Limitations to teaching children 2 + 2 = 4: typical arithmetic problems can hinder learning of mathematical equivalence

Do typical arithmetic problems hinder learning of mathematical equivalence? Second and third graders (7–9 years old; N= 80) received lessons on mathematical equivalence either with or without typical arithmetic problems (e.g., 15 + 13 = 28 vs. 28 = 28, respectively). Children then solved math equivalence problems (e.g., 3 + 9 + 5 = 6 + __), switched lesson conditions, and solved math equivalence problems again. Correct solutions were less common following instruction with typical arithmetic problems. In a supplemental experiment, fifth graders (10–11 years old; N= 19) gave fewer correct solutions after a brief intervention on mathematical equivalence that included typical arithmetic problems. Results suggest that learning is hindered when lessons activate inappropriate existing knowledge.     

从2+2=4谈起

一位聪明天真的小朋友问妈妈:"为什么2加2等于4 ?"妈妈答:"傻孩子,连这么简单的算术都不懂!"于是这位母亲伸出左手的2个指头,又伸出右手的2个指头,左右的2个指头往一起一并,说:"这就叫2加2,你数一数,看是不是4 ?"孩子勉强点头,接着又问:"可是4是什么玩意儿呢?"妈妈欲言而无语.是呀,如果母亲说这些指头的数目就叫做4,孩子再追问什么叫做999 999 999,那可就不好用指头之类的东西来比划着解释了!     

关于丢番图方程x4+mx2y2+ny4=z2

利用数论方法及Fermat无穷递降法 ,证明了丢番图方程x4 mx2 y2 ny4 =z2 在 (m ,n) =(± 6,-3 ) ,(6,3 ) ,(± 3 ,3 ) ,(-12 ,2 4) ,(± 12 ,-2 4) ,(± 6,15 ) ,(-6,-15 ) ,(3 ,6)仅有平凡整数解 ,并且获得了方程在 (-6,3 ) ,(12 ,2 4) ,(3 ,-6) ,(-6,3 3 )时的无穷多组正整数解的通解公式 ,从而完善了Aubry等人的结果     

关于丢番图方程x4+mx2y2 +ny4=z2(2)

利用 fermat无穷递降法证明了方程 x4+mx2 y2 +ny4=z2 在 (m,n) =(6 ,- 30 ) ,(- 12 ,15 6 ) ,(- 6 ,- 6 ) ,(12 ,6 0 )时均无正整数解 ,并且获得了方程在 (m ,n) =(- 6 ,± 30 ) ,(- 12 ,6 0 ) ,(12 ,- 84) ,(6 ,- 6 ) ,(12 ,15 6 )时的无穷多组正整数解的通解公式 ,从而完善了 Aubry等人的结果 .     

Middle school children's strategic behavior: Classification and relation to academic achievement and mathematical problem solving

Theories of problem solving (e.g., Verschaffelet al., 2000) hold strategic behavior centralto processing mathematical word problems. Thepresent study explores 80 sixth- andseventh-grade students' self-reported use of 14categories of strategies (Zimmerman &Martinez-Pons, 1986) and the relationship ofstrategy use to academic achievement,problem-solving behaviors, and problem-solvingsuccess. High and low achievement groupsdiffered in the number of different strategiesand categories of strategies reported but notin overall number of strategies, confidence inusing strategies, or frequency of strategy use.Students whose behaviors evidenced elaborationof the word problem's text reported moreself-evaluation; organizing and transforming;and goal setting and monitoring behavior.Implications for instructional practices thatsupport active stances toward problem solvingare discussed.     

关于丢番图方程x4+my4=z4

利用初等数论方法及Fermat无穷递降法,证明了丢番图方程x′ my′＝z′,在m＝12,—48,42,—168时均无正整数解;在m＝—12,—42,48,168时均有无穷多组正整数解,并进一步得出了其解的通解公式,从而获得了Tijdeman猜想与广义Fermat猜想的进一步结果．     

关于丢番图方程x3+y3=2z2与x3+y3=2z4

获得了丢番图方程x3+y3=2z2的通解公式,证明了方程x3+y3=2z4仅有适合(x,y)=1的整数解x=y=z=1对广义Fermat猜想的研究具有重要作用.     

Almost thinking counterfactually: children's understanding of close counterfactuals

Saying something almost happened indicates that one is considering a close counterfactual world. Previous evidence suggested that children start to consider these close counterfactuals at around 2 years of age (P. L. Harris, 1997), substantially earlier than they pass other tests of counterfactual thinking. However, this success appears to result from false positives. In Experiment 1 (N = 41), 3- and 4-year-olds could identify a character who almost completed an action when the comparison did not complete it. However, in Experiments 1 and 2 (N = 98), children performed poorly when the comparison character completed the action. In Experiment 3 (N = 28), 5- and 6-year-olds consistently passed the task, indicating that they made appropriate counterfactual interpretations of the almost statements. This understanding of close counterfactuals proved more difficult than standard counterfactuals.     

参数为ar＋2=4的距离正则图

利用距离正则图的交叉表及性质对k=10,a1=1的距离正则图的参数进行了讨论,可对其得到的结论进行分类。     

参数为ar+2=4的距离正则图

利用距离正则图的交叉表及性质对k=10,a1=1的距离正则图的参数进行了讨论,可对其得到的结论进行分类.     

方程(x4+y4+z4)2=2(x8+y8+z8)的整数解

给出了方程(x4 y4 z4)2=2(x8 y8 z8)的所有整数解(x,y,z).     

