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Descriptors:
Social Environment; Problem Solving; Financial Support; Administrator Attitudes; Childhood Attitudes; Parent Attitudes; Clergy; Participation; Observation; Interviews; Mathematical Applications; Mathematical Aptitude; Mathematical Concepts; Religion Studies; Number Concepts; Number Systems; Numeracy; Influences; Theory of Mind; Beliefs; Spiritual Development; Role of Religion

Abstract:
The purpose of this study was to examine children's mathematical understandings related to participation in tithing (giving 10% of earnings to the church). Observations of church services and events, as well as interviews with parents, children, and church leaders, were analyzed in an effort to capture the ways in which mathematical problem solving was related to the social context of tithing. I introduce the direct influence model to describe components of the practice itself and to identify supports and constraints that exist related to children's opportunities to engage in problem solving that varied by participation frequency and mathematical complexity. (Contains 2 figures and 1 footnote.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Geometric Concepts; Mathematical Applications; Young Children; Grade 2; Grade 3; Short Term Memory; Mathematics Anxiety; Longitudinal Studies; Elementary School Students; Individual Differences; Prediction; Achievement Gains

Abstract:
This study explored mathematics anxiety in a longitudinal sample of 113 children followed from second to third grade. We examined how mathematics anxiety related to different types of mathematical performance concurrently and longitudinally and whether the relations between mathematics anxiety and mathematical performance differed as a function of working memory. Concurrent analyses indicated that mathematics anxiety represents a unique source of individual differences in children's calculation skills and mathematical applications, but not in children's geometric reasoning. Furthermore, we found that higher levels of mathematics anxiety in second grade predicted lower gains in children's mathematical applications between second and third grade, but only for children with higher levels of working memory. Overall, our results indicate that mathematics anxiety is an important construct to consider when examining sources of individual differences in young children's mathematical performance. Furthermore, our findings suggest that mathematics anxiety may affect how some children use working memory resources to learn mathematical applications. (Contains 2 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:
Mathematics Education; Mathematics Instruction; Mathematics; Mathematics Activities; Mathematical Concepts; Mathematical Applications

Abstract:
The nature of mathematical objects, their various types, the way in which they are formed, and how they participate in mathematical activity are all questions of interest for philosophy and mathematics education. Teaching in schools is usually based, implicitly or explicitly, on a descriptive/realist view of mathematics, an approach which is not free from potential conflicts. After analysing why this view is so often taken and pointing out the problems raised by realism in mathematics this paper discusses a number of philosophical alternatives in relation to the nature of mathematical objects. Having briefly described the educational and philosophical problem regarding the origin and nature of these objects we then present the main characteristics of a pragmatic and anthropological semiotic approach to them, one which may serve as the outline of a philosophy of mathematics developed from the point of view of mathematics education. This approach is able to explain from a non-realist position how mathematical objects emerge from mathematical practices. 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:
Accuracy; Reaction Time; Arithmetic; Adults; Mathematics; Mathematics Instruction; Mathematics Education; Mathematical Applications; Intuition

Abstract:
This study tested the hypothesis that intuitions about the effect of operations, e.g., "addition makes bigger" and "division makes smaller", are still present in educated adults, even after years of instruction. To establish the intuitive character, we applied a reaction time methodology, grounded in dual process theories of reasoning. Educated adult participants were asked to judge the correctness of statements about the effect of operations. Their accuracy and reaction times were measured. For items where the correct answer was not in line with the assumed intuitions, more mistakes were observed; moreover, we found longer reaction times for correct responses, indicating that these intuitions interfere in participants' reasoning on these tasks, even when they provide a correct response. 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:
Grading; College Faculty; Physics; Earth Science; Chemistry; College Science; Mathematical Applications; Values; Logical Thinking; Conflict; Protocol Analysis; Interviews

Abstract:
Grading practices can send a powerful message to students about course expectations. A study by Henderson et al. ("American Journal of Physics" 72:164-169, 2004) in physics education has identified a misalignment between what college instructors say they value and their actual scoring of quantitative student solutions. This work identified three values that guide grading decisions: (1) a desire to see students' reasoning, (2) a readiness to deduct points from solutions with obvious errors and a reluctance to deduct points from solutions that might be correct, and (3) a tendency to assume correct reasoning when solutions are ambiguous. These authors propose that when values are in conflict, the conflict is resolved by placing the burden of proof on either the instructor or the student. Here, we extend the results of the physics study to earth science (n = 7) and chemistry (n = 10) instructors in a think-aloud interview study. Our results suggest that both the previously identified three values and the misalignment between values and grading practices exist among science faculty more generally. Furthermore, we identified a fourth value not previously recognized. Although all of the faculty across both studies stated that they valued seeing student reasoning, the combined effect suggests that only 49% of faculty across the three disciplines graded work in such a way that would actually encourage students to show their reasoning, and 34% of instructors could be viewed as penalizing students for showing their work. This research may contribute toward a better alignment between values and practice in faculty development. 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:
Statistical Analysis; Misconceptions; Statistics; Data; Teacher Attitudes; Teachers; Mathematical Applications; Mathematics Education; Mathematics Instruction; Mathematics; Data Analysis; Comparative Analysis; Teaching Methods

