Do individual differences in counting strategy determine arithmetic performance?
Background
Over the past several decades, cognitive development research has endeavoured to understand how and why individual differences in mathematics performance exist. A current key topic in this area of research is understanding whether choice of counting strategy influences arithmetic performance. Two main strategies commonly used for arithmetic problems are cumulative sum, encoding and sum. Cumulative sum refers to performing addition computation at each presentation of new stimuli (e.g., 2 + 3 = 5, 5 + 4 = 9, 9 + 1 = 10), whereas encoding and sum refers to retaining the presented stimuli and computing at the end (e.g., 2 + 3 + 4 + 1 = 10). These two strategies are used interchangeably depending on the difficulty of the task. This research project consisted of two smaller individual experiments. Both experiments contained (a) a simple addition task, where participants were required to sum sequences of numbers, and (b) a structural feature of the task varied to manipulate task difficulty. Of the two experiments, the first experiment involved manipulation of inter-stimulus intervals, to manipulate difficulty through processing speed, encouraging a switch to the encoding and sum strategy. Conversely, the second experiment involved manipulation of the numbers of digits to be added, to manipulate difficulty through working memory capacity, encouraging a switch to cumulative sum strategy.
Research Questions / Hypotheses
For hypotheses to hold, several requirements must be met: Firstly, poor performance should motivate individuals to change their strategy. Secondly, individuals are using either encoding and sum or cumulative sum strategies. The three hypotheses for the first experiment were: 1. Some participants would be able to do the entire task regardless of inter-stimulus interval, suggesting that they used one particular strategy (likely add at the end) which was sufficiently efficient for them to be able to perform well across all conditions. It was expected that the typical profile of performance of these participants would be characterised by fast response times and high accuracy across the entire task (i.e. little to no change in performance). 2. Some participants would not be able to do the task when inter-stimulus intervals were shorter than 400 ms as their particular counting strategy (add as you go) was inefficient for processing numbers quickly. To improve their performance, these participants would be motivated to switch their strategy to one which was less susceptible to the difficulties of fast presentation speeds (add at the end). It was expected that the typical profile of performance of these participants would be characterised by (a) poor performance (low accuracy and slow response times) on trials with inter-stimulus intervals shorter than 400 ms which motivates a switch in strategies; (b) changes in accuracy and response times as a function of switch in strategies, specifically an improvement in accuracy for trials with inter-stimulus intervals shorter than 400 ms and convergence of response times for all trials regardless of inter-stimulus interval. 3. Little to no participants would be unable to do the task regardless of their choice of counting strategy. It was expected that the typical profile of performance of these participants would be characterised by low accuracy across the entire task. The two hypotheses for the second experiment were: 1. Successful strategy change (from encoding and sum to cumulative sum) would result in a rapid increase in accuracy and a corresponding rapid decrease in response times. 2. Low accuracy for large set sizes will be associated with lower digit spans, assuming errors occur from insufficient working memory capacity to retain and manipulate the numerical information in larger set sizes.
Participants
73 REP participants completed this research project. No participant exclusions were applied.
Methods
Both experiments spanned 60 minutes and consisted of four different tasks. Both experiments involved 3 baseline tasks, specifically a visual digit span task, an odd or even task, and a symbolic sequential add-four task. Each experiment involved a slightly different main task. The main task of the first experiment involved a symbolic sequential add-four task where the number inter-stimulus intervals varied across trials (450, 400, 350 and 300 ms). The main task of the second experiment involved a symbolic sequential task where the number of digits varied across trials (addition of 4, 5, 6 or 8 digits).
Results
The results of the first experiment included an analysis of preliminary results largely in terms of raw data at a group level and an analysis of raw data at an individual level in terms of the three different proposed profiles of performance. The results of the second experiment included an analysis in terms of the raw data at a group level to determine that requirements were met. Analysis of group level data and smoothed individual data for proposed profiles of performance were inconsistent with hypothesis 1. Spearman's rank-order correlations and smoothed individual data for proposed profiles of performance were consistent with hypothesis 2.
Implications
The first experiment found that most participants achieved high accuracy and fast RTs across all four task conditions and there was no strong systematic evidence of any improvements in performance consistent with a change in strategy. These findings have important implications for improving theoretical understandings of the complex relationships between strategy choice, flexibility of strategy changes, individual threshold of an acceptable level of poor performance and overall arithmetic performance. The second experiment found that most participants had lower accuracy for the addition of more numbers, suggesting a successful manipulation of task difficulty. However, there was no strong systematic evidence of the expected profile of strategy change. These findings have important implications for the relationship between strategy choice and arithmetic performance; the role of working memory in selecting and executing strategies; and the importance of strategy flexibility in comparison to experience in using strategies. All findings were reported in two different honours theses.