From Concrete to Abstract: Teaching for Transfer of Learning when Using Manipulatives

One of the most important uses of manipulatives in a classroom is to aid a learner to make connection from tangible concrete object to its abstraction. In this paper we discuss how teacher educators can foster deeper understanding of how manipulatives facilitate student learning of math concepts by emphasizing the connection between concrete objects and math symbolization with, preservice elementary teachers, the future implementers of knowledge. We provide an example and a model, with specific steps of how teacher educators can effectively demonstrate connections between concrete objects and abstract math concepts.

The Learning of the Mathematical Models to Aid Decision Making in the High Level Engineering Schools?

At present, in the University curricula in most countries, the decision theory and the mathematical models to aid decision making is not included, as in the graduate program like in Doctored and Master´s programs. In the Technical School of High Level Agronomic Engineers of the Technical University of Madrid (ETSIA-UPM), the need to offer to the future engineers training in a subject that could help them to take decisions in their profession was felt. Along the life, they will have to take a lot of decisions. Ones, will be important and others no. In the personal level, they will have to take several very important decisions, like the election of a career, professional work, or a couple, but in the professional field, the decision making is the main role of the Managers, Politicians and Leaders. They should be decision makers and will be paid for it. Therefore, nobody can understand that such a professional that is called to practice management responsibilities in the companies, does not take training in such an important matter. For it, in the year 2000, it was requested to the University Board to introduce in the curricula an optional qualified subject of the second cycle with 4,5 credits titled " Mathematical Methods for Making Decisions ". A program was elaborated, the didactic material prepared and programs as Maple, Lingo, Math Cad, etc. installed in several IT classrooms, where the course will be taught. In the course 2000-2001 this subject was offered with a great acceptance that exceeded the forecasts of capacity and had to be prepared more classrooms. This course in graduate program took place in the Department of Applied Mathematics to the Agronomic Engineering, as an extension of the credits dedicated to Mathematics in the career of Engineering.

Graphical Approach to Teaching Compound Interest in High School Math Classes

Report published in the Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, May, 2015

A study of the relationship between educational placement and the achievement of urban low socioeconomic Hispanic middle school students with and without specific learning disabilities

Public schools traditionally have been held accountable for educating the majority of the nation’s school children, and through the years, these schools have been evaluated in a variety of ways. Currently, evaluation measures for accountability purposes consist solely of standardized test scores. In the past, only test scores of general education students were analyzed. Laws governing the education of students with disabilities, however, have extended accountability measures not only to include those students, but to report their scores in a disaggregated form (No Child Left Behind Act, 2001). The recent emphasis on accountability and compliance has resulted in the need for schools to carefully examine how programs, services, and policies impact student achievement (Bowers & Figgers, 2003). ^ Standard-based school reform and accountability systems have raised expectations about student learning outcomes for all students, including those with disabilities and minority students. Yet, overall, racial/ethnic minority students are performing well below their White non-Hispanic peers in most academic areas. Additionally, with respect to special education, there exists an enduring problem of disproportionate representation of racial/ethnic minority students (National Research Council, 2000). ^ This study examined classroom placement (inclusive versus non-inclusive) relative to academic performance of urban, low socioeconomic Hispanic students with and without disabilities in secondary content area classrooms. A mixed method research design was used to investigate this important issue using data from a local school district and results from field observations. The study compared performance levels of four middle school Hispanic student subgroups (students with disabilities in inclusive settings, students without disabilities in inclusive settings, students with disabilities in resource settings, and student without disabilities in general education settings) each in their respective placements for two consecutive years, exploring existing practices within authentic settings. ^ Significant differences were found in the relationship of educational placement and achievement between grade level and disability in the areas of math and reading. Additionally, clear and important differences were observed in student-teacher interactions. Recommendations for further researchers and stakeholders include soliciting responses from teams at the schools composed of general education and special education teachers, administrative personnel, and students as well as broadening the study across grade levels and exceptionalities. ^

The relationship between educational placement, instructional practices, and achievement gains of Black students with specific learning disabilities in secondary urban school settings

