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Teaching Realistic Mathematical Modeling in the Elementary School: A Teaching Experiment with Fifth Graders
Abstract
Recent research has convincingly documented elementary school children's tendency to neglect real-world knowledge and realistic considerations during mathematical modeling of word problems in school arithmetic. The present article describes the design and the results of an exploratory teaching experiment carried out to test the hypothesis that it is feasible to develop in pupils a disposition toward (more) realistic mathematical modeling. This goal is achieved by immersing them in a classroom culture in which word problems are conceived as exercises in mathematical modeling, with a focus on the assumptions and the appropriateness of the model underlying any proposed solution. The learning and transfer effects of an experimental class of 10- and 11-year-old pupils--compared to the results in two control classes--provide support for the hypothesis that it is possible to develop in elementary school pupils a disposition toward (more) realistic mathematical modeling.
... Mathematics provides a set of tools for describing, analyzing, and predicting systems in the real world (Verschaffel and De Corte, 1997). Mathematical skills cannot be limited to use in a specific subject matter area students learn at school; it also has to include elements to help individuals cope outside school in modern society (Haara et al., 2017;Niss and Højgaard, 2019;OECD, 2019). ...
... In light of the fact that a person's psyche is shaped by their environment (Vygotsky, 1978;Bronfenbrenner, 1979;Bronfenbrenner and Morris, 2006;Toomela, 2020), mathematical concepts and procedures can only serve as a part of an individual's problem-solving arsenal, and it is imperative that individuals possess real-world knowledge of the situation at hand (Verschaffel and De Corte, 1997). By understanding mathematical concepts and procedures, as well as real-world knowledge outside of mathematics, an individual is better equipped to achieve their goals and develop their abilities to participate in society . ...
... Third, they have to solve the model to identify its unknown element(s) as well as interpret and evaluate the outcome (and if it meets the terms of the practical situation that was the basis of the mathematical model). Finally, they must communicate the results (Verschaffel and De Corte, 1997). ...
Extra-mathematical knowledge is often overlooked when investigating mathematical skills. This study explores profiles of mathematical skills and associations with extra-mathematical knowledge and the understanding of complex sentences. The study involved 1,288 sixth-grade students (52.1% male) from 95 classes in 58 schools in Estonia. Students completed a math test as part of their regular lessons. The profiles of mathematical skills included students’ calculation skills, standard problems, and complex problems. Three distinct profiles of students emerged: students with high skill levels, students with average skill levels, and students with low skill levels. Students with high mathematical skills also had high extra-mathematical knowledge showing the crucial role of understanding the context of the math tasks in addition to having good mathematical skills.
... In Spain, Llinares and Roig (2008) used contextualized tasks to assess secondary students' construction and use of models. Belgian scholars Verschaffel and De Corte (1997) created problem-solving items to assess grade 5 students' use of realistic considerations and they modeled tasks that required interpreting solutions in relation to the real-world context. Students' work was scored using a rubric that assessed the correctness of the mathematical answer and the use of real-world considerations, with a focus on the problem-solving/modeling competence evidenced in students' solutions. ...
... The MMSA included four open-response items that aligned with multiple modeling competencies. In three of these items, students set up and operated on models and then drew on understandings of real-world situations to interpret the reasonableness of their mathematical solutions (Verschaffel & De Corte, 1997). This process can involve adjusting mathematical solutions so they make sense in the real-world scenario. ...
... We coded the first three open-response items using a two-part rubric focused on (a) the reasonableness of the answer, with four levels of codes; and (b) the use of realistic considerations, with two levels of codes (see Table 1). We modeled the rubric on (Verschaffel & De Corte, 1997). Each of the four Part 1 codes were combined with a "plus" or a "minus" code for realistic considerations, resulting in eight possible combinations of codes. ...
Mathematical modeling is a high-leverage topic, critical for college and career readiness, participation in STEM education, and civic engagement. Mathematical modeling involves connecting real-world situations, phenomenon, and/or data with mathematical models, and in this way applies across various STEM disciplines, including mathematics, engineering, and science. Although research has begun to explore mathematical modeling instruction in the elementary grades, questions remain about how to assess student learning at the elementary level. We addressed this need by designing an assessment of mathematical modeling competencies for students in grades 3 through 5. Informed by international research, our assessment includes a hybrid structure to assess mathematical modeling competencies holistically (as students engage in the complete modeling process) and atomistically (as students engage in different components of the modeling process, including making sense of phenomena and real-world situations, setting up and operating on mathematical models, and interpreting results in relation to the real-world context). We conducted student interviews, followed by two rounds of pilot testing to inform item development and ensure acceptable psychometric properties. The final assessment included 13 items (9 multiple choice, 3 open-response, and 1 complete modeling task). We describe our assessment development process, and provide sample assessment items and detailed coding rubrics. We summarize quantitative analyses which established high reliability and low standard error for our assessment, supporting its use for grades 3 to 5. Implications of our framework and assessment for mathematical modeling instruction and future research on STEM learning are discussed.
... Previous studies have investigated different factors contributing to WP difficulty. For example, in the 1980s, researchers began investigating the difficulties that students encounter when solving various WPs, starting from simple arithmetic WPs (e.g., change, combine, compare: Carpenter 1985;Cummins et al. 1988;De Corte and Verschaffel 1987;Greer 1987;Riley and Greeno 1988) and progressing to more complex WPs requiring non-routine thinking (Lee et al. 2014;Verschaffel and De Corte 1997). Based on a recent literature review by Daroczy and colleagues (2015), the factors influencing WP difficulty could be distilled into three components: linguistic factors, numerical factors, and interaction between linguistic and numerical factors (e.g., reading direction and numerical process, order of number word system). ...
... Another factor influencing WP difficulty, which can be seen as an extended aspect of a situation model, is the necessity of using realistic considerations requiring a non-direct translation of the situational model into a mathematical one. WPs that demand the use of realistic considerations were reported to be very difficult for many students (e.g., Verschaffel and De Corte 1997;Verschaffel et al. 2000). ...
... Individual items seem to have unique factors in their deeper structure that contribute to the item's difficulty level. For example, the main factor that contributes to the difficulty level of the extreme item WP13 seems to be the demand to use real-world knowledge in the modelling process (see Verschaffel and De Corte 1997), while the factor influencing WP difficulty for WP10 could be the difficulty in developing a situation model from the WP statement. One possible explanation for the finding that some of the superficially linguistically similar items appeared to be more difficult is that, in these difficult items, the deep structure is different. ...
In this study we investigated word-problem (WP) item characteristics, individual differences in text comprehension and arithmetic skills, and their relations to mathematical WP-solving. The participants were 891 fourth-grade students from elementary schools in Finland. Analyses were conducted in two phases. In the first phase, WP characteristics concerning linguistic and numerical factors and their difficulty level were investigated. In contrast to our expectations, the results did not show a clear connection between WP difficulty level and their other characteristics regarding linguistic and numerical factors. In the second phase, text comprehension and arithmetic skills were used to classify participants into four groups: skilful in text comprehension but poor in arithmetic; poor in text comprehension but skilful in arithmetic; very poor in both skills; very skilful in both skills. The results indicated that WP-solving performance on both easy and difficult items was strongly related to text comprehension and arithmetic skills. In easy items, the students who were poor in text comprehension but skilful in arithmetic performed better than those who were skilful in text comprehension but poor in arithmetic. However, there were no differences between these two groups in WP-solving performance on difficult items, showing that more challenging WPs require both skills from students.
... In addition to solving the six textbook word problems, students across all three conditions then solved three of the "problematic problems" used previously in the literature (Greer, 1993; STUDENT SENSE-MAKING ON WORD PROBLEMS 6 Verschaffel & De Corte, 1997). The "problematic problems" were presented the same across all three conditions and used as an assessment of student sense-making. ...
... For example, see Figure 3 for one "problematic problem" used in the study and the coded answers. Students' Realistic Responses (RR) were the outcome used as evidence of student sense-making, following in line with previous work in this area (Palm, 2006;Verschaffel & De Corte, 1997;Verschaffel et al., 1994). ...
... Furthermore, our review of the research on elementary mathematical modeling indicates that a great deal of existing research focuses on the upper elementary grade students, that is, grades 3, 4, and 5 (for some examples, see Cyrino & Oliveira, 2011;Peter-Koop, 2004;Petrosino, Lehrer, & Schauble, 2003;Verschaffel & De Corte, 1997;Verschaffel, De Corte, & Lasure, 1994). Hence, little is known about young children's engagement in modeling activities. ...
