reports an intervention informed by critical systems thinking. The intervention drew upon a variety of systems and operational research methods to systemically explore the problems facing housing services for older people. Stakeholders were then supported in developing a response to these problems in the form of an integrated model of user involvement and multi-agency working. The methods used in this study included Cognitive Mapping, Critical Systems Heuristics, Interactive Planning and Viable System Modelling. Following a description of the project and its outcomes, the author's practical experiences are used to reflect back on critical systems thinking. Five innovations are presented in the thesis: First a new method called 'Problem Mapping' is developed. This has five stages: (i) interviewing stakeholders to surface problems and identify further potential interviewees; (ii) listing the problems as seen through the eyes of the various stakeholders; (iii) consolidating the list by removing duplicate problems and synthesising similar problems into larger 'problem statements'; (iv) mapping the relationships between problems; and (v) presenting the results back to stakeholders to inform the development of proposals for improvement. Reflection upon the use of this method indicates that it is particularly valuable where there are multiple stakeholders who are not initially visible to researchers, each of whom sees different aspects of a problem situation. Second, Problem Mapping is used to systemically express the problems facing housing services for older people in two geographical areas in the UK. This shows how problems in the areas of assessment, information provision and planning are mutually reinforcing, making a strong case for change. Third, a process of evolving an integrated model of user involvement and multi-agency working is presented. The model was designed in facilitated workshops by managers from statutory agencies, based on specifications developed by a variety of stakeholders (including service users and carers). Fourth, the strengths and weaknesses of Cognitive Mapping (one of the methods used in the project) are discussed. Significant limitations of this method are highlighted. Fifth, contributions and reflections on the theoretical and practical basis of the research are presented. These among others focus on the theory of boundary critique, which is an important aspect of critical systems thinking. It is often assumed that boundary critique is only undertaken at the start of an intervention to ensure that its remit has been adequately defined. However, this project shows that it is both possible and desirable to use the theory of boundary critique in an on-going basis in interventions to inform the creative design of methods.
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Farlow, Brian. "Square Peg Thinking, Round Hole Problems: An Investigation of Student Thinking About and Mathematical Preparation for Vector Concepts in Cartesian and Non-Cartesian Coordinates Used in Upper-Division Physics." Diss., North Dakota State University, 2019. https://hdl.handle.net/10365/31479.
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Serbin, Kaitlyn Stephens. "Prospective Teachers' Knowledge of Secondary and Abstract Algebra and their Use of this Knowledge while Noticing Students' Mathematical Thinking." Diss., Virginia Tech, 2021. http://hdl.handle.net/10919/104563.
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I examined the development of three Prospective Secondary Mathematics Teachers' (PSMTs) understandings of connections between concepts in Abstract Algebra and high school Algebra, as well as their use of this understanding while engaging in the teaching practice of noticing students' mathematical thinking. I drew on the theory, Knowledge of Nonlocal Mathematics for Teaching, which suggests that teachers' knowledge of advanced mathematics can become useful for teaching when it first helps reshape their understanding of the content they teach. I examined this reshaping process by investigating how PSMTs extended, deepened, unified, and strengthened their understanding of inverses, identities, and binary operations over time. I investigated how the PSMTs' engagement in a Mathematics for Secondary Teachers course, which covered connections between inverse functions and equation solving and the abstract algebraic structures of groups and rings, supported the reshaping of their understandings. I then explored how the PSMTs used their mathematical knowledge as they engaged in the teaching practice of noticing hypothetical students' mathematical thinking. I investigated the extent to which the PSMTs' noticing skills of attending, interpreting, and deciding how to respond to student thinking developed as their mathematical understandings were reshaped.
There were key similarities in how the PSMTs reshaped their knowledge of inverse, identity, and binary operation. The PSMTs all unified the additive identity, multiplicative identity, and identity function as instantiations of the same overarching identity concept. They each deepened their understanding of inverse functions. They all unified additive, multiplicative, and function inverses under the overarching inverse concept. They also strengthened connections between inverse functions, the identity function, and function composition. They all extended the contexts in which their understandings of inverses were situated to include trigonometric functions. These changes were observed across all the cases, but one change in understanding was not observed in each case: one PSMT deepened his understanding of the identity function, whereas the other two had not yet conceptualized the identity function as a function in its own right; rather, they perceived it as x, the output of the composition of inverse functions.
The PSMTs had opportunities to develop these understandings in their Mathematics for Secondary Teachers course, in which the instructor led the students to reason about the inverse and identity group axioms and reflect on the structure of additive, multiplicative, and compositional inverses and identities. The course also covered the use of inverses, identities, and binary operations used while performing cancellation in the context of equation solving.
The PSMTs' noticing skills improved as their mathematical knowledge was reshaped. The PSMTs' reshaped understandings supported them paying more attention to the properties and strategies evident in a hypothetical student's work and know which details were relevant to attend to. The PSMTs' reshaped understandings helped them more accurately interpret a hypothetical student's understanding of the properties, structures, and operations used in equation solving and problems about inverse functions. Their reshaped understandings also helped them give more accurate and appropriate suggestions for responding to a hypothetical student in ways that would build on and improve the student's understanding.<br>Doctor of Philosophy<br>Once future mathematics teachers learn about how advanced mathematics content is related to high school algebra content, they can better understand the algebra content they may teach. The future teachers in this study took a Mathematics for Secondary Teachers course during their senior year of college. This course gave them opportunities to make connections between advanced mathematics and high school mathematics. After this course, they better understood the mathematical properties that people use while equation solving, and they improved their teaching practice of making sense of high school students' mathematical thinking about inverses and equation solving. Overall, making connections between the advanced mathematics content they learned during college and the algebra content related to inverses and equation solving that they teach in high school helped them improve their teaching practice.
