The Current Research

The current study investigated the influences of format and mathematics experience across a wide range of numerical functions. By taking into account individual differences related to numeracy and mathematics education, the study explored the possibility that format effects in adult number processing and manipulation could be regulated by mathematics experience.

Support for models of symbolic number representation such as the abstract code model (McCloskey, 1992) and the encoding complex model (e.g. Campbell & Clark, 1992) have mostly come from studies of arithmetic. More research is thus needed to relate processing differences between formats to early numerical

processing such as magnitude comparison or subitizing. While a small number of studies have compared the reading of arabic digits and number words, Stroop tasks investigating number–size comparisons and subitizing have not compared different formats directly, with experiments mostly focusing on either arabic digits or number words. The first three experiments in the current thesis examined such basic

numerical processing by adapting Stroop tasks to investigate the processing

differences that might emerge for arabic digits, number words and quantifier words in the English language.

In chapter 2 (Experiment 1) the original counting Stroop task was adapted to include arabic digits for comparison with number words. The increase in RT on incongruent (e.g four four four, respond ‘three’) relative to neutral (e.g. cat cat cat,

respond ‘three’) trials was investigated for each format and at high and low levels of mathematics experience, based on participants’ Irish Leaving Certificate performance and results on a numeracy test. The view was explored that greater experience with mathematics could result in an advantage for processing numerical information, and that this might further vary between arabic digits and number words. Since the Stroop task has been widely used in other individual differences domains (see Chapter 2) it seemed well-suited to the study of individual differences in numerical information processing.

Chapter 3 (Experiment 2) addressed a similar question by considering format- specific effects in terms of size congruity and symbolic distance at different levels of mathematics experience. By modifying the task developed by Tang et al. (2006; see Chapter 3), arabic digits as well as number words were investigated in physical

(e.g.

two

(e.g.

two

magnitude are processed similarly and the role that stimulus format and mathematics experience can play in this regard.

Based on the results from Experiments 1 and 2, Experiment 3 (Chapter 4) investigated whether or not greater mathematics experience can result in an advantage in extracting numerical information from language more generally. The counting Stroop task used in Experiment 1 was adapted for studying quantifier words with specific (e.g. both) and general (e.g. some) number meanings. Since quantifier

digits, it is not certain whether or not quantifier word processing follows a more numerical or linguistic processing route. The role of number knowledge in quantifier word processing has not been explored to a great extent. However, in development, number knowledge seems to be central to quantifier word knowledge. Differences in quantifier word processing related to adults’ mathematics experience were thus explored.

Overall, Experiments 1 to 3 considered basic number encoding and how mathematics experience can influence this process. Subsequent to encoding, various other functions take place, such as calculation and arithmetic fact retrieval. To investigate these processes, Experiments 4 and 5 considered the role of operand format and mathematics experience in performing mental arithmetic. In addition to this, eye-tracking and event-related potential (ERP) technology were used in

Experiments 4 and 5 respectively, as these measures have been shown to be sensitive to effects that might not be evident from behavioural measures alone (e.g. Merkley & Ansari, 2010). Different stimuli can be processed along separate routes, but can still yield similar behavioural patterns (e.g. Zhang et al., 2010; Zhou, 2011). More sensitive measures such as eye-tracking and ERP technology were therefore

employed in the second part of the thesis alongside behavioural measures of accuracy and reaction time.

As mentioned above, two opposing viewpoints exist on how the encoding and answer-retrieval stages of arithmetic relate to one another. Recent studies favour both the additive viewpoint (e.g. MCloskey’s abstract code; Zhou, 2011) of format-

answer retrieval (Campbell & Alberts, 2009). The role of mathematics experience has, however, not been explored in these studies, and it seemed an important variable to consider in the study of adult arithmetic. In Chapter 5 (Experiment 4), a study of Campbell and Alberts (2009) was replicated in order to examine the influence of operand format on the calculation strategies used in arithmetic. Campbell and Alberts (2009) investigated whether the format of the operands directly influences the

strategies that participants reported using (e.g. direct memory retrieval or calculation), or if relatively similar calculation processes take place for arabic digits and number words, with their results supporting the former view. Since shortcomings have been noted with self-reports (Kirk & Ashcraft, 2001), Experiment 4 employed eye-tracking measures to investigate if the findings of Campbell and Alberts (2009) could be supported. Specifically, the experiment tested whether or not measures of fixation count and fixation duration reflect similar interactions of format, operation and problem size as was noted in the self-reports of Campbell and Alberts’s (2009) participants.

While overall relatively little research has been conducted using eye-tracking in the study of numerical cognition, it has been a useful tool in studying information processing in reading (e.g. Inhoff, 1984, 1985; Rayner & Pollatsek, 1987) and thus seems well suited for the study of numerical processes. A recent interest in using eye-tracking to study numerical cognition specifically has also emerged (e.g. Merkley & Ansari, 2010; Moeller, Neuburger & Kaufman, 2009), as eye-tracking can provide a more extensive measure of information processing than reaction time and accuracy (Desroches, Joanisse & Robertson, 2006).

While addressing the question regarding the relationship between the different stages in arithmetic, a more in depth analysis of the interaction between the encoding and answer-retrieval stages was conducted in the final experiment. To do this, Experiment 5 replicated an event-related potentials (ERPs) study of Zhou (2011), which aimed to separate the presentation of the encoding and retrieval phases of arithmetic equations in a true/false verification task that presented addition and multiplication equations in separate blocks. In this study, the equation ‘3 + 2 = 5’, for example, was presented as ‘3’ and ‘+ 2 = 5’ on separate presentation-screens (or ‘three’ and ‘+ two = five’). This allowed the effects of operation, format and

mathematics experience at the encoding and answer-retrieval stages to be investigated separately. Zhou (2011) noted a dissociation in how addition and multiplication is mentally represented even during the encoding phase where participants only saw a single digit operand on-screen. In support of the additive view of arithmetic (e.g. McCloskey’s abstract code model), multiplication and addition operands presented in the same format are encoded differently, which allows the relevant arithmetic facts to be retrieved. If the interactive view of arithmetic (e.g. Campbell & Clark’s encoding complex model) were supported, the dissociation between arithmetic operations should only emerge subsequently to the encoding phase, since addition and multiplication operands presented in the same format should be encoded similarly (Zhou, 2011). The final experiment (Experiment 5) investigated the event-related potentials at the encoding and retrieval phases separately, while controlling for mathematics experience and presenting equations in digit as well as word format, unlike Zhou’s (2011) study which only involved arabic digit operands. By including

two formats, the specific effects of operation and mathematics experience that emerge for each format and at each level of arithmetic could be compared.

The overall objective of the current research was to investigate how numerical information is accessed from symbolic formats, and how this might differ at different levels of mathematics experience. By investigating these effects for various

numerical functions and utilising a wide range of measures, the aim was to gain a more comprehensive view of the mental representation of numbers. The early experiments (Experiments 1 to 3) investigated basic number encoding, which formed the basis for investigating format effects in more complex numerical cognition, such as calculation (Experiments 4 and 5). The following chapter (Experiment 1) set out to explore the role of mathematics experience in format-specific processing in a simple counting task. By investigating cognitive interference of arabic digits and number words, the automaticity of processing of the two formats could be directly compared.

Chapter 2

Experiment 1: Cognitive Interference in a Digit–Word Counting Task: The Role of Mathematics Experience in Format-specific Processing