The Tabular Mode: Not Just Another Way to Represent a Function

Understanding mathematical functions as systematic processes involving the covariation of related variables is foundational in learning mathematics. In this article, findings are reported from two investigations examining students' thinking processes with functions. The first study focused on seven middle school students' explorations with a dynamic physical model. Students were videotaped during the 20‐ to 45‐minute sessions occurring two or three times per week over a period of 2 months, and students' written work was collected. The second investigation included 19 preservice elementary and middle school teachers enrolled in a course focusing on a combination of mathematical content and pedagogy. Participants' written problem‐solving work and reflective writing were collected, and participants were individually interviewed in 50‐minute videotaped sessions. Results from both investigations indicated that students often relied on a table, or some variation of a table, as a cognitive link advancing the development of their reasoning about underlying function relationships.     

A false belief about fractions – What is its source?

This paper presents the results of interviews with 174 participants solving a problem of elementary mathematics, connected with the part–whole approach to fractions. The motive for the investigation was a specific kind of difficulty observed during a case study conducted to verify the elementary school student's understanding of the concept of fractions. The authors decided to examine the problem in a broader population of mathematics learners at different levels of education: from elementary school to university students and graduates of science majors. Approximately 65% of respondents reported the wrong answer immediately after reading the fraction problem taken from the fourth grade of elementary school. Detailed analysis of the respondents’ performance showed that the source of many wrong answers was a false belief about fractions: The only way to get 1/n of a given whole is to divide this whole into n equal parts, not yet described in educational literature.     

Effects of variation theory approach in teaching and learning of algebra on urban and rural students’ algebraic achievement and motivation

This study investigates the effect of utilizing variation theory approach (VTA) on students' algebraic achievement and their motivation in learning algebra. The study used quasi-experimental non-equivalent control group research design. It involved 114 Form Two students in four intact classes (two classes were from an urban school, another two classes from a rural school). The first group of students from each school learnt algebra in class which used the VTA, while the second group of students in each school learnt algebra through conventional teaching approach. Two-way analysis of covariance and two-way multivariate analysis of variance (MANOVA) were used to analyse the data collected. The result of this study indicated that the use of VTA has significant effect on both urban and rural students' algebraic achievement. There were evidences that VTA has significant effect on rural VTA students' overall motivation in its five subscales: attention, relevance, confidence, satisfaction and interest but it was not so for urban VTA students' motivation. This study provides further empirical evidence that utilization of variation theory as pedagogical guide can promote the teaching and learning of Form Two Algebra topics in urban and rural secondary school classrooms.     

The finite-to-finite strand of a learning progression for the concept of function: A research synthesis and cognitive analysis

In this paper we report validation efforts around the finite-to-finite strand of a provisional learning progression (LP) for the concept of function. We regard an LP as an empirically-verified account of how student understandings form over time and in response to instruction. The finite-to-finite strand of the LP was informed by literature on students’ thinking and learning related to functions as well as the Algebra Project’s curricular approach, which is designed for students who are traditionally underserved by mathematics education. Developing and validating an LP is a multi-step, cyclic process. Here we report on one step in this process, an item and response analysis. Data sources include 680 students’ responses to 13 multipart computer-delivered tasks. Results suggest that revisions to the items, associated scoring rubrics, and in some instances the LP are warranted. We illustrate this task, rubric, and LP revision process through an item analysis for a selected task.     

One-step and multi-step linear equations: a content analysis of five textbook series

Being able to solve one-step and multi-step linear equations is a hallmark of algebraic proficiency in the United States and abroad. The purpose of this paper is to report on a textbook comparison study regarding the treatment of linear equations in five textbook series that are popular in the United States: Center for Mathematics Education Project Algebra 1 and Algebra 2 (CME), Core-Plus Mathematics Program Courses 1–3 (CPMP), Glencoe Algebra 1 and Algebra 2, Interactive Mathematics Program Years 1–3 (IMP), and University of Chicago School Mathematics Project Algebra and Advanced Algebra (UCSMP). Data are reported for the following curriculum variables: content, cognitive behavior, real-world context, and tools (technology and manipulatives). Sequencing of the content, and links between symbolic and functions-based approaches are discussed. Based on the data presented, what students experience relative to setting up and solving one-step and multi-step linear equations is likely to be markedly different, depending on which textbook is used in their classrooms. Implications for practice and ideas for further research are discussed.     

