The study employed Ernest (2006) Theory of Semiotic Systems to investigate the use of and exposure to multiple representations in a 10th grade algebra II suburban high school class located in the southeastern region of the United States. The purpose of this exploratory case study (Yin, 2014) was to investigate the role of multiple representations in influencing and facilitating algebra II students’ conceptual understanding of piece-wise function, absolute-value functions, and quadratic functions. This study attempted to answer the following question: How does the use of and exposure to multiple representations influence algebra II students’ understanding and transfer of algebraic concepts? Furthermore, the following sub-questions assisted in developing a deeper understanding of the question: a) how does exposure to and use of multiple representations influence students’ identification of their pseudo-conceptual understanding of algebraic concepts?; b) how does exposure to and use of multiple representations influence students’ transition from pseudo-conceptual to conceptual understanding?; c) how does exposure to and use of multiple representations influence students’ transfer of their conceptual understanding to other related concepts? Understanding the notion of pseudo-conceptual understanding in algebra is significant in providing a tool for examining the veracity of algebra students’ conceptual understanding, where teachers have to consistently examine if students accurately understand the meanings of the mathematical signs that they are constantly using. The following data collection techniques were utilized: a) classroom observation, b) task based interviews, and c) study of documents. The unit of analysis was students’ verbal and written responses to task questions. Three themes emerged from the analysis of in this study: (a) re-imaging of conceptual understanding; (b) reflective approach to understanding and using mathematical signs; and (c) representational versatility in the use of mathematical signs. Findings from this study will contribute to the body of knowledge needed in research on understanding and assessing algebra students’ conceptual understanding of mathematics. In particular the findings from the study will contribute to the literature on understanding; the process of algebraic concepts knowledge acquisition, and the challenges that algebra students have with comprehension of algebraic concepts (Knuth, 2000: Zaslavsky et al., 2002).

Magister Educationis - MEd / The purpose of this paper was to examine the use of the multiple representation approach as a teaching strategy to improve learners understanding of vectors in Grade 10 Physical Sciences. The study also wanted to consider the MR approach through the lens of the learners. A sample consisting of 45 Grade 10 learners from a total of 160 Grade 10 Physical Sciences learners participated in the study. Both quantitative and qualitative data were collected and analysed. Learners were first given a pre-test to establish their initial understanding of vectors. This pre-test was followed by an intervention in the form of a lesson. The lesson was conducted in order to expose learners to learning through Multiple Representations. A post-test was then administered to determine the impact of the intervention. To gather and quantify the learners’ perceptions on the use of Multiple Representations in teaching and learning of vectors in Grade 10 Physical Sciences learners were given questionnaires to complete. The last step was interviewing of learners to triangulate the results from the three instruments. The study found that learners were struggling with understanding of vectors in their traditional chalk-and-talk lessons and their perceptions towards vectors were negative. The study also found that Multiple Representations can improve understanding and develop positive perception of learners towards the teaching and learning of vectors. This improvement occurs only if Multiple Representations is used correctly. The study further found out that when Multiple Representations is used improperly it limits deeper understanding by learners. A number of recommendations were made out of the findings of the study. Some of them were that multiple representations should be used when teaching vectors and that subject advisers and teachers should be developed on the proper use of multiple representations. The Physical Sciences textbooks must be designed to accommodate Multiple Representations.

