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Making realistic assumptions is an important part of solving open modelling problems and also a potential source of errors. But little is known about the difficulties that result from the openness of modelling problems and how they can be addressed in interventions. Here, we focus on two central solution steps that are necessary for making assumpti...
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Context 1
... mathematics classrooms, modelling problems are used to foster students' modelling competence. Figure 1 presents an example of a modelling problem. A characteristic feature of modelling problems is their openness as they often do not include all of the necessary information. ...
Context 2
... solve open modelling problems, two different solution steps are necessary ( Krawitz et al., 2018): First, students need to notice the openness of the problem, and second, they have to estimate the missing quantities. For example, in the Speaker problem (Figure 1), students need to notice that the diameter of the speaker has to be taken into account and replace the missing quantity with an estimate (e.g., about 5 cm because, in the picture, the diameter is about one fourth of the height). Prior modelling research has shown that many students have trouble understanding, structuring, and simplifying the information given in modelling problems ( Krawitz et al., 2021 openness of modelling problems and the cognitive demands of making assumptions (Ärlebäck, 2009). ...
Context 3
... sessions consisted of three stages: problem solving, stimulated-recall interview, and semi-structured interview. In the problem-solving stage, participants were first PME 45 -2022 given an open modelling problem (Shortcut Route Problem, Table 1) without information about the openness of the problem, a subsequent problem (Speaker Problem, Figure 1) with information about the openness ("To solve the problem, you must estimate the diameter of the speaker"), and finally another problem without such information (Tree Problem, Table 1). Table 1: Open modelling problems used in the study. ...
Context 4
... of the four participants noticed the openness of the problem. All of them neglected the fact that an assumption had to be made about the additional length of the support pole needed to fasten it to the ground in order to obtain a realistic solution (see Christian's solution in Figure 1). Consequently, the participants did not transfer their experience with the previous open modelling problem to the next one. ...
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Citations
... Mathematical modeling has been an explicit part of the competence-based curriculum in Germany since 2003 (Chang 2014;Chang et al. 2020). In Taiwan, mathematical modeling only recently gained curricular attention in teaching reforms (Chang et al. 2020; called modeling tasks-may need students to make assumptions or consider multiple solution methods (e.g., Krawitz et al. 2022; for the role of assumptions in modeling processes, see Galbraith and Stillman 2001). Mathematical modeling, in the sense of solving modeling tasks, received much attention in many countries, especially in Western ones. ...
Factors like the potential of tasks to support students’ mathematical learning and its use in instruction are consensually understood to be relevant for instructional quality across cultural contexts. Yet, research has also shown that perspectives on instructional quality may vary between cultural contexts. As an explanation, it is argued that such perspectives depend on instructional norms, which correspond to the expected behavior in instruction within a cultural context. Notably, research contrasting mathematics instruction from East Asian and Western cultures hints at potentially different instructional norms regarding high-quality use of task potential, but systematic evidence is lacking so far. This study addresses this gap and uses three text vignettes of instructional situations to systematically elicit and contrast instructional norms regarding the use of word problems for mathematical learning. Researchers from Germany (N = 17) and Taiwan (N = 19) evaluated the use of tasks in various instructional situations in an online survey, and their answers were qualitatively analyzed to determine possible culture-specific norms based on their reasoning. In two of the three cases, culture-specific norms in line with assumptions could be identified. In the third case, researchers in both countries referred to an interculturally shared instructional norm. Differences between the reasoning in answers from Germany and Taiwan indicate further cultural influences in line with assumptions based on prior research. We discuss the findings and their implications for the validity of intercultural research in mathematics education.
... Stellen beispielsweise eingekleidete Textaufgaben die überwiegende Menge an Aufgaben im Unterricht dar, bei denen genau jene Informationen enthalten sind, die zur Lösung benötigt werden (und somit den Einsatz einer Ersatzstrategie mit korrektem Resultat erst ermöglichen), sind Schülerinnen und Schüler von den fehlenden Angaben irritiert, sodass sie die Modellierungsaufgabe als unlösbar ansehen (Frejd & Ärlebäck, 2011;Niss & Blum, 2020). Gerade die Selektion von passenden Informationen konnte als besonders herausfordernd identifiziert werden Krawitz, Kanefke et al., 2022;Wijaya et al., 2014). ...
... By obtaining the missing values, a real model can be set up, which is then translated into mathematics (Chang et al., 2020). In some cases, the missing data can be obtained through research (e.g. on the internet) but often missing quantities have to be estimated by making assumptions (Krawitz et al., 2022). Making assumptions means estimating missing information using everyday knowledge and known comparative values (Krawitz et al., 2022). ...
... In some cases, the missing data can be obtained through research (e.g. on the internet) but often missing quantities have to be estimated by making assumptions (Krawitz et al., 2022). Making assumptions means estimating missing information using everyday knowledge and known comparative values (Krawitz et al., 2022). A distinction can be made between numerical assumptions, i.e. missing quantities, and non-numerical assumptions, concerning situational conditions (Chang et al., 2020). ...
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One characteristic feature of modelling problems is their openness. In current research, there is a lack of theoretical and empirical information on how to deal with the openness of modelling problems in the classroom. In this chapter, we summarise research on open problems and present a process model for dealing with modelling problems with a specific focus on openness. We use prior research and the theoretical process model to create a teaching method that is designed to help ninth-graders solve open modelling problems. The essential characteristics of this teaching method is that it is closely linked to the presented process model, it scaffolds students’ learning with the use of situational objects and guiding questions, and it is embedded in a learner-centred learning environment.
במאמר זה אנו מצביעים על החשיבות שבאימוץ ביקורתי ואדפטיבי של חומרי הוראה בידי מורים למתמטיקה לפני הבאתם בפני התלמידים בכיתות. מורות ומורים המביאים לכיתה בעיות שעיצבו בעצמם, מעוררים סקרנות ומוטיבציה גבוהות יותר ופותחים בפני תלמידיהם אפשרויות למידה מגוונות. כיצד ניתן לעצב או להתאים תרגילים ובעיות מספרי לימוד? במחקר רחב היקף בן שלוש שנים בחנו כיצד מורים מחברים ומעצבים מחדש בעיות מתמטיות בהקשר – בעיות המבוססות על סיטואציות מחיי היומיום. בעזרת תיאוריית הווריאציות (הָהֶגְוֵון) של הלמידה זיהינו מספר ממדים שסייעו למורים ביצירת בעיות מגוונות ומשמעותיות עבור תלמידיהם. אנו מציגים את המודל הרב־ממדי ומדגימים כיצד ניתן ליישמו.
