学生所知道的 教师所知道的——理论、工具和结果

评估——无论采用何种形式——不得不建立在对不同水平和质量的理论思考基础之上。其中至关重要的是测验内容领域要依据教与学的过程来构建。其次,任务是评估相关内容结构的核心工具。因此,在设计对数学的教育评估时,按照合理的标识和特征对任务进行分类就成为一个息息相关的问题,也是一个巨大的挑战。本文重点研究PISA(国际学生评估项目的缩写,是以数学素养为目标的国际比较研究,包括在国家层面德国所做的许多调整)和COACTIV(德国教师数学知识和数学教学知识的一个代表性研究)。两者都旨在实现区分性评估(differentiated assessment),都考虑了评价内容和任务分析的不同方面。区分性评估结果使得我们能够获取有用的信息,进一步推动课堂教与学的发展。这些信息主要表现为两个方面:评估学生不同类型的知识,这些知识类型直接指向课堂教学与学习中存在的各种问题;探索教师知识对学生学习进展的影响,表明哪种教学行为是有效的。     

Coordinating Multiple Representations in a Reform Calculus Textbook

Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions of coordination tasks by chapter and for the type of task demanded (perception vs. construction). Results suggest that different coordination tasks are used earlier and later in learning and for different topics, as well as for specific pedagogical and scaffolding purposes. For example, the algebra-to-text coordination task was more prevalent in the first chapter, suggesting that students are being eased into calculus content. By contrast, requests to construct graphs from algebraic expressions were emphasized in later chapters, suggesting that students are being pushed to think more conceptually about functions. Our nuanced look at coordination tasks in a reform textbook has implications for research in teaching and learning calculus.     

Training peer-feedback skills on geometric construction tasks: role of domain knowledge and peer-feedback levels

Peer feedback is widely used to train assessment skills and to support collaborative learning of various learning tasks, but research on peer feedback in the domain of mathematics is limited. Although domain knowledge seems to be a prerequisite for peer-feedback provision, it only recently received attention in the peer-feedback literature. In this study, preservice mathematics teachers (N = 43) were involved in a peer-feedback training in which they evaluated geometric construction tasks and were (a) trained to provide peer feedback on different levels (i.e. task, process and self-regulation) and (b) scaffolded with worked examples, feedback provision prompts and evaluation rubrics during the training. A quasi-experimental mixed design was implemented with domain knowledge as the between-subject factor and measurement occasion as the within-subject factor. Students’ peer-feedback provision skills and their beliefs about peer-feedback provision were measured before and after the training. Students with high and medium domain knowledge provided more peer feedback at the self-regulation level, whereas those low in domain knowledge provided more peer feedback at the task-level after the training. Students’ beliefs about peer-feedback provision became less positive after the training, regardless of the level of their domain knowledge.     

Alpha spectral power and coherence in the patients with mild cognitive impairment during a three-level working memory task

Objective: The functional relationship between calculated alpha band spectral power and inter-/intra-hemispheric coherence during a three-level working memory task of patients with mild cognitive impairment (MCI) was investigated. Methods: Subjects included 35 MCI patients according to the DSM-IV criteria (mean age: 62.3, SD: 6.5) and 34 healthy controls (mean age: 57.4, SD: 4.0) were selected from the community at large. All subjects performed a simple calculation and recall task with three levels of working memory load while electroencephalograph (EEG) signal was recorded. The spectral EEG power was computed over alpha1 (8.0～10.0 Hz) and alpha2 (10.5～13.0 Hz) frequency bands and was compared between rest stage and working memory processing stage by two-way ANOVA. Post hoc testing analyzed the differences between each two levels of working memory load during task processing. The inter-hemisphere EEG coherence of frontal (F3-F4), central (C3-C4), parietal (P3-P4), temporal (T5-T6) as well as occipital (O1-O2) was compared between MCI patients and normal controls. The EEG signals from F3-C3, F4-C4, C3-P3, C4-P4, P3-O1, P4-O2, T5-C3, T6-C4, T5-P3 and T6-P4 electrode pairs resulted from the intra-hemispheric action for alpha1 and alpha2 frequency bands. Result: There was significantly higher EEG power from MCI patients than from normal controls both at rest and during working memory processing. Significant differences existed between rest condition and three-level working memory tasks (P<0.001). The inter-and intra-hemispheric coherence during working memory tasks showed a "drop to rise" tendency compared to that at rest condition. There was significantly higher coherence in MCI patients than in the controls. When task difficulties increased, the cortical connectivity of intra-hemispheric diminished while the inter-hemispheric connectivity dominantly maintained the cognitive processing in MCI patients. Conclusion: The results of the present study indicate that the alpha frequency band may be the characteristic band in distinguishing MCI patients from normal controls during working memory tasks. MCI patients exhibit greater inter-hemispheric connectivity than intra-hemispheric connectivity when memory demands increase. MCI patients mobilize a compensatory mechanism to maintain the processing effectiveness while the processing efficiency is reduced.     