... This nding echoes those of English (2012), who also found that young learners were capable of developing and describing concepts such as variation during mathematical modeling activities. And, it stands in contrast to the ndings of Verschaffel and De Corte (1997), who found that elementary students tend to neglect real-world knowledge and other appropriate considerations when solving modeling problems. Lastly, and perhaps most importantly, we see the students creating age-appropriate representations and inscriptions during their efforts to mathematize the situation and to communicate their solutions to others -an important aspect of mathematical modeling in the kindergarten classroom. ...
... Volume 14 Issue 4, 2019 are in parallel with the studies of Altun (2002), Akyüz (2010), Bıldırcın (2012), Can (2012), Çakır (2013), Gelibolu (2008), Gravemeijer and Doorman (1999), Heuvel-Paunheizen (2003), Ünal (2008), Üzel (2007), Keijzer and Terwel (2004), Kwon (2002), Özdemir (2008), Rasmussen and King (2000), Searle and Barmby (2012), Verschaffel and Corte (1997) who investigated the effect of GME on mathematics achievement. ...
... According to the permanence test results, it was observed that the final test scores of the experimental and control group students were higher than the retention test scores, and it was concluded that teaching with GME positively affected the permanence of the success. This result was in parallel with the studies of Can (2012), Ersoy (2013) and Verschaffel and Corte (1997), which investigated the persistence of success with GME, and a significant difference was found in favor of GME among students' persistence levels. ...
Hayatımızın her anında var olan matematiği öğretirken ve
öğrenirken yaşadığımız güçlükler matematik başarımızın istenilen
düzeye çıkmasındaki sınırlılığımız olarak karşımızda durmaktadır.
Nitelikli bir matematik öğretimindeki unsurlardan biri öğretim
yöntemlerimizdir. Yapılan araştırmalar öğretim yöntemlerinin farklı
düzeylerde ve konularda matematik öğretimi üzerindeki etkililiğini
göstermektedir. Öğretim yöntemlerinin çeşitliliği öğrencilerin
dikkatlerini çekebilme, ön yargılarından kurtarabilme ve sonuç olarak
matematik başarılarını artırmada önemli bir unsur olmaktadır. Bu
gerekçeden yola çıkılarak, İlkokul 4. sınıflarda uzunlukları ölçme
konusu öğretiminde, Gerçekçi Matematik Eğitimi (GME)’nin öğrenci
başarısı üzerine etkisini ve öğrenilen bilgilerin kalıcılığını araştırmak
üzere gerçekleştirilen araştırmada, deneme modellerinden ön test-son
test eşitlenmemiş kontrol gruplu yarı deneysel desen kullanılmıştır.
Araştırma 2013–2014 öğretim yılı ikinci döneminde bir devlet okulunda deney grubunda GME, kontrol grubunda ise Milli Eğitim
Bakanlığı(MEB) ilkokul matematik dersi öğretim programında yer alan
etkinlikler doğrultusunda yürütülen araştırmaya gruplarda 23’er kişi
olmak üzere toplam 46 öğrenci katılmıştır. Öğrenci başarılarını ölçmek
için deney ve kontrol gruplarında işlenen “Uzunlukları Ölçme”
konusunda matematik başarı testi (MBT) kullanılmıştır. Uygulamanın
önü ve ardında ön test, son test, son testten 4 ay sonra ise kalıcılık testi gerçekleştirilmiştir. Araştırmada başarı testinden elde edilen nicel
veriler SPSS istatistik programındaki bağımlı ve bağımsız gruplar t-testi kullanılarak analiz edilmiştir. Araştırma sonucunda, GME’nin öğrenci başarısı ve başarının kalıcılığını arttırdığı görülmüştür.
... 206). Otherwise, students will try to solve the problem as if it had a camouflaged context, through a phenomenon called suspension of sense making (Greer, 1997;Schoenfeld, 1991;Verschaffel & De Corte, 1997). In agreement with the latter, Palm (2008) describes the authenticity of a realistic problem as the degree to which a class task can be transferred to the real-world, affirming that students can take advantage of those characteristics of the task that connect with characteristics of reality. ...
... It is the different ways in which students understand reality that may have led to these differences in the problem-solving processes, since in authentic realistic problems students can take advantage of those characteristics of the task that connect with characteristics of reality Palm (2008). Moreover, in some cases, it is the suspension of sense-making (Greer, 1997;Schoenfeld, 1991;Verschaffel & De Corte, 1997) that may have resulted in students failing to produce mathematical strategies to solve the problem, on not being able to link mathematical knowledge to their knowledge of the real world. ...
This paper presents a qualitative study developed with a group of 16-year-old students who were asked to estimate large numbers of elements on a bounded surface. Taking the realistic mathematics education framework as a reference, we presented the students with an activity sequence comprised of four different tasks-each one with a different realistic context, which were variations of the same estimation problem. Our data comprises the written reports that the students submitted on completing the tasks. The strategies deployed by the students to solve the tasks consisted in the incardination of procedures related to the concepts of measurement and proportionality. For each task, we present the characterization of the strategies through a visual resource in the form of a tree-like diagram. We integrated the treelike diagrams of the four tasks to generate a tree-like diagram for the general problem. This allowed us to discuss the role of context in the strategies the students reported having used.
... Other researchers have focused on changing the norms of the classroom environment to focus on consideration of context and sense-making. For example, Verschaffel and De Corte (1997) described a teaching experiment focused on bringing a mathematical modeling perspective to word problems. Students in the experimental group worked on nonroutine problems through processes of cooperative learning, and the teacher focused on establishing norms that encouraged collective sense-making and multiple interpretations of answers. ...
... In this paper, we have proposed a framework describing how levels of abstraction could be applied and discussed in elementary mathematics classrooms to help students include more consideration of context in their problem-solving processes. Admittedly, other models of instruction have been suggested that could accomplish the same goal (notably, Verschaffel and De Corte's (1997) framework for teaching from a modeling perspective, which inspired Greer's (1997) model that forms the basis of Fig. 3). However, our proposed framework has the additional advantage demonstrating a close connection between general processes of abstraction, useful in computer science, and problem solving in mathematics activities common in elementary school. ...
In many discussions of the ways in which abstraction is applied in computer science (CS), researchers and advocates of CS education argue that CS students should be taught to consciously and explicitly move among levels of abstraction (Armoni Journal of Computers in Mathematics and Science Teaching, 32(3), 265–284, 2013; Kramer Communications of the ACM, 50(4), 37–42, 2007; Wing Communications of the ACM, 49(3), 33–35, 2006). In this paper, we describe one way that attention to levels of abstraction could also support learning in mathematics. Specifically, we propose a framework for using abstraction in elementary mathematics based on Armoni’s (2013) framework for teaching computational abstraction. We propose that such a framework could address an enduring challenge in mathematics for helping elementary students solve word problems with attention to context. In a discussion of implications, we propose that future research using the framework for instruction and teacher education could also explore ways that attention to levels of abstraction in elementary school mathematics may support later learning of mathematics and computer science.
... Även om uppfattningar om matematik är relativt stabila över tid och kan vara svåra att förändra (Hannula, 2006), finns det studier som visar att elevers uppfattningar om matematiska kan förändras i en positiv riktning både med hjälp av längre, men även kortare interventioner (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998;Cobb et al., 1991;A. J. Stylianides & Stylianides, 2014;Verschaffel & De Corte, 1997;Yeager & Walton, 2011). Det finns även studier som visar att en undervisning som just bygger på lärande genom problemlösning kan förändra elevers uppfattningar om matematik på ett positivt sätt (t.ex. ...
... Det finns även studier som visar att en undervisning som just bygger på lärande genom problemlösning kan förändra elevers uppfattningar om matematik på ett positivt sätt (t.ex. Bruder & Prescott, 2013;Carpenter et al., 1998;Cobb et al., 1991;Verschaffel & De Corte, 1997). ...
In mathematics education, there is generally too much emphasis on rote learning and superficial reasoning. If learning is mostly done by rote and imitation, important mathematical competencies such as problem-solving, reasoning, and conceptual understanding are not developed. Previous research has shown that students who work with problems (i.e. constructs a new solution method to a task), to a greater extent increase their mathematical understanding than students who only solve routine tasks.
