Mathematical Modeling in the Elementary Grades: Developing and Testing an Assessment

Mathematical modeling is a high-leverage topic, critical for college and career readiness, participation in STEM education, and civic engagement. Mathematical modeling involves connecting real-world situations, phenomenon, and/or data with mathematical models, and in this way applies across various STEM disciplines, including mathematics, engineering, and science. Although research has begun to explore mathematical modeling instruction in the elementary grades, questions remain about how to assess student learning at the elementary level. We addressed this need by designing an assessment of mathematical modeling competencies for students in grades 3 through 5. Informed by international research, our assessment includes a hybrid structure to assess mathematical modeling competencies holistically (as students engage in the complete modeling process) and atomistically (as students engage in different components of the modeling process, including making sense of phenomena and real-world situations, setting up and operating on mathematical models, and interpreting results in relation to the real-world context). We conducted student interviews, followed by two rounds of pilot testing to inform item development and ensure acceptable psychometric properties. The final assessment included 13 items (9 multiple choice, 3 open-response, and 1 complete modeling task). We describe our assessment development process, and provide sample assessment items and detailed coding rubrics. We summarize quantitative analyses which established high reliability and low standard error for our assessment, supporting its use for grades 3 to 5. Implications of our framework and assessment for mathematical modeling instruction and future research on STEM learning are discussed.

     

Beyond Motivation: Exploring Mathematical Modeling as A Context for Deepening Students' Understandings of Curricular Mathematics

Views of mathematical modeling in empirical, expository, and curricular references typically capture a relationship between real-world phenomena and mathematical ideas from the perspective that competence in mathematical modeling is a clear goal of the mathematics curriculum. However, we work within a curricular context in which mathematical modeling is treated more as a venue for learning other mathematics than as an instructional goal in its own right. From this perspective, we are compelled to ask how learning of mathematics beyond modeling may occur as students generate and validate mathematical models. We consider a diagrammatic model of mathematical modeling as a process that allows us to identify how mathematical understandings may develop or surface while learners engage in modeling tasks. Through examples from prospective teachers' mathematical modeling work, we illustrate how our diagrammatic model serves as a tool to unpack the intricacies of students’ mathematical experience while engaging in modeling tasks.     

浅谈数学建模教学

探索有效的数学建模课程教学方法具有重要意义．以《数学模型》为教材,从创设良好的课堂气氛、引导学生质疑思考、深入挖掘教材内涵、鼓励学生分组讨论、构建网络教学平台等几个方面对数学建模课教学做一些探讨．同时,要将数学建模的精神,内涵及方法有机地体现到一些重要的教学课程之中．     

Creating an alternate reality: Critical,creative, and empathic thinking generated in the Global Village Playground capstone experience

The Global Village Playground (GVP) was a capstone learning experience designed to address institutional assessment needs while providing an integrated and authentic learning experience for students aimed at fostering critical and creative thinking. In the GVP, students work on simulated and real-world problems as a design team tasked with developing an alternate reality game that makes an impact on the United Nations Millennium Development Goals. Researchers employed a qualitative case study approach to evaluate participant reactions to the course, their perceptions of the instructional design methodology, what they learned in the course, and the challenges they experienced during the pilot implementation of this capstone design. This article documents what students reported to have learned in the course as a result of their experience creating an alternate reality game.     

Cutting the distance in distance education: Perspectives on what promotes positive,online learning experiences

This qualitative research study was designed to inform the development and implementation of effective online learning environments by exploring, from both teacher and student perspectives, what constitute effective online learning experiences. The study examined course content, tasks, and pedagogical approaches, as identified by students and instructors, which contributed to or hindered positive online learning experiences. Researchers interviewed 6 online course instructors and 10 adult students to understand their experiences in undergraduate and graduate level online degree programs. Using a Cognitive Apprenticeship Model to inform the analysis of data, findings revealed an emphasis on text-based content and lecture; instruction that led to disconnect between students, teachers, and course content and goals; and one innovative program that links real-world experiences with online classroom learning. Given the growing number of online programs, the study provides insight for course development and pedagogy as well as offers possibilities for additional research.     