In STEM domains, robust learning includes not only fluency with procedures, but also recognition and application of the conceptual principles that underlie them. Grounded feedback is one instructional approach proposed to help students integrate conceptual knowledge into their learning of procedures. Grounded feedback functions primarily by having students take an action in the target domain (often symbolic) and receiving feedback in a representation that is easier to reason with. This thesis defines grounded feedback and evaluates its effectiveness. I define grounded feedback with four characteristics: (1) The feedback reflects students’ inputs according to rules that are inherent to the topic of study. For example, an inputted equation with two variables may be shown as a graph. (2) The feedback facilitates selfevaluation - by examining the feedback, students can evaluate for themselves if their answers are correct or not. (3) Students do not directly manipulate the feedback representation. Instead, the inputs are in a format that matches the domain learning goals. (4) The feedback conveys information about the nature of errors, not just that a particular action was incorrect. For example, the feedback may indicate the direction or magnitude of the error. Some prior experiments on systems with the four characteristics of grounded feedback found greater learning of target procedures (Nathan 1998) and greater transfer (Mathan & Koedinger 20015), relative to robust controls. Over four studies with 4th and 5th graders, this thesis explores three tutor designs for fraction addition that incorporate visualizations of magnitude, including grounded feedback. Two studies of grounded feedback show effects of robust learning relative to correctness feedback, including greater future learning (in study 2) and transfer (in study 3). Another study found little difference between grounded feedback with and without correctness. In the last study, relative to correctness feedback, two implementations of dynamically linked concrete representations (variations on grounded feedback) showed greater robust learning (pre-test to delayed test). The correctness feedback tutor, used in three of these studies, is a high-bar control, including immediate step-level correctness feedback and adaptive on-demand hints. Indications of more robust learning with the grounded feedback tutor are promising, though not conclusive. Grounded feedback is intended to leverage concrete representations to elicit students’ prior knowledge of relevant concepts. Over two Difficulty Factor Assessments, 5th graders demonstrated difficulty incorporating magnitude information when evaluating fraction addition equations. In particular, students could generally evaluate an equation correctly when it was represented with fraction bars. However, including symbols with the bars interfered with students’ evaluations by triggering incorrect transfer from whole-number addition. Students also did not fully grasp that when two positive fractions are added, the resulting sum is bigger than each addend alone. These findings may help explain why the benefits of grounded feedback are not as strong as proponents of concrete representations might hope. Namely, the target population may not be able to take full advantage of the magnitude visualization because they lack pre-requisite knowledge of how fraction addition involves magnitude.

Den här undersökningen inom fysikdidaktik utförs enligt ramverket The Scholarship of Teaching and Learning (SoTL). Det didaktiska verktyget som ska utvärderas benämns Pictionary Physics. Studien börjar med en litteraturöversikt av multipla representationer och interaktivt engagemang. Översikten syftar till att utforska hur lärandet kan möjliggöras vid användningen av det didaktiska verktyget. Pictionary Physics användes sedan för att främja en interaktiv användning av multipla representationer i en grupp bestående av fyra studenter. Studenternas agerande studerades och analyserades kvalitativt, följt av en utvärdering av studenternas upplevelser när de använde det didaktiska verktyget. Utvärderingen visar att Pictionary Physics kan gynna konceptuell förståelse för begrepp inom fysik. Utvärderingen visar även att det didaktiska verktyget skulle kunna bidra till förbättrat studieresultat då studenter uppmuntras att använda multipla representationer på ett interaktivt sätt. Fortsatt förädling av Pictionary Physics föreslås. Detta är stommen för SoTL, där tanken är att konsekvent utöka var kunskap om lärandet av fysik. / This physics education research project is carried out following the framework of the Scholarship of Teaching and Learning (SoTL). The didactic instrument investigated is termed Pictionary Physics. The study begins with a literature review of research on multiple representation, and interactive engagement. This review is used to evaluate the learning potential of the intended didactic instrument. Pictionary Physics was then used to facilitate the multi-representational interaction of a group of four physics students. The students’ behavior was studied and analyzed qualitatively, followed by an evaluation of the students’ experiences when using this didactic instrument. The investigation shows that Pictionary Physics may promote conceptual understanding of physics phenomena. The investigation also implies that this didactic instrument can contribute to improved learning outcomes when students are encouraged to interact by using multiple representations. Continued refinement of the Pictionary Physics concept is suggested. Such refinement is the essence of SoTL, incrementally expanding our knowledge of the teaching and learning of physics.

Representations (such as pictures, diagrams, word descriptions, equations, etc.) are critical tools for learning, problem solving, and communicating in science, particularly in physics where multiple representations often serve as intermediate steps, a means to evaluate a solution, and highlight different aspects a physical phenomenon. This dissertation explores the representation choices made by students in the University Modeling Instruction (MI) courses on problems from across introductory physics content. Modeling Instruction is a two-semester introductory, calculus-based physics sequence that was designed to guide students through the process of building, testing, applying, and refining models. As a part of this modeling cycle, students have explicit instruction and practice in building, evaluating, and coordinating representations in introductory physics. Since I am particularly interested in representations across all of introductory physics, this work was situated in the second semester of MI. To address students' representation choices, the Problem Solving and Representation Use Survey (PSRUS) was developed as modified card sort survey, which asked students to simply list the representations that they would use on 25 physics questions from across introductory physics. Using non-parametric statistical tests (Mann-Whitney-Wilcox, Wilcoxon-Ranked Sign, and Cliff's Delta), I compare the number and variety of representations that students choose. Initially, students who took the first semester of MI use significantly more representations in their problem solving when compared to those who did not; however, there are significant gains in the number of representations that these students choose over the semester across the introductory physics content. After significant changes to the second semester MI curriculum, the difference between these two groups disappears, with both groups increasing their representation choices when compared to the previous semester. Using network analysis to compare students' concurrent representation choices, I also show that students use a consistent set of representations on mechanics problems; whereas, they choose a wider variety on electricity and magnetism (EM) problems. In both mechanics and EM, pictures serve as an important connecting representation between the others. I use these results to make suggestions for instructors, curriculum developers, and physics education researchers.

