From “You have to have three numbers and a plus sign” to “It’s the exact same thing”: K–1 students learn to think relationally about equations

This research shares progressions in thinking about equations and the equal sign observed in ten students who took part in an early algebra classroom intervention across Kindergarten and first grade. We report on data from task-based interviews conducted prior to the intervention and at the conclusion of each school year that elicited students’ interpretations of the equal sign and equations of various forms. We found at the beginning of the intervention that most students viewed the equal sign as an operational symbol and did not accept many equations forms as valid. By the end of first grade, almost all students described the symbol as indicating the equivalence of two amounts and were much more successful interpreting and working with equations in a variety of forms. The progressions we observed align with those of other researchers and provide evidence that very young students can learn to reason flexibly about equations.     

Success-factors in transition to university mathematics

This study examines different factors' relative importance for students' performance in the transition to university mathematics. Students' characteristics (motivation, actions and beliefs) were measured when entering the university and at the end of the first year. Principal component analysis revealed four important constructs: Self-efficacy, Motivation type, Study habits and Views of mathematics. Subsequently, orthogonal partial least squares (OPLS) analysis was used for measuring the constructs' ability to predict students' university mathematics grades. No individual constructs measured at the time of entrance predicted more than 5% of the variation. On the other hand, jointly they predicted 14%, which is almost in pair with upper secondary grades predicting 17%. Constructs measured at the end of the first year were stronger predictors, jointly predicting 37% of the variation in university grades, with Self-efficacy (21%) and Motivation (12%) being the two strongest individual predictors. In general, Study habits were not important for predicting university achievement. However, for students with low upper secondary grades, the textbook and interaction with peers, rather than internet-based resources, contributed positively to achievement. The association between Views of mathematics and performance was weak for all groups and non-existing for students with low grades.     

旱涝预测的演化建模方法

发展了一种旱涝预测的演化建模方法, 数值试验结果表明, 该方法能够从历史资料中准确地挖掘出气候要素演化的主要动力学特征, 所建立的模型不仅能够较好地模拟要素的历史演变情况, 而且还能够对未来的演变趋势进行准确的预测. 对于演化建模所得到的误差较大的模型, 对其预测误差序列进行二次演化建模可明显提高模型的预测精度.     

Gender and Racial Differences: Development of Sixth Grade Students' Geometric Spatial Visualization within an Earth/Space Unit

This study investigated sixth‐grade middle‐level students' geometric spatial development by gender and race within and between control and experimental groups at two middle schools as they participated in an Earth/Space unit. The control group utilized a regular Earth/Space curriculum and the experimental group used a National Aeronautics and Space Administration‐based curriculum. The quantitative data sources included the Lunar Phases Concept Inventory, Geometric Spatial Assessment, and the Purdue Spatial Visualization–Rotation Test. The results indicated the experimental males and females, and the students of color and white students in the experimental group showed significant gains in their understanding of geometric spatial visualization from pre‐ to post‐implementation. However, for the control group, the significant gains were limited to the males and the white students. The findings reveal that support is needed for males, females, and all racial groups to have the opportunity to develop their spatial reasoning, which in turn, increases students' scientific understanding.     

Working at scale to initiate ongoing validation of learning trajectory-based classroom assessments for middle grade mathematics

The paper reports on the design and validation argument for classroom assessments within a digital diagnostic assessment system built on learning trajectories (LTs). It consists of a learning map of nine big ideas, 25 relational learning clusters, and 62 LTs for grades 6−8 mathematics. Students take cluster assessments, and teachers use the data to adapt instruction. An ongoing validation process is presented with data for an algebra cluster. Validation among practitioners, learning scientists, and psychometricians is conceptualized as examining and adjusting inter-level, intra-level, and construct-irrelevant variation in measures of item difficulty and deploying item response theory modeling followed by sequential regressions. Using data from 37,000 assessments collected over three years at 3 middle schools, 167 potentially non-conforming items of the 676 calibrated items (24 %) were identified and revised. The paper discusses how the trajectories and map were refined through a combination of data analysis and feedback from practitioners.     

关于高等数学课程分层次教学的实践与思考

高等数学在高校课程体系中具有十分重要的地位．在高等教育进入大众化教育阶段的背景下，本文介绍了我校高等数学课程分层次教学的改革实践与经验，阐述了该教学模式对提高学生的数学素质和对教学质量的作用，同时也指出了存在的主要问题．     

基于最大灰色关联度的聚类方法分析

利用多元统计学的最短距离法原理,定义了不同类别之间的关联度,构造了一种基于最大灰色关联度的聚类方法,建立了最大灰色关联度模型,并对模型进行求解,结果表明,新的聚类方法纠正了已有的灰色关联聚类中聚类原则存在的缺陷,从而为聚类提供了更有说服力且易于计算的方法.最后通过实例验证了该方法的客观性和有效性.     

An investigation of 6th graders’ solutions of Cartesian product problems and representation of these problems using arrays

Two hour-long interviews were conducted with each of 14 sixth-grade students. The purpose of the interviews was to investigate how students solved combinatorics problems, and represented their solutions as arrays. This paper reports on 11 of these students who represented a balanced mix of students operating with two of three multiplicative concepts that have been identified in prior research (Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009). One finding of the study was that students operating with different multiplicative concepts established and structured pairs differently. A second finding is that these different ways of operating had implications for how students produced and used arrays. Overall, the findings contribute to models of students’ reasoning that outline the psychological operations that students use to constitute product of measures problems (Vergnaud, 1983). Product of measures problems are a kind of multiplicative problem that has unique mathematical properties, but researchers have not yet identified specific psychological operations that students use when solving these problems that differ from their solution of other kinds of multiplicative problems (cf. Battista, 2007).     

基于模糊证据推理及改进TOPSIS的多属性群决策方法

针对不同识别框架多属性群决策问题属性准则度量的不确定性、随机性,定义基于梯形模糊数表征的属性准则评价等级相似度量,求解专家决策权重的最优解。对公共识别框架备选方案属性准则采用模糊证据推理过程综合专家评价等级置信度信息;利用可严格区分属性准则评价等级的相似度量,改进TOPSIS方法中备选方案属性准则评价等级置信度距离因子,获取备选方案逼近正负理想解的贴近度。实例分析以某通信企业电信产品市场竞争力评估为例,说明基于模糊证据推理、改进TOPSIS的多属性群决策问题求解过程,从属性准则专家模糊评价等级置信度集中获取直观的待评估产品市场竞争力排序结果,验证该方法解决此类决策问题的可行性与有效性。     

电子光学系统质量的模糊二级综合评判

采用模糊二级综合评判来比较不同屏面位置时的电子光学系统的质量,从而找到一个最佳的屏面位置。