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Realistic considerations in mathematical modeling of school arithmetic word problems
Abstract
The aim of the present study was to collect in a systematic way empirical data about the (lack of) activation of real-world knowledge during elementary school pupils' understanding and solution of school arithmetic word problems. Ten pairs of word problems were collectively administered to 75 fifth-graders during a typical mathematics lesson. While the first item of each pair could be modeled and solved in a straightforward and unproblematic way by one or two simple arithmetic operations with the given numbers, the second problem could not be modeled and solved in such a way, at least if one seriously takes into account the realities of the context called up by the problem statement. An analysis of the pupils' reactions to these problematic word problems shows an alarmingly small number of realistic responses or additional comments based on realistic considerations.
... Still, students who have mastered the ability to solve standard mathematical tasks are not necessarily able to correctly solve the more complex word problems that include real-word components (Verschaffel et al., 2020). For example, Verschaffel et al. (1994) presented 10-to 11-year-old students with standard word problems they were able to solve using learned procedures (S-problems) and more realistic complex word-problems (P-problems) that required them to consider the real-life context. Many students who solved the standard problems correctly ignored real-life logic when solving the complex problems. ...
... The complex word problems that simulate real-life situations are often the most difficult to solve (Verschaffel et al., 2020). When creating a mathematical model of the problem situation, in addition to requiring a substantive understanding of the concepts and procedures as well as relationships to other concepts and procedures in a domain (i.e., learners can recognize, recall, group, derive information from text or tables, and use it to compute and measure; Rittle-Johnson et al., 2001;Crooks and Alibali, 2014;Lindquist et al., 2017;Niss and Højgaard, 2019), there is also a need to take the realistic problem modeling perspective into account (Verschaffel et al., 1994;Verschaffel and De Corte, 1997). ...
... Specifically, we first investigated different profiles emerging from sixth-grade students' calculation skills, standard problem-solving skills, and complex problem-solving skills. As different mathematical skills develop together (Rittle-Johnson et al., 2001;Rittle-Johnson, 2017;Niss and Højgaard, 2019) while students tend to ignore the reallife context when building a mental situation model of the problem (Verschaffel et al., 1994;Verschaffel and De Corte, 1997), we expected to find (H1a) a low skill group that exhibits limited understanding of calculation problems and standard problems and, thus, is only very poorly able to solve complex problems and (H1b) a high skill group that exhibits high mathematical skills and, thus, is better able to solve the calculation problems and standard problems correctly. ...
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.
... Extensive research spanning the last few decades has been dedicated to unraveling the intricacies of the errors children encounter during the problem-solving journey. A nuanced understanding of problem solving as a multifaceted process, encompassing phases like comprehending and defining the problem situation, constructing a mathematical model, executing computational procedures, interpreting numeric outcomes, and evaluating the model, has been outlined [6][7][8]. Scholars [6][7][8] have postulated that faulty representation of the problem situation or the inappropriate selection of mathematical operations when determining the unknown element are common sources of errors among children. Additionally, authors [9,10] emphasize that a lack of comprehension of abstract mathematical linguistic forms embedded in the problem text can impede the creation of mental representations, consequently affecting the problem-solving process. ...
... A nuanced understanding of problem solving as a multifaceted process, encompassing phases like comprehending and defining the problem situation, constructing a mathematical model, executing computational procedures, interpreting numeric outcomes, and evaluating the model, has been outlined [6][7][8]. Scholars [6][7][8] have postulated that faulty representation of the problem situation or the inappropriate selection of mathematical operations when determining the unknown element are common sources of errors among children. Additionally, authors [9,10] emphasize that a lack of comprehension of abstract mathematical linguistic forms embedded in the problem text can impede the creation of mental representations, consequently affecting the problem-solving process. ...
... Noteworthy observations by [6,11] indicate that children often neglect real-world knowledge when approaching word problems. This tendency is linked to certain features of instructional practices, such as the prevalence of stereotyped word problems and the premature imposition of a formal approach to arithmetic problem solving. ...
First-grade students often encounter challenges in understanding and solving arithmetic word problems due to their limited reading comprehension abilities. Despite these difficulties, students may employ arbitrary strategies, such as combining numbers based on specific keywords, even if they lack a full understanding of the problems. Research suggests that effective mathematical
reasoning involves the use of visual mental representations during the problem-solving process. To address this, some studies have explored methods to enhance students’ comprehension of word problems. Building on this, the current study explores the impact of first-grade pupils creating visual representations of problem situations on their comprehension and the number of correct solutions. In
a typical math class, 45 first graders received a paper-and-pencil task, and, in a visual context, they solved similar problems after reading and illustrating the situation. The findings reveal that while most participants correctly represented the problem situations through drawing, about half struggled to determine the numeric solutions. Nevertheless, the visual context led to an increase in the number
of correct problem solutions compared to the normal context, suggesting the potential benefits of incorporating visual representations in enhancing comprehension and problem-solving skills.
... Ebersbach, Van Dooren, Van den Noortgate, and Resing (2008), for example, show that children's understanding of exponential growth is hindered when they learn to count (a linear process), and over-apply this strategy. In addition, when presented with mathematical problems individuals tend to neglect relevant real-life knowledge suggesting non-linearity and instead adopt linear integration strategies (see e.g., Verschaffel, De Corte, & Lasure, 1994) and individuals tend to integrate based on a linear model in mathematical settings due to reliance on mathematical habits and expectations invited by the task-content (see e.g., De Bock, Van Dooren, Janssens, & Verschaffel, 2002;De Bock, Van Dooren, Janssens, & Verschaffel, 2007;Dewolf, Van Dooren, & Verschaffel, 2011). Relatedly, linear properties are overused also in probabilistic reasoning in school mathematics (see Van Dooren, De Bock, Depaepe, Janssens, & Verschaffel, 2003). ...
... Although there are important exceptions (e.g., pictorial cues in Albrecht et al., 2020;Hoffmann et al., 2019), it seems fair to conclude that most of the studies of multiple-cue judgment with metric cues have relied on a numerical format (Brehmer, 1994;Cooksey, 1996;Karelaia & Hogarth, 2008). The claim for the widespread use of linear additive cue integration may thus be over-stated or premature, given that many studies use a numeric format that in itself may elicit a "linear imperative" (De Bock et al., 2002;De Bock et al., 2007;Dewolf et al., 2011;Van Dooren et al., 2008;Verschaffel et al., 1994). A numeric format may affect the participants' expectations (or "priors") about the task in two different ways. ...
... Second, the numbers may elicit the expectation that the cues and criterion are governed by an underlying simple equation that can be induced ("cracked") by intense efforts at problem solving. Both of these factors are likely to encourage linear additive cue integration, becauseas we have seenmathematical education supports linear thinking as a default rule that is applied whenever it is triggered by mathematical cues in the context (De Bock et al., 2002;De Bock et al., 2007;Dewolf et al., 2011;Van Dooren et al., 2008;Verschaffel et al., 1994). ...
When people use rule-based integration of abstracted cues to make multiple-cue judgments they tend to default to linear additive integration of the cues, which may interfere with efficient learning in non-additive tasks. We hypothesize that this effect becomes especially pronounced when cues are presented numerically rather than verbally, because numbers elicit expectations about a task with a simple numerical solution that can be appropriately addressed by linear and additive integration. This predicts that, relative to a verbal format, a numerical format should be advantageous for learning in additive tasks, but detrimental for learning in non-additive tasks. In two experiments, we find support for the hypothesis that a verbal format can improve learning in non-additive tasks. The division-of-labor between cognitive processes observed in previous research (Juslin et al., 2008), with cue abstraction in additive tasks and exemplar memory in non-additive tasks, was only present in conditions with numeric information and may therefore in part be driven by the use of numeric formats. This illustrates how surface characteristic of stimuli can elicit different priors about the nature of the variables and the generative model that produced the cues and the criterion. We fitted cue-abstraction and exemplar algorithms by PNP-modeling (Sundh et al., 2021). At the end of training both cue abstraction and exemplar memory processes primarily involved exact analytic processes marred by occasional error, rather than the noisy and approximate intuitive processes typically assumed in previous studies - specifically, cue abstraction was primarily implemented by number crunching and exemplar memory by rote memorization.
... Following the analysis of classes of open problems, we explore the relationships between creativity and openness as in Silver (1997) and Haylock (1987Haylock ( , 1997. We then zoom in on problems with realistic contexts (Greer, 1993;Reusser & Stecler, 1997;Verschaffel et al., 1994Verschaffel et al., , 1997Verschaffel et al., , 2000, which are considered a special class of open problems. Studies have found that students' success when solving realistic problems is significantly lower than when solving standard problems due to students' inclination to suspend real-world knowledge and realistic considerations when solving word problems (Reusser & Stebler, 1997;Verschaffel, 1999;Verschaffel et al, 1997). ...
... Interestingly, a literature review reveals some contradictions regarding the role of realistic contexts in problem solving. Some researchers demonstrate that adding a realistic context to a problem supports students' capacity to solve the problem (Sullivan et al., 2000) whereas other studies demonstrate "suspension of real-world knowledge" that leads to difficulties in connecting realistic contexts to mathematical content Verschaffel et al., 1994). asked students to solve the "army bus problem" (depicted in Table 2) and demonstrated that incorrect interpretation of the problem context led to a situation in which only a low percent (20-40%) of students succeeded in answering the question correctly. ...
