"The purpose of the study was to determine whether there is a difference in the ability of two equally capable groups of subjects to generalize the winning strategy of a simple game when one group learns the perfect strategy for one form of the game by the discovery method and the other group learns it by reading an explanation of the strategy"--Introduction. / Typescript. / "August, 1970." / "Submitted to the Department of Mathematics Education of Florida State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: E. D. Nichols, Professor Directing Dissertation. / Includes bibliographical references.

This experimental study was intended to examine whether game-based learning (GBL) that encompasses four particular game characteristics (challenges, a storyline, rewards, and the integration of game-play with learning content) in the OpenSimulator-supported virtual reality (VR) learning environment can improve mathematical achievement and motivation for elementary school students toward math learning. In this pre- and post-test experimental comparison study, data were collected from 132 fourth graders through an achievement test, and a Short Instructional Materials Motivational Survey (SIMMS). The same tasks were provided to the experimental and control groups. Tasks for the experimental group involved the following four game characteristics: (1) challenges, (2) a storyline, (3) rewards, and (4) the integration of game-play with learning content. The control group was given the same tasks and learning environment setting (OpenSimulator-supported VR) that was used for the experimental group. The exception was that the control group tasks did not include the game characteristics: (1) challenges, (2) a storyline, (3) rewards, and (4) the integration of game-play with learning content. Analysis of covariance (ANCOVA) using a treatment (treatment vs. control) on the achievement indicated a significant effect of GBL in the VR environment on math knowledge test performance. For motivation, the results indicated that there was no significant difference on the post-test scores for the perceived motivational quality of the learning activity (MQLA) between the experimental group and the control group. / A Dissertation submitted to the Department of Educational Psychology and Learning Systems in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2015. / September 15, 2015. / fractions, Game-Based Learning, math performance, OpenSim, Real-life application skills, Virtual Reality / Includes bibliographical references. / Fengfeng Ke, Professor Directing Dissertation; Young-Suk Kim, University Representative; Allan Jeong, Committee Member; Insu Paek, Committee Member.

This study looks at the use of integrated mathematics and music lessons at the high school level. Four lessons were taught by the researcher in both a research and a control class to determine how mathematically motivated music instruction affects students understanding of operations of functions, composition of functions, inverse functions and domain and range. A pretest-posttest was used to determine the effect of these lessons and a questionnaire was used to identify differences between groups and to help determine the effect of musical applications of mathematics on students’ mathematical perceptions, self-efficacy and grit. The pretest-posttest included both a standard mathematics section and a section involving non-musical applications. A gain score approach using independent sample t tests was used to determine the impact of the integrated instruction. The research group demonstrated significantly greater gains both overall and on the applications portion of the exam. Additional qualitative analysis was done to determine how the posttests differed between groups. Three major differences were identified: the research group used function notation more frequently than the control group, the control group demonstrated confusion between composition of functions and inverse functions while the research group did not and the research group showed more mathematical work for the applications portion of the exam than the control group. Qualitative analysis was also done to identify trends in the questionnaire data. Among the major differences between groups was the increased willingness to work with mathematical applications in the future by the research group compared to the control group. The integrated instruction led to comparable and in some cases significantly better mathematics outcomes than the control group and led students to an increased willingness to work with mathematical applications both on the posttest and moving forward.

by Wong Ka-Ming. / Thesis (M.A.Ed.)--Chinese University of Hong Kong, 1989. / Bibliography: leaves 164-173.

