 Home
 AZ Publications
 Pythagoras
 Previous Issues
 Volume 36, Issue 1, 2015
Pythagoras  Volume 36, Issue 1, January 2015
Volumes & Issues

Volume 40 (2019)

Volume 39 (2018)

Volume 38 (2017)

Volume 37 (2016)

Volume 36 (2015)

Volume 35 (2014)

Volume 34 (2013)

Volume 33 (2012)

Volume 32 (2011)

Volume 2010 (2010)

Volume 2009 (2009)

Volume 2008 (2008)

Volume 2007 (2007)

Volume 2006 (2006)

Volume 12 (2006)

Volume 2005 (2005)

Volume 2004 (2004)
Volume 36, Issue 1, January 2015

Understanding students' misconceptions : an analysis of final Grade 12 examination questions in geometry : original research
Author Kakoma LunetaThe role geometry plays in real life makes it a core component of mathematics that students must understand and master. Conceptual knowledge of geometric concepts goes beyond the development of skills required to manipulate geometric shapes. This study is focused on errors students made when solving coordinate geometry problems in the final Grade 12 examination in South Africa. An analysis of 1000 scripts from the 2008 Mathematics examination was conducted. This entailed a detailed analysis of one Grade 12 geometry examination question. Van Hiele levels of geometrical thought were used as a lens to understand students' knowledge of geometry. Studies show that Van Hiele levels are a good descriptor of current and future performance in geometry. This study revealed that whilst students in Grade 12 are expected to operate at level 3 and level 4, the majority were operating at level 2 of Van Hiele's hierarchy. The majority of students did not understand most of the basic concepts in Euclidian transformation. Most of the errors were conceptual and suggested that students did not understand the questions and did not know what to do as a result. It is also noted that when students lack conceptual knowledge the consequences are so severe that they hardly respond to the questions in the examination.

Investigating learners' metarepresentational competencies when constructing bar graphs : original research
Author Michael MhloloCurrent views in the teaching and learning of data handling suggest that learners should create graphs of data they collect themselves and not just use textbook data. It is presumed realworld data creates an ideal environment for learners to tap from their pool of stored knowledge and demonstrate their meta representational competences. Although prior knowledge is acknowledged as a critical resource out of which expertise is constructed, empirical evidence shows that new levels of mathematical thinking do not always build logically and consistently on previous experience. This suggests that researchers should analyse this resource in more detail in order to understand where prior knowledge could be supportive and where it could be problematic in the process of learning. This article analyses Grade 11 learners' metarepresentational competences when constructing bar graphs. The basic premise was that by examining the process of graph construction and how learners respond to a variety of stages thereof, it was possible to create a description of a graphical frame or a knowledge representation structure that was stored in the learner's memory. Errors could then be described and explained in terms of the inadequacies of the frame, that is: 'Is the learner making good use of the stored prior knowledge?' A total of 43 learners were observed over a week in a classroom environment whilst they attempted to draw graphs for data they had collected for a mathematics project. Four units of analysis are used to focus on how learners created a frequency table, axes, bars and the overall representativeness of the graph visàvis the data. Results show that learners had an inadequate graphical frame as they drew a graph that had elements of a value bar graph, distribution bar graph and a histogram all representing the same data set. This inability to distinguish between these graphs and the types of data they represent implies that learners were likely to face difficulties with measures of centre and variability which are interpreted differently across these three graphs but are foundational in all statistical thinking.

Problematising current forms of legitimised participation in the examination papers for Mathematical Literacy : original research
Authors: Marc North, Iben M. Christiansen and Marc NorthIn this article we argue that in South Africa the current format of legitimised participation and practice in the examination papers for Mathematical Literacy restricts successful apprenticeship in the discipline of scientific mathematics and limits empowered preparation for realworld functioning. The currency of the subject, then, is brought into question. We further argue that the positioning of the subject as a compulsory alternative to Mathematics and the differential distribution of these two subjects to differing groups of learners facilitates the (re)production and sustainment of educational disadvantage. We draw on Dowling's theoretical constructs of differing domains of mathematical practice and positions and focus analysis on a collection of nationally set exemplar Grade 12 examination papers to identify legitimised forms of participation in the subject. We conclude by arguing for a reconceptualised structure of knowledge and participation in Mathematical Literacy and make preliminary recommendations in this regard.

Teacher narratives in making sense of the statistical mean algorithm : original research
Authors: Erna Lampen and Erna LampenTeaching statistics meaningfully at school level requires that mathematics teachers conduct classroom discussions in ways that give statistical meaning to mathematical concepts and enable learners to develop integrated statistical thinking. Key to statistical discourse are narratives about variation within and between distributions of measurements and comparison of varying measurements to a central anchoring value. Teachers who understand the concepts and tools of statistics in an isolated and processual way cannot teach in such a connected way. Teachers' discourses about the mean tend to be particularly processual and lead to limited understanding of the statistical mean as measure of centre in order to compare variation within data sets. In this article I report on findings from an analysis of discussions about the statistical mean by a group of teachers. The findings suggest that discourses for instruction in statistics should explicitly differentiate between the everyday 'average' and the statistical mean, and explain the meaning of the arithmetic algorithm for the mean. I propose a narrative that logically explains the mean algorithm in order to establish the mean as an origin in a measurement of variation discourse.

The relational nature of rational numbers : original research
Author Bruce BrownIt is commonly accepted that the knowledge and learning of rational numbers is more complex than that of the whole number field. This complexity includes the broader range of application of rational numbers, the increased level of technical complexity in the mathematical structure and symbol systems of this field and the more complex nature of many conceptual properties of the rational number field. Research on rational number learning is divided as to whether children's difficulties in learning rational numbers arise only from the increased complexity or also include elements of conceptual change. This article argues for a fundamental conceptual difference between whole and rational numbers. It develops the position that rational numbers are fundamentally relational in nature and that the move from absolute counts to relative comparisons leads to a further level of abstraction in our understanding of number and quantity. The argument is based on a number of qualitative, indepth research projects with children and adults. These research projects indicated the importance of such a relational understanding in both the learning and teaching of rational numbers, as well as in adult representations of rational numbers on the number line. Acknowledgement of such a conceptual change could have important consequences for the teaching and learning of rational numbers.