An integrated theory of whole number and fractions development

This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds indicate that, as with whole numbers, accuracy of fraction magnitude representations is closely related to both fractions arithmetic proficiency and overall mathematics achievement test scores, that fraction magnitude representations account for substantial variance in mathematics achievement test scores beyond that explained by fraction arithmetic proficiency, and that developing effective strategies plays a key role in improved knowledge of fractions. Theoretical and instructional implications are discussed.     

Segmenting dynamic human action via statistical structure

Human social, cognitive, and linguistic functioning depends on skills for rapidly processing action. Identifying distinct acts within the dynamic motion flow is one basic component of action processing; for example, skill at segmenting action is foundational to action categorization, verb learning, and comprehension of novel action sequences. Yet little is currently known about mechanisms that may subserve action segmentation. The present research documents that adults can register statistical regularities providing clues to action segmentation. This finding provides new evidence that structural knowledge gained by mechanisms such as statistical learning can play a role in action segmentation, and highlights a striking parallel between processing of action and processing in other domains, such as language.     

Are People Successful at Learning Sequences of Actions on a Perceptual Matching Task?

We report the results of an experiment in which human subjects were trained to perform a perceptual matching task. Subjects were asked to manipulate comparison objects until they matched target objects using the fewest manipulations possible. An unusual feature of the experimental task is that efficient performance requires an understanding of the hidden or latent causal structure governing the relationships between actions and perceptual outcomes. We use two benchmarks to evaluate the quality of subjects' learning. One benchmark is based on optimal performance as calculated by a dynamic programming procedure. The other is based on an adaptive computational agent that uses a reinforcement-learning method known as Q-learning to learn to perform the task. Our analyses suggest that subjects were successful learners. In particular, they learned to perform the perceptual matching task in a near-optimal manner (i.e., using a small number of manipulations) at the end of training. Subjects were able to achieve near-optimal performance because they learned, at least partially, the causal structure underlying the task. In addition, subjects' performances were broadly consistent with those of model-based reinforcement-learning agents that built and used internal models of how their actions influenced the external environment. We hypothesize that people will achieve near-optimal performances on tasks requiring sequences of action-especially sensorimotor tasks with underlying latent causal structures-when they can detect the effects of their actions on the environment, and when they can represent and reason about these effects using an internal mental model.     

Action–effect congruence during observational learning leads to faster action sequence learning

Common coding theory suggests that any action (pressing a piano key) is intimately linked with its resultant sensory effect (an auditory musical tone). We conducted two experiments to explore the effect of varying auditory action–effect patterns during complex action learning. In Experiment 1, participants were assigned to 1 of 4 groups, watched a silent video of a hand playing a sequence on a piano keyboard with no auditory action effect (observation) and were asked to practise and perform the sequence on an identical keyboard with varying action effects (reproduction). During reproduction, Group 1 heard no auditory tones (identical to observed video), Group 2 heard typical scale-ascending piano tones with each key press, Group 3 heard fixed but out-of-sequence piano tones with each key press, and Group 4 heard random piano tones with each key press. In Experiment two, new participants were assigned to 1 of 2 groups and watched an identical video; however, the video in this experiment contained typical, scale-ascending piano sounds. During reproduction, Group 1 heard no auditory tones while Group 2 heard typical, scale-ascending piano tones with each key press (identical to observed video). Our results showed that participants whose action–effect patterns during reproduction matched those in the observed video learned the action sequence faster than participants whose action–effect patterns during reproduction differed from those in the observed video. Additionally, our results suggest that adding an effect during reproduction (when one is absent during observation) is somewhat more detrimental to action sequence learning than removing an effect during reproduction (when one is present during observation).     

Do as I say - or as I do?! How 18- and 24-month-olds integrate words and actions to infer intentions in situations of match or mismatch

Infants use behavioral and verbal cues to infer another person’s action intention. However, it is still unclear how infants integrate these often co-occurring cues depending on the cues’ coherence (i.e., the degree to which the cues provide coherent information about another’s intention). This study investigated how 18- and 24-month-olds’ (N = 88 per age group) action selection was influenced by varying the coherence of a model’s verbal and behavioral cues. Using a between-subjects design, infants received six trials with different stimulus objects. In the conditions Congruent, Incongruent, and Failed-attempt, the model uttered a telic verb particle that was followed by a matching or contradicting goal-directed action demonstration, or by a non goal-directed slipping motion, respectively. In the condition Pseudo-word, a nonsense word was combined with a goal-directed action demonstration. Infants’ action selection indicated an adherence to the verbal cue in Congruent, Incongruent, and Failed-attempt, and this was stronger in 24- than 18-month-olds. Additionally, in Incongruent and Failed-attempt, patterns of cue integration across the six trials varied in the two age groups. Regarding the behavioral cue, infants in Congruent and Pseudo-word preferentially followed this cue in both age groups, which also suggested a rather unspecific effect of the verbal cue in Congruent. Relatively longer first action-latencies in Incongruent and Failed-attempt implied that these types of coherence elicited higher cognitive demands than in Congruent and Pseudo-word. Results are discussed in light of infants’ flexibility in using social cues, depending on the cue’s coherence and on age-related social-cognitive differences.     

