Making sense of the traditional long division algorithm

This classroom scholarship report presents a group of elementary students’ experiences learning the traditional long division algorithm. The traditional long division algorithm is often taught mechanically, resulting in the student's performance of step-by-step procedures with no or weak understanding of the concept. While noting some initial difficulties, the class episodes in this article provide examples of internalization that highlight the active role of the learner in transforming concrete representations into an abstract algorithm. Several factors encouraged students to be deeply engaged in making sense of the long division algorithm: meaningful tasks based on a theoretically well-articulated curriculum, effective pedagogical measures, and dynamic class discussions.     

The Effects of a Teacher In‐Service on Low‐Achieving Grade 7 and 8 Mathematics Students

Previous research (e.g., Woodward & Baxter, 1997 ) found that Standards‐based mathematics teaching provides marginal or no benefits for low achievers, in contrast with positive effects for middle and high ability students. A randomized quasi‐experiment in 52 Canadian schools found that low achieving grade 7 and 8 students who received support consisting of placement on a learning continuum, instruction focused on their specific learning needs, and concrete materials to represent mathematical constructs, benefited from teaching that emphasized construction over transmission of knowledge. Treatment students showed small but statistically significant improvements over controls in student achievement, and controversially, in mathematical beliefs, and attitudes. The latter finding raised issues of the appropriate balance between Type I and Type II error in educational research.     

The Effects of Non‐CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta‐Analysis

Forty‐two studies comparing students with access to graphing calculators during instruction to students who did not have access to graphing calculators during instruction are the subject of this meta‐analysis. The results on the achievement and attitude levels of students are presented. The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.     

Emergent modeling: discrete graphs to support the understanding of change and velocity

In this paper we focus on an instructional sequence that aims at supporting students in their learning of the basic principles of rate of change and velocity. The conjectured process of teaching and learning is supposed to ensure that the mathematical and physical concepts will be rooted in students’ understanding of everyday-life situations. Students’ inventions are supported by carefully planned activities and tools that fit their reasoning. The central design heuristic of the instructional sequence is emergent modeling. We created an educational setting in three tenth grade classrooms to investigate students’ learning with this sequence. The design research is carried out in order to contribute to a local instruction theory on calculus. Classroom events and computer activities are video-taped, group work is audio-taped and student materials are collected. Qualitative analyses show that with the emergent modeling approach, the basic principles of calculus can be developed from students’ reasoning on motion, when they are supported by discrete graphs.     

Isetl: A programming language for learning mathematics

This paper gives a brief history of the development of an approach to help students learn mathematical concepts at the post-secondary level. The method uses ISETL, a programming language derived from SETL, to implement instruction whose design is based on an emerging theory of learning. Examples are given of uses of this pedagogical strategy in abstract algebra, calculus, and mathematical induction. © 1996 John Wiley & Sons, Inc.     

Gender Differences in Learning Constructs,Shifts in Learning Constructs,and Their Relationship to Course Achievement in a Structured Inquiry,Yearlong College Physics Course for Life Science Majors

This study investigated differences and shifts in learning and motivation constructs among male and female students in a nonmajors, yearlong structured inquiry college physics course and examined how these variables were related to physics understanding and course achievement. Tests and questionnaires measured students' learning approaches, motivational goals, self‐efficacy, epistemological beliefs, scientific reasoning abilities, and understanding of central physics concepts at the beginning and end of the course. Course achievement scores were also obtained. The findings showed that male students had significantly higher self‐efficacy, performance goals, and physics understanding compared to females, which persisted throughout the course. Differential shifts were found in students' meaningful learning approaches, with females tending to use less meaningful learning from beginning to end of the course; and males using more meaningful learning over this time period. For both males and females, self‐efficacy significantly predicted physics understanding and course achievement. For females, higher reasoning ability was also a significant predictor of understanding and achievement; whereas for males, learning goals and rote learning were significant predictors, but in a negative direction. The findings reveal that different variables of learning and motivation may be important for females' success in inquiry physics compared to males. Instructors should be cognizant of those needs in order to best help all students learn and achieve in college physics.     

Teaching and learning of inclusive and transitive properties among quadrilaterals by deductive reasoning with the aid of SmartBoard

Learning to identify Euclidean figures is an essential content of many elementary school geometry curricula. Students often learn to distinguish among quadrilaterals, for example, by categorizing their geometric properties according to two attributes, namely the length of the edges and the size of the interior angles. But knowing how to differentiate them based on their geometric properties does not necessarily help students to develop the abstract concepts of the inclusive and transitive properties among the quadrilaterals. With the aid of dynamic geometry multimedia software in SmartBoard (SB), a kind of digital whiteboard (DWB), we enhanced the teaching and learning effectiveness by the effect of “animation-on-demand” in classrooms. This is basically a dual delivery of geometric concepts by texts, narrations and words accompanied by pictures, illustrations and animations. The preliminary results of our study on 9-year-old students’ performance in tests given after three such lessons show that those students could differentiate with reasons why a square is a rhombus (inclusion) as well as a parallelogram (transitivity).     

