Learning from instruction: the case of mathematics

Starting from a brief analysis of adaptive competence in mathematics, this article describes a series of research-based characteristics of the kind of learning processes that should be elicited in students to facilitate and support in them the progressive acquisition of such competence. Four major characteristics are discussed in some detail: learning is constructive, self-regulated, situated or contextual, and collaborative. A rather new approach to transfer of learning is then presented in which transfer is conceived as the preparation for future learning. Throughout the article it is argued that, notwithstanding the progress made in research on learning from instruction, numerous and complex issues and problems remain for continued inquiry.     

Hemispheric laterality in music and math

Hemispheric laterality may be a useful concept in teaching, learning, training, and in understanding more about human development. To address this issue, a measure of hemispheric laterality was compared to musical and mathematical ability. The Human Information Processing Survey (HIPS) instrument, designed to measure hemispheric laterality, was administered to 101 participants who were then asked to provide a measure of their conceptual aptitude for mathematics and music. Scores were then compared through canonical correlation to test the hypothesis that perceived mathematical ability may be explained by left-brain hemisphere preference and musical by the right hemisphere. A relatively strong correlation was found between music ability and right-brain hemisphere preference. A relationship between math and left-brain hemisphere preference or integrated brain processing scores was found to be marginal. Ramifications to learning are discussed.     

Individual differences in the effect of relevant concreteness on learning and transfer of a mathematical concept

The concreteness of training materials influences learning and—perhaps more importantly—transfer. Building on prior research finding abstract representations best facilitated transfer to a game task, we conducted a similar study using training figures varying in concreteness but directly assessed transfer to modular arithmetic problems. Training figures: (a) were purely abstract, (b) were abstract but with features relevant to the transfer task, or (c) included additional concrete-relevant features. We hypothesized that concreteness—or number of relevant features—would be positively correlated with learning and transfer—especially among younger and/or lower-ability students. Although there was no overall difference in initial learning, the concrete-relevant and abstract-relevant features independently facilitated near-transfer, where concrete-relevant features supported lower-reasoning students. For far-transfer, eighth-graders benefited from the abstract-relevant features, whereas sixth-graders required additional concrete-relevant features. These findings suggest that concreteness interacts with learner and task characteristics to produce learning and transfer outcomes.     

Stigma and stratification limiting the math course progression of adolescents labeled with a learning disability

Learning disability (LD) designations may produce stigma by masking the real causes of learning differences, altering perceptions, and legitimizing stratification. This study uses data on adolescents and their teachers from The Education Longitudinal Study of 2002 to show the negative effect of LD designations on adolescents' math course attainment is partially mediated by disparities in adolescents' earlier math course placements, and teachers' more negative attributions and expectations. Results indicate addressing low achievement through LD designations may reproduce disadvantage through stigma and stratification.     

Starting behind and staying behind in South Africa: The case of insurmountable learning deficits in mathematics

This study quantifies a year's worth of mathematics learning in South Africa (0.3 standard deviations) and uses this measure to develop empirically calibrated learning trajectories. Two main findings are (1) only the top 16% of South African Grade 3 children are performing at an appropriate Grade 3 level. (2) The learning gap between the poorest 60% of students and the wealthiest 20% of students is approximately three Grade-levels in Grade 3, growing to four Grade-levels by Grade 9. The paper concludes by arguing that the later in life we attempt to repair early learning deficits in mathematics, the costlier the remediation becomes.     

Motivation and students’ use of learning strategies: Evidence of unidirectional effects in mathematics classrooms

Considerable evidence indicates that student motivation and use of learning strategies are related. There is insufficient understanding, however, about their reciprocal effects—whether motivation affects strategy use, the converse, or whether the effects are bidirectional—and which components of motivation and strategies are involved. A two-wave longitudinal design was used to examine this issue among 9th grade students (N = 306) enrolled in high school mathematics classes during an academic term. A cross-lagged structural model found that students’ self-efficacy in mathematics and value predicted their reported use of learning strategies. There was no evidence, however, that learning strategy use predicted motivation and, thus, support for unidirectional effect of motivation during that time interval. Implications for models of self-regulated learning and instruction are discussed.     

