Seeking mathematics success for college students: a randomized field trial of an adapted approach

Many students enter the Canadian college system with insufficient mathematical ability and leave the system with little improvement. Those students who enter with poor mathematics ability typically take a developmental mathematics course as their first and possibly only mathematics course. The educational experiences that comprise a developmental mathematics course vary widely and are, too often, ineffective at improving students’ ability. This trend is concerning, since low mathematics ability is known to be related to lower rates of success in subsequent courses. To date, little attention has been paid to the selection of an instructional approach to consistently apply across developmental mathematics courses. Prior research suggests that an appropriate instructional method would involve explicit instruction and practising mathematical procedures linked to a mathematical concept. This study reports on a randomized field trial of a developmental mathematics approach at a college in Ontario, Canada. The new approach is an adaptation of the JUMP Math program, an explicit instruction method designed for primary and secondary school curriculae, to the college learning environment. In this study, a subset of courses was assigned to JUMP Math and the remainder was taught in the same style as in the previous years. We found consistent, modest improvement in the JUMP Math sections compared to the non-JUMP sections, after accounting for potential covariates. The findings from this randomized field trial, along with prior research on effective education for developmental mathematics students, suggest that JUMP Math is a promising way to improve college student outcomes.     

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.     

Elementary teachers’ mathematical beliefs and mathematics anxiety: How do they shape instructional practices?

This quantitative study investigated the relationships among practicing elementary teachers’ (N = 153) beliefs about mathematics and its teaching and learning, mathematics anxiety, and instructional practices in mathematics. When viewed singly, the findings reveal the teachers with higher levels of mathematics anxiety tend to use less standards‐based instruction and those with beliefs oriented toward a problem‐solving view of mathematics reported more standards‐based teaching. A combined analysis shows that after controlling for mathematical beliefs, teaching longevity, and educational degree attainment, there is no relationship between teachers’ mathematics anxiety and instructional practices. These findings suggest a spurious relationship between anxiety and practices, with beliefs having the strongest relationship with practices. Several suggestions for positively influencing the mathematical beliefs and affect in general of elementary teachers while learning mathematics are offered.     

Three women's understandings of algebra in a precalculus course integrated with the graphing calculator

This study investigated the role of function in a precalculus classroom which incorporated the graphing calculator in the instructional process. Perspectives were taken from students, teachers, and textbooks. Emphasis was placed on choice of functional symbol system when thinking and problem solving, connections across symbol systems, the role of the instructor and the textbook in learning, affective components, and the effect of the graphing calculator.The study starts with a defination of the concept of structure as it relates to function. The account of a semester-long qualitative study on students' concept images of function and its role in problem solving follows. It was found that the students involved in the study entered the graph-intensive course with predominantly symbolic notions of algebra, in part due to prior instruction. The students also possessed highly procedural views of algebraic content. These preconceptions and expectations resulted in the students' inability to effectively coordinate graphic and symbolic notions of algebra, both in procedural and conceptual realms. Implications and curricular suggestions are provided.     

The interconnectedness of relational and content dimensions of quality instruction: Supportive teacher–student relationships in urban elementary mathematics classrooms

Scholars assert that the often-impoverished instructional practices found in urban schools are tied to teachers’ negative relationships with African American and Latin@¹ students (Ferguson, 1998, McKown and Weinstein, 2002, McKown and Weinstein, 2008, Morris, 2005, Stiff and Harvey, 1988). However, measures of mathematics instructional quality rarely measure relational elements of instruction. This study responds to such shortcomings by analyzing relational interactions in urban elementary mathematics classrooms in tandem with content instruction of teachers who engage in supportive relationships with African American and Latin@ students. This study identified teachers with high quality student performance, content instruction, and supportive relationships as defined through relational interactions. After selecting two teachers, the results detail relational interactions that show how these teachers established supportive relationships with students vis-à-vis their mathematics instruction. Therefore, these findings offer insight into the ways in which relational interactions add to our understanding of quality content instruction for African American and Latin@ students.     

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.     

Integrating Science,Mathematics, and Technology in Middle School Technology‐Rich Environments: A Study of Implementation and Change

The GTECH project, funded through a grant from the GTE Foundation, prepared school teams of science, mathematics and technology teachers and an administrator to set goals for their local schools regarding implementation of electronic technology and integration of content across curricular areas. A variety of teacher‐centered staff development strategies were used to enable participants to achieve local school objectives, model and encourage active learning environments involving technology, develop integrated curriculum and provide training to their peers. GTECH staff provided workshops and summer institutes based on teacher feedback and classroom observations. Data from the Stages of Concerns Questionnaire assisted the staff in designing effective staff development activities. Over the 2‐year period, teacher teams developed and implemented integrated instructional materials and developed skills in using HyperStudio, PowerPoint, telecommunications applications, and instructional resources from the Internet. They also linked instruction to new state and national standards in science, mathematics, and technology. GTECH teachers reported that their students have expanded their knowledge and skill in problem solving, teamwork, technical expertise, and creativity.     

