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.

Latent class analysis is a useful tool to deal with discrete multivariate response data. Croon (1990) proposed the ordered latent class model where latent classes are ordered by imposing inequality constraints on the cumulative conditional response probabilities. Taking stochastic ordering of latent classes into account in the analysis of data gives a meaningful interpretation, since the primary purpose of a test is to order students on the latent trait continuum. This study extends Croon's model to ordered latent class regression that regresses latent class membership on covariates (e.g., gender, country) and demonstrates the utilities of an ordered latent class regression model in educational assessment using data from Trends in International Mathematics and Science Study (TIMSS). The benefit of this model is that item analysis and group comparisons can be done simultaneously in one model. The model is fitted by maximum likelihood estimation method with an EM algorithm. It is found that the proposed model is a useful tool for exploratory purposes as a special case of nonparametric item response models and cross-country difference can be modeled as different composition of discrete classes. Simulations is done to evaluate the performance of information criteria (AIC and BIC) in selecting the appropriate number of latent classes in the model. From the simulation results, AIC outperforms BIC for the model with the order-restricted maximum likelihood estimator.

Given the reality that active and competitive participation in the 21st century requires American students to deepen their scientific and mathematical knowledge base, the National Research Council (NRC) proposed a new conceptual framework for K-12 science education. The framework consists of an integration of what the NRC report refers to as the three dimensions: scientific and engineering practices, crosscutting concepts, and core ideas in four disciplinary areas (physical, life and earth/spaces sciences, and engineering/technology). The Next Generation Science Standards (NGSS), which are derived from this new framework, were released in April 2013 and have implications on teacher learning and development in Science, Technology, Engineering, and Mathematics (STEM). Given the NGSS's recent introduction, there is little research on how teachers can prepare for its release. To meet this research need, I implemented a self-study aimed at examining my teaching practices and classroom outcomes through the lens of the NRC's conceptual framework and the NGSS. The self-study employed design-based research (DBR) methods to investigate what happened in my secondary classroom when I designed, enacted, and reflected on units of study for my science, engineering, and mathematics classes. I utilized various best practices including Learning for Use (LfU) and Understanding by Design (UbD) models for instructional design, talk moves as a tool for promoting discourse, and modeling instruction for these designed units of study. The DBR strategy was chosen to promote reflective cycles, which are consistent with and in support of the self-study framework. A multiple case, mixed-methods approach was used for data collection and analysis. The findings in the study are reported by study phase in terms of unit planning, unit enactment, and unit reflection. The findings have implications for science teaching, teacher professional development, and teacher education.

In recent years, Mainland Chinese teams have been the dominant participants in the two COMAP-sponsored mathematical modeling competitions: the Mathematical Contest in Modeling (MCM) and the Interdisciplinary Contest in Modeling (ICM). This study examines five factors that lead to the Chinese teams' dramatic increase in participation rate and performance in the MCM and ICM: the Chinese government's support, pertinent organizations' efforts, support from initiators of Chinese mathematical modeling education and local resources, Chinese teams' preferences in selecting competition problems to solve, and influence from the Chinese National College Entrance Examination (NCEE). The data made clear that (1) the policy support provided by the Chinese government laid a solid foundation in popularizing mathematical modeling activities in China, especially in initial stages of the development of mathematical modeling activities. (2) Relevant organizations have been the main driving force behind the development of mathematical modeling activities in China. (3) Initiators of mathematical modeling education were the masterminds of Chinese mathematical modeling development; support from other local resources served as the foundation of mathematical modeling popularity in China. (4) Chinese teams have revealed a preference for discrete over continuous mathematical problems in the Mathematical Contest in Modeling. However, in general, the winning rates of these two problem types have been shown to be inversely related to their popularity — while discrete problems have traditionally had higher attempt rates, continuous problems enjoyed higher winning rates. (5) The NCEE mathematics examination seems to include mathematical application problems rather than actual mathematical modeling problems. Although the extent of NCEE influence on students' mathematical modeling ability is unclear, the content coverage suggests that students completing a high school mathematics curriculum should be able to apply what they learned to simplified real-world situations, and pose solutions to the simple models built in these situations. This focus laid a solid mathematics foundation for students' future study and application of mathematics.