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Schillinger, Tammy. "Mathematical Instructional Practices and Self-Efficacy of Kindergarten Teachers." ScholarWorks, 2016. https://scholarworks.waldenu.edu/dissertations/2101.
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A local urban school district recently reported that 86% of third graders did not demonstrate proficiency on the Math Standardized Test, which challenges students to solve problems and justify solutions. It is beneficial if these skills are developed prior to third grade. Students may be more academically successful if kindergarten teachers have moderate to high self-efficacy when teaching lessons that focus on justifying solutions. Bandura's self-efficacy theory was incorporated into this study as the conceptual framework lens. Research questions were designed to investigate kindergarten teachers' instruction in mathematics that focused on justifying solutions, their self-efficacy in challenging students to justify solutions, and the identification of professional development. Voluntary participants for this study were selected from the 11 elementary schools in the district. Within the 11 elementary schools, there were 33 lead teachers who were invited to participate in the study and 7 agreed to participate in interviews and observations. The data were analyzed using both situation and strategy coding. The analysis of the data revealed a connection between professional development, self-efficacy, and instructional strategies. A relationship was identified between professional development and the teachers' ability to challenge students to problem solve and justify solutions. These findings may be valuable for early childhood stakeholders within the education field. Professional development tends to improve the self-efficacy of teachers and the instructional strategies they incorporate.
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Graves, Barbara, and Christine Suurtamm. "Disrupting linear models of mathematics teaching|learning." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79920.
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In this workshop we present an innovative teaching, learning and research setting that engages beginning teachers in mathematical inquiry as they investigate, represent and connect mathematical ideas through mathematical conversation, reasoning and argument. This workshop connects to the themes of teacher preparation and teaching through problem solving. Drawing on new paradigms to think about teaching and learning, we orient our work within complexity theory
(Davis & Sumara, 2006; Holland, 1998; Johnson, 2001; Maturana & Varela, 1987; Varela, Thompson & Rosch, 1991) to understand teaching|learning as a complex iterative process through which opportunities for learning arise out of dynamic interactions. Varela, Thompson and Rosch, (1991) use the term co-emergence to understand how the individual and the environment inform each other and are “bound together in reciprocal specification and selection” (p.174). In particular we are interested in the conditions that enable the co-emergence of teaching|learning collectives that support the generation of new mathematical and pedagogical ideas and understandings. The setting is a one-week summer math program designed for prospective elementary teachers to deepen particular mathematical concepts taught in elementary school. The program is facilitated by recently graduated secondary mathematics teachers to provide them an opportunity to experience mathematics teaching|learning through rich problems. The data collected include
questionnaires, interviews, and video recordings. Our analyses show that many a-ha moments of mathematical and pedagogical insight are experienced by both groups as they work together throughout the week. In this workshop we will actively engage the audience in an exploration of the mathematics problems that we pose in this unique teaching|learning environment. We will present our data on the participants’ mathematical and pedagogical responses and open a discussion of the implications of our work.
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Lo, Steson Koon-Siu. "Thinking Digitally: Individual Differences in Mental Representations of Number and Computational Algorithms Associated with Exact Quantities." Thesis, The University of Sydney, 2016. http://hdl.handle.net/2123/16328.
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The present research explored mental representations of number, the computational algorithms used to solve multi-digit arithmetic tasks, and how mental manipulation of exact quantities are modulated by individual differences in working memory capacity and method of instruction through six experiments. The first three experiments focused on holistic or decomposed mental representation of 2-digit numbers in Australian university students using a magnitude judgement and number bisection task. Since holistic or decomposed magnitude representations are associated with different multi-digit addition algorithms, Experiment 4 investigated individual differences in orthographic precision and strategies for multi-digit addition using a same-different mental arithmetic task. Experiment 5 explored the relationship between working memory and construction of novel strategies in a mathematical problem solving task using Diophantine equations. Individual differences in mental representations were further explored in Experiment 6 by comparing Australian university students with Japanese children who were skilled users of a ‘mental abacus’. Although the Japanese children’s performance was superior to the university students on virtually all tasks, both groups appeared to use qualitatively similar strategies of computing the total of 2-digit numbers by decomposing each of the operands according to unit and decade positions. Why are some people better at maths than others? The results suggest that computational speed and accuracy were associated with larger working memory capacity, stronger decomposed mental representations of number, and an educational approach that involved plenty of practice. All of these attributes facilitate application of exact multi-digit algorithms. Aside from the theoretical implications of these findings for the field of numerical cognition, the results are also practically important for teachers and policy makers.
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Taylor, Carol H. "Promoting Mathematical Understanding through Open-Ended Tasks; Experiences of an Eighth-Grade Gifted Geometry Class." Digital Archive @ GSU, 2008. http://digitalarchive.gsu.edu/msit_diss/36.