Factors Influencing Prospective Teachers’ Recommendations to Students: Horizons,Hexagons, and Heed

This article examines pre-service secondary school teachers’ responses to a learning situation that presented a student's struggle with determining the area of an irregular hexagon. Responses were analyzed in terms of participants’ evoked concept images as related to their knowledge at the mathematical horizon, with attention paid toward the influence of one on the other. Specifically, our analysis attends to common features in participants’ understanding of the mathematical task, and explores the interplay between participants’ personal solving strategies and approaches and their identified preferences when advising a student. We conclude with implications for mathematics teacher education research and pedagogy.     

Detecting Students' Experiences of Discontinuities Between Middle School and High School Mathematics Programs: Learning During Boundary Crossing

Transitions from middle school to high school mathematics programs can be problematic for students due to potential differences between instructional approaches and curriculum materials. Given the minimal research on how students experience such differences, we report on the experiences of two students as they moved out of an integrated, problem-based mathematics program in their middle school into a high school mathematics program that emphasized procedural fluency. We conducted an average of two interviews per year for two and a half years with participants and engaged in participant-observation at their high school. In this study, we illustrate an analytic process for detecting discontinuities between settings from participants' perspectives. We determined that students experienced a discontinuity if they reported meaningful differences between settings (frequent mention, in detail, with emphasis terms) and concurrently reported a change in attitude. Additionally, these students' experiences illustrate two opportunities to learn during boundary-crossing experiences: identification and reflection.     

A comparative analysis of linear functions in Korean and American standards-based secondary textbooks

This paper compares sections on functions and linear functions from two Korean textbooks and an American standards-based textbook (University of Chicago School Mathematics Project [UCSMP] Algebra) to understand differences and similarities among these textbooks through horizontal and vertical analyses. We found that these textbooks provide different opportunities to learn (OTL). UCSMP Algebra places strong emphasis on real-life applications of linear functions rather than on pure mathematics and mathematical algorithms. Also, compared to Korean textbooks, UCSMP Algebra offers more OTL for students to solve, explain, and reason about higher level cognitively demanding mathematics problems than Korean secondary textbooks. Contradictory results, compared to previous studies about East Asian mathematics textbooks indicate the need for further study to compare secondary textbooks from more East Asian countries. Also, with the results of this study, we need to understand the results of international assessments more carefully.     

Improving Science Teaching for All Students

An inservice program designed to enhance the knowledge and skills of elementary school teachers with respect to science content, effective teaching strategies, and gender equity was implemented as a semester-long course. During the course, teachers explored new science content in chemistry and physics and then collaboratively developed lesson plans from it based on hands-on, discovery-centered learning, enmeshed in strategies that could maximize female student interest and participation in science. Teachers tried out their lessons between course sessions in their own classrooms and then collaboratively reflected on their progress and problems in subsequent sessions. Program results were positive for both teachers and students. Teachers reported significant increases in both their level of knowledge of and their confidence in teaching chemistry and physics concepts, as well as in their knowledge of strategies for addressing gender inequities. Project students' attitudes, particularly those of the girls, improved for some dimensions, remained stable for others, and declined for one; the girls also increased their level of active participation in science activities. Overall, the project seems to have had a positive impact on science teaching content and pedagogy, and on student (especially girls') interest and active participation in science.     

Mathematical modelling in engineering: an alternative way to teach Linear Algebra

Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic classroom approach in which students modelled real-world problems and turn gain a deeper knowledge of the Linear Algebra subject. Considering that most students are digital natives, we use the e-portfolio as a tool of communication between students and teachers, besides being a good place making the work visible. In this article, we present an overview of the design and implementation of a project-based learning for a Linear Algebra course taught during the 2014–2015 at the ‘ETSEIB'of Universitat Politècnica de Catalunya (UPC).     

