Learning artificial number symbols with ordinal and magnitude information.

The question of how numerical symbols gain semantic meaning is a key focus of mathematical cognition research. Some have suggested that symbols gain meaning from magnitude information, by being mapped onto the approximate number system, whereas others have suggested symbols gain meaning from their ordinal relations to other symbols. Here we used an artificial symbol learning paradigm to investigate the effects of magnitude and ordinal information on number symbol learning. Across two experiments, we found that after either magnitude or ordinal training, adults successfully learned novel symbols and were able to infer their ordinal and magnitude meanings. Furthermore, adults were able to make relatively accurate judgements about, and map between, the novel symbols and non-symbolic quantities (dot arrays). Although both ordinal and magnitude training was sufficient to attach meaning to the symbols, we found beneficial effects on the ability to learn and make numerical judgements about novel symbols when combining small amounts of magnitude information for a symbol subset with ordinal information about the whole set. These results suggest that a combination of magnitude and ordinal information is a plausible account of the symbol learning process.

Counting many as one: Young children can understand sets as units except when counting.

Young children frequently make a peculiar counting mistake. When asked to count units that are sets of multiple items, such as the number of families at a party, they often count discrete items (i.e., individual people) rather than the number of sets (i.e., families). One explanation concerns children's incomplete understanding of what constitutes a unit, resulting in a preference for discrete items. Here we demonstrate that children's incomplete understanding of counting also plays a role. In an experiment with 4- and 5-year-old children (N = 43), we found that even if children are able to name sets, group items into sets, and create one-to-one correspondences with sets, many children are nevertheless unable to count sets as units. We conclude that a nascent understanding of the abstraction principle of counting is also a cause of some children's counting errors.

Stereotype threat, gender and mathematics attainment: A conceptual replication of Stricker & Ward.

Stereotype threat has been proposed as one cause of gender differences in post-compulsory mathematics participation. Danaher and Crandall argued, based on a study conducted by Stricker and Ward, that enquiring about a student's gender after they had finished a test, rather than before, would reduce stereotype threat and therefore increase the attainment of women students. Making such a change, they argued, could lead to nearly 5000 more women receiving AP Calculus AB credit per year. We conducted a preregistered conceptual replication of Stricker and Ward's study in the context of the UK Mathematics Trust's Junior Mathematical Challenge, finding no evidence of this stereotype threat effect. We conclude that the 'silver bullet' intervention of relocating demographic questions on test answer sheets is unlikely to provide an effective solution to systemic gender inequalities in mathematics education.

Finding the subitizing in groupitizing: Evidence for parallel subitizing of dots and groups in grouped arrays.

'Groupitizing' refers to the observation that visually grouped arrays can be accurately enumerated much faster than can unstructured arrays. Previous research suggests that visual grouping allows participants to draw on arithmetic abilities and possibly use mental calculations to enumerate grouped arrays quickly and accurately. Here, we address how subitizing might be involved in finding the operands for mental calculations in grouped dot arrays. We investigated whether participants can use multiple subitizing processes to enumerate both the number of dots and the number of groups in a grouped array. We found that these multiple subitizing processes can take place within 150 ms and that dots and groups seem to be subitized in parallel and with equal priority. Implications for research on mechanisms of groupitizing are discussed.

Cognitive performance in pain is predicted by effort, not goal desire.

BACKGROUND: Pain's disruptive effects on cognition are well documented. The seminal goal-pursuit account of pain suggests that cognitive disruption is less likely if participants are motivated to attended to a focal goal and not a pain goal. OBJECTIVES: Existing theory is unclear about the conceptualisation and operationalisation of 'focal goal'. This study aims to clarify how goals should be conceptualised and further seeks to test the theory of the goal-pursuit account. METHODS: In a pre-registered laboratory experiment, 56 participants completed an arithmetic task in high-reward/low-reward and pain/control conditions. Pain was induced via cold-water immersion. RESULTS: High levels of reported effort exertion predicted cognitive-task performance, whereas desire for rewards did not. Post-hoc analyses further suggest that additional effort in the pain condition compensated for pain's disruptive effects, but when this extra effort was not exerted, performance deficits were observed in pain, compared to control, conditions. CONCLUSION: Results suggest that 'motivation', or commitment to a focal goal, is best understood as effort exertion and not as a positive desire to achieve a goal. These results solidify existing theory and aid researchers in operationalising these constructs.

People in pain make poorer decisions.

Chronic pain affects 1 in 5 people and has been shown to disrupt attention. Here, we investigated whether pain disrupts everyday decision making. In study 1, 1322 participants completed 2 tasks online: a shopping-decisions task and a measure of decision outcomes over the previous 10 years. Participants who were in pain during the study made more errors on the shopping task than those who were pain-free. Participants with a recurrent pain condition reported more negative outcomes from their past decisions than those without recurrent pain. In study 2, 44 healthy participants completed the shopping-decisions task with and without experimentally induced pain. Participants made more errors while in pain than while pain-free. We suggest that the disruptive effect of pain on attending translates into poorer decisions in more complex and ecologically valid contexts, that the effect is causal, and that the consequences are not only attentional but also financial.