... Additionally, they stated that students were writing down answers that they believed would be considered correct by their teachers, and therefore many of them did not expose their own reasoning and way of thinking. Greer (1993) and Verschaffel et al. (1994) took another approach, comparing students' responses to typical textbook mathematical problems with responses to contextually rich mathematical problems, the solutions to which require mathematical modeling of a given situation. Problems that required using realistic contexts were considered "problematic"; these were presented in contrast to standard problems with straightforward arithmetic calculations. ...
The notion of open mathematical problems that appears in the mathematics education literature includes a variety of mathematical questions and tasks. Observation of the distinction between open-end and open-start problems leads us to draw a distinction between Multiple Solution Strategies Tasks (MSTs) and Multiple Solution Outcomes Tasks (MOTs). Whereas MSTs are inherently open, MOTs are not necessarily open ended. Their openness depends on the formulation of a question. MOTs can either be open or require completeness of solution sets. In this chapter, we discuss MOTs from the point of view of their openness and completeness and draw a connection between MOTs and “sense making” tasks, the complexity of which is thought to be rooted in the lack of realistic considerations applied by students and teachers. We argue that the complexity of these problems is linked to their multiple possible outcomes and the requirement to find a complete solution, both of which are unconventional, and to the fact that the solutions of these problems are insight based.KeywordsMultiple solution outcomes tasks (mots)Sense makingOpen tasksCompleteness of solution spaceUnconventional tasksInsight
... We know from the literature and our experience as teacher that for students open mathematical problems (including realistic problems) can be very problematic to deal with. Students, regardless from their age, have difficulties with recognition of the realistic content of mathematical tasks (Verschaffel, de Corte, & Lasure, 1994). They give stereotyped answers for tasks in most of the case on mathematics lessons ignoring their open nature. ...
... We agree with Verschaffel et al. (1994) that stereotyped word problems in school mathematics "exclude real-world knowledge and realistic considerations from the different stages of their solution processes, i.e. the initial understanding of the problem, the construction of a mathematical model, the actual computational activities, and the interpretation and evaluation of the outcome of these computations". (Verschaffel, de Corte, & Lasure, 1994, p. 273). ...
... In order to classify students' reactions to the 'Four children problem' we adapted the coding system of Verschaffel et al. (1994). We kept three main categories (EA, RA, OA, see in what follows) that have similar meaning. ...
There are well-known differences between problem solving competencies of novices and experts. It is also known from the literature that students, regardless of age, typically give the expected answer, ignoring the openness of the problem. In our study, we use a problem that is open at the starting point nevertheless it has closed solution. We analyze how experts and novices manage the openness of an elementary problem based on a survey prepared for 7 th and 8 th graders.
... Researchers have studied this phenomenon of nonsensical answers by presenting students "problematic problems" (Greer, 1993). These "problematic problems" were designed to parallel traditional word problems, except "mathematical modeling assumptions were deliberately made problematic" (Verschaffel et al., 1994). That is, if students follow the straightforward use of one operation, the resulting answer is nonsensical in the problem context. ...
... For the three "problematic problems" used for the sense-making assessment, the same coding process was used as by Verschaffel and colleagues (Verschaffel et al., 1994). Students' answers were coded as one of the following: Expected Answer (EA), Realistic Response (RR), or Technical Error (TE). ...
... 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). ...
... Similarly, recognition tasks may provide a way to probe participants' representation of the problems (e.g. Mani & Johnson-Laird, 1982;Verschaffel, 1994), to assess whether their interpreted structures differed depending on the problem statements. ...
... Baruk found that children overwhelmingly replied to such problems by using the provided numerical values, while only a minority of participants pointed out the inadequacy of the question and the impossibility to find the answer. The idea that children could manifest a "suspension of sense-making" (Verschaffel, Van Dooren, Greer, & Mukhopadhyay, 2010, p.12) and exclude real-world knowledge from their responses to mathematical word problems was further investigated through the creation of "problematic problems" (called P-items) requiring the use of common sense to be solved (Greer, 1993;Verschaffel, De Corte, & Lasure, 1994). Contrary to the captain's problem, those P-items admitted sensible answers, although unusual ones that would not normally be expected in a mathematical classroom. ...
... Replying with the algorithm 4 × 2.5 = 10 was deemed a "non-realistic reaction", whereas a "realistic reaction" was either to provide the correct answer (4 × 2 = 8) or to reply that the solution was not straightforward and acknowledge the unrealistic nature of the 4 × 2.5 algorithm. Greer (1993) and Verschaffel et al. (1994) showed that the vast majority of children tended to display non-realistic reactions to P-items, whereas they had realistic reactions to non-problematic control problems. In other words, students tended to ignore real-world knowledge as well as realistic considerations from the answers they gave: they replied by applying the mathematical procedures they learned, regardless of their relevance in the described situation. ...
With its context-independent rules valid in any setting, mathematics is considered to be the champion of abstraction, and for a long time human mathematical reasoning was thought to follow nothing but the laws of logic. However, the idea that mathematics is grounded in nature has gained traction over the past decades, and the context-independency of mathematical reasoning has come to be questioned. The thesis we defend concerns the role played by general, non-mathematical knowledge on individuals' understanding of numerical situations. We propose that what we count has a crucial impact on how we count, in the sense that human's representation of numerical information is dependent on the semantic context in which it is embedded. More specifically, we argue that general, non-mathematical knowledge about the entities described in a mathematical word problem can shape its interpretation and foster one of two representations: either a cardinal encoding, or an ordinal encoding. After introducing a new framework of arithmetic word problem solving accounting for the interactions between mathematical knowledge and world knowledge in the encoding, recoding and solving of arithmetic word problems, we present a series of 16 experiments assessing how world knowledge about specific quantities can promote one of two problem representations. Using isomorphic arithmetic word problems involving either cardinal quantities (weights, prices, collections) or ordinal quantities (durations, heights, number of floors), we investigate the pervasiveness of the cardinal-ordinal distinction in a wide range of activities, including problem categorization, problem comparison, algorithm selection, problem solvability assessment, problem recall, sentence recognition, drawing production and transfer of strategies. We gather data using behavioral measures (success rates, algorithm use, response times) as well as eye tracking (fixation times, saccades, pupil dilation), to show that the difference between problems meant to foster either a cardinal or an ordinal encoding has a far-reaching influence on participants from diverse populations (N = 2180), ranging from 2nd graders and 5th graders to lay adults, expert mathematicians and math teachers. We discuss the general educational implications of these effects of semantic (in)congruence, and we propose new directions for future research on this crucial issue. We conclude that these findings illustrate the extent to which human reasoning is constrained by the content on which it operates, even in domains where abstraction is praised and trained.
... 본 연구에서는 Verschaffel et al. (1994), Yoshida et al. (1997), Heo (2008) (Carpenter et al., 1983;Inoue, 2005). ...
... (Carpenter et al., 1983) (2) 12마리의 양과 13마리의 염소가 배 위에 있다. 이 배의 선장은 몇 살일까? (Baruk, 1989;Inoue, 2005, 재인용) 위의 (1)번 문제에 대해 13세 학생들의 약 70%가 "37대, 나머지 18대" 또는 "37.6대"라고 답했고 (Carpenter et al., 1983), (2)번 문제에 대해 1학년과 2학년의 대다수의 학생들이 "25세" (12+13=25, 선장의 나이를 동물들이 결정할 수 있는 것처럼)라고 답했다 (Baruk, 1989;;Inoue, 2005, 재인용 (Greer, 1993;Verschaffel et al., 1994). ...
... 10초 후에 4 cm의 높이만큼 물 이 채워진다고 했을 때, 30초 후 물의 높이는 얼마일까? (Verschaffel et al., 1994) 위의 (3)번 문제에 대해 13세와 14세 학생들의 약 94%가 "170초" (17×10=170, 사람이 1 km 동안 100 m 달리 기의 최고 기록을 유지할 수 있는 것처럼)라고 답했고 (Greer, 1993), (4)번 문제에 대해 초등학교 5학년 학생들 75명 중 66명이 플라스크의 모양을 고려하지 않은 채 표준적인 해법(4 cm×3=12 cm)에 따라 답을 구했다 (Verschaffel et al., 1994 (Heo, 2008;Verschaffel et al., 1994). ...
This study aims to explore how ‘sensitively’ Korean middle school students consider realistic contexts while solving problematic problems. Six pairs of word problems were collectively administered to 85 students in third grade during a mathematics lesson, and follow-up interviews were conducted with 11 students. The survey found that the proportion of students who solved the problems by sensitively considering real-world knowledge was not high (22%). In the follow-up interviews, the students reported that they would think of mathematics separately from real-world situations. An analysis of the students’ unrealistic reactions to real-life problems has implications for the practice of school mathematics regarding real-life word problems.
... First, the real-life problems involved contexts that were often familiar to problem-solvers and presented "dilemma-driven" and "goal-directed" situations (Stanic & Lester, 1989). Second, problem-solvers in the school aimed to solve mathematics problems by applying procedures and algorithms they had been taught while problem-solvers outside the school aimed to make decisions based on invented solution methods through anticipating or testing real-life consequences (Masingila et al., 1996;Verschaffel et al., 1994). Third, out-of-school problem-solvers were more flexible than in-school problem-solvers regarding the solution methods they employed (Masingila et al., 1996). ...