The aim of the thesis was to further understand why teaching is dominated by rote learning and imitation of procedures and investigate how opportunities for students to solve tasks through problem-solving could be improved. This was done through the following studies. (1) Investigating the relation between types of solution strategy required, used, and the rate of correct task solutions in students’ textbook task-solving. (2) Studying the relationship between students’ beliefs and choice of solution strategy when working on problems. (3) Conducting a textbook analysis of mathematics textbooks from 12 countries, to determine the proportions of tasks that could be solved by mimicking available templates and of tasks where a solution had to be constructed without guidance from the textbook. (4) Conducting a literature review in order to characterize teaching designs intended to enhance students to develop mathematical understanding through problem solving and reasoning. (5) Conducting an intervention study were a teacher guide, structured in line with central tenets of formative assessment, was developed, tested, and evaluated in real classroom settings. The teacher guide was designed to support teachers in their support of students’ in their problem-solving process. Studies I, II and V were conducted in Swedish upper secondary school settings.
The students’ opportunities to solve tasks through problem-solving were limited: by the low proportion of problems among the easier tasks in the textbooks; by the students' choice of using imitative solution strategies; and by the guidance of solution methods that students received from other students and their teachers. The students’ opportunities were also limited by the students' beliefs of mathematics and the fact that a solution method of problem tasks was not always within reach for the students, based on the students' knowledge. In order to improve students’ opportunities, teachers should allow students to work with more problems in a learning environment that lets students engage in problem-solving and support students' work on problems by adapting their support to students' difficulties. The results also give implications for the construction and use of textbooks and how the use of the teacher guide could be part of teachers’ professional development and a tool that teacher students may meet within their education.
... As shown, in each intervention condition, children were taught a domain-specific strategy for constructing text-level representations (situation and problem models). The strategy was designed to address demands on working memory, attentive behavior, analytical reasoning, and text-structure vocabulary knowledge (Perfetti et al., 2008;Rapp et al., 2007;Verschaffel & De Corte, 1997;J. P. Williams et al., 2016). ...
Reading comprehension (RC) and word-problem solving (WPS) both involve text processing. Yet, despite evidence that RC text-structure intervention (RC.INT) improves RC, transfer to WPS has not been investigated. Similarly, despite evidence that WPS text-structure intervention (WP.INT) improves WPS, transfer to RC has not been examined. The purpose of this randomized controlled trial was to assess effects of single-domain text-structure intervention (RC or WPS intervention) for simultaneously improving RC and WPS. Second-grade children with comorbid learning difficulty across RC and WPS were randomly assigned to three conditions: RC.INT, WP.INT, or the standard school program (the control group). Inferential tests for acquisition effects in multilevel models were significant. RC.INT’s effect sizes (ESs) (vs. control) on RC were g = 1.16 and 0.83; WP.INT’s ESs (vs. control) on WPS were g = 1.36 and 1.05. Inferential tests for cross-domain transfer effects in multilevel models were also significant. RC.INT’s ESs (vs. control) on WPS were g = 0.41 and 0.68; WP.INT’s ES (vs. control) on RC were g = 0.91 and 0.54. Children’s text-structure knowledge in passages and word problems mediated acquisition and cross-domain transfer effects in multilevel models. Findings suggest that text-structure intervention may help address the complex needs of children with comorbid RC and WPS difficulty.
... The first four questions respectively assess four different sub-abilities of mathematical modeling (simplifying, mathematising, interpreting, and validating), while the fifth question (consisting of four sub-questions) evaluates the students' global modeling competencies. An example of the mathematical modeling competencies test items for this study are shown in Appendix B. In contrast to standardized math achievement tests, mathematical modeling competencies test items revolve around realworld contexts and are more deeply integrated with real life (Verschaffel & De Corte, 1997;Turner et al., 2022). Whereas standardized mathematics tests are more oriented towards calculations and proofs of abstract mathematical problems. ...
The relevance of metacognition and mathematical modeling competencies to the development of good mathematics achievement throughout schooling is well-documented. However, few studies have explored the longitudinal relationship among metacognition, mathematical modeling competencies, and mathematics achievement. More importantly, the existing research has mostly focused on unidirectional effects with metacognition typically modelled as antecedents of mathematical modeling competencies and mathematics achievement. Nevertheless, the relationships among metacognition, mathematical modeling competencies, and mathematics achievement may be dynamic, and variables might reciprocally influence each other. Hence, we conducted a longitudinal study examining the reciprocal associations between metacognition, mathematical modeling competencies, and mathematics achievement. To this end, we recruited 408 seventh-grade students to complete a metacognition-related questionnaire and a mathematical modeling competencies test concurrently. This procedure was repeated one year later. A cross-lagged panel analysis showed four main findings: (a) metacognition in Grade 7 longitudinally predicted mathematical modeling competencies in Grade 8; (b) mathematical modeling competencies in Grade 7 longitudinally predicted metacognition and mathematics achievement; (c) higher levels of mathematics achievement drive the subsequent shaping of metacognition and mathematical modeling competencies; (d) There were no gender differences among metacognition, mathematical modeling competencies, and mathematics achievement. Finally, theoretical and practical implications are discussed.
... When the literature is examined, the results of the studies on GME show that GME is effective in increasing academic achievement (Lee, 2006;Özdemir & Üzel, 2011;Çakır, 2013;Nama Aydın, 2014;Kaylak, 2014;Özçelik, 2015;Cansız, 2015;Çilingir, 2015;Özkaya, 2015;Demir, 2017;Yetim et al., 2017;Korkmaz & Tutak, 2017;Zakaria & Syamaun, 2017;Erdoğan, 2018;Özkürkçüler, 2019), increased retention (Özçelik, 2015;Demir, 2017;Özkürkçüler, 2018), improved students' problem-solving skills (Verschaffel & De Corte, 1997;Fauzan, 2002;Bonotto, 2005;Çilingir, 2015), improved their attitude towards mathematics (Özkaya, 2015; Özçelik, 2015; Özkürkçüler, 2019) and increased their motivation (Çakır, 2013). In addition, studies are showing that GME has a positive effect on reducing students' mathematical anxiety (Demir, 2017), improves their estimation skills (Bonotto, 2005), eliminates the disconnect between theory and practice (Korthagen & Russel, 1999), provides easy learning (Yorulmaz & Doğan, 2019), improves concept understanding ability (Hadi, 2002;Barnes, 2004;Lestari & Surya, 2017) and communication skills (Cansız, 2015). ...
This research was conducted to examine the effects of digital stories prepared according to Realistic Mathematics Education (RME) on the mathematics achievement, anxiety and attitudes of 4th grade students. In the research, a quasi-experimental design with pretest-posttest control group, which is one of the quantitative research designs, was used. The research was carried out with 69 fourth grade students studying in a public primary school in the Tarsus district of Mersin province in the second term of the 2021-2022 academic year. In the experimental group, the teaching was carried out by the researcher using digital stories prepared according to RME, the teaching in one of the control group was carried out by the researcher according to the current curriculum, and the teaching in the other control group was carried out by the classroom teacher according to the current curriculum for 7 weeks. Two-factor ANOVA (mixed ANOVA) was used for mixed measurements in the analysis of the data obtained from the study. As a result of the statistical analyzes, it was determined that digital stories prepared according to RME made a significant difference in the increase of students' academic achievement and mathematics attitudes of students, but did not create a significant difference in mathematics anxiety.
... Indeed, many researcher insights serve as foundational studies that researchers and practitioners draw on each day. These include research on the importance of focusing on student conceptual explanations [10] and the various types of student mathematical justifications [11], the student schemas that support calculus success [12] and those that children draw on when solving word problems [13], the impact of instructional interventions on student learning with ratios [14] and nonroutine mathematical situations [15], and the challenges that students with learning disabilities face in the learning of mathematics [16] as well as the specific challenges that all students face when learning mathematical topics, such as fractions [17]. These research findings are useful because they provide frameworks for analyzing and building student mathematical understanding. ...
This manuscript provides a theoretical framing of a collaborative research design effort among mathematics education and special education researchers. To gain insight into the current state of research on mathematics learning, we drew on how researchers in mathematics education and special education have defined and operationalized the term ‘mathematical concept’ related to the learning of fractions. Using this information, we designed a future study that focuses on and connects prior research in mathematics and special education. We conclude by discussing the implications of such collaborative research efforts.
