Arithmetic,Mathematical Intuition,and Evidence

Abstract

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic (Oxford: Oxford University Press, 2011). The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics.     

Assessing the Approximate Number System: no relation between numerical comparison and estimation tasks

Whether our general numerical skills and the mathematical knowledge that we acquire at school are entwined is a debated issue, which many researchers are still striving to investigate. The findings reported in the literature are actually inconsistent; some studies emphasized the existence of a relationship between the acuity of the Approximate Number System (ANS) and arithmetic competence, while some others did not observe any significant correlation. One potential explanation of the discrepancy might stem from the evaluation of the ANS itself. In the present study, we correlated two measures used to index ANS acuity with arithmetic performance. These measures were the Weber fraction (w), computed from a numerical comparison task and the coefficient of variation (CV), computed from a numerical estimation task. Arithmetic performance correlated with estimation CV but not with comparison w. We further investigated the meaning of this result by taking the relationship between w and CV into account. We expected a tight relation as both these measures are believed to assess ANS acuity. Crucially, however, w and CV did not correlate with each other. Moreover, the value of w was modulated by the congruity of the relation between numerical magnitude and non-numerical visual cues, potentially accounting for the lack of correlation between the measures. Our findings thus challenge the overuse of w to assess ANS acuity and more generally put into question the relevance of correlating this measure with arithmetic without any deeper understanding of what they are really indexing.     

The production and verification tasks in mental addition: An empirical comparison

We respond to A. Baroody's comment (1984, Developmental Review, 4, 148–156) with an empirical comparison of the production and verification tasks. With the exception of performance at the first grade level, the two tasks yield essentially identical conclusions. The results of an adjunct task, in which the rate of mental counting was assessed, suggest that children as young as second grade are relying on memory retrieval to a significant degree. In contrast to Baroody's speculation, there appear to be no widespread difficulties associated with results from the verification task. Furthermore, the task permits a more analytic examination of performance and underlying mental process than is afforded by the production task. We conclude by reiterating the empirical support for a model of fact retrieval, and suggesting that accessibility of the arithmetic facts is the basic factor which underlies both fact retrieval and procedural knowledge performance.     

Memory and metacognition for piano melodies: Illusory advantages of fixed- over random-order practice

Some learning schedules can foster an illusion of competence, whereby the learner feels that the skill will be retained better than it actually will be. Consider fixed-order practice, in which a person practices a task repeatedly before switching to the next task (e.g., task order A, A, B, B); in contrast, in random-order practice, a person randomly alternates among two or more tasks (e.g., task order C, D, D, C). In the present experiment, participants (n = 25) who had formal training in piano practiced melodies under fixed- or random-order conditions (within-subjects), and then returned for a retention test 2 days later. Initially, the participants performed faster on melodies practiced in a fixed-order. However, on a retention test 2 days later, participants were faster with melodies from the random-order condition. Despite the within-subjects design, which facilitated the comparison of practice conditions, participants’ metacognitive judgments indicated an illusion of competence, whereby they erroneously believed that fixed-order practice would result in faster retention performance. Our results suggest that even some trained musicians may use ease of acquisition as a heuristic for predicting future performance.     

Costs and Benefits of Bullying in the Context of the Peer Group: A Three Wave Longitudinal Analysis

Whereas previous research has shown that bullying in youth is predictive of a range of negative outcomes later in life, the more proximal consequences of bullying in the context of the peer group at school are not as clear. The present three-wave longitudinal study followed children (N?=?394; 53 % girls; M _age?=?10.3 at Time 1) from late childhood into early adolescence. Joint trajectory analyses were used to examine the dynamic prospective relations between bullying on the one hand, and indices tapping perceived popularity, peer-reported social acceptance, self-perceived social competence, and internalizing symptoms on the other. Results show that although young bullies may be on a developmental path that in the long run becomes problematic, from the bullies’ perspective in the shorter term personal advantages outweigh disadvantages. High bullying is highly positively related to high social status as indexed by perceived popularity. Although bullies are not very high in peer-rated social acceptance, most are not very low either. Moreover, bullies do not demonstrate elevated internalizing symptoms, or problems in the social domain as indexed by self-perceived social competence. As bullying yields clear personal benefits for the bullies without strong costs, the findings underscore the need for interventions targeting mechanisms that reward bullying (198 words).     

