Immediate and delayed effects of meta-cognitive instruction on regulation of cognition and mathematics achievement

The present study addressed two research questions: (a) the extent to which students who were exposed to meta-cognitive instruction are able to implement meta-cognitive processes in a delayed, stressful situation, in our case—being examined on the matriculation exam; and (b) whether students preparing themselves for the matriculation exam in mathematics, attain a higher level of mathematics achievement and meta-cognitive awareness (knowledge about cognition and regulation of cognition) as a result of being exposed to meta-cognitive instruction. Participants were 61 Israeli high school students who studied mathematics for four-point credit on the matriculation exam (middle level). About half of the students (N = 31) were assigned to meta-cognitive instruction, called IMPROVE, and the others (N = 30) studied with no explicit meta-cognitive guidance (control group). Analyses included both quantitative and qualitative methods. The later was based on students’ interviews, conducted about a couple of months after the end of the intervention, immediately after students completed the matriculation exam in mathematics. Results indicated that IMPROVE students outperformed their counterparts on mathematics achievement and regulation of cognition, but not on knowledge about cognition. Furthermore, during the matriculation exam, IMPROVE students executed different kinds of cognitive regulation processes than the control students. The theoretical and practical implications of the study are discussed.     

First-graders’ knowledge of multiplicative reasoning before formal instruction in this domain

Our study investigated children’s knowledge of multiplicative reasoning (multiplication and division) at the end of Grade 1, just before the start of formal instruction on multiplicative reasoning in Grade 2. A large sample of children (N = 1176) was assessed in a relatively formal test setting, using an online test with 28 multiplicative problems of different types. On average, the children correctly answered more than half (58%) of the problems, including several bare number problems. This indicates that before formal instruction on multiplicative reasoning, children already have a considerable amount of knowledge in this domain, which teachers can build on when teaching them formal multiplication and division. Using analysis of variance and cross-classified multilevel regression analysis, we identified several predictors of children’s pre-instructional multiplicative knowledge. With respect to the characteristics of the multiplicative problems, we found that the problems were easiest to solve when they included a picture involving countable objects, and when the multiplicative situation was of the equal groups semantic structure (e.g., 3 boxes of 4 cookies). Regarding student characteristics, pre-instructional multiplicative knowledge was higher for children with higher-educated parents. Finally, the mathematics textbook used in school appeared to have influenced children’s pre-instructional multiplicative knowledge.     

Making sense of graphs: does metacognitive instruction make a difference on students' mathematical conceptions and alternative conceptions?

The present study investigates the differential effects of cooperative learning with or without metacognitive instruction on making sense of graphs. Participants were 196 eighth-graders who studied in six classrooms. Data were analyzed by using quantitative and qualitative methods. Results indicated that students who were exposed to the metacognitive instruction within cooperative learning (COOP + META) significantly outperformed their counterparts who were exposed to cooperative learning with no metacognitive instruction (COOP). The positive effects of COOP + META were observed on both graph interpretation and graph construction (transfer task) with regard to alternative conceptions. Furthermore, observations indicated differential characteristics of discourse behaviors during small group interaction under these methods. The practical implications of the study are discussed.     

The relative effects and equity of inquiry‐based and commonplace science teaching on students' knowledge,reasoning, and argumentation

We conducted a laboratory‐based randomized control study to examine the effectiveness of inquiry‐based instruction. We also disaggregated the data by student demographic variables to examine if inquiry can provide equitable opportunities to learn. Fifty‐eight students aged 14–16 years old were randomly assigned to one of two groups. Both groups of students were taught toward the same learning goals by the same teacher, with one group being taught from inquiry‐based materials organized around the BSCS 5E Instructional Model, and the other from materials organized around commonplace teaching strategies as defined by national teacher survey data. Students in the inquiry‐based group reached significantly higher levels of achievement than students experiencing commonplace instruction. This effect was consistent across a range of learning goals (knowledge, reasoning, and argumentation) and time frames (immediately following the instruction and 4 weeks later). The commonplace science instruction resulted in a detectable achievement gap by race, whereas the inquiry‐based materials instruction did not. We discuss the implications of these findings for the body of evidence on the effectiveness of teaching science as inquiry; the role of instructional models and curriculum materials in science teaching; addressing achievement gaps; and the competing demands of reform and accountability. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 47:276–301, 2010     