Studying the classroom implementation of tasks: High-level mathematical tasks embedded in ‘real-life’ contexts

Mathematical tasks embedded in real-life contexts have received increased attention by educators, in part due to the considerable levels of student engagement often triggered by their motivational features. Nevertheless, it is often challenging for teachers to implement high-level (i.e., cognitively demanding), real-life tasks in ways that exploit their motivational features without overshadowing the mathematics involved. This paper proposes an analytic framework for describing and explaining the classroom implementation of different kinds of tasks, and uses this framework to analyse a classroom episode where a secondary teacher implemented with low fidelity a high-level, real-life mathematical task. Implications for research are discussed.     

The impact of task-based professional development on teachers’ practices and beliefs: a design research study

The paper describes a design research study with experienced mathematics teachers of post-16 low-attaining students. The study explored the use of generic collaborative mathematics tasks in promoting professional development. I describe the theoretical basis for the design of the tasks, the tasks themselves and a professional development programme in which they were used by teachers drawn from 44 colleges. Teachers that used many of the tasks reported profound changes to their practices and this was confirmed student reports and classroom observation. Teachers’ beliefs about mathematics, teaching and learning both constrained the ways the tasks were implemented and were challenged by the affective and cognitive outcomes. The variation in these outcomes are examined and explained from the evidence.     

Research on mathematics teachers as partners in task design

Mathematical tasks and tools, including tasks in the form of digital tools, are key resources in mathematics teaching and in mathematics teacher education. Even so, the ‘design’ of mathematical tasks is perceived in different ways: sometimes seen as something distinct from the teaching and learning process, and sometimes as integral to it. Whilst task design has often been carried out by designers or mathematicians (perhaps as textbook authors), the focus for this review article is research that has involved mathematics teachers as partners in the design of tasks. The article provides a state-of-the-art review of relevant literature and is presented under three headings that consider, in turn, the role of mathematical ‘tasks’; the nature of ‘task design’; and the notion of ‘partnerships for task design’ in mathematics education. Subsequently, we present current research that is providing new insights into tasks, task design, and task design partnership. Based on this, we argue that ‘task design’ needs to pay particular attention to what to design, which tools are necessary or beneficial, and under what conditions; digital tools and task resources offer particular affordances that traditional resources cannot provide; and not only do teachers benefit from being partners in task design (in terms of their professional learning) but without their involvement some aspects of task design would most likely be neglected.     

A Model for Assessing Higher Order Thinking in Statistics

ABSTRACT

As in other areas of the school curriculum, the teaching, learning and assessment of higher order thinking in statistics has become an issue for educators following the appearance of recent curriculum documents in many countries. These documents have included probability and statistics across all years of schooling and have stressed the importance of higher order thinking across all areas of the mathematics curriculum. This paper reports on a pilot project which applied the theoretical framework for cognitive development devised by Biggs and Collis to a higher order task in data handling in order to provide a model of student levels of response. The model will assist teachers, curriculum planners and other researchers interested in increasing levels of performance on more complex tasks. An interview protocol based on a set of 16 data cards was developed, trialed with Grade 6 and 9 students, and adapted for group work with two classes of Grade 6 students. The levels and types of cognitive functioning associated with the outcomes achieved by students completing the task in the two contexts will be discussed, as will the implications for classroom teaching and for further research.     

建构数学学习发展性评价体系的实践与思考

发展性评价体系中一个重要的工作就是开发多样化的评价工具和方法。在课题研究中我们重点开展了如下七个方面的评价工具的开发与实践:日常观察,等级和分项测试,成长记录袋,表现性任务,通过浓缩孩子成长特点的评价,二次评价,通过家长、学校合作,取得了较好的效果。     

Opportunities to Learn and Achievement in Mathematics in a Sample of Sixth Grade Students in Lima, Peru

In recent years, the mathematics achievement of Peruvian students has been evaluated in three national and two international studies. The results in all cases suggest very poor learning. A similar situation is found in many developing countries. In this study, we analyzed the opportunities to learn (OTL) mathematics of sixth grade students from 22 public schools in Lima, Peru. OTL were defined in this study as curriculum coverage, cognitive demand of the tasks posed to the students, percent of mathematical exercises that were correct and quality of feedback. These variables were coded in the workbooks and notebooks of the students, which were gathered at the end of the school year (at the same time, the mathematics achievement test was administered). The results show that: (a) regarding OTL, less than half of the exercises available in the workbooks were solved, teachers overemphasize some topics of the national curriculum ({i.e.} related to Number and Number Sense), they pose tasks that are at very low levels of cognitive demand, and it is common to find mistakes in the students' answers to problems that have no feedback (or even worse, the feedback is wrong); (b) students in relatively poorer, multigrade classrooms have less OTL; (c) OTL, as defined earlier, is positively associated with achievement.A preliminary version of this paper received the Research Medal in the category Knowledge, Education andTechnology at the Annual Meeting of the Global Development Network (GDN) in Cairo, in January 2003.     

