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A study with prospective teachers without prior mathematical modeling experience sheds light on how their newly developed conceptual understanding of modeling manifested itself in their work on the final task of a modeling module within a pedagogy course in secondary mathematics curriculum and assessment. The main purpose of the module was to provi...
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... We classified the contextualization of their model results as high level or low level of proficiency displayed. A low level of proficiency in contextualization refers to limiting the interpretation of the mathematical results to a description of the model or its variables with little connection to the situation context and without discussing the implications of the results. We coded one prospective teacher's contextualization as incomprehensible. A high level of proficiency in contextualization discusses connection to the situation context and the implications of the results on the original problem. Example cases are displayed in Figure ...
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... In order to improve students' mathematical literacy, researchers are increasingly focusing on students' mathematical modeling skills (Anhalt et al. 2018;Schukajlow et al. 2018). Mathematical modeling skills are defined as the capacity to comprehensively apply mathematical knowledge and methods to transform real-world problems into mathematical models for solution and validation (Blum and Ferri 2009;Blomhøj and Jensen 2003;Kaiser et al. 2006). ...
This study was to investigate the relationship between metacognition and the mathematical modeling skills of high school students, as well as the mediating role of computational thinking. A cluster sampling method was adopted to investigate 661 high school students, using the metacognition scale, computational thinking scale, and mathematical modeling skill test questions. The results showed that metacognitive knowledge and metacognitive monitoring had a direct and positive correlation with high school students’ mathematical modeling skills. Additionally, the critical thinking dimension of computational thinking mediated the relationship between metacognitive knowledge, experience, monitoring, and mathematical modeling skills. These findings indicated that sufficient metacognition could improve the critical thinking of high school students’ computational thinking and enhance their mathematical modeling skills.
... Kaiser, 2007), undergraduate students (e.g. Haines et al., 2001), and teachers (Anhalt et al., 2018;Aydin-Güc & Baki, 2019;Durandt & Lautenbach, 2020). When focusing on PSTs (e.g. ...
... When focusing on PSTs (e.g. Aydin-Güc & Baki, 2019; Haines et al., 2001;Kaiser, 2007), these studies did not explicitly investigate changes but described PSTs' competencies at one point in their projects, such as their engagement with a holistic modelling task (Anhalt et al., 2018;Govender, 2020), or word problems (Winter & Venkat, 2013). ...
This study draws on quantitative reasoning research to explain how secondary mathematics preservice teachers’ (PSTs) modelling competencies changed as they participated in a teacher education programme that integrated modelling experience. Adopting a mixed methods approach, we documented 110 PSTs’ competencies in Vietnam using an adapted Modelling Competencies Questionnaire. The results show that PSTs improved their real-world-problem-statement, formulating-a-model, solving-mathematics, and interpreting-outcomes competencies. Showing their formulating-a-model and interpreting-outcomes competencies, PSTs enhanced their quantitative reasoning by properly interpreting the quantities and their relationships using different representations. In addition, the analysis showed a statistically significant correlation between PSTs’ modelling competencies and quantitative reasoning. Suggestions for programme design to enhance modelling competencies are included.
... To reiterate, much of the work on posing mathematical modeling problems is situated in elementary-level contexts. Although research has studied how secondary PSTs and undergraduate mathematics majors solve modeling tasks [e.g., 4,43] and reflect on this experience as future educators [3], few studies have explored how undergraduate mathematics majors and secondary PSTs develop their own mathematical modeling lessons, and particularly how they attend to issues of social justice (or not) in their lessons. ...
... I et al.'s [21] framework does include one criterion that does not align with Galbraith's [15] design principles: a Shareable Approach. This criterion aligns well with a mathematical modeling competency, reporting out, which is one of seven modeling competencies within the modeling process [4,42]. I et al. [21] provide a criterion for differentiating mathematical modeling problems that address a social justice issue from other tasks: "the context involves unjust situations of the real world and encourages learners to be an agent of change by identifying mathematical conflicts and resolving conflicts" (page 891). ...
