Parsing the notion of algebraic thinking within a cognitive perspective

There is a growing consensus that algebra is an important aspect of mathematics teaching and learning and several abilities are required in order students to have successful performance in algebra. The present study uses insights from the domain of psychology to enrich what is currently known in the domain of mathematics education about the relationship of algebraic thinking with abilities involved in fundamental cognitive processes. In total, 190 students between the ages of 13–17 years old were tested through two tests. The first test addressed four types of cognitive systems which are responsible for the representation and processing of different types of relations in the environment: the spatial-imaginal, the causal-experimental, the qualitative-analytic and the verbal-propositional. The second test addressed algebraic thinking. The results support the key role of the four types of cognitive processes in students’ algebraic thinking. The results also suggest that abilities involved in the four types of cognitive processes predict algebraic thinking abilities, irrespective of the age of the students.     

状态方程的约当化方法与分析

在自动控制及计算机应用研究中往往遇到提出的不用特征向量,直接根据方阵A的特征值的代数重数、退化度和其特征值指数构造其相似约当阵,再根据线性变换关系求出变换阵P,进而对状态方程进行约当化的方法,经理论和实践证明,该方法简单有效。     

一类曲线的定义域（英文）

设K为代数闭域k的有限生成扩域.C:f(x)=ayn为K上曲线,其中f是k上至少有3个单零点的多项式且n>3是正整数,n不是域k的特征的倍数,再设a■Kn,那么曲线C不能定义在k上,即曲线C:(x)=ayn不会k(a)同构于一条k上的曲线.     

Macro-Situation and Numerical Knowledge Building: The Role of Pupils’Didactic Memory in Classroom Interactions

This paper is based on a long-term didactic engineering about division problems (only in a numerical setting) at primary school. Situations and students’ work are analyzed by means of a double theoretical framework: the theory of situations and the theory of conceptual fields (Vergnaud 1991). The analysis focuses mainly on classroom interactions and on the didactic memory from both the teacher perspective and the learner perspective: in particular, it not only investigates how didactic memory is managed by the teacher, but also how students recall past events or reread those events in a-didactic situations.     

Estimating the globally attractive set and positively invariant set of a unified chaotic system

By constructing two suitable generalized Lyapunov functions, we derived a generalized ellipsoidal estimate of the globally attractive set and positively invariant set of the unified chaotic system with the parameters α=1/29 and 1/29〈α〈2/29, respectively, which extends some related results of Li, et al. [Li DM, Lu JA, Wu XQ, Chen GR, Estimating the global basin of attraction and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications, 2006, 323（2）： 844-853]. The theoretical results obtained in this paper will find wide application in chaos control and synchronization.     

仿射不变量在初等几何中的应用

简要叙述克莱因(F.Klein)几何观点,介绍几个仿射不变量,探讨如何利用仿射变换中的不变量解决一些初等几何问题,体现了高等几何对初等几何有重要的指导意义。     

平面射影几何中二维射影变换不变元素的分布

基于平面射影几何中二维射影变换不变元素的存在性及其求法,讨论了二维射影变换不变点与不变直线结合关系的分布特征,从理论上分析了二维射影变换不变元素的结构,并给出了不变元素的分布对射影变换的影响．     

求解三维泊松方程的新瀑布型代数二重网格法

文章针对三维泊松方程,讨论四面体有限元离散格式下的代数多重网格法.通过采用一种快速的粗化算法,提出了一种新的插值算子和迭代终止条件,设计了一种新的瀑布型代数二重网格法.数值实验表明这种新算法计算时间更短.     

Keep DRAGging ON: Is solving more problems in DragonBox 12+ associated with higher mathematical performance during the COVID-19 pandemic?

Prior research has shown that game-based learning tools, such as DragonBox 12+, support algebraic understanding and that students' in-game progress positively predicts their later performance. Using data from 253 seventh-graders (12–13 years old) who played DragonBox as a part of technology intervention, we examined (a) the relations between students' progress within DragonBox and their algebraic knowledge and general mathematics achievement, (b) the moderating effects of students' prior performance on these relations and (c) the potential factors associated with students' in-game progress. Among students with higher prior algebraic knowledge, higher in-game progress was related to higher algebraic knowledge after the intervention. Higher in-game progress was also associated with higher end-of-year mathematics achievement, and this association was stronger among students with lower prior mathematics achievement. Students' demographic characteristics, prior knowledge and prior achievement did not significantly predict in-game progress beyond the number of intervention sessions students completed. These findings advance research on how, for whom and in what contexts game-based interventions, such as DragonBox, support mathematical learning and have implications for practice using game-based technologies to supplement instruction.

Practitioner notes

What is already known about this topic

• DragonBox 12+ may support students' understanding of algebra but the findings are mixed.
• Students who solve more problems within math games tend to show higher performance after gameplay.
• Students' engagement with mathematics is often related to their prior math performance.

What this paper adds

• For students with higher prior algebraic knowledge, solving more problems in DragonBox 12+ is related to higher algebraic performance after gameplay.
• Students who make more in-game progress also have higher mathematics achievement, especially for students with lower prior achievement.
• Students who spend more time playing DragonBox 12+ make more in-game progress; their demographic, prior knowledge and prior achievement are not related to in-game progress.

Implications for practice and/or policy

• DragonBox 12+ can be beneficial as a supplement to algebra instruction for students with some understanding of algebra.
• DragonBox 12+ can engage students with mathematics across achievement levels.
• Dedicating time and encouraging students to play DragonBox 12+ may help them make more in-game progress, and in turn, support math learning.