Mastery of the logic of natural numbers is not the result of mastery of counting: evidence from late counters

To master the natural number system, children must understand both the concepts that number words capture and the counting procedure by which they are applied. These two types of knowledge develop in childhood, but their connection is poorly understood. Here we explore the relationship between the mastery of counting and the mastery of exact numerical equality (one central aspect of natural number) in the Tsimane’, a farming‐foraging group whose children master counting at a delayed age and with higher variability than do children in industrialized societies. By taking advantage of this variation, we can better understand how counting and exact equality relate to each other, while controlling for age and education. We find that the Tsimane’ come to understand exact equality at later and variable ages. This understanding correlates with their mastery of number words and counting, controlling for age and education. However, some children who have mastered counting lack an understanding of exact equality, and some children who have not mastered counting have achieved this understanding. These results suggest that understanding of counting and of natural number concepts are at least partially distinct achievements, and that both draw on inputs and resources whose distribution and availability differ across cultures.     

Soritic thinking, vagueness, and weakness of will

Soritic thinking based on reasoning that is involved in the sorites paradox plays a crucial role in some forms of weakness of will. Such soritic reasoning leads to failures of behavior, but cannot be shown to be irrational by standard means. Thus weakness of will appears to be rational, whereas strength of will is irrational when viewed soritically. The puzzle is how to undermine weakness of will and expose it as irrational. Even though such weakness of will is not moral, moral-type reasoning involving the principle of equality can be brought to bear. Weakness of will can also be seen to be analogous to free-rider problems and the prisoner's dilemma.     

Cosmopolitan Justice and Equalizing Opportunities

This paper defends a global principle of equality of opportunity, which states that it is unfair if some have worse opportunities because of their national or civic identity. It begins by outlining the reasoning underpinning this principle. It then considers three objections to global equality of opportunity. The first argues that global equality of opportunity is an inappropriate ideal given the great cultural diversity that exists in the world. The second maintains that equality of opportunity applies only to people who are interconnected in some way and infers from this that it should not be implemented at the global level. The third, inspired by Rawls's The Law of Peoples , maintains that it is inappropriate to thrust liberal ideals (like global equality of opportunity) on nonliberal peoples. Each of these challenges, I argue, is unpersuasive.     

Children's understanding of the relationship between addition and subtraction

In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12 + 9 − 9 yields 12. Here, we investigate whether preschool children’s approximate number knowledge nevertheless supports understanding of this relationship. Five-year-old children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding.     

An investigation of cognitive test performance across conditions of silence,background noise and music as a function of neuroticism

The present study investigates the role of trait neuroticism on cognitive performance under distraction. Seventy participants were given a personality test and then undertook a number of different cognitive tasks in silence, in the presence of popular music and in background noise. It was predicted that performance on a general intelligence test, a test of abstract reasoning, and a mental arithmetic task would be adversely affected by background sounds. It was predicted that neuroticism would be negatively correlated with performance on the mental arithmetic task but only when the individuals were working in the presence of background sound. Stable vs. unstable participant's performance on a mental arithmetic task during noise was significantly higher as predicted. The results provided partial support for the hypotheses and are discussed with respect to previous findings in the literature on personality (particularly introversion–extraversion) and distraction on cognitive task performance. Limitations are noted.     

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.     

A Dissociation between Symbolic Number Knowledge and Analogue Magnitude Information

Semantic understanding of numbers and related concepts can be dissociated from rote knowledge of arithmetic facts. However, distinctions among different kinds of semantic representations related to numbers have not been fully explored. Working with numbers and arithmetic requires representing semantic information that is both analogue (e.g., the approximate magnitude of a number) and symbolic (e.g., what / means). In this article, the authors describe a patient (MC) who exhibits a dissociation between tasks that require symbolic number knowledge (e.g., knowledge of arithmetic symbols including numbers, knowledge of concepts related to numbers such as rounding) and tasks that require an analogue magnitude representation (e.g., comparing size or frequency). MC is impaired on a variety of tasks that require symbolic number knowledge, but her ability to represent and process analogue magnitude information is intact. Her deficit in symbolic number knowledge extends to a variety of concepts related to numbers (e.g., decimal points, Roman numerals, what a quartet is) but not to any other semantic categories that we have tested. These findings suggest that symbolic number knowledge is a functionally independent component of the number processing system, that it is category specific, and that it is anatomically and functionally distinct from magnitude representations.     

