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.     

Fostering creativity through instruction rich in mathematical problem solving and problem posing

Although creativity is often viewed as being associated with the notions of “genius” or exceptional ability, it can be productive for mathematics educators to view creativity instead as an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population. In this article, it is argued that inquiry-oriented mathematics instruction which includes problem-solving and problem-posing tasks and activities can assist students to develop more creative approaches to mathematics. Through the use of such tasks and activities, teachers can increase their students’ capacity with respect to the core dimensions of creativity, namely, fluency, flexibility, and novelty. Because the instructional techniques discussed in this article have been used successfully with students all over the world, there is little reason to believe that creativity-enriched mathematics instruction cannot be used with a broad range of students in order to increase their representational and strategic fluency and flexibility, and their appreciation for novel problems, solution methods, or solutions.     

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.     

Japanese and American teachers' evaluations of mathematics lessons: A new technique for exploring beliefs

In our study, we use a novel technique to explore the beliefs of Japanese and American elementary school teachers. Four American and four Japanese teachers watched a mathematics lesson—videotaped in either Nagano, Japan or Chicago, Illinois—and commented on the lesson's strengths and weaknesses. The major pedagogical issues that differentiated the teachers' comments were: what students should do during a lesson, how instructors should use language, how instructors should pace lessons and address ability differences, and how instructional materials should be used. The specific beliefs of the American and Japanese teachers in this study mapped easily onto common instructional practices in elementary school mathematics classes in the United States and Japan. We conclude that, at least for the teachers in this sample, beliefs are linked to practices and they may help to tie teachers to their culturally preferred method of mathematics instruction.     

Characterizing exemplary mathematics instruction in Japanese classrooms from the learner’s perspective

This paper aims to examine key characteristics of exemplary mathematics instruction in Japanese classrooms. The selected findings of large-scale international studies of classroom practices in mathematics are reviewed for discussing the uniqueness of how Japanese teachers structure and deliver their lessons and what Japanese teachers value in their instruction from a teacher’s perspective. Then an analysis of post-lesson video-stimulated interviews with 60 students in three “well-taught” eighth-grade mathematics classrooms in Tokyo is reported to explore the learners’ views on what constitutes a “good” mathematics lesson. The co-constructed nature of quality mathematics instruction that focus on the role of students’ thinking in the classroom is discussed by recasting the characteristics of how lessons are structured and delivered and what experienced teachers tend to value in their instruction from the learner’s perspective. Valuing students’ thinking as necessary elements to be incorporated into the development of a lesson is the key to the approach taken by Japanese teachers to develop and maintain quality mathematics instruction.     

Solution representations of percentage change problems: the pre-service primary teachers’ mathematical thinking and reasoning

Solution representations can reveal how problem solvers communicate mathematical thinking and reasoning in problem-solving process. The present study examined the solution representations used by 20 pre-service teachers for the percentage change problems. The pre-service teachers were invited to solve a combination of simple and complex percentage change problems. The score for the majority of simple problems was 75% or above, but the score for the complex problems was below 75%. The highest percentage error occurred when the pre-service teachers encountered a percentage greater than 100% in the percentage change problems. Irrespective of their level of mathematics qualifications, the equation approach demonstrating two-step problem-solving process was the predominant strategy adopted by the pre-service teachers. The equation approach imposes low cognitive load and, therefore, is more accessible and efficient than the unitary approach. A few pre-service teachers used the unitary approach. The findings indicate that the pre-service teachers possessed relevant mathematical knowledge for percentage change problems. Furthermore, the inclusion of the equation approach in mathematics textbooks would provide an alternative perspective regarding the teaching and learning of percentage change problems.     

Controlling Choice: Teachers,Students, and Manipulatives in Mathematics Classrooms

This research study examines the instructional practices of 10 middle grades teachers related to their use of manipulatives in teaching mathematics and their control of mathematics tools during instruction. Through 40 observations of teaching, 30 interviews, and an examination of 67 written documents (including teachers' plans and records), profiles were developed that describe how teachers used and controlled manipulatives during instruction. Results showed that teachers used a variety of manipulatives and other mathematics tools over the course of the year‐long study. Teachers reported using a mathematics tool (manipulative, calculator, or measuring device) in 70% of their lessons, and this self‐report was verified by observations in which teachers used mathematics tools in 68% of their lessons. During a 3‐ to 4‐month period of “free access,” in which students had some measure of control in their selection and use of the mathematics tools, the students used manipulatives spontaneously and selectively. During free access, teachers exhibited various behaviors, including posting lists of items on containers, assigning group leaders to manage tools, and negotiating the control of the mathematics tools during instruction.     

