Mathematical Problem-Solving Processes and Performance: Translation Among Symbolic Representations

1990 ◽  
Author(s):  
Noreen M. Webb ◽  
Karen Gold ◽  
Sen Qi
Keyword(s):  
Problem Solving ◽  
Mathematical Problem ◽  
Mathematical Problem Solving ◽  
Symbolic Representations ◽  
And Performance

Defragmenting structures of students’ translational thinking in solving mathematical modeling problems based on CRA framework

2020 ◽  
Vol 13 (2) ◽  
pp. 130-151
Author(s):  
Kadek Adi Wibawa ◽  
I Putu Ade Andre Payadnya ◽  
I Made Dharma Atmaja ◽  
Marius Derick Simons
Keyword(s):  
Mathematical Modeling ◽  
Problem Solving ◽  
Undergraduate Students ◽  
Mathematical Problem ◽  
Mathematical Problem Solving ◽  
Graph Representations ◽  
Symbolic Representations ◽  
Mathematics Educators ◽  
Three Stages ◽  
Algebraic Forms

 [English]: The fragmentation of thinking structure is a failed construction existing in students’ memory due to disconnections on what they have learned. It makes students undergo difficulties and errors in solving mathematical modeling problems. There is a need to prevent permanent fragmentations. The problem-solving involving modeling problems requires translational thinking, changing from source representations to targeted representations. This research aimed to formulate undergraduate students’ effort in restructuring their fragmented translational thinking (defragmentation of translational thinking structure). The defragmentation was mapped through the CRA framework (checking, repairing, ascertaining). The subjects were three of eighty-five 4th and 6th-semester students. Data were analyzed through three stages; categorization, reduction, and conclusion. The analysis resulted in three types of defragmentation of translational thinking structure: from verbal representations to graph representations, from graph representations to symbolic representations (algebraic forms), and from the graph and symbolic representations to mathematical models. The finding shows that it is essential for mathematics educators to allow students to manage their thinking structures while experiencing difficulties and errors in mathematical problem-solving. Keywords: Thinking structure, Fragmentation, Defragmentation, Translational thinking, CRA framework  [Bahasa]: Fragmentasi struktur berpikir merupakan kegagalan konstruksi yang terjadi di dalam memori akibat dari konsep-konsep yang dipelajari tidak terkoneksi dengan baik. Hal ini membuat mahasiswa sering mengalami kesulitan dan kesalahan dalam memecahkan masalah pemodelan matematika. Untuk itu, perlu dilakukan upaya agar tidak terjadi fragmentasi struktur berpikir yang permanen. Dalam memecahkan masalah pemodelan matematika, mahasiswa perlu melakukan berpikir translasi, yaitu mengubah representasi sumber menjadi representasi yang ditargetkan. Penelitian ini bertujuan untuk merumuskan upaya mahasiswa dalam melakukan penataan fragmentasi struktur berpikir translasi yang terjadi (defragmentasi struktur berpikir translasi) dalam memecahkan masalah pemodelan matematika. Defragmentasi yang dilakukan mahasiswa dipetakan melalui kerangka CRA (checking, repairing, dan ascertaining). Subjek penelitian adalah mahasiswa semester 4 dan 6 yang terdiri dari 3 orang dipilih dari 85 mahasiswa. Analisis data dilakukan melalui tiga tahap, yaitu pengategorian data, reduksi data, dan penarikan kesimpulan. Penelitian ini menemukan tiga jenis defragmentasi struktur berpikir translasi: defragmentasi dari representasi verbal ke grafik, dari representasi grafik ke simbol (bentuk aljabar), dan representasi grafik dan simbol (bentuk aljabar) ke model matematika. Penelitian ini menunjukkan pentingnya pengajar matematika memberikan kesempatan kepada mahasiswa dalam menata struktur berpikirnya ketika mengalami kesulitan dan kesalahan dalam memecahkan masalah matematika. Kata kunci: Struktur berpikir, Fragmentasi, Defragmentasi, Berpikir translasi, Kerangka CRA

scholarly journals Integration of Autograph in Improving Mathematical Problem Solving and Mathematical Connection Ability Using Cooperative Learning Think-Pair-Share

