What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <s> This paper is the Discussion Documentfor a forthcoming ICMI Study on Applications and Modelling in MathematicsEducation. As will be well-known, fromtime to time ICMI (the InternationalCommission on Mathematical Instruction)mounts specific studies in order toinvestigate, both in depth and in detail,particular fields of interest inmathematics education. The purpose of thisDiscussion Document is to raise someimportant issues related to the theory andpractice of teaching and learningmathematical modelling and applications,and in particular to stimulate reactionsand contributions to these issues and tothe topic of applications and modelling asa whole (see Section 4). Based onthese reactions and contributions, alimited number (approximately 75) ofparticipants will be invited to aconference (the Study Conference)which is to take place in February 2004 inDortmund (Germany). Finally, using thecontributions to this conference, a bookwill be produced (the Study Volume)whose content will reflect thestate-of-the-art in the topic ofapplications and modelling in mathematicseducation and suggest directions for futuredevelopments in research and practice.The authors of this Discussion Document are themembers of the International ProgrammeCommittee for this ICMI Study. Thecommittee consists of 14 people from 12countries, listed at the end of Section4. The structure of the Document isas follows. In Section 1, we identifysome reasons why it seems appropriate tohold a study on applications and modelling.Section 2 sets a conceptual frameworkfor the theme of this Study, and Section3 contains a selection of importantissues, challenges and questions related tothis theme. In Section 4 we describepossible modes and ways of reacting to theDiscussion Document, and in the finalSection 5 we provide a shortbibliography relevant to the theme of thisStudy. <s> BIB001 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <s> In this article we present, illustrate, test and refine a framework developed by Galbraith, Stillman, Brown and Edwards (2006) for identifying student blockages whilst undertaking modelling tasks during transitions in the modelling process. The framework was developed with 14~15 year old students who were engaging in their first experiences of modelling at the secondary level. <s> BIB002 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <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> Future Research <s> This paper begins by describing teachers’ knowledge as the creation and development of increasingly sophisticated models or ways of interpreting the tasks of teaching. One study illuminates several ways that pre-service teachers perceive the processes of modelling and the limits of their experiences with stochastic models. Results from a second study indicate that teachers need to have a broad and deep understanding of the diversity of approaches that students might take with modeling tasks. The second study also suggests a reversal in the usual roles of teachers and students by engaging students as evaluators of models. <s> BIB004 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <s> This article presents one approach to addressing ::: the important issue of interdisciplinarity in the primary ::: school mathematics curriculum, namely, through realistic ::: mathematical modelling problems. Such problems draw ::: upon other disciplines for their contexts and data. The ::: article initially considers the nature of modelling with ::: complex systems and discusses how such experiences ::: differ from existing problem-solving activities in the primary mathematics curriculum. Principles for designing ::: interdisciplinary modelling problems are then addressed, ::: with reference to two mathematical modelling problems— ::: one based in the scientific domain and the other in the ::: literary domain. Examples of the models children have ::: created in solving these problems follow. A reflection on ::: the differences in the diversity and sophistication of these ::: models raises issues regarding the design of interdisciplinary modelling problems. The article concludes with suggested opportunities for generating multidisciplinary projects within the regular mathematics curriculum. <s> BIB005 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <s> The importance of reflective metacognitive activity during mathematical modelling activity has been recognised by scholars and researchers over the years. The metacognitive activity (or lack of it) of secondary students associated with transitions between stages in the modelling process – especially in relation to the identification and release of blockages to progress – is considered. Productive metacognitive acts are identified as occurring at three levels. Routine metacognition together with metacognitive responses to Goos’ red flag situations are elaborated together with the notion of meta-metacognition being engaged in by teachers trying to foster students’ development of independent modelling competencies especially their metacognitive competencies. <s> BIB006 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <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> BIB007 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <s> Many mathematics teachers in Singapore are not familiar with designing, implementing and facilitating modelling activities in the classroom. In this chapter, we report an experiment in which a teacher with no experience in mathematical modelling made an attempt to conduct a modelling activity in his mathematics class at a local secondary school. Data in the form of students’ work and feedback, lesson videos and interviews were collected. We examine the reactions from students, and discuss the lessons gained from this experience. From the data collected, it appears that while students were generally motivated in carrying out the task, they had some difficulties handling the mathematics involved and the modelling process. The teacher was able to design a reasonable task, but had some difficulty with classroom implementation of the activity. <s> BIB008 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <s> With the adoption of the Common Core State Standards for Mathematics (CCSSM), the process of mathematical modeling has been given increased