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Promoting Mathematical Understanding Through Open-Ended Tasks; Experiences of an Eighth-Grade Gifted Geometry Class by Carol H. Taylor Gifted students of mathematics served through acceleration often lack the opportunities to engage in challenging, complex investigations involving higher-level thinking. This purpose of this study was to examine the ways mathematically gifted students think about and do mathematics creatively as indicators of deep understanding through collaborative work on four open-ended tasks with high-level cognitive demand. The study focused on the mathematical thinking involved in students’ construction of mathematical understanding through the social interaction of group problem solving. This case study used ethnographic methodology within a social constructivist frame with gifted education and sociocultural contextual influences. Participants were 15 gifted students in an 8th-grade gifted geometry class. Data collection included field notes, student artifacts, student journal entries, audio recordings, and reflections. Transcribed audio recordings were segmented (Tesch, 1990) into phases of interaction, coded by function, then coded by levels of exhibited mathematical thinking from observable cognitive actions (Dreyfus, Hershkowitz, & Schwarz, 2001; Williams, 2000; Wood, Williams, & McNeal, 2006), and analyzed for maintenance or decline of high-level cognitive demand (Stein, Smith, Henningsen, & Silver, 2000). Interpretive data analysis was connected to data analysis of transcribed recordings. Results indicated social interaction among students enabled them to talk through the mathematics to understand mathematical concepts and relationships, to construct more complex meaning, and exhibit mathematical creativity, inventiveness, flexibility, and originality. Students consistently exhibited these characteristics indicating mathematical thinking at the levels of building-with analyzing, building-with synthetic-analyzing, building-with evaluative-analyzing, constructing synthesizing, and occasionally constructing evaluating (Dreyfus et al., 2001; Williams, 2000; Wood et al., 2006). The results of the study support the claim of a relationship between mathematical giftedness and the ability to abstract and generalize (Sriraman, 2003), provide evidence that given the opportunity, students can construct deep mathematical understanding, and indicate the importance of social interaction in the construction of knowledge. This study adds to the body of knowledge needed in research on gifted education, problem solving, small-group interaction, mathematical thinking, and mathematical understanding, through empirically assessed classroom practice (Friedman-Nima et al., 2005; Good, Mulryan, & McCaslin, 1992; Hiebert & Carpenter, 1992; Lester & Kehle, 2003; Phillipson, 2007; Wood, Williams, & McNeal, 2006).
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Cheng, Chun Chor Litwin. "Basic knowledge and Basic Ability: A Model in Mathematics Teaching in China." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79584.
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This paper aims to present a model of teaching and learning mathematics in China. The model is “Two Basic”, basic knowledge and basic ability. Also, the paper will analyze some of the background of the model and why it is efficient in mathematics education. The model is described by a framework of “slab” and based on a model of learning cycle, allow students to work with mathematical thinking. Though the model looks of demonstration and practice looks very traditional, the explanation behind allows us to understand why Chinese students achieved well in many international studies in mathematics. The innovation of the model is the teacher intervention during the learning process. Such interventions include repeated practice, and working on group of selected related questions so that abstraction of the learning process is possible and student can link up mathematical expression and process. Examples used in class are included and the model can be applied in teaching advanced mathematics, which is usually not the case in some many other existing theories or framework.
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Rao, Rashmi Jayathirtha. "Modeling learning behaviour and cognitive bias from web logs." The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1492560600002105.
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Nascimento, Anderson de Araújo. "Análise dos tipos de provas matemáticas e pensamento geométrico de alunos do 1º ano do Ensino Médio." Universidade Estadual da Paraíba, 2017. http://tede.bc.uepb.edu.br/jspui/handle/tede/2907.
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Previous issue date: 2017-08-21<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>The present research work investigated the level of geometric thinking and the types of mathematical proofs by 1st year high school students from the application of a Didactic Proposal. This research was constituted as a qualitative one, and as case study, having instruments of the application an essay with the theme Proofs and Mathematical Demonstrations, Didactic Proposal developed by a team of five members who worked collaboratively, inserted in the Project CAPES/OBEDUC/UFMS/UEPB/UFAL Edital 2012, participant observation and audio recording. We developed the didactic proposal with 18 activities, divided into four parts, which stimulated students to reflect, justify, prove and demonstrate. The application of this proposal occurred in June 2015 for 1st year high school students in a public school in the city of Areia, Paraíba. Our research took place in three moments. In the first moment, we apply the essay on the subject mathematical proofs and demonstrations. In the second moment we did a didactic intervention approaching definitions, theorems, proofs and mathematical demonstrations with the objective of taking to the students this knowledge. In the third moment, Part I and II of the Didactic Proposal were applied, involving activities to conjecture and demonstrate the Pythagorean Theorem, Internal Angle Sum Theorem and External Angle Theorem. This proposal helped in the investigation of the mathematical knowledge of the 1st year high school students, divided into 8 pairs and one trio, chosen freely. The two pairs of students who achieved the best performance in our Didactic Proposal were chosen for our case study and the one of better performance had its dialogue recorded and transcribed as a source of evidence of our case study. In our research we analyzed the answers given by the two pairs on Activities 1 and 3 (Part II) and Activity 2 (Part III), totaling in 3 questions. We used the data triangulation method for our case study. Firstly, we draw the profile of the two pairs of students in relation to Proofs and Mathematical Demonstrations. Next, we investigate the types of mathematical proofs used by them and their geometric thinking. To do so, we use discussions about the levels of geometric thinking proposed by Van Hiele and the types of evidence. From our results we can conclude that the pairs of students were able to develop informal justifications, that is, informal proofs. Thus, the pairs presented pragmatic evidence and the types of evidence Pragmatic Justification and Crucial Example. Regarding the geometric thinking proposed by Van Hiele, only one pair could be classified in one of the levels of development of geometric thinking, Level 3, informal deduction. Therefore, we come to the end of this research convinced that it is necessary to start working mathematical proofs and demonstrations in the basic education level, adapting its teaching to the degree of maturity and to the mathematical knowledge of the students, since our results point out that this subject is not approached properly in the classroom.