Made available in DSpace on 2014-12-17T14:36:30Z (GMT). No. of bitstreams: 1 GeorgianeAS_TESE.pdf: 2906729 bytes, checksum: 517705591ff207d241daa57f0bb21381 (MD5) Previous issue date: 2013-03-15 / The thesis presents a systematic description about the meaning, as Skemp, relational understanding and understanding instrumental, in the context of mathematics learning, being that we had as a guide his understanding of the schema. Especially, we analyze some academic productions, in the area of Mathematics Education, who used the categories of understanding relational and instrumental understanding how evaluative instrument and we see that in most cases the analysis is punctual. Being so, whereas the inherent understanding relational schema has a network of connected ideas and non-insulated, we investigated if the global analysis, where it is the understanding of the diversity of contributory concepts for formation of the concept to be learned, is more appropriate than the punctual, where does the understanding of concepts so isolated. For this, we apply a teaching module, having as main content the Quaternos Pythagoreans using History of Mathematics and the work of Bahier (1916). With the data we obtained the teaching module to use the global analysis and the punctual analysis, using research methodology the Case Study, and consequently we conduct our inferences about the levels of understanding of the subject which has made it possible for us to investigate the ownership of global analysis at the expense of punctual analysis. On the opportunity, we prove the thesis that we espouse in the course of the study and, in addition, we highlight as a contribution of our research evidence of need for a teaching of mathematics that entices the relational understanding and that evaluation should be global, being necessary to consider the notion of schema and therefore know the schematic diagram of the concept that will be evaluated / O presente trabalho apresenta uma descri??o sistem?tica acerca do que significa, segundo Skemp (1980,1989), compreens?o relacional e compreens?o instrumental, no ?mbito da aprendizagem matem?tica, sendo que tivemos como norte ? sua compreens?o sobre esquema. Sobretudo, analisamos alguns trabalhos acad?micos, na ?rea de Educa??o Matem?tica, que fizeram uso das categorias de compreens?o relacional e compreens?o instrumental enquanto instrumento avaliativo e detectamos que na maioria dos casos a an?lise ? pontual. Diante disso, considerando que o esquema inerente ? compreens?o relacional apresenta uma rede de ideias conectadas e n?o isoladas, investigamos se a an?lise global, na qual considera-se a compreens?o da diversidade de conceitos contributivos ? forma??o do conceito a ser aprendido, ? mais apropriada que a pontual, na qual considera a compreens?o de conceitos de modo isolado. Para tanto, aplicamos um m?dulo de ensino, tendo como conte?do principal os quaternos pitag?ricos utilizando a Hist?ria da Matem?tica e a obra de Bahier (1916). Com o referido m?dulo de ensino obtivemos os dados para realizarmos tanto a an?lise global quanto a pontual, utilizando como modalidade de pesquisa o Estudo de Caso, e consequentemente realizamos nossas infer?ncias acerca dos n?veis de compreens?o apresentados pelos sujeitos o que nos possibilitou investigarmos a apropria??o da an?lise global em detrimento da an?lise pontual. Na oportunidade, comprovamos a tese que defendemos no decorrer do estudo e, al?m disso, apontamos como contribui??o da nossa pesquisa a evid?ncia da necessidade de um ensino de Matem?tica que promova a compreens?o relacional e que avalia??o a ser realizada deve ser global, sendo necess?rio levar em considera??o a no??o de esquema e consequentemente conhecer o diagrama esquem?tico do conceito a ser avaliado

Syftet med den här studien är att belysa faktorer som möjliggör eller hindrar förståelsen av likhetstecknet hos elever och dess övergång mellan aritmetik och algebra. Med en systematisk litteraturstudie som metod söktes vetenskapliga artiklar som sammanställdes i fem olika kategorier. Svårigheter i tidig algebra, matematikens språk, relationell/instrumentell förståelse samt relationella tolkningar och till sist lärarperspektivet. Resultatet från den systematiska litteraturstudien bekräftade att elever inte har en relationell förståelse av likhetstecknet. Artiklarna visade framför allt att traditionell aritmetikundervisning hindrar elevers utveckling i algebraiskt tänkande. Uppgifter med operationer skrivna i vänsterled visade sig stärka elevers instrumentella förståelse av likhetstecknet. Ett instrumentellt och relationellt språk (både skriftligt och verbalt) framträdde också som en avgörande faktor för elevers förståelse. Utifrån artiklarnas resultat undersöktes vilka faktorer av likhetstecknet som elever får möjlighet att öva i digitala spel med hjälp av en innehållsanalys. Det visade sig att de digitala spelen i den här studien inte är utformade för att stödja elevers relationella förståelse av likhetstecknets betydelse utan fortsätter att stärka den instrumentella förståelsen hos elever. / The purpose of this study is to illustrate factors concerning the concept of the equal sign and its impact on the transition from arithmetic to algebra. Using a systematic literature study as a method, scientific articles were compiled into five different categories. Difficulties linked to early algebra, The language of Mathematics, Relational/instrumental understanding as well as relational interpretations and finally the Teacher perspective. Our systematic literature review confirmed that students do not have a relational understanding of the equal sign. Indeed, it highlighted that traditional arithmetic teaching methods tended to hinder student development in algebraic thinking through tasks, such as operations being on the left side of the equal sign and the answer as an outcome on the right side of the equal sign. This has been shown to strengthen students' instrumental understanding of the sign rather than dismantling it. Further, instrumental, and relational language (both written and verbal) proved to be decisive factors in students’ learning. Based on the results of our scientific articles, we undertook a content analysis of digital resources and examined whether the equal sign strengthens students instrumental or relational understanding of the equal sign. It turned out that the digital games analysed in this study are not designed to support students’ relational understanding of the equal sign but continues to strengthen their instrumental understanding.

This text discusses final results from empirical studies, scientific articles and literature from the period 1990-2023, concerning teachers’ knowledge, students’ misconceptions and various teaching methods related to geometric objects. The results highlight challenges for both teachers and students in understanding and defining geometric shapes and figures. The work elucidates students in grade 4-6 difficulties and knowledge in identifying geometric objects in varied positions, as well as their understanding of the properties of geometric shapes and figures. The teacher’s understanding is crucial för imparting accurate information to students in instruction. Therefore, effective teaching methods such as practical activities, everyday connection and Van Hiele’s instructional model are suitable to apply in practice. This instructional model has been effective in students education and is therefore a good example for a teaching method.  In summary, the results are based on the abilities of teachers and students and their abilities can be enhanced by adapting teaching methods related to geometric objects, and how misconceptions are something that should be taken seriously and be prevented.

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)