2015 ◽  
Vol 5 (1) ◽  
pp. 83-95
Author(s):  
Ida Karnasih ◽  
Mariati Sinaga
Keyword(s):  
Problem Solving ◽  
Cooperative Learning ◽  
Mathematical Problem ◽  
Attitude Scale ◽  
Repeated Measure ◽  
Mathematical Problem Solving ◽  
Dynamic Software ◽  
And Performance ◽  
Teaching Learning ◽  
Mathematical Connection

The aim of this study was to investigate students’ mathematical problem solving and mathematical connection ability in cooperative learning setting using Dynamic Software Autograph. This experimental study was conducted at high school in learning statistics. The collection of the data was done using observation sheets, documentation, attitude scale, and performance tests. Repeated measure tests were delivered to students for four times. The result of the analysis showed that: (1) Using Dynamic Software Autograph in teachinglearning statistics with cooperative learning Think-Pair-Share improved students’ problem solving and mathematical connection ability; (2) Students’ activity during teaching learning processes continuously improved; (3) The result of analysis of the questionnaire showed that most students like learning statistics using cooperative learning with dynamic software Autograph; (4) Students were very active and showed positive attitude toward learning using cooperative learning Think-Pair-Share using dynamic software Autograph.

scholarly journals Relation between Pupils’ Mathematical Self-Efficacy and Mathematical Problem Solving in the Context of the Teachers’ Preferred Pedagogies

2020 ◽  
Vol 12 (23) ◽  
pp. 10215
Author(s):  
Vlastimil Chytrý ◽  
Janka Medová ◽  
Jaroslav Říčan ◽  
Jiří Škoda
Keyword(s):  
Problem Solving ◽  
Self Efficacy ◽  
Primary Schools ◽  
Mathematical Problem ◽  
Mathematical Problem Solving ◽  
Significant Difference ◽  
Montessori Schools ◽  
Mathematical Problems ◽  
The Difference ◽  
And Performance

In research focused on self-efficacy it is usually teacher-related phenomena that are studied, while the main aspects related to pupils are rather neglected, although self-efficacy itself is perceived as a belief in one’s own abilities. Evidently, this strongly influences the behavior of individuals in terms of the goal and success in mathematical problem-solving. Considering that alternative teaching methods are based on the principle of belief in one’s own ability (mainly in the case of group work), higher self-efficacy can be expected in the pupils of teachers who use predominantly the well-working pupil-centered pedagogies. A total of 1133 pupils in grade 5 from 36 schools in the Czech Republic were involved in the testing of their ability to solve mathematical problems and their mathematical self-efficacy as well. Participants were divided according to the above criteria as follows: (i) 73 from Montessori primary schools, (ii) 332 pupils educated in mathematics according to the Hejný method, (iii) 510 pupils from an ordinary primary school, and (iv) 218 pupils completing the Dalton teaching plan. In the field of mathematical problem-solving the pupils from the Montessori primary schools clearly outperformed pupils from the Dalton Plan schools (p = 0.027) as well as pupils attending ordinary primary schools (p = 0.009), whereas the difference between the Montessori schools and Hejný classes was not significant (p = 0.764). There is no statistically significant difference in the level of self-efficacy of pupils with respect to the preferred strategies for managing learning activities (p = 0.781). On the other hand, correlation between mathematical problem-solving and self-efficacy was confirmed in all the examined types of schools. However, the correlation coefficient was lower in the case of the pupils from the classes applying the Hejný method in comparison with the pupils attending the Montessori schools (p = 0.073), Dalton Plan schools (p = 0.043), and ordinary primary schools (p = 0.002). Even though the results in mathematical problem-solving are not consistent across the studies, the presented results confirm better performance of pupils in some constructivist settings, particularly in the case of individual constructivism in the Montessori primary schools. The factors influencing lower correlation of self-efficacy and performance in mathematical problem-solving ought to be subject to further investigation.

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