<br>A presente pesquisa investigou o nível do pensamento geométrico e os tipos de provas matemáticas de alunos do 1º ano do Ensino Médio a partir da aplicação de uma Proposta Didática. Esta pesquisa se constituiu como qualitativa, e estudo de caso, tendo como instrumentos a aplicação de uma redação com o tema Provas e Demonstrações Matemáticas, Proposta Didática desenvolvida por uma equipe de cinco membros que trabalhou de forma colaborativa, inserida no Projeto CAPES/OBEDUC/UFMS/UEPB/ UFAL Edital 2012, observação participante e gravação em audio do diálgo de umas das duplas participantes da pesquisa. Elaboramos uma proposta didática com 18 atividades, dividida em quatro partes, que estimulavam aos alunos refletirem, justificarem, provarem e demonstrarem. A aplicação dessa proposta se deu em junho de 2015 para alunos do 1º ano do Ensino Médio de uma escola pública da cidade de Areia, Paraíba. Nossa pesquisa se deu em três momentos. No primeiro momento, aplicamos a redação sobre o tema provas e demonstrações matemáticas. No segundo momento realizamos uma intervenção didática abordando definições, teoremas, provas e demonstrações matemáticas com o objetivo de levar aos alunos esses conhecimentos. No terceiro momento foi aplicado a Parte I e II da Proposta Didática, envolvendo atividades de conjecturar e demonstrar o Teorema de Pitágoras, Teorema da Soma dos Ângulos Internos e Teorema dos Ângulo Externo. Essa proposta auxiliou na investigação do conhecimento matemático dos alunos do 1º ano do Ensino Médio, divididos em 8 duplas e um trio, escolhidos livremente. As duas duplas de alunos que obteveram melhores desempenhos em nossa Proposta Didática foram escolhidas para o nosso estudo de caso e a de melhor desenpenho teve seu diálogo gravado e transcrito como fonte de evidência de nosso estudo de caso. Em nossa pesquisa analisamos as respostas dadas pelas duas duplas sobre Atividades 1 e 3 (Parte II) e Atividade 2 (Parte III), totalizando em 3 questões. Utilizamos o método de triângulação de dados para nosso estudo de caso. Primeiramente, traçamos o perfil das duas duplas de alunas com relação às Provas e Demonstrações Matemáticas. Em seguida, investigamos os tipos de provas matemáticas utilizadas por elas e o seu pensamento geométrico. Para tanto, utilizamos as discussões sobre os níveis do pensamento geométrico proposto por Van Hiele e os tipos de provas. A partir de nossos resultados pudemos concluir que as duplas de alunas conseguiram desenvolver justificativas informais, ou seja, provas informais. Assim, as duplas apresentaram provas pragmáticas e os tipos de provas Justificativa Pragmática e Exemplo Crucial. Com relação ao pensamento geométrico proposto por Van Hiele, apenas uma dupla pôde ser classificada em um dos níveis de desenvolvimento do pensamento geométrico, o Nível 3, dedução informal. Portanto, chegamos ao final desta pesquisa convictos de que é preciso iniciar o trabalho das provas e demonstrações matemáticas na Educação Básica, adequando seu ensino ao grau de maturidade e aos conhecimentos matemáticos dos alunos, visto que nossos resultados apontam que esse tema não é abordado adequadamente em sala de aula.
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Atz, Dafne. "A análise combinatória no 6º Ano do Ensino Fundamental pormeio da resolução de problemas." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2017. http://hdl.handle.net/10183/164618.
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Esta dissertação apresenta o desenvolvimento de uma pesquisa referente ao ensino da Análise Combinatória, por meio da Resolução de Problemas, em uma turma de 6º ano do Ensino Fundamental. Para isso, elaborou-se uma sequência didática que buscava proporcionar aos educandos um contato com esse conteúdo antes do Ensino Médio. A partir dessa sequência analisou-se como a Resolução de Problemas, segundo Onuchic e Allevato, auxiliou os alunos a compreender os conceitos iniciais de Análise Combinatória, buscando também como referencial teórico o estudo referente ao Pensamento Matemático, de David Tall. Concluímos que a Resolução de Problemas auxiliou a expandir e modificar as Imagens dos Conceitos que os alunos possuíam com relação à Análise Combinatória.<br>This dissertation shows the development of research related to teaching Combinatorics, through Problem Solving, at a 6th grade level. A lesson plan was prepared and aimed to confront students of middle school with problems involving Combinatorics, allowing them to work with such concepts before high school. Based on this lesson plan, our intent was to verify how Problem Solving, according to Onuchic e Allevato, helped the students to understand initial concepts of Combinatorics. Also, using David Tall’s studies about Mathematical Thinking as reference. We could verify that the Problem Solving Theory helped the students to expand and modify their Concept Images related to Combinatorics.
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Bango, Siduduzile. "An investigation into Grade 7 learners’ knowledge of ratios." Diss., University of Pretoria, 2002. http://hdl.handle.net/2263/78505.
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Ratio is one of the key mathematics concepts included in the South African Mathematics curriculum. It is applied in other topics of the Grade 7 curriculum, including geometry, functions and relationships, algebra, similarity and congruency.
The aim of this qualitative research study was to explore the difficulties that learners experience in learning ratio. The primary research question for the study was: What is Grade 7 learners’ knowledge of ratio? This research question was answered through the following secondary research questions: How do learners solve problems involving ratio? What is learners’ conceptual knowledge of ratio? And what learning difficulties do learners experience when learning about ratio?
The study was informed by Kilpatrick, Swafford and Findell’s (2001) five strands of mathematical proficiency; however, the focus was on conceptual and procedural knowledge of ratio. The interpretivist paradigm and the single exploratory case study design were used to gain insight into the learning of ratio. Data was collected from Grade 7 learners (23 of the 35 learners originally sampled) through a self-developed test that followed the prescripts of the Grade 7 Mathematics curriculum in South Africa and through semi-structured interviews. The test scripts were analysed using the Atlas.tiTM windows coding system and the results were used to construct questions for the semi-structured interviews. The interviews were used to corroborate data emerging from the test.