... ;Greer, 1993;Verschaffel & De Corte, 1997;Wyndhamn & Säljö, 1997;Reusser & Stebler, 1997). Αποτυγχάνουν δηλαδή να προσδιορίσουν τη μη αναλογική φύση του προβλήματος και επομένως εφαρμόζουν αναλογικές στρατηγικές για την επίλυσή του, χωρίς να λαμβάνουν υπόψη τους ρεαλιστικούς περιορισμούς του. ...
Η έννοια της αναλογίας είναι μια από τις πιο σημαντικές έννοιες των
μαθηματικών, ενώ παράλληλα ο αναλογικός συλλογισμός αποτελεί έναν από
τους πιο σπουδαίους μηχανισμούς της γνωστικής ανάπτυξης του ατόμου.
Ωστόσο, η παγκόσμια ευρεία χρήση της αναλογίας στα προγράμματα
σπουδών έχει ως άμεσο επακόλουθο την δημιουργία παρανοήσεων και
συγκεκριμένα ότι το αναλογικό μοντέλο μπορεί να εφαρμοστεί παντού,
ακόμη και σε μη αναλογικές καταστάσεις. Το φαινόμενο αυτό αναγράφεται
στην βιβλιογραφία ως ψευδαίσθηση της αναλογίας και συναντάται σε
διάφορους τομείς των μαθηματικών όπως στην άλγεβρα, στις πιθανότητες
και στην γεωμετρία. Ειδικότερα, στον τομέα της γεωμετρίας έχει
παρατηρηθεί από ένα ευρύ πλήθος ερευνών ότι οι μαθητές τείνουν συνεχώς
να αντιμετωπίζουν τις σχέσεις μεταξύ μήκους και εμβαδού ή μεταξύ μήκους
και όγκου ως γραμμικές αντί ως τετραγωνικές ή κυβικές αντίστοιχα.
Στην παρούσα ερευνητική εργασία γίνεται μια προσπάθεια μελέτης
αυτού του φαινομένου σε 10 μαθητές διαφορετικών τάξεων από το Δημοτικό
μέχρι και το Λύκειο, με κύριο σκοπό να εξακριβωθεί ανάλογα με τις
επιδόσεις των μαθητών σε αναλογικά και μη έργα μέσα από έξι διαφορετικές
φάσεις αν το φαινόμενο της ψευδαίσθησης της αναλογίας είναι ανεξάρτητο
της ηλικίας των μαθητών και να διαπιστωθούν πιθανοί τρόποι
καταπολέμησης αυτού του επίπονου φαινομένου. Τα αποτελέσματα
καταδεικνύουν και επιβεβαιώνουν ότι η ηλικία των μαθητών δεν παίζει
κάποιο ουσιαστικό ρόλο, καθώς αυτή η παρανόηση εντοπίζεται τόσο σε
μαθητές Δημοτικού όσο και σε μαθητές Γυμνασίου και Λυκείου.
Παράλληλα, διαπιστώνεται ότι η χρήση αναπαραστάσεων, όπως
διαγραμμάτων σε τετραγωνισμένο χαρτί, καθώς και η μέθοδος των
πολλαπλών τρόπων λύσεων και η συμπλήρωση κενών αποτελούν μια πιθανή
διέξοδο και αντιμετώπισης του φαινομένου.
... However, it was observed that these studies were mostly carried out on RME within the scope of academic achievement. In addition, when the literature was reviewed, it was noticed that studies on WME were mostly planned experimentally (Akyüz, 2010;Bıldırcın, 2012;Can, 2012;Çakır, 2013;Çilingir, 2015;Gelibolu, 2008;Verschaffel & de Corte, 1997). In the studies, the realistic mathematics education approach and the traditional approach were compared. ...
The purpose of this study is to investigate how to improve middle school 6th grade students' skill of associating mathematics with real life through Realistic Mathematics Education, which was reinforced with educational games, to determine the problems possible to be encountered in practice, and how to solve these problems in details. This study, designed as an action research, was carried out in mathematics lessons with totally 25 sixth grade students studying at the first and second semesters of 2018-2019 academic year in an official public secondary school in Adana-Turkey. In the analysis of quantitative data, t-test was used for dependent samples, and inductive and descriptive analysis within the scope of content analysis were performed for analyzing the qualitative data. According to the results, it was determined that Realistic Mathematics Education reinforced with educational games was efficient on increasing the skill of associating mathematics with real life.
... Such an approach is consistent with calls (Catts & Kamhi, 2017;Ukrainetz, 2017) to connect an instructional focus on oral language to specific RC task demands, and some evidence (Fuchs et al., 2015(Fuchs et al., , 2018 suggests that WPS may be treated as a form of text comprehension. A coordinated approach focused across RC intervention and WPS intervention thus includes embedding language instruction on vocabulary that spans the processing of reading passages and word problems, while including methods to assist students in constructing explicit text-level representations with close reading and text-connecting inferencing (Perfetti et al., 2008;Rapp et al., 2007;Verschaffel & De Corte, 1997;Williams et al., 2016). This serves to build the situation model and problem model (i.e., schema) of reading passages or word-problem statements. ...
Analyses were conducted with second graders, drawn from an ongoing multi-cohort randomized controlled trial (RCT), who had been identified for RCT entry based on comorbid reading comprehension and word-problem solving difficulty. To estimate pandemic learning loss, we contrasted fall performance for 3 cohorts: fall of 2019 (pre-pandemic; n = 47), 2020 (early pandemic, when performance was affected by the truncated preceding school year; n = 35), versus 2021 (later pandemic, when performance was affected by the truncated 2019 to 2020 school year plus the subsequent year's ongoing interruptions; n = 75). Across the 2 years, declines (SDs below expected growth) were approximately 3 times larger than those reported for the general population and for students in high-poverty schools. To estimate the promise of structured remote intervention for addressing such learning loss during extended school closures, we contrasted effects in the RCT's 2018 to 2019 cohort (entirely in-person intervention delivery; n = 66) against the same intervention's effects in the 2020 to 2021 cohort (alternating periods of remote and in-person delivery; n = 29). Large intervention effects were not moderated by pandemic status, suggesting potential for structured remote intervention to address student needs during extended school closures.
... The most essential reason for considering modeling and EL together is that MM activities consist of real life problems and allow students to learn experientially. At this point, the results of research on RME, which is the common point of EL and MM, reveal the importance of real life situations and learning by experience (Cobb et al. 2008;Fauzan et al. 2002;Gravemeijer, 1994;Laurens et al. 2017;Mulbar & Zaki, 2018;Riyanto & Putri, 2017;Van den Heuvel-Panhuizen & Drijvers, 2020Verschaffel & De Corte, 1997Wubbels et al. 1997) and these are considered evidence that forms the basis of research. We aim to evaluate the mathematical modeling activity on perimeter and area topics of 4th-grade students within the framework of the EL theory. ...
The aim of the study is to evaluate the mathematical modeling activities of 4th-grade students within the framework of the experiential learning theory. One of the reasons for considering modeling and experiential learning together is that mathematical modeling activities consist of real life problems and allow students to learn experientially. The research conducted as a qualitative case study and whole process fulfilled online. 13 fourth grade students determined with the appropriate sampling method. Modeling activity story, modeling evaluation rubric, observation form and semi-structured interview forms were used as data collection tools. During the implementation process, the students were shown a digital story created by the researchers and the problem situation was embedded in the story. The story about a virus facing humanity asked to the students to find a solution of the problem. In the context of this story, the students carried out a mathematical modeling activities about perimeter and area. At the end of the implementation process, a semi-structured interview form was used to get students' opinions about the activity. As a result, students have achieved on mathematical and mental modeling while they made low score on mathematical results which subject area.
... Öğrencilerin günlük yaşam ve matematik ilişkisini kurduğu, derin ve anlamlı bir matematik anlayışı geliştirmeleri önemlidir. Bu nedenle matematiksel modelleme eğitimi ilkokul döneminden itibaren başlamalıdır (Carlson ve diğ., 2016;Watters, English & Mahoney, 2004;Verschaffel & De Corte, 1997 (Reiners, 2012). Bu bağlamda, araştırma odağındaki konuyu deneyimleyen ve bu deneyimlerini paylaşabilecek ya da yansıtabilecek kişi veya gruplar ile görüşmeler yapılır (Büyüköztürk, Kılıç Çakmak, Akgün, Karadeniz, & Demirel, 2016). ...
