Are beliefs believable? An investigation of college students’ epistemological beliefs and behavior in mathematics

College students’ epistemological belief in their academic performance of mathematics has been documented and is receiving increased attention. However, to what extent and in what ways problem solvers’ beliefs about the nature of mathematical knowledge and thinking impact their performances and behavior is not clear and deserves further investigation. The present study investigated how Taiwanese college students espousing unlike epistemological beliefs in mathematics performed differently within different contexts, and in what contexts these college students’ epistemological beliefs were consistent with their performances and behavior. Results yielded from the survey of students’ performances on standardized tests, semi-open problems, and their behaviors on pattern-finding tasks, suggest mixed consequences. It appears that beliefs played a more reliable role within the well-structured context but lost its credibility in non-standardized tasks.     

The cultural dimension of beliefs: an investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries

Beliefs constitute a central part of a person’s professional competencies and are crucial to the perception of situations as they influence our choice of actions. This paper focuses on epistemological beliefs about the nature of mathematics of future primary teachers from an international perspective. The data reported are part of a larger sample originating from the TEDS-M study which compares primary mathematics teacher education in 15 countries. In this paper we examine the pattern of beliefs of future teachers aiming to teach mathematics at primary level. We explore whether and to what extent beliefs concerning the nature of mathematics are influenced by cultural factors, in our case the extent to which a country’s culture can be characterized by an individualistic versus collectivistic orientation according to Hofstede’s terminology. In the first part of the paper, the literature on epistemological beliefs is reviewed and the role of culture and individualism/collectivism on the formation of beliefs concerning the nature of mathematics will be discussed. In the empirical part, means and distributions of belief ratings will be reported. Finally, multilevel analyses explore how much of the variation of belief preferences between countries can be explained by the individualistic orientation of a country.     

Changing students’ beliefs about the relevance of mathematics in an advanced secondary mathematics class

ABSTRACT

This study shows that using authentic contexts for learning differential equations in a differentiation-by-interest setting can enhance students’ beliefs about the relevance of mathematics. The students in this study were studying advanced mathematics (wiskunde D) at upper secondary school in the Netherlands. These students are often not aware of the relevance of the mathematics they have to learn in school. More insights into the application of mathematics in other sciences can be beneficial for these students in terms of preparation for their future study and career. A course differentiating by student interest with new context-rich curriculum materials was developed in order to enhance students’ beliefs about the relevance of mathematics. The intervention aimed at teaching differential equations through guided small-group tasks in scientific, medical or economical contexts. The results show that students’ beliefs about the relevance of mathematics improved, and they appreciated experiencing how the mathematics was applied in real-life situations.     

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.     

Inquiry-based learning in mathematics and science: a comparative baseline study of teachers’ beliefs and practices across 12 European countries

In the European educational context, reports by expert groups have identified the necessity of a renewed pedagogy in schools to overcome deficits in science and mathematics teaching and to raise the standards of scientific and mathematical literacy. Inquiry-based learning (IBL) is considered the method of choice. However, it remains open to what extent IBL is actually used in day-to-day teaching. In the study presented here we elaborate—from the perspective of teachers—the current status of IBL in day-to-day teaching. Further, we explore what problems teachers anticipate when implementing IBL. In order to gain insight into the wide spectrum of practices in mathematics and science teaching in relation to IBL, a baseline study using teacher questionnaires was carried out in the 12 participating countries. We present selected results from this study that for the first time provides an overview of teachers’ beliefs and their reports on the current use of IBL practices in a European context. The results facilitate a cross-cultural comparison on the potentials and challenges of implementing IBL from the perspective of practicing teachers. Furthermore, the study reveals considerable differences between the teaching of mathematics and science subjects. The findings of the baseline study can serve as a reference line against which the impact of interventions to improve the quality of teaching and learning can be evaluated.     