数学课堂教学任务认知水平的评价模式探究简

数学课堂教学任务认知水平是对学生学习效果进行过程性评价的重要手段,通过方差分析及回归分析影响不同学业成绩、年级、性别学生数学课堂教学任务认知水平的各种因素及其重要程度.不同学业成绩和性别的学生在任务认知水平及多数影响因素方面差异显著;不同年级的学生在任务认知水平及多数影响因素方面差异不显著;影响任务认知水平最重要的三种因素是数学教学情境、数学课堂交流、数学知识应用,构建科学的评价任务认知水平的模式,为测量学生任务认知水平和实施高水平认知要求的教学任务提供可操性方法.     

Children's self‐regulated learning profile in language and mathematics: The role of task value beliefs

This study explored the self‐regulated learning (SRL) profile of upper elementary (fifth and sixth grade) school children who were differentiated in their task value beliefs (low and high) in language and mathematics. Students' SRL profile involved their teachers' ratings of achievement outcomes and SRL behaviors. The subscale of task value beliefs from the Motivational Self‐Regulated Learning Questionnaire (MSLQ) was administered to the children, and their teachers completed a battery of scales and measurements concerning students' achievement and SRL behaviors. The results confirm previous evidence indicating that SRL involves high levels of motivation, metacognition, and strategic action. Furthermore, the results support the domain‐specific character of task value beliefs. Differences in teachers' evaluations about the achievement outcomes and SRL behaviors regarding the two groups of students (low and high in task value beliefs) were found significant mostly in the domain of mathematics. Students with high value beliefs in mathematics were described as more cognitively, metacognitively, and motivationally competent learners as compared to students with lower value beliefs. The results suggest that future intervention studies should focus on strengthening task value beliefs in “threatening” school subjects, such as mathematics, from the elementary school years. © 2010 Wiley Periodicals, Inc.     

Imagery deficits in nonverbal learning disabilities

This study reports the observations gathered from 11 children referred to consulting services because of learning difficulties at school and diagnosed with nonverbal learning disabilities (NVLD). These children had an average verbal IQ, but a WISC-R performance IQ lower than the verbal IQ by at least 15 points and experienced difficulties especially in mathematics and drawing. The children completed a battery of four tasks requiring visuospatial working memory and visual imagery: a memory task composed of pictures and their positions (Pictures task), a task that required them to memorize the positions filled in a matrix (Passive Matrix task), a task that required them to imagine a pathway along a matrix (Active Matrix task) and a task that required them to learn groups made up of three words, using a visual interactive imagery strategy (TV task). In comparison to a control group of 49 children, children with NVLD scored lower in all the tasks, showing deficits in the use of visuospatial working memory and visual imagery. By contrasting subgroups of children of different ages in the control group, it was possible to show that some tasks did not show a clear developmental trend. Thus the deficits shown by the children with NVLD cannot simply be attributed to a developmental delay of these children, but seem to reflect a more severe disability.     

Poor Performance in Mathematics: Is there a basis for a self‐worth explanation for women?

The self‐worth theory of achievement motivation holds that in situations in which poor performance is likely to reveal low ability, certain students (known as self‐worth protective students) intentionally withdraw effort in order to avoid the negative implications of poor performance in terms of damage to self‐worth. In this study, evidence of self‐worth protection was sought where it is perhaps least likely to be seen: in women in mathematics, where the negative effects of stereotype threat are suggested. Links between self‐worth protection and performance on three maths tasks (a computational task, an algebraic task, and the Tower of Hanoi, a game‐like task tapping into mathematical skills yet devoid of mathematical signs and symbols) and a verbal task (a remote associates task) were examined in an intellectually evaluative situation (following failure) and in an intellectually non‐evaluative situation (following success). Relative to low maths self‐worth protective students, high maths self‐worth protective students performed poorly on both mathematical tasks, but not on either the Tower of Hanoi or the remote associates test. Collectively, findings are consistent with the conclusion that a self‐worth explanation can be applied to the poor performance of women in mathematics. Critical eliciting conditions appear to be mathematical signs and symbols and the presence of evaluative threat.     

Extended analyses: Promoting mathematical inquiry with preservice mathematics teachers

This paper describes the implementation of extended analysis tasks (EATs) in a required mathematics course for prospective high school mathematics teachers, and investigates the mathematical discovery promoted through engagement with these tasks. Extended analysis tasks are designed to move students beyond the problem context to the underlying mathematical structure of the problem. The results of this study revealed that with careful implementation, these tasks were effective for generating powerful mathematical thinking in preservice mathematics teachers, improving their problem solving, and generally enhancing the habits-of-mind exhibited by successful mathematicians (Carlson & Bloom, 2005).     