The aim of the research is to determine the views of elementary school teachers and pre-service elementary school teachers about mathematical modeling. The method of the research was determined as the descriptive phenomenology approach, one of the qualitative research designs. The study group of the research consists of elementary school teachers and pre-service elementary school teachers. The data of the research were obtained by interview method. Content analysis method was used in the analysis of the data of the research. In line with the findings obtained, there are similarities in the views of elementary school teachers and pre-service elementary school teachers towards the mathematical model. However, it was concluded that elementary school teachers perceive mathematical modeling not as a process, but as a modeling mathematics, while pre-service elementary school teachers perceive mathematical modeling as a process.
... Öğrencilerin günlük yaşam ve matematik ilişkisini kurduğu, derin ve anlamlı bir matematik anlayışı geliştirmeleri önemlidir. Bu nedenle matematiksel modelleme eğitimi ilkokul döneminden itibaren başlamalıdır (Carlson ve diğ., 2016;Watters, English & Mahoney, 2004;Verschaffel & De Corte, 1997 (Reiners, 2012). Bu bağlamda, araştırma odağındaki konuyu deneyimleyen ve bu deneyimlerini paylaşabilecek ya da yansıtabilecek kişi veya gruplar ile görüşmeler yapılır (Büyüköztürk, Kılıç Çakmak, Akgün, Karadeniz, & Demirel, 2016). ...
Araştırmanın amacı, sınıf öğretmenlerinin ve sınıf öğretmeni adaylarının matematiksel modellemeye ilişkin farkındalıklarını belirlemektedir. Araştırmanın yöntemi, nitel araştırma desenlerinden betimleyici fenomenoloji yaklaşımı olarak belirlenmiştir. Araştırmanın çalışma grubunu sınıf öğretmeni ve sınıf öğretmeni adayları oluşturmaktadır. Araştırmanın verileri görüşme yöntemi ile elde edilmiştir. Yapılandırılmış görüşme alma formu sınıf öğretmenlerine ve sınıf öğretmeni adaylarına gönderilerek, formda yer alan sorulara yazılı olarak cevap vermeleri istenmiştir. Yapılandırılmış görüşme alma formu, on açık uçlu sorudan oluşmaktadır. Araştırmanın verilerinin analizinde, içerik analizi (tümevarımsal analiz) yöntemi kullanılmıştır. Elde edilen bulgular doğrultusunda sınıf öğretmenleri ve sınıf öğretmeni adaylarının matematiksel modele yönelik algılarında benzerlikler bulunmaktadır. Ancak sınıf öğretmenlerinin matematiksel modellemeyi süreç olarak değil matematiği modelleme olarak; sınıf öğretmeni adaylarının ise matematiksel modellemeyi bir süreç olarak algıladıkları sonucuna ulaşılmıştır.
... The problems of teaching the basics of mathematical modeling are closely related to the issues of applied orientation of mathematical (fundamental, natural sciences) disciplines. In this regard we note the researches (Blomhøj and Kjeldsen, 2006;Blomhøj and Jensen, 2007;Burghes, 1980;Finlay and King, 1986;Flores et al., 2016;Geiger et al., 2010;Kaiser et al., 2011;Kapur, 1982;Klymchuk et al., 2008;Lofgren, 2016;Oke, 1980;Schukajlow et al., 2018;Soloviev et al., 2019;Temur, 2012;Verschaffel and De Corte, 1997;Vos, 2011). There are some applied issues of mathematical modeling in agroengineering are considered in the textbook (Flehantov, 2006). ...
... Blum y Niss (1991) señalan que los problemas verbales consisten en un modelo de la realidad simplificado y preestablecido, que exige la aplicación de una determinada estructura matemática pero no un análisis de la situación planteada. Distintas investigaciones (Verschaffel, De Corte, y Lasure, 1994;Verschaffel y De Corte, 1997;Verschaffel, Greer, y De Corte, 2000) sugieren que la mayoría de estudiantes no utilizan para resolver este tipo de problemas ningún tipo de conocimiento del contexto real en el que se sitúan los problemas, dándose a menudo una suspensión del sentido de la realidad que hace que den resultados de naturaleza incompatible con la realidad (números decimales como respuesta a preguntas que requieren solución entera, o dando como solución magnitudes claramente desproporcionadas, por ejemplo). El papel activo del contexto real en el proceso de resolución de un problema es la diferencia entre un problema de enunciado verbal y un problema de modelización, que es una clase de problemas de contexto real sobre la que profundizaremos en la siguiente sección. ...
The aim of the thesis is to study the performance of prospective teachers in solving a type of modelling problems involving estimation: Fermi problems, which we will call real-context estimation problems. The use of modelling activities in the classroom is an effective way of connecting Mathematics with the real world. Real-context estimation problems are accessible tasks that allow modelling to be introduced in primary school. However, their implementation is a challenge for primary school teachers, because shortcomings have been detected in their specialised knowledge of mathematical content for teaching, in particular, in their proficiency in problem solving. There is consensus that the flexible use of various types of resolution is a component of problem-solving proficiency. It is therefore of interest to study the flexibility of pre-service teachers in solving real-context estimation problems, and to analyse possible relationships with their performance. In order to address these aspects linked to the flexibility and performance of prospective teachers in solving real-context estimation problems, the research design is complex: two sequences of four problems and two questionnaires are designed, and the research is divided into three parts: the first part is the central one, and is composed of two experiences, in which the N = 224 pre-service teachers involved solve a sequence of real-context estimation problems, first individually and schematically ( resolution plan), and then as a group and performing measurements at the problem site (group and on-site resolution). The second part is based on an alternative sequence of problems in order to validate the results of the previous one with another sample of N= 87 prospective teachers, although it is also proposed to study the effect of syntactic structure on success in solving the problems. The third part deals with the implementation of a questionnaire answered by N = 81 experts in Mathematics and/or its didactics to determine adaptability criteria (what is the best solution) in this type of problems. An analysis of the resolution plans and the group and on-site resolutions, combining qualitative and quantitative techniques, leads to address the research objectives of the thesis: to categorise the productions of prospective teachers and to establish a significant relationship between certain context characteristics and the type of resolution; to categorise and analyse specific errors in real-context estimation problem solving, defining performance levels based on the errors made; to analyse inter-task flexibility (understood as the ability to change the type of resolution from one problem to another in the sequence, depending on the characteristics of the context) and to find relationships between the level of flexibility and performance; to compare individual resolution plans and group and on-site resolutions; to define adaptability criteria for this type of problem and to analyse the adaptability of pre-service teachers. The results offer the opportunity to design problem sequences that promote flexibility and learning from errors, which will contribute to improve the initial training of prospective teachers and enrich their specialised knowledge of mathematical content for teaching. More: https://roderic.uv.es/handle/10550/81850
... In this article, we investigate strategy instructions, namely, drawing instructions, as a form of support that is aimed at promoting students' modeling performance while maintaining their autonomy (i.e., in making assumptions about reality or in choosing a mathematical model). Case studies have indicated that constructing a drawing for a modeling problem can be an effective way to facilitate students' modeling processes (Rellensmann, 2019), and it has been included in educational programs that have been found to enhance students' modeling competencies (Schukajlow et al., 2015;Verschaffel & De Corte, 1997). ...
Making drawings can help students avoid and overcome difficulties they encounter when solving real‐world (or modelling) problems. However, many students do not make drawings spontaneously. In the present study, 132 students in Grades 9 and 10 were randomly assigned to the experimental conditions “with situational or mathematical drawing instructions” or to the control condition “without drawing instructions.” Multilevel path analyses showed that mathematical drawing instructions had a positive total effect on performance, and the effect was transmitted via the use of drawings. Also, strategic knowledge about drawing had a positive total effect on performance via the use of drawings. Further, strategic knowledge about drawing moderated the effect of mathematical drawing instructions on students’ use of drawings. An in‐depth analysis of an exemplary modelling problem provided us with additional hypotheses about how drawing instructions and the use of drawings affect modelling performance on the problem level. This article is protected by copyright. All rights reserved.
