Realistic Mathematics Education Theory Research Articles

Promover el razonamiento proporcional mediante la tecnología digital

This article explores alternative teaching methods that promote proportional reasoning in Mexican students aged fourteen to fifteen using digital technology. To this end, a sequence of didactic activities has been designed to give meaning to the concepts of ratio and proportion in their different representations from the approach of linear functions. For the design of the tasks, elements of the theory of Realistic Mathematics Education and the Didactics of Cuevas-Pluvinage were used, coinciding with several didactic points. The proposed learning objectives and activities have been organized through a hypothetical learning trajectory. The analysis and detailed evaluation of the responses allowed us to identify: the didactic advantages of the design, the learning difficulties, and the vital role played by technology in the learning process.

Realistic mathematics education and authentic learning: A combination of teaching mathematics in high schools

Realistic mathematics education (RME) has been gradually asserting its influence on the development of many mathematics education in the world, including Vietnam. Through decades of application and development, RME is increasingly proving itself to be an effective approach in improving the quality of teaching and learning mathematics. The article focuses on clarifying some similarities between RME theory and authentic education method. The combination of the main ideas of these two approaches can help teachers in designing and selecting appropriate learning tasks to improve the effectiveness of teaching and learning mathematics in the new curriculum context.

From informal to formal understandings: analysing the development of proportional reasoning and its retention

This study is a case study within a three-year-long design research study. The purposes of this study are twofold. The first purpose includes investigating whether there is a significant difference in seventh-grade students’ proportional reasoning skills before and after a teaching experiment conducted in line with the perspectives of Hypothetical Learning Trajectories and the theory of Realistic Mathematics Education and assess retention of those skills by means of quantitative analysis. The second purpose includes describing the changes and patterns in students’ levels of ratio understandings and related uses of informal, pre-formal, and formal representations through the analysis of students’ written work in the achievement tests and task-based interview results. Findings of the study showed that the intervention proved successful in enhancing students’ proportional reasoning skills and supporting the shift in students’ levels of ratio understanding and representation use from informal ones to more formal ones. Students’ levels of ratio understanding and representation use are described in detail with several sample student responses from the pre-, post- and retention assessments, which also delineate the shift in students’ proportional reasoning skills from informal to formal ones. Several discussions are made to discuss the findings and their potential implications.

USING VIRTUAL LEARNING ENVIRONMENT ON REALISTIC MATHEMATICS EDUCATION TO ENHANCE SEVENTH GRADERS’ MATHEMATICAL MODELING ABILITY

Many research studied that realistic mathematics education (RME) can be an alternative solution to students’ difficulties in learning mathematics. Various forms of technology additionally are further employed to support students' mathematical achievements. However, research on the implementation of virtual learning environments (VLE) with the RME approach is still lacking. The main goals of this research were to create an instructional process of virtual learning environments on realistic mathematics education to improve seventh graders' mathematical modeling abilities and to examine the effect of designs on mathematical modeling ability. Theory of realistic mathematics education and virtual learning environment literature were integrated. The design model developed was verified by experts to be tested. The pre-test / post-test test method was carried out to see the effectiveness of the design. The sixty-seventh graders from a secondary school in North Sumatera were selected as samples. The instructional process developed consists of four stages, namely (1) purposing contextual problems, (2) defining situations from contextual problems, (3) solving problems individually or in groups, and (4) reviewing and comparing solutions. The developed virtual learning environment consists of 5 components, namely (1) users management, (2) content and activities management, (3) resources management, (4) visualization and communication management, and (5) evaluation and assessment management. The mathematical modeling ability concerning experimental group students is significantly higher after being taught through a realistic mathematics education instructional process via a virtual learning environment. Comparison of the experimental group with the control group also showed the same results.

Integrating Indigenous Knowledge and Culturally based Activities in South African Mathematics Classrooms

Culturally based activities embedded within indigenous knowledge, in general, may be used to support the teaching of mathematics in multicultural classes. The article reflects on research that has been conducted with twenty-five post-graduate students studying Mathematics Education at one university in KwaZulu-Natal, South Africa. These post-graduate students were also practicing mathematics teachers at schools. The study explored the use of indigenous knowledge and culturally based activities by post-graduate students in schools while teaching mathematical concepts. The theory of Realistic Mathematics Education framed this qualitative, interpretive study which used a questionnaire, lesson observations and semi-structured interviews to generate data. Qualitative data were analysed inductively and thematically. The findings reveal that the participants needed to understand indigenous knowledge to integrate culturally based activities in mathematics lessons. Secondly, culturally based activities established on indigenous knowledge scaffolded mathematics lessons and promoted the understanding of mathematical concepts to make learning more meaningful and relevant. Thirdly, this study provides examples of good practice to support teachers in integrating classroom activities and activities outside the classroom, ensuring that mathematical concepts learned in classrooms are not done in isolation but take into account learners’ authentic experiences in various settings. Finally, by integrating indigenous knowledge and culturally based activities in the mathematics curriculum, learners interacted and engaged more freely within the educational context. Similar studies could be conducted at universities internationally. Implications for mathematics teachers, mathematics teacher educators and mathematics curriculum developers globally are discussed.

