What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Methods <s> Abstract Qualitative data analysis was used to investigate the impact of prior knowledge of task context on senior secondary students' approaches to applications tasks. Prior knowledge was classified according to its source, namely (a) vicarious experiences in other academic subject areas, (b) general encyclopaedic knowledge of the world, or (c) truly experiential knowledge developed from personal experiences outside school or in practical school subjects. These have been referred to as (a) academic, (b) encyclopaedic, and (c) episodic prior knowledge, respectively. This classification proved useful in examining the nature of the prior knowledge reported, which was task rather than student specific. The study highlights the largely unpredictable differential nature of the effects of prior knowledge on student solutions. However, episodic prior knowledge was found to be more likely to have a positive influence than academic or encyclopaedic. It is suggested that the salience of cues, such as problem constraints, is mediated by how knowledge is acquired with experiential knowledge derived through observation, proving less effective than that derived through action. <s> BIB001 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Methods <s> The research reported here describes similarities and differences between (a) modeling cycles that students typically go through during 60–90 min solutions to a class of problems thast we refer to as model-eliciting activities, and (b) stages of development that students typically go through during the “natural” development of constructs (conceptual systems, cognitive structures) that cognitive psychologists consider to be relevant to these specific problems. Examples of relevant constructs include those that underlie children’s developing ways of thinking about fractions, ratios, rates, proportions, or other elementary, but deep mathematical ideas. Results show that, when problem solvers go through an iterative sequence of testing and revising cycles to develop productive models (or ways of thinking) about a given problem solving situation, and when the conceptual systems that are needed are similar to those that underlie important constructs in the school mathematics curriculum, then these modeling cycles often appear to be local or situated versions of the general stages of development that developmental psychologists and mathematics educators have observed over time periods of several years for the relevant mathematics constructs. Furthermore, the processes that contribute to local conceptual development in model-eliciting activities are similar in many respects to the processes that contribute to general cognitive development. Applying principles from developmental psychology to problem solving—and vice versa—is a relatively new phenomenon in mathematics education (Lesh & MATHEMATICAL THINKING AND LEARNING, 5(2&3), 157–189 Copyright © 2003, Lawrence Erlbaum Associates, Inc. <s> BIB002 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Methods <s> This chapter reports on the development of test instruments for the assessment of modelling skills. It summarises the progress made in the last few years and gives a rationale for the latest test used with students at the University of Ulster at the beginning of the academic year 2002/03. Results obtained from testing undergraduates in the honours course in Mathematics, Statistics and Computing are analysed and inferences are drawn about how students understand the various stages in the modelling cycle. <s> BIB003 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Methods <s> In this paper interim findings of an empirical study on the effects of context are presented. The study focusses on the question how upper secondary students deal individually with contextual aspects. This research project is based on a qualitative approach Triangulation of methods is applied in order to get a broader access to the field. It becomes clear that the context is neither an objective nor an invariable feature of the task. Studies deal very individually with the context, and it can be an object of change during the solving process. Four types of dealing with the context are gained from the analysis of the empirical data. These types can be embedded in and explained by the concept of sociomathematical norms and the theory of situated learning. <s> BIB004 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Methods <s> This paper focuses on notions of function Year 9 students hold as they begin to study functions. As these notions may be fragile, the questions, tasks, and ways of interacting orchestrated by the teacher to elicit depth of understanding, or allow observation of changing notions, are of interest. Extended tasks where students were required to make choices about solution paths provided opportunities for students to develop and consolidate their concept images. Discussion between small groups provided the best evidence of developing and stable conceptions held by students in contrast to written scripts where the strength of these understandings was not evident. <s> BIB005 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Methods <s> Two sets of students with weak content skills and no prior experience were introduced to mathematical modelling. Contrasting instructional approaches were used — exposure to a variety of models, versus constructing the model themselves. Both groups demonstrated understanding of aspects of modelling and model structures. The students actively modelling the situation statistically out-performed the other group on four evaluation parameters. <s> BIB006 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Methods <s> This study reports on an experimental teaching program to foster students’ thinking to promote modelling which consisted of nine lessons for Japanese 9 th grade students. In this program, the following three teaching principles were emphasized;(TP1) Conflicting Situations: The teacher presents meaningful conflicting situations arising from particular modelling problem so that students can derive key ideas from these situations, (TP2) Repeated Connections: he teacher makes repeated connections between students’ thinking that promotes modelling and (TP3) Spiral Reflections: The teacher encourages students to reflect on their thinking throughout the series of nine lessons. As a pre- and post-test, the (PISA) Heartbeat problem and a general question on modelling were used before and after the teaching program to examine what transfigurations or shifts in students’ knowledge of modelling can be observed. This general