Systems Thinking for Knowledge

The capacity to engage in systems thinking is often viewed as being a product of being able to understand complex systems due to one's facility in mastering systems theories, methods, and being able to adeptly reason. Relatively little attention is paid in the systems literature to the processes of learning from experience and creating knowledge as a direct consequence of individuals engaging systems thinking itself over time. In fact, the potential efficacy of systems thinking to improve performance normally seen as only contingent on a priori knowledge, rather than knowledge created via learning from experience. Such newly create knowledge often results from engaging in modeling efforts and systemic forms of inquiry. This article proposes a model for creating new knowledge by coupling systems modeling with a pragmatic approach to knowledge-creation. This approach is based on a foundation of the pragmatic concepts first proposed by the American philosopher/scientist Charles Sanders Peirce over a century ago. This model offers systems practitioners a framework to engage in knowledge-intensive systems thinking (KIST) for addressing complex problematic issues.     

Conscious awareness of action potentiates sensorimotor learning

Many everyday skills are unconsciously learned through repetitions of the same behaviour by binding independent motor acts into unified sets of actions. However, our ability to be consciously aware of producing newly and highly trained motor skills raises the question of the role played by conscious awareness of action upon skill acquisition. In this study we strengthened conscious awareness of self-produced sequential finger movements by way of asking participants to judge their performance in terms of maximal fluency after each trial. Control conditions in which participants did not make any judgment or performance-unrelated judgments were also included. Findings indicate that conscious awareness of action, enhanced via subjective appraisal of motor efficiency, potentiates sensorimotor learning and skilful motor production in optimising the processing and sequencing of action units, as compared to the control groups. The current work lends support to the claim that the learning and skilful expression of sensorimotor behaviours might be grounded upon our ability to be consciously aware of our own motor capability and efficiency.     

The trajectory of mathematics skills and working memory thresholds in girls with fragile X syndrome

Fragile X syndrome is a common genetic disorder associated with executive function deficits and poor mathematics achievement. In the present study, we examined changes in math performance during the elementary and middle school years in girls with fragile X syndrome, changes in the working memory loads under which children could complete a cognitive switching task, and the association between these two areas of function, in girls with fragile X syndrome relative to their peers. Our findings indicate that the trajectory of math and executive function skills of girls with fragile X differs from that of their peers and that these skills contribute to predicting math achievement and growth in math performance over time. Also, changes in math performance were associated with incremental increases in working memory demands, suggesting that girls with fragile X have a lower threshold for being able to perform under increasing task demands. Still, we found improvement in executive function performance between 10 and 12 years in girls with fragile X rather than a performance plateau as has been reported in other studies. The findings implicate the importance of early intervention in mathematics for girls with fragile X that addresses poor calculation skills, the supporting numerical skills, and deficits in executive functions, including working memory.     

Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes

This study presents evidence that humans have intuitive, perceptually based access to the abstract fraction magnitudes instantiated by nonsymbolic ratio stimuli. Moreover, it shows these perceptually accessed magnitudes can be easily compared with symbolically represented fractions. In cross-format comparisons, participants picked the larger of two ratios. Ratios were presented either symbolically as fractions or nonsymbolically as paired dot arrays or as paired circles. Response patterns were consistent with participants comparing specific analog fractional magnitudes independently of the particular formats in which they were presented. These results pose a challenge to accounts that argue human cognitive architecture is ill-suited for processing fractions. Instead, it seems that humans can process nonsymbolic ratio magnitudes via perceptual routes and without recourse to conscious symbolic algorithms, analogous to the processing of whole number magnitudes. These findings have important implications for theories regarding the nature of human number sense – they imply that fractions may in some sense be natural numbers, too.     

Making the uncontrollable seem controllable: The role of action in the illusion of control

It is well known that certain variables can bias judgements about the perceived contingency between an action and an outcome, making them depart from the normative predictions. For instance, previous studies have proven that the activity level or probability of responding, P(R), is a crucial variable that can affect these judgements in objectively noncontingent situations. A possible account for the P(R) effect is based on the differential exposure to actual contingencies during the training phase, which is in turn presumably produced by individual differences in participants' P(R). The current two experiments replicate the P(R) effect in a free-response paradigm, and show that participants' judgements are better predicted by P(R) than by the actual contingency to which they expose themselves. Besides, both experiments converge with previous empirical data, showing a persistent bias that does not vanish as training proceeds. These findings contrast with the preasymptotic and transitory effect predicted by several theoretical models.     

