The Benefits of an Interdisciplinary Approach to Mathematics Education on Issues Around Computation in School.

We present arguments in favor of an interdisciplinary approach in mathematics education. As an instance, we briefly recall how cognitive neuropsychologists promoted intense finger gnosis acquisition, i.e., acquiring the ability to mentally represent one's fingers, at an early age. Mathematics educators definitely recommended the development of finger gnosis but examined its limits. They also presented arguments in favor of developing flexible mental calculation as a goal of arithmetical instruction in elementary school. In this context we describe the training of "Zahlenblick" as a way to foster flexible mental calculation and connect it with concepts from the theory of metacognition. We illustrate how precisely this branch of metacognition demands further interdisciplinary research. In our analysis, "Zahlenblick" extends to acquiring an eye for proportions, beyond just whole numbers. We illustrate how useful it would be to better understand the neural underpinnings responsible for the advantages of so-called natural frequencies, compared with percentages or probabilities, and of icon arrays for representing them. Such natural frequencies are adequate formats for the early confrontation with decision-making under risk.

Finger gnosis predicts a unique but small part of variance in initial arithmetic performance.

Recent studies indicated that finger gnosis (i.e., the ability to perceive and differentiate one's own fingers) is associated reliably with basic numerical competencies. In this study, we aimed at examining whether finger gnosis is also a unique predictor for initial arithmetic competencies at the beginning of first grade-and thus before formal math instruction starts. Therefore, we controlled for influences of domain-specific numerical precursor competencies, domain-general cognitive ability, and natural variables such as gender and age. Results from 321 German first-graders revealed that finger gnosis indeed predicted a unique and relevant but nevertheless only small part of the variance in initial arithmetic performance (∼1%-2%) as compared with influences of general cognitive ability and numerical precursor competencies. Taken together, these results substantiated the notion of a unique association between finger gnosis and arithmetic and further corroborate the theoretical idea of finger-based representations contributing to numerical cognition. However, the only small part of variance explained by finger gnosis seems to limit its relevance for diagnostic purposes.

Natural frequencies improve Bayesian reasoning in simple and complex inference tasks.

Representing statistical information in terms of natural frequencies rather than probabilities improves performance in Bayesian inference tasks. This beneficial effect of natural frequencies has been demonstrated in a variety of applied domains such as medicine, law, and education. Yet all the research and applications so far have been limited to situations where one dichotomous cue is used to infer which of two hypotheses is true. Real-life applications, however, often involve situations where cues (e.g., medical tests) have more than one value, where more than two hypotheses (e.g., diseases) are considered, or where more than one cue is available. In Study 1, we show that natural frequencies, compared to information stated in terms of probabilities, consistently increase the proportion of Bayesian inferences made by medical students in four conditions-three cue values, three hypotheses, two cues, or three cues-by an average of 37 percentage points. In Study 2, we show that teaching natural frequencies for simple tasks with one dichotomous cue and two hypotheses leads to a transfer of learning to complex tasks with three cue values and two cues, with a proportion of 40 and 81% correct inferences, respectively. Thus, natural frequencies facilitate Bayesian reasoning in a much broader class of situations than previously thought.

Children's sequential information search is sensitive to environmental probabilities.

We investigated 4th-grade children's search strategies on sequential search tasks in which the goal is to identify an unknown target object by asking yes-no questions about its features. We used exhaustive search to identify the most efficient question strategies and evaluated the usefulness of children's questions accordingly. Results show that children have good intuitions regarding questions' usefulness and search adaptively, relative to the statistical structure of the task environment. Search was especially efficient in a task environment that was representative of real-world experiences. This suggests that children may use their knowledge of real-world environmental statistics to guide their search behavior. We also compared different related search tasks. We found positive transfer effects from first doing a number search task on a later person search task.

Effects of finger counting on numerical development - the opposing views of neurocognition and mathematics education.

