How to Teach Difficult Stochastic Problems. A Case Study on the "Three-Door-Problem"

Stochastik ist die mathematische Teildisziplin mit den berühmtesten "kognitiven Illusionen". Bereits das Konzept der Wahrscheinlichkeit selbst kann zu Verwirrungen führen, insbesondere wenn bedingte Wahrscheinlichkeiten ins Spiel kommen. Am Beispiel des "Drei-Türen-Problems" wird gezeigt, wie durch multiple Repräsentationen ein Verständnis für schwierige stochastische Probleme erreicht werden kann. Die Annahme ist dabei, dass die Schwierigkeit einer in Text eingekleideten mathematischen Aufgabe nicht durch die zugrundeliegende Gleichung determiniert ist, sondern dass die Repräsentation der Information die entscheidende Rolle spielt. In einer Untersuchung mit 139 Gymnasiastinnen und Gymnasiasten zeigte sich, dass durch geeignete Repräsentationen, die zu Lösungshinweisen ausgebaut werden, sogar schon Schüler in die Lage versetzt werden können, das "Drei-Türen-Problem" verständnisvoll zu bearbeiten und die zugrundeliegende mathematische Struktur kognitiv zu erfassen. (DIPF/Orig.)Probability theory is the mathematical subdiscipline with the most notorious „cognitive illusions". Already the concept of probability itself can lead to confusion, especially if conditional probabilities come into play. In the article we focus on a particular probability brain-teaser that has dumbfounded even nobel prize winners and high ranking mathematicians: The „Three-Door-Problem" or „Monty Hall Dilemma". We show how this problem can be represented in an intuitive way by applying tools of cognitive psychology. The assumption is that the difficulty of a mathematical task is not determined by the underlying equation but by the appropriate representation of information. In a survey of secondary school (Gymnasium) students we show that after extending such representations to problem solving tips even students can fully understand the mathematical structure of the tenacious „Three-Door-Problem". (DIPF/Orig.

Prediction processes during multiple object tracking (MOT):involvement of dorsal and ventral premotor cortices

BACKGROUND: The multiple object tracking (MOT) paradigm is a cognitive task that requires parallel tracking of several identical, moving objects following nongoal-directed, arbitrary motion trajectories. AIMS: The current study aimed to investigate the employment of prediction processes during MOT. As an indicator for the involvement of prediction processes, we targeted the human premotor cortex (PM). The PM has been repeatedly implicated to serve the internal modeling of future actions and action effects, as well as purely perceptual events, by means of predictive feedforward functions. MATERIALS AND METHODS: Using functional magnetic resonance imaging (fMRI), BOLD activations recorded during MOT were contrasted with those recorded during the execution of a cognitive control task that used an identical stimulus display and demanded similar attentional load. A particular effort was made to identify and exclude previously found activation in the PM-adjacent frontal eye fields (FEF). RESULTS: We replicated prior results, revealing occipitotemporal, parietal, and frontal areas to be engaged in MOT. DISCUSSION: The activation in frontal areas is interpreted to originate from dorsal and ventral premotor cortices. The results are discussed in light of our assumption that MOT engages prediction processes. CONCLUSION: We propose that our results provide first clues that MOT does not only involve visuospatial perception and attention processes, but prediction processes as well

Prognostic and predictive value of estrogen receptor 1 expression in completely resected non-small cell lung cancer

Adjuvant chemotherapy (ACT) leads to a modest improvement in survival among patients with completely resected non-small cell lung cancer (NSCLC) but molecular predictors are still rare. Publicly available gene microarray, clinical and follow-up data from two different studies on early-stage NSCLC were used to determine the expression of estrogen receptor 1 (ESR1). Expression values were calculated against clinical and survival data in a training set (n = 138) and a test set (subpopulation from the adjuvant JBR.10 study) allowing the determination of the prognostic effect of ESR1 in the observational arm as well as the predictive effect of ESR1 regarding ACT. Data were well balanced in terms of ESR1 expression. ESR1 high expression was of significant positive prognostic value in the training set and this could be confirmed in the test set cohort (hazard ratio for overall survival 0.248, 95% confidence interval: 0.088-0.701; p = 0.008). Additionally, ESR1 low tumors showed a benefit from ACT in terms of 5-year survival (33.3% observation arm and 77.8% ACT arm; p = 0.003), whereas patients with ESR1 high tumors did not have any benefit from ACT (test of interaction p = 0.024). ESR1 is an independent positive prognostic factor for survival in early-stage NSCLC patients. Patients with ESR1 high tumors did not benefit from ACT. What's new? While non-small cell lung cancer (NSCLC) patients are at a high risk of tumor relapse in spite of curative resection and also adjuvant chemotherapy, some remain relapse-free even with no adjuvant chemotherapy. Prognostic and predictive markers are therefore of utmost importance for an adequate therapeutic strategy. The authors show for the first time that estrogen receptor 1 (ESR1) expression in NSCLC is of prognostic value and might be predictive of the benefit of adjuvant chemotherapy. If the results are confirmed, ESR1 expression could help stratify NSCLC patients into different prognostic groups and inform decision-making regarding the use of adjuvant chemotherapy

Action co-representation: The joint SNARC effect

Contains fulltext : 73600.pdf (publisher's version ) (Closed access)Traditionally, communication has been defined as the intentional exchange of symbolic information between individuals. In contrast, the mirror system provides a basis for nonsymbolic and nonintentional information exchange between individuals. We believe that understanding the role of the mirror system in joint action has the potential to serve as a bridge between these two domains. The present study investigates one crucial component of joint action: the ability to represent others' potential actions in the same way as one's own in the absence of perceptual evidence. In two experiments a joint spatial numerical association of response codes (SNARC) effect is demonstrated, providing further evidence that individuals form functionally equivalent representations of their own and others' potential actions. It is shown that numerical (symbolic) stimuli that are mapped onto a spatially arranged internal representation (a mental number line) can activate a co-represented action in the same way as spatial stimuli. This generalizes previous results on co-representation. We discuss the role of the mirror system in co-representation as a basis for shared intentionality and communication

The joint flanker effect: sharing tasks with real and imagined co-actors

Contains fulltext : 99810.pdf (publisher's version ) (Open Access)The Eriksen flanker task (Eriksen and Eriksen in Percept Psychophys 16:143-149, 1974) was distributed among pairs of participants to investigate whether individuals take into account a co-actor's S-R mapping even when coordination is not required. Participants responded to target letters (Experiment 1) or colors (Experiment 2) surrounded by distractors. When performing their part of the task next to another person performing the complementary part of the task, participants responded more slowly to stimuli containing flankers that were potential targets for their co-actor (incompatible trials), compared to stimuli containing identical, compatible, or neutral flankers. This joint Flanker effect also occurred when participants merely believed to be performing the task with a co-actor (Experiment 3). Furthermore, Experiment 4 demonstrated that people form shared task representations only when they perceive their co-actor as intentionally controlling her actions. These findings substantiate and generalize earlier results on shared task representations and advance our understanding of the basic mechanisms subserving joint action