Constructs of engagement emerging in an ethnomathematically-based teacher education course

This paper presents a case study, in which we apply and develop theoretical constructs to analyze motivating desires observed in an unconventional, culturally contextualized teacher education course. Participants, Israeli students from several different cultures and backgrounds (pre-service and in-service teachers, Arabs and Jews, religious and secular) together studied geometry through inquiry into geometric ornaments drawn from diverse cultures, and acquired knowledge and skills in multicultural education. To analyze affective behaviors in the course we applied the methodology of engagement structures recently proposed by Goldin and his colleagues. Our study showed that engagement structures were a powerful tool for examining motivating desires of the students. We found that the constructivist ethnomathematical approach highlighted the structures that matched our instructional goals and diminished those related to students’ feelings of dissatisfaction and inequity. We propose a new engagement structure “Acknowledge my culture” to embody motivating desires, arising from multicultural interactions, that foster mathematical learning.     

Dimensions of students’ views of themselves as learners of mathematics

Students’ views of themselves as learners of mathematics are a decisive parameter for their engagement and success in school. We are interested in students’ experiences with mathematics encompassing cognitive, emotional and motivational aspects. In particular, we focus on capturing the structural properties of affect related to mathematics. Participants in our study were 1,436 randomized chosen students of secondary schools from overall Finland. In the Finnish upper secondary school, there are two different syllabi for mathematics: the general and the advanced one. Schools were invited to organize the survey by one of their year 2 general syllabus courses and one of their year 2 advanced syllabus courses in grade 11. By means of factor analysis, we obtained seven dimensions in which students’ hold beliefs and emotions about mathematics partly intertwined with their motivational orientations. These dimensions are described by reliable scales, which allow outlining an average image of Finnish students’ views of themselves as learners of mathematics. Moreover, we analyzed relations between the seven dimensions and what kind of structure they generate. Thereby, a core of three high correlating dimensions could be identified, yielding different accentuations with regard to course choice.     

Factors affecting the process of proficient mental addition and subtraction: case studies of flexible and inflexible computers

The relationship between mental computation and number sense is complex: mental computation can facilitate number sense when students are encouraged to be flexible, but flexibility and number sense is neither sufficient nor necessary for accuracy in mental computation. It is possible for familiarity with a strategy to compensate for a lack of number sense and inefficient processes. This study reports on six case studies exploring Year 3 students’ procedures for and understanding of mental addition and subtraction, and understanding of number sense and other cognitive, metacognitive, and affective factors associated with mental computation. The case studies indicate that the mental computation process is composed of four stages in which cognitive, metacognitive and affective factors operate differently for flexible and inflexible computers. The authors propose a model in which the differences between computer types are seen in terms of the application of different knowledges in number facts, numeration, effect of operation on number, and beliefs and metacognition on strategy choice and strategy implementation.     

Student and Teacher Perspectives Across Mathematics and Science Classrooms: The Importance of Engaging Contexts

Substantial recent focus has been placed upon the competitiveness of American students in increasingly global economies and entrepreneurial enterprises. As concerns center on students’ educational preparedness and their efforts at continued learning, researchers acknowledge the importance of student engagement with school. In order to foster engaged learners, teachers must be able to determine and monitor their students’ levels of engagement. The current study examined the alignment of perceptions of engagement by students, teachers, and outside observers across middle and high school mathematics and science classrooms. Results indicated significant teacher‐student differences in perceptions of student cognitive engagement across mathematics and science classrooms with teachers consistently perceiving higher levels than students. Moreover, most effect sizes were moderate to large. A subsequent multi‐level analysis indicated that while teacher perceptions of student cognitive engagement were somewhat predictive of student reported cognitive engagement, academic engagement ratings by outside observers were not.     

Proving as problem solving: The role of cognitive decoupling

This paper discusses the process of proving from a novel theoretical perspective, imported from cognitive psychology research. This perspective highlights the role of hypothetical thinking, mental representations and working memory capacity in proving, in particular the effortful mechanism of cognitive decoupling: problem solvers need to form in their working memory two closely related models of the problem situation – the so-called primary and secondary representations – and to keep the two models decoupled, that is, keep the first fixed while performing various transformations on the second, while constantly struggling to protect the primary representation from being “contaminated” by the secondary one. We first illustrate the framework by analyzing a common scenario of introducing complex numbers to college-level students. The main part of the paper consists of re-analyzing, from the perspective of cognitive decoupling, previously published data of students searching for a non-trivial proof of a theorem in geometry. We suggest alternative (or additional) explanations for some well-documented phenomena, such as the appearance of cycles in repeated proving attempts, and the use of multiple drawings.     

