Relationships between emotional dispositions and mathematics achievement moderated by instructional practices: analysis of TIMSS 2015

ABSTRACT

This research is a secondary analysis with Korean students’ data collected in the TIMSS 2015 to describe the moderation effects of instructional practices on the relationships between students’ emotional dispositions toward mathematics and mathematics achievement. From the TIMSS 2015 database, we collected mathematics achievement scores, a student-level contextual scale for students’ emotional disposition, and teacher-level contextual scales representing teachers’ instructional practices. We applied hierarchical linear modelling to construct multilevel models. The findings showed that the achievement gap between emotional dispositions – like and dislike – became smaller when teachers more frequently implemented certain instructional practices like asking students to complete challenging exercises, decide their own problem-solving procedures, and express their ideas in class. Students who disliked mathematics were likely to have higher scores as their teachers implemented each of those practices more frequently. Findings provide important implications to teachers regarding: It is important to encourage students to reason through instructional practices like asking them to decide their own problem-solving procedures and to solve challenging problems.     

Mathematics teachers’ support and retention: using Maslow's hierarchy to understand teachers’ needs

As part of a larger study, four mathematics teachers from diverse backgrounds and teaching situations report their ideas on teacher stress, mathematics teacher retention, and their feelings about the needs of mathematics teachers, as well as other information crucial to retaining quality teachers. The responses from the participants were used to develop a hierarchy of teachers’ needs that resembles Maslow's hierarchy, which can be used to better support teachers in various stages of their careers. The interviews revealed both non content-specific and content-specific needs within the hierarchy. The responses show that teachers found different schools foster different stress levels and that as teachers they used a number of resources for reducing stress. Other mathematics-specific ideas are also discussed such as the amount of content and pedagogy courses required for certification.     

The functions of function discourse – university mathematics teaching from a commognitive standpoint

This paper addresses a topic within university mathematics education which has been somewhat underexplored: the teaching practices actually used by university mathematics teachers when giving lectures. The study investigates the teaching practices of seven Swedish university teachers on the topic of functions using a discursive approach, the commognitive framework of Sfard. In the paper a categorization of the construction and substantiation routines used by the teachers is presented, for instance various routines for constructing definitions and examples, and for verifying whether an example satisfies a given definition. The findings show that although the overall form of the lectures is similar, with teachers using ‘chalk talk’, and overt student participation limited to asking and answering questions, there are in fact significant differences in the way the teachers present and do mathematics in their lectures. These differences present themselves both on the level of discursive routines and on a more general level in how the process of doing mathematics is made visible in the teachers’ teaching practices. Moreover, I believe that many of the results of the study could be relevant for investigating the teaching of other mathematical topics.     

Conception of expert mathematics teacher: a comparative study between Hong Kong and Chongqing

This paper reports the similarities and differences in how “expert mathematics teacher” is conceptualized by mathematics educators in Hong Kong and Chongqing, two cities in China which share similar but different cultural and social backgrounds. Thirty-seven mathematics education researchers, school principals with mathematics education background, and mathematics teachers were interviewed on their perceptions of expert mathematics teacher. It is found that in both cities an expert mathematics teacher should have a profound knowledge base in mathematics, teaching, and students; strong ability in teaching; and a noble personality and a spirit of life-long learning. As for differences, an expert mathematics teacher should have the ability to conduct research, mentor other teachers, and have profound knowledge of examination and educational theories in Chongqing. These attributes were not found in Hong Kong. These similarities and differences are discussed, and relevant social and cultural factors in the two contexts are examined.     

Explanation,motivation and question posing routines in university mathematics teachers' pedagogical discourse: a commognitive analysis

This paper investigates the teaching practices used by university mathematics teachers when lecturing, a topic within university mathematics education research which is gaining an increasing interest. In the study, a view of mathematics teaching as a discursive practice is taken, and Sfard's commognitive framework is used to investigate the teaching practices of seven Swedish university mathematics teachers on the topic of functions. The present paper looks at the discourse of mathematics teaching, presenting a categorization of the didactical routines into three categories – explanation, motivation and question posing routines. All of these are present in the discourses of all seven teachers, but within these general categories, a number of different sub-categories of routines are found, used in different ways and to different extent by the various teachers. The explanation routines include known mathematical facts, summary and repetition, different representations, everyday language, and concretization and metaphor; the motivation routines include reference to utility, the nature of mathematics, humour and result focus; and the question posing routines include control questions, asking for facts, enquiries and rhetorical questions. This categorization of question posing routines, for instance, complements those already found in the literature. In addition to providing a valuable insight into the teaching of functions at the university level, the categorizations presented in the study can also be useful for investigating the teaching of other mathematical topics.     