Online discussions in a doctoral research methods course: ‘Like a text by many authors’

In this mixed-methods study, we investigated how doctoral students used asynchronous online discussions that were added to a face-to-face research methods course. We focused specifically on what sociocognitive tools the students used and what they thought about the value of the online discussions for their learning. Our data included three online whole-class discussions from the course and students’ self-evaluations, which they completed three times during the semester. Our findings support the usefulness of online discussions in providing a site where students could draw on a rich repertoire of tools and strategies as well as on the knowledge of others to enhance their thinking and understanding of theory and research.     

Leading Undergraduate Research Projects in Mathematical Modeling

Abstract

In this article, we provide some useful perspectives and experiences in mentoring students in undergraduate research (UR) in mathematical modeling using differential equations. To engage students in this topic, we present a systematic approach to the creation of rich problems from real-world phenomena; present mathematical models that are derived from calculus, but can be suitably modified to solve more complex problems; and finally provide pointers on how to train students through the UR process that can help to impact student’s future academic and non-academic careers.     

Connections as a Focus for Model-Building Learning in Engineering

Learning to create and use models of real-world situations is an important goal for under-graduate engineers. Engineers today use computer models to explore design problems and test solutions before actually building anything. Learning to create and use models of real-world sit-uations is a challenging task. Most efforts to teach modeling are essentially a means of teach-ing students programming, which is recognized as difficult. In the approach described in this paper, we sought to simplify the task as much as possible away from programming, while still requiring students to learn to use the mathematical models necessary to understand a given sit-uation. We worked in the domain of Chemical Engineering with a tool called DEVICE (Dynamic Environment for Visualization in Chemical Engineering). What we found through an iterative development effort is that emphasizing connections between the real world and the mathematical model led to learning and transfer benefits.     

Weighing up the Influence of Context on Judgements of Mathematical Literacy

Mathematical literacy, viewed as a set of ideas involving applications of mathematics to real-world contexts, has recently featured in curricular discussions about the aims for mathematics education. This article explores the effect that differences in the way that a mathematical task is contextualised can have on students’ mathematical arguments and, therefore, on their perceived levels of mathematical literacy. Seventy-two students’ responses to three similar measurement tasks are described according to Kaiser and Willander’s levels of mathematical literacy. The arguments used for assigning each level of mathematical literacy are then investigated for the presence of specific macro- and micro-linguistic features. The context of the task affects what students perceive to be most relevant approaches to use, which are reflected in the arguments they give; this, in turn, affects external judgements of their level of mathematical literacy.     

高职院校开设数学建模课程的实践研究

随着数学建模在社会各领域的影响力逐渐扩大,数学建模竞赛在全国高校学生中越来越受到青睐,在高职院校开设数学建模课程也成为必要。阐述高职院校开设数学建模课程的必要性,对我学院开设数学建模选修课的情况作了实践研究,提出了高职院校开设数学建模课程的益处和有待改进的问题,为高职院校全面开设数学建模课程提供了参考。     

数学建模对学生综合素质的影响分析

本文从建模及竞赛分析数学建模在学生素质培养中所起的作用,对学生解决实际问题的能力的影响,阐述了数学建模训练是人们解决实际问题必要的训练和准备的观点,说明数学建模对学生的影响是质变的过程,参加过建模训练的学生必定是“终生难忘,终生受益”。     

Flounders unlimited

The aim of this paper is to provide an introduction to higher mathematical concepts and techniques as applied to real-world problems. An interesting case study is the 200-mile limit for fisheries presented in Wan (1989, Mathematical Models and their Analysis (New York: Harper and Row)). It is commented upon, re-examined and solved in a different manner inspired by Turing's (1952, Philosophical Transactions of the Royal Society London , B237 , 37-72) paper as a suitable classroom example for undergraduate students taking a first course on mathematical modelling. Emphasis is made not only on the mathematics--always kept as elementary as possible--but also on interpretations, explanations and translation into plain language of the mathematical expressions and results.     