... Numerous studies have shown that mathematical modeling is challenging for many students (Anhalt et al., 2018;Corum and Garofalo, 2019;Czocher, 2017;Kannadass et al., 2023). Metacognitive competencies improve students' modeling abilities (Galbraith, 2017;Vorhölter, 2019;Wendt et al., 2020). ...
... Integrating mathematical modeling across subject areas can give students a more meaningful and context-rich understanding of mathematics. Numerous studies have shown that many students find mathematical modeling difficult and complex (Anhalt et al., 2018;Corum and Garofalo, 2019;Czocher, 2017). For example, some students have difficulty translating real-world problems into mathematical terms, while others have difficulty finding appropriate mathematical models to represent complex systems and phenomena. ...
Mathematical modeling is indeed a versatile skill that goes beyond solving real-world problems. Numerous studies show that many students struggle with the intricacies of mathematical modeling and find it a challenging and complex task. One important factor related to mathematical modeling is metacognition which can significantly impact expert and student success in a modeling task. However, a notable gap of research has been identified specifically in relation to the influence of metacognition in mathematical modeling. The study’s main goal was to assess whether the different sub-dimensions of metacognition can predict the sub-constructs of a student’s modeling competence: horizontal and vertical mathematization. The study used a correlational research design and involved 538 participants who were university students studying mathematics education in Riau Province, Indonesia. We employed structural equation modeling (SEM) using AMOS version 18.0 to evaluate the proposed model. The measurement model used to assess metacognition and modeling ability showed a satisfactory fit to the data. The study found that the direct influence of awareness on horizontal mathematization was insignificant. However, the use of cognitive strategies, planning, and self-checking had a significant positive effect on horizontal mathematization. Concerning vertical mathematization, the direct effect of cognitive strategy, planning, and awareness was insignificant, but self-checking was positively related to this type of mathematization. The results suggest that metacognition, i.e., awareness and control over a person’s thinking processes, plays an important role in modeling proficiency. The research implies valuable insights into metacognitive processes in mathematical modeling, which could inform teaching approaches and strategies for improving mathematical modeling. Further studies can build on these findings to deepen our understanding of how cognitive strategies, planning, self-assessment, and awareness influence mathematical modeling in both horizontal and vertical contexts.
... Hankeln (2020) found that some students encountered significant barriers when they were missing information needed to solve the problem and put a lot of time and effort toward finding out what information they needed to solve the problem. The barriers occurred not only at the beginning but also later in the solution process when students were setting up a mathematical model or while they were validating the model and results (Anhalt et al., 2018;Czocher, 2018). These difficulties are typical of novices, who tend to apply the mathematical procedures immediately. ...
... Some students identify a quantity that requires them to make an assumption but do not proceed further in the solution process, or they make inappropriate assumptions . For example, while solving a problem that required the growth of tree leaves to be modeled, preservice teachers used inadequate assumptions that led to unrealistic solutions by assuming that the leaves fall off a tree after exactly 87 days (Anhalt et al., 2018). One important reason for inadequate numerical assumptions is a lack of knowledge about the real-world phenomenon described in the task (Krawitz, 2020). ...
... Positive effects of instructional prompts indicate that difficulties in identifying unknown quantities and making realistic numerical assumptions are considerable barriers that students face when solving open modelling problems. These results explain students' difficulties in making assumptions observed in prior studies, which found out that some students do not notice the openness, some students notice it but conclude that such problems cannot be solved, and other students make unrealistic assumptions (Anhalt et al., 2018;Chang et al., 2020;Dewolf et al., 2017;Galbraith & Stillman, 2001;Hankeln, 2020;Reusser & Stebler, 1997). Providing students with instructional prompts Content courtesy of Springer Nature, terms of use apply. ...