Reasoning and normative beliefs: not too sophisticated

Does reasoning to a certain conclusion necessarily involve a normative belief in support of that conclusion? In many recent discussions of the nature of reasoning, such a normative belief condition is rejected. One main objection is that it requires too much conceptual sophistication and thereby excludes certain reasoners, such as small children. I argue that this objection is mistaken. Its advocates overestimate what is necessary for grasping the normative concepts required by the condition, while seriously underestimating the importance of such concepts for our most fundamental agential capacities. Underlying the objection is the observation that normative thoughts do not necessarily cross our minds during reasoning. I show that proponents of the normative belief condition can accommodate this observation by taking the required normative belief to guide the reasoning process and offer a novel account of what such guidance consists in.     

Girls’ Spatial Skills and Arithmetic Strategies in First Grade as Predictors of Fifth-Grade Analytical Math Reasoning

This study investigated longitudinal pathways leading from early spatial skills in first-grade girls to their fifth-grade analytical math reasoning abilities (N = 138). First-grade assessments included spatial skills, verbal skills, addition/subtraction skills, and frequency of choice of a decomposition or retrieval strategy on the addition/subtraction problems. In fifth grade, girls were given an arithmetic fluency test, a mental rotation spatial task, and a numeric and algebra math reasoning test. Using structural equation modeling, the estimated path model accounted for 87% of the variance in math reasoning. First-grade spatial skills had a direct pathway to fifth-grade math reasoning as well as an indirect pathway through first-grade decomposition strategy use. The total effect of first-grade spatial skills was significantly higher in predicting fifth-grade math reasoning than all other predictors. First-grade decomposition strategy use had the second strongest total effect, while retrieval strategy use did not predict fifth-grade math reasoning. It was first-grade spatial skills (not fifth-grade) that directly predicted fifth-grade math reasoning. Consequently, the results support the importance of early spatial skills in predicting later math. As expected, decomposition strategy use in first grade was linked to fifth-grade math reasoning indirectly through first-grade arithmetic accuracy and fifth-grade arithmetic fluency. However, frequency of first-grade decomposition use also showed a direct pathway to fifth-grade arithmetic reasoning, again stressing the importance of these early cognitive processes on later math reasoning.     

Preschoolers prior formal mathematics education engage numerical magnitude representation rather than counting principles in symbolic ±1 arithmetic: Evidence from the operational momentum effect

In numerical cognition research, the operational momentum (OM) phenomenon (tendency to overestimate the results of addition and/or binding addition to the right side and underestimating subtraction and/or binding it to the left side) can help illuminate the most basic representations and processes of mental arithmetic and their development. This study is the first to demonstrate OM in symbolic arithmetic in preschoolers. It was modeled on Haman and Lipowska's (2021) non-symbolic arithmetic task, using Arabic numerals instead of visual sets. Seventy-seven children (4–7 years old) who know Arabic numerals and counting principles (CP), but without prior school math education, solved addition and subtraction problems presented as videos with one as the second operand. In principle, such problems may be difficult when involving a non-symbolic approximate number processing system, whereas in symbolic format they can be solved based solely on the successor/predecessor functions and knowledge of numerical orders, without reference to representation of numerical magnitudes. Nevertheless, participants made systematic errors, in particular, overestimating results of addition in line with the typical OM tendency. Moreover, subtraction and addition induced longer response times when primed with left- and right-directed movement, respectively, which corresponds to the reversed spatial form of OM. These results largely replicate those of non-symbolic task and show that children at early stages of mastering symbolic arithmetic may rely on numerical magnitude processing and spatial-numerical associations rather than newly-mastered CP and the concept of an exact number.     

Preschoolers doing arithmetic: the concepts are willing but the working memory is weak.