attention in mathematics education. This article reports on a study intended to inform the implementation of modeling in classroom contexts by examining students’ interactions with the process of creating mathematical models and analyzing the types of models so generated. Results indicate students’ tendencies to oversimplify or overcomplicate the modeling process. Implications of the study are discussed, especially for understanding which aspects of the modeling cycle might be most helpful for teachers to focus on in order to develop students’ modeling abilities. <s> BIB009 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <s> upper-elementary teachers’ mathematical content knowledge and their students’ mathematics achievement, after controlling for studentand teacher-level characteristics. Further, the mathematical content and pedagogical knowledge of middlegrades teachers were each directly and positively related to their students’ mathematics achievement, with and without teacher-level controls. Significant interactions emerged between teachers’ perceptions and knowledge influencing student achievement. Teachers’ claimed awareness of their students’ dispositions toward mathematics interacted with upper-elementary teachers’ content knowledge; middlegrades teachers’ beliefs regarding modeling mathematical solutions and organizing instruction to support incremental mastery of skills interacted with both content and pedagogical knowledge. Findings provide evidence of the relevance of teacher knowledge and perceptions for teacher preparation and professional development programs. <s> BIB010 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Future Research <s> While engineering mathematics curricula often prescribe a fixed body of mathematical knowledge, this study takes a different approach; second-year engineering students are additionally required to investigate and document an aspect of mathematics used in engineering practice. A qualitative approach is used to evaluate the impact that students’ investigations have on their mathematics learning and whether this approach creates greater value for students compared to curriculum mathematics learning. This article contains an account of students’ engagement with and their emotional responses to their investigations of professional engineers’ mathematics usage. The study illustrates the positive changes in students’ mathematics attitudes arising from their insights into engineering practice. Students have acquired a sense of wanting to learn mathematics: they now see themselves as learners preparing for their future careers. This study also introduces students to the concept of communicating mathematics. <s> BIB011
There are several other areas of research that can be undertaken to broaden the research base for mathematical modelling. Several studies stated the importance of teachers' knowledge and experience with mathematical modelling BIB009  BIB003 . Future research can focus on linking teachers' content knowledge and experiences specific to mathematical modelling BIB004 with student outcomes. In general, there is research that shows that teachers' mathematical content knowledge is positively related to students' mathematics achievement BIB010 . More research could be done though specific to mathematical modelling and the impact of teachers on the effectiveness of mathematical modelling. This is important because teachers with limited experience with mathematical modelling can implement it poorly leading to negative student outcomes BIB008 . Research from mathematical modelling at the elementary level with multi-tiered teaching experiements can be used to guide this work. A quality example of this is English's (2009) three-year longitudinal study of four classes of 3rd grade students. At the first level of a multi-tiered teaching experiment students create models in their work. At the second and third level the teachers work with the researcher in designing and implementing the modelling problems. The modelling activities challenge the teachers as they consider the mathematical ideas that students might use and the best ways to respond to students' ideas. Researchers make sense of both the students' models and the teachers' implementation of the modelling activities. A related approach that has been used at the secondary level is in the investigation of meta-metacognition BIB006 . Here teachers think about and explore students' metacognition while the researcher can make sense of both the students' metacognition and the teachers' meta-metacognition. In general, some form of professional development or training for teachers in studies should be provided to assist teachers in proper implementation of mathematical modelling (e.g.  BIB005 . Integrating STEM education with mathematical modelling is a natural fit that deserves more attention. While there were several studies that discussed the impact of technology with mathematical modelling BIB002 Galbraith et al., 2007) , students' interactions with technology were the not the main focus of the studies. More research can be done on effective technological tools that benefit students during mathematical modelling and are worth the time investment for students to learn their usage. A question to guide this work is -In what cases is technology a crucial need in modelling in the classroom? Are there circumstances (if any) where modelling processes can't be developed without technology? BIB001 . Two main considerations for technology integration are will the technology allow students to do something that they could not have done before and also will the technology allow students to do something they could have done before but in a better way. Many of the modelling activities had real world science contexts, for example non-renewable energy sources and animal populations, that make for natural connections for students to learn science content or concepts along with the mathematics content. Mathematical modelling is often also a natural way for students to learn more about engineering. Mathematical modelling and predictive analysis are essential to engineering, and engineers must be comfortable applying mathematics (National Academy of Engineering, 2009). Future research can focus on how engineering can be best integrated with mathematical modelling BIB011 BIB007 .