Facilitating parental engagement in school mathematics and science through inquiry-based learning: an examination of teachers’ and parents’ beliefs

This study examined teachers’ and parents’ beliefs on the implementation of inquiry-based modeling activities as a means to facilitate parental engagement in school mathematics and science. The study had three objectives: (a) to describe teachers’ beliefs about inquiry-based mathematics and science and parental engagement; (b) to describe parents’ beliefs about inquiry-based mathematics and science and their engagement in inquiry-based problem solving; and (c) to explore the impact of an inquiry-based learning environment comprising a model-eliciting activity and Twitter. The research involved three sixth-grade teachers and 32 parents from one elementary school. Teachers and parents participated in workshops, followed by the implementation of a model-eliciting activity in two classrooms. Three teachers and six parents participated in semi-structured interviews. Teachers reported positive beliefs on parental engagement in the mathematics and science classrooms and the potential positive role of parents in implementing innovative problem-solving activities. Parents expressed strong beliefs on their engagement and welcomed the inquiry-based modeling approach. Based on the results of this aspect of a four-year longitudinal design, implications for parental engagement in inquiry-based mathematics and science teaching and learning and further research are discussed.     

Examining epistemic beliefs across conceptual and procedural knowledge in statistics

The purpose of this paper was to examine whether students’ epistemic beliefs differed as a function of variations in procedural versus conceptual knowledge in statistics. Students completed Hofer’s (Contem Edu Psychol 25:378–405, 2000) Discipline-Focused Epistemological Beliefs Questionnaire five times over the course of a semester. Differences were explored between students’ initial beliefs about statistics knowledge and their specific beliefs about conceptual knowledge and procedural knowledge in statistics. Results revealed differences across these contexts; students’ beliefs differed between procedural versus conceptual knowledge. Moreover, students’ initial beliefs about statistics knowledge were more similar to their beliefs about conceptual knowledge rather than procedural knowledge. Finally, regression analyses revealed that students’ beliefs about the justification of knowledge, attainability of truth and source of knowledge were significant predictors of examination performance, depending on the examination. These results have important theoretical, methodological and pedagogical implications.     

Supporting Middle School Teachers’ Implementation of STEM Design Challenges

We describe and analyze a professional development (PD) model that involved a partnership among science, mathematics and education university faculty, science and mathematics coordinators, and middle school administrators, teachers, and students. The overarching project goal involved the implementation of interdisciplinary STEM Design Challenges (DCs). The PD model targeted: (a) increasing teachers’ content and pedagogical content knowledge in mathematics and science; (b) helping teachers integrate STEM practices into their lessons; and (c) addressing teachers’ beliefs about engaging underperforming students in challenging problems. A unique aspect involved low‐achieving students and their teachers learning alongside each other as they co‐participated in STEM design challenges for one week in the summer. Our analysis focused on what teachers came to value about STEM DCs, and the challenges in and affordances for implementing DCs. Two significant areas of value for the teachers were students’ use of scientific, mathematical, and engineering practices and motivation, engagement, and empowerment by all learners. Challenges associated with pedagogy, curriculum, and the traditional structures of the schools were identified. Finally, there were four key affordances: (a) opportunities to construct a vision of STEM education; (b) motivation to implement DCs; (c) ambitious pedagogical tools; and, (d) ongoing support for planning and implementation. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

Mathematics teacher education advanced methods: an example in dynamic geometry

We present a research work about an innovative national teacher training program in France: the Pairform@nce program, designed to sustain ICT integration. We study here training for secondary school teachers, whose objective is to foster the development of an inquiry-based approach in the teaching of mathematics, using investigative potentialities of dynamic geometry environments. We adopt the theoretical background of the documentational approach to didactics. We focus on the interactions between teachers and resources: teachers’ professional knowledge influences these interactions, which at the same time yield knowledge evolutions, a twofold process that we conceptualise as a documentational genesis. We followed in particular the work of a team of trainees; drawing on the data collected, we analyse their professional development, related with the training. We observe intertwined evolutions and stabilities, consistent with ongoing geneses.     