The study of early mathematical development provides important insights into young children's emerging academic competencies and, potentially, a basis for adapting instructional methods. We presented nonverbal forms of two- and three-term arithmetic problems to 4-year-olds to determine (a) the extent to which certain information-processing demands make some problems more difficult than others and (b) whether preschoolers use arithmetic concepts spontaneously when solving novel problems. Children's accuracy on simple arithmetic problems (a + b and a - b) was strongly related (r2 = .88) to representational set size, the maximum number of units that need to be held in working memory to solve a given problem. Some children also showed spontaneous use of procedures based on the arithmetic principle of inversion when solving problems of the form a + b - b. These results highlight the importance of identifying information-processing and conceptual characteristics in the early development of mathematical cognition.     

Core information processing deficits in developmental dyscalculia and low numeracy

There are two different conceptions of the innate basis for numerical abilities. On the one hand, it is claimed that infants possess a 'number module' that enables them to construct concepts of the exact numerosities of sets upon which arithmetic develops (e.g. Butterworth, 1999; Gelman & Gallistel, 1978). On the other hand, it has been proposed that infants are equipped only with a sense of approximate numerosities (e.g. Feigenson, Dehaene & Spelke, 2004), upon which the concepts of exact numerosities are constructed with the aid of language (Carey, 2004) and which forms the basis of arithmetic (Lemer, Dehaene, Spelke & Cohen, 2003). These competing proposals were tested by assessing whether performance on approximate numerosity tasks is related to performance on exact numerosity tasks. Moreover, performance on an analogue magnitude task was tested, since it has been claimed that approximate numerosities are represented as analogue magnitudes. In 8-9-year-olds, no relationship was found between exact tasks and either approximate or analogue tasks in normally achieving children, in children with low numeracy or in children with developmental dyscalculia. Low numeracy was related not to a poor grasp of exact numerosities, but to a poor understanding of symbolic numerals.     

Metaphorical motion in mathematical reasoning: further evidence for pre-motor implementation of structure mapping in abstract domains

The theory of computation and category theory both employ arrow-based notations that suggest that the basic metaphor “state changes are like motions” plays a fundamental role in all mathematical reasoning involving formal manipulations. If this is correct, structure-mapping inferences implemented by the pre-motor action planning system can be expected to be involved in solving any mathematics problems not solvable by table lookups and number line manipulations alone. Available functional imaging studies of multi-digit arithmetic, algebra, geometry and calculus problem solving are consistent with this expectation.     

Arithmetic knowledge from the spontaneous attention to relations

Children's knowledge of arithmetic principles is a key aspect of early mathematics knowledge. Knowledge of arithmetic principles predicts how children approach solving arithmetic problems and the likelihood of their success. Prior work has begun to address how children might learn arithmetic principles in a classroom setting. Understanding of arithmetic principles involves understanding how numbers in arithmetic equations relate to another. For example, the Relation to Operands (RO) principle is that for subtracting natural numbers (A ? B = C), the difference (C) must be smaller than the minuend (A). In the current study we evaluate if individual differences in arithmetic principle knowledge (APK) can be predicted by the learners' spontaneous attention to relations (SAR) and if feedback can increase their attention to relations. Results suggest that participants’ Spontaneous Attention to Number (SAN) does not predict their knowledge of the RO principle for symbolic arithmetic. Feedback regarding the attention to relations did not show a significant effect on SAR or participants’ APK. We also did not find significant relations between reports of parent talk and the home environment with individual differences in SAN. The amount of parent's talk about relations was not significantly associated with learner's SAR and APK. We conclude that children's SAR with non‐symbolic number does not generalize to attention to relations with symbolic arithmetic.     

Forget about equality.

Justice is widely thought to consist in equality. For many theorists, the central question has been: Equality of what? The author argues that the ideal of equality distorts practical reasoning and has deeply counterintuitive implications. Moreover, an alternative view of distributive justice can give a better account of what egalitarians should care about than can any of the competing ideals of equality.     