Integrating learning theories and application-based modules in teaching linear algebra

The research team of The Linear Algebra Project developed and implemented a curriculum and a pedagogy for parallel courses in (a) linear algebra and (b) learning theory as applied to the study of mathematics with an emphasis on linear algebra. The purpose of the ongoing research, partially funded by the National Science Foundation, is to investigate how the parallel study of learning theories and advanced mathematics influences the development of thinking of individuals in both domains. The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted, in some students, a rich understanding of both domains and that had a mutually reinforcing effect. Furthermore, there is evidence that the deeper insights will contribute to more effective instruction by those who become high school math teachers and, consequently, better learning by their students. The courses developed were appropriate for mathematics majors, pre-service secondary mathematics teachers, and practicing mathematics teachers. The learning seminar focused most heavily on constructivist theories, although it also examined socio-cultural and historical perspectives. A particular theory, Action-Process-Object-Schema (APOS) [10], was emphasized and examined through the lens of studying linear algebra. APOS has been used in a variety of studies focusing on student understanding of undergraduate mathematics. The linear algebra courses include the standard set of undergraduate topics. This paper reports the results of the learning theory seminar and its effects on students who were simultaneously enrolled in linear algebra and students who had previously completed linear algebra and outlines how prior research has influenced the future direction of the project.     

Science as a Learner and as a Teacher: Measuring Science Self‐Efficacy of Elementary Preservice Teachers

Academic science achievement of U.S. students has raised concerns regarding our ability as a nation to compete in a global economy. Additionally, research has shown that many elementary teachers have weak science content backgrounds and had poor/negative experiences as students of science, resulting in a lack of confidence regarding teaching science. However, efforts to increase science self‐efficacy (SE) in preservice teachers can help to combat these issues. This study looked at a sample of preservice elementary teachers engaged in a semester‐long science content course, using Bandura's concept of SE as a conceptual framework. Our quantitative data showed significant increases in science SE on both subscales (personal efficacy and outcome expectancy). Our qualitative data showed that students communicated an increased sense of confidence with regard to the discipline of science. In addition, students reported learning science pedagogy through the instructor's modeling. Combining our findings resulted in several meta‐inferences, one of which showed students growing as both confident learners of science and teachers of science simultaneously. We created a construct new to the literature to describe this phenomenon: “teacher‐learner,” for students are both learning science and learning to teach science simultaneously through the content course experience, resulting in increased science SE.     

Epistemological beliefs concerning the nature of mathematics among teacher educators and teacher education students in mathematics

Beliefs constitute a central part of a person’s professional competences as beliefs are crucial to the perception of situations and as they influence our choice of actions. The present article focuses on epistemological beliefs about the nature of mathematics among future teachers and their educators at university and post-university teacher-training institutions in Germany. The data reported are part of a larger sample originating from the MT21 study [supported by the National Science Foundation through a grant to W. S. Schmidt and M. T. Tatto (REC-0231886). MT21 started in 2003] which explores and compares mathematics teacher education in Bulgaria, Germany, Mexico, South Korea, Taiwan, and the United States. In this article, we examine the structure and level of beliefs concerning the nature of mathematics for teacher education students in Germany both at the beginning (n = 368) and the end of their education (n = 286) as well as their educators (n = 77) in three academic disciplines (mathematics, mathematics pedagogy and general pedagogy). In the first part of the article, the literature on epistemological beliefs and their structure will be reviewed. In the empirical part, analyses on the level and the structure of beliefs for our samples and subsamples will be presented. Relations between educators’ and students’ beliefs will be explored.     

Turkish high school students’ definitions for parallelograms: appropriate or inappropriate?

The aim of this study was to investigate the appropriateness of high school students' definitions. The participants in this study were 269 high school students from a public school in Ordu city, which is on the Black Sea coast of Turkey. The participants were asked to write their definitions with no time constraints. In the analysis of the definitions, students' ability to distinguish necessary and sufficient conditions and their ability to use appropriate mathematical terminology were taken into account. The task used in this study enabled us to mirror students' difficulties and inadequacies about their definitions of a parallelogram. The findings indicated that most of the students defined parallelogram inappropriately because they had used incomplete or incorrect statements. On the other hand, for the appropriate definitions, it was found that the number of uneconomical definitions was almost the same as the number of economical ones. At the end of the study, it was suggested that defining activities should be integrated into curriculums explicitly and should be given importance in our mathematic lessons.     