... Third, out-of-school problem-solvers were more flexible than in-school problem-solvers regarding the solution methods they employed (Masingila et al., 1996). Hence, using realistic contexts in school mathematics was particularly important not for applying previously learned mathematical concepts and procedures but for developing conceptual understanding through mathematization of the context (Bonotto, 2004;Gravemeijer, 1999;Masingila et al., 1996;Verschaffel et al., 1994). Furthermore, Civil (2002) highlighted that contextual problems gave students more control over their actions: ...
... Although characteristics of the problems had a significant role in actualizing a rich problem-solving experience, some studies reported students' tendencies to use school mindset in solving contextual problems and ask for the teacher's validation of an answer (e.g., Lave, 1988;Verschaffel et al., 1994). Even though word problems were designed to connect mathematical ideas and real-life situations, students might not go beyond computational work and might not utilize realistic considerations and assumptions to solve the problem (see Verschaffel et al., 2000 for review of studies with similar results). ...
This study investigated the middle school preservice mathematics teachers’ conceptualization of what it means for a problem to be mathematically rich and contextually realistic and how their conceptions evolved during a series of professional development activities. Fifteen middle school preservice mathematics teachers were involved in this design-based research. The professional development activities were designed based on L. Shulman and J. Shulman’s (Shulman and Shulman, Journal of Curriculum Studies 36:257–271, 2004) reflective and communal clusters with psychological roots in Schön’s notion of “reflective practitioners” and Vygotsky’s social development theory. These professional development activities were integrated into the methods and field experience courses of the teacher education program. The audio records of the whole group and small group discussions and all written products (i.e., problems, criteria list, and reflection memos) were analyzed to understand preservice teachers’ conceptions. This study revealed that the professional development activities helped preservice teachers produce the problems that would be interesting and encouraging for students to develop their own solution methods. Preservice teachers’ conceptions also evolved in a collective and reflective community. In this respect, this study presented a way to design professional development for teachers that nurtures their understanding of the components supporting mathematical richness and contextual meaningfulness of a problem. Hence, this study provided implications for teacher education practices and future research on mathematics teacher preparation.
... Researchers and mathematics educators have attempted to promote a new concept of WPs more analogous to real-life problems. Problematic WP (P-item) is an example of a new conception of WPs, developed by Greer (1993) and Verschaffel et al. (1994), comprising actual situations and is more analogous to real-life problems. Rather than performing a routine operation, solving P-items requires realistic consideration and real-world knowledge. ...
... Seminal studies have been conducted by Greer (1993), and Verschaffel et al. (1994) confirmed these beliefs. Through their empirical studies, the researchers proposed standard word problem-solving (S-items) and the problematic word problem-solving (P-items) to investigate students' responses to WPs. ...
Annotation. The main purpose of this study is to investigate students' reactions when doing realistic word problems based on their implicit beliefs and based on their personal factors. Our study revealed that students tended to choose non-realistic responses by ignoring real-world knowledge and excluding realistic considerations when doing realistic mathematics tasks. There were no significant differences in students' reactions to word problems according to their attitude, grade, and gender.
... However, it is often observed that learners disregard the real-world context when working on problems with a close connection to reality, and that they just arbitrarily choose numbers and operations without checking for plausibility (e.g., Greer, 1997;Reusser & Stebler, 1997;Verschaffel et al., 1994Verschaffel et al., , 1999 for an overview, see Krawitz et al., 2022). Even student teachers show a strong tendency to exclude knowledge from the real world (Verschaffel et al., 1997). ...
... We expand on previous research on students' difficulties to provide realistic solutions (e.g., Verschaffel et al., 1994Verschaffel et al., , 2010 by showing that drawing attention to certain types of considerations does or does not influence whether or not people give realistic solutions. Our findings are especially meaningful because we did not solely focus on finding interventions to improve realistic solutions, but rather tried to also identify aspects that might impair realistic solutions. ...
In this study, effects of asking participants to make different types of considerations when solving a realistic word problem were investigated. A two-factorial experiment with the factors “addressing realistic considerations” (with vs. without) and “addressing mathematical operations” (with vs. without) was conducted. It was assumed that reality-based considerations would lead to reality-based problem-solving strategies, thus fostering real-life solutions, while considering mathematical operations would lead to problem-solving strategies usually promoted in school, which were expected to impair realistic solutions. Analyses are based on N = 165 participants. The results showed that being asked to make reality-based considerations did not significantly affect realistic solutions ( F (1, 161) = 2.43, p = 0.121, η p ² = 0.015), while being asked to consider appropriate mathematical operations significantly impaired realistic solutions ( F (1, 161) = 8.54, p = 0.004, η p ² = 0.050). These findings suggest that inducing typical school problem-solving strategies may be detrimental when it comes to solving mathematical problems in a realistic way.
... Πρώτα από όλα, όπως έχει ήδη αναφερθεί, παρατηρείται σε έρευνες που ασχολούνται με την χρήση και την επίλυση ρεαλιστικών λεκτικών προβλημάτων, όποτε το φαινόμενο έρχεται στην επιφάνεια μέσα από τις μη ρεαλιστικές απαντήσεις των μαθητών. Αυτές οι έρευνες (Verschaffel et al, 1994(Verschaffel et al, , 2000Greer, 1997) έχουν ως πρωταρχικό στόχο να ανακαλύψουν τον τρόπο με τον οποίο τα λεκτικά προβλήματα, όπως χρησιμοποιούνται γενικά στην κουλτούρα της μαθηματικής εκπαίδευσης, προωθούν την «απομάκρυνση του νοήματος» από την επίλυση προβλήματος. Για παράδειγμα στο πρόβλημα: «Ο καλύτερος χρόνος του John για να τρέξει 18 100 μέτρα είναι 17 δευτερόλεπτα. ...
... Για παράδειγμα στο πρόβλημα: «Ο καλύτερος χρόνος του John για να τρέξει 18 100 μέτρα είναι 17 δευτερόλεπτα. Πόσο χρόνο θα του πάρει για να τρέξει 1 χλμ;» οι περισσότεροι μαθητές έδωσαν την μη ρεαλιστική απάντηση των 17x10=170 δευτερολέπτων, δηλαδή το ενός απλού πολλαπλασιασμού με βάση την υποτιθέμενη άμεση αναλογικότητα μεταξύ των ποσοτήτων που εμπλέκονταν στο πρόβλημα, χωρίς να λάβουν υπόψιν την κούραση που πολύ πιθανόν θα είχε ο αθλητής (Verschaffel et al, 1994). Αυτό το πρόβλημα ήταν ένα τυπικό παράδειγμα διατύπωσης προβλήματος που φαινομενικά απαιτούσε λύση που βασίζονταν σε ευθέως αναλογικό συλλογισμό. ...
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.
... More complicated real word problems require students to use their imageries rather than regular operations; namely, problematic word problems (socalled P-items) developed by Verschaffel et al. (1994) required students to deviate from the usual superficial solution strategy. In this type of word problem, students must understand the semantic structure and dynamic network of implicit rules (Greer et al., 2003). ...
... Students need to recognize that there is not a single solution, or the task cannot be solved. A study on Pitems by Verschaffel et al. (1994) revealed that students tend to use nonrealistic reactions when solving word problems. This study inspired other researchers in various countries to analyze students' approaches in solving mathematics. ...
The purpose of the present study is to provide empirical evidence on the adaptation of the mathematics-related beliefs questionnaire (MRBQ) supplemented by info-communication technology-related items. Besides, we also investigate the relationship between their beliefs about mathematics and their ability in problem-solving mathematics. 234 grade eight students from five schools in Java, Indonesia, participated in this study. The questionnaire has appropriate reliability. To examine the validity of the questionnaire, factor analysis was applied, and regression analysis was conducted to explore students’ beliefs and their relation to their performance. Factor analysis revealed a good fit of the model; therefore, confirming the validity of MRBQ in the Indonesian context. Descriptive statistics showed students’ tendency to follow the nonrealistic approach when doing word problems. Regression analysis indicated the significant role of beliefs in mathematics predicting students’ performance on mathematical word problems.
Keywords:
... Brian lives 17 km away from school and Sylvia 8 km. How many km apart do Brian and Sylvia live?. " These studies show that experienced students tend to answer these problems in a superficial way by selecting the most likely operation and inserting the numbers in the slots (17-8 = 9 in the example), without making realistic considerations such as that Brian and Sylvia could also live on different sides of the school (Verschaffel et al., 1994(Verschaffel et al., , 2020. In the words of Verschaffel et al. (2000, p. 13), students used "the rules of the game of word problem solving. ...
... Consequently, Dutch students may have encountered a wider variety of word problems than students from countries with other instructional approaches. Further studies could investigate how Dutch students solve other types of non-standard word problems such as the non-routine problems from Verschaffel et al. (1994) or problems with more than one piece of irrelevant information. ...