Examining the unique contributions and developmental stability of individual forms of relational reasoning to mathematical problem solving

Relational reasoning, a higher-order cognitive ability that identifies meaningful patterns among information streams, has been suggested to underlie STEM development. This study attempted to explore the potentially unique contributions of four forms of relational reasoning (i.e., analogy, anomaly, antinomy, and antithesis) to mathematical problem solving. Two separate samples, fifth graders (n = 254) and ninth graders (n = 198), were assessed on their mathematical problem solving ability and the different forms of relational reasoning ability. Linear regression analysis was conducted, with participants’ age, working memory, and spatial skills as covariates. The results showed that analogical and antithetical reasoning abilities uniquely predicted mathematical problem solving. This pattern demonstrated developmental stability across a four-year time frame. The findings clarify the unique significance of individual forms of relational reasoning to mathematical problem solving and call for a shift of research direction to reasoning abilities when exploring dissimilarity-based relations (opposites in particular).     

Taking the relational structure of fractions seriously: Relational reasoning predicts fraction knowledge in elementary school children

Understanding and using symbolic fractions in mathematics is critical for access to advanced STEM concepts. However, children and adults consistently struggle with fractions. Here, we take a novel perspective on symbolic fractions, considering them within the framework of relational structures in cognitive psychology, such as those studied in analogy research. We tested the hypothesis that relational reasoning ability is important for reasoning about fractions by examining the relation between scores on a domain-general test of relational reasoning (TORR Jr.) and a test of fraction knowledge consisting of various types of fraction problems in 194 s grade and 145 fifth grade students. We found that relational reasoning was a significant predictor of fractions knowledge, even when controlling for non-verbal IQ and fractions magnitude processing for both grades. The effects of relational reasoning also remained significant when controlling for overall mathematics knowledge and skill for second graders but was attenuated for fifth graders. These findings suggest that this important subdomain of mathematical cognition is integrally tied to relational reasoning and opens the possibility that instruction targeting relational reasoning may prove to be a viable avenue for improving children’s fractions skills.     

Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving

This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms. We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof.

Early development of quantity to number-word linkage as a precursor of mathematical school achievement and mathematical difficulties: Findings from a four-year longitudinal study

This article reports results of a four-year longitudinal study that investigated the impact of specific and non-specific precursors on mathematical school achievement. Preschool quantity-number competencies (QNC) predicted mathematical achievement in primary school. Furthermore, basic arithmetic fact retrieval in Grade 1 had an impact on early mathematics school achievement. The influence of socioeconomic status and number naming speed, assessed in kindergarten, became especially important at the end of Grade 4. Particularly, a subgroup of mathematically low-achieving children in Grade 4 had already performed more poorly than normal children in tasks assessing preschool QNC, number naming speed, and basic arithmetic fact retrieval, as well as nonverbal intelligence and socioeconomic status.     

Cognitive underpinnings of moral reasoning in adolescence: The contribution of executive functions

Adolescence is a developmental period characterized by intense changes, which impact the interaction between individuals and their environments. Moral reasoning (MR) is an important skill during adolescence because it guides social decisions between right and wrong. Identifying the cognitive underpinnings of MR is essential to understanding the development of this function. The aim of this study was to explore predictors of MR in typically developing adolescents (n = 92, 33 males, M = 16.3 years, SD = 2.2 years) and the specific contribution of higher order cognitive processing using an innovative visual MR assessment tool and measures of executive functioning and intelligence. MR maturity was correlated with four executive functions (cognitive flexibility, feedback utilization, conceptual reasoning, verbal fluency) and was predicted by four variables: age, intelligence, nonverbal flexibility and verbal fluency. Overall, these results contribute to a better understanding of MR during adolescence and highlight the importance of using innovative tools to measure social cognition.     

