Impediments to mathematical creativity: Fixation and flexibility in proof validation

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

Students’ mathematical beliefs and motivation in the context of inquiry-based mathematics teaching

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

Using pathologies as starting points for inquiry-based mathematics education: The case of the palindrome

Source at http://matematikdidaktik.org/Inquiry-based mathematics education (IBME) is an increasingly important ingredient of the mathematics education in the Nordic countries. The central principle of IBME is that the students are to work in ways similar to how professional mathematicians work. In this qualitative case study, we investigate whether mathematical pathologies induce students to work like mathematicians, and thus if pathologies are suitable starting points for IBME. We based our investigations on a little-known pathology: multiplication problems that can be mirrored about the equal sign without altering the answer

What are the characteristics of mathematical creativity? An empirical and theoretical investigation of mathematical creativity.

The main research question in this research project was "what are the characteristics of mathematical creativity?" It was investigated both qualitatively and quantitatively and the findings are presented in four articles. In the first article, three high achieving students were given an unfamiliar trigonometric problem. The findings indicate strongly that the student’s way of thinking is strongly linked with imitative reasoning and only when received some form of guidance, were they able to display creative and flexible reasoning. In the second article, a theoretical model for optimizing creativity was investigated empirically using ANCOVA. Intrinsic motivation and an aesthetic sense of mathematics were found to be significant predictors of mathematical creativity. The data also revealed a strong relationship between mathematical achievement and mathematical creativity. The third article looked at differences in mathematical creativity between 8th grade students and 11th grade students. Older students were found to score significantly higher on the mathematical creativity test. A proposed explanation for this is that the older students have three more years of schooling and might have developed a stronger connectedness of their mathematical knowledge base. Article four is a commentary and synthesis of three articles on mathematical creativity, giftedness and ability. The synthesis revealed that mathematical creativity is strongly linked with mathematical ability, spatial cognitive style and high intelligence

I hvilken grad påvirker omvendt undervisning elevenes matematikkunnskap og oppfatninger om matematikk?

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, conception

Creativity in problem solving: Integrating two different views of insight

Even after many decades of productive research, problem solving instruction is still considered inefective. 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 frst 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 frst 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 diferent 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 diferent, but important, aspects of the problem solving process. We pursue this aim by studying how expert and novice mathematics students worked on two ill-defned 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

Coherence through inquiry based mathematics education

International audienceSUM is a four-year research and developmental project with the aim of contributing to coherence in children’s and students’ motivation for, activities in, and learning of mathematics throughout the educational system from kindergarten to higher education. The concept of inquiry is key in the project, and it involves the implementation of different types of theories and methods related to inquiry based mathematics teaching (IBMT) at three systemic levels: (1) The students’ inquiry in and with mathematics. (2) The teachers’ use of inquiry based mathematics teaching as a means for supporting the students’ learning and inquiry into their own practice across a particular transition. (3) Inquiry into the interplay between the development of teaching practice and research in IBMT. In this paper, based on the project design and preliminary findings from the implementation, we discuss the project as an implementation of theory at these three levels, with a particular focus on level 2

Immigrant and "Alien" Reactions to Obama 's Educational Policy:Disposition of Authenticity or the Politics of The Emperor's New Clothes

The article by Stella Batagiannis brings into sharp relief issues confronting education in general in the United States through the narrative of an immigrant inside the system. In particular. Ms. Batagiannis addresses current policy level rhetoric and subsequent decision making that has led to increased polarization of views about the nature and purpose of public schools, the role of teachers, and measured accountability through high stakes testing as the shibboleth for society. This reaction, in the form of voices is composed by another immigrant to the United States (Bharath Sriraman), and two Norwegian aliens1 to the system (Haavold Per 0ystein and Gunnar Kristiansen)

Inquiry-based mathematics teaching in practice: a case of a three-phased didactical model

International audienceDuring the latest decades, inquiry-based teaching (IBMT) has become one of the top issues at the agenda for educational politics. IBMT is seen as having a potential for enhancing the students' motivation for and appreciation of mathematics as a field of activity and as a tool for understanding the world. IBMT can be conceptualized and operationalized in different ways. In this paper, we focus on a three-phased didactical model for IBMT, which can frame the students' inquiry in and with mathematics, and support the teachers' planning and implementation of an inquiry based activity. More specifically, we present a case study of how the use of the didactical model can facilitate the implementation of IBMT, and what are the challenges that remain and need to be addressed