Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative

Borji, Vahid; Ahadi, Hassan; Radmehr, Farzad

doi:10.29333/ejmste/91451

Cited by 19 publications

(24 citation statements)

References 35 publications

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“…Students in the first year of the university face difficulties in solving the calculus questions where input information is given only through graphical or numerical representations (Borji, Font, Alamolhodaei, & Sánchez, 2018;Baker et al, 2000;Cooley et al, 2007). Many research have recommended that, in order to achieve a conceptual understanding of a mathematical concept, it is better to start learning that concept in its various representations and connections between them (Baker et al, 2000;Borji, Alamolhodaei, & Radmehr, 2018;Dreher & Kuntze, 2015;Goldin & Shteingold, 2001;Kendal & Stacey, 2003;Özmantar et al, 2010;Pape & Tchoshanov, 2001;Ronda, 2015). Students usually learn derivative and integral concepts in high school for the first time.…”

Section: Discussionmentioning

confidence: 99%

“…Many researchers (e.g., Baker, et al, 2000;Borji, Alamolhodaei, & Radmehr, 2018;Dreher & Kuntze, 2015;Goldin & Shteingold, 2001;Kendal & Stacey, 2003;Özmantar et al, 2010;Pape & Tchoshanov, 2001;Ronda, 2015) have explored the importance of representations in developing mathematical understanding. Goldin and Shteingold (2001) explained that representations (algebraic, graphical, numerical, and textual) are important for the learning of mathematical concepts because of the coherent structure contained within each representation.…”

Section: / 12mentioning

confidence: 99%

“…Calculus students usually have acceptable performance in solving routine questions that are represented algebraically (Baker, Cooley, & Trigueros, 2000). Their difficulties begin when they see questions in the form of other representations, such as graphical and numerical representations (Baker, et al, 2000;Borji, Alamolhodaei, & Radmehr, 2018;Sahin, Erbas, & Yenmez, 2015). A number of students do not have a conceptual understanding of graphical and numerical representations of calculus concepts (Kendal & Stacey 2003).…”

Section: Introductionmentioning

confidence: 99%

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An Exploratory Analysis of the Representations of Functions in the University Entrance Exam in Spain and Iran

Borji

Sánchez

2019

EURASIA J MATH SCI T

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The University Entrance Exam (UEE) has an important impact on how math instructors teach mathematics to prepare students for this exam as well as how their students learn mathematics. Since the UEE is like a bridge between high school and university, its importance becomes twofold. However, research regarding this issue is very limited in mathematics education. The purpose of this research is to conduct an exploratory analysis of mathematics questions from the UEE in Spain and Iran from the point of view of representations of functions (i.e., algebraic, graphical, numerical and explanations with words). For this, the mathematics questions related to derivatives and integrals, which are two of the most important concepts in mathematics, from the last five years of the UEE in these two countries were considered to analyze them from the point of view of different representations. The results showed that most of the derivative and integral questions in the UEE in both countries were related to the algebraic representation. However, a very small percentage of the derivative and integral questions were related to the graphical and numerical representations and the relationships between the representations. At the end, suggestions to the curriculum planners and designers of mathematics questions of the UEEs are given.

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Section: Discussionmentioning

confidence: 99%

Section: / 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An Exploratory Analysis of the Representations of Functions in the University Entrance Exam in Spain and Iran

Borji

Sánchez

2019

EURASIA J MATH SCI T

Self Cite

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“…It should be noted that a GD is not unique, one mathematical concept can has more than one GD. A GD is as a useful cognitive model, as evidenced by the results of several empirical studies that show the effectiveness of the APOS as an efficient tool for design and analysis of instruction (Borji, Alamolhodaei, & Radmehr 2018;Weller, Arnon, & Dubinsky, 2011).…”

Section: Apos Theorymentioning

confidence: 99%

Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA

Borji

Font

2019

EURASIA J MATH SCI T

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Our goal in this paper is to study students' understanding of integration by parts based on two theories, APOS and OSA. We make an epistemic configuration (EC) of primary objects that a student activate for solving tasks in relation to the integration by parts, and then we design a genetic decomposition (GD) of mental constructions that he/she might need to learn the integration by parts. We then describe the EC and GD in terms of the levels of development of Schema (i.e., intra, inter and trans). Three tasks in a semi-structured interview were used to explore twenty three first-year students' understanding of integration by parts and classify their schemas. Results showed that students had difficulties in integration by parts, especially in using this technique to obtain a simpler integral than the one they started with. Using APOS and OSA gave us a clear insight about students' difficulties and helped us to better describe students' understanding of integration by parts.

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“…Mathematical software and graphing calculator (Chien, 2019;Sevimli, 2016) are among the technology tools that have been surveyed. Despite the positive results reported, some researchers have argued about its effectiveness to accommodate students of different learning styles, pace, and engagement levels (Borji, Alamolhodaei & Radmehr, 2018). Researchers commented that technology will never replace the role and effectiveness of hands-on learning approach, mainly due to the implementation difficulties in terms of the level of integrating the technology, and its didactical nature in terms of the usage (Hong & Thomas, 2015).…”

Section: Introductionmentioning

confidence: 99%

Bridging the Gap Between the Derivatives And Graph Sketching in Calculus

Bakri

Liew

Chen

et al. 2021

AJUE

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Sketching the graph of mathematical functions using derivatives is a challenging task for undergraduate students who enrol for the first level of calculus course. Before graph plotting, students are required to perform a thorough function analysis using the concepts learned in differentiation. They are then expected to solicit the results obtained to sketch the graph. Nevertheless, the students face great difficulties in achieving this goal; and fail to relate the results obtained in the analysis and their representation in a graph. Their performance is thus negatively affected eventually. To overcome this cognitive gap, an innovative board game named Graph Puzzle (GP) is developed. It is intended to function as a manipulator to facilitate students in comprehending the inter-related algebraic, symbolic, and graphic representation of a function under the applications of derivatives, forming the corresponding procedural and conceptual knowledge. To measure the effectiveness of this board game, 84 undergraduate students who took this calculus course were given pre- and post-test before and after the learning session. An ANCOVA test conducted reveals a significant difference (F (1, 81) = 12.182, p = 0.001) between pre and post-test score in solving polynomial functions, whereas students’ performance in solving rational functions indicates no meaningful difference (F (1, 81) = 0.04, p = 0.841) of post-scores between control and treatment groups. From this standpoint, it is shown that GP has the potential to serve as a solution to the difficulties faced on graph sketching in calculus, particularly when dealing with polynomial functions. Keywords: Visualization, Calculus, Embodied learning, Game-Based Learning, Graph Sketching

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Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative

Cited by 19 publications

References 35 publications

An Exploratory Analysis of the Representations of Functions in the University Entrance Exam in Spain and Iran

An Exploratory Analysis of the Representations of Functions in the University Entrance Exam in Spain and Iran

Exploring Students’ Understanding of Integration by Parts: A Combined Use of APOS and OSA

Bridging the Gap Between the Derivatives And Graph Sketching in Calculus

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