Abstract
The chapter explores the diverse interpretations related to the learning of mathematics for learners at lower grade levels. At the core of understanding the dynamics of teaching and assessment of mathematics tasks for lower grade learners lie the general perceptions on the subject and how these have shaped educator and learner approaches to the subject as well as associated nuances of cognitive development. The chapter also explores how educators deal with inclusivity in the context of the curriculum and cognitive capabilities of learners in teaching and assessment. Different interpretations to assessments are also explored. A different approach to assessment that locates learners on a continuum of current proficiency is advocated. A flexible teaching philosophy is proposed which rather than cut the class into a pass/fail dichotomy, acknowledges each learner's right to progress in particular through the use of targeted teaching for challenged learners.
TopIntroduction
The mantra of researchers, educators in South Africa is that “our learners do not know any mathematics” and that they are far behind their international counterparts (Long, 2011). Often these statements, with supporting evidence from test results, or classroom observations, or anecdote, are explained by socio-economic context, by language difficulties, by poor teaching and by lack of resources (Sfard, 1991 & Long, 2011). A generalizing claim that learners can hardly understand mathematics, can however be challenged from a number of perspectives, the first of which is that there is a natural engagement of a child with his environment, and this engagement is almost certain to engage mathematical thinking through a constructivist process involving the learner drawing from their previously held knowledge.
One of the key features of this chapter is to explore how the teaching and learning of mathematics has been viewed as a complex process beyond the ability of many individuals. This is in particular with regard to the subject having continuously attracted controversy and sparked debate across many sections of society. What has been agreed among the different scholars however has been that the teaching and learning of mathematics across the different phases has been found to be demanding for both learners and educators (Sfard, 1991 & Long, 2011). Sfard (1991) thus adds that specific details related to mathematical thinking, the omnipresent and perceivably overwhelming difficulties experienced by learners of mathematics are as bewildering as they are noticeable. Whilst the aforementioned views seem to be alarming when it comes to the learning of mathematics, the noticeable challenges that usually accompany the teaching and assessment of the subject is equally concerning. With the learning of mathematics being a two-way process in which factors that are linked to both the learners and educator could influence the process, it becomes important to ensure that any focus on the subject of learning mathematics embraces a holistic approach. In this regard, a holistic approach will thus assist in exploring how issues that relate to the school environment such as educator competences and the curriculum on the one hand and external factors linked to the home environment of the learner contribute towards the learning of mathematics. In the South African context, it is important to note that inclusivity has been associated with a curriculum that has been transformed and founded on liberating epistemologies. It follows that despite the links to liberating epistemologies and the envisaged inclusiveness especially in their foundations in constructivism, a curriculum such as curriculum 2005, had short comings at the implementation phase. It can thus be argued that notwithstanding the progressive pedagogies where more attention to the conditions of responsiveness involved educators being alerted to differences in the ways in which learners engaged and responded to the content and context of learning, a serious limitation presented itself in particular with respect to the educational role of the educator. Of further concern was the curriculum’s lack of clarity on the theoretical foundations of the key elements of a pedagogy informed by a constructivist approach to learning and its relation to learning mathematics in particular.
Key Terms in this Chapter
Mathematics: This entails the subject which is broadly seen as a science of numbers and other abstract concepts with numerical significance.
Abled: This entails a classificatory procedure where educators classify learners into the less and more abled category to the effect that some have tended to see it fit to exercise exclusion and inclusion based on the classification.
Rasch Model: This is a model that is grounded on measuring assessments by predicting the probability of success of an individual on an issue depending on the responsible individual’s aptitude and the issue’s level of complexity whilst also taking into consideration the construct being measured.
Cognitive Capabilities: Learner cognitive capabilities can be explained within a constructivist framework where learners are viewed as having capacity to be mathematics doers and thinkers with an ability of constructing mathematical knowledge.
Foucaultian Approach: A post structural approach perspective propounding the view that although being capable of fulfilling certain goals, mathematics education can equally be viewed as generally riddled with power dynamics to the extent that it can be liberating whilst also oppressing in other ways.
Assessment: These are evaluation processes and protocols that form an integral part in measuring learner achievement. They are essential for the learner, educator, parents, and other stakeholders involved in the education process.
Curriculum: This is a policy related document that provides guidelines on the content that educators are expected to deliver within a specific duration with learners expected to acquire it whilst they subsequently get assessed on how well they would have learnt what educators would have taught them.
Inclusivity: Involves exploring ways in which mathematics teaching and learning is defined from a more holistic angle broader than policy directives whilst appreciating that learners with different abilities are put together within a classroom setting with targeted teaching being used to ensure that their performance is optimized.
Measurements: The chapter provides a broader conceptualisation of measurement that whilst acknowledging the involvement of validation through quantitative means and qualifying through value based frameworks also calls for going beyond the technical validity of measurements, in order to establish whether the intended issues have been measured as well as ensuring that there is a clear understanding of the influence of values in educational decisions.
Educator Beliefs: These refer to the perceptions and interpretations of educators regarding mathematics as a subject which includes learner knowledge acquisition as well as teaching and assessment approaches an educator would prefer adopting.