Mathematics and Science Teachers' Use of and Confidence in Empirical Reasoning: Implications for STEM Teacher Preparation

The recent trend to unite mathematically related disciplines (science, technology, engineering, and mathematics) under the broader umbrella of STEM education has advantages. In this new educational context of integration, however, STEM teachers need to be able to distinguish between sufficient proof and reasoning across different disciplines, particularly between the status of inductive and deductive modes of reasoning in mathematics. Through a specific set of mathematical conjectures, researchers explored differences between mathematics (n = 24) and science (n = 23) teachers' reasoning schemes, as well as the confidence they had in their justifications. Results from the study indicate differences between the two groups in terms of their levels of mathematical proof, as well as correlational trends that inform their confidence across these levels. Implications particularly for teacher training and preparation within the context of an integrated STEM education model are discussed.     

Transforming the mathematical practices of learners and teachers through digital technology*

ABSTRACT

This article argues that mathematical knowledge, and its related pedagogy, is inextricably linked to the tools in which the knowledge is expressed. The focus is on digital tools and the different roles they play in shaping mathematical meanings and in transforming the mathematical practices of learners and teachers. Six categories of digital tool-use that distinguish their differing potential are presented: (1) dynamic and graphical tools; (2) tools that outsource processing power; (3) tools that offer new representational infrastructures for mathematics; (4) tools that help to bridge the gap between school mathematics and the students’ world; (5) tools that exploit high-bandwidth connectivity to support mathematics learning; and (6) tools that offer intelligent support for the teacher when their students engage in exploratory learning with digital technologies. Following exemplification of each category, the article ends with some reflections on the progress of research in this area and identifies some remaining challenges.     

STEM Teachers' Planned and Enacted Attempts at Implementing Engineering Design‐Based Instruction

This study investigates grades 5 and 6 science, technology, engineering, and mathematics (STEM) teachers' planned and actualized engineering design‐based instruction, the instruments used to characterize their efforts, and the implications this work has for teachers' implementations of an integrated approach to STEM education. Participants included 23 STEM teachers from six schools (three rural, two suburban, and one urban). Data were gathered via lesson implementation plans and classroom observations. Teachers demonstrated strength in planning for standards‐ and engineering design‐based lessons, incorporating engineering practices within their respective implementation plans, and aligning their plans with content and design process standards. Missing from their plans was attention to science concepts and their placement, use, and application within a design task. Classroom observations indicated that the teacher participants gave priority to “front loading,” the design process by concentrating more of their instructional time on problem identification and planning and less time on testing designs, communicating performance results, and redesigning. Measures utilized in this study provided insight into the content of teachers' planning and subsequent instruction and suggest potential for capturing content planning in the context of classrooms in which teachers are attempting to integrate novel curriculum, such as the new standards for engineering practices.     

Problematizing mathematics and its pedagogy through teacher engagement with history-focused and classroom situation-specific tasks

We explore the conjecture that engaging teachers with activities which feature mathematical practices from the past (history-focused tasks) and in today’s mathematics classrooms (mathtasks) can promote teachers’ problematizing of mathematics and its pedagogy. Here, we sample evidence of discursive shifts observed as twelve mathematics teachers engage with a set of problematizing activities (PA) – three rounds of history-focused and mathtask combinations – during a four–month postgraduate course. We trace how the commognitive conflicts orchestrated in the PA triggered changes in the teachers’ narratives about: mathematical objects (such as what a function is); how mathematical objects come to be (such as what led to the emergence of the function object); and, pedagogy (such as what value may lie in listening to students or in trialing innovative assessment practices). Our study explores a hitherto under-researched capacity of the commognitive framework to steer the design, evidence identification and impact evaluation of pedagogical interventions.     

Elementary preservice teachers' integration of engineering into STEM lesson plans

Science, technology, engineering, and mathematics (STEM) integration is a desired outcome according to Next Generation Science Standards. However, learning to teach integrated STEM content has been challenging for teachers. Consequently, the purpose of this qualitative study was to describe how 16 preservice teachers enrolled in a mathematics methods course created integrated STEM lesson plans that incorporated an authentic engineering problem. Content analysis of the completed integrated STEM lesson plans used the Quality K-12 Engineering Education Framework to identify any characteristics of engineering. We found that 15 of 16 preservice teachers demonstrated at least an emerging ability to create an integrated STEM lesson that contained an engineering problem, constraints, a prototype or model, model testing, and data collection and analysis related to the model. We concluded that giving preservice teachers opportunities to experience engineering design problems could better prepare them to design and implement integrated STEM education in their classrooms. The findings from this study have practical implications for mathematics methods teacher educators who teach the pedagogy behind STEM education. This study also has theoretical implications because socially situated learning theory was extended to Model-Eliciting Activities and connected them to the K-12 Framework for Quality Engineering Education.     

