Profiles of algebraic competence

The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map algebraic symbols onto word problems was evaluated. Two weeks later, unassisted algebraic problem-solving tests were completed. Cluster analyses revealed four meaningful, relatively well-ordered, arithmetic/algebraic competence profiles reflecting partial knowledge states associated with the acquisition of algebraic understanding. The findings are discussed in terms of the conceptual changes associated with the acquisition of algebraic competence.     

Mathematical practices in a technological workplace: the role of tools

This paper investigates the role of tools in the formation of mathematical practices and the construction of mathematical meanings in the setting of a telecommunication organization through the actions undertaken by a group of technicians in their working activity. The theoretical and analytical framework is guided by the first-generation activity theory model and Leont’ev’s work on the three-tiered explanation of activity. Having conducted a 1-year ethnographic research study, we identified, classified, and correlated the tools that mediated the technicians’ activity, and we studied the mathematical meanings that emerged. A systemic network was generated, presenting the categories of tools such as mathematical (communicative, processes, and concepts) and non-mathematical (physical and written texts). This classification was grounded on data from three central actions of the technicians’ activity, while the constant interrelation and association of these tools during the working process addressed the mathematical practices and supported the construction of mathematical meanings that this group developed from the researchers’ perspective. Technicians’ emerging mathematical meanings referred to place value, spatial, and algebraic relations and were expressed through personal algorithms and metaphorical and metonymic reasoning. Finally, the educational implications of the findings are discussed.     

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.

     

G’/G-展开法及其在Burgers-KdV方程精确解中的应用

本文以数学机械化思想为指导,以计算机代数系统软件Maple为工具,提出了用G’/G-展法来构造非线性孤子方程的行波解。为了验证方法的有效性和优越性,将其应用到Burgers-KdV方程,获得了具有一般形式的新的精确解,其中包括新的双曲函数解以及三角函数解。     

平面解析几何

在解析几何中,人们建立了几何与代数之间的对应关系．几何中的基本概念及定理可以代数地描述和证明;代数中的基本概念和过程可以几何地解释．当一个几何问题看起来比较困难时,可考虑相应的代数问题．如果在这个特殊情况下,代数工具更加有效的话,我们就先代数地解决这个问题,而后把结果翻译成几何语言．但常常是沿相反的方向进行的．     

The promises of numeracy

The aim of this study is to better understand the notion of early algebraic thinking by describing differences in grade 4–7 students’ thinking about basic algebraic concepts. To achieve this goal, one test that involved generalized arithmetic, functional thinking, and modeling tasks, was administered to 684 students from these grades. Quantitative analysis of the data yielded four distinct groups of students demonstrating a wide range of performance in these tasks. Qualitative analysis of students’ solutions provided further insight into their understanding of basic algebraic concepts, and the nature of the processes and forms of reasoning they utilized. The results showed that students in each group were able to solve different number and types of tasks, using different strategies. Results also indicated that students from all grades were present in each group. These findings suggest the presence of a consistent trend in the difficulty level across early algebraic tasks which may support the existence of a specific developmental trend from more intuitive types of early algebraic thinking to more sophisticated ones.     

Revisiting the pressure-volume law in history-what can it teach us about the emergence of mathematical relationships in science?

In this article the pressure-volume law is reviewed from the point of view of its historical emergence from 1644–1662 and its application in the science classroom. It is contended that mathematical laws in science have value as rich conceptual tools in addition to their role in computation. A classification scheme for algebraic mathematical expressions, based on their historical context, is proposed as a means of assigning significance to the mathematical expressions commonly used in science.     

概念图及其在数学学习中的现实意义

概念图是用来组织与表征知识的工具.在数学学习中概念图具有重要的现实意义,它可以激励学习者主动建构概念,培养元认知知识和自主性,激发猜想并检验猜想等.学习制作概念图时,要注意:布置任务的形式由结构化逐渐转向弱结构化;概念图制作时应首先集中在一个主题,然后再与其它概念图或主题相连接.     

Sense making in the context of algebraic activities

This article concerns student sense making in the context of algebraic activities. We present a case in which a pair of middle-school students attempts to make sense of a previously obtained by them position formula for a particular numerical sequence. The exploration of the sequence occurred in the context of two-month-long student research project. The data were collected from the students’ drafts, audiotaped meetings of the students with the teacher and a follow-up interview. The data analysis was aimed at identification and characterization of the algebraic activities in which the students were engaged and the processes involved in the students’ sense-making quest. We found that sense-making process consisted of a sequence of generational and transformational algebraic activities in the overarching context of a global, meta-level activity, long-term problem solving. In this sense-making process, the students: (1) formulated and justified claims; (2) made generalizations, (3) found the mechanisms behind the algebraic objects (i.e., answered why-questions); and (4) established coherence among the explored objects. The findings are summarized as a suggestion for a four component decomposition of algebraic sense making.     

