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Mathematical abilities and mathematical memory during problem solving and some aspects of mathematics education for gifted pupils
Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.ORCID-id: 0000-0002-5423-5580
2017 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: Department of Mathematics and Science Education, Stockholm University , 2017. , s. 139
Serie
Doctoral thesis from the department of mathematics and science education ; 17
Nyckelord [en]
mathematical abilities, mathematical memory, high-achieving students, problem solving, mathematics education for gifted pupils
Nationell ämneskategori
Didaktik
Forskningsämne
matematikämnets didaktik
Identifikatorer
URN: urn:nbn:se:su:diva-146542ISBN: 978-91-7649-948-1 (tryckt)ISBN: 978-91-7649-949-8 (digital)OAI: oai:DiVA.org:su-146542DiVA, id: diva2:1143981
Disputation
2017-11-10, Högbomsalen, Geovetenskapens hus, Svante Arrhenius väg 12, Stockholm, 10:00 (Engelska)
Opponent
Handledare
Anmärkning

At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 4: In press.

Tillgänglig från: 2017-10-18 Skapad: 2017-09-24 Senast uppdaterad: 2017-10-04Bibliografiskt granskad
Delarbeten
1. Matematikundervisning för begåvade elever – en forskningsöversikt
Öppna denna publikation i ny flik eller fönster >>Matematikundervisning för begåvade elever – en forskningsöversikt
2017 (Svenska)Ingår i: Nordisk matematikkdidaktikk, ISSN 1104-2176, Vol. 22, nr 1, s. 21-44Artikel, forskningsöversikt (Refereegranskat) Published
Abstract [sv]

Artikeln redovisar de huvudsakliga pedagogiska och organisatoriska metoder relaterade till begåvade elevers matematikundervisning som fokuseras i forskningslitteraturen – även könsskillnader, motivation och matematiskt begåvade elevers sociala situation i klassrummet diskuteras. Översikten visar att det finns åtgärder – t.ex. frivillig acceleration i ämnet där undervisningen är anpassad till elevens förkunskaper och kapacitet eller arbete med utmanande uppgifter i prestationshomogena grupper – som antas ha goda effekter på begåvade elevers kunskapsutveckling i matematik. Analysen visar också att det kan uppfattas som problematiskt att vara begåvad i matematik samt att begåvade flickor upplever vissa aspekter av matematikundervisningen annorlunda jämfört med motsvarande grupp pojkar.

Abstract [en]

The present article offers an overview of those main methodological and pedagogical approaches associated with gifted pupils’ education in mathematics which are focused in the research literature. Furthermore, the article discusses gender differences, motivation and some central aspects of mathematically gifted pupils’ social situation in the classroom. The analysis shows that there are some pedagogical and organizational approaches, e.g. voluntary acceleration where the teaching is adapted to the knowledge and the capacity of the participants or working with challenging mathematical problems in performance-homogenous groups, which may have good effects on gifted pupils’ mathematical achievement. The overview also indicates that mathematically gifted adolescents are facing difficulties in their social interaction and that gifted female and male pupils are experiencing certain aspects of their mathematics education differently.

Nyckelord
begåvade elever, matematik, undervisning
Nationell ämneskategori
Didaktik
Forskningsämne
matematikämnets didaktik
Identifikatorer
urn:nbn:se:su:diva-146529 (URN)
Anmärkning

Access to the two most recent volumes are password protected.

Tillgänglig från: 2017-08-31 Skapad: 2017-08-31 Senast uppdaterad: 2017-09-24Bibliografiskt granskad
2. Examining the interaction of mathematical abilities and mathematical memory: A study of problem-solving activity of high-achieving Swedish upper secondary students
Öppna denna publikation i ny flik eller fönster >>Examining the interaction of mathematical abilities and mathematical memory: A study of problem-solving activity of high-achieving Swedish upper secondary students
2017 (Engelska)Ingår i: The Mathematics Enthusiast, ISSN 1551-3440, Vol. 14, nr 1-3, s. 141-159Artikel i tidskrift (Refereegranskat) 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.

