Indicators of middle school students' mathematics enjoyment and confidence

A study involving 916 students spanning grades 5–8 was conducted to investigate indicators that may contribute to enjoyment and confidence in mathematics. For this group of middle school students, mathematics enjoyment, mathematics confidence, and attitudes toward school were found to generally decline across grade levels. Mathematics enjoyment and confidence were also found to differ significantly based upon student preferences for future careers. Gender differences as well as similarities regarding predictors of mathematics enjoyment and confidence were identified. Specifically, the best predictor for end of year mathematics enjoyment for males was beginning of year attitude toward school, while the most significant predictor of end of year enjoyment for females was beginning of year dispositions toward mathematics (semantic perception). A strong relationship between mathematics enjoyment and confidence overall was found, with indications that activities that increase mathematics enjoyment contribute to increased confidence, and also that activities that increase mathematics confidence contribute to increased enjoyment.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

Opportunities for learning: the use of variation to analyse examples of a paradigm shift in teaching primary mathematics in England

We demonstrate the power of Variation Theory as an analytical tool used to understand the underlying conceptual structure of mathematics lessons taught by English primary school teachers. We study excerpts of three lessons that are posted on a professional website. We show how lesson analysis using variation allows us to focus on what is made available to be learnt in the lesson excerpts. We identify some differences in their use of dimensions of variation and the associated ranges of change and discuss how suitable patterns of variation and invariance might differ according to the nature of the learning focus. We reflect on the value of our analytical approach.     

The mathematical beliefs and behavior of high school students: Insights from a longitudinal study

There is a documented need for more research on the mathematical beliefs of students below college. In particular, there is a need for more studies on how the mathematical beliefs of these students impact their mathematical behavior in challenging mathematical tasks. This study examines the beliefs on mathematical learning of five high school students and the students’ mathematical behavior in a challenging probability task. The students were participants in an after-school, classroom-based, longitudinal study on students’ development of mathematical ideas funded by the United States National Science Foundation. The results show that particular educational experiences can alter results from previous studies on the mathematical beliefs and behavior of students below college, some of which have been used to justify non-reform pedagogical approaches in mathematics classrooms. Implications for classroom practice and ideas for future research are discussed.     

Conceptions of mathematics and student identity: implications for engineering education

Lecturers of first-year mathematics often have reason to believe that students enter university studies with naïve conceptions of mathematics and that more mature conceptions need to be developed in the classroom. Students’ conceptions of the nature and role of mathematics in current and future studies as well as future career are pedagogically important as they can impact on student learning and have the potential to influence how and what we teach. As part of ongoing longitudinal research into the experience of a cohort of students registered at the author's institution, students’ conceptions of mathematics were determined using a coding scheme developed elsewhere. In this article, I discuss how the cohort of students choosing to study engineering exhibits a view of mathematics as conceptual skill and as problem-solving, coherent with an accurate understanding of the role of mathematics in engineering. Parallel investigation shows, however, that the students do not embody designated identities as engineers.     

Mathematical tasks,study approaches,and course grades in undergraduate mathematics: a year-by-year analysis

Students approach learning in different ways, depending on the experienced learning situation. A deep approach is geared toward long-term retention and conceptual change while a surface approach focuses on quickly acquiring knowledge for immediate use. These approaches ultimately affect the students’ academic outcomes. This study takes a cross-sectional look at the approaches to learning used by students from courses across all four years of undergraduate mathematics and analyses how these relate to the students’ grades. We find that deep learning correlates with grade in the first year and not in the upper years. Surficial learning has no correlation with grades in the first year and a strong negative correlation with grades in the upper years. Using Bloom's taxonomy, we argue that the nature of the tasks given to students is fundamentally different in lower and upper year courses. We find that first-year courses emphasize tasks that require only low-level cognitive processes. Upper year courses require higher level processes but, surprisingly, have a simultaneous greater emphasis on recall and understanding. These observations explain the differences in correlations between approaches to learning and course grades. We conclude with some concerns about the disconnect between first year and upper year mathematics courses and the effect this may have on students.     