Online educational videos according to specific didactics: the case of mathematics / Los Vídeos educativos en línea desde las didácticas específicas: el caso de las matemáticas

Abstract

The vast number of online educational videos available at the moment has generated an emerging area of research concerning their level of suitability. This study considers the epistemic quality of educational videos on mathematics, focusing on the specific content of directly proportional distributions. A qualitative study is used, based on the application of theoretical and methodological tools from the onto-semiotic approach to knowledge and mathematics instruction, principally the notion of epistemic suitability and the identification of algebraic levels. The sample consists of the 31 most popular videos in Spanish on YouTube? on directly proportional distributions. Analysis reveals interesting results on these kinds of resources. In general, it is observed that they are weak in epistemic suitability, which does not seem to affect their level of popularity. Moreover, the existence of videos with inaccurate arguments or incorrect procedures, together with the diversity of algebraic levels used, indicates that teachers should be careful when selecting them and only recommend those that better suit their students’ needs.     

Closing the gap on the map: Davydov’s contribution to current early algebra discourse in light of the 1960s Soviet debates over word-problem solving

Recent scholarship around teaching elementary mathematics supports the learning of early algebra with 5- to 12-year olds. However, in spite of the recognition of the affordances of early algebra, issues about how to introduce it remain open. Within this context, Davydov’s work is often cited as a source of impressive demonstration of young learners’ capacity for algebraic thinking. This work requires further exploration in order to yield a clearer picture of a very particular teaching approach, which focuses on early abstractions and symbolic language. We argue that in order to fully understand how Davydov’s work contributes to current conversations and what Davydov was trying to do, we need to shed light on the context- and time-specific discourse of the 1960 Soviet educational reforms that made it possible for Davydov to develop his vision about algebraic thinking and to set in motion appropriate teaching approaches for young learners. In this paper, we look back to the Soviet debates that unfolded in Russia on the integration of early algebra in elementary school word-problem solving. Drawing on these debates and the results of Davydov’s school experiments, we lay out the developmental axes of capacity building. This can be done by emphasizing ascent from the abstract to the concrete using a variety of representational modeling tools to support the emergence of algebraic thinking while targeting particular habits of mind within carefully designed learning activities. We conclude with some insights about current arithmetic-algebra debates, and how these could be enriched and deepened by Davydov’s work, which yet remains open to future discussion and reflection.

     

指标为3的自对偶拟循环码

设l为非零自然数,R=Fq[x]/〈xm-1〉,这里Fq为有限域.视拟循环码为代数Rl上的一个子模,利用模上的Grbner基理论及拟循环码的代数结构作为工具,得到了两个主要定理:在l=3的情况下,把一个关于rPOT项序的Grbner基生成集转化为一个关于POT项序的既约Grbner基生成集;指标为3的拟循环码是自对偶码的充要条件.     

ENACTED TYPES OF ALGEBRAIC ACTIVITY IN DIFFERENT CLASSES TAUGHT BY THE SAME TEACHER

The aim of this study was to examine how teachers enact the same written algebra curriculum materials in different classes. The study addresses this issue by comparing the types of algebraic activity (Kieran, 2004) enacted in two 7th grade classes taught by the same teacher, using the same textbook. Data sources include lesson observations and an interview with the teacher. The findings show that students in the two classes were offered somewhat different algebraic experiences. At one school, more emphasis was placed on global/meta-level activities (activities that are not exclusive to algebra and suggest general mathematical processes), whereas at the other school, more emphasis was placed on transformational activities (“rule-based” algebraic activities). Analysis of the sources of the differences related to the ways in which the teacher used and enacted the curriculum materials in the two classes revealed that these were linked to the teacher’s attempts to be attentive to the students in the class and to the nature of the students’ work.     

Providing a computer based framework for algebraic thinking

This paper describes and presents the findings of a study which aimed to trace the development of pupils' use and understanding of algebraic ideas within a Logo programming context relating this to their use and understanding of similar ideas within a non-computational context. The research consisted predominantly of a three year longitudinal case study of four pairs of pupils (aged 11–14) programming in Logo during their normal school mathematics lessons. The data included video recordings of all the case study pulils' Logo sessions, and individually presented Logo and algebra structured interviews. The overall conclusion of this research is that Logo experience does enhance pupils' understanding of algebraic ideas, but the links which pupils make between Logo and algebra depend very much on the nature and extent of their Logo experience.     

