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Descriptors:
Cognitive Ability; Profiles; Children; Reaction Time; Number Concepts; Multivariate Analysis; Individual Differences; Comparative Analysis; Arithmetic; Young Children; Foreign Countries; Stimuli; Correlation; Algebra; Mathematical Concepts; Problem Solving; Statistical Analysis; Learning Disabilities; Mathematics Achievement; Mathematics

Abstract:
Dot enumeration (DE) and number comparison (NC) abilities are considered markers of core number competence. Differences in DE/NC reaction time (RT) signatures are thought to distinguish between typical and atypical number development. Whether a child's DE and NC signatures change or remain stable over time, relative to other developmental signatures, is unknown. To investigate these issues, the DE and NC RT signatures of 159 children were assessed 7 times over 6 years. Cluster analyses identified within-task and across-age subgroups. DE signatures comprised 4 parameters: (a) the RT slope within the subitizing range, (b) the RT slope for the counting range, (c) the subitizing range (indicated by the point of slope discontinuity), and (d) the overall average DE RT response. NC RT signatures comprised 2 parameters (NC intercept and slope) derived from RTs comparing numbers 1 to 9. Analyses yielded 3 distinct DE and NC profiles at each age. Within-age subgroup profiles reflected differences in 3 of the 4 DE parameters and only 1 NC parameter. Systematic changes in parameters were observed across ages for both tasks, and both tasks broadly identified the same subgroups. Sixty-nine percent of children were assigned to the same subgroup across age, even though parameters themselves changed across age. Subgroups did not differ in processing speed or nonverbal reasoning, suggesting that DE and NC do not tap general cognitive abilities but reflect individual differences specific to the domain of numbers. Indeed, both DE and NC subgroup membership at 6 years predicted computation ability at 6 years, 9.5 years, and 10 years. (Contains 4 figures and 11 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Epistemology; Mathematics Education; Educational Practices; Mathematics; Mathematics Instruction; Mathematical Concepts

Abstract:
Educational practices are to be based on proven scientific knowledge, not least because the function science has to perform in human culture consists of unifying practical skills and general beliefs, the episteme and the techne (Amsterdamski, 1975, pp. 43-44). Now, modern societies first of all presuppose regular and standardized ways of organizing both our concepts and our institutions. The explanatory schemata resulting from this standardization tend to destroy individualism and enchantment. But mathematics education is in fact the only place in which to treat the human subject's relationship with mathematics. And that is what mathematics education is all about: make the human subject grow intellectually and as a person by means of mathematics. At first sight, mathematics, in its formal guise, seems the opposite of philosophy, because philosophy constructs concepts (meanings), whereas mathematics deals with extensions of concepts (sets). We shall, however, turn this problem into an instrument, using the complementarity of intensions and extensions of theoretical terms as our main device for discussing the relationship between philosophy and mathematics education. The complementarity of the "how" and the "what" of our representations outlines, in fact, the terrain on which epistemology and education are to meet. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Probability; Mathematics Education; Cooperative Learning; Mathematics; Mathematics Instruction; Mathematical Aptitude; Mathematical Logic; Secondary School Mathematics; High School Students; Problem Solving; Longitudinal Studies; Teaching Methods; Mathematical Concepts

Abstract:
The purpose of this study is to contribute insights into how collaborative activity can help promote students' mathematical understanding. A group of six high school students (15- to 16-year olds) worked together on a challenging probability task as part of a larger, after-school, longitudinal study on students' development of mathematical ideas in problem-solving settings. The students solved the problem and produced a valid justification of their solution. This study shows that collaborative activity can help promote students' mathematical understanding by providing opportunities for students to critically reexamine how they make claims from facts and also enable them to build on one another's ideas to construct more sophisticated ways of reasoning. Implications for classroom teaching and ideas for future research are also discussed. The study helps address a documented need for a better understanding of how mathematical learning evolves in social settings. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Arithmetic; Elementary School Students; Grade 2; Grade 3; Mathematical Concepts; Fundamental Concepts; Mathematics Instruction; Elementary School Mathematics; Organization; Drills (Practice); Homework; Comprehension; Pretests Posttests; Instructional Effectiveness

Abstract:
This experiment tested the hypothesis that organizing arithmetic fact practice by equivalent values facilitates children's understanding of math equivalence. Children (M age = 8 years 6 months, N = 104) were randomly assigned to 1 of 3 practice conditions: (a) equivalent values, in which problems were grouped by equivalent sums (e.g., 3 + 4 = 7, 2 + 5 = 7, etc.), (b) iterative, in which problems were grouped iteratively by shared addend (e.g., 3 + 1 = 4, 3 + 2 = 5, etc.), or (c) no extra practice, in which children did not receive any practice over and above what they ordinarily receive at school and home. Children then completed measures to assess their understanding of math equivalence. Children who practiced facts organized by equivalent values demonstrated a better understanding of math equivalence than children in the other 2 conditions. Results suggest that organizing arithmetic facts into conceptually related groupings may help children improve their understanding of math equivalence. (Contains 4 tables and 1 figure.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Problem Solving; Program Effectiveness; Foreign Countries; Grade 6; Arithmetic; Students; Comprehension; Task Analysis; Elementary Education; Mathematics; Word Problems (Mathematics)

