Developmental Variation in Children's Creative Mathematical Thinking as a Function of Schooling,Age, and Knowledge

ABSTRACT: Prior researchers reported that children's creativity development displays a nonlinear trajectory. This article investigated the association of age, years of schooling, and domain-specific knowledge in the development of children's creativity in mathematics. DISCOVER math assessment was used to measure mathematical knowledge; originality, flexibility, and elaboration (OFE); and fluency as indexes of students' creativity. Participants included 841 first- to fifth-grade students from 4 schools. Hierarchical regression analysis indicated that domain knowledge was progressively associated with fluency and OFE from lower to upper grades, whereas age was so associated only in lower grades. Multivariate analysis of variance showed that years of schooling significantly contributed to students' creativity even after domain knowledge was partialed out. Students displayed peaks and slumps as a function of age and domain knowledge, but not as a function of grade. Knowledge at the level of 2 SDs above the mean was found to be the threshold for creativity at the level of 1 SD above the mean.     

Educational Intervention and the Development of Young Art Students' Talent and Creativity

This study focuses on behavior associated with young art students' developing artistic talent (skills and art‐making behavior) and creativity (personal expressions of visual information). The study examines the role of personal expertise in a student's development of problem finding, domain‐specific technical skill, perseverance, evaluation, and creative ideation. The study compares 30 experienced art students' artistic processing and products with those of 29 novice art students. Both groups are 7‐ through 11‐year‐olds. The author recorded participants' behavior as they created drawings in two contexts — from imagination and from life — and three adult artists then assessed the technical skill and creativity revealed in the drawings. Multivariate analyses of the variables associated with the drawing products and processes offer evidence of the changes related to the students' developing expertise in both novice and experienced groups. This study finds that the drawing situation (life or imagination) interacts clearly with the relationships among hypothesized components of creativity, gender, and predictors of expertise. Technical skill, perseverance, modifications, and creativity in drawings from life were significant predictors of expertise. Modifications, efficient problem finding, and creativity in drawings from imagination were additional significant predictors of expertise. Gender was found to be a measurable factor in both the artistic process and the assessments of drawings from imagination. The findings are discussed within the context of three conceptions: artistic talent, developing creativity, and art education.     

How Much Does Creative Teaching Enhance Elementary School Students' Achievement?

This study examined the relationship between creative teaching and elementary students' achievement gains. Forty‐eight upper elementary school teachers' classroom instruction was observed and evaluated over the course of 8 different lessons throughout the year. For each teacher, during each lesson, both a creative teaching frequency score and a quality score were derived. These scores were then used as predictor variables in a structural equation model to determine the magnitude of the relationship between creative teaching and classroom achievement gains in reading, language, and mathematics. Our results demonstrated that (a) the majority of teachers do not implement any teaching strategies that foster student creativity; (b) teachers who elicit student creativity turn out students that make substantial achievement gains; and (c) classrooms with high proportions of minority and low‐performing students receive significantly less creative teaching.     

The Structure of Creativity in Primary Education: An Empirical Confirmation of the Amusement Park Theory

In this study, we explored the structure of pupils’ creativity in primary education following the Amusement Park Theory, by investigating undiscovered linkages between the domains of writing, mathematics, and drawing. More specifically, we examined: (a) whether some domains and general thematic areas are more closely related to each other than to others, (b) whether literacy and mathematical ability are specific underlying traits of creativity in writing and mathematics, respectively, and (c) whether intelligence and divergent thinking are related to creativity in all domains. The sample consisted of 331 Dutch 4th grade pupils. For each research question, a model was analyzed using structural equation modeling. We found creativity in mathematics and creativity in writing to be most similar, followed by creativity in mathematics and creativity in drawing, with creativity in writing and creativity in drawing being least similar. Additionally, we found evidence for several underlying traits (i.e., literacy ability and mathematical ability) and initial requirements of creativity (i.e., intelligence and divergent thinking), none of which were important for creativity in only one domain, and of which only intelligence was important for creativity in all domains. Herewith, our study provides insights regarding the complexity of the structure of creativity in primary education.     

Reliability and Validity of a Newly Constructed Multiple-Choice Creativity Instrument

An alternative method for testing creativity was investigated. A multiple-choice paper-and-pencil test called the Abedi–Schumacher Creativity Test (CT) was developed in an attempt to shorten the amount of time required for the administration and scoring of creativity tests. This instrument was translated into Spanish and was administered with the Torrance Tests of Creative Thinking (TTCT) and the Villa and Auzmendi Creativity Test (VAT) to 2,270 students in Spain. Teacher ratings of student creativity were also examined. Significant but low correlations were found between the 4 CT subscale scores and the students' academic achievement measures and TTCTsubscale scores. The reliability coefficient of the CT subscale scores was at an acceptable level; however, the correlation coefficients between the CT and the TTCT and VAT were moderate or higher. The results of this study have influenced further modification of the CT test items.     

