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Examining the interaction of mathematical abilities and mathematical memory: A study of problem-solving activity of high-achieving Swedish upper secondary students
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.ORCID iD: 0000-0003-3679-9187
2017 (English)In: The Mathematics Enthusiast, ISSN 1551-3440, Vol. 14, no 1-3, 141-159 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we investigate the abilities that six high-achieving Swedish upper secondary students demonstrate when solving challenging, non-routine mathematical problems. Data, which were derived from clinical interviews, were analysed against an adaptation of the framework developed by the Soviet psychologist Vadim Krutetskii (1976). Analyses showed that when solving problems students pass through three phases, here called orientation, processing and checking, during which students exhibited particular forms of ability. In particular, the mathematical memory was principally observed in the orientation phase, playing a crucial role in the ways in which students' selected their problem-solving methods; where these methods failed to lead to the desired outcome students were unable to modify them. Furthermore, the ability to generalise, a key component of Krutetskii's framework, was absent throughout students' attempts. These findings indicate a lack of flexibility likely to be a consequence of their experiences as learners of mathematics.

Place, publisher, year, edition, pages
2017. Vol. 14, no 1-3, 141-159 p.
Keyword [en]
mathematical ability, non-routine problem solving, Krutetskii, mathematical memory, abstraction, generalization, high achieving students, Swedish upper secondary
National Category
Other Mathematics
Research subject
Mathematics Education
Identifiers
URN: urn:nbn:se:su:diva-139538ISI: 000396454000009OAI: oai:DiVA.org:su-139538DiVA: diva2:1072485
Available from: 2017-02-08 Created: 2017-02-08 Last updated: 2017-09-24Bibliographically approved
In thesis
1. Mathematical abilities and mathematical memory during problem solving and some aspects of mathematics education for gifted pupils
Open this publication in new window or tab >>Mathematical abilities and mathematical memory during problem solving and some aspects of mathematics education for gifted pupils
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis reports on two different investigations.

The first is a systematic review of pedagogical and organizational practices associated with gifted pupils’ education in mathematics, and on the empirical basis for those practices. The review shows that certain practices – for example, enrichment programs and differentiated instructions in heterogeneous classrooms or acceleration programs and ability groupings outside those classrooms – may be beneficial for the development of gifted pupils. Also, motivational characteristics of and gender differences between mathematically gifted pupils are discussed. Around 60% of analysed papers report on empirical studies, while remaining articles are based on literature reviews, theoretical discourses and the authors’ personal experiences – acceleration programs and ability groupings are supported by more empirical data than practices aimed for the heterogeneous classroom. Further, the analyses indicate that successful acceleration programs and ability groupings should fulfil some important criteria; pupils’ participation should be voluntary, the teaching should be adapted to the capacity of participants, introduced tasks should be challenging, by offering more depth and less breadth within a certain topic, and teachers engaged in these practices should be prepared for the characteristics of gifted pupils.

The second investigation reports on the interaction of mathematical abilities and the role of mathematical memory in the context of non-routine problems. In this respect, six Swedish high-achieving students from upper secondary school were observed individually on two occasions approximately one year apart. For these studies, an analytical framework, based on the mathematical ability defined by Krutetskii (1976), was developed. Concerning the interaction of mathematical abilities, it was found that every problem-solving activity started with an orientation phase, which was followed by a phase of processing mathematical information and every activity ended with a checking phase, when the correctness of obtained results was controlled. Further, mathematical memory was observed in close interaction with the ability to obtain and formalize mathematical information, for relatively small amounts of the total time dedicated to problem solving. Participants selected problem-solving methods at the orientation phase and found it difficult to abandon or modify those methods. In addition, when solving problems one year apart, even when not recalling the previously solved problem, participants approached both problems with methods that were identical at the individual level. The analyses show that participants who applied algebraic methods were more successful than participants who applied particular methods. Thus, by demonstrating that the success of participants’ problem-solving activities is dependent on applied methods, it is suggested that mathematical memory, despite its relatively modest presence, has a pivotal role in participants’ problem-solving activities. Finally, it is indicated that participants who applied particular methods were not able to generalize mathematical relations and operations – a mathematical ability considered an important prerequisite for the development of mathematical memory – at appropriate levels.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics and Science Education, Stockholm University, 2017. 139 p.
Series
Doctoral thesis from the department of mathematics and science education, 17
Keyword
mathematical abilities, mathematical memory, high-achieving students, problem solving, mathematics education for gifted pupils
National Category
Didactics
Research subject
Mathematics Education
Identifiers
urn:nbn:se:su:diva-146542 (URN)978-91-7649-948-1 (ISBN)978-91-7649-949-8 (ISBN)
Public defence
2017-11-10, Högbomsalen, Geovetenskapens hus, Svante Arrhenius väg 12, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 4: In press.

Available from: 2017-10-18 Created: 2017-09-24 Last updated: 2017-10-04Bibliographically approved

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