Student Mathematical Thinking Research Articles

A Marking Scheme Rubric: To Assess Students' Mathematical Knowledge for Applied Algebra Test

Students' ability in mathematics mainly relies on their performance in the assessment task such as tests, quizzes, assignments and final examinations. However, the grading process depends on the respective mathematics teacher who sets a marking scheme in assessing students' learning. How do these teachers assign grades to their students' problem solving work? What does it mean by five marks or ten marks for a mathematics problem? How does a teacher evaluate a student's mathematical knowledge and skills based on the grades? These questions address the vagueness of the grading process that gives no concrete evidence about a student's mathematical thinking. Hence, this paper aims to discover the effectiveness of using a marking scheme rubric to assess students' mathematical knowledge. The paper begins by reviewing different types of scoring rubrics in assessing mathematical problem solving tasks. A marking scheme rubric was proposed to assess samples of actual students' problem solving work in an applied algebra test. The rubric serves as an assessment instrument to gather information about students' achievement level in demonstrating both knowledge and skills in the test. Based on the findings, the score reflected the quality of the students’ work rather than just a numerical representation. It showed the students’ comprehension of adapting the mathematical concepts and problem solving strategies. In a nutshell, the implementation of rubric marking scheme has improved the consistency in grading and made the scoring points as a meaningful figure that describes the quality of a students' performance.

TIMSS 2011 결과에 나타난 초등학교 4학년 학생들의 수학에 대한 정의적 태도와 교사 변인과의 관계 분석

The purpose of this study is to examine the effects of teacher on fourth-grade students' attitudes toward mathematics using data from TIMSS 2011. Students' attitudes toward mathematics included interest in learning mathematics, interest in mathematics lessons, and confidence in their mathematics ability. Teacher factors included mathematics professional development, confidence in teaching mathematics, teacher-centered mathematics instruction, and enhancing student mathematical thinking. The two level Hierarchical Linear Model was employed to analyze the relationship between teacher factors and student attitudes. Results showed that teacher-centered mathematics instruction significantly and positively predicted students' confidence about their mathematics ability. The findings suggest that school systems and mathematics educators need to provide teachers with the curriculum, assessment, and research-based practices and knowledge to overcome the obstacles to change their mathematics classroom.

Implementasi Model Pembelajaran Search, Solve, Create And Share (Sscs) Untuk Meningkatkan Kemampuan Berpikir Kreatif Matematis Dan Dampaknya Terhadap Disposisi Matematis Siswa SMA

The main purpose of this resaerch was to conduct a study focusing on the use SSCS model i s assumed capable to increase students mathematical creative thinking ability and see the impact on mathematical disposition, based on student of all and Prior Mathematical Ability (PMA) students (high and low). The method of this research is a mixed methods type of Embedded Design. The population of th is research is X graders of SMA Pasundan 3 Bandung, and the samples are the students in X - 1 as the experimental group and student in X - 2 as the control group. This research used two kinds of instruments: PMA test, mathematical creative thinking ability test, mathematical disposition scale, observation sheet and interview. This research used Independent Sample t - Test and Two - Way ANOVA. Conclusion of this research are : 1) based on students of all, the enhancement mathematical creative thinking ability of students who were given SSCS model are better than students who were given expository learning. When based on PMA category , increase mathematical creative thinking ab ility of students high and low who were given SSCS model are better than students who were given expository learning; 2) based on students of all, the enhancement mathematical disposition of students who were given SSCS model are better than students who w ere given expository learning. When based on PMA category , increase mathematical disposition of students high and low who were given SSCS model are better than students who were given expository learning; 3) Mathematical disposition to support the success of learning mathematics, in this case the mathematical creative thinking ability.. Students who have high disposition to be more persistent, persevere and interested to explore new things. This Allows students to have more knowledge than the students who did not show such behavior. Knowledge is what causes students to have Certain abilities. 4) There is a positive impact on the mathematical creative thinking ability of student to mathematical disposition