Identifying children with persistent low math achievement: The role of number-magnitude mapping and symbolic numerical processing

Although an increasing number of research studies have investigated the cognitive deficits related to difficulties in learning mathematics, little is known about whether these cognitive deficits longitudinally predict low mathematics achievement over time. The current 6-year longitudinal study was conducted to address this issue. A sample of 101 students was tested on various numerical and cognitive competencies when they were in kindergarten and in Grade 1. They were then followed until they were in Grade 6, and their mathematics achievement was assessed bi-annually. A group of persistent low mathematics achievers (PLA) who scored consistently below the 25th percentile was identified. This group of PLA showed difficulties in most of the numerical tasks as early as kindergarten. More importantly, three of the early predictors correctly identified 79% of the PLAs. The current findings provide valuable information concerning the core cognitive deficits underlying difficulties in learning mathematics as well as an important tool for educators for identifying children who are at risk of persistent math learning difficulties in the elementary school years.     

Effectiveness of grade retention: A systematic review and meta-analysis

Research on the effectiveness of grade retention has a long history, yet, has seen an upsurge during the last decade. In this study, we review 84 recent, methodologically sound studies estimating effects of retention in grades K-12 on repeaters' and nonrepeaters' development, in a variety of countries across the world, disentangling grade and age comparison results. Based on vote counting analysis and three-level metaregression analysis we find grade retention to have an average zero effect, indicating that repeaters and non-repeaters seem to show a similar development, on average. At the same time, we find grade retention effects to differ according to some specific effect and study characteristics. More specifically, grade retention seems less effective in countries applying a mixture of grade retention and tracking to tackle student heterogeneity, and when repeaters are compared with non-repeaters of the same age. Conversely, grade retention seems more effective in countries using strategies such as ability grouping, setting, and streaming to deal with student heterogeneity. Positive effects also seem to arise when studying students' psychosocial functioning, when investigating short-run effects, when comparing repeaters with their younger non-retained grade-mates, and when evaluating effects via a regression discontinuity method.     

Beyond conceptual change learning in science education: focusing on transfer,durability and metacognition

The frequently encountered problems of students being unable to utilize schoollearned science in different contexts, and of students forgetting what they have learned in a short time after initial instruction, are two important problems for classroom practitioners. This paper advocates a shift in focus of conceptual change learning research in order to address these problems. It draws upon four overlapping areas: conceptual change learning is the broad subject area that sets the epistemological background; transfer and durability of scientific conceptions are the two problem areas under scrutiny, while metacognition is seen as potential mediator of improvement. The paper offers a brief review of existing literature on the four areas; it proposes confronting the two problems by incorporating metacognitive instruction in the learning environment of primary school science; and it reports on recently completed research.     

Developing a thirst for knowledge: How uncertainty in the classroom influences curiosity,affect, learning,and transfer

Uncertainty is a key variable in fostering curiosity, which, in turn, is associated with learning. Yet, research in educational contexts rarely takes uncertainty into account, and rarely explores uncertainty and curiosity in the context of complex instructional activities. One concern with uncertainty is that it can provoke negative affect. Providing learners with expectations of future uncertainty may attenuate their feelings of negative affect. In a study with 138 middle school students learning physics concepts, we examined the relationship between uncertainty, curiosity, learning, transfer, and affect. Some students were given an inherently uncertain form of instruction, called Invention, in which information on how to solve the problem was initially withheld, while others were given direct instruction with all the necessary information to solve similar problems beforehand (No Uncertainty condition). Some of the students receiving uncertain instruction were given expectations about feeling uncertain (Expected Uncertainty condition), and some were not (Unexpected Uncertainty condition). Students in the unexpected uncertainty condition were the most curious, while students in the no uncertainty condition were the least curious. However, giving expectations of uncertainty reduced students’ negative affect. All groups learned the content equally well, but the expected and unexpected uncertainty groups exhibited greater transfer. Further, positive affect predicted learning, above and beyond condition, and curiosity predicted transfer, but not above and beyond condition. This study extends existing research on uncertainty and curiosity by studying these constructs in real classrooms, in the context of an exploratory learning paradigm, and by considering curiosity’s effect on transfer, rather than just learning. This work also demonstrates a practical approach for educators to foster students’ curiosity and transfer.     