Black students, in general, are underserved academically (Darling-Hammond, 2000; Townsend, 2002) and overrepresented in special education (Donovan & Cross, 2002). Black students with disabilities are further overrepresented in more restrictive educational environments (Skiba, Poloni-Staudinger, Gallini, Simmons & Feggins-Azziz, 2006). Although the National Longitudinal Transition Study 2 (NLTS2) revealed that the academic performance of students with learning disabilities is positively related to the percentage of courses taken in the general education setting (Newman, 2006), the research specifically on placement of Black students with disabilities, particularly at the secondary level, as it relates to academic achievement is lacking. While previous studies have sought to determine which placement is better for students with disabilities, no study was found that specifically examined the impact of placement specific to Black students with specific learning disabilities (SLD) in urban settings (Fore, III, Hagan-Burke, Burke, Boon & Smith, 2008; Rea, McLaughlin & Walther-Thomas, 2002). This study examined educational placement, instructional best practices, and achievement gains of Black students with SLD in urban secondary settings using an ex post facto research design. Achievement, placement, and demographic data were collected and analyzed on approximately 314 Black eighth grade students with SLD. The Teacher Instructional Practices Survey was developed and used to collect and analyze data from the teachers of 78 of these students as it relates to instructional best practices. Results indicate no significant difference in reading but a significant difference in math gains of students served in inclusive settings as compared to resource settings with a small effect size. Also, no significant relationship was found between achievement gains and the reported use of instructional best practices. However, there was a relationship between educational placement and the use of instructional best practices. The results implied that there is a need for training with both general and special education teachers on instructional best practices for SWD and that there should be certain IEP team considerations when making placement decisions for this population of students with disabilities. It is recommended that future research in this area include classroom observations and factors other than test scores to measure growth in achievement.

Metacognitive Strategies and Learning

During the past decade, metacognition has been identified not only as a component of cognition but also as an important factor in learning. This practitioner proposes that educators and educational researchers should focus on the development and implementation of metacognitive learning strategies. The existing metacognitive studies have concentrated on several areas. One area centers on the continuing efforts to identify all the elements of metacognition. Another area concentrates on the roles that metacognition plays in specific learning behaviors that occur at various ages and levels of complexity. The third area investigates the relationships of metacognition to specific content areas of learning by focusing on the effects of metacognitive learning strategies. The most common areas of study have been reading comprehension, math skills, writing skills, and applying metacognitive strategies to learn various subjects using the computer. Directly or indirectly, the existing studies relate to the expanding applications of the relationships and relevancies of metacognition to learning. Considerable evidence confirms that when students use metacognitive strategies they often experience a higher level of learning. This practitioner believes that experiencing higher levels of learning gives students the confidence they need to construct knowledge which promotes lifelong learning.

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

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

Math 412 - Topology with Applications

Highlights of Data Expedition: • Students explored daily observations of local climate data spanning the past 35 years. • Topological Data Analysis, or TDA for short, provides cutting-edge tools for studying the geometry of data in arbitrarily high dimensions. • Using TDA tools, students discovered intrinsic dynamical features of the data and learned how to quantify periodic phenomenon in a time-series. • Since nature invariably produces noisy data which rarely has exact periodicity, students also considered the theoretical basis of almost-periodicity and even invented and tested new mathematical definitions of almost-periodic functions. Summary The dataset we used for this data expedition comes from the Global Historical Climatology Network. “GHCN (Global Historical Climatology Network)-Daily is an integrated database of daily climate summaries from land surface stations across the globe.” Source: https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/ We focused on the daily maximum and minimum temperatures from January 1, 1980 to April 1, 2015 collected from RDU International Airport. Through a guided series of exercises designed to be performed in Matlab, students explore these time-series, initially by direct visualization and basic statistical techniques. Then students are guided through a special sliding-window construction which transforms a time-series into a high-dimensional geometric curve. These high-dimensional curves can be visualized by projecting down to lower dimensions as in the figure below (Figure 1), however, our focus here was to use persistent homology to directly study the high-dimensional embedding. The shape of these curves has meaningful information but how one describes the “shape” of data depends on which scale the data is being considered. However, choosing the appropriate scale is rarely an obvious choice. Persistent homology overcomes this obstacle by allowing us to quantitatively study geometric features of the data across multiple-scales. Through this data expedition, students are introduced to numerically computing persistent homology using the rips collapse algorithm and interpreting the results. In the specific context of sliding-window constructions, 1-dimensional persistent homology can reveal the nature of periodic structure in the original data. I created a special technique to study how these high-dimensional sliding-window curves form loops in order to quantify the periodicity. Students are guided through this construction and learn how to visualize and interpret this information. Climate data is extremely complex (as anyone who has suffered from a bad weather prediction can attest) and numerous variables play a role in determining our daily weather and temperatures. This complexity coupled with imperfections of measuring devices results in very noisy data. This causes the annual seasonal periodicity to be far from exact. To this end, I have students explore existing theoretical notions of almost-periodicity and test it on the data. They find that some existing definitions are also inadequate in this context. Hence I challenged them to invent new mathematics by proposing and testing their own definition. These students rose to the challenge and suggested a number of creative definitions. While autocorrelation and spectral methods based on Fourier analysis are often used to explore periodicity, the construction here provides an alternative paradigm to quantify periodic structure in almost-periodic signals using tools from topological data analysis.