... There are many national and international studies on the RME approach (Verschaffel & De Corte, 1997;De Corte, 2004;Afthina, Mardiyana, & Pramudya, 2017;Palinussa, 2013;Saleh, Prahmana, Isa, & Murni, 2018;Aydın Ünal, 2008;Aydın Ünal & İpek, 2009;Ayvalı, 2013;Bıldırcın, 2012;Can, 2012;Cansız, 2015;Cihan, 2017;Çakır, 2011;Çakır, 2013;Çilingir & Dinç Artut, 2016). Studies conducted in our country (Bintaş, Altun & Arslan, 2003;Çakır, 2013;Çilingir, 2015;Demirdöğen, 2007;Demirdöğen & Kaçar, 2010;Ersoy, 2013;Özdemir & Üzel, 2011;Üzel, 2007;Yazgan, 2007;Korkmaz & Tutak, 2017;Erdoğan, 2018) reveal that there are studies investigating the effect of the RME approach on student achievement. ...
... The long division algorithm is surely the most used algorithm for integer division worldwide. On the other hand, students' difficulties in managing this algorithm and its meanings are well known, as shown by the extensive literature published over the last decades (Fischbein, 1994;Lee, 2007;Teule-Sensacq & Vinrich, 1982;Treffers, 1987;Verschaffel & De Corte, 1997;Wu, 2011). Boero, Ferrari, and Ferrero (1989) have discussed students' lack of awarenessalso in the case of good performersabout why this traditional algorithm works, bringing the attention to the way these algorithms are introduced and taught in primary school. ...
During their education cycle in mathematics, students are exposed to algorithms as early as primary school. Several studies show how students frequently learn to perform these algorithms without controlling the mathematical meanings behind them. On the other hand, several National Standards have highlighted the need to construct meanings in mathematics from the first cycle of education. In this paper we focus on division algorithms, investigating to what extent 6th graders can be guided to understanding the whys behind an algorithm, through the comparison of two different algorithms for integer division. Our results suggest, on the one hand, that “it could work!”, and on the other hand, that the exposure to different algorithms for the same mathematical operation seems particularly significant for bringing out the whys behind such algorithms, as well as for capturing the difference between a mathematical operation and algorithms for calculating the result of such an operation.
... In today's world, individuals who solve the complex problems of life are needed in every stage of daily life. Studies showed that problem solving is a teachable skill (Verschaffel & De Corte;Educational Policy Analysis and Strategic Research, V16, N2, 2021© 2021INASED 315 1997Yazgan & Bintaş, 2005;Altun & Arslan, 2006;Cankoy & Darbaz, 2010). In this sense, training individuals who can overcome the problems they may encounter in life is seen as one of the primary goals of education (Soylu & Soylu, 2006). ...
Problem solving is one of the basic skills needed by individuals today. An individual who has attained problem-solving skills can both overcome difficulties in daily life and be successful in professional business life. In this respect, problem solving is considered as a skill that children should attain from an early age. Many variables are effective in the process of teaching problem solving skills. In this study, 4 th grade students' perceptions of problem-solving skills were assessed according to the variables of sex, receiving preschool education, parental education status, family type and person helping with school work. Employing the quantitative research method, the study used the descriptive survey design. The Problem-Solving Inventory developed for elementary school children was used to collect data. The inventory is a 5-point Likert type scale consisting of 24 items and three dimensions. The study universe was 4 th grade students attending public schools in the central districts of Gaziantep. Because of the size of the universe, transportation, economic difficulties and lack of time, the study was conducted with a sample determined by simple random sampling method, one of the random sampling methods. 744 4 th grade students participated in the study. The data of the research were analyzed with the SPSS 15.0 program. According to the study results, no difference was found in students' problem-solving perceptions according to the sex variable. However, students' problem-solving perceptions differ according to the variables of receiving preschool education, mother's education level, father's education level and family type. Also, it was concluded that having someone who helps students with school work makes a significant difference in their problem-solving perceptions. The study results were discussed within the framework of the relevant literature, and late various recommendations were given.
Keywords: Problem Solving, Student, Perception, Variable, Assessment
... Si el contexto ha de desempeñar el papel de promotor del desarrollo del conocimiento, debe proporcionar todos los elementos necesarios para situar al estudiante en un papel apropiado para resolver el problema. De lo contrario, los estudiantes tratarán de resolver el problema como si tuviera un contexto camuflado, a través de un fenómeno llamado suspensión del sentido (Greer, 1997;Schoenfeld, 1991;Verschaffel & De Corte, 1997). En palabras de Sriraman y Knott (2009), al analizar el contexto de un problema, "la palabra realista se refiere no sólo a la conexión con el mundo real, sino también a las situaciones de resolución de problemas que son reales en la mente del estudiante" (p. ...
Análisis de los factores de complejidad en planes de resolución individuales y resoluciones grupales de problemas de estimación de contexto real Analysis of the complexity factors in individual solution plans and group developed solutions for real context estimation problems Resumo. Los problemas de estimación en contexto real pueden utilizarse como iniciación en la elaboración de modelos matemáticos. En este estudio se recogieron los planes de resolución individuales de estudiantes (N=224) del grado de Maestro/a en Educación Primaria que se enfrentaron a una secuencia de problemas de estimación contextualizados. Posteriormente, esos mismos estudiantes trabajando en grupos (N=63), resolvieron los mismos problemas realizando mediciones y estimaciones in situ. Así, el estudio se centra en los denominados factores de complejidad, referidos a los aspectos de la resolución con los que se pretende obtener una estimación más precisa. El objetivo es determinar qué factores de complejidad enriquecen los planes de resolución individuales y cuáles son los que enriquecen las resoluciones grupales. Además, se iden-tifica qué características del contexto real promueven que los estudiantes incluyan determinados factores de complejidad, tanto en los planes de resolución individual como en las resoluciones grupales. Los resultados permiten identificar el impacto del trabajo de campo en el proceso de resolución de tareas de estimación formuladas en contextos reales cercanos; en relación al conocimiento del profesor en formación, los resultados permiten identificar carencias en el conocimiento de las tareas matemáticas para la enseñanza.
... Eğitim yılları, doğru cevaplara ulaşmak için egzersizler yaparak geçiren öğrencinin, aniden matematiği, dünyayı anlamlandırmasına yardımcı olacak bir araç olarak görmesini beklemenin bir anlamı yoktur (Carlson ve diğerleri, 2016). Bu nedenle matematiksel modelleme eğitimi ilkokuldan itibaren başlamalıdır (Carlson ve diğerleri, 2016;Verschaffel ve De Corte, 1997;Watters, English ve Mahoney, 2004). ...
... Eğitim yılları, doğru cevaplara ulaşmak için egzersizler yaparak geçiren öğrencinin, aniden matematiği, dünyayı anlamlandırmasına yardımcı olacak bir araç olarak görmesini beklemenin bir anlamı yoktur (Carlson ve diğerleri, 2016). Bu nedenle matematiksel modelleme eğitimi ilkokuldan itibaren başlamalıdır (Carlson ve diğerleri, 2016;Verschaffel ve De Corte, 1997;Watters, English ve Mahoney, 2004). ...
İlkokul matematik dersi öğretim programı, çevresinde ve okul ortamı içerisinde öğrendiklerini anlamlandırabilen ve kendine ait anlamlar oluşturabilen, oluşturduğu anlamları günlük yaşam içerisinde karşılaştığı durumlara uygulayabilen bireyler yetiştirmeyi hedeflemektedir. Bu bağlamda matematik eğitimi, matematik ve günlük yaşam arasında anlamlı ilişkilerin kurulduğu uygulamalara yönelmiştir. Matematik öğretiminde matematik ve günlük yaşam ilişkisinin önem kazanmasıyla birlikte matematiksel model ve modelleme süreçleri, öğretme ve öğretme süreçlerinde karşımıza çıkmaktadır. Bu nedenle “İlkokul döneminde matematiksel modelleme nasıl inşa edilmeli ve uygulanmalı?” sorusunun yanıtlanması gerekliliği ortaya çıkmaktadır. Bu gereklilik göz önüne alındığında matematiksel modelleme etkinliklerinin ilkokul döneminde yaygınlaştırılması ve sınıf öğretmenlerine öğrenme-öğretme sürecinde kullanabilecekleri bir öğretim çerçevesi ve örnek etkinlik uygulamalarının sağlanmasının önemli olduğu düşünülmektedir. Bu bağlamda araştırma kapsamında ilkokul döneminde matematiksel modellemenin uygulanabilmesi için Carlson, Wickstrom, Burroughs ve Fulton (2016) tarafından ortaya konulan öğretim çerçevesi açıklanmış ve bu çerçeve doğrultusunda bir modelleme etkinliği örneğine yer verilmiştir.