The results of the study indicated that Grade 7 learners can do simple and routine manipulations of ratio as well as non-proportional ratio problems but struggle to solve problems that require multiplicative thinking and proportional reasoning skills. Although there could be other factors contributing to learners’ struggle to tackle proportional ratio problems requiring multiplication and proportional reasoning, a lack of conceptual knowledge seemed to contribute significantly.<br>Dissertation (MEd)--University of Pretoria, 2020.<br>Science, Mathematics and Technology Education<br>MEd<br>Unrestricted
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Szyjka, Sebastian. "Cognitive And Attitudinal Predictors Related To Graphing Achievement Among Pre-Service Elementary Teachers." OpenSIUC, 2009. https://opensiuc.lib.siu.edu/dissertations/43.
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The purpose of this study was to determine the extent to which six cognitive and attitudinal variables predicted pre-service elementary teachers' performance on line graphing. Predictors included Illinois teacher education basic skills sub-component scores in reading comprehension and mathematics, logical thinking performance scores, as well as measures of attitudes toward science, mathematics and graphing. This study also determined the strength of the relationship between each prospective predictor variable and the line graphing performance variable, as well as the extent to which measures of attitude towards science, mathematics and graphing mediated relationships between scores on mathematics, reading, logical thinking and line graphing. Ninety-four pre-service elementary education teachers enrolled in two different elementary science methods courses during the spring 2009 semester at Southern Illinois University Carbondale participated in this study. Each subject completed five different instruments designed to assess science, mathematics and graphing attitudes as well as logical thinking and graphing ability. Sixty subjects provided copies of primary basic skills score reports that listed subset scores for both reading comprehension and mathematics. The remaining scores were supplied by a faculty member who had access to a database from which the scores were drawn. Seven subjects, whose scores could not be found, were eliminated from final data analysis. Confirmatory factor analysis (CFA) was conducted in order to establish validity and reliability of the Questionnaire of Attitude Toward Line Graphs in Science (QALGS) instrument. CFA tested the statistical hypothesis that the five main factor structures within the Questionnaire of Attitude Toward Statistical Graphs (QASG) would be maintained in the revised QALGS. Stepwise Regression Analysis with backward elimination was conducted in order to generate a parsimonious and precise predictive model. This procedure allowed the researcher to explore the relationships among the affective and cognitive variables that were included in the regression analysis. The results for CFA indicated that the revised QALGS measure was sound in its psychometric properties when tested against the QASG. Reliability statistics indicated that the overall reliability for the 32 items in the QALGS was .90. The learning preferences construct had the lowest reliability (.67), while enjoyment (.89), confidence (.86) and usefulness (.77) constructs had moderate to high reliabilities. The first four measurement models fit the data well as indicated by the appropriate descriptive and statistical indices. However, the fifth measurement model did not fit the data well statistically, and only fit well with two descriptive indices. The results addressing the research question indicated that mathematical and logical thinking ability were significant predictors of line graph performance among the remaining group of variables. These predictors accounted for 41% of the total variability on the line graph performance variable. Partial correlation coefficients indicated that mathematics ability accounted for 20.5% of the variance on the line graphing performance variable when removing the effect of logical thinking. The logical thinking variable accounted for 4.7% of the variance on the line graphing performance variable when removing the effect of mathematics ability.
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Carvalho, Liliane Maria Teixeira Lima de. "O papel dos artefatos na construÃÃo de significados matemÃticos por estudantes do ensino fundamental II." Universidade Federal do CearÃ, 2008. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=2617.
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CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior<br>A pesquisa investiga se diferentes formas de conceber o papel dos artefatos e apresentaÃÃo da informaÃÃo influenciam a construÃÃo de significados matemÃticos por estudantes de 11 a 14 anos. A cogniÃÃo humana à concebida como processo mediado pela tradiÃÃo cultural e histÃrica das representaÃÃes enquanto artefatos, inserindo-se essa anÃlise no Ãmbito do raciocÃnio matemÃtico. Utilizou-se o mÃtodo experimental aliado a uma pesquisa-aÃÃo envolvendo o design intencional de tarefas. Explorou-se o papel mediacional das tarefas, desde a sua confecÃÃo e introduÃÃo na sala de aula de matemÃtica, atà o seu uso pelos estudantes. Essa abordagem se concretizou por meio de seis experimentos, dos quais participaram 922 estudantes: 598 oriundos do key Stage Three (corresponde em idade ao 7Â, 8 e 9 anos do Ensino Fundamental II no Brasil) de quatro escolas inglesas, e 324 oriundos do 7Â, 8 e 9 anos de duas escolas brasileiras. O Experimento 1 investiga se grÃficos, tabelas ou casos isolados influenciam o raciocÃnio dos estudantes sobre variÃveis discretas. O Experimento 2 verifica se diferentes informaÃÃes sobre variÃveis contÃnuas influenciam a interpretaÃÃo grÃfica dos estudantes. O Experimento 3 analisa se interaÃÃes de aspectos visuais e conceituais da informaÃÃo sobre variÃveis contÃnuas influenciam a interpretaÃÃo grÃfica dos estudantes. O Experimento 4 investiga se grÃficos, tabelas ou a combinaÃÃo de ambas as representaÃÃes influencia interaÃÃes de aspectos visuais e conceituais da informaÃÃo. Esses quatro experimentos foram realizados nas escolas inglesas. As tarefas usadas no primeiro e quarto experimentos foram aplicadas nas escolas brasileiras, sendo designados Experimentos 5 e 6, respectivamente. As tarefas foram potencialmente facilitadoras ao uso de conteÃdos matemÃticos. Os Experimentos 1 e 5 oferecem evidÃncias de que estudantes jà familiarizados com representaÃÃes em tabelas e grÃficos para representar variÃveis discretas nÃo se beneficiam em atividades em que eles precisam organizar os dados por eles mesmos. Estudantes ingleses tiram proveito igualmente de tabelas e grÃficos. Estudantes brasileiros nÃo se beneficiam do uso de tabelas. Os Experimentos 2 e 3 confirmam resultados de estudos prÃvios de que informaÃÃes grÃficas sobre variÃveis contÃnuas possuem diferentes nÃveis de complexidade. Ler pontos à significativamente mais fÃcil do que interpretar problemas globais. Os Experimentos 2 e 3 tambÃm confirmam a hipÃtese de que os problemas de inferÃncia inversa explicam as dificuldades com informaÃÃes globais. Essa dificuldade à acentuada em grÃficos com inclinaÃÃo negativa. O Experimento 4 mostra que a forma de apresentaÃÃo da informaÃÃo nÃo afeta o desempenho dos estudantes na resoluÃÃo de problemas sobre variÃveis contÃnuas. O raciocÃnio dos estudantes sobre variÃveis contÃnuas, no entanto, à influenciado pela forma de apresentaÃÃo da informaÃÃo. A pesquisa sugere a necessidade de uma discriminaÃÃo da informaÃÃo nÃo apenas quanto ao tipo de variÃvel, discreta ou contÃnua, ou tipo de relaÃÃo proporcional, direta ou inversa, mas tambÃm quanto ao tipo de inferÃncias requeridas dos estudantes
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Mulder, Isabella Dorothea. "Graad 12-punte as voorspeller van sukses in wiskunde by 'n universiteit van tegnologie / I.D. Mulder." Thesis, North-West University, 2011. http://hdl.handle.net/10394/10338.