Who succeeds in STEM? Elementary girls' attitudes and beliefs about self and STEM

With ongoing underrepresentation of women in STEM fields, it is necessary to explore ways to maintain girls' STEM interest throughout elementary and middle school. This study is situated within the context of Designs in STEM (pseudonym), an out-of-school program that engages urban youth in authentic STEM experiences. Participants were 30 girls attending Designs in STEM in grades four and five. Participants were interviewed about their STEM interest, out-of-school versus in-school STEM learning experiences, and how gender relates to STEM success. Several key findings emerged. First, although students' prior school experiences with mathematics resulted in less positive dispositions toward mathematics than other STEM disciplines, their experiences at Designs in STEM revealed that mathematics could be fun and valuable when used for real-world purposes. Second, students found Designs in STEM to be more engaging and inspiring due to the context and pedagogies employed by Designs in STEM instructors. Third, despite observing girls' behavior that was more aligned with academic success, participants still identified STEM advantages for boys. Finally, participants defined success and intelligence in STEM based on speed and tracking. Discussion focuses on the need to consider how school-based mathematics instruction may serve as a barrier to girls' STEM interest and involvement.     

Teachers’ beliefs towards teaching calculus

This paper focuses on the belief systems towards teaching calculus of 29 upper secondary mathematics teachers. Firstly, we discuss different educational trends in teaching calculus, and the theoretical approach based on the construct of belief systems. Afterwards we describe the method of our study used to analyse the belief systems of the calculus teachers. Referring to findings of our research, we firstly focus on central and peripheral beliefs and thus discuss the structure of the teachers’ belief systems. Further, we compare the teachers’ belief systems to established educational trends of teaching calculus. Finally, we conclude the paper by reflecting on our main findings and by discussing possible directions of further research.     

Impacting prospective teachers’ beliefs about mathematics

This study investigated: (1) the changes in the beliefs about mathematics held by 25 prospective elementary teachers as they went through a university mathematics course that aimed, among other things, to promote a problem-solving view about mathematics; and (2) the possible factors that accounted for the observed changes. The course incorporated specific features that prior research suggested reflect successful mechanisms for belief change (e.g., cognitive conflict). The data included students’ reflections, and responses to prompts and interview questions. Analysis of the data revealed the following major trends: (1) a movement towards a problem-solving view from the more traditional Platonist and instrumentalist views; and (2) no change in students’ initial views. Activities creating cognitive conflict, as well as the implementation of instruction valuing group collaboration and explanations, appear to have played important roles in the process of belief change. The findings have implications for research on teacher beliefs and teacher education.     

Role of operator semirings in characterizing Γ-semirings in terms of fuzzy subsets

The operator semirings of a ??-semiring have been brought into use to study ??-semiring in terms of fuzzy subsets. This is accomplished by obtaining various relationships between the set of all fuzzy ideals of a ??-semiring and the set of all fuzzy ideals of its left operator semiring such as lattice isomorphism between the sets of fuzzy ideals of a ??-semiring and its operator semirings.     

Initiation of Cracks in Griffith’s Theory: An Argument of Continuity in Favor of Global Minimization

The initiation of a crack in a sound body is a real issue in the setting of Griffith’s theory of brittle fracture. If one uses the concept of critical energy release rate (Griffith’s criterion), it is in general impossible to initiate a crack. On the other hand, if we replace it by a least energy principle (Francfort–Marigo’s criterion), it becomes possible to predict the onset of cracking in any circumstance. However this latter criterion can appear too strong. We propose here to reinforce its interest by an argument of continuity. Specifically, we consider the issue of the initiation of a crack at a notch whose angle ω is considered as a parameter. The result predicted by the Griffith criterion is not continuous with respect to ω, since no initiation occurs when ω>0 while a crack initiates when ω=0. In contrast, the Francfort–Marigo’s criterion delivers a response which is continuous with respect to ω, even though the onset of cracking is necessarily brutal when ω>0. The theoretical analysis is illustrated by numerical computations.     

Fifth-grade students�� approaches to and beliefs of mathematics word problem solving: a large sample Hungarian study

Mathematical word problems used in Verschaffel et al.??s (Learning and Instruction 7:339?C359, 1994) study were applied in several follow-up studies. The goal of the present study was to replicate and extend the results of this line of research in a large sample of Hungarian students using an alternative set of data-gathering and data-analysis techniques. 4,037 students forming a nationwide representative sample of the Hungarian fifth-grade student population (aged 10?C11) completed the test. The test contained five word problems from the list of 10 P(??problematic)-items from Verschaffel et al.??s test. In contrast to all previous research in this domain, we used a multiple-choice format, where three options were given for each task: (a) routine-based, non-realistic answer, (b) numerical response that does take into account realistic considerations, (c) a realistic solution stating that the task cannot be solved. The hypotheses of this study were: (1) Students?? responses will confirm previous results, i.e. upper elementary school students prefer to respond to P-items by means of the routine-based answer; (2) Most students will demonstrate a more or less consistent preference for a given answer type (a, b or c) over problems; (3) Students?? school math marks will have low correlation indices with students?? achievement on these word problems. Our results confirm student??s overall tendency to follow non-realistic approaches when doing school word problem solving. The tendency even holds when confronting students with various kinds of realistic answers. Our results show that students demonstrate response patterns over problems, and that the correlation with math school performance is significant but small.     