Nyckelord
mathematical ability, non-routine problem solving, Krutetskii, mathematical memory, abstraction, generalization, high achieving students, Swedish upper secondary
Nationell ämneskategori
Annan matematik
Forskningsämne
matematikämnets didaktik
Identifikatorer
urn:nbn:se:su:diva-139538 (URN)000396454000009 ()
Tillgänglig från: 2017-02-08 Skapad: 2017-02-08 Senast uppdaterad: 2019-12-12Bibliografiskt granskad
3. Uncovering the Relationship Between Mathematical Ability and Problem Solving Performance of Swedish Upper Secondary School Students
Öppna denna publikation i ny flik eller fönster >>Uncovering the Relationship Between Mathematical Ability and Problem Solving Performance of Swedish Upper Secondary School Students
2018 (Engelska)Ingår i: Scandinavian Journal of Educational Research, ISSN 0031-3831, E-ISSN 1470-1170, Vol. 62, nr 4, s. 555-569Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

In this paper, we examine the interactions of mathematical abilities when 6 high achieving Swedish upper-secondary students attempt unfamiliar non-routine mathematical problems. Analyses indicated a repeating cycle in which students typically exploited abilities relating to the ways they orientated themselves with respect to a problem, recalled mathematical facts, executed mathematical procedures, and regulated their activity. Also, while the nature of this cyclic sequence varied little across problems and students, the proportions of time afforded the different components varied across both, indicating that problem solving approaches are informed by previous experiences of the mathematics underlying the problem. Finally, students’ whose initial problem formulations were numerical typically failed to complete the problem, while those whose initial formulations were algebraic always succeeded.

Nyckelord
high achieving students, mathematical abilities, mathematical problem solving, Swedish upper secondary school
Nationell ämneskategori
Didaktik
Forskningsämne
matematikämnets didaktik
Identifikatorer
urn:nbn:se:su:diva-146532 (URN)10.1080/00313831.2016.1258671 (DOI)000435015900005 ()
Tillgänglig från: 2017-08-31 Skapad: 2017-08-31 Senast uppdaterad: 2018-07-23Bibliografiskt granskad
4. Mathematical memory revisited: mathematical problem solving by high achieving students
Öppna denna publikation i ny flik eller fönster >>Mathematical memory revisited: mathematical problem solving by high achieving students
2017 (Engelska)Ingår i: Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1-5, 2017) / [ed] Thérèse Doole, Ghislaine Gueudet, Dublin: DCU Institute of Education, ERME , 2017, s. 1202-1209Konferensbidrag, Publicerat paper (Refereegranskat)
Abstract [en]

The present study deals with the role of the mathematical memory in problem solving. To examine that, two problem-solving activities of high achieving students from secondary school were observed one year apart - the proposed tasks were non-routine for the students, but could be solved with similar methods. The study shows that even if not recalling the previously solved task, the participants’ individual ways of approaching both tasks were identical. Moreover, the study displays that the participants used their mathematical memory mainly at the initial phase and during a small fragment of the problem-solving process, and indicates that students who apply algebraic methods are more successful than those who use numerical approaches.

Ort, förlag, år, upplaga, sidor
Dublin: DCU Institute of Education, ERME, 2017
Nyckelord
high-achievers, mathematical memory, mathematical abilities, problem solving
Nationell ämneskategori
Didaktik
Forskningsämne
matematikämnets didaktik
Identifikatorer
urn:nbn:se:su:diva-146536 (URN)978-1-873769-73-7 (ISBN)
Konferens
Tenth Congress of the European Society for Research in Mathematics Education, Dublin Ireland, 1 – 5 February, 2017
Tillgänglig från: 2017-08-31 Skapad: 2017-08-31 Senast uppdaterad: 2018-03-23Bibliografiskt granskad

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