Developing the Practice of Teacher Questioning through a K-2 Elementary Mathematics Field Experience

This article presents findings from research on a field experience designed to help elementary preservice teachers learn the practice of teacher questioning during formal and informal interviews to analyze student mathematical thinking in K-2 classrooms. The practice of teacher questioning is framed as choosing a mathematical goal, analyzing student thinking about that goal, and then deciding on a question or prompt. Preservice teachers were specific and accurate in identifying a mathematical goal and analyzing student thinking when responding to a video-taped interview, but were less successful in their explicit discussion about the rationales for particular teacher questions in response to student thinking. More research is needed to understand tasks for preservice teachers in early field experiences that will help them develop intentional use of teacher questioning to facilitate student thinking about mathematics.

Examining the Impact of a Videocase-based Mathematics Methods Course on Secondary Preservice Teachers’ Skills at Analyzing Students’ Strategies

This paper focuses on results from a study conducted with two cohorts of preservice teachers (PSTs) in a videocase-based mathematics methods course at a large midwestern university in the US. The motivation for this study was to look beyond whether or not PSTs pay attention to mathematical thinking of students, as shown by previous studies when engaging with video, and, in turn, characterize at a more specific level areas in which PSTs’ responses change. Our findings show that regarding PSTs anticipation of strategies, both cohorts showed a significant increase in the overall number of strategies PSTs were able to anticipate, and a significant increase in the mathematical depth of the anticipated strategies. However, there was no change in terms of PSTs identification and description of high school students’ strategies as displayed in video given that both cohorts of PSTs performed equally proficient at both pre- and post- tests.

Examining How Teachers Use Graphs to Teach Mathematics during a Professional Development Program

There are urgent calls for more studies examining the impact of Professional Development (PD) programs on teachers’ instructional practices. In this study, we analyzed how grades 5-9 mathematics teachers used graphs to teach mathematics at the start and end of a PD program. This topic is relevant because while many studies have investigated students’ difficulties with graphs, there is limited research on how teachers use graphs in their classrooms and no research on how PD impacts the way teachers use graphs in class to teach mathematics. Participant teachers took three graduate level semester-long courses focused on mathematics and student mathematical thinking. The program provided teachers with multiple opportunities for exploration and discussion, systematic feedback, contexts for collaboration and collegial sharing, and extended follow-up support. We analyzed all lessons where teachers used graphs in class at the start and end of the program, finding that teachers’ use of graphs was qualitatively more sophisticated in the end lessons. Results suggest that the features of the PD program had a positive effect on teachers’ classroom practices regarding the use of graphs.

Aplicación de Actividades de modelización matemática en la educación secundaria costarricense

This paper describes the experience gained in implementing mathematical modeling activities as a methodological strategy in teaching issues such as proportions, with a group of eighth year of an academic-day-school, located in the province of San Jose, Costa Rica in 2012. Different techniques for gathering information were applied, such as participant observation and questionnaires. Among the relevant results are the cyclical development of mathematical thinking of students in the stages of mathematical modeling (description, manipulation, prediction and validation) for solving the problem; developing of teamwork skills; and appreciation of mathematics as a useful and effective discipline. To resolve the activities proposed in this study, social interactions such as sharing information, thoughts and ideas, were generated, stimulating the zone of proximal development of the participating students. Likewise, the mathematical modeling activities allowed students to have a positive role in mathematics classes, stimulating, in turn, a different attitude compared to regular classes.