Motivation as a Mediator of Social Disparities in Academic Achievement

The present study aimed at contributing to the understanding of social disparities in relation to students' academic achievement in the science, technology, engineering and mathematics domains. A sample of n = 321 German 11th graders completed measures of their family socio‐economic status (SES), general intelligence, domain‐specific ability self‐concepts and subjective scholastic values in math, physics and chemistry. Students' grades in these subjects received four months after testing served as criteria. Significant mediation effects were found for all motivational variables between fathers' SES and students' achievement, whereas for mothers' SES, only children's academic self‐concept in chemistry was a significant mediator. These results also held when students' general intelligence was controlled. Additionally, we controlled for students' grades before testing to investigate which variables mediated the influence of SES on change in school performance. Motivational variables significantly mediated the influence of fathers' SES on change in school performance in math but not in chemistry and physics. Intelligence significantly mediated the influence of fathers' SES on change in school performance in physics and chemistry but not in mathematics. The impact of mothers' SES on change in grades in chemistry was mediated by intelligence. Among others, the reasons potentially accounting for the differential influences of fathers' and mothers' SES are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Mathematical Creativity: A Combination of Domain-general Creative and Domain-specific Mathematical Skills

Creativity is an understudied topic in elementary school mathematics research. Nevertheless, we argue that creativity plays an important role in mathematics, but that more research is needed to understand this relation. Therefore, this study aimed to investigate this relation, specifically between domain-general creativity, domain-specific mathematical creativity, and mathematical ability. Measures for these constructs were administered to 342 Dutch fourth graders. In order to examine the nature of the relation between creativity and mathematics, two competing models were tested, using Structural Equation Modeling. The results indicated that models in which general creativity and mathematical ability both predict mathematical creativity fitted the data better than models in which mathematical and general creativity predict mathematical ability. This study showed that both general creativity and mathematical ability are important to think creatively in mathematics.     

Racing dragons and remembering aliens: Benefits of playing number and working memory games on kindergartners' numerical knowledge

Sources that contribute to variation in mathematical achievement include both numerical knowledge and general underlying cognitive processing abilities. The current study tested the benefits of tablet‐based training games that targeted each of these areas for improving the mathematical knowledge of kindergarten‐age children. We hypothesized that playing a number‐based game targeting numerical magnitude knowledge would improve children's broader numerical skills. We also hypothesized that the benefits of playing a working memory (WM) game would transfer to children's numerical knowledge given its important underlying role in mathematics achievement. Kindergarteners from diverse backgrounds (n = 148; 52% girls; M_age = 71.87 months) were randomly assigned to either play a number‐based game, a WM game, or a control game on a tablet for 10 sessions. Structural equation modeling was used to model children's learning gains in mathematics and WM across time. Overall, our results suggest that playing the number game improved kindergarten children's numerical knowledge at the latent level, and these improvements remained stable as assessed 1 month later. However, children in the WM group did not improve their numerical knowledge compared to children in the control condition. Playing both the number game and WM game improved children's WM at the latent level. Importantly, the WM group continued to improve their WM for at least a month after playing the games. The results demonstrate that computerized games that target both domain‐specific and domain‐general skills can benefit a broad range of kindergarten‐aged children.     

Multi-Party,Whole-Body Interactions in Mathematical Activity

This study interrogates the contributions of multi-party, whole-body interactions to students' collaboration and negotiation of mathematics ideas in a task setting called walking scale geometry, where bodies in interaction became complex resources for students' emerging goals in problem solving. Whole bodies took up overlapping roles representing geometric objects, contributing to the communication and negotiation of problem-solving strategies and engaging as mathematical instruments for representation.     

A Latent-Variable Modeling Approach to Assessing Reliability and Validity of a Creativity Instrument

ABSTRACT: A multiple-choice paper-and-pencil test was developed based on the constructs from the Torrance Test of Creative Thinking (TTCT). The rationale of this new instrument was to shorten the amount of time required for the administration and scoring of creativity tests. This instrument was translated into Spanish and administered along with the TTCT and the Villa and Auzmendi Creativity Test (VAT) to 2,270 students in Spain. Teachers' ratings of student creativity were also obtained. Using a traditional concurrent validation approach from an earlier study, we found low but significant correlations between the scores of the four subscales of the new test and the same subscales of the VAT and TTCT, as well as the students' academic achievement measures. In terms of reliability, mostly moderate (internal consistency) reliability coefficients were found for the new test. As an alternative to the traditional approach, this study employed structural equation modeling (SEM) with multiple indicators to examine the validity and reliability of the new creativity test. The results indicated that substantial improvements in the validity and reliability estimation over the traditional approaches can be made by using SEM with multiple indicators.     