“数学模型”课程教学体会

数学模型对提高学生的综合数学素质具有重要意义。本文在实践的基础上,论述了数学建模在培养学生的综合应用能力方面的重要作用。以提高教学质量和加强学生应用数学能力培养为目标,通过科学选取教学内容,运用互动式教学方式,加强实践力度,重视数学建模竞赛等方面阐述了笔者在数学建模教学中的体会和认识。     

Different ways in which students handle chance encounters in the explorative setting of a dice game

The purpose of this study is to investigate the ways in which Swedish seventh grade students (12 and 13 years old) handle chance encounters. Four groups of students working in pairs participated in the study. In the group discussions, which were tape-recorded and fully transcribed, the students were encouraged to explore strategies for winning a specifically designed dice game based on the sum of two dice. The dice game included four different set-ups of dice designed to bring to the fore different aspects of probability modelling and to offer the student the opportunity to encounter small differences in the mathematical structure of the sample space and of the probability distribution between the four different set-ups. The study describes strategies that the students use when confronted with these different set-ups, what their activities imply in terms of resources in handling random phenomena and what the dice game offers in terms of opportunities for learning probability. In order to explain such meaning-making processes the students’ activities are viewed from a perspective that takes into consideration how the students’ understanding varies with their interpretations of the situation they are confronted with, i.e., how they contextualize the different set-ups of the dice game. The results show how the students, during the course of the game, reorganize their interpretations of the mathematical content confronting them, and how a variation of guiding principles becomes the object of exploration. Approaches of extremes and a number model are described as a means for the students to identify and assign probabilities for the total of two dice.     

Effects of an augmented reality-based educational game on students' learning achievements and attitudes in real-world observations

Augmented reality (AR) has been recognized as a potential technology to help students link what they are observing in the real world to their prior knowledge. One of the most challenging issues of AR-based learning is the provision of effective strategy to help students focus on what they need to observe in the field. In this study, a competitive gaming approach is proposed to support AR-based learning activities conducted in real-world contexts. An experiment has been conducted on an elementary school ecology course to explore the effectiveness of the proposed approach in comparison with the conventional AR-based mobile learning approach in field trips. The experimental results show that the AR-based gaming approach can improve not only students' learning attitudes, but also their learning performance on the field trip. Accordingly, discussions and some suggestions for future work are provided.     

Student Perceived and Determined Knowledge of Biology Concepts in an Upper-Level Biology Course

Students who lack metacognitive skills can struggle with the learning process. To be effective learners, students should recognize what they know and what they do not know. This study examines the relationship between students’ perception of their knowledge and determined knowledge in an upper-level biology course utilizing a pre/posttest approach. Significant differences in students’ perception of their knowledge and their determined knowledge exist at the beginning (pretest) and end (posttest) of the course. Alignment between student perception and determined knowledge was significantly more accurate on the posttest compared with the pretest. Students whose determined knowledge was in the upper quartile had significantly better alignment between their perception and determined knowledge on the pre- and posttest than students in the lower quartile. No difference exists between how students perceived their knowledge between upper- and lower-quartile students. There was a significant difference in alignment of perception and determined knowledge between males and females on the posttest, with females being more accurate in their perception of knowledge. This study provides evidence of discrepancies that exist between what students perceive they know and what they actually know.     

A Combinatorics Course with One Goal: Authentic Mathematical Inquiry

Abstract

This article shares an example of a course in Combinatorics, taught at Adelphi University in Fall 2012, designed with a primary goal of engaging students in pursuing mathematics as mathematicians do. The course went beyond usual applications of inquiry-based learning in that students were also charged with the responsibility of posing the mathematical statements that would be examined as part of the course. In this way, students decided what was interesting to study and were responsible for the creation of mathematics within their classroom community.

This article looks at the syllabus policies for the course, describes some of the work that students did as part of the course, and reflects upon the outcomes of the course.     

探析数学建模课程的ARCS学习动机的教学模式

ARCS学习动机(成功引导)教学模式是"引导学生主动地去寻找有意义、有价值的学习活动,并努力从中获得预期的学习益处的教学方法".在数学建模教学中运用此教学模式既有利于培养和提高学生的综合素质,又有利于增强数学建模课程的教学实效性.ARCS学习动机(成功引导)教学模式在数学建模课中的运用需注意:此教学法需要和别的教学模式相结合并且应该在实践教学起到更大的作用;教师要扮演好教学活动的组织者、引导者、促进者、监督者的角色.ARCS教学模式在数学建模课程的运用中存在考核机制缺失和师生能力不足的突出问题,全面推广此教学法还需进一步试点,同时完善相应机制.