Developing a Learning Progression for Curriculum,Instruction, and Student Learning: An Example from Mathematics Education

Learning progressions have been demarcated by some for science education, or only concerned with levels of sophistication in student thinking as determined by logical analyses of the discipline. We take the stance that learning progressions can be leveraged in mathematics education as a form of curriculum research that advances a linked understanding of students learning over time through careful articulation of a curricular framework and progression, instructional sequence, assessments, and levels of sophistication in student learning. Under this broadened conceptualization, we advance a methodology for developing and validating learning progressions, and advance several design considerations that can guide research concerned with engendering forms of mathematics learning, and curricular and instructional support for that learning. We advance a two-phase methodology of (a) research and development, and (b) testing and revision. Each phase involves iterative cycles of design and experimentation with the aim of developing a validated learning progression. In particular, we gathered empirical data to revise our hypothesized curricular framework and progression and to measure change in students. thinking over time as a means to validate both the effectiveness of our instructional sequence and of the assessments designed to capture learning. We use the context of early algebra to exemplify our approach to learning progressions in mathematics education with a focus on the concept of mathematical equivalence across Grades 3-5. The domain of work on research on learning over time is evolving; our work contributes a broadened role for learning progressions work in mathematics education research and practice.     

数学学习策略的实验研究

本研究在对北京市六所中学437名初二年级学生数学学习动机和策略调查研究的基础上,选取普通中学257名初二学生为被试,考察了数学学习策略训练的不同方法、不同时间和不同性别对训练效果的影响。结果表明:(1)不同的策略训练方法(讨论法、学生出声思维与实验人员指导相结合、教师随堂渗透讲授)影响训练效果;(2)策略训练时间的因素影响训练效果;(3)性别因素也对训练效果有一定影响。     

Pathways to Reading Competence: Emotional Self-regulation,Literacy Contexts,and Embodied Learning Processes

Abstract

Reading competence is one of the main gateways to learning and serves as the foundation for nearly all academic subjects, but reading is not a natural skill. For beginning and struggling readers, the process of learning to read is often fraught with frustration. Thus, abilities to manage affect or emotions and maintain attention or focus (i.e. emotional self-regulation processes) are critical for literacy development and reading competence. Building on bio-social-ecological systems and contextual-developmental frameworks, we present a model of reading competence to integrate multidisciplinary empirical research on the fit between children’s emotional self-regulation processes and their literacy contexts and how these person-in-context dynamics influence reading competence through reading motivation and engagement. We present empirical research in support of the pathways in this model of reading competence, and call for increased multidisciplinary research that takes into consideration children’s literacy contexts and their neurobiological and behavioral assets as well as vulnerabilities in order to better understand the dynamical cognitive-emotional-motivational processes that underlie the development of reading competence from early childhood through young adulthood, including the timing and mechanisms of change to target for reading interventions to have optimal impact.     

Learning Logical Tolerance: Hans Hahn on the Foundations of Mathematics

Hans Hahn's long-neglected philosophy of mathematics is reconstructed here with an eye to his anticipation of the doctrine of logical pluralism. After establishing that Hahn pioneered a post-Tractarian conception of tautologies and attempted to overcome the traditional foundational dispute in mathematics, Hahn's and Carnap's work is briefly compared with Karl Menger's, and several significant agreements or differences between Hahn's and Carnap's work are specified and discussed.     

Hand Gesture and Mathematics Learning: Lessons From an Avatar

A beneficial effect of gesture on learning has been demonstrated in multiple domains, including mathematics, science, and foreign language vocabulary. However, because gesture is known to co‐vary with other non‐verbal behaviors, including eye gaze and prosody along with face, lip, and body movements, it is possible the beneficial effect of gesture is instead attributable to these other behaviors. We used a computer‐generated animated pedagogical agent to control both verbal and non‐verbal behavior. Children viewed lessons on mathematical equivalence in which an avatar either gestured or did not gesture, while eye gaze, head position, and lip movements remained identical across gesture conditions. Children who observed the gesturing avatar learned more, and they solved problems more quickly. Moreover, those children who learned were more likely to transfer and generalize their knowledge. These findings provide converging evidence that gesture facilitates math learning, and they reveal the potential for using technology to study non‐verbal behavior in controlled experiments.     

Mindful Learning: A Moderator of Gender Differences in Mathematics Performance

Past research has demonstrated that males outperform females in mathematics (Hyde, J. S., Fennema, E., & Lamon, S. J., Psychol Bull 107:139–155, 1990a). Research has also shown that encouraging mindful learning–learning information in a conditional rather than an absolute way–can increase mathematics performance in females (Ritchhart, R., & Perkins, D. N., J Social Issues 56:27–47, 2000). This paper examines the moderating role of mindful learning for gender differences, by manipulating mindful learning for females’ and males’ performance on a novel math task. The results from this study show that males performed better than females when mindful learning was not encouraged (absolute instruction), but males and females performed equally well when mindful learning was encouraged (conditional instruction). Thus we find that mindful learning moderates gender differences in math performance.     