An investigation of preservice teachers’ use of guess and check in solving a semi open-ended mathematics problem

Open-ended problems have been regarded as powerful tools for teaching mathematics. This study examined the problem solving of eight mathematics/science middle-school teachers. A semi-structured interview was conducted with (PTs) after completing an open-ended triangle task with four unique solutions. Of particular emphasis was how the PTs used a specific heuristic strategy. The results showed that the primary strategy PTs employed in attempting to solve the triangle problem task was guess and check; however, from the PTs’ reflections, we found there existed misapplications of guess and check as a systematic problem-solving strategy. In order to prepare prospective teachers to effectively teach, teacher educators should pay more attention to the mathematical proficiency of PTs, particularly their abilities to systematically and efficiently use guess and check while solving problems and explain their solutions and reasoning to middle-school students.     

Teaching-Learning of Evaluative Criteria for Mathematical Arguments Through Classroom Discourse: A Cross-National Study

Videotaped lessons of 5th graders on equivalent fractions from 7 American and 6 Japanese classrooms were analyzed in terms of a recurrent pattern in public discourse among a teacher and students. This pattern—called inquiry, response, feedback—occurs when a teacher initiates discourse (mostly with an inquiry), a student or students respond (often with an answer to the teacher inquiry), and the teacher provides feedback to the student's response. We found2 approaches to the teaching-learning of the criteria for evaluating mathematical arguments. In the Japanese classroom, students were encouraged to offer their own argument to the whole class and evaluate arguments proposed by other students. They seldom were given direct evaluation by their teacher. In contrast, American teachers often gave individual elaboration as well as direct evaluation to the student's responses, and some of the teachers offered their own opinions about mathematics, about valid ways of argumentation, or about both. The Japanese approach would help students acquire evaluative criteria indirectly through participating in mathematical discourse, whereas the American approach would help students learn modes of arguments through direct instruction.     

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.     

Teaching-Learning of Evaluative Criteria for Mathematical Arguments Through Classroom Discourse: A Cross-National Study

Videotaped lessons of 5th graders on equivalent fractions from 7 American and 6 Japanese classrooms were analyzed in terms of a recurrent pattern in public discourse among a teacher and students. This pattern—called inquiry, response, feedback—occurs when a teacher initiates discourse (mostly with an inquiry), a student or students respond (often with an answer to the teacher inquiry), and the teacher provides feedback to the student's response. We found2 approaches to the teaching-learning of the criteria for evaluating mathematical arguments. In the Japanese classroom, students were encouraged to offer their own argument to the whole class and evaluate arguments proposed by other students. They seldom were given direct evaluation by their teacher. In contrast, American teachers often gave individual elaboration as well as direct evaluation to the student's responses, and some of the teachers offered their own opinions about mathematics, about valid ways of argumentation, or about both. The Japanese approach would help students acquire evaluative criteria indirectly through participating in mathematical discourse, whereas the American approach would help students learn modes of arguments through direct instruction.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.     

Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.     

The ability of students and teachers to use counter-examples to justify mathematical propositions: a pilot study in South Korea and Hong Kong

Counter-examples, which are a distinct kind of example, have a functional role in inducing logically deductive reasoning skills in the learning process. In this investigation, we compare the ability of students and prospective teachers in South Korea and Hong Kong to use counter-examples to justify mathematical propositions. The results highlight that South Korean students performed better than Hong Kong students at justifying propositions using counter-examples in algebra problems, but both did equally well in geometry problems. In terms of the prospective teachers’ ability to justify propositions using counter-examples in two particular topics, properties of the absolute value function and parallelogram, Hong Kong prospective teachers performed relatively weakly in the absolute value problem but better in the parallelogram problem compared with South Korean prospective teachers. The weaknesses and strengths of students and prospective teachers in generating counter-examples associated with logical reasoning in mathematical contexts in the two regions indicate ways of improving the effectiveness of learning and teaching mathematics through the use of counter-examples.     