Solving arithmetic word problems requires constructing a situation model based on the problem text and translating that into a mathematical model. As such, word problem solving makes demands on students’ language comprehension and their domain-general cognitive resources. These demands may decrease when students get more experienced and use strategies that do not require fully understanding the situation presented in the problem. The current study aims to address this hypothesis. Students (N=444) from third to sixth grade solved a paper-and-pencil task with 48 mathematics problems, comprising symbolic arithmetic problems and standard word problems, as well as more complex word problems that involve two arithmetic steps or include irrelevant numerical information. Their performance was analyzed with multilevel logistic regression analyses. Results showed that within each grade, performance on the different problem types did not differ, suggesting that already in third-grade students seem helped nor hindered by presenting arithmetic problems in a story, even if that story contains irrelevant numerical information. Non-verbal reasoning was more important in standard word problems than in arithmetic problems in symbolic format in one-step arithmetic, and reading comprehension was more important in solving two-step arithmetic word problems than in one-step arithmetic word problems.
... Nonetheless, the effectiveness of using arithmetic word problems in schools is debated (Chapman, 2006;Verschaffel, 2002). For example, Verschaffel et al. (1994) warned about such problems promoting "a strong tendency to exclude real-world knowledge and realistic considerations" (p. 273) in students, echoing the notion of the "suspension of sense-making," in which students concentrate on structural features and prioritize mechanical calculations while solving word problems (Chapman, 2006;de Corte et al., 2000). ...
Posing purposeful questions is one of the most effective teaching practices (NCTM in Principles to actions: Ensuring mathematics success for all. National Council of Teachers of Mathematics, 2014). Although the types and functions of teacher questioning have been abundantly studied, research on the role of teacher questioning in students’ contextualization process as they solve word problems is rather scarce. This study was conducted to investigate the function of six elementary preservice teachers’ questioning, its impact on students’ contextualization, as well as the successes and difficulties of enacting questioning. The collected data were analyzed using thematic analysis. The findings indicated that the implementation of task clarification (TC) moves effectively enhanced contextualization only when students possessed a relatively strong sense of agency while solving word problems. Furthermore, when students’ attentional focus was not appropriately redirected by the functional moves, including procedural understanding (PU), making connections (MC), the rationale behind a strategy (RA), and an alternative strategy (AS), their understanding of the contextual features and construction of mathematical relationships in word problem solving could not be refined. Implications for field experience design and future research on the quality of teacher questioning in mathematics teacher education programs are discussed.
... Students' preference for formulas can be interpreted as an attempt to memorize the answers. A similar situation has been found in several studies (Greer, 1997;Stacey, 1989;Verschaffel et al., 1994). In addition, the fact that the desired shape was not a regular shape led the students to transform the shape into different forms. ...
In studies over recent years, there has been an increasing interest in teachers’ predicting middle school students’ thinking processes. However, as far as we are aware, there are no studies examining students’ thinking in terms of mathematical thinking components. This study primarily aimed to determine the mathematical thinking of middle school students. Therefore, the study examined how six mathematics teachers and 24 preservice mathematics teachers (from first to fourth grade) predicted the mathematical thinking of 96 middle school students. In this context, the predictions were categorized according to the sub-components of mathematical thinking: conjecturing, specializing, justifying and convincing, and generalizing. Regarding the conjecturing, the teachers explained students’ prediction of their mathematical thinking in more detail than preservice teachers. Regarding the specializing, the study, both groups of teachers could not predict that the students could express different situations in their problem solutions. Within the scope of the justifying and convincing, the preservice teachers had different perspectives on problem solving compared to the teachers. In regard to the generalizing, teachers and preservice teachers made similar predictions but all groups from first to fourth grade lack experience for this component. It can be stated that preservice teachers’ interaction with more students will be effective in predicting students’ mathematical thinking. The same is true for teachers, as it is believed that greater experience will be beneficial.
... How old is the captain?"). Verschaffel reported that only 17% of the answers to mathematics problems were realistic (Verschaffel et al., 1994). This means children provide a calculated answer to the mathematics problems without being aware of, or without pointing out, the non-sensical nature of the problem. ...
Critical thinking is a recurrent educational ambition. At the same time, it is not self‐evident how that ambition can be realised. This is partly due to the different perspectives from which Critical Thinking can be approached. The literature on critical thinking is extensive and diverse, different meanings and aspects of critical thinking have been explored. However, there is agreement among several researchers that critical thinking entails both ability and attitudinal components. Research in psychology on different types of cognitive processing has similarly pointed to the importance of both skills and attitudes. This article builds on a tripartite notion of disposition that has been proposed in the context of education. The tripartite dispositional perspective on which we elaborate highlights the importance of ability, inclination and sensitivity. We describe and discuss an educational protocol aligned with the tripartite conceptualisation of disposition. The protocol identifies characteristics of powerful learning environments. We propose that the proposed educational protocol—aligned to Critical Thinking education goals, conditions and interventions—can be used for fostering critical thinking. More specifically, the use of four types of interventions are recommended: (1) modelling, (2) inducing, (3) declaring and (4) surveillance. Finally, we underscore that there is a need for further research on the use of the educational protocol.
... How many 1.5 m long pieces of rope should one tie together to connect both poles? ' (DeWolf et al., 2014;Verschaffel et al., 1994). In this example, children need to understand the problem situation in the real world to acknowledge that tying knots and going around the poles will take some of the length of the rope which makes the problem more complex. ...
The aim of this study was to investigate individual differences in mathematical problem-solving among 3- to 5-year-old children (N = 328; n3-year-olds = 115, n4-year-olds = 167, n5-year-olds = 46). First, we examined the extent to which children in this age group were able to solve open and closed non-routine mathematical problems representing a variety of mathematical domains. Second, we investigated the extent to which underlying academic and cognitive skills (i.e., expressive and receptive language, visuospatial, and early numeracy skills) were associated with individual differences in mathematical problem-solving concurrently and longitudinally (i.e., one year later). The results showed that 4- to 5-year-olds were able to solve a variety of non-routine mathematical problems. However, though 3-year-olds were also able to solve a variety of problems, the mathematical problem-solving measure did not meet the reliability criteria, resulting in excluding 3-year-olds from further analyses. Expressive and receptive language, visuospatial, and early numeracy skills were associated with mathematical problem-solving concurrently among 4-year-olds. Among 5-year-olds, only visuospatial and early numeracy skills were associated with mathematical problem-solving. Furthermore, only prior mathematical problem-solving skills and early numeracy skills predicted mathematical problem-solving skills longitudinally. These findings indicate that preschoolers are able to solve open and closed non-routine mathematical problems representing a variety of mathematical domains. Additionally, individual differences may stem not only from differences in mathematical problem-solving skills but also from early numeracy.
... This finding supports Hypothesis 1, which holds that beliefs are associated with achievements, attitudes, and motivation.Interestingly, the direct association between beliefs and attitudes is the strongest relation among other variables (motivation and achievements). It means that students' beliefs about mathematics would drive their attitudes toward an object (Kim & Keller, 2010), which may affect their achievements (Csíkos, 2011;Verschaffel et al., 1994). This finding is in line with the prior research (Caprara et al., 2003;Rarujanai et al., 2022), which consistently indicated the association between the two. ...
Investigating factors affecting students' academic performance seems a hard job for researchers on the empirical front. Beliefs, parents' educational background, motivation, and attitudes have been proven significantly influence achievement. However, concurrent research on the relationship among these variables seems scarce. Therefore , to contribute to this gap in knowledge, the purpose of this study is to examine the structural relationships among beliefs, parents' educational level, attitude, motivation, and achievement in mathematics learning. We selected 30 classes randomly from six schools in Surabaya, Indonesia. This study involved 894 fifth-and sixth-grade students (448 boys and 446 girls). Structural equation modeling results showed that this model predicts students' achievement in mathematics (R 2 = 0.49). Beliefs are positively associated with students' achievement (β = 0.20, p < 0.001), attitude (β = 0.82, p < 0.001), and motivation (β = 0.68, p < 0.001). Parents' educational level is positively associated with achievements (β = 0.17, p < 0.001) and motivation (β = 0.07, p = 0.04). Beliefs were indirectly associated with achievements through attitude (β = 0.31, p < 0.001) and motivation (β = 0.08, p = 0.01). The indirect association between par-ents' educational level and achievement through motivation was insignificant. This study is valuable because it helps unpack the relationship between beliefs, parents' educational level, attitudes, motivation, and achievement.
... The task is based on a task presented in Verschaffel et al. (1994). Its solution presumes students' understanding that the two groups of friends can have common participants. ...