Influence of interconnecting knowledge in mathematics teaching on development of mathemathical thinking

This paper is on the basic features of integration of knowledge in mathematics teaching and contribution of this process to development of mathematical thinking of students. The standpoint presented here is that it is of great importance for students in mathematics able, with help of appropriate teaching content, to adopt a system of interconnected and conditioned mathematical knowledge and concepts. The knowledge system students can form in the teaching of mathematics, under certain conditions, if the selection of teaching contents allow its formation in the learning process represents more stable and logically consistent system of knowledge in comparison to any other area of learning in the classroom. The basis of the connectivity of knowledge in mathematics, according to the basic assumptions of the theory of developmental teaching, is the discovery of the subject of starting basic mathematical concepts in mathematics teaching, such as the relations among size, number, number system, set and others. Practicing different cognitive activities of acquiring mathematical concepts and interconnecting knowledge in mathematics teaching contribute to the development of mathematical thinking in students.

Conceptualizing Mathematically Significant Pedagogical Opportunities to Build on Student Thinking

The mathematics education community values using student thinking to develop mathematical concepts, but the nuances of this practice are not clearly understood. We conceptualize an important group of instances in classroom lessons that occur at the intersection of student thinking, significant mathematics, and pedagogical opportunities—what we call Mathematically Significant Pedagogical Opportunities to Build on Student Thinking. We analyze dialogue to illustrate a process for determining whether a classroom instance offers such an opportunity and to demonstrate the usefulness of the construct in examining classroom discourse. This construct contributes to research and professional development related to teachers' mathematically productive use of student thinking by providing a lens and generating a common language for recognizing and agreeing on a critical core of student mathematical thinking that researchers can attend to as they study classroom practice and that teachers can aspire to notice and build upon when it occurs in their classrooms.

The “MOST” Productive Student Mathematical Thinking

Identify student thinking that has potential to support significant mathematical discussion and pedagogical opportunity.

곱셈적 구조에 대한 2, 4, 6학년 학생들의 수학적 사고의 연결성 분석

This study investigated the connections of mathematical thinking of students at the second, fourth, and sixth grades with regard to multiplication, fraction, and proportion, all of which have multiplicative structures. A paper-and-pencil test and subsequent interviews were conducted. The results showed that mathematical thinking including vertical thinking and relational thinking was commonly involved in multiplication, fraction, and proportion. On one hand, the insufficient understanding of preceding concepts had negative impact on learning subsequent concepts. On the other hand, learning the succeeding concepts helped students solve the problems related to the preceding concepts. By analyzing the connections between the preceding concepts and the succeeding concepts, this study provides instructional implications of teaching multiplication, fraction, and proportion.

A cognitive analysis of Cauchy’s conceptions of function, continuity, limit and infinitesimal, with implications for teaching the calculus

In this paper, we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It highlights the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern set-theoretic formulation of mathematical analysis. This offers a re-evaluation of the relationship between the natural geometry and algebra of elementary calculus that continues to be used in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of non-standard analysis using infinitesimal concepts. It suggests that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students, and to make an effort to understand the rationale of differing theoretical viewpoints.

Assessing the Effects of Collaborative Professional Learning: Efficacy Shifts in a Three-Year Mathematics Study

Researchers examine the outcomes of professional collaborative inquiry in mathematics on teacher efficacy in a three-year study of teacher professional learning in Canada. The study applies a mixed methods approach involving over 200 teachers and 1000 students as well as case study sites in English and French. The collaborative inquiry-based professional learning program (called CIL-M) focuses on teacher collaboration, mathematics knowledge for teaching, and student mathematical thinking. The program, which was refined annually based on research report recommendations, was found to increase teacher efficacy, student achievement and positive student beliefs.

The marriage of place and value

Students say some amazing things. Back Talk highlights the learning of one or two students and their approach to solving a math problem. Each article includes the prompt used to initiate the discussion, a portion of dialogue, student work samples (when applicable), and teacher insights into the mathematical thinking of students. This article investigates a student's understanding of the base-ten number system and how she uses her conceptions of place value to learn about the base-two number system.

How Many Dogs? How Many Chickens?