From formal standards to everyday practice of mathematics learning

This paper uses the example of six Japanese teachers and their mathematics lessons to illustrate how clear, high standards for mathematics instruction are combined with teachers' holistic concern for students. We draw upon data from the Third International Math and Science Study Case Study Project in Japan that was designed to elucidate the context behind the high achievement of Japanese students. Using everyday examples of classroom practice, we illustrate both flexibility in teachers' approach to teaching and adherence to Monbusho's (Ministry of Education, Science, Sports, and Culture)Course of Study. Our purpose is to emphasize how flexibility and attention to individual needs by Japanese teachers combine with quality mathematics instruction based on the detailed Japanese curricula. Six teachers' characteristics and lessons (two teachers at each educational level—elementary, junior high, and high school) are described in order to show the variety of teachers who exist in Japan. These teachers use their understanding of theCourse of Study and are supported by their school environment to enhance their students' conceptual understanding of the fundamentals of mathematics. Characteristics of their teaching include: 1) involving the whole class in learning. 2) using extremely focused curriculum guidelines that expect mastery of concepts at each grade level, 3) thoroughly covering mathematics units in an organized and in-depth manner, 4) leading classes as facilitators or guides more often than as lecturers, and 5) focusing on problem solving with the primary goal of developing students' ability to reason, especially to reason inductively. The examples in this paper show how these methods develop in individal classrooms.     

Mathematics classroom instruction excellence through the platform of teaching contests

In this study, we aimed to examine features of mathematics classroom instruction excellence identified and valued through teaching contests in the Chinese mainland. By taking a case study approach, we focused on a prize-winning lesson as an exemplary lesson that was awarded the top prize in teaching contests at both the district and the city level. The analyses of the exemplary lesson itself revealed important features on the lesson’s content treatment, students’ engagement, and the use of multiple methods to facilitate students’ learning. These features are consistent with what the contest evaluation committees valued and what seven other mathematics expert teachers focused in their comments. The Chinese teaching culture in identifying and promoting classroom instruction excellence is then discussed in a broader context.     

Gesture in collaborative mathematics problem-solving

In this paper we examine the significance of gestures in a setting where two students try to make sense of, and solve, mathematical problems involving speed and time. We are particularly interested in exploring the claims that gesture serves various signaling functions in collaborative problem-solving communication generally and in mathematics problem-solving more specifically, and that gesture has a diagnostic role for the collaborators and for teachers. The overall purpose of our paper is to illustrate the integral role of gesture in dyadic communication where core problem domain concepts may be difficult to explicate.     

Teaching mathematics for Korean language learners based on ELL education models

The classroom culture of Korean schools has recently been changing as the population of linguistically and culturally diverse students increases. Students with multicultural backgrounds as well as Korea-born students returning from long residences in foreign countries have difficulties adjusting to Korean public schools due to a lack of Korean language proficiency and knowledge of Korean school culture. This study defines these students as Korean language learners (KLLs) and investigates both teacher and student perspectives on effective mathematics education for them. Cummins’ Quadrant model and the sheltered instruction observation protocol model, which were developed and used for English language learners (ELLs), are the frameworks used. The study explores various pedagogies for language learners and discusses the effectiveness and feasibility of ELL education models in a Korean school context based on the survey results of Korean elementary teachers and interviews of KLLs.     

Enhancing pre-service teachers’ fraction knowledge through open approach instruction

This study explores whether using the open approach instruction in teaching mathematics has a positive effect for enhancing pre-service teachers’ fraction knowledge. The test consisted of 32 items that were designed to examine pre-service teachers’ procedural and conceptual knowledge of fractions before and after receiving open approach instruction. The study was undertaken among students in four mathematics content and methods courses for the Elementary School Education program in a mid-western public university. The findings show that most of the teachers achieved improved learning outcomes through the open approach instruction.     

Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education

This article focuses on a form of instructional design that is deemed fitting for reform mathematics education. Reform mathematics education requires instruction that helps students in developing their current ways of reasoning into more sophisticated ways of mathematical reasoning. This implies that there has to be ample room for teachers to adjust their instruction to the students' thinking. But, the point of departure is that if justice is to be done to the input of the students and their ideas built on, a well-founded plan is needed. Design research on an instructional sequence on addition and subtraction up to 100 is taken as an instance to elucidate how the theory for realistic mathematics education (RME) can be used to develop a local instruction theory that can function as such a plan. Instead of offering an instructional sequence that “works,” the objective of design research is to offer teachers an empirically grounded theory on how a certain set of instructional activities can work. The example of addition and subtraction up to 100 is used to clarify how a local instruction theory informs teachers about learning goals, instructional activities, student thinking and learning, and the role of tools and imagery.