The Math-Key program described and characterized in this chapter integrates Multiple Solution-Strategies Tasks (MSTs) and Multiple Outcomes Tasks (MOTs). We demonstrate that MSTs are inherently open tasks while, in contrast, MOTs can either be open or can require attaining completeness of a solution set. We argue that a multiplicity of solutions both in MOTs and MSTs increases both the complexity of the task and the mathematical curiosity of school students, making Math-Key tasks inherently mathematically challenging. In addition, Math-Key tasks require a change in socio-mathematical norms, and thus, the program is didactically challenging. To provide scaffolds for teaching and learning processes Math-Key tasks are accompanied by exploratory and task-directed dynamic applets (DA). The exploratory nature of the DA enables solvers of Math-Key tasks to understand the problem structure and to support the teachers’ orchestration of classroom teachers. We characterize Math-Key tasks using several examples and explain the task directness of the DA. Integration of the Math-Key program within the regular curricular sequence is a part of the recommended curricular change suggested in this chapter.KeywordsMath-Key TasksDynamic appletsMultiple solution strategiesMultiple solution outcomesSolution spacesOpen-start problemsOpen-end problemsSolution completeness
... When the literature on mathematics education is examined, students' tendency to illusion of linearity have been proven in different learning areas of mathematics such as geometry and probability (De Bock, Van Dooren, Janssens and Verschaffel, 2002;De Bock, et al., 1998;De Bock, Verschaffel et al., 2002;De Bock, Verschaffel, Janssens, Van Dooren and Claes, 2003;Van Dooren et al., 2007;Van Dooren et al., 2003;Van Dooren et al., 2008;Fischbein, 1999;Fischbein and Schnarch, 1997) in missing value problem types (Cramer et al., 1993; Van Dooren, De Bock, Hessels, Janssens and Verschaffel, 2005;Verschaffel, De Corte and Lasure, 1994), different mathematics subjects such as patterns (Esteley et al., 2010;Stacey, 1989) and graph drawing (Leinhardt, Zaslavsky and Stein, 1990;Hadjidemetriou and Williams, 2002) and in university-level courses such as calculus (Esteley et al., 2010). Geometry and measurement are the areas where the best-known examples of students' illusion of linearity are seen and at the same time this phenomenon is mostly investigated (De Bock, Van Dooren et al., 2002;De Bock et al., 1998;De Bock, Verschaffel et al., 2002;De Bock et al., 2003;Van Dooren et al., 2007;Vlahović-Štetića, Pavlin-Bernardića and Rajtera, 2010). ...
... The routine and non-routine problem tests consisted of six routine problems and six non-routine problems based on previous studies (Cai, 2000;Verschaffel et al., 1994;Xin et al., 2007). It consisted of open-ended questions including numbers, fractions, proportions, patterns, generalization, areas, and lengths. ...
This study examined the relationship between secondary school students’ problem-solving success and perceptions using a relational survey model. This study investigated 378 students (212 girls and 166 boys) in the sixth, seventh, and eighth grades between 11 and 14 years old using the convenience sampling method. The problem-solving inventory for children, routine and non-routine problem tests, and problem evaluation rubric were used for data collection. Descriptive and inferential analyzes were utilized. The results indicated that students tended to avoid the problem-solving process. Variables of trust, self-control, and avoidance regarding problem-solving perceptions significantly predicted students’ success in solving routine and non-routine problems.
... Additional expertise can also add complexity in the area of mathematics. This is in part because mathematical text problems often represent simplifications of real-world scenarios and learners usually do not include realistic considerations in their representation of a problem (Verschaffel et al., 1994). For example, a teacher might give the following task: 'Steve has bought 4 planks of 2.5 m each. ...
Background & Aims
Cognitive load theory assumes that the higher the learner's prior knowledge (i.e., the more expert the learner), the lower the intrinsic cognitive load (complexity) experienced for a given problem. While this is the case in many scenarios, there can be cases in which the converse is also true, resulting in more expert learners reporting higher intrinsic cognitive load than novices for the same problem. This can occur in relation to problems involving complex systems (e.g., ecological systems), for which novices' problem representations may underestimate problem complexity and therefore report lower intrinsic load than experts. This finding is borne out in the current paper.
Samples, Methods & Results
In Study 1 with 118 participants from the Black Forest area in Germany, participants with higher levels of forestry and ecological expertise evaluated a problem relating to the restructuring of the Black Forest to adapt to climate change as more complex than did novices. In Study 2 (within-subjects design, n = 66 primary-school students), we conceptually replicated this finding in a domain more typical of cognitive load theory studies, mathematics. We found that higher prior knowledge also reduced the underestimation of the complexity of ‘tricky’, but frequently used, mathematics word problems.
Conclusion
Our findings suggest that cognitive load theory's assumptions about intrinsic load and prior knowledge should be refined, as there seems to exist a sub-set of problem-solving tasks for which the traditional relationship between prior knowledge and reported ICL is reversed.
... Verschaffel L from KU Leuven in Belgium is the most prolific researcher, with 34 articles. Professor Verschaffel L has published numerous articles on many different topics of mathematics education such as "problem-solving, conceptual change, strategy choice and strategy change, metacognitive and affective aspects of learning especially in early and elementary mathematics education" (e.g., Verschaffel et al., 1994;Verschaffel et al., 1999). The authors with the second greatest number of these top papers are Clements DH and Rittle-Johnson B from the US. ...
The aim of the study is to identify and assess the 500 most-cited articles in mathematics education research for the period 1970-2020, and then discuss the impact on the evolution of the historical platform by using bibliometric citation analysis. The USA is the most productive country in terms of having the highest percentage of the top 500 papers, researchers, and institutions. The most productive institution is The University of Wisconsin located in the United States. Verschaffel L from KU Leuven in Belgium is the most prolific researcher, with 34 articles. The Journal for Research in Mathematics Education (JRME) is the leading journal. Mathematics achievement is the most studied topic that has garnered the greatest interest in the top 500 articles. Mathematics education research has its origins mainly in the literature of mathematics education, followed by general education, psychology, educational technology, cognition and brain research, and educational psychology. This study also reveals that although all the classic laws of bibliometrics were theorized about 40 years ago, these are still capable of clarifying the current evolution of mathematics education research. Based on these important results, the study portrays a comprehensive understanding of mathematics education research by using bibliometric citation analysis.
... as an example, consistent with Bahmaei [20] in elementary faculties normally, early arithmetic teaching cantered on process skills. Further, it's conjointly supported by the assertion that the results of ancient approaches in arithmetic square measure mechanistic or reminding of solutions to story drawbacks/word problem [21,22]. moreover, consistent with Kulkarni [23] in ancient programs, objectives square measure classified as low-level goals supported the abilities of formulas, straightforward algorithms, and definitions. ...
We proposed an approach using multiple regression analysis to develop a mathematical model that represents a dynamic manufacturing system. Simulation data are specifically analyzed using this multiple regression analysis approach to obtain a data pattern. The aim of the approach is to reduce the gap between theory and real-time data of the system. To evaluate the effectiveness of the proposed mathematical mode, simulation model was first validated using real-time data. The applicability of the proposed mathematical model was evaluated by testing with real-time data. The outcome +positively demonstrated that the develop mathematical model based on multiple regression analysis approach can be used to make predictions in the dynamic manufacturing environment with an acceptable error percentage range. The mathematical development in this field will enhance the future establishment of a decision-making model using a spreadsheet in the management field.
... The first subtest comprised of five tasks based on a seminal work by Verschaffel et al. (1994). In that work, ten pairs of simple arithmetic word problems were collected and referred to as either standard (S) or problematic (P). ...
In this research we aim to reproduce and extend the positive results about the effect of humor in solving mathematical word problems. We examined whether solving humorous word problems has some effect on either word problems demanding realistic consideration or word problems of the well-known routine type. Fifth- and sixth-grade students (N=1,153) solved different types of tasks including humorous, routine and “problematic” word problems, answered further questions concerning their attitude towards mathematics. Our results suggest that while sixth-graders outperformed their fifth-grade peers, the positive effect of humorous word problems proved to be even more relevant than the school grade. However, there has been some possible negative effect of humor when solving routine word problems. Furthermore, students expressed a positive attitude toward humorous word problems by selecting these as their favorite tasks compared to other types of word problems. The results obtained provide further evidence on the desirability and feasibility of introducing (more) humor in mathematics education.
... The importance of realistic problems that connect school mathematics with students' daily lives is strongly advocated (Brenner et al., 2002;Verschaffel et al., 1994Verschaffel et al., , 2000. Suggestions include using real-life connections (Lee, 2012) and cultural artefacts (Bonotto, 2013;Bonotto & Dal Santo, 2015) as sources of problems. ...
An essential task for mathematics teachers is posing problems. Selecting mathematics problems that develop mathematical proficiency and engage students in desirable mathematical practices is a critical decision-making process. We present the Framework for Posing Elementary Mathematics Problems (F-PosE) developed to focus prospective teacher noticing on desirable features of mathematics problems and inform decision-making processes around the selection of problems for use in elementary classrooms. Development of the framework was informed by a three-phase design research process consisting of an extensive review of the literature, document content analysis and successive testing of mathematics problems in elementary classrooms in partnership with teachers and children. Consequently, it draws from emergent practice informed by the collective endeavour of a community of educators. The framework consists of eight indicators: use of a motivating and engaging context, clarity in language and cultural context, curriculum coherence, attention to cognitive demand, an appropriate number of solution steps to support reasoning, a variety of solution strategies, facilitating multiple solutions and opportunity for success. This F-PosE provides a critical focusing lens for prospective teachers when creating and selecting mathematics problems specifically for use in elementary classrooms.
... At this point, some theoretical and experimental studies showed that students' unrealistic perspective on mathematics could be affected by the culture of courses. Schoenfeld (1991) and Verschaffel et al. (1994) claimed that students suspended this perception in getting used to the culture of their mathematics course and problem-solving. More specifically, Chen et al. (2011) stated that students' neglect of their thoughts about real-life was caused by the teaching practices and two aspects of class culture: (1) the stereotypical and unrealistic nature of the word problems used in the textbooks and classroom, and (2) the way these problems were designed and handled by the teachers in mathematics courses (Verschaffel et al., 1999). ...