An Investigation of Teachers' Beliefs of Students' Algebra Development

Elementary, middle, and high school mathematics teachers (N = 105) ranked a set of mathematics problems based on expectations of their relative problem-solving difficulty. Teachers also rated their levels of agreement to a variety of reform-based statements on teaching and learning mathematics. Analyses suggest that teachers hold a symbol-precedence view of student mathematical development, wherein arithmetic reasoning strictly precedes algebraic reasoning, and symbolic problem-solving develops prior to verbal reasoning. High school teachers were most likely to hold the symbol-precedence view and made the poorest predictions of students' performances, whereas middle school teachers' predictions were most accurate. The discord between teachers' reform-based beliefs and their instructional decisions appears to be influenced by textbook organization, which institutionalizes the symbol-precedence view. Because of their extensive content training, high school teachers may be particularly susceptible to an expert blindspot, whereby they overestimate the accessibility of symbol-based representations and procedures for students' learning introductory algebra.     

Exploring Collective Mathematical Creativity in Elementary School

This study combines theories related to collective learning and theories related to mathematical creativity to investigate the notion of collective mathematical creativity in elementary school classrooms. Collective learning takes place when mathematical ideas and actions, initially stemming from an individual, are built upon and reworked, producing a solution which is the product of the collective. Referring to characteristics of individual mathematical creativity, such as fluency, flexibility, and originality, this paper examines the possibility that collective mathematical creativity may be similarly characterized. The paper also explores the role of the teacher in fostering collective mathematical creativity and the possible relationship between individual and collective mathematical creativity. Many studies have investigated ways of characterizing, identifying, and promoting mathematical creativity. Haylock (1997), for example, and more recently, Kwon, Park, and Park (2006) assessed students' mathematical creativity by employing open‐ended problems and measuring divergent thinking skills. Leikin (2009) explored the use of multiple solution tasks in evaluating a student's mathematical creativity. These studies focused on an individual's mathematical creativity as it manifests itself in the solving of various problems. Yet students, acting in a classroom community, do not necessarily act on their own. Ideas are interchanged, evaluated, and built‐upon, often with the guidance of the teacher. The resultant mathematical creativity of an individual may be a product of collective community practice. The question which then arises is: Who is being mathematically creative, the individual or the community? This study focuses on the collective, not as the aggregation of a few individuals, but as a unit of study. Although some of the studies mentioned above acknowledged the effect of classroom culture on the development of mathematical creativity, and others considered the creative range of a group of students, those studies did not necessarily investigate mathematical creativity as a collective process or as the product of participating in a collective endeavour. This study combines theories related to collective learning and theories related to mathematical creativity to investigate the notion of collective mathematical creativity. The notion of collective creativity has been used to investigate creativity in several contexts including the work place (Hargadon & Bechky, 2006) and the global community (Family, 2003). In those cases, collective creativity was considered to occur when the social interactions between individuals yielded new interpretations that the individuals involved, thinking alone, could not have generated. Can the notion of collective creativity also be applied to the classroom community?     

Who Benefits from Diagrams and Illustrations in Math Problems? Ability and Attitudes Matter

How do diagrams and illustrations affect mathematical problem solving? Past research suggests that diagrams should promote correct performance. However, illustrations may provide a supportive context for problem solving, or they may distract students with seductive details. Moreover, effects may not be uniform across student subgroups. This study assessed the effects of diagrams and illustrations on undergraduates' trigonometry problem solving. We used a 2 (Diagram Presence) × 2 (Illustration Presence) within‐subjects design, and our analysis considered students' mathematics ability and attitudes towards mathematics. Participants solved problems more accurately when they included diagrams. This effect was stronger for students who had more positive mathematics attitudes, especially when there was an illustration present. Illustrations were beneficial for students with high mathematics ability but detrimental for students with lower ability. Considering individual differences in ability and attitude is essential for understanding the effects of different types of visual representations on problem solving. Copyright © 2017 John Wiley & Sons, Ltd.     

