Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning,learning without teaching,or helpful learning-path teaching?

This article provides an overview of some perspectives about special issues in classroom mathematical teaching and learning that have stemmed from the huge explosion of research in children's mathematical thinking stimulated by Piaget. It concentrates on issues that are particularly important for less-advanced learners and for those who might be having special difficulties in learning mathematics. A major goal of the article is to develop a framework for understanding what effective mathematics teaching and learning is, because doing so is so important for struggling students and for research about them. Piaget's research had a fundamental influence on the on-going tension between understanding and fluency in the classroom, supporting efforts toward increasing understanding. But in some countries, misinterpretations of Piaget led to practices that are counterproductive for children, especially struggling learners. Such misinterpretations are identified and a more balanced approach that also draws on Vygotsky is described—a learning-path developmentally-appropriate learning/teaching approach.     

Numerical representation: abstract or nonabstract?

Until now it has been a commonly held view that numbers are represented abstractly in the human brain. However, a recent imaging study challenged the existence of an abstract representation at least of digits and number words, at the brain level, and argued that previous studies and paradigms were not sensitive enough to detect deviations from abstract representation at the behavioural level. The current study addressed this issue with an analysis of distance and sequential effects in magnitude classification. Previous studies that used this paradigm did not find deviation from abstract representation for digits and number words (e.g., Dehaene, 1996; Schwarz & Ischebeck, 2000). However, in the current study a short stimulus-response interval was used, which reduced subjective expectancy and increased automatic processing. The current results showed deviation from abstract representation in both reaction time and accuracy and therefore support the idea that nonabstract representations of numbers do exist.     

Effects of a secondary task on "implicit" sequence learning: learning or performance?

Traditionally, implicit learning has been defined in terms of a lack of awareness of the process and products of learning. In the face of a number of conceptual and empirical difficulties with this definition, it has recently been suggested instead that the critical feature of implicit learning is that it proceeds without making any demands on attentional resources. As disconfirmatory evidence for this, we describe the results of two experiments which each used a sequential reaction time task. With a tone-counting secondary task, measures of sequence learning were significantly affected by whether training occurred under single- or dual-task conditions, regardless of whether testing took place under single- or dual-task conditions.     

Accounting for Cognitive Aging: Context Processing,Inhibition or Processing Speed?

ABSTRACT

Age-related deficits in context processing were examined in relationship to two predominant theories of cognitive aging (the Inhibitory Deficit and Processing Speed Models). Older and younger adults completed a measure of context processing (AX Continuous Performance Test (CPT) task) as well as a computerized battery of inhibitory tasks: Stroop, garden path sentences, go no-go, and the stop-signal paradigm. Participants also completed a simple processing speed task. After controlling for baseline differences in processing speed, age effects were detected on the AX-CPT. Smaller, but significant age effects were noted on the Stroop and stop-signal tasks, but no significant age effects were found on the garden path sentence and go no-go tasks. Intertask correlations were weak, providing little evidence for a homogenous or uniform construct of inhibition. The sensitivity of the AX-CPT to cognitive aging is discussed in the context of existing theories of cognitive aging. The authors suggest that deficits in context processing and utilization may underlie cognitive aging phenomena.     

Processing redundant signals: coactivation, divided attention, or what?

The evidence for and against a redundancy gain in reaction time (RT) when the target is repeated in the visual display is reviewed. We consider the relevance of redundancy gains under these circumstances to the question of whether attention can be simultaneously directed to separate locations in the visual field. In the present experiments, two capital letters were the target stimuli in a two-alternative forced-choice RT paradigm. In addition to the usual conditions of single-target trials, trials on which the target is repeated in the display, and trials on which the target occurs with a noise letter, we introduced the innovation of a condition in which both targets occur in the display. In our two experiments, RT was fastest with single-target displays and slowest with displays containing a target and a noise letter. There was no significant difference in RT to displays in which the target was repeated and displays in which both targets were presented. Both conditions showed a redundancy gain when compared with displays containing a target and a noise letter. The lack of response competition in the both-targets condition and the overall pattern of the results were well explained by a unitary attentional focus that serially processed the letters in the display. Analyses of minima and maxima RTs were consistent with this interpretation.     

Naming without knowing and appearance without associations: Evidence for constructive processes in semantic memory?

Abstract

This study describes a patient (SE) with temporal lobe injury resulting from Herpes Simplex Encephalitis, who displayed a previously unreported impairment in which his knowledge of associative and functional attributes of animals was disproportionately impaired by comparison with his knowledge of their sensory attributes (including their visual properties and characteristic sounds). His knowledge of man-made objects was preserved. A striking aspect of the present case was that the patient remained able to name many animals from their pictures, despite making gross errors in generating associative information about these same animals. This suggests that a semantic representation incorporating stored sensory knowledge may be sufficient for naming (at least for biological categories) and associative information may be unnecessary. Semantic knowledge may normally incorporate more information than is necessary for identification. SE's errors were found to be confabulatory and reconstructive in nature and it is argued that this aspect of his performance challenges passive conceptions of semantic memory couched in terms of a catalogue of stored representations. It is proposed that the patient's disorder affects a dynamic, constructive, and inferential component of his knowledge base, and that this component is sensitive to semantic category.     