The goals of this study were to research the learning difficulties among a group of four pre-university introductory calculus students who were mainly international students studying English as a Second Language (M). The intention was to create a constructivist-style classroom environment in order to determine if it could improve students` knowledge about the use and management of multiple representations (that is, graphical, rum~ symbolic, pictorial, linguistic or diagrammatic approaches for problem representation), increase their classroom communication as a means to improving ability in the modelling of calculus word problems, and to develop, implement and evaluate a teaching package that encouraged the use of multiple representations as a means of improving conceptual understanding. The achievement of these goals was sought by means of the development, implementation and evaluation of a number of calculus extended tasks that encouraged the use of multiple representations. These activities facilitated the compilation of a menu of approaches to the solution of mathematical problems, while the longitudinal nature of the study allowed for the monitoring of student changes in their preferred approach. A traditional calculus curriculum was used for the study, but the instructional emphasis was based more on students' understanding of concepts in a classroom environment utilising a constructivist approach rather than on their memorising computational techniques. Reading, writing, and discussion were emphasised m small group settings to develop language skills and to foster an appreciation of the alternative solution strategies of individual students. / The study was conducted at an International College north of Perth in Western Australia, and the majority of students in the sample were from Non-English-Speaking-Backgrounds (NESB). A range of methods was used to collect qualitative and quantitative data in order to increase the credibility of the research. These methods included audio recordings of structured task-based interviews with each of the four students in the sample; teacher analysis of student worksheets; my classroom observations; the analysis of alternative student conceptions on assessment tasks obtained through post-test interviews, and my personal reflections. Quality controls were employed to ensure the credibility of the data collected. As classroom teacher and principal researcher, it was possible for me to treat each of the four students involved as an individual case study. Descriptive questionnaires were used in order to gain information regarding the course and the use of graphics calculators. The results are applicable to ESL introductory calculus students only, and the nature of the sample implies a number of study limitations detailed in Chapter Five. There was extensive evidence of the benefits of the use of a multi-representational mode and evidence also of the benefits of encouraging the use of a diversity of modes of classroom instruction. Outcomes of the study were qualified by the difficulties ESL students face in coordinating conflicting information and interpreting the language demands of problem presentation. It is expected that this study will assist m extending the knowledge and understanding of the learning difficulties faced by ESL students in the am of pre-university calculus. / Results of this study suggest that instructional material has an important influence on ESL students’ use and management of multiple representations. However, there are often limitations to the influence of the material due to student preferences, mathematical ability and firmly held beliefs as well as on the amount of detail presented in a problem Secondly, small group learning environments based on a constructivist approach were found to influence student ability to model calculus word problems in a positive manner, provided there is teacher support to overcome cognitive obstacles. Finally, it was established that an effective teaching package could be developed to assist ESL students in calculus learning. The teaching package's evaluation highlighted the need for matching language use in problem presentation with the current mathematical language register of each student.

The purpose of this study was to determine eighth grade students&rsquo / skills of translating among different representations / graphic, table, equation, and verbal sentence / of algebraic concepts. Moreover, it was also aimed to investigate if there is any gender difference regarding the translation skills of students translating multiple representations, and their most common errors in making these translations. For data collection, 18 schools were selected randomly from 103 elementary schools in &Ccedil / ankaya district of Ankara. Then all of the eighth grade students in each school were selected as sample. In total 705 eighth grade students were participated in the study. To assess students&rsquo / translation skills &ldquo / Translation among different representations of algebraic concepts test&rdquo / (TADRACT) was developed by researcher. Descriptive statistics were obtained to understand students&rsquo / achievement in translation process. To compare mean scores of female and male students, the statistical analysis of Independent Samples t-test was used. Every question were examined in detail to determine any misconceptions, and most frequent errors students made in translating among different algebraic representations. The results of test indicated that 8th grade students had poor skill in translations of four different representations / verbal statement, equation, table, graphic / in algebraic concepts. There was no significant difference between mean scores of girls and mean scores of boys. The most problematic translations were from other representations / equation, table, graphic / to verbal statement, and translations from other three representations / verbal statement, equation, graphic / to table were the easiest translations.

