The Overgeneralization of Linear Models among University Students' Mathematical Productions: A Long-Term Study

Esteley, Cristina; Villarreal, Mónica; Alagia, Humberto

doi:10.1080/10986060903465988

Cited by 16 publications

(13 citation statements)

References 6 publications

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“…Empirically, students' strong tendency to engage in linear overgeneralizations has been supported by a large amount of research that has included different age groups ranging from primary school (Van Dooren et al, 2005) to university students (Esteley et al, 2010) and has referred to different mathematical domains such as arithmetic word problems (Van Dooren et al, 2005, algebraic patterns (Stacey, 1989), geometry (De Bock et al, 1998Ayan and Bostan, 2018), and probability (Van Dooren et al, 2003). More specifically, linear overgeneralizations seemed to increase after linear problems were taught in class (Van Dooren et al, 2005), supporting the assumption that students' experiences with linear concepts in the mathematics classroom are responsible for their strong tendency to engage in linear overgeneralizations.…”

Section: Linear Overgeneralizationsmentioning

confidence: 99%

When Can Making a Drawing Hinder Problem Solving? Effect of the Drawing Strategy on Linear Overgeneralizations and Problem Solving

Krawitz

Schukajlow

2020

Front. Psychol.

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The strategy of making a drawing has been claimed to facilitate mathematical problem solving. However, De Bock et al. (2003) surprisingly found that drawing negatively affected performance in solving non-linear geometry problems, in which the area or volume of similar figures or solids had to be determined by a given scaling factor. The authors suggested that making a drawing increased the number of overgeneralizations and negatively affected students' performance. Our study involves a partial replication and also an important validation and extension of this study by addressing two factors: low-quality drawing strategy and poor visual monitoring, both of which might explain the negative effect of drawing. First, we expected that improving the quality of the drawing strategy by prompting students to highlight important information in their drawings would diminish the negative effect of the drawing strategy. Second, we expected that fostering visual monitoring while drawing, by offering problems with small scaling factors, would diminish the negative effect of the drawing strategy. We conducted a randomized controlled trial with 180 students (ninth-to eleventh-graders) to investigate the effects of drawing and visual monitoring on solving non-linear geometry problems. Our results replicated the previous finding that drawing negatively affects performance. We confirmed that linear overgeneralizations are a prevalent reason for this finding. Elaborating on previous findings revealed that the quality of the drawing strategy but not visual monitoring was responsible for the effect of the drawing strategy on linear overgeneralizations. Furthermore, an exploratory analysis of students' awareness of linear overgeneralizations indicated that improving the quality of drawing strategy and enhancing visual monitoring did not lead to a greater awareness of the mistakes learners made because of linear overgeneralizations. We conclude that the way the drawing strategy is used determines whether it is useful or damaging, and more efforts are essential to enable students to apply it appropriately.

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Section: Linear Overgeneralizationsmentioning

confidence: 99%

When Can Making a Drawing Hinder Problem Solving? Effect of the Drawing Strategy on Linear Overgeneralizations and Problem Solving

Krawitz

Schukajlow

2020

Front. Psychol.

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“…For example, Greer [25] summarized how students worldwide tend to apply linear thinking to non-linear settings in mathematical concepts ranging from kindergarten to university level. Esteley, Villarreal, and Alagia [26] detailed this phenomenon with university-level students applying linear thinking to trigonometric relationships, specifically the cosine function.…”

Section: Quantitative Reasoning Frameworkmentioning

confidence: 99%

Promoting uncertainty to support preservice teachers’ reasoning about the tangent relationship

Glassmeyer

Brakoniecki

Amador

2018

International Journal of Mathematical Education in Science and

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Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers' thinking about a trigonometric relationship was impacted by a series of tasks that prompted uncertainty. Using dynamic geometry software, we asked preservice teachers to compare angle measures of lines on a coordinate grid to their slope values, beginning by investigating lines whose angle measures were in a near-linear relationship to their slopes. After encountering and resolving the uncertainty of the exact relationship between the values, preservice teachers connected what they learned to the tangent relationship and demonstrated new ways of thinking that entail quantitative and covariational reasoning about this trigonometric relationship. We argue that strategically using uncertainty can be an effective way for promoting preservice teachers' reasoning about the tangent relationship.

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“…Siguiendo a Blum [6], los procesos de modelización se pueden estructurar en cinco fases principales: Estos procesos no siempre son sencillos para los estudiantes, y se han documentado dificultades en la modelización como la presencia excesiva de modelos lineales en situaciones que no lo requieren (Esteley, Villarreal i Alagia [13]).…”

Section: Modelizaciónunclassified

Problemas de estimación de magnitudes no alcanzables: una propuesta de aula a partir de los modelos generados por los alumnos

Albarracín

Gorgorió

2013

Model. sci. educ. learn.

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En este artículo presentamos una propuesta didáctica para introducir la modelización en las aulas de Educación Secundaria. La propuesta presentada se basa en el uso de un conjunto de problemas de Fermi orientados a estimar el valor de grandes cantidades. Para concretarla se utilizan los resultados de un estudio previo en el que se describen las estrategias propuestas por los alumnos a este tipo de problemas. In this paper we present a proposal to introducing modeling in Secondary School classrooms. The proposal is based on the use of a set of Fermi problems related to estimating large quantities. It uses the results of a previous study which describes the strategies proposed by the students to solve this kind of problems.

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The Overgeneralization of Linear Models among University Students' Mathematical Productions: A Long-Term Study

Cited by 16 publications

References 6 publications

When Can Making a Drawing Hinder Problem Solving? Effect of the Drawing Strategy on Linear Overgeneralizations and Problem Solving

When Can Making a Drawing Hinder Problem Solving? Effect of the Drawing Strategy on Linear Overgeneralizations and Problem Solving

Promoting uncertainty to support preservice teachers’ reasoning about the tangent relationship

Problemas de estimación de magnitudes no alcanzables: una propuesta de aula a partir de los modelos generados por los alumnos

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