Investigating students' levels of engagement with mathematics: critical events,motivations, and influences on behaviour

Universities invest significant resources in the provision of mathematics tuition to first year students, through both traditional and non-traditional means. Research has shown that a significant minority of students do not engage with these resources appropriately. This paper presents findings from a study of two groups of students at Maynooth University. Both groups had similar mathematical backgrounds on entry to university. The first group consisted of seven students who had failed first year mathematics and had very low levels of engagement with available supports. The second group consisted of nine students who had passed first year mathematics and had engaged with the supports to a significant extent. It emerged that while both groups initially displayed similar tactics and encountered similar difficulties, their levels of reaction to a number of critical events in their mathematical education were key to their engagement levels and their subsequent progression. Further analysis revealed aspects of the students' behaviour which caused them to approach or avoid difficulties. The reasons behind the different student behaviours were investigated, and the main categories of influence on student behaviour which emerged from the interview data were fear, social factors, and motivation.     

The role of prediction in the teaching and learning of mathematics

The prevalence of prediction in grade-level expectations in mathematics curriculum standards signifies the importance of the role prediction plays in the teaching and learning of mathematics. In this article, we discuss benefits of using prediction in mathematics classrooms: (1) students’ prediction can reveal their conceptions, (2) prediction plays an important role in reasoning and (3) prediction fosters mathematical learning. To support research on prediction in the context of mathematics education, we present three perspectives on prediction: (1) prediction as a mental act highlights the cognitive aspect and the conceptual basis of one's prediction, (2) prediction as a mathematical activity highlights the spectrum of prediction tasks that are common in mathematics curricula and (3) prediction as a socio-epistemological practice highlights the construction of mathematical knowledge in classrooms. Each perspective supports the claim that prediction when used effectively can foster mathematical learning. Considerations for supporting the use of prediction in mathematics classrooms are offered.     

A multidimensional approach to training mathematics students at a university: improving the efficiency through the unity of social,psychological and pedagogical aspects

The article deals with social, psychological and pedagogical aspects of teaching mathematics students at universities. The sociological portrait and the factors influencing a career choice of a mathematician have been investigated through the survey results of 198 first-year students of applied mathematics major at 27 state universities (Russia). Then, psychological characteristics of mathematics students have been examined based on scientific publications. The obtained results have allowed us to reveal pedagogical conditions and specific ways of training mathematics students in the process of their education at university. The article also contains the analysis of approaches to the development of mathematics education both in Russia and in other countries. The results may be useful for teaching students whose training requires in-depth knowledge of mathematics.     

Differentiation of teaching and learning mathematics: an action research study in tertiary education

Diversity and differentiation within our classrooms, at all levels of education, is nowadays a fact. It has been one of the biggest challenges for educators to respond to the needs of all students in such a mixed-ability classroom. Teachers’ inability to deal with students with different levels of readiness in a different way leads to school failure and all the negative outcomes that come with it. Differentiation of teaching and learning helps addressing this problem by respecting the different levels that exist in the classroom, and by responding to the needs of each learner. This article presents an action research study where a team of mathematics instructors and an expert in curriculum development developed and implemented a differentiated instruction learning environment in a first-year engineering calculus class at a university in Cyprus. This study provides evidence that differentiated instruction has a positive effect on student engagement and motivation and improves students’ understanding of difficult calculus concepts.     

Development of an attitude-towards-using-mathematics scale for high-school students and an analysis of student attitudes

ABSTRACT

This research has been carried out in two stages and has two main objectives. The first aim of the study is to develop a Likert-type scale which is used to determine the attitudes towards the use of mathematics in real life. The second aim is to examine the attitudes of high school students about the use of mathematics in real life according to different variables used in the developed scale. The research was carried out according to the correlational research method, and the participants comprise the sample of 340 and 356 students for the scale development and implementation stages of the study, respectively. As a result of the research, a structure consisting of 23 items and three sub-factors was determined for the scale. In the second stage of the study, it was observed that the student attitudes were at the level corresponding to the ‘undecided’ option of the scale, and they differed significantly according to gender and grade level variables. In addition, it was found that there was a positive and significant relationship between the students’ attitudes towards the use of mathematics and their mathematics achievement.     

Loop invariants,exploration of regularities,and mathematical games

Loop invariants are assertions of regularities that characterize the loop components of algorithms. They are fundamental components of computerprograms verification, but their relevance goes beyond verification—they can be significantly utilized for algorithm design and analysis. Unfortunately, they are only modestly introduced in the teaching of programming and algorithms. One reason for this is an unjustified notion that loop invariants are ‘tied to formality’, hard to illustrate, and difficult to comprehend. In this paper a novel approach is presented for illustrating on a rather intuitive level the significance of loop invariants. The illustration is based on mathematical games, which are attractive examples that require the exploration of regularities via problemsolving heuristics. Throughout the paper students' application of heuristics is described and emphasis is placed on the links between loop invariants, heuristic search activities, recognition of regularities and design and analysis of algorithms.     