On mathematical understanding: perspectives of experienced Chinese mathematics teachers

Researchers have long debated the meaning of mathematical understanding and ways to achieve mathematical understanding. This study investigated experienced Chinese mathematics teachers’ views about mathematical understanding. It was found that these mathematics teachers embrace the view that understanding is a web of connections, which is a result of continuous connection making. However, in contrast to the popular view which separates understanding into conceptual and procedural, Chinese teachers prefer to view understanding in terms of concepts and procedures. They place more stress on the process of concept development, which is viewed as a source of students’ failures in transfer. To achieve mathematical understanding, the Chinese teachers emphasize strategies such as reinventing a concept, verbalizing a concept, and using examples and comparisons for analogical reasoning. These findings draw on the perspective of classroom practitioners to inform the long-debated issue of the meaning of mathematical understanding and ways to achieve mathematical understanding.     

判别式的整体结构:b2-4ac=(2ax+b)2(下)

在上面的内容中，我们已经看到了判别式法在例2、例3中的“巧思妙解”．下面，我们换一个角度，再以2004年的一道竞赛题的判别式处理来揭示：巧思妙解不是特殊技巧的神秘操作，其背后是对数学知识的深层认识，是题目结构特征的充分挖掘。     

关于不定方程x^2＋64=4y^13的解

利用代数数论的方法,证明了不定方程x^2＋64=4y^13无整数解.     

Chemistry problem solving instruction: a comparison of three computer-based formats for learning from hierarchical network problem representations

Within the cognitive load theory framework, we designed and compared three alternative instructional solution formats that can be derived from a common static hierarchical network representation depicting problem structure. The interactive-solution format permitted students to search in self-controlled manner for solution steps, static-solution format displayed all solutions steps, and no-solution format did not have solution steps. When we matched instructional time across the formats, in relation to the complex molarity problems rather than the dilution problems, differential transfer performance existed between the static-solution or no-solution formats and the interactive-solution format, but not between the static-solution format and no-solution format. The manner in which learners interact with the static-solution and no-solution formats depends on their level of expertise in the chemistry domain. With considerable learner expertise, provision of solution steps may be redundant incurring extraneous cognitive load. Absence of the solution steps may not have left sufficient cognitive capacity for germane cognitive load as some beginning learners lacked the prior knowledge to deduce the solution steps. Searching for solution steps presumably incurred extraneous cognitive load which interfered with learning and hence, in the interactive-solution format, it outweighed the benefit of engaging in self-regulated interaction with the content. Hence, cognitive load theory is a promising tool to predict the mental load associated with learning from the three alternative computer-based instructional formats.     

Components of young children's trait understanding: behavior-to-trait inferences and trait-to-behavior predictions

Trait attribution is central to people's na?ve theories of people and their actions. Previous developmental research indicates that young children are poor at predicting behaviors from past trait-relevant behaviors. We propose that the cognitive process of behavior-to-behavior predictions consists of two component processes: (1) behavior-to-trait inferences and (2) trait-to-behavior predictions. Experiment 1 demonstrates that 4-, 5-, 7-, and 9-year-olds can infer trait labels from behaviors. Experiment 2 demonstrates that 4-, 5-, and 7-year-olds can predict behaviors from trait labels but not from past behaviors. Experiment 3 demonstrates that 4- and 5-year-olds understand traits as predictive and stable over time. Taken together, these three studies show that young children, in possessing component trait-reasoning processes, have a nascent understanding of traits.     

Breaking the habit: engineering students’ understanding of mathematical creativity

ABSTRACT

In higher education, engineering students have to be prepared for their future jobs, with knowledge but also with several soft skills, among them creativity. In this paper, we present a study carried on with 128 engineering undergraduate students on their understanding of mathematical creativity. The students were in the first year of different engineering first degrees in a north-eastern Portuguese university and we analysed the content of their texts for the question ‘What do you understand by mathematical creativity?’. Data collection was done in the first semester of the academic years 2014/2015 and 2016/2017 in a Linear Algebra course. The results showed that ‘problem solving’ category had the majority of the references in 2014/2015, but not in the academic year 2016/2017 were ‘involving mathematics’ category had the majority. This exploratory study pointed out for ‘problem solving’ and ‘involving mathematics’ categories and gave us hints for teaching mathematics courses in engineering degrees.     

Developing children's conceptual understanding of area measurement: A curriculum and teaching experiment

The present study examined the effectiveness of three instructional treatments which had different combinations of mathematical elements regarding 2-dimensional (2-D) geometry and area measurement for developing 4th-grade children's understanding of the formulas for area measurement and their ability to solve area measurement problems. Participants were 120 fourth graders. The results showed that the enriched curriculum, involving the geometry motions and area measurement connections effectively facilitated children's mathematical judgments and explanations demanding high-level conceptual understanding. The instructional curricula accentuating only 2-D geometry or numerical calculations for area measurement did not exhibit such effectiveness. Interview data revealed that the geometric operations of superimposition, decomposition, re-composition as well as the concept of congruence were deemed essential by children for the conceptualization of the formulas for area measurement.     

“2＋4＋4”人才培养模式探析

人才培养是高校的根本任务,而培养什么样的人、怎样培养人是由人才培养模式所决定的。人才培养模式从根本上规定了人才的特征,集中地反映了一所高校的教育思想、办学理念、人才观、质量观和办学风格。北京的高等教育从大众化阶段进入普及化阶段后,学生知识水平的差距逐渐拉大。