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Problems with students’ performance in Mathematics at tertiary level are common in South Africa − as it is worldwide. Pass rates at the university of technology where the researcher is a lecturer, are only about 50%. At many universities it has become common practice to refer students who do not have a reasonable chance to succeed at university level, for additional support to try to rectify this situation. However, the question is which students need such support? Because the Grade 12 marks are often not perceived as dependable, it has become common practice at universities to re-test students by way of an entrance exam or the "National Benchmark Test"- project. The question arises whether such re-testing is necessary, since it costs time and money and practical issues make it difficult to complete timeously. Many factors have an influence on performance in Mathematics. School-level factors include articulation of the curriculum at different levels, insufficiently qualified teachers, not enough teaching time and language problems. However, these factors also affect performance in most other subjects, but it is Mathematics and other subjects based on Mathematics that are generally more problematic. Therefore this study focused on the unique properties of the subject Mathematics. The determining role of prior knowledge, the step-by-step development of mathematical thinking, and conative factors such as motivation and perseverance were explored. Based on the belief that these factors would already have been reflected sufficiently in the Grade 12 marks, the correlation between the marks for Mathematics in Grade 12 and the Mathematics marks at tertiary level was investigated to assess whether it was strong enough for the marks in Grade 12 Mathematics to be used as a reliable predictor of success or failure at university level. It was found that the correlation between the marks for Mathematics Grade 12 and Mathematics I especially, was strong (r = 0,61). The Mathematics marks for Grade 12 and those for Mathematics II produced a correlation coefficient of rs = 0,52. It also became apparent that failure in particular could be predicted fairly accurately on the basis of the Grade 12 marks for Mathematics. No student with a Grade 12 Mathematics mark below 60% succeeded in completing Mathematics I and II in the prescribed two semesters, and only about 11% successfully completed it after one repetition. The conclusion was that the reliability of the prediction based on the marks for Grade 12 Mathematics was sufficient to refer students with a mark of less than 60% to receive some form of additional support.<br>MEd, Learning and Teaching, North-West University, Vaal Triangle Campus, 2011
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Bilous, Olena Anatoliivna, Елена Анатольевна Белоус та Олена Анатоліївна Білоус. "Міжпредметні зв`язки при вивченні математичних дисциплін". Thesis, Видавництво СумДУ, 2010. http://essuir.sumdu.edu.ua/handle/123456789/4270.
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Zulatto, Rúbia Barcelos Amaral. "A natureza da aprendizagem matemática em um ambiente online de formação continuada de professores /." Rio Claro : [s.n.], 2007. http://hdl.handle.net/11449/102133.
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Orientador: Miriam Godoy Penteado<br>Banca: João Pedro Mendes da Ponte<br>Banca: Marcelo de Carvalho Borba<br>Banca: Maria Elizabeth Bianconcini Trindade Morato Pinto de Almeida<br>Banca: Vani Moreira Kenski<br>Resumo: A presente pesquisa analisa a natureza da aprendizagem matemática em um curso online de formação continuada de professores, denominado Geometria com Geometricks. Nele, alunos-professores de uma mesma rede de escolas, situadas em diferentes localidades do país, desenvolveram atividades de Geometria utilizando-se do software Geometricks, e se encontravam para discuti-las. Esses encontros aconteceram a distância, em tempo real, por chat ou videoconferência. Nessa proposta pedagógica, a telepresença condicionou a comunicação e oportunizou o estar-junto-virtual-com-mídias. De modo singular, os recursos da videoconferência permitiram que construções geométricas fossem compartilhadas visualmente e realizadas por todos os envolvidos, fomentando a interação e a participação ativa, constituindo, por meio do diálogo, uma comunidade virtual de aprendizagem. Os resultados levam a inferir que, nesse contexto, a aprendizagem matemática teve natureza colaborativa, na virtualidade das discussões, tecidas a partir das contribuições de todos os participantes; coletiva, na medida em que a produção matemática era condicionada pelo coletivo pensante de seres-humanos-com-mídias; e argumentativa, uma vez que conjecturas e justificativas matemáticas se desenvolveram intensamente do decorrer do processo, contando para isso com as tecnologias presentes na interação ocorrida de forma constante e colaborativa.<br>Abstract: This study was conducted to analyze the nature of mathematical learning in an online continuing education course for teachers entitled Geometry with Geometricks. Teachers employed in a nation-wide network of privately-supported schools developed geometry activities using the software Geometricks and discussed them in virtual meetings, in real time, via chat or video-conference. In this pedagogical proposal, tele-presence conditioned the communication and provided the opportunity for virtual-togetherness-with-media. In a unique way, the resources of the videoconference made it possible for everyone to participate in and visually share geometrical constructions, encouraging interaction and active participation and constituting a virtual learning community through dialogue. The results indicate that, in this context, mathematical learning nature was characterized by: collaboration, in the virtual discussions that were woven from the contributions of all the participants; collectivity, to the degree to which mathematical production was conditioned by the humans-with-media thinking collective; and argumentation, as the development of mathematical conjectures and justifications was intense throughout the process, aided by the technologies that were present in the constant, collaborative interaction.<br>Doutor
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Wielewski, Sergio Antonio. "Pensamento instrumental e pensamento relacional na educação matemática." Pontifícia Universidade Católica de São Paulo, 2008. https://tede2.pucsp.br/handle/handle/11325.
