What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Students' Beliefs <s> This paper reports on a project that incorporated mathematical modelling within a pre-university subject at a secondary college. The program emphasised modelling rather than models and extended over two years. A qualitative methodology was used to evaluate the program with information collected in a variety of forms including student written reports, oral interviews, video reports, diaries, questionnaire data, as well as informal observation. The program was evaluated in terms of five criteria derived from the stated aims. Erom the synthesis and triangulation of the data it was inferred that the program aims were achieved to an encouraging degree. The paper describes and illustrates key aspects of the program and its evaluation. <s> BIB001 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Students' Beliefs <s> Attention to the beliefs of teachers and teacher candidates should be a focus of educational research and can inform educational practice in ways that prevailing research agendas have not and cannot. The difficulty in studying teachers’ beliefs has been caused by definitional problems, poor conceptualizations, and differing understandings of beliefs and belief structures. This article examines the meaning prominent researchers give to beliefs and how this meaning differs from that of knowledge, provides a definition of belief consistent with the best work in this area, explores the nature of belief structures as outlined by key researchers, and offers a synthesis of findings about the nature of beliefs. The article argues that teachers’ beliefs can and should become an important focus of educational inquiry but that this will require clear conceptualizations, careful examination of key assumptions, consistent understandings and adherence to precise meanings, and proper assessment and investigation of spec... <s> BIB002 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Students' Beliefs <s> The paper reports on University seminars of mathematical modelling at school, that were held together by the departments of mathematics and mathematics education and some schools in Hamburg. Prospective teachers together with students in upper secondary level carried out modelling examples either in ordinary lessons or special afternoon groups. They tackled authentic problems proposed by applied mathematicians working in industry. In this paper we consider three of these examples in more detail and describe students' attempts at solving the problem. <s> BIB003 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Students' Beliefs <s> The paper deals with the gap between the relevance of applications and modelling in didactical discussions and its minor importance in everyday mathematics teaching. Results of our own empirical studies that describe mathematical beliefs of teachers and students as being central obstacles are presented. Further, the studies demonstrate the possibility to change these beliefs as well as ways to promote modelling competencies. <s> BIB004 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Students' Beliefs <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> BIB005 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Students' Beliefs <s> Modeling and applications are regarded as important components of mathematics education within the didactical discussion. Little however is known about the impacts of integrating modeling and application into day-to-day teaching practice. A qualitative study has been carried out which investigated the effects on students during a 15 months intervention. This paper deals with the effects on student’s beliefs. <s> BIB006 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Students' Beliefs <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> Students' Beliefs <s> In this study which was part of the DISUM-project, 224 ninth graders from 14 German classes from middle track schools (Realschule) were asked about their enjoyment, interest, value and self-efficacy expectations concerning three types of mathematical problems: intra-mathematical problems, word problems and modelling problems. Enjoyment, interest, value and self-efficacy were assessed before and after a ten-lesson teaching unit promoting modelling competency related to the topics “Pythagoras’ theorem” and “linear functions”. The study aimed to answer the following research questions: (1) Do students’ enjoyment, value, interest and self-efficacy expectations differ depending on the type of task? (2) Does the treatment of modelling problems in classroom instruction influence these variables? (3) Are there any differential effects for different ways of teaching modelling problems, including a “directive”, teacher-centred instruction and an “operative-strategic”, more student-centred instruction emphasising group work and strategic scaffolding by the teacher? The findings show that there were no differences in students’ enjoyment, interest, value and self-efficacy between the three types of tasks. However, teaching oriented towards modelling problems had positive effects on some of the student variables, with the student-centred teaching method producing the most beneficial effects. <s> BIB008 </s> What Is Known about Secondary Grades Mathematical Modelling --A Review <s> Students' Beliefs <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> BIB009
Twelve studies looked at students' beliefs in regards to mathematical modelling and applications, mathematics, and mathematics teaching. The research has found that students have mostly positive views towards mathematical modelling after multiple mathematical modelling experiences BIB007 Dunne & Galbraith, 2003; BIB004 BIB006 . Two classes of German students, age 13-14, participated in six modelling units over 15 months time. Students with an application oriented or dynamic belief system had positive attitudes towards modelling while students with a static or more formalism-oriented belief system rejected mathematical modelling BIB004 . BIB007 used five modelling tasks with one class of 14-15 year old German students and found that boys and girls enjoyed modelling, but girls' interest decreased slightly over the five modelling tasks. The context of the modelling problems could have affected this. In a study of several hundred age 14-15 year old German students paired with university students as mentors it was found that students had mainly positive views on mathematical modelling. The students found the tasks to be well-structured, realistic, difficult, and interesting . Studies that have looked at students' beliefs about mathematics after mathematical modelling experiences have shown mainly that students come to see mathematics as more real life and applicable BIB001 BIB003 BIB006 BIB009 . Four hundred and sixty 7 th -9 th grade Japanese students participated in a problem to model the population of Bluefin tuna. After the activity the students realized that mathematics is helpful for the real world BIB009 . In Germany, two-hundred eighty nine students age 16-18 completed a 1 week modelling problem and became more aware of the relevance of mathematics in their life . In contrast a year long study with a modelling treatment class and a traditional control class of 12 year old students in Germany found little change in beliefs about mathematics for either class. The beliefs about mathematics that were measured by a Likert survey  were related to enjoyment, motivation, importance, and freedom from fear. The total scores could range from 24 (positive) to 72 (negative). The pre-post means for the treatment class were 36.7 and 35.8; while it was 30.8 and 31.1 for the control class. Little information is given about the modelling intervention or the teacher's experience with modelling (Dunne & Galbraith, 2003) . Another study that was done in Germany with 224 ninth grade students used a pre and post Likert survey to assess the students' beliefs on enjoyment, value, interest, and self-efficacy for modelling problems, word problems, and mathematics with no real world context. It was found there was no difference in students' beliefs for the three problem types. However, the study lasted only two weeks and all three problem types were done during the two weeks, not solely modelling BIB008 . Two studies that were done outside of the regular school setting investigated if students would like to have mathematical modelling included in their regular mathematics classes. found that students wanted to have mathematical modelling included in their regular school teaching. Another study with the same lead author found that students felt that the skills they learned during the mathematical modelling experience could be used in many fields. However, while the majority of students wanted modelling in their regular mathematics classes, some students did not because they felt it was too time consuming . Students' beliefs are an important construct as beliefs are the best indicators of the decisions that individuals make throughout their lives BIB002 . Realistic contexts of modelling activities can lead to student engagement but more research is needed on the realistic contexts that positively engage a diverse range of students BIB005 . A few questions that can be researched connected to beliefs include: What is the potential of mathematical modelling to support both students and teachers in their development of appropriate beliefs about and attitudes towards mathematics? How do attitudes and beliefs affect the models that students create? This is important because when students design models, they tend to do more than simply engage logical-mathematical systems. They also engage feelings, values, and beliefs making these integral parts of models (English, Lesh, & Fennewald, 2008) .