Progressive, classical or balanced— A look at mathematical learning environments in Swiss-German lower-secondary schools

In a national supplement to TIMSS, lower-secondary school teachers (N=102) and their students (N=975) reported on mathematics instruction by means of a teacher questionnaire (teaching-learning methods, instructional sub-goals, facilitated student activities, achievement assessment, teacher role) and a student questionnaire (teachers' instructional proficiency, classroom climate). A cluster analysis performed on the ratings of teaching-learning methods yielded a solution with three clusters referred to as progressive, classical, and balanced learning environment. Cluster-related differences in facilitated student activities, achievement evaluation and preferred teacher role were found but not in instructional sub-goals. Students from different learning environments equally approved teachers' instructional proficiency and classroom climate and also had similar TIMSS mathematics scores. The results of this study provide empirical evidence that in addition to classical teacher-centered learning environments there seem to exist more diversified and studentcentered learning environments that address the needs for students to direct their own learning, communicate and work with others, and develop ways of dealing with complex problems. In line with the research literature it was also found that high mathematics achievement is not restricted to a certain type of learning environment.     

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.     

Varied Meanings and Engagement in School Mathematics: Cross‐Case Analysis of Three High‐Achieving Young Adolescent Girls

This paper presents an in‐depth cross‐case analysis of three high‐achieving young adolescent girls who had contrasting mathematics learning experiences during the first year of middle school. In particular, this study examines the foundation for their motivation, as well as the dominant mode of learning and academic engagement in relation to three sociocultural factors, family background, the role of peers, and the level of teachers' understanding of the students and instructional support provided. Our data analysis revealed that the three girls possessed motivation structures and learning dispositions that are more or less prone to conceptual or procedural understanding in mathematics. This resulted in a significant variation in the mode of their academic engagement with the subject, and this provided a different set of challenges in each girl's pursuit of higher level of mathematics learning.     

Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students’ progressively sophisticated ways of reasoning and instructional design heuristics that can facilitate this process. In this article we analyze the case of student reasoning with analytic expressions as they reinvent solutions to systems of two differential equations. The significance of this work is twofold: it includes an elaboration of the Realistic Mathematics Education instructional design heuristic of emergent models to the undergraduate setting in which symbolic expressions play a prominent role, and it offers teachers insight into student thinking by highlighting qualitatively different ways that students reason proportionally in relation to this instructional design heuristic.     

Controlling Choice: Teachers,Students, and Manipulatives in Mathematics Classrooms

This research study examines the instructional practices of 10 middle grades teachers related to their use of manipulatives in teaching mathematics and their control of mathematics tools during instruction. Through 40 observations of teaching, 30 interviews, and an examination of 67 written documents (including teachers' plans and records), profiles were developed that describe how teachers used and controlled manipulatives during instruction. Results showed that teachers used a variety of manipulatives and other mathematics tools over the course of the year‐long study. Teachers reported using a mathematics tool (manipulative, calculator, or measuring device) in 70% of their lessons, and this self‐report was verified by observations in which teachers used mathematics tools in 68% of their lessons. During a 3‐ to 4‐month period of “free access,” in which students had some measure of control in their selection and use of the mathematics tools, the students used manipulatives spontaneously and selectively. During free access, teachers exhibited various behaviors, including posting lists of items on containers, assigning group leaders to manage tools, and negotiating the control of the mathematics tools during instruction.     

Construction of Teacher Knowledge in Context: Preparing Elementary Teachers to Teach Mathematics and Science

The purpose of this study was to further the understanding of how preservice teachers construct teacher knowledge and pedagogical content knowledge of elementary mathematics and science in a school‐based setting and the extent of knowledge construction. Evidence of knowledge construction (its acquisition, its dimensions, and the social context) was collected through the use of a qualitative methodology. The methods course was content‐specific with instruction in elementary mathematics and science. Learning experiences were based on national standards with a constructivist instructional approach and immediate access to field experiences. Analysis and synthesis of data revealed an extensive acquisition of teacher knowledge and pedagogical content knowledge. Learning venues were discovered to be the conduits of learning in a situated learning context. As in this study, content‐specific, school‐based experiences may afford preservice teachers greater opportunities to focus on content and instructional strategies at deeper levels; to address anxieties typically associated with the teaching of elementary mathematics and science; and to become more confident and competent teachers. Gains in positive attitudes and confidence in teaching mathematics and science were identified as direct results of this experience.     

