Information processing by intellectually gifted pupils solving mathematical problems

What differences may be found in the way gifted pupils, as compared with average pupils from the second form of lower secondary education, process information while solving mathematical problems? Gifted pupils apparently solved the problems better, faster and needed less assistance than average pupils. A global distinction of the subprocesses orientation, execution and evaluation also allows for the conclusion that in most cases the gifted pupils processed information in a different fashion. The yield of this line of research should be the improvement of education in two respects. For one, the results may induce teachers to adapt their teaching to gifted pupils. Secondly they may try to teach average pupils to process information like gifted pupils typically would.     

Visual Chunking Skills of Hong Kong Children

This study investigated the development of visual chunking skills in the processing of Chinese characters among Hong Kong pupils. One-hundred-seventy-nine primary school students from first, second and fourth grades were administered a character copying task. Children as young as 6 years of age were aware of character units and were able to apply visual chunking strategies when processing characters. Children in higher grades performed better than those in lower grades on every character type, and the types of errors they made revealed that their chunking level was higher than that of younger children. Differences between ability groups emerged in second grade and vanished in fourth grade, suggesting that children with a lower reading ability are slower to develop advanced chunking skills.     

Individual differences in mathematical competence modulate brain responses to arithmetic errors: An fMRI study

Data from both neuropsychological and neuroimaging studies have implicated the left inferior parietal cortex in calculation. Comparatively less attention has been paid to the neural responses associated with the commission of calculation errors and how the processing of arithmetic errors is modulated by individual differences in mathematical competence. Do more competent individuals exhibit a different brain response to errors than less mathematically able individuals? These outstanding questions were addressed in the present functional Magnetic Resonance Imaging (fMRI) study through an investigation of which brain regions respond more to erroneously versus correctly solved arithmetic problems while a group of 24 adult participants with varying levels of mathematical competence solved problems of all four arithmetic operations. Despite high levels of accuracy, a robust main effect of accuracy (incorrect vs. correct) was observed in both medial and lateral regions of the prefrontal cortex. These regions have frequently been associated with both the detection of errors and the deployment of cognitive control following an error. Furthermore, mathematical competence was found to modulate the activation of an area in the right dorsolateral prefrontal cortex. Specifically, individuals with relatively higher mathematical competence (n = 12) were found to activate this region more for incorrectly solved trials than their less mathematically competent peers (n = 12). Taken together, these findings suggest that the commission of arithmetic errors modulates responses of prefrontal regions and, moreover, that activation of the right lateral prefrontal cortex during arithmetic errors is affected by individual differences in mathematical competence. In view of the evidence associating the lateral prefrontal cortex with the implementation of cognitive control, we suggest that individuals with relatively high mathematical competence may exhibit greater awareness of calculation mistakes and implement greater control following the commission of errors.     

Nous écouter,nous soutenir,nous apprendre 1 : a comparative study of pupils' perceptions of the pedagogic process

In many countries around the world there is a current focus on the restructuring of education systems in a bid to increase the quality of the educational experience for pupils in order to raise their academic achievement. However, the defiition of quality as expressed through policy may not always accord with the aims and aspirations of individual teachers or, perhaps more importantly, match the constructions given to the concept of quality by pupils. The rhetoric and intent expressed in policy texts may even have the potential to restrict the quality of what teachers do and what pupils experience. This paper draws on the findings of the ENCOMPASS project to illustrate the concepts of quality as expressed by the pupils themselves. It looks at what pupils in England, France and Denmark had to tell us about motivation, engagement and the conditions necessary for effective teaching and learning. It proposes some reflections on questions such as: What do young people see as the purpose of schooling? What motivates young people to learn? What do young people expect from their teachers in order to enhance their learning?     

Proof schemes and learning strategies of above-average mathematics students

What patterns can be observed among the mathematical arguments above-average students find convincing and the strategies these students use to learn new mathematical concepts? To investigate this question, we gave task-based interviews to eleven female students who had performed well in their college-level mathematics courses, but who differed in the number of proof-oriented courses each had taken. One interview was designed to elicit expressions of what students find convincing. These expressions were categorized according to the proof schemes defined by Harel and Sowder (1998). A second interview was designed to elicit expressions of what strategies students use to learn a mathematical concept from its definition, and these expressions were classified according to the learning strategies described by Dahlberg and Housman (1997). A qualitative analysis of the data uncovered the existence of a variety of phenomena, including the following: All of the students successfully generated examples when asked to do so, but they differed in whether they generated examples without prompting and whether they successfully generated examples when it was necessary to disprove conjectures. All but one of the students exhibited two or more proof schemes, with one student exhibiting four different proof schemes. The students who were most convinced by external factors were unsuccessful in generating examples, using examples, and reformulating concepts. The only student who found an examples-based argument convincing generated examples far more than the other students. The students who wrote and were convinced by deductive arguments were successful in reformulating concepts and using examples, and they were the same set of students who did not generate examples spontaneously but did successfully generate examples when asked to do so or when it was necessary to disprove a conjecture.     