Pedagogical reasoning: the foundation of the professional knowledge of teaching

ABSTRACT

Much of the debate about that which comprises teachers’ professional knowledge has been important in the academic literature but does not necessarily reflect the reality of how they think as they construct the knowledge that underpins their practice. Typically, teachers are not encouraged to spend time talking about teaching in ways that are theoretically robust, or to unpack their teaching in order to show others what they know, how and why. Because they are busy ‘doing teaching’ they are not commonly afforded opportunities to ‘unpack’ their practice to explore and articulate the reasoning underpinning what they do. This paper argues that the essence of teachers’ professional knowledge is bound up in the teaching procedures they employ and that knowledge is accessible and demonstrable through the pedagogical reasoning that underpins their decision-making, actions and intents; all of which come to the fore when their pedagogical reasoning is examined. If teaching is to be more highly valued, it is important to more closely examine the nature of teachers’ pedagogical reasoning as it offers a window into the complex and sophisticated knowledge of practice that influences what they do, how and why.     

教学策略、方法与结构——以知识与技能教学为例

教学策略是使学生掌握学习结果并形成互动的主要媒介。教学方法是教学策略的一个组成要素,也是传递学习内容并提供学习指导以保证学生掌握所学知识技能的过程。教学策略和方法本身很难说有什么好坏优劣之分,只不过不同的人适合不同的策略,而选择合适的教学策略的目的就是为了充分发挥其功效而不至于浪费或误用。课堂教学结构是指在教学中发挥特定功能的可资辨别的要素。它是教学过程中的基本构成板块,也是充分发挥教学功能,提高教学效率所必需的。课堂教学结构中的核心成分包括:明确教学目标,导入新课,聚焦核心知识点,提供举例或说明,开展练习和给予反馈。各类教学策略或者方法都可以在课堂教学结构中得到运用。     

The role of contextual,conceptual and procedural knowledge in activating mathematical competencies (PISA)

This paper analyses the difficulties which Spanish student teachers have in solving the PISA 2003 released items. It studies the role played by the type and organisation of mathematical knowledge in the activation of competencies identified by PISA with particular attention to the function of contextual knowledge. The results of the research lead us to conclude that the assessment of the participant’s mathematical competencies must include an assessment of the extent to which they have school mathematical knowledge (contextual, conceptual and procedural) that can be productively applied to problem situations. In this way, the school knowledge variable becomes a variable associated with the PISA competence variable. This paper is based on a research project funded by a grant awarded in 2003 by the General Directorate for Research of the Spanish Ministry of Education and Science (BSO-2003-7133).     

The role of rational number density knowledge in mathematical development

Many students still have not developed a robust understanding of rational number concepts at the end of primary school, despite several years of instruction on the topic. The present study aims to examine the patterns, predictors, and outcomes of the development of rational number knowledge in lower secondary school. Latent transition analysis revealed that rational number development from primary to lower secondary school (N = 362) appears to follow similar patterns as in younger students. In particular, a majority of students had poor knowledge of the density of the rational number set. Whole number magnitude knowledge appeared to be an important predictor of the development of rational number size knowledge, but not density knowledge. Finally, fraction density knowledge appeared to be related to concurrent algebra knowledge. Together these results point to an important role for density knowledge in mathematical development.     

Metacognitive knowledge of effort, personality factors, and mood state: their relationships with effort-related metacognitive experiences

The present study aimed at identifying the effects of mood treatment, personality factors, and metacognitive knowledge of effort–i.e., conceptualization of effort and perceptions of effort regulation–on metacognitive experiences of students, particularly on their reported feeling of difficulty and estimate of effort. The sample comprised 474 students of 5th and 7th grade of both genders. The participants were tested in two phases. In the first phase, they were asked to respond to questionnaires measuring (a) metacognitive knowledge of effort, (b) maths self-concept, (c) goal orientations, and (d) a test of maths ability. In the second phase, participants were subjected to mood treatment–neutral, positive, and negative– and were asked to solve a mathematical problem. They also rated their prospective metacognitive experiences before solving the problem and the retrospective ones after solving it. Mood treatment interacted with gender in the case of performance but it had no effect on metacognitive experiences. A series of regression analyses showed that positive mood, personality factors, and feeling of difficulty predicted the prospective estimate of effort. Only feeling of difficulty and performance predicted the retrospective estimate of effort. No effect of metacognitive knowledge of effort on estimate of effort was found.     