Data-driven sequence learning or search: What are the prerequisites for the generation of explicit sequence knowledge?

In incidental sequence learning situations, there is often a number of participants who can report the task-inherent sequential regularity after training. Two kinds of mechanisms for the generation of this explicit knowledge have been proposed in the literature. First, a sequence representation may become explicit when its strength reaches a certain level (Cleeremans, 2006), and secondly, explicit knowledge may emerge as the result of a search process that is triggered by unexpected events that occur during task processing and require an explanation (the unexpected-event hypothesis; Haider & Frensch, 2009). Our study aimed at systematically exploring the contribution of both mechanisms to the generation of explicit sequence knowledge in an incidental learning situation. We varied the amount of specific sequence training and inserted unexpected events into a 6-choice serial reaction time task. Results support the unexpected-event view, as the generation of explicit sequence knowledge could not be predicted by the representation strength acquired through implicit sequence learning. Rather sequence detection turned out to be more likely when participants were shifted to the fixed repeating sequence after training than when practicing one and the same fixed sequence without interruption. The behavioral effects of representation strength appear to be related to the effectiveness of unexpected changes in performance as triggers of a controlled search.     

What matters in implicit task sequence learning: perceptual stimulus features, task sets, or correlated streams of information?

Implicit task sequence learning may be attributed to learning the order of perceptual stimulus features associated with the task sequence, learning a series of automatic task set activations, or learning an integrated sequence that derives from 2 correlated streams of information. In the present study, our purpose was to distinguish among these 3 possibilities. In 4 separate experiments, we replicated and extended a previous study by Heuer, Schmidtke, and Kleinsorge (2001). The presence or absence of a sequence of tasks, as well as that of a sequence of different task-to-response mappings, was manipulated independently within experiments. Evidence of implicit sequence learning was found only when correlated sequences of tasks and mappings were present. No sequence learning effects were found when only a single task sequence or a single mapping sequence was present, even when the structure of the single sequence was identical to the structure of the integrated sequence of task-mapping combinations. These results suggest that implicit task sequence learning is not dependent on either perceptual learning of stimulus features or automatic task-set activation per se. Rather, it appears to be driven by correlated streams of information.     

Incidental task sequence learning: perceptual rather than conceptual?

In four experiments we investigated whether incidental task sequence learning occurs when no instructional task cues are available (i.e. with univalent stimuli). We manipulated task sequence by presenting three simple binary-choice tasks (colour, form or letter case decisions) in regular repeated or random order. Participants were required to use the same two response keys for each of the tasks. We manipulated response sequence by ordering the stimuli so as to produce either a regular or a random order of left versus right-hand key presses. When sequencing in both, or either, separate stream (i.e. task sequence and/or response sequence) was changed to random, only those participants who had processed both sequences together showed evidence of sequence learning in terms of significant response time disruption (Experiments 1-3). This effect disappeared when the sequences were uncorrelated (Experiment 4). The results indicate that only the correlated integration of task sequence and response sequence produced a reliable incidental learning effect. As this effect depends on the predictable ordering of stimulus categories, it suggests that task sequence learning is perceptual rather than conceptual in nature.     

Re-examining selective adaptation: Fatiguing feature detectors,or distributional learning?

When a listener hears many good examples of a /b/ in a row, they are less likely to classify other sounds on, e.g., a /b/-to-/d/ continuum as /b/. This phenomenon is known as selective adaptation and is a well-studied property of speech perception. Traditionally, selective adaptation is seen as a mechanistic property of the speech perception system, and attributed to fatigue in acoustic-phonetic feature detectors. However, recent developments in our understanding of non-linguistic sensory adaptation and higher-level adaptive plasticity in speech perception and language comprehension suggest that it is time to re-visit the phenomenon of selective adaptation. We argue that selective adaptation is better thought of as a computational property of the speech perception system. Drawing on a common thread in recent work on both non-linguistic sensory adaptation and plasticity in language comprehension, we furthermore propose that selective adaptation can be seen as a consequence of distributional learning across multiple levels of representation. This proposal opens up new questions for research on selective adaptation itself, and also suggests that selective adaptation can be an important bridge between work on adaptation in low-level sensory systems and the complicated plasticity of the adult language comprehension system.     

Processing alternative explanations of behavior: correction or integration?

Three experiments investigated how activation of knowledge about situational forces affects discounting in dispositional inference tasks. Each experiment varied a different knowledge activation factor--salience, accessibility, or specificity of situational information. In addition, all 3 experiments varied situational demands and cognitive load. The results showed that cognitive load eliminated discounting when situational information was low in salience, accessibility, or specificity. However, when situational information was more salient, accessible, or specific, it produced strong discounting effects even when perceivers were under cognitive load. These results are discussed in terms of correction and integration models of dispositional inferences from behavior.     

Processing Pitch and Temporal Structures in Music Reading: Independent or Interactive Processing Mechanisms?

