What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> Contents: Preface. Part I: Introduction to a Models and Modeling Perspective. R. Lesh, H.M. Doerr, Foundations of a Models and Modeling Perspective on Mathematics Teaching, Learning, and Problem Solving. R. Lesh, K. Cramer, H.M. Doerr, T. Post, J.S. Zawojewski, Model Development Sequences. R. Lehrer, L. Schauble, Origins and Evolution of Model-Based Reasoning in Mathematics and Science. R. Lesh, G. Carmona, Piagetian Conceptual Systems and Models for Mathematizing Everyday Experiences. P.E. Kehle, F.K. Lester, Jr., A Semiotic Look at Modeling Behavior. Part II: A Models and Modeling Perspective on Teacher Development. H.M. Doerr, R. Lesh, A Modeling Perspective on Teacher Development. R.Y. Schorr, R. Lesh, A Modeling Approach for Providing Teacher Development. K.K. Clark, R. Lesh, A Modeling Approach to Describe Teacher Knowledge. K. McClain, Task-Analysis Cycles as Tools for Supporting Students' Mathematical Development. M. van Reeuwijk, M. Wijers, Explanations Why? The Role of Explanations in Answers to (Assessment) Problems. Part III: Models and Modeling as Viewed by Heavy Users of Mathematics. R. Lesh, J.S. Zawojewski, G. Carmona, What Mathematical Abilities Are Needed for Success Beyond School in a Technology-Based Age of Information? W. Oakes, A.G. Rud, Jr., The EPICS Model in Engineering Education: Perspective on Problem-Solving Abilities Needed for Success Beyond Schools. G. Kardos, The Case for Cases. C.D. Aliprantis, G. Carmona, Introduction to an Economic Problem: A Models and Modeling Perspective. T. Johnson, R. Lesh, A Models and Modeling Perspective on Technology-Based Representational Media. M.J. Dark, A Models and Modeling Perspective on Skills for the High Performance Workplace. Part IV: Models and Modeling in Problem Solving and Learning. L. English, R. Lesh, End-in-View Problems. J.S. Zawojewski, R. Lesh, A Models and Modeling Perspective on Problem Solving. J.S. Zawojewski, R. Lesh, L. English, A Models and Modeling Perspective on the Role of Small Group Learning Activities. G. Harel, R. Lesh, Local Conceptual Development of Proof Schemes in a Cooperative Learning Setting. R. Lesh, F.K. Lester, Jr., M. Hjalmarson, A Models and Modeling Perspective on Metacognitive Functioning in Everyday Situations Where Problem Solvers Develop Mathematical Constructs. J.A. Middleton, R. Lesh, M. Heger, Interest, Identity, and Social Functioning: Central Features of Modeling Activity. Part V: Models and Modeling Before and After Middle School. S.J. Lamon, Beyond Constructivism: An Improved Fitness Metaphor for the Acquisition of Mathematical Knowledge. K. Cramer, Using a Translation Model for Curriculum Development and Classroom Instruction. M. Carlson, S. Larsen, R. Lesh, Integrating a Models and Modeling Perspective With Existing Research and Practice. B. Shternberg, M. Yerushalmy, Models of Functions and Models of Situations: On the Design of Modeling-Based Learning Environments. Part VI: Next Steps. F.K. Lester, Jr., P.E. Kehle, From Problem Solving to Modeling: The Evolution of Thinking About Research on Complex Mathematical Activity. R. Lesh, H.M. Doerr, In What Ways Does a Models and Modeling Perspective Move Beyond Constructivism? <s> BIB001 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> A modeling approach to the teaching and learning of mathematics shifts the focus of the learning activity from finding a solution to a particular problem to creating a system of relationships that is generalizable and reusable. In this article, we discuss the nature of a sequence of tasks that can be used to elicit the development of such systems by middle school students. We report the results of our research with these tasks at two levels. First, we present a detailed analysis of the mathematical reasoning development of one small group of students across the sequence of tasks. Second, we provide a macrolevel analysis of the diversity of thinking patterns identified on two of the problem tasks where we incorporate data from multiple groups of students. Student reasoning about the relationships between and among quantities and their application in related situations is discussed. The results suggest that students were able to create generalizable and reusable systems or models for selecting, ranking, and weighting data. Furthermore, the extent of variations in the approaches that students took suggests that there are multiple paths for the development of ideas about ranking data for decision making. <s> BIB002 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> Abstract Modelling and application are regarded as an important topic for maths lessons. But so far the concept 'modelling competencies” has not been described in a comprehensive manner. The aim of this paper is to supplement former descriptions of modelling competencies based on empirical data. An empirical study was carried out which aimed at showing the effects of the integration of modelling tasks into day-to-day maths classes. Central questions of this study were - among others: How far do math lessons with focus on modelling enable students to carry out modelling processes on their own? What are modelling competencies? The paper describes the theoretical and the methodological approach, the classroom settings, and the results of the study. <s> BIB003 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> This study analyzed the processes used by students when engaged in modeling activities and examined how students' abilities to solve modeling problems changed over time. Two student populations, one experimental and one control group, participated in the study. To examine students' modeling processes, the experimental group participated in an intervention program consisting of a sequence of six modeling activities. To examine students' modeling abilities, the experimental and control groups completed a modeling abilities test on three occasions. Results showed that students' models improved as they worked through the sequence of problem activities and also revealed