question had the following sub-questions: What processes are required to build up a functional model? What ideas are important for developing an effective functional model? A graduated coding analysis was applied to assess students’ responses to the Specific (Heartbeat) question and to the General question. A graduated coding analysis was applied to assess students’ responses to the Specific (Heartbeat) question and to the General question. Significant gains in students` post-test codes/scores on both questions were obtained. This and the resulting discussion provide strong evidence that a teaching program based on the above three teaching principles is substantially effective in fostering students’ thinking to promote modelling. <s> BIB007 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Methods <s> This paper presents an extensive overview of preschool to 12th grade (P-12) engineering education literature published between 2001 and 2011. Searches were conducted through education and engineering library engines and databases as well as queries in established publications in engineering education. More than 50 publications were found, including books, articles, and reviews that discussed P-12 engineering education research efforts. With the synthesis of literature and distinct criteria to classify the literature, a rubric was developed. This rubric allowed authors to synthesize retrieved publications in seven ways: (1) overarching agenda, (2) nature of the engineering program or intervention, (3) assessment methods, (4) object of the study/unit of analysis, (5) population/ sample of interest, (6) informing theory, and (7) standards addressed. Discussions of the current levels of research based on recommendations of National Agencies are presented along with a set of recommendations for the advancement of the P-12 engineering education research. <s> BIB008 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Methods <s> This chapter explores an emerging framework on mathematical models and modelling using the theoretical perspective of commognition to analyse and discern discursive objects in a dialogue between two students engaged in a modelling activity. The results, partly presented as realization trees, show a variety of signifiers from different discourses coming into play during the modelling, and examples are given of the activity of negotiation, which plays an important role in any modelling activity. In addition, it is argued that the framework has potential to bridge different research perspectives on mathematical models and modelling. <s> BIB009
The results of this paper stem from a literature search for empirical studies involving middle and high school mathematical modelling. Articles that were theoretical or that described mathematical modelling activities without description of a research study connected with the activities were not included. Adequate papers were identified through a targeted and a general literature search. The targeted literature search involved looking at the edited books arising from the International Community of Teachers of Mathematical Modelling and Applications (ICTMA) conferences; as well as the conference proceedings of the thematic working groups in the Congresses of the European Society for Research in Mathematics Education (CERME). Also looked at was the International Commission on Mathematical Instruction (ICMI) 14 study on modelling and applications in mathematics education. Using a method similar to BIB008 , who summarized engineering education articles, all empirical studies that involved mathematical modelling in middle and high school that also meet the essential elements of mathematical modelling were summarized using the categories of assessment data collected, unit of analysis studied, population, and effectiveness. There were sixty-seven publications identified. Table 1 describes the number of articles identified from each journal, book, or conference. Once this was completed, each category was quantitatively summarized by looking at similarities in assessment data, unit of analysis or population. Areas of future research were identified throughout the process of organizing and categorizing the literature review. BIB009 . With multiple MEAs BIB002 used Piaget's stages of development for proportional reasoning to show how students can progress through the stages during one modelling session. A couple of studies used multiple frameworks.  used the Piagetian notion of abstraction (Piaget, Inhelder, & Szeminksa, 1960) ,  conceptual framework of proof scheme, and  outline of local conceptual development. Brown and Edwards (2011) also used multiple frameworks: BIB004 four types for how students deal with reality and mathematics, BIB001 classifications of prior knowledge as academic, encyclopaedic, or episodic, and BIB005 higher order thinking framework. Rubrics were also used to assess students' mathematical modelling competencies and effectiveness of the steps of the modelling process. Biccard and Wessels (2011) used a scale from 0 to 3 to assess modelling competencies from students' work on complete modelling problems. BIB007 used a rubric with individual questions targeted at specific steps of the modelling process. Another rubric involved twenty performance goals for the assessment of modelling and used a scale of 0 to 5 for each performance goal BIB006 . One of the goals did not seem to fit with mathematical modelling, stay with the plan previously made; as modelling often involves iterations that can involve students trying a new approach or idea. Students also may proceed done on an unproductive path and in these cases a new plan should be tried. Dunne and Galbraith (2003) also used a rubric to assess students' steps of the modelling process but used multiple complete modelling problems to do this. MaaB and Mischo (2011) combined both multiple choice questions targeted at specific steps of the modelling process along with an open-ended complete modelling problem to assess students' effectiveness of the modelling process. The complete modelling problem was scored based on a rubric for each step of being not comprehensible, partly comprehensible, and comprehensible. A couple of tests were used to measure modelling competencies. MaaB (2006) used a ten task modelling pre test and a slightly modified post-test version that was designed to assess all the sub-competencies of modelling. Another test that assessed sub-competencies involved eight multiple choice items. Each question asked for the next appropriate step at a specific phase in a modelling problem . This test was based on existing modelling tests designed and reworked by  and BIB003 .