The development of students’ use of additive and proportional methods along primary and secondary school

Fernández, Ceneida; Llinares, Salvador; Dooren, Wim Van; Bock, Dirk De; Verschaffel, Lieven

doi:10.1007/s10212-011-0087-0

Cited by 42 publications

(33 citation statements)

References 10 publications

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“…Despite the paramount importance of proportionality, mastering it remains a challenge for school curriculum (Tourniaire and Pulos, 1985; Lamon, 2012). In particular, students experience difficulty in developing fluency with proportions that build upon – yet are differentiated from – simpler non-multiplicative concepts (e.g., additive constructions), notations, terminology, and procedures (Karplus et al, 1983; Tourniaire and Pulos, 1985; Lamon, 2007; Fernández et al, 2012). …”

Section: Theoretical Backgroundmentioning

confidence: 99%

Touchscreen Tablets: Coordinating Action and Perception for Mathematical Cognition

et al. 2017

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Proportional reasoning is important and yet difficult for many students, who often use additive strategies, where multiplicative strategies are better suited. In our research we explore the potential of an interactive touchscreen tablet application to promote proportional reasoning by creating conditions that steer students toward multiplicative strategies. The design of this application (Mathematical Imagery Trainer) was inspired by arguments from embodied-cognition theory that mathematical understanding is grounded in sensorimotor schemes. This study draws on a corpus of previously treated data of 9–11 year-old students, who participated individually in semi-structured clinical interviews, in which they solved a manipulation task that required moving two vertical bars at a constant ratio of heights (1:2). Qualitative analyses revealed the frequent emergence of visual attention to the screen location halfway along the bar that was twice as high as the short bar. The hypothesis arose that students used so-called “attentional anchors” (AAs)—psychological constructions of new perceptual structures in the environment that people invent spontaneously as their heuristic means of guiding effective manual actions for managing an otherwise overwhelming task, in this case keeping vertical bars at the same proportion while moving them. We assumed that students’ AAs on the mathematically relevant points were crucial in progressing from additive to multiplicative strategies. Here we seek farther to promote this line of research by reanalyzing data from 38 students (aged 9–11). We ask: (1) What quantitative evidence is there for the emergence of AAs?; and (2) How does the transition from additive to multiplicative reasoning take place when solving embodied proportions tasks in interaction with the touchscreen tablet app? We found that: (a) AAs appeared for all students; (b) the AA-types were few across the students; (c) the AAs were mathematically relevant (top of the bars and halfway along the tall bar); (d) interacting with the tablet was crucial for the AAs’ emergence; and (e) the vast majority of students progressed from additive to multiplicative strategies (as corroborated with oral utterances). We conclude that touchscreen applications have the potential to create interaction conditions for coordinating action and perception into mathematical cognition.

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Section: Theoretical Backgroundmentioning

confidence: 99%

Touchscreen Tablets: Coordinating Action and Perception for Mathematical Cognition

et al. 2017

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“…Examples of the misuse of linearity can be found at different age levels and in various mathematical and scientific domains (Fernández, Llinares, Van Dooren, De Bock, & Verschaffel, 2012; for a review, see Van Dooren, De Bock, Janssens, & Verschaffel, 2008). For instance, in a study on arithmetic word problems solving by Cramer, Post and Currier (1993) De Bock, Hessels, Janssens, and Verschaffel (2005) observed that Flemish primary school pupils' performance on linear word problems considerably improved from 3rd to 6th grade.…”

Section: The Illusion Of Linearitymentioning

confidence: 99%

“…Problems were formulated in a missing-value format, just like in previous investigations on students' improper linear reasoning (De Bock et al, 1998;Fernández et al, 2012) and we asked for the perimeter or the area in an indirect way, i.e. by using a variable that is proportionally related to the perimeter or area.…”

Section: Design Of the Paper-and-pencil Testmentioning

confidence: 99%

Do students confuse dimensionality and “directionality”?

Fernández

Bock

Verschaffel

et al. 2014

The Journal of Mathematical Behavior

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“…Children making this error answer "45 laps" to the aforementioned runner problem (i.e., 3×5=15, so 9×5=45). This error has been found mainly in upper primary education (Fernández, Llinares, Van Dooren, De Bock, & Verschaffel, 2012;, and in missingvalue problems wherein the numbers form integer number ratios (Van Dooren et al, 2009;. As for the inverse mistake of erroneously solving multiplicative word problems additively, the internal ratio has been found to have the largest impact here too.…”

Section: The Multiplicative Error In Additive Missing-value Word Probmentioning

confidence: 88%

To add or to multiply? An investigation of the role of preference in children's solutions of word problems

Degrande

Verschaffel

Dooren

2019

Learning and Instruction

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Previous research has shown that upper primary school children frequently erroneously solve additive word problems multiplicatively, while younger children frequently erroneously solve multiplicative word problems additively. It has been suggested that children's preference for additive or multiplicative relations explains these errors, besides their lacking skills, but this claim has not been tested empirically yet. Therefore, we administered four test instruments (a word problem test, a preference test, and two tests measuring additive and multiplicative computation and discrimination skill) to 246 third to sixth graders. Previous research results on errors in word problems, as well as on preference were replicated and systematized. Further, they were extended by explaining this erroneous word problem solving behavior by preference, for those children who unmistakably had acquired the necessary computation and discrimination skills. This finding provides strong evidence for the unique additional role of children's preference in erroneous additive or multiplicative word problem solving behavior.

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The development of students’ use of additive and proportional methods along primary and secondary school

Cited by 42 publications

References 10 publications

Touchscreen Tablets: Coordinating Action and Perception for Mathematical Cognition

Touchscreen Tablets: Coordinating Action and Perception for Mathematical Cognition

Do students confuse dimensionality and “directionality”?

To add or to multiply? An investigation of the role of preference in children's solutions of word problems

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