Mental computation: The identification of associated cognitive, metacognitive and affective factors

The purpose of the study was to develop an explanation why some children are better at addition and subtraction mental computation than others. For the purposes of this thesis, mental computation was defined as "the process of carrying out arithmetic calculations without the aid of external devices" (Sowder, 1988, p.182). To reflect current views of mental computation as calculating with the head, rather than merely, in the head, the definition was extended to calculating using strategies with understanding (Anghileri, 1999). Thus, proficiency was not confined to accuracy, but also included flexibility of strategy choice.

The study investigated the part played by number sense knowledge (e.g., numeration, number facts, estimation and effects of operations on number), metacognition, affects (e.g., beliefs, attitudes), and memory.

The study showed that students proficient in mental computation (accurate and flexible) possessed integrated understandings of number facts (speed, accuracy, and efficient number facts), numeration, and number and operation. These proficient students also exhibited some metacognitive strategies and possessed reasonable short term memory and executive functioning. Where there was less knowledge and fewer connections between knowledge, students compensated in different ways, depending on their beliefs and what knowledge they possessed. Accurate and inflexible students used the teacher taught strategy of mental image of pen and paper algorithm in which strong beliefs were held. Combined with fast and accurate number facts and some numeration understanding, their familiarity with this strategy enabled the students to complete the mental computation tasks with accuracy. Working memory was sufficient to use an inefficient mental strategy accurately. The visuospatial scratchpad was used as a visual memory aid. The inaccurate and flexible students compensated for their poor number facts and minimal and disconnected knowledge base by using a variety of mental strategies in an endeavour to find one that would enable them to complete the calculation. Although their limited numeration understanding and memory (including central executive) were sufficient to support the development of some alternative strategies, these were not high level strategies. Finally, the inaccurate and inflexible students who exhibited deficient and disconnected understanding tried to compensate by using teacher-taught procedures (similar to the strategy employed by accurate and inflexible students), but they were unsuccessful, as they possessed no procedural understanding and also had poor working memory.

Detailed analysis of students' knowledge was used to develop frameworks, which explained children's proficiency in addition and subtraction mental computation. The theoretical frameworks explained the influence of contributing factors and the relationships (if any) between them. The frameworks formed the basis of flowcharts, which explained the process in mental computation for each group of students.

The importance of connected knowledge for proficient mental computation demonstrates the need for teaching practices to focus on the development of an extensive and integrated knowledge base. Students can and do formulate their own strategies, but do not always use them accurately. Therefore, students should be encouraged to formulate their own strategies but in a supportive environment that assists them to use strategies appropriately. Because of memory load, students should be permitted to use external memory aids (e.g. pen and paper) to assist mental computation. This has a second payoff in that efficient mental strategies are, at times, also efficient written strategies. By having students formulate mental strategies, they have to call upon number sense knowledge, thus acquiring connected knowledge while they develop computational procedures. This is in contrast to students using teacher-taught procedures, which require little connected knowledge.