What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Construction of Knowledge <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> Construction of Knowledge <s> This paper describes a variety of sharable multi-purpose research tools which have evolved from recent studies in which models & modeling perspectives (Lesh & Doerr, 2003) were used to investigate what it means for students to develop meaningful understandings of foundation-level concepts and abilities in an introductory statistics course. Such a course typically is intended to “cover” topics ranging from basic measures of probability, centrality, and spread, to more advanced topics such as analysis of variance, hypothesis testing, or regression and correlation. Although MMP studies have involved students from grade school through graduate school, as well as professionals in fields ranging from education to engineering, the students referred to in this paper were sophomore elementary education majors at Indiana University. This group is identified simply to give readers a frame of reference for observations made - not because it served as a treatment group or a control group of any kind… These students had sufficiently strong academic records to be admitted to a university program with high academic standards; but, compared with peers in other fields, none considered themselves to be outstanding in mathematics. The tools described here were designed mainly to investigate the following questions. (i) What are the most important “big ideas” that should be emphasized in a given mathematics topic area (e.g. statistics)? (ii) What does it mean to “understand” these ideas? (iii) How do these understandings develop? In fields like engineering, or in other “design sciences” which are more mature than mathematics education, researchers in leading research communities often devote significant portions of their time and efforts toward the development of tools and artifacts for their own use. Whereas, in mathematics education, our research community has spent relatively little time developing such tools and resources. One result of this neglect is that mathematics educators are unable to reliably observe, document, or measure either students’ or teachers’ levels of development for nearly any of the deeper or higher-order achievements that current theories hypothesize to be important. For this reason, this paper focuses on theory-based tools and tool development; and, to begin, main elements of MMP theoretical foundations are described in enough detail so that the tools will be useful to other researchers. Throughout this paper, bulleted questions or statements indicate issues about which a great deal is known – but that still should be investigated more thoroughly in order to more thoroughly understand their meanings and implications. <s> BIB002 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Construction of Knowledge <s> This chapter presents types of mathematical knowledge application during a design-based interdisciplinary project as displayed through the work of student-group cases from grades 7 and 8 (aged 13–14) in two educational streams across three Singapore government secondary schools. It was found that the students did not apply all the expected mathematical knowledge and skills afforded by the project. They also lacked awareness of the purpose of scale and displayed limited activation of real-world knowledge for mathematical decision making. Findings presented have implications on the facilitation of quality mathematical application in contextualised tasks. <s> BIB003 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Construction of Knowledge <s> Functions provide powerful tools for describing change, but research has shown that students find difficulty in using functions to create and interpret models of changing phenomena. In this study, ... <s> BIB004
Eleven studies looked at students' knowledge construction during mathematical modelling and mainly found that students are able to construct developing mathematical understandings BIB004 Dunne & Galbraith, 2003; Hitt & Gonzalez-Martin, 2015;   BIB003  . Six of the studies involved MEAs done in the U.S. in which students were able to reason with concepts of average rate of change with exponential functions BIB004 , inverse variation , proportionality  , geometry and measurement  , and transformational proof scheme that involves pictorial anticipation of an action not yet performed . Five of the studies were done with average or low ability middle school students age 11-13. A limitation of these studies however is that four of the studies involved only 3 students and one did not specify the number of students. Other studies have had larger sample sizes but were done with medium or high mathematics ability students. A study with fifty-one students that were going to enroll as first year university engineering majors found over the course of six weeks that students could work with exponential functions as well as reason with directionality in terms of speed, velocity, and average rate of change but had greater difficulty with negative rate of change .  showed that a mathematical modelling activity supported conceptualization in calculus through various representations and iterations of thinking with one class of high ability 8 th grade students in South Korea. Another study (Dunne & Galbraith, 2003) with 12 year-old students in Australia compared the basic mathematics content and skill knowledge of a medium mathematics ability modelling treatment group with a high mathematics ability control group over the course of a year. The modelling treatment class used modelling problems to introduce concepts and then had traditional instruction after that. The modelling class had statistically significant higher mathematics achievement after the year compared to the high ability control group. However, a year later both classes of students had a traditional class and then the high ability students outscored the modelling treatment class. The research on construction of knowledge with mathematical modelling can be expanded. Both  and suggested research focused on mathematical modelling with additional mathematical concepts. This can be done by having studies use a modelling treatment group and a control group to further document the effectiveness of mathematical modelling. The modelling treatment group can employ a method similar to the -island‖ approach  in which mathematical modelling is integrated as a formative assessment or a supplement to normal teaching. Promising results have been shown with this approach (Dunne & Galbraith, 2003) but more research can be done with different aged students and with larger samples. Key questions to guide this work include: What would be an appropriate balance-in terms of attention, time and effort---between applications and modelling activities and other mathematical activities in mathematics classrooms at the secondary level? BIB001 -When applications and modelling are included at different places in mathematics curricula, how can it be guaranteed that basic modelling skills and competencies are acquired systematically and coherently?‖ BIB001 . In order for this to occur it is imperative to identify -big ideas‖ that modelling curricula can be built around. BIB002 stated three main questions that deserve focus for how students construct mathematical knowledge through modelling: (a) -What are the most important ‗big ideas' that should be emphasized in a given mathematics topic area?, (b) What does it mean to ‗understand' these ideas? and (c) How do these understandings develop?‖ (p.16). In the U.S. Common Core State Standards for Mathematics [CCSSM] , 2010) sixteen high school content standards are highlighted as modelling standards. This can be beneficial for teachers and researchers to have a specific focus on concepts that best lend themselves to be developed through modelling.