What Is Known about Secondary Grades Mathematical Modelling --A Review <s> modeling cycle <s> In this paper uses an analytical tool refereed to as the MAD (Modelling Activity Diagram) framework adapted from Schoenfeld's parsing protocol coding scheme to address the issues of how to introduce mathematical modelling to upper secondary students. The work of three groups of students engaged in solving so called realistic Fermi problems were analysed using this framework, and it was observed that the processes involved in a typical mathematical modelling cycle were richly represented in the groups' solving processes. The importance of the social interactions within the groups was noted, as well as the extensive use of extra- mathematical knowledge used by the students during the problem solving session. <s> BIB001 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> modeling cycle <s> In literature about mathematical modeling a diversity can be seen in ways of presenting the modeling cycle. Every year, students in the Bachelor’s program Applied Mathematics of the Eindhoven University of Technology, after having completed a series of mathematical modeling projects, have been prompted with a simple three-step representation of the modeling cycle. This representation consisted out of 1) problem translation into a mathematical model, 2) the solution to mathematical problem, and 3) interpretation of the solution in the context of the original problem. The students’ task was to detail and complete this representation. Their representations also showed a great diversity. This diversity is investigated and compared with the representations of the students’ teachers. The representations with written explanations of 77 students and 20 teachers are analyzed with respect to the presence of content aspects such as problem analysis, worlds/models/knowledge other than mathematical, verification, validation, communication and reflection at the end of the modeling process. Also form aspects such as iteration and complexity are analyzed. The results show much diversity within both groups concerning the presence or absence of aspects. Validation is present most, reflection least. Only iteration (one is passing the modeling cycle) more than once is significantly more present in the teachers’ group than in the students’ group. While accepting diversity as a natural phenomenon, the authors plea for incorporating all aspects mentioned into mathematical modeling education. <s> BIB002 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> modeling cycle <s> Mathematical modelling has often been emphasized at the secondary level, but more research is needed at the elementary level. This paper serves to summarize what is known about elementary mathematical modelling to guide future research. A targeted and general literature search was conducted and studies were summarized based on five categories: content of mathematical modelling intervention, assessment data collected, unit of analysis studied, population, and effectiveness. It was found that there were three main units of analysis into which the studies could be categorized: representational and conceptual competence, models created, and student beliefs. The main findings from each of these units of analysis are discussed along with future research that is needed. <s> BIB003
Another essential element is the inclusion of key questions that stem from the real world problem. A key question can guide the solution and work of a mathematical modelling activity. The key question provides direction for the problem to be solved. An example of a key question is, how big is someone based on his or her footprint and stride length?  . A key question can serve to focus work and is often the way that people approach problems in their jobs. BIB002 cycle has a similar structure to  cycle but provides more detail and highlights three more essential elements of mathematical modelling. Both clear verbal and written communication is paramount while students work on a mathematical modelling task and detail their solution. Students should also reflect on the modelling process in order to make explicit the mathematics that they used and how well they understood it. This cycle also highlights the iterative nature of mathematical modelling. Revisions of a solution should take place and students may progress in different ways through the modelling cycle before developing an adequate solution. For this to occur modelling activities should be open-ended. A modelling cycle that appears often in the literature is from and connects to several essential elements. Again there is a distinction between the real world and mathematics. It can be seen that students must make sense of the problem with mathematics and ensure that the model developed makes sense in the realistic context. Though not shown we know that the modelling process is iterative in nature (e.g. BIB001 ). In summary, there are seven essential elements of mathematical modelling: (a) start with a real world problem, (b) work from key questions (c) make sense of the problem with mathematics often involving assumptions and approximations, (d) ensure the mathematics is accurate and makes sense in the realistic situation, (e) goal of clear verbal and written communication throughout often including multiple representations, (f) modelling is an iterative process that involves open-ended problems, and (g) reflection on mathematics used or the modelling process BIB003 . There are different theoretical perspectives for mathematical modelling-realistic, contextual, educational, socio-critical, epistemological, and cognitive (Kaiser & Sriraman, 2006) and the essential elements are common ground for the different persepectives. The main mathematical modelling theoretical perspective for contextual modelling, The Models and Modelling Perspective, is discussed more in the following section because the most frequent modelling curricula that was used in the articles were Model-Eliciting Activities that are based on the Models and Modelling Perspective.