Can action support cognition? Can direct touch support performance? Embodied interaction involving digital devices is based on the theory of grounded cognition. Embodied interaction with gestural interfaces involves more of our senses than traditional (mouse-based) interfaces, and in particular includes direct touch and physical movement, which are believed to help retain the knowledge that is being acquired. There is growing evidence that spontaneous gestures affect thought and possibly learning. The author was interested to explore whether designed gestures (for gestural interfaces) affect thought. It was hypothesized that the use of congruent gestures helps construct better mental representations and mental operations to solve problems (Gestural Conceptual Mapping). There is also evidence that physical manipulation of objects can benefit cognition and learning; it was therefore also hypothesized that manipulating objects through direct touch on the screen supports performance. These hypotheses were addressed by observing children's performance in arithmetic and numerical estimation. Arithmetic is a discrete task, and should be supported by discrete rather than continuous actions. Estimation is a continuous task, and should be supported by continuous rather than discrete actions. Children used either a gestural interface (multi-touch, e. g., iPad) or a traditional mouse interface. The actions either mapped congruently to the cognition (continuous action for estimation and discrete action for arithmetic), or not. If action supports cognition, children who use continuous actions for estimation or discrete actions for addition should perform better than children for whom the action-cognition mapping is less congruent. In addition, if manipulating the objects by touching them directly on the screen could yield a better performance, children who use a touch interface should perform better than children who use a mouse interface. The results confirmed the predictions.

This dissertation traces the history of the teaching of elementary trigonometry in United States colleges and universities from 1776 to 1900. This study analyzes textbooks from the eighteenth and nineteenth centuries, reviews in contemporary periodicals, course catalogs, and secondary sources. Elementary trigonometry was a topic of study in colleges throughout this time period, but the way in which trigonometry was taught and defined changed drastically, as did the scope and focus of the subject. Because of advances in analytic trigonometry by Leonhard Euler and others in the seventeenth and eighteenth centuries, the trigonometric functions came to be defined as ratios, rather than as line segments. This change came to elementary trigonometry textbooks beginning in antebellum America and the ratios came to define trigonometric functions in elementary trigonometry textbooks by the end of the nineteenth century. During this time period, elementary trigonometry textbooks grew to have a much more comprehensive treatment of the subject and considered trigonometric functions in many different ways. In the late eighteenth century, trigonometry was taught as a topic in a larger mathematics course and was used only to solve triangles for applications in surveying and navigation. Textbooks contained few pedagogical tools and only the most basic of trigonometric formulas. By the end of the nineteenth century, trigonometry was taught as its own course that covered the topic extensively with many applications to real life. Textbooks were full of pedagogical tools. The path that the teaching of trigonometry took through the late eighteenth and nineteenth centuries did not always move in a linear fashion. Sometimes trigonometry education stayed the same for a long time and then was suddenly changed, but other times changes happened more gradually. There were many international influences, and there were many influential Americans and influential American institutions that changed the course of trigonometry instruction in this country. This dissertation follows the path of those changes from 1776 to 1900. After 1900, trigonometry instruction became a topic of secondary education rather than higher education.

How do children use physical and virtual tools to develop new numerical knowledge? While concrete instructional materials may support the delivery of novel information to learners, they may also over-simplify the task, unintentionally reducing learners' performance in recall and transfer tasks. This reduction in testing performance may be mitigated by embedding physical incongruencies in the design of instructional materials. The effort of resolving this incongruency can foster a richer understanding of the underlying concept. In two experiments children were trained on a computerized number line estimation task, with a novel scale (0-180), and then asked to perform a series of posttest number line estimation tasks that varied spatial features of the training number line. In experiment 1, during training with feedback, children either received a ruler depicting endpoint and quartile magnitudes (i.e., 0, 45, 90, 135, 180) that physically matched the on-screen number line (congruent ruler), a proportionally-similar ruler scaled 33% larger than the on-screen number line (incongruent ruler), or no ruler. Children were trained to criterion before proceeding to posttest. Results indicated that while children who used the congruent ruler performed well during training, their performance at posttest was less accurate than the other two conditions. On the other hand, by increasing the difficulty of the learning task, while providing relevant landmark information, children in the incongruent ruler condition produced the highest accuracy at posttest. In experiment 2, controlling for learning task duration, the incongruent ruler and congruent ruler conditions were compared directly. Posttest results confirmed an advantage for children in the more complex, incongruent ruler condition. These results are interpreted to suggest that landmarks representations are an important and accessible means of developing a mature numerical representation of the number line. Furthermore, the results confirm that desirable difficulties are an essential component of the learning process. Potential implications for the design of learning activities that balance instructional support with conceptual challenge are discussed.