a number of factors, such as students' grade, experiences with modeling activities, and modeling abilities that influenced their modeling processes. The study proposes a three-dimensional theoretical model for examining students' modeling behavior, with ubsequent implications for the teaching and learning of mathematical problem solving. <s> BIB004 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> This chapter argues for a future-oriented, inclusion of Engineering Model Eliciting Activities (EngMEAs) in elementary mathematics curricula. In EngMEAs, students work with meaningful engineering problems that capitalise on and extend their existing mathematics and science learning, to develop, revise and document powerful models, while working in groups. The models developed by six groups of 12-year students in solving the Natural Gas activity are presented. Results showed that student models adequately solved the problem, although student models did not take into account all the data provided. Student solutions varied to the extent students employed the engineering context in their models and to their understanding of the mathematical concepts involved in the problem. Finally, recommendations for implementing EngMEAs and for further research are discussed. <s> BIB005 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> Modelling is a mathematical competence and a means of learning significant mathematics. Preliminary findings on the nature of competence and modelling competencies, as well as the development of mathematical modelling competencies of Grade 7 students are presented in this chapter. Twelve students solved three model-eliciting tasks in groups over a period of 12 weeks. The development of competencies is considered across the group as a whole. Analysis of the data shows that modelling competencies are activated and do develop when students take part in well-orchestrated modelling activities. <s> BIB006 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> This chapter introduces a long-term modelling project that has been conducted at Goethe-Gymnasium-Germersheim with 14-/15-year-old students. The main question of this project was if (and how) it is possible to integrate real-world modelling tasks into regular math lessons in a way that demands for the application of knowledge from selected topics of a whole school year. Moreover we wanted to know if the pupils accept the intended frequency of five modelling phases as a convenient diversion or if they consider them not worthwhile (concerning the effort)? Finally, we were curious to learn to which extent we can expect pupils to learn mathematical modelling through frequent repetition. The students had to deal with five realistic modelling tasks and one final comparison task. Solutions developed by the students as well as the concept of the questionnaires used for evaluation are presented. <s> BIB007 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> The meaning of mathematical modelling and modelling competencies has been discussed frequently and, in parts also controversially, recently. There is still a debate about how to measure modelling competency and a comprehensive agreement is not yet in sight. In this chapter we applied six stages of mathematical modelling competency to classify student solutions according to their modelling performance and investigated 234 student solution approaches from grades 6 to 11 for the modelling task, Restringing a Tennis Racket. We identified four main solution approaches. In the study we analysed the correlation of modelling competency stage and students’ grade as well as gender. The study shows that mathematical modelling competency is independent of gender issues in this specific case, and gives insights into the modelling behaviour of students of different grades. <s> BIB008 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> Extra curricular modelling events for school students are increasing in popularity. The experience from the perspective of participating Australian and Singaporean Year 10 and 11 students in one such event, the A. B. Paterson College Mathematical Modelling Challenge, is investigated. The aim is to gauge whether their experience meets the intended purposes of mentors that the whole event be considered an inherently valuable learning experience about modelling and application of mathematics to real situations. Findings from a questionnaire and other data included motivation for choice of situation to model, decision making about approach, and reporting of findings were mainly goal oriented rather than performance oriented, despite the Challenge ostensibly being a competition. <s> BIB009 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Modelling Competencies/ability <s> Nature and the constructed world in which students live have provided suitable sources of tasks for junior high school students in the past. The decreasing number of bluefin tuna was considered another appropriate context. Students were asked to provide a mathematical solution to this problem, utilising data on the number of bluefin and the catch of bluefin in the past. Three different mathematical approaches, appropriate to the educational level of the students involved, were taken to considering the decline in bluefin tuna. Year 7 students used a proportional model, Year 8 students used linear function models, and in Year 9 students used recurring formula models. Each set of models was appropriate to the mathematics in the curricula at the level. The aim of this mathematical modelling activity was to develop people with the skills to lead the way in our society. <s> BIB010