Preservice mathematics teachers' conceptions and enactments of modeling standards

Mathematical modeling has been highlighted recently as Common Core State Standards for Mathematics (CCSSM) included Model with Mathematics as one of the Standards for Mathematical Practices (SMP) and a modeling strand in the high school standards. This common aspect of standards across most states in the United States intended by CCSSM authors and policy makers seems to mitigate the diverse notions of mathematical modeling. When we observed secondary mathematics preservice teachers (M‐PSTs) who learned about the SMP and used CCSSM modeling standards to plan and enact lessons, however, we noted differences in their interpretations and enactments of the standards, despite their attendance in the same course sections during a teacher preparation program. This result led us to investigate the ways the M‐PSTs understood modeling standards, which could provide insights into better preparing teachers to teach mathematical modeling. We present the contrasting ways in which M‐PSTs presented modeling related to their conceptions of mathematical modeling, choices of problems, and enactments over an academic year, connecting their practices to extant research. We consider this teaching and research experience as an opportunity to make significant changes in our instruction that may result in our students enhanced implementation of mathematical modeling.     

An “inverse” relationship between mathematics identities and classroom practices among early career elementary teachers: The impact of accountability

This qualitative case study guided by portraiture examines the relationships between three early career elementary teachers’ beliefs about themselves in relation to mathematics (mathematics identities) and their classroom practices. Through autobiographical inquiry, reflective practice, classroom observations, interviews, and artifacts, findings show that all three second grade teachers appeared to have an “inverse” relationship between their mathematics identities and their classroom practices. In this relationship, as negative as they felt about themselves with regards to mathematics, they expended that much more effort to ensure that their students would have positive experiences with it and not be stigmatized by it as they had been. Accountability to schools, students, and parents, to increase student achievement appeared to play an important role in this relationship. Implications for preservice teacher education, inservice professional development, and research on beliefs and practices are discussed.     

Approximations of e and π: an exploration

ABSTRACT

Fractional approximations of e and π are discovered by searching for repetitions or partial repetitions of digit strings in their expansions in different number bases. The discovery of such fractional approximations is suggested for students and teachers as an entry point into mathematics research.     

Three modes of STEM integration for middle school mathematics teachers

Of the four subjects in an integrated science, technology, engineering, and mathematics (STEM) approach, mathematics has not received enough focus. This could be in part because mathematics teachers may be apprehensive or unsure about how to implement integrated STEM education in their classrooms. There are benefits to integrated STEM in a mathematics classroom though, including increased motivation, interest, and achievement for students. This article discusses three methods that middle school mathematics teachers can utilize to integrate STEM subjects. By focusing on open‐ended problems through engineering design challenges, mathematical modeling, and mathematics integrated with technology middle school students are more likely to see mathematics as relevant and valuable. Important considerations are discussed as well as recent research with these approaches.     

The interconnectedness of relational and content dimensions of quality instruction: Supportive teacher–student relationships in urban elementary mathematics classrooms

Scholars assert that the often-impoverished instructional practices found in urban schools are tied to teachers’ negative relationships with African American and Latin@¹ students (Ferguson, 1998, McKown and Weinstein, 2002, McKown and Weinstein, 2008, Morris, 2005, Stiff and Harvey, 1988). However, measures of mathematics instructional quality rarely measure relational elements of instruction. This study responds to such shortcomings by analyzing relational interactions in urban elementary mathematics classrooms in tandem with content instruction of teachers who engage in supportive relationships with African American and Latin@ students. This study identified teachers with high quality student performance, content instruction, and supportive relationships as defined through relational interactions. After selecting two teachers, the results detail relational interactions that show how these teachers established supportive relationships with students vis-à-vis their mathematics instruction. Therefore, these findings offer insight into the ways in which relational interactions add to our understanding of quality content instruction for African American and Latin@ students.     