Epistemological beliefs concerning the nature of mathematics among teacher educators and teacher education students in mathematics

Beliefs constitute a central part of a person’s professional competences as beliefs are crucial to the perception of situations and as they influence our choice of actions. The present article focuses on epistemological beliefs about the nature of mathematics among future teachers and their educators at university and post-university teacher-training institutions in Germany. The data reported are part of a larger sample originating from the MT21 study [supported by the National Science Foundation through a grant to W. S. Schmidt and M. T. Tatto (REC-0231886). MT21 started in 2003] which explores and compares mathematics teacher education in Bulgaria, Germany, Mexico, South Korea, Taiwan, and the United States. In this article, we examine the structure and level of beliefs concerning the nature of mathematics for teacher education students in Germany both at the beginning (n = 368) and the end of their education (n = 286) as well as their educators (n = 77) in three academic disciplines (mathematics, mathematics pedagogy and general pedagogy). In the first part of the article, the literature on epistemological beliefs and their structure will be reviewed. In the empirical part, analyses on the level and the structure of beliefs for our samples and subsamples will be presented. Relations between educators’ and students’ beliefs will be explored.     

Locally logical mathematics: An emerging teacher honoring both students and mathematics

This case study explores the mathematics engagement and teaching practice of a beginning secondary school teacher. The focus is on the mathematical opportunities available to her students (the classroom mathematics) and how they relate to the teacher's personal capacity and tendencies for mathematical engagement (her personal mathematics). We use a mathematical process-and-action approach to analyze mathematical engagement and then employ the teaching triad—mathematical challenge, sensitivity to students, and management of learning—to situate mathematical engagement within the larger context of teaching practice. The article develops the construct of locally logical mathematics to underscore the cogency of mathematical engagement in the classroom as part of a coherent mathematical system that is embedded within a teaching practice. Contributions of the study include the process-and-action approach, especially in tandem with the teaching triad, as a tool to understand nuances of mathematical engagement and differences in demand between written and implemented tasks.     

Attitude towards mathematics: a bridge between beliefs and emotions

Recent research in the field of affect has highlighted the need to theoretically clarify constructs such as beliefs, emotions and attitudes, and to better investigate the relationships among them. As regards the definition of attitude, in a previous study we proposed a characterization of attitude towards mathematics grounded in students’ experiences, investigating how students express their own relationship with mathematics. The data collected suggest a three-dimensional model of attitude towards mathematics that includes students’ emotional disposition, their vision of mathematics, and their perceived competence. In this paper, we discuss the relationship between beliefs and emotions, investigating the interplay among the three dimensions in the proposed model of attitude, as emerging in the students’ essays.     

A Comparison of Preservice Elementary Teachers' Beliefs About Mathematics and Teaching Mathematics: 1968 and 1998

The study replicates Collier's (1972) work. It focuses on the beliefs of a large sample of elementary education students at four stages of teacher preparation, about both the nature of and the teaching of mathematics. The instrument measures what Collier termed a “formal‐informal” dimension of belief. The data suggest that initially the 1998 students held significantly more informal (constructivist) beliefs than did their 1968 counterparts. In both years, students moved toward more informal beliefs during the course of their programs, with the most significant changes occurring in their beliefs about how mathematics should be taught. However, apparent contradictions in belief structures were observed both at the start and at the end of their programs. Thus, it appears that though many students acquired new, more informal beliefs during the course of their programs, they did not develop robust, consistent philosophies of mathematics education.     

Effects of proactive decision making on life satisfaction

Proactive decision making, a concept recently introduced to behavioral operational research and decision analysis, addresses effective decision making during its phase of generating alternatives. It is measured on a scale comprising six dimensions grouped into two categories: proactive personality traits and proactive cognitive skills. Personality traits are grounded on theoretical constructs such as proactive attitude and proactive behavior; cognitive skills reflect value-focused thinking and decision quality. These traits and skills have been used to explain decision satisfaction, although their antecedents and other consequences have not yet been the subject of rigorous hypotheses and testing.This paper embeds proactive decision making within a model of three possible consequences. We consider—and empirically test—decision satisfaction, general self-efficacy, and life satisfaction by conducting three studies with 1300 participants. We then apply structural equation modeling to show that proactive decision making helps to account for life satisfaction, an explanation mediated by general self-efficacy and decision satisfaction. Thus proactive decision making fosters greater belief in one's abilities and increases satisfaction with one's decisions and with life more generally. These results imply that it is worthwhile to help individuals enhance their decision-making proactivity.Demonstrating the positive effects of proactive decision making at the individual level underscores how important the phase of generating alternatives is, and it also highlights the merit of employing “decision quality” principles and being proactive during that phase. Hence the findings presented here confirm the relevance of OR, and of decision-analytic principles, to the lives of ordinary people.     