Students say some amazing things. Back Talk highlights the learning of one or two students and their approach to solving a math problem. Each article includes the prompt used to initiate the discussion, a portion of dialogue, student work samples (when applicable), and teacher insights into the mathematical thinking of students. This article describes the thought processes of two third-grade students who were given a problem involving finding the number of dogs and chickens on a farm when they know only the total number of legs. Their solution strategies include standard mathematical operations as well as pictorial representations.

Characterizing pivotal teaching moments in beginning mathematics teachers’ practice

Although skilled mathematics teachers and teacher educators often “know” when interruptions in the flow of a lesson provide an opportunity to modify instruction to improve students’ mathematical understanding, others, particularly novice teachers, often fail to recognize or act on such moments. These pivotal teaching moments (PTMs), however, are key to instruction that builds on student thinking about mathematics. Video of beginning secondary school mathematics teachers’ instruction was analyzed to identify and characterize PTMs in mathematics lessons and to examine the relationships among the PTMs, the teachers’ decisions in response to them, and the likely impacts on student learning. These data were used to develop a preliminary framework for helping teachers learn to identify and respond to PTMs that occur during their instruction. The results of this exploratory study highlight the importance of teacher education preparing teachers to (a) understand the mathematical terrain their students are traversing, (b) notice high-leverage student mathematical thinking, and (c) productively act on that thinking. This preparation would improve beginning teachers’ abilities to act in ways that would increase their students’ mathematical understanding.

Eliciting Mathematical Thinking of Students through Realistic Mathematics Eucation

This paper focuses on an implementation a sequence of instructional activities about addition of fractions that has been developed and implemented in grade four of primary school in Surabaya, Indonesia. The theory of Realistic Mathematics Education (RME) has been applied in the sequence, which aims to assist low attaining learners in supporting students’ thinking in the addition of fractions. Based on the premise that eliciting and addressing learners’ alternative conceptions in mathematics is beneficial in assisting them to improve their understanding, the paper seeks to explore the role that RME plays pertaining to this particular supposition. The paper presents and discusses examples of learners’ responses to contextual problems given to them during the course of the instructional activities. Keywords: Realistic Mathematics Education, mathematical thinking, a sequence of instructional activities  DOI : http://dx.doi.org/10.22342/jme.3.1.620.55-70

Supporting Students Mathematical Thinking in the Learning of Two-Variable Functions Through Blended Learning

Abstract In multivariable calculus, the concept of multivariable function is one of the most difficult for undergraduate students to study. The main objective of this study is to establish a model of teaching and learning to support students’ mathematical thinking in the learning of two-variable functions through a blended learning environment. The impact of this environment on students’ learning of two-variable functions and in overcoming students’ obstacles are put forward. Findings revealed that blended learning supports students’ mathematical thinking and helps in overcoming additional obstacles in mathematical learning. Poor mathematical knowledge of single-variable functions and poor algebraic manipulations are still the main reasons behind student obstacles in learning two-variable functions.

Fostering Mathematical Thinking in the Learning of Multivariable Calculus Through Computer-Based Tools

Calculus as a prerequisite course to other advanced mathematics courses is one of the important and difficult courses for undergraduate students in many fields of study. Mathematical thinking is an important method to support students in the learning of calculus and specifically multivariable calculus. Researchers endeavour to support student's mathematical thinking in calculus with or without computer-based tools. The main goal of this paper is to illustrate the importance of using computer-based tools for fostering student's mathematical thinking to overcome their obstacles in multivariable calculus.

Reflection of primary school 6th grade mathematics activities on the development of students mathematical thinking

The purpose of this study is to investigate 6th grade math class activities with students in classroom interactions in terms of development of mathematical thinking respectively.Qualitative research methods, case study was used to work. The case has been appointed as the classroom activities. In this context, a primary school 6th grade mathematics teacher's video recordings were examined for 120minutes. Classroom communication is analyzed separately for each activity with regard to the categories. Results indicated that in learning and teaching process students and teacher carried out 8 sub-categories which belong to “Organize their mathematical thinking through communication” category.