Teaching for understanding and/or self-regulated learning? A video-based analysis of reform-oriented mathematics instruction in Switzerland

In mathematics instruction, can a teacher implement surface features of instruction that foster self-regulated learning as well as achieve quality at the deeper level of instruction, that is, focus on higher-order thinking, problem-solving, and mathematical modeling? An educational reform effort in Switzerland, which is based on constructivist and sociocultural theories of mathematics learning, targets both these dimensions: self-regulated learning and conceptual understanding. We examined the realization of the two dimensions in classroom instruction in a video-based study of 79 eighth-grade math classes using three kinds of data: videotapes of mathematics lessons, student and teacher questionnaires, and achievement tests. As to the surface level of instruction, teachers reported how frequently they provided opportunities for self-regulated learning. With regard to the deeper level of instruction, teachers reported how frequently they provided opportunities for independent problem solving. In addition, we examined the extent to which teachers’ pedagogical beliefs reflected a constructivist orientation. The results showed that teachers implemented the two dimensions relatively independently of one another. Teachers’ constructivist-oriented beliefs influenced only opportunities provided for independent problem solving and did not affect opportunities for self-regulated learning. Opportunities for self-regulated learning had a positive effect on students’ learning experience. Professional development should encourage teachers to take greater account of both surface-level and deeper-level (quality) features of instruction.     

Supplemental instruction in introductory biology I: enhancing the performance and retention of underrepresented minority students

Supplemental instruction classes have been shown in many studies to enhance performance in the supported courses and even to improve graduation rates. Generally, there has been little evidence of a differential impact on students from different ethnic/racial backgrounds. At San Francisco State University, however, supplemental instruction in the Introductory Biology I class is associated with even more dramatic gains among students from underrepresented minority populations than the gains found among their peers. These gains do not seem to be the product of better students availing themselves of supplemental instruction or other outside factors. The Introductory Biology I class consists of a team-taught lecture component, taught in a large lecture classroom, and a laboratory component where students participate in smaller lab sections. Students are expected to master an understanding of basic concepts, content, and vocabulary in biology as well as gain laboratory investigation skills and experience applying scientific methodology. In this context, supplemental instruction classes are cooperative learning environments where students participate in learning activities that complement the course material, focusing on student misconceptions and difficulties, construction of a scaffolded knowledge base, applications involving problem solving, and articulation of constructs with peers.     

论高职院校经济数学课程教学的改革与思考

通过对目前高职院校经济数学的教学现状、高职院校经济数学课程教学改革的必要性进行深入分析,并结合个人教学实践,提出了经济数学课程改革的具体思路,使高职院校经济数学的教学更能适应素质拓展的要求,不断提高高职学生经济数学的应用能力和创新能力.     

Developing strategic readers: a multimodal analysis of a primary school teacher's use of speech,gesture and artefacts

Research on implementing reading strategy instruction has primarily focused on teachers' verbal communication with limited attention to other semiotic resources such as gesture and artefacts. In this paper, we construct a ‘telling case’ on the basis of how one primary teacher from the United States used speech, gesture and artefacts as a means of communication while instructing her students in reasons to predict when reading. Data sources for this case study consisted of field notes, artefacts and digital video. We analysed the teacher's use of gesture, speech and artefacts from a social semiotic multimodal perspective. Findings indicate that the teacher created meaning by interweaving multiple modes in the communicative contexts of strategy instruction using speech, deictic gestures, metaphoric gestures and artefacts. These findings are important to reading strategy instruction because much of the research and discussion of practice to date has centred on the instruction of reading strategies using teacher and student speech and not attending to the use of semiotic resources beyond speech.     

Motivation and transfer in professional training: A meta-analysis of the moderating effects of knowledge type, instruction, and assessment conditions

This meta-analysis (148 studies, k = 197, N = 31,718) examined the relationship between motivation and transfer in professional training. For this purpose, motivation was conceptualized in the following nine dimensions: motivation to learn, motivation to transfer, pre- and post-training self-efficacy, mastery orientation, performance orientation, avoidance orientation, expectancy, and instrumentality. Population correlation estimates ranged between −0.11 and 0.52. Three moderator effects were estimated. First, correlations were higher when the training focused on declarative and self-regulatory, rather than on procedural, knowledge. Second, learner-centered environments tended to show greater numbers of positive correlations than did knowledge-centered environments. Third, when compared with external, supervisory, or peer assessment, self-assessment of transfer produced upwardly biased population estimates irrespective of the transfer criterion. These findings are discussed in terms of their implications for theories of training effectiveness and their significance for the practice of training evaluation.     