Preparing elementary prospective teachers to teach early algebra

Researchers have argued that integrating early algebra into elementary grades will better prepare students for algebra. However, currently little research exists to guide teacher preparation programs on how to prepare prospective elementary teachers to teach early algebra. This study examines the insights and challenges that prospective teachers experience when exploring early algebraic reasoning. Results from this study showed that developing informal representations for variables and unknowns and learning about the two interpretations of the equal sign were meaningful new insights for the prospective teachers. However, the prospective teachers found it a conceptual challenge to identify the relationships contained in algebraic expressions, to distinguish between unknowns and variables, to bracket their knowledge of formal algebra and to represent subtraction from unknowns or variables. These findings suggest that exploring early algebra is non-trivial for elementary prospective teachers and likely necessary to adequately prepare them to teach early algebra.     

Focusing on informal strategies when linking arithmetic to early algebra

In early algebra students often struggle with equation solving. Modeled on Streefland's studies of students' own productions a prototype pre-algebra learning strand was designed which takes students' informal (arithmetical) strategies as a starting point for solving equations. In order to make available the skills and tools needed for manipulating equations, the students are stimulated and guided to develop suitable algebraic language, notations and reasoning. One of the results of the study is that reasoning and symbolizing appear to develop as independent capabilities. For instance,students in grades 6 and 7 can solve equations at both a formal and an informal level, but formal symbolizing has been found to be a major obstacle. This revised version was published online in August 2006 with corrections to the Cover Date.     

JUSTIFICATIONS AND EXPLANATIONS IN ISRAELI 7TH GRADE MATH TEXTBOOKS

This study analyzes six seventh grade Israeli mathematics textbooks, examining (1) the extent to which students are required to justify and explain their mathematical work, and (2) whether students are asked to justify a mathematical claim that is stated by the textbook or a mathematical claim that they themselves generated when solving a problem. Two different units of analysis were used to analyze two central topics in the seventh grade curriculum as follows: (1) equation solving in algebra and (2) triangle properties in geometry. The findings indicate that all six textbooks had considerably larger percentages of geometric tasks than algebraic tasks, which required students to justify or explain their mathematical work. Moreover, considerable differences were found among the six textbooks regarding the percentages of tasks that required students to justify and explain in both topics, but more so with the algebraic topic. Analysis of whether the textbook tasks required students to justify a mathematical claim that is stated by the textbook or a mathematical claim that the students themselves generated also revealed substantial differences among the textbooks. These findings are discussed, as well as the research methods used, in light of relevant literature.

     

Perspectives on the Origins of Life in Science Textbooks from a Christian Publisher: Implications for Teaching Science

This study analyzes six seventh grade Israeli mathematics textbooks, examining (1) the extent to which students are required to justify and explain their mathematical work, and (2) whether students are asked to justify a mathematical claim that is stated by the textbook or a mathematical claim that they themselves generated when solving a problem. Two different units of analysis were used to analyze two central topics in the seventh grade curriculum as follows: (1) equation solving in algebra and (2) triangle properties in geometry. The findings indicate that all six textbooks had considerably larger percentages of geometric tasks than algebraic tasks, which required students to justify or explain their mathematical work. Moreover, considerable differences were found among the six textbooks regarding the percentages of tasks that required students to justify and explain in both topics, but more so with the algebraic topic. Analysis of whether the textbook tasks required students to justify a mathematical claim that is stated by the textbook or a mathematical claim that the students themselves generated also revealed substantial differences among the textbooks. These findings are discussed, as well as the research methods used, in light of relevant literature.     

BCK-代数的Ω-模糊代数理想

给定一个集合Ω,引入了有界可交换的BCK-代数的Ω-模糊代数理想之概念,研究了它的一些相关性质,并给出了它几个特征,讨论了有界可交换的BCK-代数的Ω-模糊理想与Ω-模糊代数理想的关系．讨论了有界可交换的BCK-代数的模糊代数理想与Ω-模糊代数理想的相互构造．     

拉格朗日的代数方程求解理论及其影响

拉格朗日的代数方程求解理论是整个代数方程求解史中不可或缺的一部分,并且该理论对以后的代数学家产生了重要的影响。为展示拉格朗日代数方程求解理论的内容,说明该理论产生的深远影响,从原始文献出发,叙述了拉格朗日的代数方程求解理论的内容,重点阐述了该理论产生的重要影响。因此,清楚拉格朗日的代数方程求解理论不仅有利于了解辅助方程理论、置换思想的内涵,更有利于清楚整个代数方程的求解历史。