Realistic Mathematics Education Principles for Designing a Learning Sequence on Number Patterns

The number pattern is one of mathematics topics taught for junior high school students that relate between arithmetic and algebra domain. This topic bridges arithmetical and algebraic thinking. Therefore, the learning for this topic should be designed meaningfully. This research aims to design a learning sequence on the number patterns using principles of Realistic Mathematics Education (RME). To do this, we used design research method, particularly the preliminary design phase, with the following three steps. First, literature study was conducted to collect student difficulties in the learning of number patterns, relevant studies, and the theory of RME. Second, we observed Indonesian mathematics textbooks addressing the number patterns to see a learning sequence for this topic. Finally, we designed a learning sequence for the number patterns using the RME principles, particularly the reality principle, level principle, and intertwinement principle. The result of this research includes the learning sequence for the number patterns according to the RME principles, which consists of three activities: relationship between patterns and numbers; exploration of numbers; and generalization of number patterns. We conclude that the three principles of RME are fruitful for designing a meaningful learning sequence for the topic of number patterns.

Contextual problem in mathematical problem solving: core ability in Realistic Mathematics Education

Problem solving is one of the ability existed in the component of Realistic Mathematics Education. This ability becomes more important for a student to understand, analyse and formulate the daily-life problem into formal mathematics, such as real-life problem, problem-based specific context, and the problem itself. Even though problem solving is an important component that must be mastered by the student in learning mathematics, not every student has a certain skill to understand and solve the problems occured in daily life. Many students do neither master the basic mathematics concepts nor know the specific criteria in problem solving ability. Regarding the fulfilment, this paper aims to investigate how to optimize problem solving ability using Realistic Mathematics Education as a part of student’s learning. The method used in this article is literature review in which the data collected through a study of several theories and the results of relevant scientific research such as mathematics learning, problem solving concept, Realistic Mathematics Education theory and the others. The result shows that the ability of problem solving and Realistic Mathematics Education interconnected each other, which is Realistic Mathematics Education has the important role in optimizing problem solving, especially in solving contextual problem. According to the result, this article contribute to improving teachers and students mathematics proficiency and deepening the learning theories especially the theories of mathematics problem solving to be more skilful and well-experienced in learning mathematics.

PHÂN TÍCH SÁCH GIÁO KHOA MÔN TOÁN DỰA TRÊN LÍ THUYẾT GIÁO DỤC TOÁN THỰC (REALISTIC MATHEMATICS EDUCATION) VÀ MỘT SỐ KHUYẾN NGHỊ

Writing and evaluating textbooks under the Vietnam new general education curriculum (2018) is a topic of interest to society as well as researchers. This study aims to evaluate the practical elements in current Mathematics textbooks (according to the previous Mathematics Education Curriculum). In order to conduct research, the author offers two frameworks for analyzing textbooks, basically built on the approach to Realistic Mathematics Education theory (RME). This study found that real contexts, real tasks, historical-cultural images, integrated or interdisciplinary elements are presented quite a lot in current Math textbooks. However, it is also necessary to innovate, update, supply real tasks, and pay attention on the structure and level of real tasks, cultural, historical and integrated elements in the mathematics textbooks in order to implement and develop the new math Mathematics Education Curriculum in Vietnam new general education curriculum. The new textbook analytical framework mentioned in this research can also be used to evaluate the “practicality” of the new mathematics textbooks.

