Position of Correct Option and Distractors Impacts Responses to Multiple-Choice Items: Evidence From a National Test.

Even though the impact of the position of response options on answers to multiple-choice items has been investigated for decades, it remains debated. Research on this topic is inconclusive, perhaps because too few studies have obtained experimental data from large-sized samples in a real-world context and have manipulated the position of both correct response and distractors. Since multiple-choice tests' outcomes can be strikingly consequential and option position effects constitute a potential source of measurement error, these effects should be clarified. In this study, two experiments in which the position of correct response and distractors was carefully manipulated were performed within a Chilean national high-stakes standardized test, responded by 195,715 examinees. Results show small but clear and systematic effects of options position on examinees' responses in both experiments. They consistently indicate that a five-option item is slightly easier when the correct response is in A rather than E and when the most attractive distractor is after and far away from the correct response. They clarify and extend previous findings, showing that the appeal of all options is influenced by position. The existence and nature of a potential interference phenomenon between the options' processing are discussed, and implications for test development are considered.

Exploring basic numerical capacities in children with difficulties in simple arithmetical achievement / Exploración de las capacidades numéricas básicas en niños con dificultades en el rendimiento en aritmética básica

Abstract Introduction: Current cognitive theories suggest that mathematical learning disabilities may be caused by a dysfunction in the ability to represent non-symbolic numerosity (non-symbolic skills), by impairments in the ability to associate symbolic numerical representations with the underlying analogic non-numerical magnitude representation (symbolic and numerical mapping skills), or by a combination of both deficits. The aim of this study was to contrast the number sense hypothesis and the access deficit hypothesis, to identify the possible origin of the varying degrees of arithmetical difficulties. Method: We compared the performance of children with very low arithmetic achievement (VLA), children with low arithmetical achievement (LA), and typically achieving peers (TA), in non-symbolic, symbolic and numerical mapping tasks. Intellectual capacity and working memory were also evaluated as control variables. The sample comprised 85 Chilean children (3rd to 6th grades) from the Public General Education System. Data were included in several covariance analyses to identify potentially different behavioural profiles between groups. Results: The results showed deficits in both non-symbolic numerosity processing and number-magnitude mapping skills in children with VLA, whereas children with LA exhibited deficits in numerical mapping tasks only. Conclusions: These findings support the hypothesis of impaired non-symbolic numerical representations as the cognitive foundation of severe arithmetical difficulties. Low arithmetical achievement, in contrast, seems to be better explained by defective numerical mapping skills, which fits the access deficit hypothesis. The results presented here provide new evidence regarding the cognitive mechanisms underlying the different behavioural profiles identified in children with varying degrees of arithmetical difficulties.

Resumen Introducción: Teorías cognitivas actuales sugieren que las dificultades en el aprendizaje de las matemáticas pueden ser causadas por una disfunción en la habilidad de representar las numerosidades no-simbólicas (habilidades no-simbólicas), por dificultades en la habilidad de asociar los números con representaciones analógicas, no-simbólicas, subyacentes a la magnitud (habilidades simbólicas y de mapeo) o por una combinación de ambos déficits. El objetivo de este estudio fue contrastar la hipótesis de un déficit en el sentido del número y la hipótesis del déficit en el acceso, para identificar el posible origen de los diferentes grados de dificultades en aritmética. Método: Se comparó el desempeño de niños con muy bajo rendimiento en aritmética (VLA), niños con bajo rendimiento en aritmética (LA) y pares con rendimiento típico (TA), en tareas numéricas no-simbólicas, simbólicas y de mapeo. También se evaluaron la capacidad intelectual y la memoria de trabajo como variables de control. La muestra estuvo conformada por 85 niños chilenos (de 3ero a 6to grado) del Sistema de General de Educación Pública. Los datos fueron incluidos en varios análisis de covarianza para identificar posibles perfiles conductuales diferentes entre grupos. Resultados: Los resultados mostraron que los niños con VLA tienen déficits tanto en el procesamiento no-simbólico de la numerosidad como en las habilidades de mapeo entre los símbolos numéricos y la magnitud analógica que estos representan. Los niños con LA solo mostraron déficits en las habilidades de mapeo. Conclusiones: Estos hallazgos sustentan la hipótesis de que un daño en las representaciones numéricas no-simbólicas subyace a las dificultades severas en aritmética. Por el contrario, el bajo rendimiento en aritmética parece explicarse por deficientes habilidades de mapeo, lo cual se ajusta mejor a la hipótesis del déficit en el acceso. Los anteriores resultados, ofrecen nuevas evidencias respecto a los mecanismos cognitivos que subyacen a los perfiles conductuales identificados en los niños con diferentes grados de dificultades en aritmética.

A Study on Congruency Effects and Numerical Distance in Fraction Comparison by Expert Undergraduate Students.