Open mathematical modelling problems that can be solved with multiple methods and have multiple possible results are an important part of school curricula in mathematics and science. Solving open modelling problems in school should prepare students to apply their mathematical knowledge in their current and future lives. One characteristic of these problems is that information that is essential for solving the problems is missing. In the present study, we aimed to analyze students’ cognitive barriers while they solved open modelling problems, and we evaluated the effects of instructional prompts on their success in solving such problems. A quantitative experimental study (N = 263) and a qualitative study (N = 4) with secondary school students indicated that identifying unknown quantities and making numerical assumptions about these quantities are important cognitive barriers to solving open modelling problems. Task-specific instructional prompts helped students overcome these barriers and improved their solution rates. Students who were given instructional prompts included numerical assumptions in their solutions more often than students who were not given such prompts. These findings contribute to theories about solving open modelling problems by uncovering cognitive barriers and describing students’ cognitive processes as they solve these problems. In addition, the findings contribute to improving teaching practice by indicating the potential and limitations of task-specific instructional prompts that can be used to support students’ solution processes in the classroom.
... Durandt et al. [7] wrote that structuring a model helps students understand real problems. However, some countries, including Malaysia, struggle to implement mathematical modelling competencies [1], [8], [9]. Prospective teachers were trained to solve word problems with correct answers [8]. ...
... However, some countries, including Malaysia, struggle to implement mathematical modelling competencies [1], [8], [9]. Prospective teachers were trained to solve word problems with correct answers [8]. However, they struggled to create a winning strategy due to their mathematical modelling ignorance. ...
... Making rational statements about mathematics is another subconstruct in this study that had the lowest value, which is consistent with Leong and Tan's [1] assertion that study participants had difficulty making rational statements about how to solve problems, make assumptions, and use maths to solve problems. Anhalt [8] explains this by defending the claim that a cohort of pre-service mathematics teachers was prepared to deal with answer-dependent word problems. Regrettably, their inadequate comprehension of mathematical modelling hindered their ability to formulate an effective strategy. ...
Participation in modelling activities significantly facilitates the development of mathematical skills. By utilizing the concept of mathematical modelling, students may be able to develop a more grounded understanding of mathematics. The objective of this research was to explore how computational thinking and critical thinking are connected to the mathematical modelling proficiency of pre-service teachers. Correlational quantitative research was conducted on 140 pre-service mathematics teachers from the Institute of Teacher Education, Penang and the Institute of Teacher Education, Ipoh, using a correlational research design. Using cluster random sampling, the Institute of Teacher Education was selected at random. The result revealed that pre-service mathematics teachers exhibited a strong aptitude for computational and critical thinking, but demonstrated a limited level of proficiency in mathematical modelling. In terms of modelling proficiency, the results indicated a significant correlation between computational thinking and critical thinking.
... The individual's previous life and scholastic experiences (Matsuzaki, 2011;Thompson & Yoon, 2007) 2. The individuals have different thinking styles Borromeo Ferri, 2010; 3. Having experience in mathematical modeling (Author, 2018) 4. The individuals' experiencing some difficulties in the modeling process (inability to understand the situation, inability to construct the real model, inability to pose the mathematical model) (Blum & Leiß, 2007;Galbraith & Stillman, 2001) Upon examination of the studies, it was revealed that students encountered challenges in several areas. Specifically, they experienced difficulties understanding the task, constructing a real model by defining the pertinent variables of the situation and making assumptions, generating a mathematical model, conducting mathematical calculations, and verifying the models (Abay & Gökbulut, 2017;Anhalt, Cortez and Bennett, 2018;Bukova Guzel, 2011;Deniz & Akgün, 2018;Deniz & Yıldırım, 2018;Galbraith & Stillman, 2006;Maaß, 2006;Schaap, Vos & Goedhard, 2011;Tekin Dede, 2016). Especially, it was observed that many pre-service teachers have a habit of solving problems without creating a model (Özer & Bukova-Güzel, 2020). ...
