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.
Even after many decades of productive research, problem solving instruction is still considered ineffective. In this study we address some limitations of extant problem solving models related to the ...phenomenon of insight during problem solving. Currently, there are two main views on the source of insight during problem solving. Proponents of the first view argue that insight is the consequence of analytic thinking and a sequence of conscious and stepwise steps. The second view suggests that insight is the result of unconscious processes that come about only after an impasse has occurred. Extant models of problem solving within mathematics education tend to highlight the first view of insight, while Gestalt inspired creativity research tends to emphasize the second view of insight. In this study, we explore how the two views of insight—and the corresponding set of models—can describe and explain different aspects of the problem solving process. Our aim is to integrate the two different views on insight, and demonstrate how they complement each other, each highlighting different, but important, aspects of the problem solving process. We pursue this aim by studying how expert and novice mathematics students worked on two ill-defined mathematical problems. We apply both a problem solving model and a creativity model in analyzing students’ work on the two problems, in order to compare and contrast aspects of insight during the students’ work. The results of this study indicate that sudden and unconscious insight seems to be crucial to the problem solving process, and the occurrence of such insight cannot be fully explained by problem solving models and analytic views of insight. We therefore propose that extant problem solving models should adopt aspects of the Gestalt inspired views of insight.
In this exploratory study, a theoretical model proposed by Sriraman (2005) consisting of five theoretical principles for optimizing creativity in a K–12 setting was investigated empirically. This was ...accomplished in two steps. In the first study, the five principles were operationalized by generating a questionnaire consisting of 45 items intended to capture the dimension of each principle. An exploratory maximum‐likelihood factor analysis indicated a relatively robust five factor structure that corresponded with the theoretical model. In the second study, the five factor model was validated using a confirmatory factor analysis. The model was then investigated using a two‐level linear mixed model with a random intercept. The results revealed that motivation and mathematical achievement were significant predictors of mathematical creativity.
In this paper, we investigate the learning experiences, beliefs and motivations of students in classes where the mathematics teachers have received support for using inquiry-based learning ...activities. Data were collected from 248 students in the age-range 11-16 using electronic questionnaires. Our results show that key features of inquiry-based mathematics were only moderately reflected in these students' beliefs about the subject, their dispositions towards mathematics were less positive across the transition from primary to secondary school, and with respect to motivation this decline was stronger for girls than for boys. Furthermore, medium to strong correlations between belief- and motivation subdomains were found, for instance, students who view mathematics as a creative subject and/or have a growth mindset of mathematics also tend to find this subject enjoyable and perceive it as useful. Finally, our results indicate that inquiry-based teaching has a potential for fostering positive dispositions towards mathematics, as students who often experience inquiry-related activities in class also tend to see mathematics as a creative and interesting subject that will be useful for them in the future.
De siste årene har omvendt undervisning, eller flipped classroom, vært mye omtalt i både norsk og utenlandsk skoledebatt. I en gjennomgang av relevant litteratur konkluderer Estes, Ingram og Liu ...(2014) at omvendt undervisning kan ha en positiv læringseffekt. I denne studien ble et kvasieksperiment gjennomført på tre videregående skoler for å undersøke i hvilken grad omvendt undervisning påvirket læringsutbyttet i matematikk, sammenlignet med tradisjonell undervisning. Det ble også undersøkt om omvendt undervisning kan påvirke elevers oppfatninger om matematikk. På én av de tre skolene var omvendt undervisning innført. Elevenes matematikkunnskap og oppfatninger om matematikk ble testet ved starten og slutten av et skoleår. Elevenes besvarelser ble deretter analysert for å se om det var statistiske forskjeller i endring av læringsutbyttet og oppfatninger om matematikk mellom elever som hadde hatt omvendt undervisning og elever som hadde hatt tradisjonell undervisning. Analysene viste at elevene som fikk omvendt undervisning, hadde en større faglig fremgang enn elevene som fikk tradisjonell undervisning. Analysene viste også at elevene som fikk omvendt undervisning, endret sine oppfatninger om matematikk i større grad enn elever som fikk tradisjonell undervisning. Dette kan tyde på at omvendt undervisning er et tiltak som kan være med på å styrke elevenes læringsutbytte i skolematematikk. Men i denne studien ble ikke selve undervisningen observert. Det betyr at også andre faktorer kan ha påvirket resultatene. For å undersøke omvendt undervisning i matematikk fremover, vil det derfor være nødvendig å undersøke selve undervisningen nærmere.Nøkkelord: omvendt undervisning, læringsutbytte, kvasieksperiment, oppfatningerTo what extent does flipped classroom affect students’ mathematical knowledge and conceptions of mathematics?AbstractFlipped classroom is a popular trend in education. In a review of relevant literature, Estes, Ingram and Liu (2014), conclude that flipped classroom can have a positive effect on students’ learning. In this study, a quasi-experiment was carried out in three upper secondary schools to investigate to what extent flipped classroom can affect students’ learning outcome in mathematics, com-pared to traditional teaching. The study also investigated whether flipped classroom can affect students’ conceptions of mathematics. Flipped classroom was introduced in one of the three schools. Students’ mathematical knowledge and conceptions of mathematics were tested at the start and finish of one school year. The students’ responses were then analyzed to see if there were statistical differences in change of learning outcome between students in flipped classrooms and students in traditional classrooms. The analyses showed that students in flipped classrooms had a larger increase in mathematical knowledge and larger change of conceptions of mathematics than students in traditional classrooms. This indicates that flipped classroom can have a positive effect on students’ learning outcomes, compared to traditional classrooms. However, the actual teaching was not observed. Other variables may therefore have had an effect on the results. Future investigations of flipped classroom in mathematics should therefore also focus on the teaching itself.Keywords: flipped classroom, learning outcome, quasi-experiment, conceptions