Phenomenology of Points Mathematics

<p style="text-align: justify;">This is a preliminary paper about a large research project on social mathematics. It proposes points mathematics, a variant of social mathematics, as a viable context for teaching mathematics to adults. Points mathematics, focuses on observing, representing and investigating patterns, regularities and quantitative relationships stemming from convertible points, that businesses offer to their customers/clients for the purpose of encouraging loyalty and for boosting up sales in competitive markets. Using ten illustrative examples, the paper asserts that points mathematics provides practical, realistic context for teaching fundamental mathematics concepts and skills to adult students. These include, but not limited to, the four operations of mathematics (addition, division, subtraction and multiplication), variable, linear equation, graph, rates, percent, ratio, patterns and proportion. The paper is grounded in the theory of realistic mathematics education (RME), that posits that the teaching and learning of mathematics should be contextually-based; entails explaining and solving contextual problems; and establishing high-level interactive relationship between learning and teaching. The paper concludes with three recommendations to guide mathematics teachers of adults who want to implement points mathematics as part of their mathematics curriculum. However, the paper is the first phase of a large research project that explores social mathematics and how it could be integrated in mathematics curricular contents for adult students.</p>

Leveraging the design heuristics of realistic mathematics education and culturally responsive pedagogy to create a richer flipped classroom calculus curriculum

While the number of studies of flipped classrooms has increased, they have primarily addressed the efficacy of using such an approach on student outcomes, often failing to account for the classroom activities and learning theories used to design the curriculum. This study begins to fill this gap in the literature by uniting the at-home video and in-class curricular components of the flipped classroom via design heuristics that empower students to critically think about mathematical problems individually before engaging with the task in a collective environment. To that end, we illustrate how elements of the instructional design theory of Realistic Mathematics Education and Culturally Responsive Pedagogy influenced the written and hidden curriculum and how those considerations were then experienced by the students as part of the enacted components of the curriculum. The context of the study is a 2-week classroom teaching experiment covering topics in trigonometry and vectors for 27 calculus students at a Norwegian university.

Designing an algebra learning sequence: the case of operations on algebraic expressions

This study aims to design a learning sequence for algebra learning on the topic of operations on algebraic expressions—which is taught for junior high school students. To do so, we used design research method, particularly the preliminary design phase. First, we compiled student difficulties in the topic of algebraic expressions from previous relevant studies. This compilation is used as a consideration for designing a learning sequence. Second, we designed the learning sequence based on the theory of realistic mathematics education. This theory is used because it offers meaningful mathematics for students. Finally, we conducted a focus group discussion to revise the design in order to produce a ready-use learning sequence for the teaching experiment. The result of this study included a sequence of algebra learning material on the topic of operations on algebraic expressions. In sum, we consider that the designed learning sequence for this algebra topic is more meaningful than the conventional learning sequence for students.

Developing Realistic and Child-friendly Learning Model for Teaching Mathematics

This study aims to develop a learning model based on the theory of realistic mathematics education and child-friendly learning for teaching mathematics in junior high school. This research is conducted using Plomp’s educational design research that consists of four development phases: 1) preliminary investigation, 2) designing, 3) realization, and 4) revision, evaluation. In addition, this study also tests the mathematics instructional model prototype being developed and validated by an expert. Based on the analysis, it can be concluded that: 1) the teacher’s learning management in the implementation of realistic and child-friendly learning model can be classified “very good”, 2) the prototype model is categorized “very good” in improving students’ activities , and 3) the students’ responses toward the developed mathematics instructional model are "positive”. Moreover, the syntaxes of realistic and child-friendly learning model for teaching mathematics in junior high school have been formulated, namely: 1) explaining learning objectives and motivating students, 2) providing contextual problems students familiar with, or horizontal mathematization, 3) processing abstraction or vertical mathematization, 4) devising strategies, 5) communicating solution in a discussion, and 6) giving inferences of mathematics subject-matter.

Modeling as a Means to Develop New Ideas: The Case of Reinventing a Bifurcation Diagram

This article provides an overview of a modeling sequence that culminates in student reinvention of a bifurcation diagram. The sequence is the result of years of classroom-based research and curriculum development grounded in the instructional design theory of Realistic Mathematics Education. The sequence of modeling tasks and examples of student work tendered offers a fresh perspective on modeling and how an inquiry-oriented sequence of tasks can lead to the reinvention of significant mathematics.

Teachers' Understandings of Realistic Contexts to Capitalize on Students' Prior Knowledge

The theory of realistic mathematics education establishes that framing mathematics problems in realistic contexts can provide opportunities for guided reinvention. Using data from a study group, I examine geometry teachers' perspectives regarding realistic contexts during a lesson study cycle. I ask the following. (a) What are the participants' perspectives regarding realistic contexts that elicit students' prior knowledge? (b) How are the participants' perspectives of realistic contexts related to teachers' instructional obligations? (c) How do the participants draw upon these perspectives when designing a lesson? The participants identified five characteristics that are needed for realistic contexts: providing entry points to mathematics, using “catchy” and “youthful” contexts, selecting personal contexts for the students, using contexts that are not “too fake” or “forced,” and connecting to the lesson's mathematical content. These characteristics largely relate to the institutional, interpersonal, and individual obligations with some connections with the disciplinary obligation. The participants considered these characteristics when identifying a realistic context for a problem‐based lesson. The context promoted mathematical connections. In addition, the teachers varied the context to increase the relevance for their students. The study has implications for supporting teachers' implementation of problem‐based instruction by attending to teachers' perspectives regarding the obligations shaping their work.

