Abstract
It is widely documented that the South African education system is not only experiencing a Literacy crisis but a Numeracy crisis as well. South Africa is one of the lowest performing countries in mathematics in the world compared to its international counterparts and the rest of Sub-Saharan Africa. At the core of this under performance is the educator’s teaching practices and selection of mathematical communication tools, such as representations. This is of great concern as educators’ understanding of content. As well as how content it is delivered to learners ultimately influences their lifelong mathematical understanding and performance. The aim of the study is thus, to find out how four grade two educators used representations to teach addition and subtraction of whole numbers and to explore to what extent the use of representations developed learners’ understanding.
Adopting a mixed methods approach, qualitative and quantitative data was collected through tests and lesson observations. In this mixed methods case study , pre-and post-tests were conducted to determine to what extent representations facilitated learners’ understanding of addition and subtraction of whole numbers. Twelve lessons in total were observed, as each educators taught three lessons. The lessons were video-recorded and then transcribed. Ensor et al. ’s framework on representations was used to identify the representations used by the four teachers and how they used them. Descriptive statistics was used to analyse the pre-and post-tests.
The findings indicated that educators’ progression and use of multiple modes of representations facilitate learners’ understanding of addition and subtraction of whole numbers. Furthermore, the use of multiple representations allows for conceptualisation of mathematical concepts. This study concluded that using representations in mathematics lessons plays a pivotal role in ensuring understanding addition and subtraction of whole numbers.
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Abbreviation
LFIN- Learning Framework in Number. NCTM- National Council of Teachers of Mathematics.
PCK- Pedagogical Content Knowledge.
CAPS -Curriculum Assessment Policy Statement ATP- Annual Teaching Plan.
List of terms
Representations: In this study refers to the way in which educators communicate mathematical ideas to learners. This includes concrete objects such as beads, to less concrete such as drawings, tallies, and charts and more abstract forms such as symbols and eventually no representations.
Concrete representations- In this study it refers to the tools used to communicate mathematical ideas and operations. This includes beads, beans, learners’ fingers, unifix cubes, abacus, these tools allow learners to embody the operations given within a mathematical task and enhanced learners mathematical understanding
Internal representations- In this study refers to how the neuro subsystems work within the brain to make the connections and promote mathematical reasoning; develop mathematical skills and develop problem solving skills.
External representations- In this study include concrete and abstract forms of representations. This is a reflection or interpretation of learners’ as well as educators’ internal representations.
Abstract representations- In this study, refers to tools used to communicate mathematics ideas and concepts. These forms of representations include symbols.
Pedagogical Content Knowledge: In this study refers to the educators’ content knowledge of addition and subtraction. This is knowledge which educators’ have acquired to apply strategies and application. Furthermore, this includes knowledge of prior misconceptions to adapt content knowledge to a more conceptually accessible version to learners.
Subject matter knowledge- In this study, it refers to the correct application of mathematical concepts, facts , and procedures. It includes knowledge of how to relay the reasoning behind mathematical procedures and the relation between mathematical
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concepts.
Knowledge of pedagogy- In this study refers to educators having acquired the knowledge to plan and organise lessons effectively. This knowledge includes utilising teaching strategies effectively.
Curriculum Knowledge-In this study refers to knowledge about the learning goals and objectives of a particular grade. This knowledge requires educators to have an understanding of what learners are required to know in the different terms within the academic year.
Procedural knowledge- In this study refers to the knowledge and understanding of mathematics rules and key principles which an educator exposes learners to solve additive mathematics tasks with efficiently and with flexibility. This is often automated through drill work.
Conceptual knowledge -In this study refers to the knowledge displayed by the educator of concepts and principles and their interrelation within mathematics.
Mathematical proficiency-In this study refers to the skills, reasoning and problem solving. Mathematical proficiency requires learners to have begun to develop conceptual understanding, procedural fluency, strategic competence , adaptive reasoning and problem solving.
Counting strategies- In this study, it refers to the ability to solve additive mathematical tasks by counting all, counting on; counting down from and counting down to. This strategy often includes the use of concrete representations.
Calculating strategies- In this study refers to basic recall, drilled facts and draws heavily on learners’ mathematical proficiency. This strategy does not require concrete representations and deals with more abstract ways of thinking and representing numbers.
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Keywords: Mathematical proficiency; concrete representations, abstract representations, pedagogical content knowledge, curriculum knowledge, subject knowledge counting strategies , calculating strategies, representation.