Abstract:
South African learners underachieve in mathematics. The many different factors that influence this
underachievement include mathematics teachers' role in teaching mathematics with
understanding. The question arises as to how teachers' mathematical content knowledge states
can be transformed to positively impact learners' achievement in mathematics.
In this study, different kinds of teachers' knowledge needed for teaching mathematics were
discussed against the background of research in this area, which included the work of Shulman,
Ma and Ball. From this study an important kind of knowledge, namely mathematical content
knowledge for teaching (MCKfT), was identified and a teacher's ability to unpack mathematical
knowledge and understanding was highlighted as a vital characteristic of MCKfT.
To determine further characteristics of MCKfT, the study focussed on the nature of mathematics,
different kinds of mathematical content knowledge (procedural and conceptual), cognitive
processes (problem solving, reasoning, communication, connections and representations) involved
in doing mathematics and the development of mathematical understanding (instrumental vs.
relational understanding). The influence of understanding different problem contexts and teachers'
ability to develop reflective practices in teaching and learning mathematics were discussed and
connected to a teacher's ability to unpack mathematical knowledge and understanding. In this
regard, the role of teachers' prior knowledge or current mathematical content knowledge states
was discussed extensively. These theoretical investigations led to identifying the characteristics of
MCKfT, which in turn resulted in theoretical criteria for the development of MCKfT.
The theoretical study provided criteria with which teachers' current mathematical content
knowledge states could be analysed. This prompted the development of a diagnostic instrument
consisting of questions on proportional reasoning and functions. A qualitative study was
undertaken in the form of a diagnostic content analysis on teachers' current mathematical content
knowledge states. A group of secondary school mathematics teachers (N=128) involved in the
Sediba Project formed the study population. The Sediba Project is an in-service teacher training
program for mathematics teachers over a period of two years. These teachers were divided into
three sub-groups according to the number of years they had been involved in the Sediba Project at
that stage.
The teachers' current mathematical content knowledge states were analysed with respect to the
theoretically determined characteristics of and criteria for the development of MCKfT. These
criteria led to a theoretical framework for assessing teachers' current mathematical content
knowledge states. The first four attributes consisted of the steps involved in mathematical problem
solving skills, namely conceptual knowledge (which implies a deep understanding of the problem),
procedural knowledge (which is reflected in the correct choice of a procedure), the ability to
correctly execute the procedure and the insight to give a valid interpretation of the answer.
Attribute five constituted the completion of these four attributes. The final six attributes were an
understanding of different representations, communication of understanding in writing, reasoning
skills, recognition of connections among different mathematical ideas, the ability to unpack
mathematical understanding and understanding the context a problem is set in. Quantitative
analyses were done on the obtained results for the diagnostic content analysis to determine the
reliability of the constructed diagnostic instrument and to search for statistically significant
differences among the responses of the different sub-groups.
Results seemed to indicate that those teachers involved in the Sediba Project for one or two years
had benefited from the in-service teacher training program. However, the impact of this teachers'
training program was clearly influenced by the teachers' prior knowledge of mathematics. It
became clear that conceptual understanding of foundation, intermediate and senior phase school
mathematics that should form a sound mathematical knowledge base for more advanced topics in
the school curriculum, is for the most part procedurally based with little or no conceptual
understanding. The conclusion was that these teachers' current mathematical content knowledge
states did not correspond to the characteristics of MCKfT and therefore displayed a need for the
development of teachers' current mathematical content knowledge states according to the
proposed criteria and model for the development of MCKfT.
The recommendations were based on the fact that the training that these teachers had been
receiving with respect to the development of MCKfT is inadequate to prepare them to teach
mathematics with understanding. Teachers' prior knowledge should be exposed so that training
can focus on the transformation of current mathematical content knowledge states according to the
characteristics of MCKfT. A model for the development of MCKfT was proposed. The innermost
idea behind this model is that a habit of reflective practices should be developed with respect to the
characteristics of MCKfT to enable a mathematics teacher to communicate and unpack
mathematical knowledge and understanding and consequently solve mathematical problems and
teach mathematics with understanding.
Key words for indexing: school mathematics, teacher knowledge, mathematical content
knowledge, mathematical content knowledge for teaching, mathematical knowledge acquisition,
mathematics teacher education