Student Strategy Choices on a Constructed Response Algebra Problem

A central goal of secondary mathematics is for students to learn to use powerful algebraic strategies appropriately. Research has demonstrated student difficulties in the transition to using such strategies. We examined strategies used by several thousand 8th‐, 9th‐, and 10th‐grade students in five different school systems over three consecutive years on the same algebra problem. We also analyzed connections between their strategies and their success on the problem. Our findings suggest that many students continued to struggle with algebraic problems, even after several years of instruction in algebra. Students did not reflect the anticipated growth toward the consistent use of efficient strategies deemed appropriate in solving this problem. Instead a surprisingly large number of students continued to rely on strategies such as guessing and checking, or offered solutions that were unintelligible or meaningless and not useful to the researchers. Even those students who used algebraic strategies consistently did not show the anticipated improvement of performance that would be expected from several years of continuing to study mathematics.     

Effective collaboration in the productive failure process

Productive failure is a learning design that encompasses problem solving prior to instruction and the learning that occurs during and after this process. In the mathematics education literature, there is a need for analyses of students’ interactions that occur as they collaborate during the productive failure process. In this paper, we contribute to this area by taking a closer look at students’ interactions that characterize an effective productive failure process. In analyzing video footage of two different groups of students working on invention tasks in a flipped mathematics classroom, we observed that the productive failure process seemed to work best in groups of students among whom the instructional design evoked students’ intellectual need and curiosity. These students also developed a set routine for solving problems whose solutions are difficult to find without prior direct instruction on the topic, which proved valuable on follow-up in-class and posttest problems.     

Re‐creating a Recipe for Science Instructional Programs: Adding Learning Progressions,Scaffolding, and a Dash of Reading Variety

The purpose of this article is to focus on the development and refinement of a science instructional design program arguing for the feasibility and usability of integrated reading and science instruction as implemented in third‐ and fourth‐grade science classrooms to meet the learning needs of diverse learners. These instructional components are easily inserted into existing programs that build students' science background knowledge and abilities to apply learning through scaffolded activities focused on (1) providing structured opportunities for students to engage in hands‐on activities; (2) increasing vocabulary knowledge and understanding of concept‐laden terms, and (3) reading paired narrative and informational science texts. Extensive research shows that as students transition from third to fourth grade and beyond, they are often challenged in science by new vocabulary coupled with new concepts. Active ingredients of our reconceptualized science instructional design program are narrative informational texts, hands‐on science activities, and science textbook(s).     

Expanding the Role of K‐5 Science Instruction in Educational Reform: Implications of an Interdisciplinary Model for Integrating Science and Reading

Addressed is the current practice in educational reform of reducing time for science instruction in favor of traditional reading/language arts instruction. In contrast, presented is an evidence‐based rationale for increasing instructional time for K‐5 science instruction as an educational reform initiative. Overviewed are consensus interdisciplinary research and complementary multi‐year findings of the Science IDEAS model demonstrating the effectiveness of integrating conceptually‐relevant reading within science instruction in improving student achievement in both science and reading comprehension. Based on research summarized, increasing time for integrated K‐5 science is advocated as a meaningful reform‐based approach to science learning and reading comprehension proficiency that, in turn, better prepares students for subsequent success in science and content‐area reading comprehension across upper elementary and middle school grades (3–8).     

An investigation of teachers’ intentions and reflections about using Standards-based and traditional textbooks in the classroom

This study analyzed teachers’ intentions for and reflections on their use of Standards-based [Connected Mathematics Program (CMP)] textbooks and traditional (non-CMP) mathematics textbooks to guide instruction. In this investigation of the interplay between textbooks and instruction, we focused on learning goals, instructional tasks, teachers’ anticipation of students’ difficulties, and their perceptions of students’ achievement of learning goals. All of these are aspects of teachers’ intentions and reflections that have proved fruitful in comparing the roles of the CMP and non-CMP mathematics textbooks in our Longitudinal Investigation of the Effect of Curriculum on Algebra Learning project. Whereas the cognitive level of the teachers’ intended learning goals appeared generally to reflect the emphases of their respective textbooks, we found that the CMP teachers’ intended learning goals were not as well aligned with the CMP textbooks as the non-CMP teachers’ learning goals were aligned with their non-CMP textbooks. The CMP and non-CMP teachers’ implementations of the lessons seemed to reduce the degree of difference between the cognitive levels of their intended goals. Even so, we found that significantly more CMP lessons than non-CMP lessons were implemented at a high level of cognitive demand. Although the non-CMP teachers’ intended learning goals were better aligned with their textbook’s learning goals, we found that the CMP teachers were more likely than the non-CMP teachers to follow the guidance of their textbooks in designing and selecting instructional tasks for a lesson. Future research should consider other aspects of teachers’ intentions and reflections that may shed a broader light on the role of textbooks and curriculum materials in teachers’ crafting of instructional experiences for their students.