Justification as a teaching and learning practice: Its (potential) multifacted role in middle grades mathematics classrooms

Justification is a core mathematics practice. Although the purposes of justification in the mathematician community have been studied extensively, we know relatively little about its role in K-12 classrooms. This paper documents the range of purposes identified by 12 middle grades teachers who were working actively to incorporate justification into their classrooms and compares this set of purposes with those documented in the research mathematician community. Results indicate that the teachers viewed justification as a powerful practice to accomplish a range of valued classroom teaching and learning functions. Some of these purposes overlapped with the purposes in the mathematician community; others were unique to the classroom community. Perhaps surprisingly, absent was the role of justification in verifying mathematical results. An analysis of the relationship between the purposes documented in the mathematics classroom community and the research mathematician community highlights how these differences may reflect the distinct goals and professional activities of the two communities. Implications for mathematics education and teacher development are discussed.     

Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions,domain restrictions,and the arcsine function

Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers’ instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter.     

Handling the difficulties of technical school students in the construction and interpretation of graphic representations

In this work, an attempt is made to evaluate the errors that have to do with the interpretation and construction of graphic representations. Although the students are studying in the second year of technical high school (secondary education), i.e. in schools with an emphasis in technical subjects (post junior secondary), it is observed that they find difficult to handle the graphic representations. This is an attempt to overcome the difficulties that exist by putting the students to work on electrical machines. They read, construct or verify the graphic representations of the electrical machines using electrical instruments with which they carry out measurements and take values. It is evaluated after the end of each activity and also after a suitable interval of time that there is an improvement in the students’ abilities (interpretation and construction) in graphic representations.     

Geometric quantization, reduction and decomposition of group representations

We consider a Hamiltonian action of a connected group G on a symplectic manifold (P, ω) with an equivariant momentum map and its quantization in terms of a K?hler polarization which gives rise to a unitary representation of G on a Hilbert space . If O is a co-adjoint orbit of G quantizable with respect to a K?hler polarization, we describe geometric quantization of algebraic reduction of J ⁻¹(O). We show that the space of normalizable states of quantization of algebraic reduction of J ⁻¹(O) gives rise to a projection operator onto a closed subspace of on which is unitarily equivalent to a multiple of the irreducible unitary representation of G corresponding to O. This is a generalization of the results of Guillemin and Sternberg obtained under the assumptions that G and P are compact and that the action of G on P is free. None of these assumptions are needed here. Dedicated to Vladimir Igorevich Arnold     

Applications of a sequence of points in teaching linear algebra,numerical methods and discrete mathematics

Based on a sequence of points and a particular linear transformation generalized from this sequence, two recent papers (E. Mauch and Y. Shi, Using a sequence of number pairs as an example in teaching mathematics. Math. Comput. Educ., 39 (2005), pp. 198–205; Y. Shi, Case study projects for college mathematics courses based on a particular function of two variables. Int. J. Math. Educ. Sci. Techn., 38 (2007), pp. 555–566) have presented some interesting examples which can be used in teaching high school and college mathematics classes. In this article, we further discuss a few interesting ways to apply this sequence of points in teaching college mathematics courses such as linear algebra, numerical methods in computing, and discrete mathematics. In addition to using them in individual courses, these studies may also be combined together to offer seminars or workshops to college mathematics students. Studies like these are likely to promote student interests and get students more involved in the learning process, and therefore make the learning process more effective.     

Behavior settings and eco-behavioral science: a new arena for mathematical social science permitting a richer and more coherent view of human activities in social systems,part II: Relationships to established disciplines,and needs for mathematical development

Part I of this paper presented the basic concepts of behavior settings and eco-behavioral science originated by the psychologist Roger Barker, showed how they could be linked with standard economic data systems, and suggested their use as a basis for time-allocation matrices and social system accounts. Part II discusses the relationships of behavior settings and eco-behavioral science to established disciplines, describes applications of mathematics to the new concepts by Fox and associates, and points out some major areas in need of mathematical and theoretical development. These areas include representation and measurement of patterns of relationships among roles within behavior settings, relationships among behavior settings within communities and organizations, and the evolution of large, heterogeneous populations of behavior settings over time. We hope some readers will be motivated to participate in this new scientific enterprise.