Models and Modeling

Most teachers use lecture method frequently, giving the students little chances for more interaction, discovery approaches, team efforts, and experimentation in the classroom and relying on traditional mathematics textbooks which mostly provide single and straightforward solution problems at which students only apply a ready-made formula to reach a solution. This leads the students to believe the subject mathematics as a mere transmission of resolution techniques, being rather than a tool in another area of knowledge. On the contrary, students working on modeling activities focus on analyzing a problematic situation, setting and testing conjectures and model construction. This paper describes the main implications of modeling in the teaching of mathematics by designing and conducting in a short teaching intervention via model-eliciting activities. The data was collected through class observation, review of textbook, questionnaires, interviews, and students’ final reports detailing the processes used in developing the model for the model-eliciting activities. The analysis of the data revealed the following results: the participating students were able to work effectively with model-eliciting activities that enable them to discover the meaning of the mathematical concepts, few modeling tasks were incorporated in the textbook, students’ report showed that their prior experience gained from the first modeling activity helped them to develop a better model and interpret the solutions back to reality, and teachers and students believed that teaching and learning via model-eliciting activities could improve the teaching-learning of mathematics.

This article presents a review of literature exploring five different perspectives on mathematical modeling in mathematics education. Because there is not a single agreed-on definition of what mathematical modeling is or how it should be done, a focused and extensive review of mathematical modeling is essential. The five broad classifications discussed in this article include realistic modeling, educational modeling, models and modeling perspective, socio-critical modeling, and epistemological modeling. For each perspective, we present (a) the goals of mathematical modeling, (b) the definition of a mathematical model, (c) the mathematical modeling cycle, (d) the design of the modeling task, and (e) key researchers and research foci. In this article, we aim to present the different perspectives of mathematical modeling in an organized way so as to (1) demonstrate the rich and diverse background of mathematical modeling and (2) clarify theoretical foundations of seminal works in mathematical modeling.

Though modeling is a popular topic in mathematics education, the field's definition of model is diverse. Less is known about what teachers identify as mathematical models, even though it is teachers who ultimately enact modeling activities in the classroom. We asked nine middle school teachers with a variety of academic backgrounds and teaching experience to collect data related to one familiar physical phenomenon, cooling liquid. We then asked each to construct a model of that phenomenon, describe why it was a model, and identify whether a variety of artifacts representing the phenomenon also counted as models during a semi-structured interview. We sought to identify: What do mathematics teachers attend to when describing what constitutes a model? And, how do their attentions shift as they engage in different activities related to models? Using content analysis, we documented what features and purposes teachers attended to over the course of the interview. When constructing their own model, they focused on the visual form of the model and what quantitative information it should include. When deciding whether particular representational artifacts constituted models, they focused on how those representations reflected the system under study, and whether those representations could help to further understand the system. These findings suggest the teachers had multiple understandings of models, which were active at different times and reflected different perspectives toward modeling. This has implications for research, teacher education, and professional development.

PNA, 2014

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers' ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers' written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds.

2003

Models and the modeling process are at the heart of mathematics. The paper discusses the importance of developing pupils’ modeling abilities and skills in the context of school mathematics and focuses in particular on the content, structure and the educational exploitation of a set of activities constructed to serve this purpose in a computational modeling environment.

The Models and Modeling Working Group has provided participants with a setting to reflect on models and modeling perspectives to understand how students and teachers learn and reason about real life situations encountered in a mathematics and science classroom. From these perspectives, a model is considered as a conceptual system that is expressed by using external representational media, and that is used to construct, describe, or explain the behaviors of other systems. There are different types of models that students and teachers develop (explicitly) to construct, describe, or explain mathematically significant systems that they encounter in their everyday experiences, as these models are elicited through the use of model-eliciting activities (Lesh, Hoover, Hole, Kelly, & Post, 2000). During the workshop we will continue to explore these aspects of learning, teaching, and research. New directions for Models and Modeling Perspectives will be the topic of discussion and disseminati...

Contemporary Issues in Technology and Teacher …, 2003

Journal of Applied Developmental Psychology, 2000

It is essential to base instruction on a foundation of understanding of children's thinking, but it is equally important to adopt the longer-term view that is needed to stretch these early competencies into forms of thinking that are complex, multifaceted, and subject to development over years, rather than weeks or months. We pursue this topic through our studies of model-based reasoning. We have identified four forms of models and related modeling practices that show promise for developing model-based reasoning. Models have the fortuitous feature of making forms of student reasoning public and inspectable-not only among the community of modelers, but also to teachers. Modeling provides feedback about student thinking that can guide teaching decisions, an important dividend for improving professional practice.

2016

Mathematical Modeling from the Teacher’s Perspective Christopher J. Huson Applying mathematics to real world problems, mathematical modeling, has risen in priority with the adoption of the Common Core State Standards for Mathematics (National Governors Association and the Council of Chief State School Officers, 2010). Teachers are at the core of the implementation of the standards, but resources to help them teach modeling are relatively undeveloped. This multicase study explored the perspectives of teachers regarding mathematical modeling pedagogy (the modeling cycle), instructional materials, and professional collaboration, with the assumption that understanding teachers’ views will assist authors, publishers, teacher educators, and administrators to develop better support for modeling instruction. A purposeful sample of six high school mathematics teachers from a variety of school settings across the country was interviewed using a semi-structured protocol. A conceptual framework...

Mathematics Education in the Digital Era