Text in Afrikaans, abstract in Afrikaans and English / Hierdie studie is daarop gemik om te bepaal tot welke mate wiskundeleerlinge op skool
en onderwysstudente in wiskunde, onderrig in logika ontvang as agtergrond vir strenge
bewysvoering. Die formele aspek van wiskunde op hoerskool en tersiere vlak is
besonder belangrik. Leerlinge en studente kom onvermydelik met hipotetiese argumente
in aanraking. Hulle leer ook om die kontrapositief te gebruik in bewysvoering. Hulle
maak onder andere gebruik van bewyse uit die ongerymde. Verder word nodige en
voldoende voorwaardes met stellings en hulle omgekeerdes in verband gebring. Dit is
dus duidelik dat 'n studie van logika reeds op hoerskool nodig is om aanvaarbare
wiskunde te beoefen.
Om seker te maak dat aanvaarbare wiskunde beoefen word, is dit nodig om te let op die
gebrek aan beheer in die ontwikkeling van 'n taal, waar woorde meer as een betekenis
het. 'n Kunsmatige taal moet gebruik word om interpretasies van uitdrukkings eenduidig
te maak. In so 'n kunsmatige taal word die moontlikheid van foutiewe redenering
uitgeskakel. Die eersteordepredikaatlogika, is so 'n taal, wat ryk genoeg is om die
wiskunde te akkommodeer. Binne die konteks van hierdie kunsmatige taal, kan wiskundige toeriee geformaliseer word. Verskillende bewystegnieke uit die eersteordepredikaatlogika word geidentifiseer,
gekategoriseer en op 'n redelik eenvoudige wyse verduidelik. Uit 'n ontleding van die
wiskundesillabusse van die Departement van Onderwys, en 'n onderwysersopleidingsinstansie,
volg dit dat leerlinge en studente hierdie bewystegnieke moet gebruik.
Volgens hierdie sillabusse moet die leerlinge en studente vertroud wees met logiese
argumente. Uit die gevolgtrekkings waartoe gekom word, blyk dit dat die leerlinge en
studente se agtergrond in logika geheel en al gebrekkig en ontoereikend is. Dit het tot
gevolg dat hulle nie 'n volledige begrip oor bewysvoering het nie, en 'n gebrekkige insig
ontwikkel oor wat wiskunde presies behels.
Die aanbevelings om hierdie ernstige leemtes in die onderrig van wiskunde aan te
spreek, asook verdere navorsingsprojekte word in die laaste hoofstuk verwoord. / The aim of this study is to determine to which extent pupils taking Mathematics at
school level and student teachers of Mathematics receive instruction in logic as a
grounding for rigorous proof. The formal aspect of Mathematics at secondary school
and tertiary levels is extremely important. It is inevitable that pupils and students
become involved with hypothetical arguments. They also learn to use the contrapositive
in proof. They use, among others, proofs by contradiction. Futhermore, necessary and
sufficient conditions are related to theorems and their converses. It is therefore
apparent that the study of logic is necessary already at secondary school level in order
to practice Mathematics satisfactorily.
To ensure that acceptable Mathematics is practised, it is necessary to take cognizance
of the lack of control over language development, where words can have more than one
meaning. For this reason an artificial language must be used so that interpretations can
have one meaning. Faulty interpretations are ruled out in such an artificial language.
A language which is rich enough to accommodate Mathematics is the first-order
predicate logic. Mathematical theories can be formalised within the context of this artificial language.
Different techniques of proof from the first-order logic are identified, categorized and
explained in fairly simple terms. An analysis of Mathematics syllabuses of the
Department of Education and an institution for teacher training has indicated that pupils
should use these techniques of proof. According to these syllabuses pupils should be
familiar with logical arguments. The conclusion which is reached, gives evidence that
pupils' and students' background in logic is completely lacking and inadequate. As a
result they cannot cope adequately with argumentation and this causes a poor perception
of what Mathematics exactly entails.
Recommendations to bridge these serious problems in the instruction of Mathematics,
as well as further research projects are discussed in the final chapter. / Curriculum and Institutional Studies / D. Phil. (Wiskundeonderwys)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:umkn-dsp01.int.unisa.ac.za:10500/17764 |
| Date | 04 1900 |
| Creators | Van Staden, P. S. (Pieter Schalk) |
| Contributors | Alderton, Ian William, 1952-, Barnard, J. J. (John James), 1957- |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 1 online resource (viii, 297 leaves) |
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