论法律推理的定义及特征

分析了几种典型的法律推理定义，给出了自己新的定义，即法律推理是特定主体在法律实践过程中，以已知的法律和事实材料为前提推导和论证法律结论的过程。这一定义反映了法律推理的基本特征。法律推理是一个知识创新的过程，具有内在的逻辑性、实践性。     

Biology instruction using a generic framework of scientific reasoning and argumentation

We investigated biology instruction—using a generic framework of scientific reasoning and argumentation (SRA) with eight epistemic activities—on how to foster student learning in biological literacy which had not been clarified in previous studies. Our analysis of videotaped biology lessons and student achievement showed varying frequencies in using these activities and their effects on achievement. Those students taught with more epistemic activities had higher achievement. We believe that the SRA framework can be a worthwhile methodical tool for teaching biology to foster student learning. Therefore, we draw practically orientated implications for educational research, practitioners, teacher educators, and curriculum developers.     

Fluid reasoning,working memory and planning ability in assessment of risk for mathematical difficulties

ABSTRACT

The demands on mathematical problem-solving have increased in almost all school systems internationally and may constitute a barrier for children with special educational needs (SEN). This study explored the role of fluid reasoning (FR), working memory (WM) and complex executive function of planning (EF) in children (N = 62) referred for assessment of SEN, and specifically of risk for mathematical difficulties (MD). Performances on FR, WM and complex EF of planning were used to predict risk for MD. Results showed that planning ability predicted children at risk for MD, beyond FR or WM ability, when comparing with children not at risk for MD. It was concluded that assessing the complex EF of planning in addition to FR and WM ability is crucial in identifying children at risk for MD. The importance of understanding how planning ability affects children’s mathematical problem-solving is discussed, in relation to assessment and teaching practices.     

Designing computer learning environments for engineering and computer science: The scaffolded knowledge integration framework

Designing effective curricula for complex topics and incorporating technological tools is an evolving process. One important way to foster effective design is to synthesize successful practices. This paper describes a framework called scaffolded knowledge integration and illustrates how it guided the design of two successful course enhancements in the field of computer science and engineering. One course enhancement, the LISP Knowledge Integration Environment, improved learning and resulted in more gender-equitable outcomes. The second course enhancement, the spatial reasoning environment, addressed spatial reasoning in an introductory engineering course. This enhancement minimized the importance of prior knowledge of spatial reasoning and helped students develop a more comprehensive repertoire of spatial reasoning strategies. Taken together, the instructional research programs reinforce the value of the scaffolded knowledge integration framework and suggest directions for future curriculum reformers.Portions of this paper were presented at the American Psychological Association Meeting, Ontario, Canada, August 22, 1993. under the title of Cognition and instruction in higher education: Applications of advanced technologies. The title of the symposium was New Fellows in Educational Psychology-The Implications of Their Work for University-Level Instruction. This material is based upon research supported by the National Science Foundation under grant MDR-8954753. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and not necessarily reflect the views of the National Science Foundation.     

Longitudinal assessment of progress in reasoning capacity and relation with self-estimation of knowledge base

The aim of the study was to investigate progress in reasoning capacity and knowledge base appraisal in a longitudinal analysis of data from summative evaluation throughout a medical problem-based learning curriculum. The scores in multidisciplinary discussion of a clinical case and multiple choice questionnaires (MCQs) were studied longitudinally for 213 students from years 2 to 5. The capacity of core knowledge delimitation was calculated as the difference between the levels of average ascertainment degrees given for correct and incorrect answers at MCQ. For both multidisciplinary discussion of a clinical case evaluation and self-estimation of core knowledge, the capacity increases throughout the curriculum. The reasoning capacity assessed through multidisciplinary discussion of a clinical case is positively correlated with MCQ scores and the capacity to discriminate the mastered core knowledge. In conclusion, this study indicates that self-estimation of core knowledge is associated with an increase in reasoning performance through a well-organised knowledge base. Since that ability is related to success or failure, it is suggested that student awareness about delimitation of mastered core knowledge is considered as part of learning.