A framework to categorize students as learners based on their cognitive practices while learning differential equations and related concepts

There are numerous theories that offer cognitive processes of students of mathematics, all documenting various ways to describe knowledge acquisition leading to successful transitions from one stage to another, be it characterized by Dubinsky's encapsulation, Sfard's reification or Piaget's equilibration. We however are interested in the following question. Who succeeds at making the leap and can we describe the attributes that set them apart from the ones that do not? In this article, we offer a framework to categorize students as learners based on their individual approaches towards learning concepts in differential equations and related concepts – as demonstrated by their efforts to resolve a conflict, conserve and rebuild their cognitive structures.     

Impact of Middle School Student Energy Monitoring Activities on Climate Change Beliefs and Intentions

The Going Green! Middle Schoolers Out to Save the World project aims to direct middle school students' enthusiasm for hands‐on activities toward interest in science and other STEM areas while guiding them to solve real‐world problems. Students in this project are taught by their teachers to use energy monitoring equipment to audit standby power consumed by electronic devices in their homes and communities. Major findings were: (a) Beliefs in climate change increased more for students in the treatment than comparison group, pre to post; and (b) For girls there was a larger positive impact on climate change beliefs than for boys. These and additional findings presented in this paper provide evidence that a hands‐on engaged‐learning curriculum can have a positive influence on climate change beliefs and intentions and strengthen the association between the two constructs.     

The Epistemic Relevance of Morphological Content

Morphological content is information that is implicitly embodied in the standing structure of a cognitive system and is automatically accommodated during cognitive processing without first becoming explicit in consciousness. We maintain that much belief-formation in human cognition is essentially morphological: i.e., it draws heavily on large amounts of morphological content, and must do so in order to tractably accommodate the holistic evidential relevance of background information possessed by the cognitive agent. We also advocate a form of experiential evidentialism concerning epistemic justification—roughly, the view that the justification-status of an agent’s beliefs is fully determined by the character of the agent’s conscious experience. We have previously defended both the thesis that much belief-formation is essentially morphological, and also a version of evidentialism. Here we explain how experiential evidentialism can be smoothly and plausibly combined with the thesis that much of the cognitive processing that generates justified beliefs is essentially morphological. The leading idea is this: even though epistemically relevant morphological content does not become explicit in consciousness during the process of belief-generation, nevertheless such content does affect the overall character of conscious experience in an epistemically significant way: it is implicit in conscious experience, and is implicitly appreciated by the experiencing agent.     

Authentic and Simulated Professional Development: Teachers Reflect What is Modeled

Science is a dynamic discipline, representative of the nature of science. Yet, young science students continue to think everything is already discovered. In this study, we examine why students are not actively doing science. From professional development to student engagement, how are classrooms and students changing as we increase teachers' content knowledge? Teaching practices modeled in professional development can change what occurs in the classroom. Our study was designed to probe differences in two different types of professional development programs both focused on content knowledge. We found that what is modeled by the professional developers has a profound effect on the direction of the classroom. This matched controlled study found that teachers reflect the teaching practice modeled by professional developers through their individual classroom teaching practices. A significant difference was found in cognitive activities and questioning skills between teachers in a professional development program modeling authentic inquiry versus the teachers in a professional development modeling simulated inquiry. While both groups increased the amount of overall inquiry used in the classroom, students whose teachers were in authentic inquiry professional development were engaged in higher cognitive activities and questioning skills. If students are engaged in dynamic classrooms, searching for answers to students' questions, perhaps they will understand that science is a dynamic discipline.     

Isosingular Sets and Deflation

This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations constructed by taking the closure of points with a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which, in a different context, are the sequences arising in Thom–Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic set that can be investigated with methods from numerical algebraic geometry.