Socially shared metacognition of dyads of pupils in collaborative mathematical problem-solving processes

This study investigated how metacognition appears as a socially shared phenomenon within collaborative mathematical word-problem solving processes of dyads of high-achieving pupils. Four dyads solved problems of different difficulty levels. The pupils were 10 years old. The problem-solving activities were videotaped and transcribed in terms of verbal and nonverbal behaviours as well as of turns taken in communication (N = 14 675). Episodes of socially shared metacognition were identified and their function and focus analysed. There were significantly more and longer episodes of socially shared metacognition in difficult as compared to moderately difficult and easy problems. Their function was to facilitate or inhibit activities and their focus was on the situation model of the problem or on mathematical operations. Metacognitive experiences were found to trigger socially shared metacognition.     

The use of mathematical symbolism in problem solving: An empirical study carried out in grade one in the French community of Belgium

This article relates to an empirical study based on the use of mathematical symbolism in problem solving. Twenty-five pupils were interviewed individually at the end of grade one; each of them was asked to solve and symbolize 14 different problems. In their classical curriculum, these pupils have received a traditional education based on a “top-down” approach (an approach that is still applied within the French Community of Belgium): conventional symbols are presented to the pupils immediately with an explanation of what they represent and how they should be used. Teaching then focuses on calculation techniques (considered as a pre-requisite for solving problems). The results presented here show the abilities (and difficulties) demonstrated by the children in making connections between the conventional symbolism taught in class and the informal approaches they develop when faced with the problems that are put to them. The limits of the “top-down” approach are then discussed as opposed to the more innovative “bottom-up” type approaches, such as those developed by supporters of Realistic Mathematics Educations in particular.     

Strategic development of multiplication problem solving: Patterns of students' strategy choices

Low mathematics achievement is a persistent problem in the United States, and multiplication is a fundamental area in which many students manifest learning difficulties. This study examined the strategic developmental levels of multiplication problem solving among 121 elementary school students in Grades 3 through 5. A latent class analysis modeling was used to identify three valid groups representing different patterns of strategy choices for each of three types of multiplication problems. Findings indicated intra-group variability for problem-solving accuracy, for frequency of using different strategies, and for accuracy of executing direct retrieval/algorithm (DR/AG) strategies. Students demonstrated relative consistency in their strategy choices for solving the three problem types. Students who used DR/AG strategies most frequently showed the highest problem-solving accuracy and the highest accuracy of executing the DR/AG strategies. Students who most frequently relied on incorrect operations or who indicated they did not know how to solve problems demonstrated the lowest problem-solving accuracy among the three groups; the number of students in this group increased with problem difficulty levels. Implications are discussed in terms of identifying students' strategic developmental levels and providing differentiated instruction based on the identified levels.     

The formal setting as context for cognitive activities. An empirical study of arithmetic operations under conflicting premisses for communication

The general concern of the present article is to contribute to an understanding of the contextual determination of cognitive activities. More specifically, the focus of the empirical research reported has been to study how pupils define and deal with cognitive tasks in situations that are recognised as pedagogical in character. Within the context of their everyday mathematics teaching, 206 twelve year old primary school pupils were given work sheets containing elementary arithmetic problems. The experimental treatment consisted of introducing (through headings and instructions) pedagogical definitions of problems that were in conflict with the nature of the problems themselves. The results indicate that the predefinitions of cognitive activities typical of educational contexts have a strong impact on the way problems are dealt with. Clear differences could be discerned between groups at different achievement levels in the extent to which the cues present in pedagogical contexts were used in defining the problem. A crucial aspect of what are conventionally conceived as differences in mathematical ability seems, judging from the present results, to have more to do with the capacity to decipher ambiguous communicative situations than with the mastery of a mathematical algorithm per se.     

Using pupil perspective research to inform teacher pedagogy: what Caribbean pupils with dyslexia say about teaching and learning

The transformative potential of pupils' voices is well documented in past research by Pedder and McIntyre; and Cooper and McIntyre. In this qualitative research, I utilise a social constructivist framework by Vygotsky to ask pupils with dyslexia about the kinds of teacher strategies that they find helpful to their learning at secondary school in Barbados. This study utilised direct observations and individual interviews as part of a multiple case study strategy of 16 pupils with dyslexia from two secondary schools in Barbados. Findings suggest that there are regular teachers' strategies like more detailed explanations, demonstrations, drama and role play, storytelling, asking questions and enquiry‐based approaches that pupils find facilitative of their learning. This research is guided by the following questions: (1) what do pupils mean when they refer to teacher strategies as helpful?; and (2) what pedagogical approaches do pupils with dyslexia find helpful to their learning at secondary school?     