Abstract:
As a consequence of the increased use of data in workplace environments, there is a need to understand the demands that are placed on users to make sense of such data. In education, teachers are being increasingly expected to interpret and apply complex data about student and school performance, and, yet it is not clear that they always have the appropriate knowledge and experience to interpret the graphs, tables and other data that they receive. This study examined the statistical literacy demands placed on teachers, with a particular focus on box plot representations. Although box plots summarise the data in a way that makes visual comparisons possible across sets of data, this study showed that teachers do not always have the necessary fluency with the representation to describe correctly how the data are distributed in the representation. In particular, a significant number perceived the size of the regions of the box plot to be depicting frequencies rather than density, and there were misconceptions associated with outlying data that were not displayed on the plot. As well, teachers' perceptions of box plots were found to relate to three themes: attitudes, perceived value and misconceptions. 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; Secondary School Students; Secondary School Mathematics; Mathematics Instruction; Computation; Mathematics Activities; Mathematical Enrichment; Mathematical Applications; Teaching Methods; Educational Experiments; Repetition; Problem Solving

Abstract:
This article will describe two activities in which students conduct experiments with random numbers so they can see that having at least one repeated birthday in a group of 40 is not unusual. The first empirical approach was conducted by author Cauto in a secondary school methods course. The second empirical approach was used by author Flores with in-service teachers. In both, the participants pretended to be secondary school students. In a third activity, students use a calculator program to determine the theoretical probability. A common misconception about random samples is that outcomes are "spread out" more or less evenly among possible results. However, students need to develop an understanding that sometimes results within a given random sample are clustered or repeated. Students must also understand that the distribution within a particular sample is not always uniformly or symmetrically distributed among the possible results. Performing simulations like the ones described here can help students develop such understanding. (Contains 2 tables and 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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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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Descriptors:
Mathematics Education; Number Concepts; Number Systems; Numbers; Numeracy; Mathematical Applications; Mathematical Models; Problem Solving; Problem Sets; Teaching Methods

Abstract:
A facility with signed numbers forms the basis for effective problem solving throughout developmental mathematics. Most developmental mathematics textbooks explain signed number operations using absolute value, a method that involves considering the problem in several cases (same sign, opposite sign), and in the case of subtraction, rewriting the problem as an addition problem. This method works, of course, but involves quite a bit of chalk to explain. It is neither a mathematically elegant way of understanding signed numbers, nor is it the method that experienced mathematics educators use themselves. "Crossroads in Mathematics: Standards for Introductory College Mathematics before Calculus" (American Mathematical Association of Two-Year Colleges, 1995) recommends developing student knowledge of mathematics by emphasizing intuitive justifications for mathematical principles and procedures. The topic of signed numbers provides an excellent context in which to develop fundamental mathematical conceptual understanding, explore properties of mathematics, and motivate students to understand higher-level structure of mathematics. This can be achieved using ring theory, which imparts a more organic understanding of numbers and illustrates why number systems are forced to operate the way they do. 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; Problem Solving; Mathematical Applications; Discovery Learning; Direct Instruction; Elementary School Students; Grade 2; Grade 3; Grade 4

Abstract:
Both exploration and explicit instruction are thought to benefit learning in many ways, but much less is known about how the two can be combined. We tested the hypothesis that engaging in exploratory activities prior to receiving explicit instruction better prepares children to learn from the instruction. Children (159 second- to fourth-grade students) solved relatively unfamiliar mathematics problems (e.g., 3 + 5 = 4 + [?]) before or after they were instructed on the concept of mathematical equivalence. Exploring problems before instruction improved understanding compared with a more conventional ''instruct-then-practice'' sequence. Prompts to self-explain did not improve learning more than extra practice. Microgenetic analyses revealed that problem exploration led children to more accurately gauge their competence, attempt a larger variety of strategies, and attend more to problem features-better preparing them to learn from instruction. (Contains 1 figure and 4 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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