A computer-oriented task analysis method for mathematics instruction

Components of a computer solution for fraction problems in arithmetic have an analog in a new approach to educational task analysis. The components may be called goal-setting or planning elements, in contrast to the goal-satisfying or behavioral steps emphasized in much contemporary task analysis. A hypothetical tutorial dialogue in which a student is asked to tell a tutor what step to perform next is presented as an example of the emphasis of the planning side of computation. Explicit identification of goal-setting elements of a task is illustrated by analysis of the task of converting a pair of fractions to a new pair with a common denominator. The decomposition thus produces is simulated with two computer programs: (1) a LISP program using a set of production rules consisting of conditions to be met and subsequent actions to be taken and (2) a PROLOG program stating goals and solving goals explicitly.     

Neighborhood consistency and memory for number facts

Verguts and Fias (Memory & Cognition 33:1-16, 2005a) proposed a new model of memory for simple multiplication facts ( $ 2 \times 3 = 6 $ ; $ 8 \times 7 = 56 $ ) in which learning and performance is governed by the consistency of a problem’s correct product with neighboring products in the times table. In the present study, to directly investigate effects of neighborhood consistency, participants memorized a set of 16 novel “pound” arithmetic equations. The pound arithmetic table included eight tie equations with repeated operands (e.g., 4 # 4 = 29) and eight nontie equations (e.g., 5 # 4 = 39). In the consistent problem set, tie and nontie answers in adjacent columns and rows shared a common decade or unit value. In the inconsistent problem set, neighboring tie and nontie problems did not share a common decade or unit. Across 14 study–test blocks, memorization of the pound arithmetic table presented a robust effect of neighborhood consistency, with the rate of learning nearly doubling that of the inconsistent condition. An analysis of error types showed that consistency fostered the development of a categorical structure based on problem operands and that tie problems were encoded as a distinct subcategory of problems. There was also a substantial learning advantage for tie problems relative to nonties both with consistent and inconsistent neighbors. The results indicate that neighborhood consistency can have a major impact on memory for number facts.     

Associations of non‐symbolic and symbolic numerical magnitude processing with mathematical competence: a meta‐analysis

Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non‐symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta‐analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two‐level random‐effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non‐symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non‐symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school‐aged children and for adults.     

Analogous selection processes in declarative and procedural working memory: N-2 list-repetition and task-repetition costs

Working memory (WM) holds and manipulates representations for ongoing cognition. Oberauer (Psychology of Learning and Motivation, 51, 45–100, 2009) distinguishes between two analogous WM sub-systems: a declarative WM which handles the objects of thought, and a procedural WM which handles the representations of (cognitive) actions. Here, we assessed whether analogous effects are observed when participants switch between memory sets (declarative representations) and when they switch between task sets (procedural representations). One mechanism assumed to facilitate switching in procedural WM is the inhibition of previously used, but currently irrelevant task sets, as indexed by n-2 task-repetition costs (Mayr & Keele, Journal of Experimental Psychology: General, 129(1), 4–26, 2000). In this study we tested for an analogous effect in declarative WM. We assessed the evidence for n-2 list-repetition costs across eight experiments in which participants switched between memory lists to perform speeded classifications, mental arithmetic, or a local recognition test. N-2 list-repetition costs were obtained consistently in conditions assumed to increase interference between memory lists, and when lists formed chunks in long-term memory. Further analyses across experiments revealed a substantial contribution of episodic memory to n-2 list-repetition costs, thereby questioning the interpretation of n-2 repetition costs as reflecting inhibition. We reanalyzed the data of eight task-switching experiments, and observed that episodic memory also contributes to n-2 task-repetition costs. Taken together, these results show analogous processing principles in declarative and procedural WM, and question the relevance of inhibitory processes for efficient switching between mental sets.     

Improving arithmetic performance with number sense training: An investigation of underlying mechanism

A nonverbal primitive number sense allows approximate estimation and mental manipulations on numerical quantities without the use of numerical symbols. In a recent randomized controlled intervention study in adults, we demonstrated that repeated training on a non-symbolic approximate arithmetic task resulted in improved exact symbolic arithmetic performance, suggesting a causal relationship between the primitive number sense and arithmetic competence. Here, we investigate the potential mechanisms underlying this causal relationship. We constructed multiple training conditions designed to isolate distinct cognitive components of the approximate arithmetic task. We then assessed the effectiveness of these training conditions in improving exact symbolic arithmetic in adults. We found that training on approximate arithmetic, but not on numerical comparison, numerical matching, or visuo-spatial short-term memory, improves symbolic arithmetic performance. In addition, a second experiment revealed that our approximate arithmetic task does not require verbal encoding of number, ruling out an alternative explanation that participants use exact symbolic strategies during approximate arithmetic training. Based on these results, we propose that nonverbal numerical quantity manipulation is one key factor that drives the link between the primitive number sense and symbolic arithmetic competence. Future work should investigate whether training young children on approximate arithmetic tasks even before they solidify their symbolic number understanding is fruitful for improving readiness for math education.     