Co-development of math anxiety,math self-concept,and math value in adolescence: The roles of parents and math teachers

Despite the well-documented negative implications of math anxiety on math learning, a scarcity of theory-guided, long-term longitudinal research limits knowledge about how math anxiety develops over time. Guided by the Control-Value Theory of Achievement Emotions (Pekrun, 2006), the present study addresses this gap by examining (1) how math anxiety develops in tandem with the development of control and value appraisals across secondary schooling, and (2) how these three constructs co-develop in relation to characteristics of home and school contexts. We used growth mixture modeling to investigate how math anxiety, math self-concept (a frequently examined indicator of control appraisal), and math utility value (one dimension of math value) develop in parallel in a sample of 3116 adolescents, who were assessed annually across middle and high school. We identified three trajectory classes: a stable class, characterized by stably modest math anxiety, high math self-concept, and high math utility value, a linear change class, characterized by increasing math anxiety and decreasing math self-concept and utility value, and a fluctuating class, characterized by curvilinear changes in math anxiety, math self-concept, and math utility value. Parental academic support and teacher bias differentiated the stable class from the fluctuating class at the transition to middle school, and from the linear change class at the transition to high school. Our findings point to the heterogeneous contributions of control and value appraisals towards the development of math anxiety and highlight the importance of investigating multiple dimensions of the socio-ecological context at different stages of math anxiety development.     

大学数学学习障碍的成因与对策

高等教育大众化以来,感觉数学学习困难的大学生人数逐渐增多,部分学生产生厌学情绪。理清这些大学生学习困难的原因,制定相应的教育策略,营造良好的学习氛围,进行因材施教,是当前大学数学教学改革的一项重要课题。导致学生学习障碍的主要原因：一是教学内容的衔接问题,二是教学方式的转变问题,三是思维方式的转变问题,四是外部环境的影响问题。大学数学教学要以学生为中心,不断改进教学方法,加强与中学内容的衔接,将数学思想、数学方法和数学建模思想融入课堂教学,构建和谐的数学教学文化理念,将数学文化融入课堂,激发学生的学习积极性,着力提高大学数学教的水平和学生学的质量,真正为学生的全面发展奠定良好的基础。     

Longitudinal investigation of the curricular effect: An analysis of student learning outcomes from the LieCal Project in the United States

In this article, we present the results from a longitudinal examination of the impact of a Standards-based or reform mathematics curriculum (called CMP) and traditional mathematics curricula (called non-CMP) on students’ learning of algebra using various outcome measures. Findings include the following: (1) students did not sacrifice basic mathematical skills if they are taught using a Standards-based or reform mathematics curriculum like CMP; (2) African American students experienced greater gain in symbol manipulation when they used a traditional curriculum; (3) the use of either the CMP or a non-CMP curriculum improved the mathematics achievement of all students, including students of color; (4) the use of CMP contributed to significantly higher problem-solving growth for all ethnic groups; and (5) a high level of conceptual emphasis in a classroom improved the students’ ability to represent problem situations. (However, the level of conceptual emphasis bears no relation to students’ problem solving or symbol manipulation skills.)     

Examining the dynamics of mathematics anxiety,perceived cost,and achievement: A control-value theory approach

Concerns about the influence of students’ perceived negative consequences of engagement in a task (i.e., cost) on their emotions, motivation, and cognition have increased in the last decade. The use of longitudinal models is needed to provide new insights into the role of perceived cost in mathematics learning. Grounded in the control-value theory, this study examined cross-lagged relations of mathematics anxiety, perceived cost, and mathematics achievement. The participants (N = 335) reported their mathematics anxiety and perceived cost four times during Grades 7 and 8, and their mathematics grades were attained from their school records. Cross-lagged panel model analysis revealed evidence of a long-term positive reciprocal relationship between mathematics anxiety and effort/emotional cost, a gradually diminished relationship between effort/emotional cost and mathematics performance, and a positive achievement to anxiety link during the transition between grade levels. Moreover, mathematics performance is a distal predictor of mathematics anxiety through effort/emotional cost rather than a proximal predictor or an outcome of anxiety. This study also clarified the distinction in the central role of effort/emotional versus opportunity cost in the interrelatedness of mathematics anxiety and performance, where the latter failed to demonstrate significant paths. Specific timing for interventions was discerned. Early cost prevention interventions along with considerations of academic achievement to alleviate both anxiety and perceived effort/emotional are highlighted as crucial for a positive high school mathematics experience.