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.

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.

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.

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'.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.