Examination of Bivariate Data Tasks in US High School Textbooks Through the Statistical Investigation and Cognitive Demands Frameworks

Through the lenses of statistical investigations and cognitive demands, we examined bivariate data tasks offered in US high school mathematics textbook series—a popular representative of three curriculum types: traditional, integrated, and hybrid. We developed a framework grounded in literature of association topics for the inclusion and exclusion of tasks. Using the Guidelines for Assessment and Instruction of Statistics Education (GAISE) framework, textbook tasks were coded for four investigation components (formulate questions, collect data, analyze data, and interpret results) and levels of statistical sophistication, as well as levels of cognitive demand as suggested by the Mathematical Complexity framework. Across the three series 582 statistical association tasks, all components of statistical investigation were evident with different levels of treatment: (a) all questions for statistical investigations were provided by textbook authors; (b) tasks rarely afforded student opportunities to collect data; and (c) nearly all of the tasks required students to analyze data and most required them to interpret results. Tasks in the integrated series were more numerous (n = 246) and required higher levels of mathematical complexity and statistical sophistication than tasks in the traditional and hybrid series. The vast majority of tasks were coded at the GAISE Level B for analyze data and interpret results and moderate level for mathematical complexity. Further analyses show the concordance between the developmental levels for statistical sophistication and mathematical complexity. Suggestions for curriculum development, content analysis, and future research are provided.     

Reliability and validity of a mathematics performance assessment

This study evaluated the reliability and validity of a performance assessment designed to measure students' thinking and reasoning skills in mathematics. The QUASAR Cognitive Assessment Instrument (QCA1) was administered to over 1.700 sixth and seventh grade students of various ethnic backgrounds in six schools that are participating in the QUASAR project. The consistency of students' responses across tasks and the validity for inferences drawn from the scores on the assessment to the more broadly-defined construct domain were examined. The intertask consistency and the dimensionality of the assessment was assessed through the use of polychoric correlations and confirmatory factor analysis, and the generalizability of the derived scores was examined through the use of generalizability theory. The results from the confirmatory factor analysis indicate that a one-factor model fits the data for each of the four QCAI forms. The major findings from the generalizability studies (person x task and person x rater x task) indicate that, for each of the four forms, the person x task variance component accounts for the largest percentage of the total variability and the percentage of variance accounted for by the variance components that include the rater effect is negligible. The variance components that-include the rater effect were negligible. The generalizability and dependability coefficients for the person x task decision studies (n_t, = 9) range from .71-.84. These results indicate that the use of nine tasks may not be adequate for generalizing to the larger domain of mathematics for individual student level scores. The QUASAR project, however, is interested in assessing mathematics achievement at the program level not the student level; therefore, these coefficients are not alarmingly low.     

Teaching primary mathematics with challenging tasks: How should lessons be structured?

Engaging students in a challenging (cognitively demanding) task and launching a mathematics lesson with a task before instruction are two characteristics of a reform-oriented approach to mathematics instruction often considered together. The authors systematically contrasted teaching with challenging tasks using a task-first lesson structure with that of a discussion-first lesson structure to three composite classes of first- and second-grade students (n = 73). Subsequent assessments of mathematical performance revealed that the discussion-first lesson structure was somewhat more efficacious in improving fluency performance but both structures similarly improved problem-solving performance. The findings suggest there is more than one way of incorporating challenging tasks into mathematics lessons to produce sizeable learning gains.     

Undergraduate students' understanding of the contraposition equivalence rule in symbolic and verbal contexts

Literature suggests that the type of context wherein a task is placed relates to students' performance and solution strategies. In the particular domain of logical thinking, there is the belief that students have less difficulty reasoning in verbal than in logically equivalent symbolic tasks. Thus far, this belief has remained relatively unexplored in the domain of teaching and learning of mathematics, and has not been examined with respect to students' major field of study. In this study, we examined the performance of 95 senior undergraduate mathematics and education majors in symbolic and verbal tasks about the contraposition equivalence rule. The selection of two different groups of participants allowed for the examination of the hypothesis that students' major may influence the relation between their performance in tasks about contraposition and the context(symbolic/verbal) wherein this is placed. The selection of contraposition equivalence rule also addressed a gap in the body of research on undergraduate students' understanding of proof by contraposition. The analysis was based on written responses of all participants to specially developed tasks and on semi-structured interviews with 11 subjects. The findings showed different variations in the performance of each of the two groups in the two contexts. while education majors performed significantly better in the verbal than in the symbolic tasks, mathematics majors' performance showed only modest variations. The results call for both major- and context- specific considerations of students' understanding of logical principles, and reveal the complexity of the system of factors that influence students' logical thinking.