Ways of talking and ways of positioning: Students’ beliefs in an inquiry-oriented differential equations class

As part of developmental research for an inquiry-oriented differential equations course, this study investigates the change in students’ beliefs about mathematics. The discourse analysis has identified two different types of perspective modes - i.e., discourse of the third-person perspective and discourse of the first-person perspective - in the students’ mathematical narratives, depending on their ways of positioning themselves with respect to mathematics. In the third-person perspective discourse, the students positioned themselves as passive recipients of mathematics that has been established by some external authority. In the first-person perspective discourse, the students positioned themselves as active mathematical inquirers and produced mathematics by interweaving their own mathematical ideas and experiences. Over the semester, students’ mathematical discourse changed from third-person perspective narratives to first-person perspective narratives. This change in their discourse pattern is interpreted as an indication of change in their beliefs about mathematics. Finally, this article discusses the instructional features that promote the change.     

基于“动态生成”的大学数学课堂教学

"动态生成"教学观的建立旨在摆脱课堂教学中以教师为中心、强调知识传授的传统教学模式的缺陷,从根本上正确理解课堂教学的复杂性和动态性,构建充满生命活力的大学数学课堂教学生态环境.在用动态生成的视角审视当前大学数学课堂教学中存在的问题的基础上,文章对在课堂教学中如何有效地进行动态生成提出了一些具体的策略:更新教学观念,精心预设弹性化的数学课堂教学方案;根据学生课堂反馈情况及时调整预设,并及时捕捉可利用的动态资源,为学生的生成创造可能的机会;加强教学研究,不断提高课堂教学智慧.     

Fostering creativity through instruction rich in mathematical problem solving and problem posing

Although creativity is often viewed as being associated with the notions of “genius” or exceptional ability, it can be productive for mathematics educators to view creativity instead as an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population. In this article, it is argued that inquiry-oriented mathematics instruction which includes problem-solving and problem-posing tasks and activities can assist students to develop more creative approaches to mathematics. Through the use of such tasks and activities, teachers can increase their students’ capacity with respect to the core dimensions of creativity, namely, fluency, flexibility, and novelty. Because the instructional techniques discussed in this article have been used successfully with students all over the world, there is little reason to believe that creativity-enriched mathematics instruction cannot be used with a broad range of students in order to increase their representational and strategic fluency and flexibility, and their appreciation for novel problems, solution methods, or solutions.     

A study of early career teachers’ practices related to language and language diversity during mathematics instruction

The role of language in mathematics teaching and learning is increasingly highlighted by standards and reform movements in the US. However, little is known about teachers’, and especially early career teachers’ (ECTs) practices and understandings related to language in mathematics instruction. This multiple case study explored the language-related understandings and practices of six ECTs in diverse elementary classrooms. Using iterative cycles of analysis, we found that all ECTs regularly attended to students’ mathematical vocabulary use and development. Yet, there was variability in ECTs’ focus on how to teach mathematical vocabulary, expectations for students’ precise use of mathematical terminology, and the use of multiple languages during instruction. These findings indicate that ECTs need more targeted support during teacher preparation and early career teaching in order to better support all students’ language development in the mathematics classroom.     

Observation of Reform Teaching in Undergraduate Level Mathematics and Science Courses

This paper reports on initial results from an ongoing evaluation study of a National Science Foundation project to implement reform‐oriented teaching practices in college science and mathematics courses. The purpose of this study was to determine what elements of reform teaching are being utilized by college faculty members teaching undergraduate science and mathematics courses, including a qualitative estimate of the frequency with which they are used. Participating instructors attended summer institutes that modeled reform‐based practices and fostered reflection on current issues in science, mathematics, and technological literacy for K‐16 teaching, with an explicit emphasis on the importance of creating the best possible learning experience for prospective K‐12 science and mathematics teachers. Utilizing a unique classroom observation protocol (the Oregon‐Teacher Observation Protocol) and interviews, the authors (a) conclude that some reform‐oriented teaching strategies are evident in undergraduate mathematics and science instruction and (b) suggest areas in which additional support and feedback are needed in order for higher education faculty members to adopt reform‐based instructional methodology.     