Aligning General Chemistry Laboratory With Lecture at a Large University

The laboratory and lecture components of general chemistry are commonly offered as two separate courses, with lecture typically meeting two or three times per week and laboratory scheduled to meet only once per week. The concepts, content, and relationships presented in lecture may be disjointed and asynchronous with respect to those encountered in laboratory experiments. In addition, traditional laboratory experiments tend to be confirmation labs, in which students are aware of the “right” answer before beginning the lab. Students enrolled in a specific lecture section do not necessarily meet for the same laboratory section. As such, learning experiences in laboratory do little to help the students construct an understanding of chemical concepts, content, or relationships. The goal of this project was to develop an inquiry‐based approach to curriculum and instruction in first‐semester general chemistry at the University of North Carolina at Charlotte. A major objective of the project was to develop a laboratory curriculum that meshes intimately with lecture. This objective was accomplished by (a) creating a laboratory course that met for 80‐minutes twice a week immediately following the lecture, (b) involving students in laboratory experiments that related to the material presented during lecture, and (c) using laboratory observations and data in lecture to help students construct an understanding of chemical phenomena.     

Inquiry Teaching and Learning: The Best Math Class Study

This research reports on prospective middle school teachers' perceptions of a “best mathematics class” during their involvement in an inquiry‐designed mathematics content course. Grounded in the prestigious Glenn Commission report ( U.S. Department of Education, 2000 ), the study examined the prospective teachers' perceptions of effective mathematics instruction both prior to and after completing the inquiry course. Pre‐essay analysis revealed that students could be grouped into one of two categories: the Watch‐Learn‐Practice view and the Self as Initiator view. Post‐essay analysis indicated that over two thirds of all students involved in the study changed their views of a best math class after the inquiry courses. The Watch‐Learn‐Practice group's changes focused on developing reasoning skills and learning how one “knows” in mathematics. The Self as Initiator group noted expanded roles for the students, particularly emphasizing the importance of going beyond basic requirements to think deeply about the why and how of mathematics and expanded views of the benefits of group learning.     

Learning events in the life of a community of mathematics educators

The paper examines some characteristics of learning events of a community of mathematics educators. Participation in the community entailed gaining familiarity with agreed upon conventions, goals, and forms of communication. The case discussed herein is an attempt to convey the complexities underlying learning in such a community through (re)negotiation of practices and goals. The notion ofreflective discourse is borrowed to describe a group discussion involving collective reflection that constituted an occasion for meaningful learning.     

Factors affecting student success in a first-year mathematics course: a South African experience

In spite of sustained efforts tertiary institutions implement to try and improve student academic performance, the number of students succeeding in first-year mathematics courses remains disturbingly low. For most students, the gap between their mathematical capability and the competencies they are expected and need to develop to function effectively in these courses persists even after course instruction. In this study, an instrument for identifying and examining factors affecting student performance and success in a first-year Mathematics university course was developed and administered to 86 students. The overall Cronbach's Alpha coefficient for the questionnaire was found to be 0.916. Having identified variables from prior research known to affect student performance, factor analysis was used to identify variables exhibiting the greatest impact on student performance. The variables included prior academic knowledge, workload, student approaches to learning, assessment, student support teaching quality, methods and resources. From the analysis, students' perceptions of their workload emerged as the factor having the greatest impact on student's performance, followed by the matriculation examination score. The findings are discussed and strategies that can be used to improve teaching and contribute to student success in a first-year mathematics course in a South African context are presented.     

The Relationship of Teacher‐Facilitated,Inquiry‐Based Instruction to Student Higher‐Order Thinking

Commissions, studies, and reports continue to call for inquiry‐based learning approaches in science and math that challenge students to think critically and deeply. While working with a group of middle school science and math teachers, we conducted more than 100 classroom observations, assessing several attributes of inquiry‐based instruction. We sorted the observations into two groups based on whether students both explored underlying concepts before receiving explanations and contributed to the explanations. We found that in both math and science classrooms, when teachers had students both explore concepts before explanations and contribute to the explanations, a higher percent of time was spent on exploration and students were more frequently involved at a higher cognitive level. Further, we found a high positive correlation between the percent of time spent exploring concepts and the cognitive level of the students, and a negative correlation between the percent of time spent explaining concepts and the cognitive level. When we better understand how teachers who are successful in challenging students in higher‐order thinking spend their time relative to various components of inquiry‐based instruction, then we are better able to develop professional development experiences that help teachers transition to more desired instructional patterns.     