Developmental differences in approaches to nonsymbolic comparison tasks.

Nonsymbolic comparison tasks are widely used to measure children's and adults' approximate number system (ANS) acuity. Recent evidence has demonstrated that task performance can be influenced by changes to the visual characteristics of the stimuli, leading some researchers to suggest it is unlikely that an ANS exists that can extract number information independently of the visual characteristics of the arrays. Here, we analysed 124 children's and 120 adults' dot comparison accuracy scores from three separate studies to investigate individual and developmental differences in how numerical and visual information contribute to nonsymbolic numerosity judgements. We found that, in contrast to adults, the majority of children did not use numerical information over and above visual cue information to compare quantities. This finding was consistent across different studies. The results have implications for research on the relationship between dot comparison performance and formal mathematics achievement. Specifically, if most children's performance on dot comparison tasks can be accounted for without the involvement of numerical information, it seems unlikely that observed correlations with mathematics achievement stem from ANS acuity alone.

The measurement of approximate number system acuity across the lifespan is compromised by congruency effects.

Recent studies have highlighted the influence of visual cues such as dot size and cumulative surface area on the measurement of the approximate number system (ANS). Previous studies assessing ANS acuity in ageing have all applied stimuli generated by the Panamath protocol, which does not control nor measure the influence of convex hull. Crucially, convex hull has recently been identified as an influential visual cue present in dot arrays, with its impact on older adults' ANS acuity yet to be investigated. The current study therefore investigated the manipulation of convex hull by the Panamath protocol, and its effect on the measurement of ANS acuity in younger and older participants. First, analyses of the stimuli generated by Panamath revealed a confound between numerosity ratio and convex hull ratio. Second, although older adults were somewhat less accurate than younger adults on convex hull incongruent trials, ANS acuity was broadly similar between the groups. These findings have implications for the valid measurement of ANS acuity across all ages, and suggest that the Panamath protocol produces stimuli that do not adequately control for the influence of convex hull on numerosity discrimination.

Is the ANS linked to mathematics performance?

Leibovich et al. argue persuasively that researchers should not assume that approximate number system (ANS) tasks harness an innate sense of number. However, some studies have reported a causal link between ANS tasks and mathematics performance, implicating the ANS in the development of numerical skills. Here we report a p-curve analysis, which indicates that these experimental studies do not contain evidential value.

Congruency effects in dot comparison tasks: convex hull is more important than dot area.

The dot comparison task, in which participants select the more numerous of two dot arrays, has become the predominant method of assessing Approximate Number System (ANS) acuity. Creation of the dot arrays requires the manipulation of visual characteristics, such as dot size and convex hull. For the task to provide a valid measure of ANS acuity, participants must ignore these characteristics and respond on the basis of number. Here, we report two experiments that explore the influence of dot area and convex hull on participants' accuracy on dot comparison tasks. We found that individuals' ability to ignore dot area information increases with age and display time. However, the influence of convex hull information remains stable across development and with additional time. This suggests that convex hull information is more difficult to inhibit when making judgements about numerosity and therefore it is crucial to control this when creating dot comparison tasks.

Dot comparison stimuli are not all alike: the effect of different visual controls on ANS measurement.

The most common method of indexing Approximate Number System (ANS) acuity is to use a nonsymbolic dot comparison task. Currently there is no standard protocol for creating the dot array stimuli and it is unclear whether tasks that control for different visual cues, such as cumulative surface area and convex hull size, measure the same cognitive constructs. Here we investigated how the accuracy and reliability of magnitude judgements is influenced by visual controls through a comparison of performance on dot comparison trials created with two standard methods: the Panamath program and Gebuis & Reynvoet's script. Fifty-one adult participants completed blocks of trials employing images constructed using the two protocols twice to obtain a measure of immediate test-retest reliability. We found no significant correlation between participants' accuracy scores on trials created with the two protocols, suggesting that tasks employing these protocols may measure different cognitive constructs. Additionally, there were significant differences in the test-retest reliabilities for trials created with each protocol. Finally, strong congruency effects for convex hull size were found for both sets of protocol trials, which provides some clarification for conflicting results in the literature.

Indexing the approximate number system.

Much recent research attention has focused on understanding individual differences in the approximate number system, a cognitive system believed to underlie human mathematical competence. To date researchers have used four main indices of ANS acuity, and have typically assumed that they measure similar properties. Here we report a study which questions this assumption. We demonstrate that the numerical ratio effect has poor test-retest reliability and that it does not relate to either Weber fractions or accuracy on nonsymbolic comparison tasks. Furthermore, we show that Weber fractions follow a strongly skewed distribution and that they have lower test-retest reliability than a simple accuracy measure. We conclude by arguing that in the future researchers interested in indexing individual differences in ANS acuity should use accuracy figures, not Weber fractions or numerical ratio effects.