School mathematics comprises a diversity of concepts whose cognitive complexity is still poorly understood, a chief example being fractions. These are typically taught in middle school, but many students fail to master them, and misconceptions frequently persist into adulthood. In this study, we investigate fraction comparison, a task that taps into both conceptual and procedural knowledge of fractions, by looking at performance of highly mathematically skilled young adults. Fifty-seven Chilean engineering undergraduate students answered a computerized fraction comparison task, while their answers and response times were recorded. Task items were selected according to a number of mathematically and/or cognitively relevant characteristics: (a) whether the fractions to be compared shared a common component, (b) the numerical distance between fractions, and (c) the applicability of two strategies to answer successfully: a congruency strategy (a fraction is larger if it has larger natural number components than another) and gap thinking (a fraction is larger if it is missing fewer pieces than another to complete the whole). In line with previous research, our data indicated that the congruency strategy is inadequate to describe participants' performance, as congruent items turned out to be more difficult than incongruent ones when fractions had no common component. Although we hypothesized that this lower performance for congruent items would be explained by the use of gap thinking, this turned out not to be the case: evidence was insufficient to show that the applicability of the gap thinking strategy modulated either participants' accuracy rates or response times (although individual-level data suggest that there is an effect for response times). When fractions shared a common component, instead, our data display a more complex pattern that expected: an advantage for congruent items is present in the first experimental block but fades as the experiment progresses. Numerical distance had an effect in fraction comparison that was statistically significant for items without common components only. Altogether, our results from experts' reasoning reveal nuances in the fraction comparison task with respect to previous studies and contribute to future models of reasoning in this task.

Reliability and Validity of Nonsymbolic and Symbolic Comparison Tasks in School-Aged Children.

Basic numerical processing has been regularly assessed using numerical nonsymbolic and symbolic comparison tasks. It has been assumed that these tasks index similar underlying processes. However, the evidence concerning the reliability and convergent validity across different versions of these tasks is inconclusive. We explored the reliability and convergent validity between two numerical comparison tasks (nonsymbolic vs. symbolic) in school-aged children. The relations between performance in both tasks and mental arithmetic were described and a developmental trajectories' analysis was also conducted. The influence of verbal and visuospatial working memory processes and age was controlled for in the analyses. Results show significant reliability (p < .001) between Block 1 and 2 for nonsymbolic task (global adjusted RT (adjRT): r = .78, global efficiency measures (EMs): r = .74) and, for symbolic task (adjRT: r = .86, EMs: r = .86). Also, significant convergent validity between tasks (p < .001) for both adjRT (r = .71) and EMs (r = .70) were found after controlling for working memory and age. Finally, it was found the relationship between nonsymbolic and symbolic efficiencies varies across the sample's age range. Overall, these findings suggest both tasks index the same underlying cognitive architecture and are appropriate to explore the Approximate Number System (ANS) characteristics. The evidence supports the central role of ANS in arithmetic efficiency and suggests there are differences across the age range assessed, concerning the extent to which efficiency in nonsymbolic and symbolic tasks reflects ANS acuity.

Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge.

Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called "flexibility" whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena.

Reliability and validity of nonsymbolic and symbolic comparison tasks in school-aged children

Basic numerical processing has been regularly assessed using numerical nonsymbolic and symbolic comparison tasks. It has been assumed that these tasks index similar underlying processes. However, the evidence concerning the reliability and convergent validity across different versions of these tasks is inconclusive. We explored the reliability and convergent validity between two numerical comparison tasks (nonsymbolic vs. symbolic) in school-aged children. The relations between performance in both tasks and mental arithmetic were described and a developmental trajectories’ analysis was also conducted. The influence of verbal and visuospatial working memory processes and age was controlled for in the analyses. Results show significant reliability (p < .001) between Block 1 and 2 for nonsymbolic task (global adjusted RT (adjRT): r = .78, global efficiency measures (EMs): r = .74) and, for symbolic task (adjRT: r = .86, EMs: r = .86). Also, significant convergent validity between tasks (p < .001) for both adjRT (r = .71) and EMs (r = .70) were found after controlling for working memory and age. Finally, it was found the relationship between nonsymbolic and symbolic efficiencies varies across the sample’s age range. Overall, these findings suggest both tasks index the same underlying cognitive architecture and are appropriate to explore the Approximate Number System (ANS) characteristics. The evidence supports the central role of ANS in arithmetic efficiency and suggests there are differences across the age range assessed, concerning the extent to which efficiency in nonsymbolic and symbolic tasks reflects ANS acuity (AU)

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Long-distance neural synchrony correlates with processing strategies to compare fractions.

Adults use different processing strategies to work with fractions. Depending on task requirements, they may analyze the fraction components separately (componential processing strategy, CPS) or consider the fraction as a whole (holistic processing strategy, HPS). It is so far unknown what is the brain coordination dynamics underlying these types of fraction processing strategies. To elucidate this issue, we analyzed oscillatory brain activity during a fraction comparison task, presenting pairs of fractions either with or without common components. Results show that CPS induces a left frontal-parietal alpha phase desynchronization after the onset of fraction pairs, while HPS induces an increase of phase synchrony on theta and gamma bands, over frontal and central-parietal sites, respectively. Additionally, the HPS evokes more negative ERPs around 400 ms over the right frontal scalp than the CPS. This ERP activity correlates with the increase of Theta phase synchrony. Our results reveal the emergence of different functional neural networks depending on the kind of cognitive strategy used for processing fractions.