... Konkretno, on razvija vještine rješavanja problema, pruža mogućnosti za preciznije procjene u računanju, doprinosi razumijevanju pojma broja i dekadskog sistema. Mentalne strategije mogu se definisati kao "pametne" metode računanja zasnovane na razumijevanju osnovnih karakteristika sistema brojeva i aritmetičkih operacija, kao i dobro razvijenom osjećaju za brojeve (Verschaffel, De Corte, 1997). One se u suštini razlikuju od algoritma cifarskog računanja u sljedećem: Desno napisani brojevi predstavljaju vrijednost svake od kuglica koja se nalazi na liniji ispred tog broja. ...
Nastava matematike predstavlja izazov za svakog savremenog učitelja. Odabir načina rada i metoda poučavanja ne samo da treba da zadovolji cilj i ishode učenja već i da učenicima olakša usvajanje apstraktnih matematičkih pojmova, kao i računskih operacija. S tim u vezi, učitelji su stalno u potrazi za novim metodama čiji je cilj da se što više učenici aktivno uključe u nastavu matematike. Mentalna aritmetika je metoda, ali i sredstvo pomoću koje se djeca fokusiraju, razmišljaju, slušaju i razumiju ono što čuju, način kojim se podstiče koncentracija, informativni proces, način kojim se ulazi u srž problema. Primjena abakusa za računanje, a kasnije i mentalno računanje utječe na razvoj kako kognitivnih tako i intelektualnih sposobnosti. Kada se sagledaju sve prednosti mentalne aritmetike, abakus i računanje na abakusu trebao bi biti temelj savremenog obrazovanja. Postoje mnogobrojna istraživanja koja su bez imalo sumnje dokazala značaj ranog učenja rada na abakusu. Cilj istraživanja je ispitati i utvrditi da li je učiteljima razredne nastave poznat efekt rada na abakusu i da li smatraju da se treba vratiti u redovnu nastavu. Dobijeni rezultati će se iskoristiti u kreiranju rada s učenicima razredne nastave.
Ključne riječi: nastava matematike, mentalna aritmetika, abakus, učenici
... The results of the present study add to previous research about the effects of enjoyment, anxiety, and prior performance on the use of the drawing strategy (Fiorella & Mayer, 2016;Fiorella & Zhang, 2018;Goetz et al., 2006;Uesaka & Manalo, 2017;Van Meter & Garner, 2005;Wu & Rau, 2019), on factors that affect the use of deep-level processing strategies (Ahmed et al., 2013;Muis et al., 2015;Obergriesser & Stoeger, 2020;Pekrun, Goetz, Titz, & Perry, 2002b;Taub et al., 2014;Winne & Hadwin, 1998), and on achievement-related variables such as modelling performance (Krawitz, Schukajlow, & Van Dooren, 2018;Lai, Zhu, Chen, & Li, 2015;Mohler, 2007;Ramirez et al., 2016;Schukajlow & Rakoczy, 2016;Verschaffel & De Corte, 1997). ...
The use of self-generated drawings has been found to be a powerful strategy for problem solving. However, many students do not engage in drawing activities. In this study, we investigated the effects of the enjoyment of the drawing strategy, anxiety about the drawing strategy, and prior intramathematical performance on the use of the drawing strategy and modelling performance. We explored the role of the drawing strategy as a mediator between emotions and modelling and whether intramathematical performance moderated the effects of emotions (N = 220, mean age 14.5 years). Enjoyment and anxiety with respect to generating drawings and intramathematical performance predicted the use of the drawing strategy. Enjoyment positively affected modelling performance indirectly via the use of the drawing strategy. Anxiety negatively affected modelling performance via the use of the drawing strategy for students with lower intramathematical performance. Our findings demonstrate that experiencing activating emotions (i.e., enjoyment and anxiety) with respect to strategies and prior intramathematical performance are important for strategy use and modelling performance. Implications for the theory of self-generated drawing and the control-value theory of achievement emotions and practical implications for training and supporting the drawing strategy are discussed.
... It has usually been noticed in mathematics classes, particularly at lower levels, that to overcome the language barrier some students mainly depend on 'guessing' to adopt their solution strategy (for supportive view see Verschaffel & De Corte, (1997) in chapter 3, p.34). They suppose 'altogether' stands for addition, 'left' or 'more' for subtraction and so on so forth. ...
The study recognizes that word problems are the necessary part and a key component of mathematics education. Knowing that mathematics, language as a means, and the situation context are never separable, the study was designed to identify the effect of language (L1, L2) and the context on problem solving in mathematics for 2nd, 3rd, and 4th grader English as a second language (ESL) learners. For this, four achievement tests with possible variations of language and context were utilized as instrument to investigate three research questions. 867 students from three existing scenarios of school mathematics learning in Pakistan participated in the study. The data were analyzed through SPSS utilizing both descriptive as well as inferential methods. The results revealed that language and context have significant effect on problem solving. The study exposed that mathematical problem-solving assessments cannot be called valid if the factors of language and context are not taken into consideration. Learners’ first language was strongly recommended for teaching mathematics at low levels. This study will uniquely contribute to understanding and determining the due role of language in mathematics learning, performance, and assessments in all educational contexts.
... Nossa segunda escolha didática apóia-se em princípios tomados da 'Educação Matemática Realista (EMR)', desenvolvida pelo Instituto Freudenthal da Universidade de Utretch -Holanda 1 , nos meados da década de 70, como uma reação aos efeitos da matemática moderna, em particular quanto a ênfase que começou a ser dada às estruturas e formalismos no ensino da matemática escolar. Um dos princípios da EMR é que a matemática deve ser descoberta e reinventada pelos alunos, devendo ser vivida como uma atividade humana, para que se torne então um conhecimento pleno de significado (Verschaffel e Corte, 1997). ...
Este artigo discute uma proposta didática implementada que trata da introdução ao
pensamento algébrico através da dimensão funcional, na 6ª série do Ensino Fundamental.
Na viabilização da proposta foi de grande importância a utilização do objeto de
aprendizagem ‘Máquinas Algébricas’, pelas suas possibilidades de concretização de
relações funcionais em interface adequada para alunos de 6ª série. A pesquisa realizada,
dentro dos princípios da Engenharia Didática, documenta que os alunos trabalharam com
o pensamento algébrico via o conceito de função, sabendo expressá-lo em tabela, gráfico
e lei algébrica.
... For this reason, it might be interesting to deepen mathematical modeling development at early ages as a way of developing mathematical competence. Few studies on mathematical modeling with students under 10 years of age have been carried out in the last few decades (English 2006(English , 2012Greer 1993;Lehrer and Schauble 2000;Verschaffel and De Corte 1997). Consequently, some aspects of mathematical modeling at early ages have not been properly explored, such as the real, concrete contexts that enable students to incorporate their own knowledge about the real world into a modeling activity (Stohlmann and Albarracín 2016). ...
This study presents a teaching experiment in which second-grade primary school students compared a city and a town according to population estimates. The activities were presented to them as Fermi problems and required an analysis of the reality to identify sub-problems that students could deal with. The students' products were analyzed from the perspective of mathematical modeling and it was observed that the students developed their own mathematical methods to make the required estimates. Learning opportunities were identified when students needed concepts and procedures that they had not yet worked with. The results show that estimation acts as a facilitator for the generation of mathematical models that enable students to connect their real-world knowledge to new mathematical learning. In the conclusions, there is a discussion of the implications of the use of Fermi problem sequences to initiate mathematical modeling and link up mathematical and extra-mathematical knowledge about social problems in the first years of primary education.
... Many scientists have been studying problems related to teaching Mathematical Modeling in different countries. For instance, Burghes and Huntley (1982) presented the teaching mathematical modeling on reflections and advice while Verschaffel and De Corte (1997) provided approaches in teaching realistic mathematical modeling in the elementary school with teaching experiment on some fifth-grade students. Also, Abrams (2001) discussed the teaching mathematical modeling and provided some skills of representing mathematical modeling. ...
Matrix theory plays a very important role in teaching Mathematics and solving mathematical problems. Studying the theory of matrix can help academics, practitioners, and students solve many problems in Engineering, Econometrics, Finance, Economics, Optimization, Decision Sciences, and many other areas. To review the matrix theory with applications, in this paper we first review the theory of matrix. We then discuss how to build up some mathematical, financial, economic, and statistical models by using matrix theory and discuss the applications by using the theory of matrix in Decision Sciences and other related areas like Mathematics, Economics, Finance, Statistics, and Education with real-life examples.