The background impact in HEI math lectures: an experience in math in the 1st year of Accounting and Management Degree

It is a fact, and far from being a new one, that students have been entering Higher Education courses with many different backgrounds in terms of secondary school programs they attended. The impact of these basic skills is a general and worldwide challenge, fundamentally when facing some specific “constructive” subjects like foreign languages and Mathematics. Working with students with an extensive variety of Math qualifications is an outrageous challenge when they enter an advanced Math course, leading to an almost generalized expectations’ failure - from students enrolled in course and from their teachers, who feel powerless in trying to monitor knowledge construction from completely different “starting points”. If teachers’ "haste" is average, more than half of the students do not “go along” and give up, even before experiencing any kind of evaluation procedure. On the contrary, if the “speed” is too low, others are discouraged (feeling not progressing at all) and the teacher runs the risk of not meeting the minimum objectives (general and specific) of its course, which may have a negative impact on students’ future training development. Failure in Mathematics, despite being a recurrent and global issue, does not have any “magical solution”, however, in general, teachers in this area seem untiring, searching, investigating, trying and implementing new and old “recipes” to tackle and demystify this subject. In this article we describe a project developed in a Math course, with the first year students from an Accounting and Management bachelor degree, and its outcomes since it was brought to practice, revealing its impact in students’ success, from approval to dropout rates, in this course. We will shortly describe students’ differentiated Math backgrounds, their results in a pre-assessment analysis and how we try to deal with these differences and level them up, having in mind the same “finish line”. One should never forget that all these students where officially accepted in higher education institutions, so they are ones’ reality, the reality of institutions whose name one should value and strive to defend.

The development of an eBook for teaching and learning mathematics

Mobile devices, smartphones, phablets and tablets, are widely avail‐ able. This is a generation of digital natives. We cannot ignore that they are no longer the same students for which the education system was designed tradition‐ ally. Studying math is many times a cumbersome task. But this can be changed if the teacher takes advantage of the technology that is currently available. We are working in the use of different tools to extend the classroom in a blended learning model. In this paper, it is presented the development of an eBook for teaching mathematics to secondary students. It is developed with the free and open standard EPUB 3 that is available for Android and iOS platforms. This specification supports video embedded in the eBook. In this paper it is shown how to take advantage of this feature, making videos available about lectures and problems resolutions, which is especially interesting for learning mathematics.

Transformations of teaching and learning through digital technologies

This chapter is a critical synthesis of research related to the transformations that take place when digital technologies are incorporated into teaching and learning practices. In developing this synthesis, research from all levels of education was reviewed with a focus on the opportunities digital technologies offer for cognitive, pedagogical, affective and professional change. The chapter is structured in alignment with Pierce and Stacey’s (Pierce and Stacey, Int J Comput Math Learn 15(1):1–20 2010) map of pedagogical opportunities in which three dimensions for educational transformation were identified: tasks, classroom, and subject. A discussion of future directions for research into technology enhanced mathematics education concludes the review.