Autonomic impairment in Borderline Personality Disorder: A laboratory investigation

Recent research suggests that emotional dysfunction in psychiatric disorders can be reflected in autonomic abnormalities. The present study examines sympathetic and parasympathetic autonomic nervous system activity in individuals with Borderline Personality Disorder (BPD) before, during, and following a social stressor task. Data were obtained from an analogue sample of participants screening positive for BPD (n = 12) and healthy controls (n = 28). In general, BPD participants exhibited increased sympathetic activity (indexed by Cardiac Sympathetic Index, CSI; Toichi et al., 1997) and decreased parasympathetic activity (indexed by Respiratory Sinus Arrhythmia, RSA) compared to controls. During the stressful task, BPD and control participants exhibited different trajectories of sympathetic activation: estimates of sympathetic activity increased for BPD participants and decreased for controls. Furthermore, BPD participants reported the task (but not baseline or recovery phases) to be more frustrating than controls. Findings are interpreted in the context of Polyvagal theory.     

Autonomic impairment in Borderline Personality Disorder: A laboratory investigation

Recent research suggests that emotional dysfunction in psychiatric disorders can be reflected in autonomic abnormalities. The present study examines sympathetic and parasympathetic autonomic nervous system activity in individuals with Borderline Personality Disorder (BPD) before, during, and following a social stressor task. Data were obtained from an analogue sample of participants screening positive for BPD (n = 12) and healthy controls (n = 28). In general, BPD participants exhibited increased sympathetic activity (indexed by Cardiac Sympathetic Index, CSI; Toichi et al., 1997) and decreased parasympathetic activity (indexed by Respiratory Sinus Arrhythmia, RSA) compared to controls. During the stressful task, BPD and control participants exhibited different trajectories of sympathetic activation: estimates of sympathetic activity increased for BPD participants and decreased for controls. Furthermore, BPD participants reported the task (but not baseline or recovery phases) to be more frustrating than controls. Findings are interpreted in the context of Polyvagal theory.     

Deflationism beyond arithmetic

The conservativeness argument poses a dilemma to deflationism about truth, according to which a deflationist theory of truth must be conservative but no adequate theory of truth is conservative. The debate on the conservativeness argument has so far been framed in a specific formal setting, where theories of truth are formulated over arithmetical base theories. I will argue that the appropriate formal setting for evaluating the conservativeness argument is provided not by theories of truth over arithmetic but by those over subject matters ‘richer’ than arithmetic, such as set theory. The move to this new formal setting provides deflationists with better defence and brings a broader perspective to the debate.

     

4-5岁幼儿数数行为的规则性与策略化应用特点的研究

本研究的主要目的是考察4-5岁儿童数数行为的规则性和策略化应用特点。研究选取了104名幼儿园中班儿童(年龄范围4-5岁),男女各半,采用现场实验法进行了逐一测查,测查任务为标准计数任务和数字复制任务。结果发现:4-5岁儿童的数数行为已经遵循一些数数规则,如固定顺序原则、一一对应原则和基数原则;但该年龄段的儿童还不会自发地把数数作为问题解决的策略应用于比较两组物体数目的问题上。     

33-month-old children succeed in a false belief task with reduced processing demands: A replication of Setoh et al. (2016)

A recent low-inhibition false belief task showed a high success rate with 33-month-old children when response-generation demands were reduced [Setoh, Scott, & Baillargeon (2016). Proceedings of the National Academy of Sciences, 113(47), 13360–13365]. We found correct responding in 74% of N = 58 33-month-old children, replicating the original findings. Within the same sample, we compared this performance with performance in a concurrent measure of false belief understanding which has previously produced competence in children below the age of 3 years [Hughes & Ensor (2007). Developmental Psychology, 43(6), 1447–1459]. Contrasting sharply with findings from the low-inhibition false belief task, we found partial competence in 15%, and full competence in only 5% of the same sample. These results show that the paradigm by Setoh and colleagues generates reliable findings in a different lab and a different language. We discuss this pattern of results in relation to theoretical considerations of early false belief understanding.     