Abstract:
Many factors influence a student's performance in word (or textbook) problem solving in class. Among them is the comprehension process the pupils construct during their attempt to solve the problem. The comprehension process may include some less formal representations, based on pupils' real-world knowledge, which support the construction of a "situation model". In this study, we examine some factors related to the pupil or to the word problem itself, which may influence the comprehension process, and we assess the effects of the situation model on pupils' problem solving performance. The sample is composed of 750 pupils of grade 6 elementary school. They were selected from 35 classes in 17 Francophone schools located in the province of Quebec, Canada. For this study, 3 arithmetic problems were developed. Each problem was written in 4 different versions, to allow the manipulation of the type of information included in the problem statement. Each pupil was asked to solve 3 problems of the same version and to complete a task that allowed us to evaluate the construction of a situation model. Our results show that pupils with weaker arithmetic skills construct different representations, based on the information presented in the problem. Also, pupils who give greater importance to situational information in a problem have greater success in solving the problem. The situation model influences pupils' problem solving performance, but this influence depends on the type of information included in the problem statement, as well as on the arithmetic skills of each individual pupil. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Mathematics Instruction; Grade 7; Middle School Students; Schemata (Cognition); Schematic Studies; Experimental Groups; Control Groups; Pretests Posttests; Problem Solving; Word Problems (Mathematics); Intervention; Hierarchical Linear Modeling; Mathematical Concepts; Faculty Development; Instructional Effectiveness

Abstract:
This study examined the effect of schema-based instruction (SBI) on 7th-grade students' mathematical problem-solving performance. SBI is an instructional intervention that emphasizes the role of mathematical structure in word problems and also provides students with a heuristic to self-monitor and aid problem solving. Using a pretest-intervention-posttest-retention test design, the study compared the learning outcomes for 1,163 students in 42 classrooms who were randomly assigned to treatment (SBI) or control condition. After 6 weeks of instruction, results of multilevel modeling indicated significant differences favoring the SBI condition in proportion problem solving involving ratios/rates and percents on an immediate posttest (g = 1.24) and on a 6-week retention test (g = 1.27). No significant difference between conditions was found for a test of transfer. These results demonstrate that SBI was more effective than students' regular mathematics instruction. (Contains 7 tables.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Teacher Effectiveness; Problem Solving; Mathematical Models; Mathematics Instruction; Teaching Methods; Mathematical Concepts; Concept Formation; Cognitive Processes

Abstract:
The links between the mathematical and cognitive models that interact during problem solving are explored with the purpose of developing a reference framework for designing problem-posing tasks. When the process of solving is a successful one, a solver successively changes his/her cognitive stances related to the problem via transformations that allow different levels of description of the initial wording. Within these transformations, the passage between successive phases of the problem-solving process determines four operational categories: decoding (transposing the text into more explicit relations among the data and the operating schemes, induced by the constraints of the problem), representing (transposing the problem via a generated mental model), processing (identifying an associated mathematical model based on the mental configurations suggested by the problem and own mathematical competence), and implementing (applying identified mathematical techniques to the particular situation of the problem, with the purpose of drafting a conventional solution). The study of this framework in action offers insights for more effective teaching and can be used in problem posing and problem analysis in order to devise questions more relevant for deep learning. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Physics; Calculus; Mathematics Instruction; Mathematical Concepts; Cognitive Processes; Problem Solving; Concept Formation

Abstract:
Researchers are currently investigating how calculus students understand the basic concepts of first-year calculus, including the integral. However, much is still unknown regarding the "cognitive resources" (i.e., stable cognitive units that can be accessed by an individual) that students hold and draw on when thinking about the integral. This paper presents cognitive resources of the integral that a sample of experienced calculus students drew on while working on pure mathematics and applied physics problems. This research provides evidence that students hold a variety of productive cognitive resources that can be employed in problem solving, though some of the resources prove more productive than others, depending on the context. In particular, conceptualizations of the integral as an addition over many pieces seem especially useful in multivariate and physics contexts. (Contains 2 tables and 12 figures.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Cognitive Processes; Mathematics Education; College Students; Mathematical Logic; Mathematical Concepts; Learning Activities; Concept Formation; College Mathematics; Problem Solving

Abstract:
This paper reports a classroom-based study involving investigation activities in a university numerical analysis course. The study aims to analyse students' mathematical processes and to understand how these activities provide opportunities for problem posing. The investigations were intended to stimulate students in asking questions, to trigger their thinking processes, to promote their ability to investigate and to support them in learning numerical analysis' concepts and procedures. The results show that the investigations provided opportunities for students to experience mathematical processes, including posing questions, formulating and testing conjectures and, to some extent, proving results. They also provide some understanding about the role of problem posing in these processes. Posing questions occurred mainly in an implicit way, in the interpretation of tasks and in identifying regularities, analysing graphs and testing cases. The conjectures were often based on pattern identification or data manipulation, and the students tended to accept them without testing or proving. The students also proposed alternative formulations for the initial questions and posed new problems from their explorations and attempts to refine previous conjectures. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Mathematics Instruction; Mathematical Concepts; Mathematics; Algebra; Equations (Mathematics); Mathematics Education; Mathematical Formulas; Problem Solving; Mathematical Applications

Abstract:
Sometimes, in the teaching and learning of mathematics, open-ended problems posed by teachers or students can lead to a fuller understanding of mathematical concepts--a depth of understanding that no one could have anticipated. Interesting solutions and ideas emerged unexpectedly when the authors asked prospective and in-service teachers an "old" algebra question in new ways. Their initial goal was to model the National Council of Teachers of Mathematics' (NCTM's) Process Standards in their classrooms (NCTM 2000). The result was a deeper understanding of solutions that emerge from the algorithm for solving equations involving radicals. Teachers gained new insights from students' ideas and strategies. Most important, students were engaged in an exploration of radical equations with true and extraneous solutions without simply relying on an algorithm that they had been taught. These investigations ultimately led them to a deeper understanding of the mathematical concepts. (Contains 4 figures.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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