... However, their level of competency differed. Similarly, Anhalt et al. (2018) determined that PTSs were generally able to recognize the needed assumptions and variables in the modeling cycle, but the level of proficiency they displayed varied. The PSTs who progressed to the real model and completed the process without creating a mathematical model took place in the first cycle. ...
... That the students who were unsuccessful in the modeling process could not make a connection between the real world and the world of mathematics and could not switch to the world of mathematics was also determined by Ji (2012). Since PSTs defined the assumptions and variables as more or less restrictive than needed (Anhalt et al., 2018), they might not create a mathematical model. ...
This study aims to explore the modelling cycles that emerge during the pre-service teachers’ mathematical modelling activities. A case study method was employed in the research. 119 pre-service teachers who were in their fourth class participated in the research. The pre-service teachers worked in groups. So, there were a total of 28 different groups. The modelling task was posed by considering the modelling criteria. The data collection tools consisted of the pre-service teachers’ working papers. The content analysis method was applied in the data analysis. The preservice teachers created four different modelling cycles. The first consisted of 7% of the groups and was the cycles that included the pre-service teachers who could reach the real model. The second was the cycle that consisted of 68% of the groups and included the pre-service teachers who could reach the mathematical results from the real model without posing any mathematical model. The third was the cycle that consisted of 7% of the groups and included the pre-service teachers who completed the process by reaching the mathematical model. The fourth was the cycle that consisted of 18% of the groups and included the pre-service teachers who completed the modelling cycle. It was determined that the cycle in the second group occurred the most among the modelling cycles. The pre-service teachers can be supported to pass the third and fourth modelling cycles.
... ✘ Go through a cycle of posing questions, building solutions, and validating their conclusions (Carlson et al., 2016) ✘ Direct their activities to come to a valid solution ✘ Make decisions that support their goal and appropriately model the situation ✘ Reflect upon their decisions to better understand the implications and refine their models (Maiorca & Stohlmann, 2016) Benefits of Mathematical Modeling ✘ Help students see the importance of mathematical understanding (Blum & Ferri, 2016) ✘ Deepen understanding of mathematical concepts (Anhalt et al., 2018) ✘ Raise student interest and build awareness of real-world applications (Cirillo et al., 2016) ✘ Bolster problem-solving skills, encourage creative and critical thinking (Anhalt et al., 2018;Gann et al. , 2016) ✘ Opportunities to express, modify, and refine ways of thinking (Lesh et al., 2003) ✘ Potential to incorporate additional STEM content and Standards for Mathematical Practice (Maiorca & Stohlmann, 2016) Participant Reflections on Modeling Benefits "The benefits of mathematical modeling allows the students to discuss and create questions and ideas that are realistic and feasible. It allows an academic freedom for the students to get engaged into the discussion." ...
... ✘ Go through a cycle of posing questions, building solutions, and validating their conclusions (Carlson et al., 2016) ✘ Direct their activities to come to a valid solution ✘ Make decisions that support their goal and appropriately model the situation ✘ Reflect upon their decisions to better understand the implications and refine their models (Maiorca & Stohlmann, 2016) Benefits of Mathematical Modeling ✘ Help students see the importance of mathematical understanding (Blum & Ferri, 2016) ✘ Deepen understanding of mathematical concepts (Anhalt et al., 2018) ✘ Raise student interest and build awareness of real-world applications (Cirillo et al., 2016) ✘ Bolster problem-solving skills, encourage creative and critical thinking (Anhalt et al., 2018;Gann et al. , 2016) ✘ Opportunities to express, modify, and refine ways of thinking (Lesh et al., 2003) ✘ Potential to incorporate additional STEM content and Standards for Mathematical Practice (Maiorca & Stohlmann, 2016) Participant Reflections on Modeling Benefits "The benefits of mathematical modeling allows the students to discuss and create questions and ideas that are realistic and feasible. It allows an academic freedom for the students to get engaged into the discussion." ...