Reading mathematics for understanding—From novice to expert

As students progress through the college mathematics curriculum, enter graduate school and eventually become practicing mathematicians, reading mathematics textbooks and journal articles appears to become easier and leads to increased proficiency and understanding. This study was designed to begin to understand how mathematically more advanced readers read for understanding in mathematical exposition as it appears in textbooks compared to first-year undergraduate students. Three faculty members and three graduate students participated in this study and read from a first-year graduate textbook in an area of mathematics unfamiliar to each of them. The observed reading strategies of these more mathematically advanced readers are compared to observed reading strategies of first-year undergraduate students from an earlier study. The reading methods of the faculty level mathematicians were all quite similar and were markedly different from those that have been identified for undergraduate students, as well as from those used by the graduate students in this study. A Mathematics Reading Framework is proposed based on this study and previous research documenting the strategies that first-year undergraduate students use for reading exposition in their mathematics textbooks.     

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.     

Reflections on the promise and complexity of mathematics coaching

If students are to develop mathematical proficiency, then mathematics teaching must both change and improve. In an effort to provide site-based professional development addressing the mathematical content and pedagogical demands that teachers encounter in reality of public schooling, many school districts are turning to elementary mathematics coaches. Knowledgeable coaches can have a significant positive impact on teachers, yet this study documents substantial variance in the amount of coaching delivered and in the nature of activity that coaches undertake within schools. Coaches are frequently responsive to the needs of individual teachers. If this support is primarily marked by shared teaching or provision of instructional materials, it may not transform either instruction or teacher knowledge. Similarly if coaches assume duties that primarily address an administrator’s needs, they will have less time to enhance a school’s mathematics program. Coaches need to engage teachers in fundamental dialogue about mathematical content, mathematical learning, and student understanding. It may be that this dialogue and the effectiveness of a coach’s work with individual teachers would benefit from a coach’s concurrent work with grade-level teams. When a coach leads a grade-level team through discussion of targeted goals and approaches, the coach may facilitate individual teacher learning while building collective learning. When coupled with the support of a principal, this partnership may foster instructional change across a school.     

What are We Developing? A Case Study of a College Mathematics Program

Over one‐third of all college mathematics enrollments are in courses considered to be developmental. While such courses have been the subject of a large body of research, one question that seems not to have been studied empirically is the alignment of the content of developmental and college level mathematics courses. This paper gives the results of such a study, conducted at a medium sized public liberal arts university. While the content of the developmental mathematics courses was used subsequently in the Precalculus course, most of the content of the Intermediate Algebra course was not used in other college level mathematics courses. It is hoped that the results of this paper will serve as a catalyst for other institutions to examine carefully and define the mission of both their developmental and college level mathematics courses.     

Perceived parental influence and students' dispositions to study mathematically-demanding courses in Higher Education

We explore the influence of family on adolescent students' mathematical habitus by investigating the association between students' perceptions of parental influence and their dispositions towards mathematics. A construct measuring ‘perceived parental influence’ was validated using Rasch methodology on data from 563 Cypriot students on ‘core’ and ‘advanced’ mathematics pre-university courses, and was then used to predict students' dispositions towards future study of mathematically-demanding courses at university. In most of the regression models, perceived parental influence was not associated significantly with students' dispositions towards mathematics, when other variables were included in the models. However, further statistical analysis showed that perceived parental influence is mediated by (i) the mathematics course students are studying and (ii) their mathematical inclination. We suggest that family influences on students' dispositions are significantly accounted for by students' prior choice of mathematics course and the family's inculcation of their mathematical inclination; these are important factors influencing university choices.     

Korean and American Elementary School Teachers' Beliefs about Mathematics Problem Solving

Elementary school teachers in South Korea and the United States completed a beliefs and practices questionnaire pertaining to mathematical problem-solving instruction. Although both groups of teachers shared a general approach to teaching with a focus on problem-solving strategies, many differences were apparent. Korean teachers rated themselves and their students higher in problem-solving ability than American teachers. Korean teachers perceived their mathematics textbook as a more valuable source for problem-solving instruction and word problems. Korean teachers more strongly agreed that students should know the key-word approach for solving problems. American teachers reported more frequent use of calculators, manipulatives, and small group instruction. The results indicate that American teachers may more often use instructional techniques that are aligned with current recommendations for mathematics instruction.     