... 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
... Many studies outline difficulties in the use of word problems in mathematics classrooms (Hoogland, Pepin, de Koning, Bakker, & Gravemeijer, 2018;Verschaffel, Greer, & De Corte, 2000;Verschaffel, Greer, Van Dooren, & Mukhopadhyay, 2009). Students, in fact, do not take into account common sense considerations about the problem (Greer, 1997;Verschaffel, Corte, & Lasure, 1994) and seem to have established a set of rules of which include: (i) any problem is solvable and makes sense; (ii) there is a single, correct and precise (numerical) answer which must be obtained by performing one or more arithmetical operations with numbers given in the text; (iii) violations of personal knowledge about the everyday may be ignored (Greer, Verschaffel, & Mukhopadhyay, 2007). The main consequences of this situation are an increasing gap between mathematics and the real-world (Gravemeijer, 1997) and a suspension of sense-making (Schoenfeld, 1991). ...
This paper reports on a teaching experiment on mathematical modelling in an upper secondary school class that investigates how the constructs of model eliciting and emergent modelling may be brought together to inform teaching and learning. A prominent role is covered by the process of task design, seen as a complex process that involves several steps: the explication of a learning goal, the formulation of hypotheses on students’ learning, the design of specific learning activities. The study provides a case study of how model eliciting activities that start from rich context problems could play a central role to support emergent modelling. This result can be attributed to a combination of several factors: the choice of a rich context problem that stimulated students to elaborate formal mathematical concepts mathematising their informal solving strategies, and the use of a suitable artefact, that presented mathematics as a means of interpreting and understanding reality.
... The focus group produced the qualitative data through the focus group interviews; observations from the video recordings and learners' written calculations in their exercise books. Learners were required to solve sevenword problems test items (TIs) adapted from Verschaffel, De Corte and Lasure (1994), and were immediately required to discuss and justify the solutions they derived. As researchers, we were conscious of the fact that all four basic mathematical operational signs which are addition, subtraction, multiplication and division apply in tackling word problem-solving. ...
This chapter, based on South African classroom experiences, is informed by feminist post-structuralism, focusing on how teachers engage with gendered English textbooks. It highlights the ways in which teachers negotiate gendered identities, specifically what I perceive as a deliberate disruption or endorsement of patriagraphies – over-determined phallocentric narrative constructions of critical voices that ultimately push femininity to the margins whilst privileging masculinity.
(6) (PDF) Disrupting patriagraphies in the classroom: Gendered constructions and languaging difference in South African texts. Available from: https://www.researchgate.net/publication/357822348_Disrupting_patriagraphies_in_the_classroom_Gendered_constructions_and_languaging_difference_in_South_African_texts [accessed Apr 07 2022].
... The focus group produced the qualitative data through the focus group interviews; observations from the video recordings and learners' written calculations in their exercise books. Learners were required to solve sevenword problems test items (TIs) adapted from Verschaffel, De Corte and Lasure (1994), and were immediately required to discuss and justify the solutions they derived. As researchers, we were conscious of the fact that all four basic mathematical operational signs which are addition, subtraction, multiplication and division apply in tackling word problem-solving. ...
... In an attempt to study more systematically learners' nonrealistic approach towards word problem solving, Greer (1993) and Verschaffel et al. (1994) confronted pupils aged between 10 and 14 years old with a word problem solving test consisting not only of standard problems that can be solved by means of a straightforward operation with the numbers in the problem (S-items) but also with word problems that are problematic from a realistic modeling perspective (P-items) in the sense that real-life knowledge should be taken into account to come to a reasonable or meaningful solution reaction. For example, the P-item "A man wants to have a rope long enough to stretch between two poles 12 m apart, but he has only pieces of rope 1.5 m long. ...
The study explored how mathematics learning loss took place among Turkish middle school students during the COVID-19 school closures through mathematics teachers’ self-reported practices, challenges, and efforts while they were trying to support their students’ learning. Interviews with 19 public and 9 private middle school mathematics teachers indicated that there were certain differences in teachers’ practices and revealed the existing inequalities among the schools, classrooms, and students. Students’ lack of participation, teachers’ limited use of methods to teach mathematics, the socio-economic status of families and their lack of collaboration with teachers were among the reasons for mathematics learning loss.
... With higher grade levels, the learning content with strong structure and hierarchy is more complex and abstract, as a result, the more difficulties students encounter in learning, the less success they have, which is more likely to lead to a low level of learning attitude and a decline of 21st-century skills confidence. The grade differences in learning confidence gradually increase with the improvement of the developmental level [45]. Secondly, study at the primary and middle stage is highly cumulative. ...
In recent years, STEM (science, technology, engineering, mathematics) education has received widespread attention from all over the world, and there are not many studies on STEM attitudes in China. One of the reasons is the lack of measurement tools that have been tested for reliability and validity. The Chinese version STEM attitudes scale for primary and secondary schools is a multidimensional scale that measures the STEM attitudes of primary and secondary school students. It consists of three subscales: STEM interest, 21st-century skills confidence, and STEM career interest. In order to test the reliability and validity of the scale application, as well as understand and improve the STEM attitudes of primary and secondary school students, the research team surveyed and collected 566 responses from primary and secondary school students in Zhejiang, Shanghai, Shandong, Liaoning, and other places. After exploratory factor analysis, confirmatory factor analysis, and a reliability and validity test, the scale finally retained 48 items. The scale supports a hypothetical five-factor model with good reliability and validity and can be used to assess STEM attitudes in Chinese primary and secondary schools. This research also shows that students’ STEM interests and STEM career interests showed clear variation among different genders, grades, and parental education levels.
... There is rarely one simple way to mathematically model the relations embedded in everyday situations; the web of inputs, representations, and estimations required to reason mathematically is complex. This is in direct contrast to the typical tasks required of students in the mathematics classroom, which are often concise, well-structured, and require little, if any, mathematical modelling, even when purportedly about everyday life (Pongsakdi, 2017;Verschaffel, De Corte, & Lasure, 1994). ...
Background:
Adaptive expertise is a highly valued outcome of mathematics curricula. One aspect of adaptive expertise with rational numbers is adaptive rational number knowledge, which refers to the ability to integrate knowledge of numerical characteristics and relations in solving novel tasks. Even among students with strong conceptual and procedural knowledge of rational numbers, there are substantial individual differences in adaptive rational number knowledge.
Aims:
We aimed to examine how a wide range of domain-general and mathematically specific skills and knowledge predicted different aspects of rational number knowledge, including procedural, conceptual, and adaptive rational number knowledge.
Sample:
173 6th and 7th grade students from a school in the southeastern US (51% female) participated in the study.
Methods:
At three time points across 1.5 years, we measured students' domain-general and domain-specific skills and knowledge. We used multiple hierarchal regression analysis to examine how these predictors related to rational number knowledge at the third time point.
Result:
Prior knowledge of rational numbers, general mathematical calculation knowledge, and spontaneous focusing on multiplicative relations (SFOR) tendency uniquely predicted adaptive rational number knowledge, after taking into account domain-general and mathematically specific skills and knowledge. Although conceptual knowledge of rational numbers and general mathematical achievement also predicted later conceptual and procedural knowledge of rational numbers, SFOR tendency did not.
Conclusion:
Results suggest expanding investigations of mathematical development to also explore different features of adaptive expertise as well as spontaneous mathematical focusing tendencies.
... Results showed that most of the students produced non-realistic solutions to problems. The findings of the study seem to be similar with the findings of the studies conducted in other countries (Verschaffel, De Corte &Lasure, 1994;Yoshida, Verschaffel & De Corte, 1997;Reusser & Stebler, 1997;Xin et al, 2007;Palm, 2008). The types of solutions did not vary with respect to gender but socio economic background and grade levels. ...
... As shown in Figure 1, the performance levels that the students had as a problem are poor, that is, the correct way of solving was very poor, which indicates that the students solved the problems mechanically using the known algorithms, without previously analyze the context. [19][20][21][22] Based on the above, an aspect that was observed among the students, and that explains the impoverished performance that they obtained in the initial evaluation, is a function of school practices that have already been normalized, such as repeating response patterns guided by the memory 9 applying any arithmetic operation that they consider convenient or that they know best. ...
The objective of the present study was to explore and describe the influence of a proposal that returns to Problem Based Learning for the application arithmetic principles in the solution of problems in sixth grade students of primary education. Eighty-six of sixth grade students participated with an average age of 11.0 years. The intervention strategy was carried out in nine sessions lasting approximately one hour each. The results in general, do not show statistically significant differences in the performance of the execution in the resolution of problems between pre-test and post-test conditions, however, it was observed through the students' report that they had experienced knowledge acquisition based on the content of the intervention.
... In the context of the study, the PD intervention was about supporting teachers who worked with mathematical word problems found in the CAPS curriculum. Mathematical word problems are verbal or (con)textual descriptions of a problem situation (Verschaffel et al., 1994) that incorporate mathematics tasks implanted in real-world situations. ...