On approximation of functions in terms of Φ-variation

Получен аналог теоре мы Джексона о приближ ении полиномами в метрике Ф-вариации; для некото рых классов функций (в ыпуклых, конечной вариации, с заданным модулем неп рерывности) установл ено соотношение типа сла бой эквивалентности, которое связывает ме жду собой рациональн ые аппроксимации, в этой и равномерной метриках Даны оценкиε-энтропи и в метрике Ф-вариации классов функций с заданным мо дулем Ф-абсолютной непреры вности и с заданным мо дулем непрерывности.     

Description of a Class of Solids in ℝ3

Let be a compact convex body such that for each parallel projection onto any plane no two opposite faces of Q are projected strictly inside the projection of the entire Q. Then Q is either a cone, or a frustum of a trihedral pyramid, or a prism (possibly with nonparallel bases). Bibliography: 1 title.     

Nilfactors of ℝ m and configurations in sets of positive upper density in ℝ m

We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. LetE be a measurable subset of ℝ ^m , with . LetV = {0,v ₁,...,v _k} ⊂ ℝ^m. We show that fort large enough, we can find an isometric copy oftV arbitrarily close toE. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss [FuKaW] showing a similar property form=k=2.     

Asymtotics of M-estmation in Non-linear Regression

Consider the standard non-linear regression model y_i=g(x_i,θ_0) ε_j,i=1,...,n whereg(x,θ) is a continuous function on a bounded closed region X×θ_0 is the unknown parametervector in R~p,{x_1,x_2,...,x_n} is a deterministic design of experiment and {ε_1,ε_2,...,ε_n} is asequence of independent random variables.This paper establishes the existences of M-estimates andthe asymptotic uniform linearity of M-scores in a family of non-linear regression models when theerrors are independent and identically distributed.This result is then used to obtain the asymptoticdistribution of a class of M-estimators for a large class of non-linear regression models.At the sametime,we point out that Theorem 2 of Wang (1995) (J.of Multivariate Analysis,vol.54,pp.227-238,Corrigenda.vol.55,p.350) is not correct.     

One-parameter families of operators in ℂ

We introduce classes of one-parameter families (OPF) of operators on C _c ^t8 (ℂ) which characterize the behavior of operators associated to the problem in the weighted space L² (ℂ, e^−2p) where p is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from L^q (ℂ) to itself, 1 < q < ∞, with a bound that depends on q and the degree of p but not on the parameter τ or the coefficients of p. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in τ between OPF operators of order m ≤ 2 and nonisotropic smoothing (NIS) operators of order m ≤ 2 on polynomial models in ℂ².     

Degeneracy of holomorphic curves in surfaces

Let X be a complex projective algebraic manifold of dimension 2 and let D1,…,Du be distinct irreducible divisors on X such that no three of them share a common point. Let f: C→X\(U1≤i≤uDi) be a holomorphic map. Assume that u≥4 and that there exist positive integers n1,…,nu, c such that ninj(Di.Dj) = c for all pairs i, j. Then f is algebraically degenerate, i.e. its image is contained in an algebraic curve on X.     

The Property of Bair in r-Space

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Differentiation of Integrals in RΩ

We show that the Lebesgue differentiation theorem does not hold in Ρ with the “product” Lebesgue measure. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sequences of composition operators in spaces of functions of bounded Φ-variation

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions h _n : 〈c, d〉 → 〈a, b〉, n = 1, 2, ..., to have bounded sequences of Ψ-variations {V _Ψ (〈c, d〉; f ? h _n )} _n=1 ^∞ evaluated for the compositions of an arbitrary function f: 〈a, b〉 → ? with finite Φ-variation and the functions h _n . In Theorem 2, the same is done for a sequence of functions h _n : ? → ?, n = 1, 2, ..., and the sequence of Ψ-variations {V _Ψ(〈a, b〉; h _n ? f)} _n=1 ^∞ .