Learning As Support For Organizational Innovation: Possibilities And Limitations

Abstract

The present article tells an intervention story where two collectives, from business and academia, came together to address a business problem through collaborative action research. Among other things, the project created new ways of learning and therefore, knowing about the “business problem.” The author argues that in order to talk about an organizational intervention in a learning context, it was helpful to focus observation at the level of practice, in this case the different learning practices brought to the project by the organization and the research group. The “scientific narrative” focuses on how the two practices interacted. The present story's plot revolves around the following questions: What happens when one collective—used to a particular style of learning—decides to engage with another collective with a different approach to learning and what are the consequences for organizational innovation?     

Mainstreams and Perspectives in Research on Learning (Mathematics) From Instruction

Le domaine de l’enseignement et de l’apprentissage des mathématiques est un exemple des plus représentatives de l’orientation des recherches sur l’apprentissage et l’instruction dans les différentes matières. Au cours du dernier quart de siècle écoulé, ce champ de questionnement a produit nombre d’investigations qui résultent d’une conception de l’apprentissage des mathématiques comme construction sociale de la signification et de la comprehension basée sur la modélisation de la réalité. Dans cet article, quelques tendances majeures et perspectives de recherche s.0s principalement aux Etats‐Unis et en Europe. Il y sera montré que d’une part, les investigations empiriques ont déjà produit des blocs de construction pour l’élaboration d’une théorie de l’apprentissage des mathématiques à partir d’instructions mais que, d’autre part, des questions et des problèmes majeurs requièrent d’être démêlés par la poursuite d’investigations. Ceci y sera documenté de façon exemplaire en utilisant comme cadre de référence quatre composants principaux d’une théorie de l’apprentissage des mathématiques à partir d’instructions: une théorie de l’expertise visant à analyser des compétences dans un domaine; une théorie de l’acquisition tendant à comprendre et expliquer les processus d’apprentissage qui conduisent à la réalisation de compétences; une théorie de l’intervention, focalisée sur la conception d’environnements enseignement‐apprentissage puissants pour provoquer ces processus d’acquisition; et une théorie de l’évaluation s’adressant au développement de méthodes et techniques pour la construction et l’application d’outils et instruments d’évaluation. The domain of mathematics learning and teaching is one of the most representative examples of the subject‐matter orientation in research on learning and instruction. During the last quarter of the past century this field of inquiry has produced a vast body of investigations, resulting in an enriched conception of mathematics learning as involving the (social) construction of meaning and understanding based on modeling of reality. In this article some of the main trends and perspectives in the field of research on mathematics learning and instruction are discussed, and illustrated with examples of empirical research, mainly carried out in Europe and the United States. It will be shown that, on the one hand, the available empirical investigations have already yielded substantial building blocks for the elaboration of a theory of mathematics learning from instruction, but that, on the other hand, major issues and problems need to be unravelled by continued inquiry. This will be documented in an exemplary way, using as a frame of reference four main components of a theory of learning mathematics from instruction: a theory of expertise, aiming at analysing competence in a domain; a theory of acquisition, attempting to understand and explain the processes of learning that are conducive to the attainment of competence; a theory of intervention, focusing on the design of powerful teaching–learning environments for eliciting those acquisition processes; and a theory of assessment, addressing the development of methods and techniques for the construction and application of assessment tools and instruments.     

Pulling out the intentional structure of action: the relation between action processing and action production in infancy

Adults and children readily construct action representations organized with respect to an ultimate goal. These representations allow one to predict the consequences of action, interpret and describe actions, and categorize action sequences. In this paper, we explore the ontogeny of hierarchically organized action representations, and its relation to infants' ability to produce similar sequences. To do so, we examine infants' perception and performance of a means-end sequence: pulling a cloth to retrieve a toy. Using a visual habituation paradigm, we demonstrate that 12-month-old infants understand that the initial step of the cloth-pulling sequence is directed toward the ultimate goal of attaining the toy, and use their knowledge of the causal constraints of the sequence to make this goal attribution. Ten-month-olds, however, appear transitional with respect to this understanding: their ability to identify the goal of the cloth-pulling sequence is related to their own ability to planfully solve a similar sequence. These findings are consistent with a burgeoning body of literature suggesting an intimate link between action production and perception, and suggest that this link is in place by at least 10 months of age.