This dissertation is a history of David Eugene Smith's collection of historical books, manuscripts, portraits, and instruments related to mathematics. The study analyzes surviving documents, images, objects, college announcements and catalogs, and secondary sources related to Smith's collection. David Eugene Smith (1860 - 1944) travelled the world in search of rare and interesting pieces of mathematics history. He enjoyed sharing these experiences and objects with his family, friends, colleagues, and students. Smith's collection had a remarkable journey itself. It was once part of the Educational Museum of Teachers College. This museum existed from 1899 - 1914 and was quite popular among educators and students. Smith was director of the museum beginning in 1909, although, he had a major influence on the museum from the moment he began his professorship at Teachers College in 1901. After the Educational Museum of Teachers College disbanded, the collection was exhibited in numerous venues. George A. Plimpton (1855 - 1936) created the Permanent Educational Exhibit that housed both modern educational items, as well as, historical pieces for display. Since Smith and Plimpton were great friends and fellow collectors, Smith's collection was included in the historical section of Plimpton's establishment. Unfortunately, due to the hard times of the world at this moment, the Permanent Educational Exhibit closed in 1917. Smith continued to exhibit his collection of mathematical artifacts through the Museums of the Peaceful Arts, founded by George F. Kunz (1856 - 1932), the New York Museum of Science and Industry, Teachers College, and Columbia University. Smith's research, teaching, and publications were directly influenced by his collection. Throughout most of his published works are images and photographs of items in his collection. He also believed in the importance of having primary sources included in mathematics education. This view he followed in his own teaching, which included research in his collection. David Eugene Smith's collection could never be replicated and thus is quite unique and valuable. Smith donated his collection to Columbia University's Libraries in the 1930s. Various exhibits of his collection have occurred since then, the most recent concluded in 2003. The history of Smith's mathematical collection is important to the history of mathematics education as it displays the importance of preserving mathematical books, manuscripts, portraits, and instruments for future generations to research.

In early childhood mathematics, strategy-use is an important indicator of children's conceptual understanding and is a strong predictor of later math performance. Strategy instruction is common in many national curricula, yet is virtually absent from most math software. The current study describes the design of one software activity teaching single-digit addition strategies. The study explores the effectiveness of the software in detecting the strategies first-graders use and teaching them to use more efficient strategies. Instead of a business-as-usual control group, the study explores the effects of one aspect of the software: the pedagogical agent, investigating whether multiple agents are more effective than a single agent when teaching about multiple strategies. The study finds that while children do not accurately report their own strategies, the software log is able to detect the strategies that children use and is particularly adept at detecting the effective use of an advanced strategy with a model that performs 67% better than chance. Overall, children improve in their accuracy, speed, and use of advanced strategies. Of the three teaching tools available to the children, the count on tool was most effective in encouraging use of an advanced strategy, highlighting a need to revise the other tools. Low-performers correctly used advanced strategies more frequently across the six sessions, while mid-performers improved after just one session and high-performers' correct use of an advanced strategy was consistent across the sessions. Whether a student saw lessons featuring a single agent or multiple agents did not have strong effects on performance. More research is needed to improve the strategy detection models, refine the tools and lessons, and explore other features of the software.

This study examines the impact of social capital on mathematics achievement in eight European cities. The study draws on data from the 2008 Youth in Europe survey, carried out by the Icelandic Center for Social Research and Analysis. The sample contains responses from 17,312 students in 9th and 10th grade of local secondary schools in the following cities: Bucharest in Romania, Kaunas, Klaípéda and Vilnius in Lithuania, Reykjavík in Iceland, Riga and Jurmala in Latvia and Sofia in Bulgaria. The study builds on social capital theory presented in 1988 by the American sociologist James Coleman. He argued that social capital in both family and community is a key factor in the creation of human capital, meaning that children that possess more social capital in their lives will do better in school. Several prior studies have empirically supported the theory, although most of those studies were carried out in the United States. The current study tests whether the theory of social capital holds across different cultures. The findings partly support the theory, showing that the key measures of social capital are positively correlated with mathematics achievement in all of the cities. The impact however was less in many of the cities than expected. Additionally, Coleman's key social capital variable did not positively associate with mathematics achievement in cities around Europe. The implications of that finding are discussed in the thesis.