The study aimed to explore misconceptions that Grade 9 learners at a school in Soweto had concerning the topic of fractions. The study was based on the ideas of constructivism in a bid to understand how learners build on existing knowledge as they venture deeper into the development of advanced constructions in the concept of fractions. A case study approach (qualitative) was employed to explore how Grade 9 learners describe the concept of fractions. The approach offered a platform to investigate how Grade 9 learners solve problems involving fractions, thereby enabling the researcher to discover the misconceptions that learners have/display when dealing with fractions. The research allowed the researcher to explore the root causes of the misconceptions held by learners concerning the concept of fractions. Forty Grade 9 participants from a township school were subjected to a written test from which eight were purposefully selected for an interview. The selection was based on learners’ responses to the written test. The researcher was looking for a learner script that showed application of similar but incorrect procedures under specific sections of operations of fractions, for example, multiplication of fractions. Both performance extremes were also considered, the good and the worst performers overall. The written test and the interviews were the primary sources of data in this study. The study revealed that learners have misconceptions about fractions. The learners’ definitions of what a fraction is were neither complete nor precise. For example, the equality of parts was not emphasised in their definitions. The gaps brought about by the learner conception of fractions were evident in the way problems on fractions were manipulated. The learners did not treat a fraction as signifying a specific point on the number system. Due to this, learners could not place fractions correctly on the number line. Components of the fraction were separated and manipulated as stand-alone whole numbers. Consequently, whole number knowledge was applied to work with fractions. A lack of conceptual understanding of equivalent fractions was evident as the common denominator principle was not applied. In the multiplication of fractions, procedural manipulations were evident. In mixed number operations, whole numbers were multiplied separately from the fractional parts of the mixed number. Fractional parts were also multiplied separately, and the two answers combined to yield the final solution. In the division of fractions, the learners displayed a lack of conceptual knowledge of division of fractions. Operations were made across the division sign numerators separate from the denominators. This reveals that a fraction was not taken as an outright number on its own by learners but viewed as one number put on top of the other which can be separated. Dividing across, learners rendered division commutative. A procedural attempt to apply the invert and multiply procedure was also evident in this study. Learners made procedural errors as they showed a lack of conceptual understanding of the keep-change-flip division algorithm. The study revealed that misconceptions in the concept of fraction were due to prior knowledge, over-generalisation and presentation of fractions during instruction. Constructivism values prior knowledge as the basis for the development of new knowledge. In this study, learners revealed that informal knowledge they possess may impact negatively on the development of the concept of fractions. For example, division by one-half was interpreted as dividing in half by learners. The prior elaboration on the part of a whole sub-construct also proved a barrier to finding solutions to problems that sought knowledge of fractions as other sub-constructs, namely, quotient, measure, ratio and fraction as an operator. Over generalisation by learners in this study led to misconceptions in which a procedure valid in a particular concept is used in another concept where it does not apply. Knowledge on whole numbers was used in manipulating fractions. For example, for whole numbers generally, multiplication makes bigger and division makes smaller. The presentation of fractions during instruction played a role in some misconceptions revealed by this study. Bias towards the part of a whole sub-construct might have limited conceptualisation in other sub-constructs. Preference for the procedural approach above the conceptual one by educators may limit the proper development of the fraction