The redundancy of recursion and infinity for natural language

An influential line of thought claims that natural language and arithmetic processing require recursion, a putative hallmark of human cognitive processing (Chomsky in Evolution of human language: biolinguistic perspectives. Cambridge University Press, Cambridge, pp 45–61, 2010; Fitch et al. in Cognition 97(2):179–210, 2005; Hauser et al. in Science 298(5598):1569–1579, 2002). First, we question the need for recursion in human cognitive processing by arguing that a generally simpler and less resource demanding process—iteration—is sufficient to account for human natural language and arithmetic performance. We argue that the only motivation for recursion, the infinity in natural language and arithmetic competence, is equally approachable by iteration and recursion. Second, we submit that the infinity in natural language and arithmetic competence reduces to imagining infinite embedding or concatenation, which is completely independent from the ability to implement infinite processing, and thus, independent from both recursion and iteration. Furthermore, we claim that a property of natural language is physically uncountable finity and not discrete infinity.     

Difference measurement spaces

An absolute-difference measurement space is a pair (X, e) where the real-valued function e on X² satisfies conditions which are shown in the paper to be necessary and sufficient for its representability by the absolute distance on the real line. A positive-difference measurement space is a pair (X, l), where the real-valued function l on X² satisfies conditions necessary and sufficient for its representability by positive distances on the real line. The conditions imposed on e and l make these functions extensive measurements of proximity and dominance, the two basic predicates of social enquiry. Another way of treating these conditions is to translate them to the formal language of multivalued logic. The translation is easy and the sentences obtained have plausible intuitive meanings such as reflexivity, symmetry, and transitivity. The two sets of conditions thus become formal theories of proximity and dominance. Our difference measurement spaces are relational structures for the multi-valued logic and models of the two formal theories. Thus proximity and dominance are considered dichotomous in principle and the multiple truth-values represent degrees of error. We suggest adopting multivalued logic as a framework within which the problem of measurement error can be treated together with the formal axiomatization of social and phychological theories.     

Beyond positive and negative affect: Achievement goals and discrete emotions in the elementary physical education classroom

ObjectivesBased on the Achievement Goal perspective [Dweck, C. S., Leggett, E. L. (1988). A social-cognitive approach to motivation and personality. Psychological Review, 95, 256–273; Nicholls, J. G. (1984). Achievement motivation: conceptions of ability, subjective experience, task choice, and performance. Psychological Review, 91, 328–346] and on Pekrun et al.'s [(2004). Beyond test anxiety: development and validation of the test emotions questionnaire (TEQ). Anxiety, Stress, and Coping, 17, 287–316] model of discrete class-related emotions, this study investigated the relation of achievement goals to discrete emotions in the Physical Education (PE) class.DesignCross-sectional.MethodParticipants were 319 Greek upper elementary school students who responded to a set of questionnaires assessing their achievement goal orientation, perceived competence, and the class-related emotions they experienced in the PE classes.ResultsHierarchical regression analyses showed that, after controlling for perceived competence and gender differences, task goals were positively related to positive activating emotions and negatively related to negative emotions. Ego goals exhibited a mixed picture as they were positively associated with pride and all the negative emotions. Furthermore, the relations between ego goals and emotions were qualified by an ego by task goal and by an ego by perceived competence interaction suggesting that ego goals were especially linked to emotional maladjustment when task goals were low and when competence perceptions were high rather than low.ConclusionUnique associations between task and ego goals and specific emotions were found, rendering insightful the disentanglement of positive and negative emotions into its components. The pursuit of task goals might help to counteract the emotional burden associated with ego goal pursuit whereas feeling competent to outperform when one endorses ego goals might perhaps put extra pressure on the pupils and, hence, have negative implications for their emotional adjustment.     

Causal discounting in the presence of a stronger cue is due to bias

People use information about the covariation between a putative cause and an outcome to determine whether a causal relationship obtains. When there are two candidate causes and one is more strongly related to the effect than is the other, the influence of the second is underestimated. This phenomenon is called causal discounting. In two experiments, we adapted paradigms for studying causal learning in order to apply signal detection analysis to this phenomenon. We investigated whether the presence of a stronger alternative makes the task more difficult (indexed by differences in d′) or whether people change the standard by which they assess causality (measured by β). Our results indicate that the effect is due to bias.     

Modeling simple driving tasks with a one-boundary diffusion model

A one-boundary diffusion model was applied to the data from two experiments in which subjects were performing a simple simulated driving task. In the first experiment, the same subjects were tested on two driving tasks using a PC-based driving simulator and the psychomotor vigilance test. The diffusion model fit the response time distributions for each task and individual subject well. Model parameters were found to correlate across tasks, which suggests that common component processes were being tapped in the three tasks. The model was also fit to a distracted driving experiment of Cooper and Strayer (Human Factors, 50, 893–902, 2008). Results showed that distraction altered performance by affecting the rate of evidence accumulation (drift rate) and/or increasing the boundary settings. This provides an interpretation of cognitive distraction whereby conversing on a cell phone diverts attention from the normal accumulation of information in the driving environment.