The fixity of reasons

I consider backtracking reasoning: that is, reasoning from backtracking counterfactuals such as if Hitler had won the war, he would have invaded Russia six weeks earlier. Backtracking counterfactuals often strike us as true. Despite that, reasoning from them just as often strikes us as illegitimate. A number of diagnoses have been offered of the illegitimacy of such backtracking reasoning which invoke the fixity of the past, or the direction of causation. I argue against such diagnoses, and in favor of one that invokes a principle I call the fixity of reasons. Backtracking reasoning violates the fixity of reasons. But, the fixity of reasons is a principle that must be observed in order to engage in practical reasoning at all.     

Preferences,reasoning errors,and resource egalitarianism

In this paper I aim to examine some problematic implications of the fact that individuals are prone to making systematic reasoning errors, for resource egalitarianism. I begin by disentangling the concepts of preferences, choices and ambitions, which are sometimes used interchangeably by egalitarians. Subsequently, I claim that the most plausible interpretation of resource egalitarianism takes preferences, not choices, as the site of responsibility. This distinction is salient, since preference-sensitive resource egalitarianism is faced with an important objection when applied to situations in which the empirically reasonable assumption that individuals have different degrees of computational abilities is introduced. I first show that this objection can be raised in cases involving individuals who have incomplete information, but that it ultimately fails for such cases since we can appeal to higher order insurance markets in order to mitigate any initial concerns. I further claim, however, that the objection is much more powerful in cases involving individuals who have different reasoning skills, since the appeal to higher order insurance markets is no longer tenable. Consequently, the ideal principle of justice proposed by Dworkin is met with a new feasibility challenge. Finally, I claim that the problem of reasoning errors and various forms of cognitive biases also affect Dworkin’s non-ideal principle of justice, skewing the outputs of the hypothetical insurance mechanism in an unjustifiable manner.     

STANDARDS OF EQUALITY AND HUME'S VIEW OF GEOMETRY

It has been argued that there is a genuine conflict between the views of geometry defended by Hume in the Treatise and in the Enquiry: while the former work attributes to geometry a different status from that of arithmetic and algebra, the latter attempts to restore its status as an exact and certain science. A closer reading of Hume shows that, in fact, there is no conflict between the two works with respect to geometry. The key to understanding Hume's view of geometry is the distinction he draws between two standards of equality in extension.     

Gender Attitudes and Public Opinion Towards Electoral Gender Quotas in Brazil

Research on public opinion towards affirmative action shows that citizens often support the principle of equality while simultaneously rejecting policies that promote it in a pattern described as the “principle-policy puzzle.” The scholarship also shows that ideology and prejudice towards the targeted group explain the puzzle with respect to racial affirmative action. In this article, we use unique survey questions included in the 2014 round of the AmericasBarometer in Brazil to show that citizens tend to support electoral gender quotas while rejecting gender-based egalitarianism in a reversed version of the “principle-policy puzzle.” We argue that a different type of gender attitudes, namely benevolent sexism, shapes support for gender quotas as well as for the principle of equality. While benevolent sexists tend to reject gender equality based on views about gender complementarity and stereotypes about women's purity, they also support quotas as policies to foster such values. Our findings suggest that even though the political and scholarly debates can provide sound normative reasons for the adoption of quotas across different contexts, public support for them often relies on paternalistic views and expectations about the role of women in politics.     

运算动量效应的理论解释及其发展性预测因素

了解运算偏差的形成与发展对探索算数运算系统的内在机制具有重要意义,早期的算数运算能力是儿童理解和进行复杂数学运算的基础。运算动量偏差是指个体在进行基本数学运算时倾向于高估加法运算结果而低估减法运算结果的一种运算偏差,主要包括三种理论解释,即注意转移假说、启发式解释和压缩解释。鉴于运算动量效应在成年群体中相对稳定却在不同发展阶段儿童中存在不一致的证据,数学能力的提高与空间注意的成熟可结合不同的理论解释来阐明儿童发展过程中运算动量效应的变化趋势。未来可以进一步整合多种研究任务以揭示运算动量效应的发展轨迹,考察数量表征系统与运算动量效应间的关联,探究运算动量效应在不同运算符号中的稳定性,探讨不同因素共同作用对运算动量效应的影响,并设计有关数学能力的干预措施以减少运算动量效应这一运算偏差。