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Previous issue date: 2008-09-08<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior<br>This doctoral thesis contains theoretical discussions as well as results of an
empirical study. The general starting point has been the thesis that our
mathematical thinking is largely ruled by certain dualities or complementarities
of which that between the representational and instrumental aspects of
concepts is best known. Ernst Cassirer, presents in his famous book
Substanzbegriff und Funktionsbegriff (Substance and Function) of 1910 the
general thesis that the historical development of science could be described
as a transition from merely referential Aristotelian concepts to operative
concepts or functions. The very same duality has been discussed widely in
mathematics education starting from the work of Richard Skemp. Our first
goal has consequently been to find connections between Cassirer and
Skemp. The discussion of these connections and differences leads then in a
second part of the thesis to a presentation of the results of an empirical case
study with fourteen participants. These had been confronted with a number of
problem situations and their problem solving activities have afterwards been
analyzed in terms of the aforementioned complementarity between relational
and operative thinking<br>Nesta tese estão apresentados resultados de investigação teórica e empíricos.
O alvo da pesquisa é identificação de características e análise das reflexões
relativas a dualidades inerentes ao pensamento matemático. Tomou-se como
pressuposto que o conhecimento de dualidades do pensamento matemático, e
o como se utilizar desse conhecimento, se num sentido de complementaridade,
seja relevante para o processo de ensino e aprendizagem da Matemática. A
referência inicial do estudo foi a obra de Ernst Cassirer, Substance and
Function (1910). Nessa obra é apresentado o desenvolvimento histórico da
teoria do conceito de Aristóteles ao século XIX, isto é, desenvolvimento esse
que vai das propriedades de substância à noção de função. Cassirer,como
neo-kantiano, dá forte ênfase aos aspectos operativos e instrumentais do
conceito. Na continuidade do estudo é destacado a fundamental importância de
um conceito teórico ser compreendido nos termos de uma dualidade, em seus
aspectos operativos e referencial. O trabalho didático de Richard Skemp é
outro que explora dualidade semelhante. Trata -se da dualidade de aprender e
de compreender, que Skemp chama de compreensão instrumental e relacional.
Nossa investigação centra-se então na busca de conexão entre as concepções
de Cassirer e Skemp. Para tal levamos em conta aspectos educacionais,
reflexões filosóficas e pedagógicas, postura profissional do educador, exemplos
de situações a-didáticas e didáticas. Esses aspectos, reflexões e exemplos
nortearam a exploração empírica desta tese. Esta exploração teve o caráter de
uma pesquisa qualitativa, tendo sido desenvolvidas atividades didáticas. O
objetivo dessas atividades era avaliar a utilização pelos sujeitos do pensamente
relacional e do pensamento instrumental
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Abramovitz, Buma, Miryam Berezina, Abraham Berman, and Ludmila Shvartsman. "Proofs and "Puzzles"." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79279.
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It is well known that mathematics students have to be able to understand and prove
theorems. From our experience we know that engineering students should also be able to
do the same, since a good theoretical knowledge of mathematics is essential for solving
practical problems and constructing models.
Proving theorems gives students a much better understanding of the subject, and helps
them to develop mathematical thinking. The proof of a theorem consists of a logical
chain of steps. Students should understand the need and the legitimacy of every step.
Moreover, they have to comprehend the reasoning behind the order of the chain’s steps.
For our research students were provided with proofs whose steps were either written in a
random order or had missing parts. Students were asked to solve the \"puzzle\" – find the
correct logical chain or complete the proof.
These \"puzzles\" were meant to discourage students from simply memorizing the proof of
a theorem. By using our examples students were encouraged to think independently and
came to improve their understanding of the subject.
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Wilson, Therese Maree. "Statistical reasoning at the secondary tertiary interface." Thesis, Queensland University of Technology, 2006. https://eprints.qut.edu.au/16358/3/Therese%20Wilson%20Thesis.pdf.
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Each year thousands of students enrol in introductory statistics courses at universities throughout Australia, bringing with them formal and informal statistical knowledge and reasoning, as well as a wide range of basic numeracy skills, mathematical inclinations and attitudes towards statistics, which have the potential to impact on their ability to develop statistically. This research develops and investigates measures of each of these components for students at the interface of secondary and tertiary education, and investigates the relationships that exist between them, and a range of background variables. The focus of the research is on measuring and analysing levels and abilities in statistical reasoning for a range of students at the tertiary interface, with particular interest also in investigating their basic numeracy skills and how these may or may not link with statistical reasoning allowing for other variables and factors. Information from three cohorts in an introductory data analysis course, whose focus is real data investigations, provides basis for the research. This course is compulsory for all students in degree programs associated with all sciences or mathematics.