Sampling from the mental number line: how are approximate number system representations formed?

Nonsymbolic comparison tasks are commonly used to index the acuity of an individual's Approximate Number System (ANS), a cognitive mechanism believed to be involved in the development of number skills. Here we asked whether the time that an individual spends observing numerical stimuli influences the precision of the resultant ANS representations. Contrary to standard computational models of the ANS, we found that the longer the stimulus was displayed, the more precise was the resultant representation. We propose an adaptation of the standard model, and suggest that this finding has significant methodological implications for numerical cognition research.

Advanced mathematical study and the development of conditional reasoning skills.

Since the time of Plato, philosophers and educational policy-makers have assumed that the study of mathematics improves one's general 'thinking skills'. Today, this argument, known as the 'Theory of Formal Discipline' is used in policy debates to prioritize mathematics in school curricula. But there is no strong research evidence which justifies it. We tested the Theory of Formal Discipline by tracking the development of conditional reasoning behavior in students studying post-compulsory mathematics compared to post-compulsory English literature. In line with the Theory of Formal Discipline, the mathematics students did develop their conditional reasoning to a greater extent than the literature students, despite them having received no explicit tuition in conditional logic. However, this development appeared to be towards the so-called defective conditional understanding, rather than the logically normative material conditional understanding. We conclude by arguing that Plato may have been correct to claim that studying advanced mathematics is associated with the development of logical reasoning skills, but that the nature of this development may be more complex than previously thought.

Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement.

Given the well-documented failings in mathematics education in many Western societies, there has been an increased interest in understanding the cognitive underpinnings of mathematical achievement. Recent research has proposed the existence of an Approximate Number System (ANS) which allows individuals to represent and manipulate non-verbal numerical information. Evidence has shown that performance on a measure of the ANS (a dot comparison task) is related to mathematics achievement, which has led researchers to suggest that the ANS plays a critical role in mathematics learning. Here we show that, rather than being driven by the nature of underlying numerical representations, this relationship may in fact be an artefact of the inhibitory control demands of some trials of the dot comparison task. This suggests that recent work basing mathematics assessments and interventions around dot comparison tasks may be inappropriate.

On mathematicians' different standards when evaluating elementary proofs.

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians.

Substitution and sameness: two components of a relational conception of the equals sign.

A sophisticated and flexible understanding of the equals sign (=) is important for arithmetic competence and for learning further mathematics, particularly algebra. Research has identified two common conceptions held by children: the equals sign as an operator and the equals sign as signaling the same value on both sides of the equation. We argue here that, in addition to these two conceptions, the notion of substitution is also an important part of a sophisticated understanding of mathematical equivalence. We provide evidence from a cross-cultural study in which English and Chinese children were asked to rate the "cleverness" of operational, sameness, and substitutive definitions of the equals sign. A principal components analysis revealed that the substitutive items were distinct from the sameness items. Furthermore, Chinese children rated the substitutive items as 'sort of clever' or 'very clever', whereas English children rated them as not so clever, suggesting that the notion of substitution develops differently across the two countries. Implications for developmental models of children's understanding of equivalence are discussed.

Non-verbal number acuity correlates with symbolic mathematics achievement: but only in children.

The process by which adults develop competence in symbolic mathematics tasks is poorly understood. Nonhuman animals, human infants, and human adults all form nonverbal representations of the approximate numerosity of arrays of dots and are capable of using these representations to perform basic mathematical operations. Several researchers have speculated that individual differences in the acuity of such nonverbal number representations provide the basis for individual differences in symbolic mathematical competence. Specifically, prior research has found that 14-year-old children's ability to rapidly compare the numerosities of two sets of colored dots is correlated with their mathematics achievements at ages 5-11. In the present study, we demonstrated that although when measured concurrently the same relationship holds in children, it does not hold in adults. We conclude that the association between nonverbal number acuity and mathematics achievement changes with age and that nonverbal number representations do not hold the key to explaining the wide variety of mathematical performance levels in adults.

Measuring the approximate number system.

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.

Eye movements and time-based selection: where do the eyes go in preview search?

In visual search tasks, presenting one set of distractors (previewing them) before a second set which contains the target, improves search efficiency compared to when all items appear simultaneously. It has been proposed that this preview benefit reflects an attentional bias against old information and toward new information. Here we tested directly whether there was such a bias by measuring eye movement behavior. The main findings were that fixations were biased against, and overall dwell times were shorter on, old stimuli during search in the preview condition. In addition, the initial onset of search was delayed in the preview condition and saccades made during the preview period did not disrupt the ability to prioritize new items. The data demonstrate directly that preview search results in an attentional bias toward new items and against old items.