concept as it promotes the use of algorithms without understanding. The researcher recommends the use of manipulatives to promote the understanding of the fraction concept before inductively guiding learners to come up with the algorithm. Imposing the algorithm promotes the procedural approach, thereby depriving learners of an opportunity for conceptual understanding. Not all correct answers result from the correct line of thinking. Educators, therefore, should have a closer look at learners’ work, including those with correct solutions, as there may be concealed misconceptions. Educators should not take for granted what was covered before learners conceptualised fractions as it might be a source of misconceptions. It is therefore recommended to check prior knowledge before proceeding with new instruction. / Mathematics Education / M. Ed. (Mathematics Education)

Abstract in English, Tswana and Northern Sotho / The study aimed to explore teachers’ mathematical knowledge in respect of teaching the concept of fractions to Grade 6 learners. To that end a qualitative study was done, using a case study design. Data were collected through the observation of, and interviews with, three teachers at three schools in the Capricorn South district. Rooted in the theory of constructivism, the study was supplemented by the conceptual framework of mathematical knowledge for teaching (MKT) (Ball et al., 2008) and Shulman’s (1986) notion of pedagogical knowledge for teaching (PCK). The key finding of this investigation revealed that, of the three teachers, two did not develop the concept of fractions for their learners, but merely followed the traditional method of teaching the concept by encouraging their learners to memorise rules without understanding. Only one teacher emphasised an understanding of mathematical concepts. The main observation which the researcher made, was that teachers require a great deal of knowledge and expertise, in carrying out the work of teaching subject matter related to fractions. / Maikaelelo a thutopatlisiso e ne e le go tlhotlhomisa kitso ya dipalo ya barutabana malebana le go ruta barutwana ba Mophato wa 6 mogopolo wa dikarolwana. Go fitlhelela seo, go dirilwe thutopatlisiso e e lebelelang mabaka, go dirisiwa thadiso ya thutopatlisiso ya dikgetsi. Go kokoantswe data ka go ela tlhoko le go nna le dipotsolotso le barutabana ba le bararo kwa dikolong tsa kgaolo ya Capricorn Borwa. Thutopatlisiso eno e e theilweng mo tioring ya kago ya kitso e ne e tshegeditswe ke letlhomeso la sediriswa sa tokololo sa kitso ya dipalo ya go ruta (MKT) (Ball et al. 2008) le mogopolo wa ga Shulman (1986) wa kitso e e kgethegileng ya go ruta (PCK). Phitlhelelo ya botlhokwa ya patlisiso eno e senotse gore mo barutabaneng ba le bararo, ba le babedi ga ba a tlhamela barutwana ba bona mogopolo wa dikarolwana, mme ba latetse fela mokgwa wa tlwaelo wa go ruta mogopolo ka go rotloetsa barutwana go tshwarelela melawana kwa ntle ga go tlhaloganya. Ke morutabana a le mongwe fela yo o gateletseng go tlhaloganngwa ga megopolo ya dipalo. Temogo e kgolo e e dirilweng ke mmatlisisi ke gore barutabana ba tlhoka kitso le boitseanape jo bogolo go tsweletsa tiro ya go ruta dithuto tse di amanang le dikarolwana. / Dinyakišišo di ikemišeditše go utolla tsebo ya dipalo ya baithuti mabapi le go ruta kgopolo ya dipalophatlo go baithuti ba Kreiti ya 6. Ka lebaka la se go dirilwe dinyakišišo tša boleng, go šomišwa tlhamo ya dinyakišišo tša seemo. Tshedimošo e kgobokeditšwe ka go lekodišiša, le go dira dipoledišano le, barutiši ba bararo ka dikolong tše tharo ka seleteng sa Borwa bja Capricorn. Ka ge di theilwe go teori ya gore baithuti ba itlhamela tsebo, dinyakišišo di tlaleleditšwe ke tlhako ya boikgopolelo ya tsebo ya dipalo go ruteng (MKT) (Ball le ba bangwe, 2008) le kgopolo ya Shulman (1986) ya tsebo ya diteng tša thuto (PCK). Kutollo ye bohlokwa ya dinyakišišo tše e utollotše gore, go barutiši ba bararo, ba babedi ga se ba ba le kgopolo ya dipalophatlo go baithuti ba bona, eupša fela ba no latela mokgwa wa setlwaedi wa go ruta kgopolo ye ya dipalophatlo ka go hlohleletša baithuti ba bona go tsenya melawana ye ka hlogong ka ntle le go e kwešiša. Ke fela morutiši o tee yo a gateletšego gore go swanetše go ba le kwešišo ya dikgopolo tša dipalo. Temogo e tee yeo monyakišiši a bilego le yona, ebile gore barutiši ba hloka tsebo ye kgolo le botsebi, go phethagatša mošomo wa go ruta diteng tša thuto tšeo di amanago le dipalophatlo. / Mathematics Education / M. Ed. (Mathematics Education)