Teachers’ reactions to animations as representations of geometry instruction

This study looks at future and novice teachers’ responses to and perceptions of a set of animations as representations of geometry instruction. While the future teachers responded more agreeably to the animations, discussion forum and survey data provide evidence that the animations prompted both future and novice teachers to consider how pedagogical decisions affect the classroom environment and how students think about and communicate geometric ideas. Data also suggest that the animations helped both groups refresh their content knowledge.     

借助几何图形理解高等数学中的抽象概念和结论

高等数学的很多内容比较抽象,学生不易理解.通过几个例子说明如何将抽象的数学概念和结论与几何图形有机的结合起来,加深对这些概念结论的理解,激发学生的学习兴趣.     

Nontraditional College Students and the Role of Collaborative Learning as a Tool for Science Mastery

This study investigates the way collaborative learning that occurs primarily outside the classroom affects college students' understanding of science. Collaborative learning is particularly important for the increasing number of nontraditional students who have limited time available for study groups and other peer learning activities occurring outside of class time. Using a national study of 4,644 college students of various academic majors, multiple linear regression was used to identify variables that enhance science learning. Time spent in peer learning settings, such as teaching science to peers and discussing science with peers, were the strongest predictors of understanding science; moreover, this finding was consistent even for nontraditional students who reported less frequency of engagement in such activities. The study suggests that science educators can enhance learning when they structure their courses to include peer learning that engages students with each other over science issues outside the classroom.     

Learning angles through movement: Critical actions for developing understanding in an embodied activity

Angle instruction often begins with familiar, real-world examples of angles, but the transition to more abstract ideas can be challenging. In this study, we examine 20 third and fourth grade students completing a body-based angle task in a motion-controlled learning environment using the Kinect for Windows. We present overall pre- and post-test results, showing that the task enhanced learners’ developing ideas about angles, and we describe two case studies of individual students, looking in detail at the role the body plays in the learning process. We found that the development of a strong connection between the body and the abstract representation of angle was instrumental to learning, as was exploring the space and making connections to personal experiences. The implications of these findings for developing body-based tasks are discussed.     

Teaching spline approximation techniques using maple

Using Computer Algebra Systems (CAS)-such as MAPLE-in teaching and learning mathematical concepts is a great challenge both from a didactical and a scientific point of view. We have to rewrite our traditional paper based teaching materials for interactive and living electronic worksheets, Only few statements and principles have to be acquired by the learner and the teacher from the CAS and after they can visualise, make animations modify quickly the program data, perform symbolic and numeric calculations step by step and in the whole, and verify deductions on their own. The author prepared Maple worksheets for teaching different types of function approximating techniques, such as interpolation-, least square-, spline and uniform approximation methods for post-graduate mechanical engineering students. In this paper we want to demonstrate how can we keep and improve the famous problem solving principles and rules given by G. Pólya and R. Descartes (Pólya 1962), when we use the capabilities of CAS. The education principles active learning, motivations and the successive phases are getting new meaning in the CAS. Our examples are always concerning with spline functions. Handling the formulas, calculating values and giving proofs are always in the form of Maple statements.     

Searching and exploring properties of geometric configurations using dynamic software

The decline in enrolments and interest in advanced mathematics studies is of growing concern internationally. Previous research suggests that a range of factors can influence students' academic decisions. The focus of the paper is on one of these potential sources of influence— students' perceptions of the tertiary mathematics learning environment. Data from two large-scale surveys (N = 1883) and from a smaller number of interviews (N = 71) with students enrolled in tertiary mathematics courses at five Australian universities are presented and discussed. Collectively, the survey results and the interview data reveal considerable variations in the quality of the teaching and student support available in different mathematics departments. Students' comments were constructive and offered valuable ideas for improving the existing situation, retaining current students and attracting others to mathematics.     

Mathematical tasks,study approaches,and course grades in undergraduate mathematics: a year-by-year analysis

Students approach learning in different ways, depending on the experienced learning situation. A deep approach is geared toward long-term retention and conceptual change while a surface approach focuses on quickly acquiring knowledge for immediate use. These approaches ultimately affect the students’ academic outcomes. This study takes a cross-sectional look at the approaches to learning used by students from courses across all four years of undergraduate mathematics and analyses how these relate to the students’ grades. We find that deep learning correlates with grade in the first year and not in the upper years. Surficial learning has no correlation with grades in the first year and a strong negative correlation with grades in the upper years. Using Bloom's taxonomy, we argue that the nature of the tasks given to students is fundamentally different in lower and upper year courses. We find that first-year courses emphasize tasks that require only low-level cognitive processes. Upper year courses require higher level processes but, surprisingly, have a simultaneous greater emphasis on recall and understanding. These observations explain the differences in correlations between approaches to learning and course grades. We conclude with some concerns about the disconnect between first year and upper year mathematics courses and the effect this may have on students.