Modern Views on Mathematical Knowledge for Teaching

In Tunstall’s (1993) thesis on The Structure of Knowledge for Mathematics, he highlighted a lack of a model for the structure of knowledge for mathematics that is accepted and used by mathematics educators in curriculum development and delivery. He argued that the existence of a model would guide educators in ‘dealing with the organisation of mathematical knowledge in their role as stewards of the discipline’ (p. 2).

In his attempt in coming up with a structure of knowledge for the discipline, he invented a model of three ‘faces’ of knowledge as represented by sides of a cube. The three faces of the cube comprise:

• Major Processes: A set of 11 hierarchal levels of processes which aims to describe the level of mathematical thinking, from the most simplistic level of representing mathematics (use of symbols to represent mathematical statements) to the most advanced level of adapting and applying (to derive new processes).

• Knowledge Content Areas: A division and/or union of major concepts in mathematics including logic, number theory, algebra, geometry and analysis where one sub-discipline may be ‘dependent upon parts of those that might precede it as an elementary foundation without pretending to wholly contain them’ (p. 26). For example, the sub-discipline of logic is thought to be a fundamental content for all other sub-disciplines due to its role as a key ingredient in the argument and soundness of the theories and concepts.

• Dominant Technologies: The use of three types of technologies in the understanding of the discipline — computers and calculators, tables and charts and manipulatives and tools. It is explained that these instruments of mathematics technologies are used in aiding students to construct mathematical knowledge.

Tunstall then seeks to explain how the model is valid, comprehensive and useful for mathematicians and mathematics educators to organise and make connections between the topics of mathematical knowledge and aid in the learning of mathematics for students.

Taking a generic approach to subject knowledge, Erickson (2007) derived her version of a knowledge structure which is applicable across all disciplines, not just mathematics. According to Erickson, the various levels of the structure are defined as such:

• Topics: An organisation of facts related to specific people, places, situations or things

• Facts: Specific examples of people, places, situations or things

• Concepts: Mental constructs that overarch different topical examples and meet these criteria: timeless, universal, abstract, different examples that share common attributes

• Generalisations: Two or more concepts linked in a relationship that meets these criteria— generally universal application, generally timeless, abstract, supported by different examples. Enduring essential understandings for a discipline

• Principles: Two or more concepts linked in a relationship, but they are considered the foundational truths of a discipline

• Theories: Explanations of the nature or behaviour of a specified set of phenomena based on the best evidence available

Erickson further pointed out that the structure of knowledge for mathematics is much more conceptual than that of other disciplines such as history, in the sense that ‘mathematics is an inherently conceptual language of concepts, sub-concepts and their relationships’. This means that the teaching of mathematics requires sufficient emphasis on conceptual understanding of the mathematical knowledge, and not simply focusing on procedural understanding, which can help to increase conceptual understanding but only up to a certain limit (Rittle-Johnson and Alibali 1999).

The works of Tunstall and Erickson, though different, seek to categorise how mathematical knowledge should be structured for teaching mathematics in an educational context. Tunstall in particular emphasised mastering the levels of major processes using the appropriate technologies in the various sub-disciplines such as Algebra and Analysis. Erickson’s model focuses more on how knowledge should be taught in order to form deep enduring understandings by creating linkages between factual knowledge and conceptual knowledge.

From the above recommendations on how mathematical knowledge can be organised and taught, the attention then turns to how much mathematical knowledge educators need to possess in order to be effective classroom teachers. Interestingly, research done by several scholars including Begle (as cited in Ball et al. 2001) and Monk (as cited in Ball et al. 2001) showed that teachers’ repertoire of mathematical knowledge has a threshold effect on students’ achievement in mathematics, such that advanced knowledge in mathematical knowledge does not necessarily or significantly produce a corresponding improvement in students’ achievement. Instead, the ability to impart mathematical knowledge through the use of proper pedagogies and teaching strategies is necessary to ensure that students have enduring understandings about the discipline and not simply picking up the procedural skills of solving mathematical problems. This would then seek to redress the current situation where students find mathematics complicated and difficult and is ‘no more than a set of arbitrary rules and procedures to be memorised’ (Ball et al. 2001, p. 434).