This article reports on the mathematical activity of a group of five high school students (15–16 year olds) who worked together on a series of challenging task in combinatorics and probability. The students were participants in an after-school, classroom-based, longitudinal research on students’ development of mathematical ideas and different forms of reasoning in several mathematical content strands. The purpose of the article is to contribute insights into how to promote growth of students’ mathematical understanding through problem-solving activities. In particular, the article shows that problem-solving activities involving strands of challenging tasks have the potential to promote growth of students’ mathematical understanding by providing opportunities for students to engage in reasoning by isomorphism. This is a type of reasoning whereby students rely on structural similarities, i.e., isomorphism among mathematical tasks to solve or deepen their understanding of the tasks. Implications for classroom teaching, and environmental conditions that promote reasoning by isomorphism are also discussed.
Penelitian ini bertujuan untuk memperoleh model bahan ajar matematika SMA berbasis Realistic Mathematics Education (RME) yang dapat mendukung pencapaian tujuan pengajaran matematika, yaitu pemahaman konsep, kemampuan pemecahan masalah, penalaran, komunikasi, dan sikap terhadap matematika. Metode yang digunakan adalah metode penelitian dan pengembangan. Hasil yang diperoleh adalah diperoleh model bahan ajar matematika SMA berbasis RME untuk mendukung pencapaian tujuan pengajaran matematika. Ketersetujuan responden terhadap model bahan ajar tersebut sebagai berikut: 1. Aspek materi sebesar 100% (Kelas X), 100% (Kelas XI), dan 95% (Kelas XII), 2. Aspek penyajian sebesar 95% (Kelas X), 95% (Kelas XI), dan 95% (Kelas XII), dan 3. Aspek bahasa dan keterbacaan sebesar 95% (Kelas X), 98% (Kelas XI), dan 100% (Kelas XII). Disarankan, agar guru matematika SMA di Provinsi Bengkulu menggunakan model bahan ajar matematika berbasis RME untuk mencapai tujuan pengajaran matematika.
While research suggests that mathematical modeling is accessible to elementary grade students, questions remain about how to assess modeling competency. Our research addresses this need through the systemic design and empirical testing of a Mathematical Modeling Student Assessment (MMSA) for elementary grades 3 through 5. Research questions focus on validity evidence for the MMSA based on test content, response process, internal structure, and relationship to other variables, as well as reliability evidence. Data sources included student interviews, MMSA post assessments and state level standardized assessments. We paid particular attention to evidence of internal structure to understand the extent to which a holistic versus an atomistic conceptualization of mathematical modeling competency was empirically supported by the data. We found that MMSA items provided sufficient coverage to provide accurate information about a range of modeling competencies. The MMSA had high reliability and low standard error for average modeling competency which supports its use for grades 3 to 5. Correlations between MMSA trait‐level estimates and standardized assessment scores indicate the MMSA captures overlapping but distinct competencies specific to modeling. We discuss implications for teaching modeling and directions for future research on the development of modeling assessments in the elementary grades.
Mathematics word problems provide students with an opportunity to apply what they are learning in their mathematics classes to the world around them. However, students often neglect their knowledge of the world and provide nonsensical responses (e.g., they may answer that a school needs 12.5 buses for a field trip). This study examined if the question design of word problems affects students' mindset in ways that affect subsequent sense‐making. The hypothesis was that rewriting standard word problems to introduce inherent uncertainty about the result would be beneficial to student performance and sense‐making because it requires students to reason explicitly about the context described in the problem. Middle school students (N = 229) were randomly assigned to one of three conditions. In the standard textbook condition, students solved a set of six word problems taken from current textbooks. In the modified yes/no condition, students solved the same six problems rewritten so the solution helped answer a “yes” or “no” question. In the disfluency control condition, students solved the standard problems each rewritten in a variety of fonts to make them look unusual. After solving the six problems in their assigned condition, all students solved the same three “problematic” problems designed to assess sense‐making. Contrary to predictions, results showed that students in the modified yes/no condition solved the fewest problems correctly in their assigned condition problem set. However, consistent with predictions, they subsequently demonstrated more sense‐making on the three problematic problems. Results suggest that standard textbook word problems may be able to be rewritten in ways that mitigate a “senseless” mindset.
The present study examined the phenomenon of the illusion of linearity in sixth grade students and investigated whether the presence of diagrams or a written verbal warning could have a positive effect on students' performance. 172 students were administered Test A, which consisted of 8 verbal geometrical problems-4 of the problems were proportional and involved the perimeter of geometrical f igures and the other 4 were non-proportional and involved area. At a second phase, half of the students were administered Test B, which consisted of the same problems accompanied by diagrams. The other half were administered Test C in which the problems were accompanied both by diagrams and a written warning. The results showed that students performed significantly lower on non-proportional problems compared to proportional problems in all tests. The presence of diagrams and the warning had a positive effect on students' performance on non-proportional problems. However, these manipulations had a negative effect on proportional problem solving, as the students generalized non-proportional reasoning.
Research on mathematical problem solving has a long tradition: retracing its fascinating story sheds light on its intricacies and, therefore, on its needs. When we analyze this impressive literature, a critical issue emerges clearly, namely, the presence of words and expressions having many and sometimes opposite meanings. Significant examples are the terms ‘realistic’ and ‘modeling’ associated with word problems in school. Understanding how these terms are used is important in research, because this issue relates to the design of several studies and to the interpretation of a large number of phenomena, such as the well-known phenomenon of students’ suspension of sense making when they solve mathematical problems. In order to deepen our understanding of this phenomenon, we describe a large empirical and qualitative study focused on the effects of variations in the presentation (text, picture, format) of word problems on students’ approaches to these problems. The results of our study show that the phenomenon of suspension of sense making is more precisely a phenomenon of activation of alternative kinds of sense making: the different kinds of active sense making appear to be strongly affected by the presentation of the word problem.
This research aimed to strengthen the geometric modeling of the tangent line in tenth grade students by using the GeoGebra software in a public school in Barranquilla - Colombia. For this, through a qualitative methodology, a didactic proposal was applied taking into account the weaknesses found in the students from the results obtained in the diagnostic stage (pretest), concluding that through the interaction of the students with this software, the students enhanced their abilities in geometric thinking, specifically by modeling conic sections in the GeoGebra software.
Bu araştırma, Gerçekçi Matematik Eğitimi (GME) ile desteklenen Tam Sayılar öğretiminde öğrenci başarısı ve başarının kalıcılığına etkisini incelemek amacıyla gerçekleştirilmiştir. Yarı deneysel yöntemin kullanıldığı bu araştırma deney ve kontrol grubunda 33’er öğrenci olan 6. Sınıf öğrencileri ile yürütülmüştür. Araştırma verileri, araştırmacı tarafından geliştirilmiş olan Matematik Başarı Testi (MBT) uygulanarak toplanmıştır. Bu test tüm deney ve kontrol grubu öğrencilerine uygulama öncesi ve uygulama sonrası ön test, son test olarak uygulanmıştır. Son testten 3 ay sonra da kalıcılık testi gerçekleştirilmiştir. Uygulamaların sonucunda ortaya çıkan veriler SPSS 17.0 paket programında yer alan Bağımlı Gruplar t-Testi ve Bağımsız Gruplar t-Testi ile analiz edilmiştir. Araştırmanın bulgularına göre GME ile desteklenen matematik öğretimin öğrencilerin başarısı ve başarısının kalıcılığına olumlu etkide bulunduğu sonucuna varılmıştır.
Bu araştırmada ilkokul 4. sınıflarda uzunlukları ölçme konusunun öğretiminde Gerçekçi Matematik Eğitimi yaklaşımının öğrenci başarısına etkisini belirlemek amaçlanmıştır.
The background of this article is an interest in a sociocultural perspective on cognition, in particular problems that concern the appropriation and use of cultural tools (in a Vygotskyan sense). Drawing from data of an extensive study of the use and understanding of a particular type of tool, a postage table, it is argued that the difficulties of coordinating the table with an outside reality that people have cannot be understood in terms of failures to correctly apply particular forms of reasoning (such as proportional reasoning). Rather, the use of a tool presupposes sensitivity to contextual considerations applicable for specific situations or domains.