An Investigation of the Relationship between Age,Achievement, and Creativity in Mathematics

In this exploratory study, I investigate the relationship between age, knowledge, and creativity in mathematics, by looking at to what extent does grade level, controlled for mathematical achievement, influence mathematical creativity and what characterizes the relationship between grade level, mathematical achievement and mathematical creativity. This was accomplished in two steps. In the first part, 301 students, 184 grade eight students and 117 grade eleven students, were given a creative mathematics test. A 3 × 2 ANOVA indicates that the older students were more creative; however, there was a significant interaction effect between grade level and achievement in mathematics on mathematical creativity. In the second part, an inductive content analysis was performed on the solutions of high achievers in grade eleven and grade eight. The results indicate that high achievers in grade eight are more creative than high achievers in grade eleven, but the nature of the task mediates the relationship between creativity and knowledge.     

Students' Divergent Thinking and Teachers' Ratings of Creativity: Does Gender Play a Role?

The differences between genders regarding the properties of divergent thinking and teachers' ratings of students' creativity are the issue of the present research. Data gathered from three previous experimental studies in Greek primary school students (N total = 228) was used for this purpose. In these studies, divergent thinking tasks were assigned to students and teachers' ratings were collected. The results showed that there were indeed differences in performance — except in the subscale of originality — in favor of girls who were more likely to perform better when they had a male teacher. Teachers' ratings of creativity were not related to students' gender but to teachers' gender.     

Cognitive correlates of performance in advanced mathematics

Background. Much research has been devoted to understanding cognitive correlates of elementary mathematics performance, but little such research has been done for advanced mathematics (e.g., modern algebra, statistics, and mathematical logic). Aims. To promote mathematical knowledge among college students, it is necessary to understand what factors (including cognitive factors) are important for acquiring advanced mathematics. Samples. We recruited 80 undergraduates from four universities in Beijing. Methods. The current study investigated the associations between students’ performance on a test of advanced mathematics and a battery of 17 cognitive tasks on basic numerical processing, complex numerical processing, spatial abilities, language abilities, and general cognitive processing. Results. The results showed that spatial abilities were significantly correlated with performance in advanced mathematics after controlling for other factors. In addition, certain language abilities (i.e., comprehension of words and sentences) also made unique contributions. In contrast, basic numerical processing and computation were generally not correlated with performance in advanced mathematics. Conclusions. Results suggest that spatial abilities and language comprehension, but not basic numerical processing, may play an important role in advanced mathematics. These results are discussed in terms of their theoretical significance and practical implications.     

English learners' achievement in mathematics and science: Examining the role of self-efficacy

The goals of the current study were twofold. The first goal was to describe levels of mathematics and science self-efficacy and achievement among a sample of students with varying levels of English language proficiency. The second goal was to examine the extent to which students' self-efficacy explains the relation between their English proficiency level and mathematics and science achievement. The sample consisted of 332 fifth graders (mean age = 10.46 years, SD = 0.38) and their 63 teachers in 20 schools. The student sample was linguistically diverse with parents reporting 22 different home languages. Based on district classification procedures, each student was coded into one of three English language proficiency level categories: English proficient-speaking students (English proficient), English Learner (EL) students who are reaching proficiency, yet are still being monitored (reaching proficiency), and EL students who are receiving English for Speakers of Other Languages services (ESOL; limited English proficient). Regression analyses indicated that students identified as limited English proficient consistently demonstrated lower achievement and self-efficacy across the content areas of mathematics and science as compared to their peers who were English proficient and reaching proficiency. In addition, students' self-efficacy partially explained the relation between limited English proficiency level and achievement for science, but not for mathematics. Results indicate that educators should consider variability in students' English proficiency levels as they select supports to promote both science achievement and self-efficacy. Findings also suggest promise for practices and programs that foster self-efficacy in addition to language and content skills.     

Changing levels of numeracy and other core mathematical skills among psychology undergraduates between 1992 and 2002

Teaching statistics and research methods to psychology undergraduates is a major pedagogic challenge. Knowledge of students' conceptual problems in mathematics is important in the current climate of widening access, a burgeoning interest in psychology, and fears about declining standards of numeracy and other quantitative skills. This study compared the mathematical knowledge of two cohorts of undergraduates who entered psychology a decade apart — one in 1992, the other in 2002. Six broadly defined components of mathematical thinking relevant to the teaching of statistics in psychology were examined –calculation, algebraic reasoning, graphical interpretation, proportionality and ratio, probability and sampling, andestimation. Both cohorts were also compared with a 1984 cohort on a subset of items reported in a study by Greer and Semrau (1984). Results revealed highly significant differences between the two cohorts on all six components, with 1992 students outperforming their 2002 counterparts. Males were also found to perform significantly better than females on a majority of components. Level of qualification in mathematics was found to predict overall performance. Comparison with Greer and Semrau's (1984) sample revealed an alarming decline in performance across the two decades on a selection of test items.