The research discusses and reports on the development of questionnaires to measure numeracy and statistical reasoning and the students' attitudes and reflections on their prior school experiences with statistics. Students' attitudes are found to be generally positive, particularly with regard to their self-efficacy. They are also in no doubt as to the links that exist between mathematics and statistics.
The Numeracy Questionnaire, developed to measure pre-calculus skills relevant to an introductory data analysis course which emphasises real data investigations, demonstrates that many students who have completed a basic algebra and calculus senior school subject struggle with skills which are in the pre-senior curricula. Direct examination of the responses helps to understand where and why difficulties tend to occur. Rasch analysis is used to validate the questionnaire and assist in the description of levels of skill. General linear models demonstrate that a student's numeracy score depends on the result obtained in senior mathematics, whether or not the student is a mathematics student, gender, whether or not higher level mathematics has been studied, self-efficacy and year. The research indicates that either the pre-senior curricula need strengthening or that exposure to mathematics beyond the core senior course is required to establish confidence with basic skills particularly when applied to new contexts and multi- step situations.
The Statistical Reasoning Questionnaire (SRQ) is developed for use in the Australian context at the secondary/tertiary interface. As with the Numeracy Questionnaire, detailed examination of the responses provides much insight into the range and features of statistical reasoning at this level. Rasch analyses, both dichotomous and polychotomous, are used to establish the appropriateness of this instrument as a measuring tool at this level. The polychotomous, Rasch partial credit model is also used to define a new approach to scoring a statistical reasoning instrument and enables development and application of a hierarchical model and measures levels of statistical reasoning appropriate at the school/tertiary interface.
General linear models indicate that numeracy is a highly significant predictor of statistical reasoning allowing for all other variables including tertiary entrance score and students' backgrounds and self-efficacy. Further investigation demonstrates that this relationship is not limited to more difficult or overtly mathematical items on the SRQ.
Performance on the end of semester component of assessment in the course is shown to depend on statistical reasoning at the beginning of semester as measured by the partial credit model, allowing for all other variables. Because of the dominance of the relationship between statistical reasoning (as measured by the SRQ) and numeracy on entry, some further analysis of the end of semester assessment is carried out. This includes noting the higher attrition rates for students with less mathematical backgrounds and lower numeracy.
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Wilson, Therese Maree. "Statistical reasoning at the secondary tertiary interface." Queensland University of Technology, 2006. http://eprints.qut.edu.au/16358/.
Full textAbstract:
Each year thousands of students enrol in introductory statistics courses at universities throughout Australia, bringing with them formal and informal statistical knowledge and reasoning, as well as a wide range of basic numeracy skills, mathematical inclinations and attitudes towards statistics, which have the potential to impact on their ability to develop statistically. This research develops and investigates measures of each of these components for students at the interface of secondary and tertiary education, and investigates the relationships that exist between them, and a range of background variables. The focus of the research is on measuring and analysing levels and abilities in statistical reasoning for a range of students at the tertiary interface, with particular interest also in investigating their basic numeracy skills and how these may or may not link with statistical reasoning allowing for other variables and factors. Information from three cohorts in an introductory data analysis course, whose focus is real data investigations, provides basis for the research. This course is compulsory for all students in degree programs associated with all sciences or mathematics. The research discusses and reports on the development of questionnaires to measure numeracy and statistical reasoning and the students' attitudes and reflections on their prior school experiences with statistics. Students' attitudes are found to be generally positive, particularly with regard to their self-efficacy. They are also in no doubt as to the links that exist between mathematics and statistics. The Numeracy Questionnaire, developed to measure pre-calculus skills relevant to an introductory data analysis course which emphasises real data investigations, demonstrates that many students who have completed a basic algebra and calculus senior school subject struggle with skills which are in the pre-senior curricula. Direct examination of the responses helps to understand where and why difficulties tend to occur. Rasch analysis is used to validate the questionnaire and assist in the description of levels of skill. General linear models demonstrate that a student's numeracy score depends on the result obtained in senior mathematics, whether or not the student is a mathematics student, gender, whether or not higher level mathematics has been studied, self-efficacy and year. The research indicates that either the pre-senior curricula need strengthening or that exposure to mathematics beyond the core senior course is required to establish confidence with basic skills particularly when applied to new contexts and multi- step situations. The Statistical Reasoning Questionnaire (SRQ) is developed for use in the Australian context at the secondary/tertiary interface. As with the Numeracy Questionnaire, detailed examination of the responses provides much insight into the range and features of statistical reasoning at this level. Rasch analyses, both dichotomous and polychotomous, are used to establish the appropriateness of this instrument as a measuring tool at this level. The polychotomous, Rasch partial credit model is also used to define a new approach to scoring a statistical reasoning instrument and enables development and application of a hierarchical model and measures levels of statistical reasoning appropriate at the school/tertiary interface. General linear models indicate that numeracy is a highly significant predictor of statistical reasoning allowing for all other variables including tertiary entrance score and students' backgrounds and self-efficacy. Further investigation demonstrates that this relationship is not limited to more difficult or overtly mathematical items on the SRQ. Performance on the end of semester component of assessment in the course is shown to depend on statistical reasoning at the beginning of semester as measured by the partial credit model, allowing for all other variables. Because of the dominance of the relationship between statistical reasoning (as measured by the SRQ) and numeracy on entry, some further analysis of the end of semester assessment is carried out. This includes noting the higher attrition rates for students with less mathematical backgrounds and lower numeracy.