Conducting classroom design research with teachers

Design research is usually motivated by university members with experience and interest in building theory and instructional designs in collaboration with one teacher. Typically, the teacher is considered as a member of the research team, with the primary responsibility of implementing instruction. However, in this chapter, I describe a Classroom Design Research project that was conducted by a team comprised mostly of classroom teachers. Their goal was to create a stable instructional unit for integer addition and subtraction that they could use to help students learn the topic with meaning. In this paper, I outline the basic tenets of Classroom Design Research, including the instructional theory of Realistic Mathematics Education and how it guided them in designing instruction. I introduce the construct of a classroom learning trajectory and elaborate on it with the integer project. Finally, I argue that Design Research is mutually beneficial for researchers and teachers. The team of teachers that participated in Design Research embarked on a unique professional development experience, one in which they engaged in practices that supported a new way of preparing their instruction. Reciprocally, the teachers’ unique craft and pedagogical content knowledge shaped the integer instruction theory in unique ways that do not occur in typical Design Research.

Sequence limits in calculus: using design research and building on intuition to support instruction

This article is based on research completed within an ongoing project to develop a calculus course which serves as the foundation for the mathematical education of undergraduate students who are training to become elementary teachers. Several research-based activities have been developed, tested, and refined. In this paper we discuss how the design research approach was used to create and implement an instructional task that introduces the concept of limit of a sequence using popular characters from a children’s television show. We present the intuition that students brought to the instructional sequence, the development of the tasks based on the instructional design theory of Realistic Mathematics Education, and the evolution of the intuition that students displayed after instruction. Results include the instructional task developed and student work which reveals that students use context, informal notions of limit, and the notion of “arbitrarily close” to write about their limit understandings.

Problems and solutions in students’ reinvention of a definition for sequence convergence

Little research exists on the ways in which students may develop an understanding of formal limit definitions. We conducted a study to (i) generate insights into how students might leverage their intuitive understandings of sequence convergence to construct a formal definition and (ii) assess the extent to which a previously established approximation scheme may support students in constructing their definition. Our research is rooted in the theory of Realistic Mathematics Education and employed the methodology of guided reinvention in a teaching experiment. In three 90-min sessions, two students, neither of whom had previously seen a formal definition of sequence convergence, constructed a rigorous definition using formal mathematical notation and quantification equivalent to the conventional definition. The students’ use of an approximation scheme and concrete examples were both central to their progress, and each portion of their definition emerged in response to overcoming specific cognitive challenges.

A local instructional theory for the guided reinvention of the group and isomorphism concepts

In this paper I describe a local instructional theory for supporting the guided reinvention of the group and isomorphism concepts. This instructional theory takes the form of a sequence of key steps as students reinvent these fundamental group theoretic concepts beginning with an investigation of geometric symmetry. I describe these steps and frame them in terms of the theory of Realistic Mathematics Education. Each step of the local instructional theory is illustrated using samples of students’ written work or discussion excerpts.

A local instructional theory for the guided reinvention of the quotient group concept

In this paper we describe a local instructional theory for supporting the guided reinvention of the quotient group concept. This local instructional theory takes the form of a sequence of key steps in the process of reinventing the quotient group concept. We describe these steps and frame them in terms of the theory of Realistic Mathematics Education. Each step of the local instructional theory is illustrated using example instructional tasks and either samples of students’ written work or excerpts of discussions.

A Proposed Instructional Theory for Integer Addition and Subtraction

This article presents the results of a 7th-grade classroom teaching experiment that supported students' understanding of integer addition and subtraction. The experiment was conducted to test and revise a hypothetical learning trajectory so as to propose a potential instructional theory for integer addition and subtraction. The instructional sequence, which was based on a financial context, was designed using the Realistic Mathematics Education theory. Additionally, an empty, vertical number line (VNL) is posited as a potentially viable model to support students' organizing their addition and subtraction strategies. Particular emphasis is placed on the mathematical practices that were established in this setting. These practices indicate that students can successfully draw on their experiences with assets, debts, and net worths to create meaning for integer addition and subtraction.