Music provides a useful domain in which to study how the different attributes of complex multidimensional stimuli are processed both separately and in combination. Much research has been devoted to addressing how the dimension of pitch and time are co-processed in music listening tasks. Neuropsychological studies have provided evidence for a certain degree of independence between pitch and temporal processing, although there are also many experimental reports favouring interactive models of pitch and temporal processing. Here we extended these investigations by examining the processing of pitch and temporal structures when music is presented in the visual modality (i.e. in the form of music notation). In two experiments, musician subjects were presented with visual musical stimuli containing both pitch and temporal information for a brief amount of time, and they were subsequently required to recall both the pitch and temporal information. In Experiment 1, we documented that concurrent, unattended, pitch and rhythmic auditory interference stimuli disrupted the recall of pitch, but not time. In Experiment 2, we showed that manipulating the tonal structure of the visual presentation stimuli affected the recall of pitch, but not time. On the other hand, manipulating the metrical properties of the visual stimuli affected recall of time, and pitch to a certain extent. Taken together, these results suggest that the processing of pitch is constrained by the processing of time, but the processing of time is not affected by the processing of pitch. These results do not support either strong independence or interactive models of pitch and temporal processing, but they suggest that the processing of time can occur independently from the processing of pitch when performing a written recall task.     

Can unconscious knowledge allow control in sequence learning?

This paper investigates the conscious status of both the knowledge that an item is legal (judgment knowledge) and the knowledge of why it is legal (structural knowledge) in sequence learning. We compared ability to control use of knowledge (Process Dissociation Procedure) with stated awareness of the knowledge (subjective measures) as measures of the conscious status of knowledge. Experiment 1 showed that when people could control use of judgment knowledge they were indeed conscious of having that knowledge according to their own statements. Yet Experiment 2 showed that people could exert such control over the use of judgment knowledge when claiming they had no structural knowledge: i.e. conscious judgment knowledge could be based on unconscious structural knowledge. Further implicit learning research should be clear over whether judgment or structural knowledge is claimed to be unconscious as the two dissociate in sequence learning.     

Effects of similarity and repetition on memory: registration without learning?

We investigated judgments of the frequency of test items (Y) that were highly similar to studied items (X) to test a prediction made by several memory models: that the judged frequency of Y should be proportional to the judged frequency of X. Whether stimuli were pictures or words, judged frequency of Y was bimodally distributed with 1 mode at zero, suggesting that frequency judgments involve a 2-stage process in which a zero judgment is made if there is a mismatch between retrieved information and the test item. Nonzero judgements, taken by themselves, were consistent with the prediction of proportionality. In 2 experiments, the percentage of zero judgments made to Y increased with repetition of X, but in 2 others the percentage did not change beyond frequency = 1. The percentage of "new" judgments in recognition memory followed this same pattern. Because the judged frequency of X increased even as X-Y discrimination showed no improvement, we characterize the result as "registration without learning."     

The Semantic Processing of Motion Verbs: Coercion or Underspecification?

Underspecification and coercion are two prominent interpretive mechanisms to account for meaning variability beyond compositionality. While there is plentiful evidence that natural language meaning constitution exploits both mechanisms, it is an open issue whether a concrete phenomenon of meaning variability is an instance of underspecification or coercion. This paper argues that this theoretical dispute can be settled experimentally. The test case are standard motion verbs (e.g. walk, ride) in combination with ±telic directional phrases, for which both underspecifaction and coercion analyses have been proposed in the literature. A self-paced reading study which incorporates motion verbs, directional phrases and durative/completive temporal adverbials (1) aims at determining the aspectual value of such verbs, and (2) compares the hypotheses of the Underspecification and Coercion Accounts. The results of the reading time experiment (flanked by a corpus study and a completion study) indicate that motion verbs are aspectually underspecified. They combine with ±telic directional phrases with equal ease. The combination with a mismatching temporal adverbial is an instance of coercion, causing additional processing costs.     

Prawn-in-a-tube procedure: habituation or associative learning in cuttlefish?

The prawn-in-a-tube procedure (J. B. Messenger, 1973a) has been used almost exclusively to study associative learning in cuttlefish. In two experiments, the authors sought to determine whether the decline in attack responses observed in this procedure was best accounted for by habituation or associative learning. Results of Experiment 1 revealed an asymmetrical stimulus-specificity effect that could be interpreted as either an instance of habituation or of associative learning. Results of Experiment 2 demonstrated that the response decline could not be reversed following the presentation of a dishabituatory stimulus. The combined results of these experiments support the conclusion that the response decline is best viewed as a resulting of associative processes. The authors considered whether the response decline represents extinction or passive avoidance.     

Remembering or knowing others? Person recognition and recollective experience

Using Tulving's (1985) remember/know procedure, the present research investigated the experiential concomitants of person recognition. Noting basic differences in the manner in which the mind processes expectancy-related material, it was anticipated that facial typicality would be a critical determinant of people's recollective experiences (i.e., remembering vs knowing). In particular, it was expected that whereas remember responses would be more prevalent for distinctive than typical faces, know responses would reflect the opposite pattern. The results of two experiments provided general support for these predictions. In addition, the recollective advantage for distinctive faces was traced to the availability of attentional resources at encoding. These results are considered in the context of contemporary issues in person recognition and social cognition.