The purpose of this study was to advance knowledge about the relationship between graphing calculator use and classroom norm development. An interpretive case study design incorporating qualitative and quantitative research methods was used to explore the question: What happens when an exemplary teacher uses graphing calculators in a college algebra class? The purposively selected participants were the teacher and eleven students of a Texas community college algebra course. All 29 classes of the 14-week spring 2006 semester were observed in their entirety by the researcher. The theoretical frameworks guiding the study were the affective representation system and the Multiple Representations Model of Learning and Teaching with the use of the Mathematics and Science Classroom Observation System for data collection, analysis, and profiling of classroom lessons. Originally developed for grades K-12, the use of the instrument was extended to college algebra. Triangulation of data sources using constant comparative and content analysis methods were used to support the three major findings: (1) The instructor's proactive orchestration of specialized instruction, support materials, and designed activities contributed to the establishment of graphing calculator use as an essential part of classroom norms and promoted students' independent use of the tool; (2) The dynamic and interactive features of the TI-84 Plus graphing calculator facilitated the delivery of instruction at high cognitive levels during student interactive activities providing access to, exploration of, and use of multiple representations for some mathematical concepts and solutions not easily attainable using traditional methods; and (3) Although the majority of students had never used a graphing calculator before the course, all students used the tool at appropriate times during instructional activities, self-reporting that their use of the calculator was generally beneficial for enhancing their understanding of lessons and supporting class interactions. Additionally, all students independently chose to use the calculator during major assessments and reported knowledgeable use of the tool to facilitate improved test performance. Replication of the study is limited because the norms developed in this case are unique to the teacher and students who negotiated their establishment. Suggestions are given regarding educational policies, reform practices, and research extensions.

Calculus at the university level is taken by thousands of undergraduate students each year. However, a significant number of students struggle with the subject, resulting in poor problem solving, low achievement, and high failure rates in the calculus courses overall (e.g., Kaput, 1994; Szydlik, 2000; Tall, 1985; Tall & Ramos, 2004; White & Mitchelmore, 1996). This is cause for concern as the lack of success in university calculus creates further barriers for students who require the course for their programs of study. This study examines this issue from the perspective of Tall’s Three Worlds of Mathematics (Tall, 2004a, 2004b, 2008), a theory of mathematics and mathematical cognitive development. A fundamental argument of Tall’s theory suggests that connecting between the different mathematical worlds, named the Embodied-Conceptual, Symbolic-Proceptual, and Formal-Axiomatic worlds, is essential for full cognitive development and understanding of mathematical concepts. Working from this perspective, this research examined, through the use of calculus task questions and semi-structured interviews, how fifteen undergraduate calculus students made connections between the different mathematical worlds for the calculus topics of limits and derivatives. The analysis of the findings suggests that how the students make connections can be described by eight different Response Categories. The study also found that how the participants made connections between mathematical worlds might be influenced by the type of questions that are asked and their experience in calculus courses. I infer that these Response Categories have significance for this study and offer potential for further study and educational practice. I conclude by identifying areas of further research in regards to calculus achievement, the Response Categories, and other findings such as a more detailed study of the influence of experience.

The purpose of this study was to investigate the effects of multiple representations-based instruction on seventh grade students&amp / #8217 / algebra performance, attitudes toward mathematics, and representation preference compared to the conventional teaching. Moreover, it was aimed to find out how students use multiple representations in algebraic situations and the reasons of preferring certain modes of representations. The study was conducted in four seventh grade classes from two public schools in Ankara in the 2003-2004 academic year, lasting eight weeks. For assessing algebra performance, three instruments called algebra achievement test, translations among representations skill test, and Chelsea diagnostic algebra test were used. To assess students&amp / #8217 / attitudes towards mathematics, mathematics attitude scale, to determine students&amp / #8217 / representation preferences before and after the treatment representation preference inventory were administered. Furthermore, as qualitative data, interview task protocol was prepared and interviews were carried out with the students from experimental and control classes. The quantitative analyses were conducted by using multivariate covariance analyses. The results revealed that multiple representations-based instruction had a significant effect on students&amp / #8217 / algebra performance compared to the conventional teaching. There was no significant difference between the experimental and control groups in terms of their attitudes towards mathematics. The chi square analyses revealed that treatment made a significant contribution to the students&amp / #8217 / representation preferences. The results of the interviews indicated that the experimental group students used variety of representations for algebra problems and were capable of using the most appropriate one for the given algebra problems.