De siste årene har omvendt undervisning, eller flipped classroom, vært mye omtalt i både norsk og utenlandsk skoledebatt. I en gjennomgang av relevant litteratur konkluderer Estes, Ingram og Liu ...(2014) at omvendt undervisning kan ha en positiv læringseffekt. I denne studien ble et kvasieksperiment gjennomført på tre videregående skoler for å undersøke i hvilken grad omvendt undervisning påvirket læringsutbyttet i matematikk, sammenlignet med tradisjonell undervisning. Det ble også undersøkt om omvendt undervisning kan påvirke elevers oppfatninger om matematikk. På én av de tre skolene var omvendt undervisning innført. Elevenes matematikkunnskap og oppfatninger om matematikk ble testet ved starten og slutten av et skoleår. Elevenes besvarelser ble deretter analysert for å se om det var statistiske forskjeller i endring av læringsutbyttet og oppfatninger om matematikk mellom elever som hadde hatt omvendt undervisning og elever som hadde hatt tradisjonell undervisning. Analysene viste at elevene som fikk omvendt undervisning, hadde en større faglig fremgang enn elevene som fikk tradisjonell undervisning. Analysene viste også at elevene som fikk omvendt undervisning, endret sine oppfatninger om matematikk i større grad enn elever som fikk tradisjonell undervisning. Dette kan tyde på at omvendt undervisning er et tiltak som kan være med på å styrke elevenes læringsutbytte i skolematematikk. Men i denne studien ble ikke selve undervisningen observert. Det betyr at også andre faktorer kan ha påvirket resultatene. For å undersøke omvendt undervisning i matematikk fremover, vil det derfor være nødvendig å undersøke selve undervisningen nærmere. Nøkkelord: omvendt undervisning, læringsutbytte, kvasieksperiment, oppfatninger To what extent does flipped classroom affect students’ mathematical knowledge and conceptions of mathematics? Abstract Flipped classroom is a popular trend in education. In a review of relevant literature, Estes, Ingram and Liu (2014), conclude that flipped classroom can have a positive effect on students’ learning. In this study, a quasi-experiment was carried out in three upper secondary schools to investigate to what extent flipped classroom can affect students’ learning outcome in mathematics, com-pared to traditional teaching. The study also investigated whether flipped classroom can affect students’ conceptions of mathematics. Flipped classroom was introduced in one of the three schools. Students’ mathematical knowledge and conceptions of mathematics were tested at the start and finish of one school year. The students’ responses were then analyzed to see if there were statistical differences in change of learning outcome between students in flipped classrooms and students in traditional classrooms. The analyses showed that students in flipped classrooms had a larger increase in mathematical knowledge and larger change of conceptions of mathematics than students in traditional classrooms. This indicates that flipped classroom can have a positive effect on students’ learning outcomes, compared to traditional classrooms. However, the actual teaching was not observed. Other variables may therefore have had an effect on the results. Future investigations of flipped classroom in mathematics should therefore also focus on the teaching itself. Keywords: flipped classroom, learning outcome, quasi-experiment, conceptions
First is the general lack of regard for education by policy makers, and the notion that any qualified person from industry or the private sector is capable of understanding the complexities of a ...school system and acting in a leadership role. The alien voices examine the article without the immigrant bias. ... the ensuing narrative is composed as three voices written in the first person following the style of the article.
Mathematical techniques in proof writing can be narrowed down to specific proof styles. Simply put, proofs can be direct or indirect- the latter using the Law of the Excluded Middle from logic as ...well as the axiom of Choice, to prove existence of mathematical objects. However, the thinking skills involved in writing indirect proofs are prone to errors, especially from novice proof writers such as prospective teachers. Creativity in mathematics entails the use of both direct and indirect approaches to determine the validity of a statement. In this article, I shed some light on this relationship, by reporting on some findings from a study on how students comprehend and validate direct and indirect proofs. Furthermore, I use the constructs of fixation and flexibility from creativity research to examine student approaches to direct and indirect proofs.
Mathematical techniques in proof writing can be narrowed down to specific proof styles. Simply put, proofs can be direct or indirect- the latter using the Law of the Excluded Middle from logic as ...well as the axiom of Choice, to prove existence of mathematical objects. However, the thinking skills involved in writing indirect proofs are prone to errors, especially from novice proof writers such as prospective teachers. Creativity in mathematics entails the use of both direct and indirect approaches to determine the validity of a statement. In this article, I shed some light on this relationship, by reporting on some findings from a study on how students comprehend and validate direct and indirect proofs. Furthermore, I use the constructs of fixation and flexibility from creativity research to examine student approaches to direct and indirect proofs.
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