This presentation describes the design and implementation of two sessions for pre-service teachers on mathematical modeling. Math modeling, while a standard for mathematical practice, isn’t always a main focus of pre-service teacher education. Our research seeks to share the design choices for our sessions, the benefits of mathematical modeling for engaging students in mathematics, and suggestions for design of future sessions.
... On the other hand, even though modelling courses or modules are beneficial for developing learners' certain modelling competencies and skills (e.g. Anhalt et al., 2018;Cetinkaya et al., 2016;Kertil et al., 2019;Suh & Seshaiyer, 2019), it is hard if not possible, to develop students' modelling competencies in a single course at the university level, and thus making mathematical modelling an integral part mathematics education at both tertiary and pre-tertiary levels is needed. ...
This study investigated the sources of difficulties experienced by prospective teachers in mathematical modelling in the context of a semester-long course on mathematical modelling for teachers. The data reported in this study were collected from six prospective teachers through in-depth semi-structured interviews supported by audio and videotaped classroom observations and prospective teachers’ written solutions for the modelling tasks. The results showed that the sources of difficulties experienced by prospective teachers in mathematical modelling could be subsumed under individual and contextual factors. The individual factors included four issues: lack of conceptual understanding of mathematics, the difficulty of connecting the real world and the mathematical world, the disposition to focus solely on reaching a result, and disorganized and unsystematic work. The contextual factors included two issues: insufficient previous experience with modelling tasks and time limitations. Some implications of these results are discussed.
https://www.tandfonline.com/eprint/YM8MHS99R6AYSUC9NDXC/full?target=10.1080/0020739X.2023.2171922
... 117). Maaß and others (e.g., Anhalt et al., 2018;Kaiser & Brand, 2015) categorize MM competencies in terms of phases of the MM cycle. ...
... Professional learning opportunities in MM for in-service teachers can be found in schools, at professional conferences, and in informal settings. Recent studies suggest that completing a short MM module or a teacher preparation course can offer teachers opportunities to experience and deepen foundational understandings of the MM process (Anhalt & Cortez, 2016;Anhalt et al., 2018;Cetinkaya et al., 2016) and to develop foundational ideas for teaching MM (Cetinkaya et al., 2016). ...
... For example, teacher noticing (Sherin et al., 2011), use of student thinking (Czocher, 2019;Franke et al., 2015;Staples, 2007;Webb et al., 2008), and facilitation of productive struggle (Warshauer, 2015) are parts of Attention to Students. How teachers learn to model (Anhalt & Cortez, 2016;Anhalt et al., 2018;Besser et al., 2015;Centinkaya et al., 2016) constitutes MM as Mathematics. Social Aspects of Working Together includes facilitation of group work (Borromeo Ferri, 2018), including both social and analytic scaffolding (Baxter & Williams, 2010). ...
This study explores secondary mathematics teachers’ perceptions of the experiences that contributed to their capacities to understand mathematical modeling and to facilitate students’ modeling experiences. The retrospective research methods and transformative learning theory frame used in the study honor teachers as adult learners and value their perspectives while providing a way to study the complexity of learning to model and to teach modeling. Data analysis identified triggers and knowledge dilemmas that challenged and prompted teacher learning as well as opportunities to resolve dilemmas through rational discourse and critical reflection. Patterns in teacher-identified meaningful learning experiences reveal a trajectory with strands that address aspects of doing and teaching mathematical modeling: mathematics, social aspects of learning, real-world contexts, student thinking, and curriculum. Results of this study provide a holistic view of learning to do and teach mathematical modeling, complementing studies of designed professional learning interventions that out of necessity target specific parts of the modeling process. The results both support and challenge common teacher education content and practices. The study illustrates the usefulness of retrospective methods to understand teachers as lifelong learners.