Problem-solving is of importance in the teaching and learning of mathematics. Nevertheless, a baseline investigation conducted in 2016 revealed that mathematical problem-solving is virtually missing in South African classrooms. In this regard, a two-cycle design-based research project was conducted to develop a professional development (PD) intervention that can be used to bolster Grade 9 South African teachers’ mathematical problem-solving pedagogy (MPSP). This article discusses the factors that emerged as fundamental to such a PD intervention. Four teachers at public secondary schools in Gauteng, South Africa, who were purposively selected, participated in this qualitative research study of a naturalistic inquiry. Teachers attended PD workshops for six months where PD activities that were relevant to their context were implemented. Between the PD workshops, teachers were encouraged to put into practice the new ideas on MPSP. Qualitative data were gathered through reflective interviews and classroom observations which were audio-recorded with teachers’ consent. Data were analysed through grounded theory techniques using constant comparison. The findings from the study suggested that teachers’ personal meaning, reflective inquiry, and collaborative learning are factors fundamental to their professional growth in MPSP. The major recommendation from the study is that facilitators of PD must acknowledge these factors to promote teachers’ professional growth in MPSP. If PD processes and activities are relevant to teachers’ personal meaning, reflective inquiry, and collaborative learning, teachers find the PD programme fulfilling and meaningful. This study contributes to the PD in MPSP body of knowledge by having worked with teachers in an under-researched context of historical disadvantage.
... But once more the problem 6 ÷ 2 =? creates a momentary black-out in the minds of many learners (Kouba & Franklin, 1993). We can thus draw the conclusion that the basic operations of arithmetic ought to be introduced to the beginning learner within the context of concrete terms or materials before attempts at abstraction can be made (Verschaffel, De Corte, & Lasure, 1994). ...
The way mathematics teaching and learning activities are presented to learners can make them hate or like the subject. The question of accomplishing the mathematics education of the learner from primary to post-secondary school levels is one which necessarily tasks, not only the teacher's stock of mathematical knowledge but also his skill, his method of approach and finally his handling of the processes of feedback mechanisms. This paper outlines some of the fundamental problems confronting mathematics teachers and learners, and by means of previously published work, it proposed some useful suggestions which have gained support through classroom implementation.
Students’performance in mathematics learning is closely associated withtheir engagement. Then, how can students’engagement in mathematicslearning be promoted? Social cognitive theory argues that those whoengage emotionally and behaviourally hold strong beliefs about theirability. This study investigated the role of beliefs about mathematicslearning and utility value in emotional and behavioural engagementthrough the mediating role of self-efficacy. This study revealed thatbeliefs about mathematics learning and utility value directly predictedemotional and behavioural engagement in mathematics learning. Self-efficacy positively mediated the relationship between beliefs aboutmathematics learning and students’engagement
This research aims to develop web-based mathematics learning media with a focus on enhancing students' mathematical problem-solving abilities. The background of the study is based on the urgent need to address challenges in mathematics education and leverage technology to improve learning outcomes. The research method used is a development research with the ADDIE (Analysis, Design, Development, Implementation, Evaluation) approach. Research instruments include student response questionnaires, media and material validation instruments, and mathematical problem-solving test questions. The research results show that the developed web-based learning media is valid and effective in improving students' mathematical problem-solving abilities, as the paired samples t-test yields a significance score of 0.002 (p < 0.05). Students' response to this learning media is positive, and there is a significant improvement in mathematical problem-solving test scores. The conclusion of this research is that the use of web-based learning media has the potential to enrich mathematics education and enhance student learning outcomes. The implications of the research highlight the necessity of integrating technology in education to optimize learning and meet the learning needs of students in the digital era.
In this chapter, we bring into perspective the conversation on equitable mathematical problem solving (PS) instruction in South African classrooms. Literature has documented that many South African classrooms exhibit restrictive environments where not all learners are given equal opportunities to succeed in mathematics. Debates about transforming the South African mathematics classrooms into equitable spaces have been ubiquitous in the South African mathematics education discourse. Consequently, in this qualitative exploratory study, we entered into a conversation on equitable mathematical PS instruction with 15 in-service South African secondary school teachers, whom we purposively selected. We used classroom observations and face-to-face interviews to investigate how participating teachers characterised and implemented equitable mathematical PS instruction. The classroom observations and interviews were audio-recorded, transcribed verbatim and classified into themes using thematic analysis. Our findings were that equity is a luxury in the South African mathematics classrooms, with teachers practising teacher-centred forms of instruction that limit learners’ opportunities to do mathematical PS. Power and status structures were prevalent in the classrooms, which in turn influenced learners’ participation during whole-class discussions and collaborative mathematical PS. We present these findings and offer recommendations on how teachers can implement equitable mathematical PS instruction.
Context, in this study, refers to the who, when, where and why of teachers’ professional development (PD). This study explored the contextual sources linked to Grade 9 South African teachers’ PD in mathematical problem-solving instruction (MPSI). Four teachers at public secondary schools in a district in Gauteng, South Africa, participated in the study that focused on designing and developing a PD intervention relevant to their experiences, views, and the environment they were working in. Teachers attended PD workshops for six months where activities that were meaningful to their context were implemented. After attending the PD workshops, teachers were encouraged to incorporate the new ideas on MPSI in their lessons. Data gathered from direct participant observations and in-depth interviews were analysed through thematic analysis. Two contextual sources emerged from the study as linked to teachers’ PD: context that relates to teachers’ personal practice and context that relates to teachers’ environment. The types of context are discussed in light of how they can support teachers’ PD in MPSI in resource-constrained contexts.
Mathematical problem solving in South Africa: Research and practice is written for researchers and mathematics education practitioners interested in quality research, methodological rigour and potentially transformative implications to assist them in understanding the teaching and learning of mathematical problem solving in South Africa. As such, the main aim of this book is to present current research on problem solving in the Southern African context. The secondary aim is to acknowledge the contextual nature of educational research and encourage others from different backgrounds to present what they are seeing and thinking around mathematical problem solving.
İlkokul dördüncü sınıf Fen Bilimleri dersinde günlük yaşam problemlerini çözme becerileri temelli çoklu zekâ kuramının uygulanabilirliğini ortaya koymayı amaçlayan bu araştırma nitel araştırma yöntemlerinden biri olan eylem araştırması deseniyle gerçekleştirilmiştir. Araştırmanın katılımcılarını İstanbul’daki bir devlet ilkokulunda 2022-2023 öğretim yılında 4. sınıfa devam eden 23 öğrenci ile onların velileri ve sınıf öğretmeni oluşturmuştur. Araştırmanın verileri, video kayıtları ve gözlem formlarından elde edilen gözlem verileri, araştırmacı günlükleri, öğrenci ürünleri, değerlendirme formları, öz-değerlendirme formları ile öğrenciler, veliler ve sınıf öğretmeniyle gerçekleştirilen yarı yapılandırılmış görüşmeler yoluyla toplanmıştır. Araştırmanın verileri içerik analizi ile çözümlenmiştir. Araştırmada elde edilen verilerin analizi sonucunda “süreç”, “bilişsel”, “duyuşsal”, “beceri” ve “sorunlar ve çözümler” olmak üzere beş tema ortaya konmuştur. Süreç boyutunda; Çoklu Zekâ kuramı bağlamında hazırlanan ders planlarının öğrencilere ilgili kazanımların kazandırılmasında kullanılabilecek uygun ders planları olduğu anlaşılmıştır. Bilişsel boyutta; öğrencilerin konuları anlakları, kavradıkları ve akademik başarılarının arttığı ve etkili öğrenmelerine katkı sağladığı görülmüştür. Duyuşsal boyutta öğrencilerin süreçte mutlu oldukları, keyif aldıkları, eğlendikleri, heyecan duydukları, etkinliklere istekle katıldıkları, öz-güven kazandıkları ve öz-yeterlik algısı geliştirdikleri gözlenmiştir. Beceri boyutunda öğrencilerin günlük yaşamla ilişkilendirme ve iletişim kurma becerilerini geliştirdiği ortaya çıkmıştır. Sorunlar ve öneriler boyutunda ise öğrencilerin gerçekleştirilen her etkinlikte görev almak istemeleri ve çoklu zekâya dayalı etkinliklerde daha fazla söz alma ve düşüncelerini ifade etme isteği taşımalarının sorun oluşturduğu sonucuna ulaşılmıştır.
Bu araştırmanın amacı, ortaokul matematik öğretmeni adaylarının sayıların öğretiminde kullandıkları gerçek hayat ilişkilendirmelerinin incelenmesidir. Bu amaçla, öğretmen adaylarının gerçek hayatla ilişkilendirmeyi nasıl yaptıkları, ilişkilendirme yaparken hangi bağlamları kullandıkları ve gerçek hayatla ilişkilendirme konusunda ne düşündükleri belirlenmiştir. Araştırmada nitel araştırma yöntemlerinden durum çalışması kullanılmıştır. Araştırmanın çalışma grubu bir devlet üniversitesinin ilköğretim matematik öğretmenliği programına devam eden öğretmen adaylarından oluşmaktadır. Veriler gerçek hayat ilişkilendirmelerine yönelik Gainsburg’un (2008) kodlar ve içerik analizi kullanılarak analiz edilmiştir. Araştırmanın bulgularına göre, öğretmen adayları genel olarak gerçek hayat ilişkilendirmelerinde sözel klasik problemleri ve alışveriş- ticaret bağlamlarını kullanmışlardır. Ayrıca öğretmen adaylarıyla yapılan görüşmeler sonucunda, öğretmen adaylarının sayılar konusunun gerçek hayatla ilişkilendirilerek öğretilmesi gerektiğini düşündükleri belirlenmiştir.