Collaborative Workshops and Student Academic Performance in Introductory College Mathematics Courses: A Study of a Treisman Model Math Excel Program

High failure rates in introductory college mathematics courses, particularly among underrepresented groups of students, have been of concern for many years. One approach to the problem experiencing some success has been Treisman's Emerging Scholars workshop model. The model involves supplemental workshops in which students solve problems in collaborative learning groups. This study reports on the effectiveness of Math Excel, an implementation of the Treisman model for introductory mathematics courses (college algebra, precalculus, differential calculus, and integral calculus) at Oregon State University over five academic terms. Regression analyses revealed a significant effect on achievement (.671 grade points on a 4‐point scale) favoring Math Excel students. Even after adjusting for prior mathematics achievement using linear regression with SAT‐M as predictor, Math Excel groups' grade averages were over half a grade point better than predicted (significant at the .001 level). This study provides supporting evidence that programs like Math Excel can help students in making a successful transition to college mathematics study.     

浅析对数学史教育的认识与实施

数学史教育在大学数学教育中具有重要的意义.有助于学生理解数学知识的本质,培养学生的思维能力.有必要开设数学史课程或在专业课教学中渗透数学史知识;有适合学生的专门数学史教材;进行数学史教育要联系学生的数学知识,要深入浅出;要注意数学精神的宣传.     

Pre-service science teachers’ perceptions of mathematics courses in a science teacher education programme1

Most science departments offer compulsory mathematics courses to their students with the expectation that students can apply their experience from the mathematics courses to other fields of study, including science. The current study first aims to investigate the views of pre-service science teachers of science-teaching preparation degrees and their expectations regarding the difficulty level of mathematics courses in science-teaching education programmes. Second, the study investigates changes and the reasons behind the changes in their interest regarding mathematics after completing these courses. Third, the current study seeks to reveal undergraduate science teachers’ opinions regarding the contribution of undergraduate mathematics courses to their professional development. Being qualitative in nature, this study was a case study. According to the results, almost all of the students considered that undergraduate mathematics courses were ‘difficult’ because of the complex and intensive content of the courses and their poor background mathematical knowledge. Moreover, the majority of science undergraduates mentioned that mathematics would contribute to their professional development as a science teacher. On the other hand, they declared a negative change in their attitude towards mathematics after completing the mathematics courses due to continuous failure at mathematics and their teachers’ lack of knowledge in terms of teaching mathematics.     

Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education

This article focuses on a form of instructional design that is deemed fitting for reform mathematics education. Reform mathematics education requires instruction that helps students in developing their current ways of reasoning into more sophisticated ways of mathematical reasoning. This implies that there has to be ample room for teachers to adjust their instruction to the students' thinking. But, the point of departure is that if justice is to be done to the input of the students and their ideas built on, a well-founded plan is needed. Design research on an instructional sequence on addition and subtraction up to 100 is taken as an instance to elucidate how the theory for realistic mathematics education (RME) can be used to develop a local instruction theory that can function as such a plan. Instead of offering an instructional sequence that “works,” the objective of design research is to offer teachers an empirically grounded theory on how a certain set of instructional activities can work. The example of addition and subtraction up to 100 is used to clarify how a local instruction theory informs teachers about learning goals, instructional activities, student thinking and learning, and the role of tools and imagery.     

Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education

This article focuses on a form of instructional design that is deemed fitting for reform mathematics education. Reform mathematics education requires instruction that helps students in developing their current ways of reasoning into more sophisticated ways of mathematical reasoning. This implies that there has to be ample room for teachers to adjust their instruction to the students' thinking. But, the point of departure is that if justice is to be done to the input of the students and their ideas built on, a well-founded plan is needed. Design research on an instructional sequence on addition and subtraction up to 100 is taken as an instance to elucidate how the theory for realistic mathematics education (RME) can be used to develop a local instruction theory that can function as such a plan. Instead of offering an instructional sequence that "works," the objective of design research is to offer teachers an empirically grounded theory on how a certain set of instructional activities can work. The example of addition and subtraction up to 100 is used to clarify how a local instruction theory informs teachers about learning goals, instructional activities, student thinking and learning, and the role of tools and imagery.     

深入开展数学建模活动，培养学生的综合应用素质

数学建模时大学生综合应用素质培养的一种有效途径。本文首先分析数学建模在大学生实践创新能力和综合应用素质培养中的作用,在此基础上从数学建模的数学理念更新、数学模式和教学内容设计,教学方法运用等多个方面,研究了以培养学生综合应用素质为牵引的多焊接、全方位的数学建模教学模式,提出了一些创新性的实践做法,为教学建模数学与组织竞赛活动的开展提供参考。