Factors contributing to academic achievement: a Bayesian structure equation modelling study

In Iran, high school graduates enter university after taking a very difficult entrance exam called the Konkoor. Therefore, only the top-performing students are admitted by universities to continue their bachelor's education in statistics. Surprisingly, statistically, most of such students fall into the following categories: (1) do not succeed in their education despite their excellent performance on the Konkoor and in high school; (2) graduate with a grade point average (GPA) that is considerably lower than their high school GPA; (3) continue their master's education in majors other than statistics and (4) try to find jobs unrelated to statistics. This article employs the well-known and powerful statistical technique, the Bayesian structural equation modelling (SEM), to study the academic success of recent graduates who have studied statistics at Shahid Beheshti University in Iran. This research: (i) considered academic success as a latent variable, which was measured by GPA and other academic success (see below) of students in the target population; (ii) employed the Bayesian SEM, which works properly for small sample sizes and ordinal variables; (iii), which is taken from the literature, developed five main factors that affected academic success and (iv) considered several standard psychological tests and measured characteristics such as ‘self-esteem’ and ‘anxiety’. We then study the impact of such factors on the academic success of the target population. Six factors that positively impact student academic success were identified in the following order of relative impact (from greatest to least): ‘Teaching–Evaluation’, ‘Learner’, ‘Environment’, ‘Family’, ‘Curriculum’ and ‘Teaching Knowledge’. Particularly, influential variables within each factor have also been noted.     

Three types of mathematical representational translations: Comparing empirical and theoretical results

Previous research has investigated the representational translation practices of high school students, high school teachers, and college preservice teachers in various mathematical contexts including linear functions. Findings from qualitative research has frequently led to new notions about participant work and understanding. Many quantitative research has investigated the degree to which some in these populations correctly perform these translations. However, it seems that only infrequently have empirical research investigated findings from qualitative studies and vice versa, and findings regarding one population are rarely compared with findings of another population. This study (a) empirically explores the frequency of success of preservice teachers (N = 80) regarding representational translations in the context of linear functions, (b) quantifies results from previous qualitative, literature‐based research regarding high school students and teachers, and (c) quantitatively compares the results. This study demonstrates that some mathematical representational translations are more difficult than others.     

Peirce the logician

This paper traces the influence of the Boolean school, and more specifically of Peirce and his students, on the development of modern logic. In the 1890s it was Schröder's Algebra derLogik that represented the state of the art. This work mentions Frege, but the quantifier notation it adopts (a variant of the modern notation) is credited to Peirce and his students O. H. Mitchell and Christine Ladd-Franklin. This notation was widely adopted; both Zermelo and Löwenheim wrote famous papers in Peirce-Schröder notation. Even Whitehead (in 1908, in his Universal Algebra) fails to mention Frege, but cites the “suggestive papers” by Mitchell and Ladd-Franklin. (Russell credits Frege, with many things, but nowhere credits him with the quantifer; if the quantifiers in Principia were devised by Whitehead, they probably come from Peirce). The aim of this paper is not to detract from our appreciation of Frege's great work, but to emphasize that its influence came largely after 1900 (after Russell pointed out its significance). Although Frege discovered the quantifier in 1879 and Peirce's student Mitchell independently discovered it only in 1883, it was Mitchell's discovery (as modified and disseminated by Peirce) that made the quantifier part of logic. And neither Löwenheim's theorem nor Zermelo set-theory depended on Frege's work at all, but only on the work of the Boole-Peirce school.     

A Locally Based Science Mentorship Program For High Achieving Students: Unearthing Issues That Influence Affective Outcomes

Little is known about the impact of university-situated science mentoring programs on the affect of high achieving high school students, and few science mentorship programs have been described. This study describes a university-based summer science mentorship program designed to offer participants a challenging science research experience and identifies issues that influence the affective outcomes for participants. Interview data was collected from eight participants at the 2nd and 6th weeks of participation. Student responses were summarized and iteratively searched to identify patterns within the responses for each interview. It was found that mentors played a crucial role in framing the participants' experience and influencing their affect and that careful selection and timing of research projects was critical to participants' attitude of learning and accomplishment. The implications for science mentorship program developers and for research are discussed.     

Motivating the concept of eigenvectors via cryptography

New methods of teaching linear algebra in the undergraduatecurriculum have attracted much interest lately. Most of thiswork is focused on evaluating and discussing the integrationof special computer software into the Linear Algebra curriculum.In this article, I discuss my approach on introducing the conceptof eigenvectors and eigenvalues, which I have used for the last3 years in my Linear Algebra course. I offer some examples onhow I have attracted the interest of our students via Hill ciphering,a method of cryptography. After emphasizing the effect of alinear transformation in a vector space and the importance ofeigenvectors, I show how students’ motivation and understandingtowards one of the abstract concepts in Linear Algebra; eigenvaluesand eigenvectors have grown positively.