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.     

Pre-service teachers' representations of children's understanding of mathematical concepts: conflicts and conflict resolution

The purpose of this study was to investigate the extent to which research experience could enhance pre-service teachers' understanding of children's knowledge of mathematical concepts. A group of five pre-service teachers designed and conducted a study of children's understanding of fractions. As participant-observers throughout their study we gathered evidence of change in pre-service teachers' representations of children's knowledge of fractions. The focus of our observations was on the conflicts generated by the gaps between the pre-service teachers' research findings and their representations of the children's knowledge of fractions. Results indicate that research experience is fruitful in developing an inquisitive disposition in pre-service teachers and in sensitizing future teachers to children's knowledge of a mathematical concept. Pre-service teachers became familiar with the research literature regarding children's understanding of fractions, they questioned the typical instructional sequence of teaching fractions and they scrutinized methods of assessing children's knowledge of mathematical topics.     

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.     

TALKING GEOGRAPHY: an investigation into young children's understanding of geographical terms PART 2

This research investigates the understanding by children of seven years of age of certain geographical terms and compares data collected in 1991, reported in an earlier paper, with that collected in 1993, after two years’ implementation of the English National Curriculum for Geography. The findings do not indicate any significant difference in the quality and range of response by the children. Most children, from both samples, appeared to show restricted or inconsistent understanding of most of the terms. It is clear from the research that children construct their own meanings for particular words, based on intuition and experience together with some formal teaching, but that even vernacular terms can present difficulties for them. Children have particular problems in conceptualising space. Further investigation of children's constructions, related to specific geographical concepts, is therefore urgently needed if geography teaching programmes are to be designed with adequate reference to the cognitive demands of particular aspects of the subject. We need to know more about children's views of geography.     

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).     

关于丢番图方程x2±y4=z5与x5+y5=z2

利用费尔马无穷递降法证明了丢番图方程x2+y4=z5,x4-y4=z5,x5+y5=(Z|z)均没有正整数解.     

Technology in the curriculum: A vehicle for the development of children's understanding of science concepts through problem solving

This research was carried out over a period of ten months with children in Grades 2 and 3 (aged 7 and 8) who were participating in a sequence of technology activities. Since the introduction into Victorian primary schools ofThe Technology Studies Framework P-10 (Crawford, 1988), more teachers are including technology studies in their classrooms and by so doing may assist children's understanding of science concepts. Children are being exposed to science phenomena related to the technology activities and Technology Studies may be a way of providing children with science experiences. ‘Technology Studies’ in this context refers to children carrying out practical problem solving tasks which can be completed without any particular scientific knowledge. Participation in the technology activities may encourage children to become actively involved, thereby facilitating an exploration of the related science concepts. The project identified the importance of challenge in relation to the children's involvement in the technology activities and the conference paper (available from the first author) discusses particular topics in terms of the balance between cognitive/metacognitive and affective influences (Baird et al., 1990) Specializations: science and technology education, interest and attitudinal change. Specialization: technology in the primary school.     

Taking decisions seriously: young children's understanding of conventional truth

Research suggests that young children may see a direct and one-way connection between facts about the world and epistemic mental states (e.g., belief). Conventions represent instances of active constructions of the mind that change facts about the world. As such, a mature understanding of convention would seem to present a strong challenge to children's simplified notions of epistemic relations. Three experiments assessed young children's abilities to track behavioral, representational, and truth aspects of conventions. In Experiment 1, 3- and 4-year-old children (N = 30) recognized that conventional stipulations would change people's behaviors. However, participants generally failed to understand how stipulations might affect representations. In Experiment 2, 3-, 5-, and 7-year-old children (N = 53) were asked to reason about the truth values of statements about pretenses and conventions. The two younger groups of children often confused the two types of states, whereas older children consistently judged that conventions, but not pretenses, changed reality. In Experiment 3, the same 3- and 5-year-olds (N = 42) participated in tasks assessing their understanding of representational diversity (e.g., false belief). In general, children's performance on false-belief and "false-convention" tasks did not differ, which suggests that conventions were understood as involving truth claims (as akin to beliefs about physical reality). Children's difficulties with the idea of conventional truth seems consistent with current accounts of developing theories of mind.     

关于Diophantine方程x^2＋D=4y^3

对一些d,Q（√d）是Euclid域,则在其对应的Euclid整环Q＇（√d）中算术基本定理成立．由此通过利用Z[i]整除理论来证明一类不定方程x^2＋D=4y^3有整数解的情况;且当D=11,该不定方程x^2＋D=4y^3没有整数解。     

Cognitive holding power,fluid intelligence,and mathematical achievement as predictors of children's realistic problem solving

The present study explored whether first and second order cognitive holding power perceived by children in mathematical classrooms, fluid intelligence, and mathematical achievement predicted their performance on standard problems, and especially realistic problems. A sample of 119 Chinese 4–6th graders were administered the word problem test, the cognitive holding power questionnaire, and Raven's standard progressive matrices. Results showed that: (1) children's fluid intelligence and general mathematical achievement significantly predicted their performance on both realistic and standard problems, however, second order cognitive holding power predicted their performance on realistic problems but not standard problems; (2) the relationship between first order cognitive holding power and children's correct answers to realistic problems was mediated by second order cognitive holding power; (3) children's performance on standard problems was significantly better than that on realistic problems, and children's performance on both types of problems improved with their grades.     

Dynamic assessment and teachers' knowledge of children's mathematical thinking: a case study in children's mathematics

This paper considers the kind of pedagogical knowledge and principles involved in the operationalisation of knowledge of children's mathematical thinking as a process of dynamic assessment. Using a case study of a particular child, this paper explores planning and instruction for a child determined by a detailed and informed interpretation of the child's conceptual understanding through a dynamic process. It presents as a case study the observations of a teacher who had undertaken professional development in children's mathematical thinking, theoretically informed by Cognitively Guided Instruction and Maths Recovery. The observations revealed the child's mathematical understanding and how the teacher used this knowledge dynamically to inform teaching. The paper outlines the kind of knowledge required of teachers to enact this dynamic process in mathematics teaching and argues for the centrality of this to the development of inclusive practice.     

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.