Opportunity to learn in the preparation of mathematics teachers: its structure and how it varies across six countries

Cross-national research studies such as the Program for International Student Assessment and the Third International Mathematics and Science Study (TIMSS) have contributed much to our understandings regarding country differences in student achievement in mathematics, especially at the primary (elementary) and lower secondary (middle school) levels. TIMSS, especially, has demonstrated the central role that the concept of opportunity to learn plays in understanding cross-national differences in achievement Schmidt et al., (Why schools matter: A cross-national comparison of curriculum and learning  2001). The curricular expectations of a nation and the actual content exposure that is delivered to students by teachers were found to be among the most salient features of schooling related to academic performance. The other feature that emerges in these studies is the importance of the teacher. The professional competence of the teacher which includes substantive knowledge regarding formal mathematics, mathematics pedagogy and general pedagogy is suggested as being significant—not just in understanding cross-national differences but also in other studies as well (Hill et al. in Am Educ Res J 42(2):371–406, 2005). Mathematics Teaching in the 21st Century (MT21) is a small, six-country study that collected data on future lower secondary teachers in their last year of preparation. One of the findings noted in the first report of that study was that the opportunities future teachers experienced as part of their formal education varied across the six countries (Schmidt et al. in The preparation gap: Teacher education for middle school mathematics in six countries, 2007). This variation in opportunity to learn (OTL) existed in course work related to formal mathematics, mathematics pedagogy and general pedagogy. It appears from these initial results that OTL not only is important in understanding K-12 student learning but it is also likely important in understanding the knowledge base of the teachers who teach them which then has the potential to influence student learning as well. This study using the same MT21 data examines in greater detail the configuration of the educational opportunities future teachers had during their teacher education in some 34 institutions across the six countries.     

Teachers Share Their Mathematics Backgrounds: Telling It Like It Was

This study focused on the mathematics backgrounds of 45 elementary mathematics teachers. A survey was administered which asked teachers to reflect on their elementary, secondary, and college mathematics experiences, the support they received and the confidence they have with their ability to teach elementary mathematics. Quantitative and qualitative data derived from the survey, revealed that sharing mathematics knowledge in elementary school; committed, enthusiastic teachers in high school; parental support; and practical uses of mathematics were positive influences for these teachers. Rote drill, seatwork, and ability grouping were negative influences. The majority of the teachers felt inservice staff development experiences helped make them confident teachers of elementary mathematics. The implications of the study for mathematics education are discussed.     

Elementary teachers’ mathematical beliefs and mathematics anxiety: How do they shape instructional practices?

This quantitative study investigated the relationships among practicing elementary teachers’ (N = 153) beliefs about mathematics and its teaching and learning, mathematics anxiety, and instructional practices in mathematics. When viewed singly, the findings reveal the teachers with higher levels of mathematics anxiety tend to use less standards‐based instruction and those with beliefs oriented toward a problem‐solving view of mathematics reported more standards‐based teaching. A combined analysis shows that after controlling for mathematical beliefs, teaching longevity, and educational degree attainment, there is no relationship between teachers’ mathematics anxiety and instructional practices. These findings suggest a spurious relationship between anxiety and practices, with beliefs having the strongest relationship with practices. Several suggestions for positively influencing the mathematical beliefs and affect in general of elementary teachers while learning mathematics are offered.     

Elementary mathematics specialists in “departmentalized” teaching assignments: Affordances and constraints

In this article, we describe the experiences of three Elementary Mathematics Specialists (EMS) who were part of a larger project investigating the impact of EMS certification and assignment (self-contained or “departmentalized”) on teaching practices and student achievement outcomes. All three of the teachers were “departmentalized,” in the sense that each was responsible for teaching mathematics to at least two groups of students, and accordingly, did not teach all subjects as would a typical self-contained elementary teacher. Each teacher had recently earned an Elementary Mathematics Specialist certificate through completion of a 24-credit, graduate-level program designed to build pedagogical content knowledge and leadership capacity in mathematics. Through a series of observations and interviews over the course of one school year, we examined how the teachers described and navigated specific affordances and constraints they encountered in their particular contexts. Common affordances included opportunities to revise and learn from instruction, and constraints included reduced flexibility introduced by the need to schedule multiple classes of mathematics. Despite these common features, we found important differences between the three models of departmentalization, which we describe as team approach, class swap, and grade-level mathematics teacher. For example, some of the models provided more opportunities for collaboration while others made it difficult for teachers to address potential inequities in learning opportunities across sections. Despite the constraints of their respective models, we found evidence of the EMS-certified teachers drawing on professional expertise in mathematics to meet student needs.