Generating multiple solutions is a promising approach in order to foster deep insight and understanding in mathematical contexts. However, in complex domains, students often struggle to generate just one solution, let alone more than one. This study investigates whether collaboration supports learners to generate multiple solutions, and to what extent collaboration and generating multiple solutions foster the acquisition of modeling competencies. In an experiment, students either learned alone or in dyads which were homogeneous or heterogeneous with respect to the level of prior knowledge, and they were prompted to generate one or multiple solutions in the learning phase (N = 193 seventh and eighth graders). Learning in homogeneous dyads fostered the acquisition of modeling competencies. Being prompted to generate a second solution was beneficial only for learners with high prior knowledge; overall, actually generating a second solution had positive impact.
This study investigates primary school children's practices of premising (the handling of task premises) and arguing when confronted with a dilemmatic school task as part of a group discussion. As such, the study contributes to the field of education and argumentation with new insights into how children negotiate validity and relevance in relation to the task when asked to argue their stance on an equity/equality matter. The data consist of transcripts of 247 min of video recordings capturing 13 group discussions (54 children, split into groups of four to five, and two teachers) in the German primary school subject Sachunterricht. The findings clarify how the children dynamically move in and out of the task premises, and at times question them, both implicitly and explicitly. The task's openness and ambiguity are discussed in relation to the children's arguing and premising. The teacher's role as mediator in group discussions is contrasted with other forms of classroom discourse. Furthermore, the relevance for researchers to consider the social and cultural circumstances of research in schools – for example, when interpreting children's utterances – is emphasized.
This study proposes a theoretical view for bridging mathematical modeling and word problem-solving activities. We introduce and elaborate on two theoretical ideas of the fictionality of word problems and the creation of possible fictional worlds. A world described by a word problem exists only fictionally (or potentially). A fictional world includes any imaginable world, any model for the real world, and any mathematical model. We developed a semi-open problem based on these theoretical ideas and observed Japanese eighth-grade students’ activity when solving it in an experimental lesson. Consequently, we identified a theoretically overlooked type of validation: considering the cultural relevance of solutions. The most important implication we draw from our observation is that the current definition of validation as a comparison between two stages in modeling should be extended to consider the integration of a target into a base possible fictional world.
Este estudio tiene como objetivo analizar cómo estudiantes españoles de 3º a 6º grado (8-12 años) resuelven problemas realistas, en particular, de división-medida y división-partitiva con resto. Los participantes resolvieron seis problemas y el análisis se centró en el nivel de éxito y las estrategias que utilizaron a lo largo de los grados. Los resultados muestran dificultades al resolver este tipo de problemas en todos los grados. Sin embargo, también indican una progresión en el nivel de éxito a lo largo de la educación primaria. Aunque el algoritmo fue la estrategia más utilizada por el alumnado, emplearon otras estrategias correctas como la gráfica en grados iniciales y para un tipo concreto de problemas. Nuestros resultados sugieren que la sustitución de estrategias gráficas por el algoritmo no conlleva siempre una comprensión de la situación.
Abstract: The objective of this study is to analyze how Spanish students in the third to the sixth grades (ages eight to twelve) solve realistic problems, in particular, quotative and partitive division problems with a non-zero remainder. The participants solved six problems, and the analysis focused on their level of success and the strategies they used in the different grades. The results show student difficulties in solving this type of problems, in all grades. Nevertheless, they also indicate an increase in the level of success throughout elementary education. Although the use of the algorithm is the strategy the students use the most, they also employed other correct strategies such as graphic strategies in the early years, for a specific type of problem. Our results suggest that replacing graphic strategies with the algorithm does not always imply student understanding of the situation. Palabras clave: solución de problemas; educación matemática; estrategias; educación primaria.
In this study, about 200 middle school students solved an augmented-quotient division-with-remainders problem, and their solution processes and interpretations were examined. Based on earlier research, semantic-processing models were proposed to explain students' success or failure in solving division-with-remainder story problems on the basis of the presence or absence of an adequate interpretation provided by the solver after obtaining a numerical solution. In this study, students' solutions and their attempts and failures to "make sense" of their answers were analyzed for evidence that supported or refuted the hypothesized semantic-processing models. The results confirmed that the models provide a solid explanation of students' failure to solve division-with-remainder problems in school settings. More generally, the results indicated that student performance was adversely affected by their dissociation of sense making from the solution of school mathematics problems and their difficulty in providing written accounts of their mathematical thinking and reasoning.
An analysis of everyday use of mathematics by working youngsters in commercial transactions in Recife, Brazil, revealed computational strategies different from those taught in schools. Performance on mathematical problems embedded in real-life contexts was superior to that on school-type word problems and context-free computational problems involving the same numbers and operations. Implications for education are examined.
explores the ways that mathematics is understood and used in our culture and the role that schooling plays in shaping those mathematical understandings / one of its goals is to blur the boundaries between formal and informal mathematics: to indicate that, in real mathematical thinking, formal and informal reasoning are deeply intertwined / begin, however, by briefly putting on the formalist's hat and defending formal reasoning (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Arguments are put forward in this paper that classroom word problem solving is more-and also less-than the urgent analysis of a factual structure, in the sense that it is essentially a species of a social-cognitive activity. Word-or story-problems, presented in classroom contexts, represent textual and pragmatic patterms of a certain grammaticality. To present a problem verbally to a student means to organize a fact in some way for the attention of a problem solver. There is not only the structure of the problem text itself by which situations are denoted, but there is also the stimulative nature of the social-pragmatic context which shapes the student's textbook-problem solving behavior over a long period of time.
The present paper discusses the results of several studies showing, for example, that subject matter related attitudes towards a problem frequently do not play an important part in the problem solving efforts; that students often solve problems correctly without understanding them; and that false contextual expectations can lead to abstruse errors of understanding and to peculiar solution attempts.
The studies indicate that students can become sensitive and skilful in perceiving and capitalizing on subtle textual and contextual signs pointing to the solution and anticipating its pattern. It seems that usual textbook problems let students get accustomed to certain courses of processing where a simple fact, like whether an equation works out evenly or does not, can stop the process or push it further. It is argued that the deeper reason for the observed textual and contextual influences on understanding and problem solving lies in a fundamental weakness of the student's epistemic control behavior. The psychological and instructional significance of the studies is discussed.
Results from the third national mathematics assessment indicate that the decline in performance of 17 year olds between 1973 and 1978 has leveled out over the last four years, and 13 year olds' performance improved significantly between 1978 and 1982. However, most gains were on lower-order skills. (MNS)
Assessed the degree to which 100 13- and 14-yr-olds adjusted assumptions of direct proportionality suggested by the surface structure of word problems. Results confirm that students frequently proffer answers that are unrealistic. Ss tended to respond to word problems according to stereotyped procedures assuming that the modeling of the situation described is "clean." A radical shift in perspective is proposed, whereby word problems are conceived as exercises in modeling, with proper consideration of the assumptions and appropriateness of the model underlying any proposed solution. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Presents a model of the optimal problem-solving process with respect to simple addition and subtraction word problems, focusing on the construction of the initial mental representation of the work problem. The competent problem-solving model is reviewed, and its design and techniques are described. Errors due to word problem schema deficiencies and faulty semantic representation, including errors on compare, cause/change, and combine problems, are discussed. It is concluded that findings support the importance of semantic processing in word problem solving, as maintained by J. G. Greeno (1982). (21 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
multiplication and division of positive integers and rational numbers may be considered relatively simple from a mathematical point of view / the research reviewed in this chapter, however, reveals the psychological complexity behind the mathematical simplicity / in particular, this complexity is manifested when the operations are considered not just from the computational point of view, but in terms of how they model situations
the chapter is in four parts / in the first, the range of applications of multiplication and division is set out, and the corresponding variety of external representations for the operations is illustrated / in the second part, the complexity is demonstrated further by reviewing current theoretical frameworks that treat the topic from several different perspectives / there follows an outline of a broader framework, the key points of which are that (a) multiplication and division model many distinguishable classes of situations, and (b) a fundamental conceptual restructuring is necessary when multiplication and division are extended beyond the domain of positive integers / [considers] the potential of computer representations, putting forward suggestions for the improvement of teaching about multiplication and division (PsycINFO Database Record (c) 2012 APA, all rights reserved)
We review the work on one-step word problems involving either an addition, a subtraction, a multiplication or a division. Moreover, this review focuses on studies that have been carried out since the late 1970s. By that time new ideas and methods from the information-processing approach led to the emergence of a firm new research paradigm in the area of elementary addition and subtraction word problems, which resulted in a large number of findings and conclusions.
The chapter starts with a discussion of the classifications of one-step arithmetic word problems around the different phases that are typically distinguished in mathematical problem solving in general, and in the solution of one-step arithmetic word problems in particular: the representation of the problem, the selection and execution of a solution strategy, and the interpretation and verification of the result. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
An overview of accumulating research on children's out-of-school mathematics raises critical questions about how children come to form mathematical understandings in their out-of-school activities and about the interplay between informal and school mathematics learning. The author investigates these questions in an illustrative multimethod study on Brazilian child candy sellers. Findings show that sellers with little or no schooling develop in their practice a complex mathematics that contrasts sharply with school mathematics. Further, analyses reveal an interplay between what they learn in selling candy and what they learn at school: Sellers in school use their street mathematics to work school mathematics problems, and schooled sellers use some limited aspects of their school, mathematics to solve problems in their practice.
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.
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