Factors contributing to success in mathematical estimation in preservice teachers: Types of problems and previous mathematical experience

Variables that may affect success in mathematical estimation in preservice elementary teachers were studied. A multiple regression analysis was performed in order to predict success on a 25 problem estimation test. Subjects' answer to the question, “Are you good at math?” was the best predictor of success on the estimation test. Other important variables were college grade point average, years of study of mathematics, and enjoyment of mathematics. Not only were average mathematics grades not related to estimation score, but they were negatively correlated to estimation score. Half the subjects saw problems in an applied format, and half saw the same problems in a computational format. The number of subjects answering problems correctly in the application format was significantly greater than the number of subjects answering questions correctly in the computational format. This result was contradictory to results usually found for finding exact answers with pencil and paper. A reason for greater success on problems in the application format may be related to the fact that the applied problems involved money, a very relevant type of problem in which to quickly estimate an answer.     

Is the relationship between competence beliefs and test anxiety influenced by goal orientation?

The study described here aimed to examine the relations between test anxiety, competence beliefs and achievement goals, and in particular if the relations between competence beliefs and test anxiety were moderated by achievement goals. Pupils in their first year of secondary schooling completed self-report questionnaires for test anxiety, competence beliefs and achievement goals. Results indicated that pupils with low competence beliefs in Mathematics reported more worrisome thoughts when they held a mastery-avoidance goal and female pupils with low verbal competence beliefs reported more off-task behaviours when they held a performance-approach goal. Male pupils with low verbal competence beliefs reported fewer off-task behaviours when they held a performance-approach goal. These findings may reflect how Mathematics may be uniquely related to a fear of failure among school subjects and how the gendered nature of verbal self-concept becomes important when peer comparison is a salient goal for pupils.     

Questions asked by primary student teachers about observations of a science demonstration

Teacher questioning has a central role in guiding pupils to learn to make scientific observations and inferences. We asked 110 primary student teachers to write down what kind of questions they would ask their pupils about a demonstration. Almost half of the student teachers posed questions that were either inappropriate or presupposed that the pupils would know the answer. For example, they directly asked for an explanation of the phenomenon instead of asking what inferences the pupils could make on the basis of their observations. There was a lack of questions that would draw the pupils’ attention to the variables that may cause the phenomenon to happen. Only about 15% of the student teachers formed questions such as ‘What is happening?’ or ‘How is it happening?’. All in all, primary student teachers seem to need extra practice in forming questions based on scientific observation.     

学习贯彻《总体方案》，深入探讨“三改”问题

从基层教师角度探讨《深化新时代教育评价改革总体方案》中“为什么改”“改什么”“怎么改”的“三改”问题。“三改”是改革的基本性问题。为什么改?教育评价改革是中国社会发展的历史必然;改什么?当前最需要改革的是教育评价中的行政化问题;怎么改?让教育评价回归到评价教育科研成果内容本身。     

Is procedure acquisition as unstable as it seems?

Are students’ mathematical procedures as unstable as they seem? Students often produce different errors in response to the same kind of problems on different testing occasions. This finding is puzzling. Past research has shown that students induce overly general procedures from worked-out examples during learning, which lead to a host of predictable errors on new problems. Do students create rule-based errors only to then switch between them at random? In this paper, we show that seemingly diverse errors on two different testing episodes may result from the same underlying stable procedure and are part of the same error category. These findings suggest that students’ errors are more stable on a category vs. on an individual level. The current study consists of teaching students addition in a new number system, called NewRoman, and analyzing students’ solution strategies in detail. Implications for teaching are discussed.     

Studying mathematics problem-solving classrooms. A comparison between the discourse of in-service teachers and student teachers

In the article we compare the approaches of 3 in-service teachers and 3 student teachers when they tried to solve a verbal arithmetic problem in the classroom. Each interaction was studied using a System of Analysis that takes into account the cognitive processes involved in the solution of a mathematic problem and describes the interaction at different levels showing what is done and to what degree teachers and/or pupils are responsible for what is done. The results of the study suggest that both groups of teachers are different in how they direct the student’s attention toward the essential aspects implied in the resolution of word problem. On the one hand, the in-service teachers guaranteed students’ understanding of the problem before dealing with the solution, while students teachers only did so when pupils committed errors. On the other hand, the in-service teachers allowed a high level of student participation, while student teachers took a more prominent role so children’s participation was lower.     

Analyse des Facteurs de Réussite de l'Implantation d'une Innovation Scolaire

School innovating involves risks for teachers; professional risks, but also personal risks. So, the implantation and the diffusion of schools innovations are only possible if certain facilitating conditions are fulfilled.

Considering a specific innovation developed for 15 years in Belgium (peer‐tutoring), the authors are trying to answer the following questions: ? What conditions have to be fulfilled for an innovation to take shape and to expand?

? What are the most important factors which determine the success or the failure of an innovation?

? Even though the effects on pupils remain the justification of an innovation, how can we take into account the questions the teachers ask themselves at the different stages of the innovation? What are those questions? How can all the actors of the project find what they expected?