Learning Mathematics From Examples and by Doing

This article demonstrates the feasibility and effectiveness of teaching several mathematical skills by presenting students with carefully chosen sequences of worked-out examples and problems - without lectures or other direct instruction. Thinking-aloud protocols of 20 students learning factorization by this method are analyzed to determine the kinds and depth of understanding students attained. Students did not simply memorize procedures but were able to recognize when the procedures were applicable and to apply them. Most students were also able to use their understanding of the concept of factorization to help learn the procedures and to check their results. The method of learning from examples has now been tested successfully with a class covering the entire 3-year curriculum in algebra and geometry in a Chinese middle school.     

Reversal Learning Deficits in Criminal Offenders: Effects of Psychopathy,Substance use,and Childhood Maltreatment History

Deficits in reinforcement learning are presumed to underlie the impulsive and incorrigible behavior exhibited by psychopathic criminals. However, previous studies documenting reversal learning impairments in psychopathic individuals have not investigated this relationship across a continuous range of psychopathy severity, nor have they examined how reversal learning impairments relate to different psychopathic traits, such as the interpersonal-affective and lifestyle-antisocial dimensions. Furthermore, previous studies have not considered the role that childhood maltreatment and substance use may have in this specific cognitive deficit. Using a standard reversal learning task in a sample of N = 114 incarcerated male offenders, we demonstrate a significant relationship between psychopathy severity and reversal learning errors. Furthermore, we show a significant interaction between psychopathy and childhood maltreatment, but not substance use, such that individuals high in psychopathy with an extensive history of maltreatment committed the greatest number of reversal learning errors. These findings extend the current understanding of reversal learning performance among psychopathic individuals, and highlight the importance of considering childhood maltreatment when studying psychopathy.     

Access Through Compensation: Emancipatory View of a Mathematics Learning Disability

Abstract

In this article, we provide a novel view of mathematics learning disability (MLD) by studying a student with an MLD (Dylan) who had compensated so effectively that she was able to major in statistics. We push back on the dominant deficit model used in studies of MLD, and consider issues of access and compensation from a Vygotskian theoretical frame. Through 8 videotaped interview sessions, we identified that Dylan’s primary difficulties were with mathematical notation and number sense, which resulted in issues accessing standard mathematical forms. Analysis revealed 8 compensatory strategies that Dylan used to address these issues of access. We frame our approach as emancipatory research. Dylan was involved in all phases of the study’s design, implementation, analysis, and dissemination, and is the second author. This work acknowledges that individuals with disabilities have research agendas of their own and have critical insight to share about the lived experience of their disability.     

Two Examples of How the History of Mathematics Can Inform Theology

In several of his writings, Bernard Lonergan emphasized the study of mathematics and especially its history as being important for philosophy and theology. In this article, I offer two examples of how the history of mathematics might inform theology. My first example explores the disparities between Euclid's Elements and Aquinas' Summa Contra Gentiles. My second example applies the idea of incompleteness to the “good and necessary consequence” clause of the Westminster Confession of Faith.     

Process Ontology: Conversations and Argumentations,Controversies in Mathematics and Mathematics as Socialisation

To better conceive the socializing and pragmatic aspects of mathematics, it can be useful to use a process ontology, which allows, starting from an analysis of the processes of conversations, to compare their recourse, from degree to degree, to supposedly common “virtualities”, in particular in argumentative conversations, with the construction of more complex mathematical entities that allow new symmetries, but also with controversies between mathematicians on the use of these entities.

     

认知图式与学术史观--中国美学与现代学术史的两个问题

１任何一种学术研究，都必然要具有这样两个前提条件：一是研究主体的“认知图式”，因为认识活动就是运用这一图式建构对象材料的过程，它作为康德认识论的一个理论硬核也早已为人们所熟知；马克思所谓的在再生产之前就以“表象”方式存在的“生产观念”是其更确切的表达...     

学科领域知识与数学学习的知识表征研究

根据领域知识的结构特点对初中数学学科知识作出改编和重组,形成数学学科领域知识单元进行教学实验,以215名普通初中二年级学生为被试,通过自编问卷和深度访谈考察学科领域知识对知识表征的影响,并探讨了知识表征与数学学业成绩之间的关系。研究发现:(1)学科领域知识可以显著提高学生数学陈述性知识表征全面性和总体水平;(2)学科领域知识可以显著提高学生的程序性知识表征全面性、自动化、组织性和总体水平;(3)认知结构三要素、陈述性知识表征及程序性知识表征与数学成绩显著相关;(4)学优生的陈述性知识表征准确性、总体水平和程序性知识表征组织性、总体水平显著高于中等生和学困生。