There were twenty-six studies done in this category that looked at students' range of ideas/strategies, modelling competencies, and modelling ability. The research has shown that students produce a range of strategies or models when working on mathematical modelling BIB001 BIB002   BIB008 BIB005 BIB004 BIB009 BIB010 . in working with German students age 10-16 identified three general modelling strategies: (a) reality oriented with low level mathematics, (b) focus on the realistic situation with mathematics used but not discussed, and (c) approximations and assumptions of realistic situation and correct mathematics discussed. Other studies have found different models that students have developed for a specific problem. In a study of 234 German students age 12-17, four different models were generated for a problem that asked students to determine the minimal string length of a tennis racket: direct measurement, rectangle model, functional model, and area model BIB008 . A study was done with 22 students age 16 in Spain that had students work with Fermi problems which require estimation of large quantities. The researchers found six categories of strategies that were used including population density and a grid approach . Students at different grade levels can develop different models based on their prior knowledge. BIB010 in a study of 460 Japanese students with a problem involving population of Bluefin tuna found that 7 th graders tended to use a proportional model, 8 th graders a linear model, and 9 th graders a recursive formula. This shows how students' prior knowledge which is affected by their teachers' instruction can influence students' models used. Modelling competencies involve the steps of the modelling process but go beyond this to -include skills and abilities to perform modelling processes appropriately and goal-oriented as well as the willingness to put these into action‖ (MaaB, 2006, p.117) . Modelling competencies specifically entail understanding the realistic problem and setting up a model based on reality, setting up a mathematical model from the real world, solving mathematical questions within a mathematical model, interpreting mathematical results in a real solution, validating a solution, metacognition, effective communication and argumentation, and positive attitudes toward real world application problems  . Three other studies used these descriptions for their understanding of modelling competencies BIB006 BIB003 . Taking a more narrow view of modelling competencies, BIB008 have developed 6 modelling competency stages that describe how far participants progress through the modelling cycle, based on Blum and LeiB's (2007) modelling cycle. There are several insights from the research on modelling competencies including that the competencies develop over time and that some competencies are more difficult for students to develop. A study of 12 students, including low mathematics achieving and high mathematics achieving students, in South Africa had students complete 3 MEAs over a span of 4 months. Twelve modelling competencies were evaluated based on a four scale rubric. The students had no previous modelling experience and initially were weak in all competencies. At the end of the study the modelling competencies were more developed but interpreting and validating remained weak throughout. It was also found that mathematising was dependent on understanding and simplifying BIB006 . Similarly found that 135 grade 9 students in Germany had no improvement in interpreting the solution in the realistic context using a pre and post test that assessed modelling sub-competencies. However, this study only used one modelling problem as an intervention. Overall, students significantly improved on the measure of modelling competencies. Also, similar to MaaB (2006) a positive relationship was found between mathematics ability and modelling competencies. Providing more detail into modelling competencies BIB003 found that better modellers had better metacognition about the modelling process. Studies with 135 students and 234 students BIB008 in Germany have shown that there are no gender differences in modelling competencies of students. The studies involving modelling abilities are connected to modelling competencies but looked specifically at groups' final models. Studies done with MEAs have found that most groups are able to develop acceptable final models BIB002 BIB004 BIB005 . A study of 640 U.S. and 14 Australian students age 11-13 found that students created generalizable and reusable models when working on five MEAs involving selecting, ranking, and weighting data BIB002 . In Cyprus 6 th and 8 th grade students completed 6 MEAs over 3 months and their models improved over time. The researchers also developed a 9 task modelling abilities test that presented students with a situation with constraints that required choosing among alternative decisions. This study was an experimental design. Based on the analysis of the modelling abilities tests using latent growth modelling, the MEA intervention was most effective for students with initially low modelling abilities while low modelling abilities students in the control group had low rates of change BIB004 . BIB007 had twenty-six 14-15 year old students in Germany complete 5 modelling tasks and used sixty-nine students as a comparison group. After the modelling group had done the five tasks, both groups completed two modelling tasks based on deciding which checkout line to pick at a grocery store and how to estimate the mass of a stone formation. University personnel not associated with the project evaluated the final reports for the two tasks. The modelling group finished faster, felt the problems were easier, and did better at using appropriate formal notation. They also did better at finding solutions and quality of documentation as four teams in the control group did not come up with solutions. However, comparing the deepness of the solutions on average, the comparison group did better. No explanation is provided concerning this finding nor how deepness of solution was determined. Time spent on writing and drafting a report can contribute to its effectiveness though. A study with 9 upper secondary students in Denmark found that students' reports had little discussion of students' models or their approximations. The researchers suggested that students may not have had enough time to complete this as the modelling activity required most of the time .