Children typically learn basic numerical and arithmetic principles using finger-based representations. However, whether or not reliance on finger-based representations is beneficial or detrimental is the subject of an ongoing debate between researchers in neurocognition and mathematics education. From the neurocognitive perspective, finger counting provides multisensory input, which conveys both cardinal and ordinal aspects of numbers. Recent data indicate that children with good finger-based numerical representations show better arithmetic skills and that training finger gnosis, or "finger sense," enhances mathematical skills. Therefore neurocognitive researchers conclude that elaborate finger-based numerical representations are beneficial for later numerical development. However, research in mathematics education recommends fostering mentally based numerical representations so as to induce children to abandon finger counting. More precisely, mathematics education recommends first using finger counting, then concrete structured representations and, finally, mental representations of numbers to perform numerical operations. Taken together, these results reveal an important debate between neurocognitive and mathematics education research concerning the benefits and detriments of finger-based strategies for numerical development. In the present review, the rationale of both lines of evidence will be discussed.

Transparency in risk communication: graphical and analog tools.

Why is it that the public can read and write but only a few understand statistical information? Why are elementary distinctions, such as that between absolute and relative risks, not better known? In the absence of statistical literacy, key democratic ideals, such as informed consent and shared decision making in health care, will remain science fiction. In this chapter, we deal with tools for transparency in risk communication. The focus is on graphical and analog representations of risk. Analog representations use a separate icon or sign for each individual in a population. Like numerical representations, some graphical forms are transparent, whereas others indiscernibly mislead the reader. We review cases of (1) tree diagrams for representing natural versus relative frequency, (2) decision trees for the representation of fast and frugal decision making, (3) bar graphs for representing absolute versus relative risk, (4) population diagrams for the analog representation of risk, and (5) a format of representation that employs colored tinker cubes for the encoding of information about individuals in a population. Graphs have long enjoyed the status of being "worth a thousand words" and hence of being more readily accessible to human understanding than long-winded symbolic representations. This is both true and false. Graphical tools can be just as well employed for transparent and nontransparent risk communications.

Detection of syntonies between multiple spike trains using a coarse-grain binarization of spike count distributions.

At very short timescales neuronal spike trains may be compared to binary streams where each neuron gives at most one spike per bin and therefore its state can be described by a binary variable. Time-averaged activity like the mean firing rate can be generally used on longer timescales to describe the dynamics; nevertheless, enlarging the space of the possible states up to the continuum may seriously bias the true statistics if the sampling is not accurate. We propose a simple transformation on binary variables which allows us to fix the dimensionality of the space to sample and to vary the temporal resolution of the analysis. For each time length interactions among simultaneously recorded neurons are evaluated using log-linear models. We illustrate how to use this method by analysing two different sets of data, recorded respectively in the temporal cortex of freely moving rats and in the inferotemporal cortex of behaving monkeys engaged in a visual fixation task. A detailed study of the interactions is provided for both samples. In both datasets we find that some assemblies share robust interactions, invariant at different time lengths, while others cooperate only at delimited time resolutions, yet the size of the samples is too small to allow an unbiased estimate of all possible interactions. We conclude that an extensive application of our method to larger samples of data, together with the development of techniques to correct the bias in the estimate of the coefficients, would provide significant information about the structure of the interactions in populations of neurons.

Use of simple heuristics to target macrolide prescription in children with community-acquired pneumonia.

BACKGROUND: Macrolides are the first-line antibiotic treatment of community-acquired pneumonia (CAP). Owing to alarming resistance rates among invasive Streptococcus pneumoniae isolates, particularly in young children, macrolide use should be restricted to patients infected with susceptible pathogens, eg, Mycoplasma pneumoniae. OBJECTIVE: To develop a simple clinical prediction rule for identifying M pneumoniae as the cause of CAP in children. DESIGN AND SETTING: Prospective cohort study in 253 children with radiologically confirmed CAP in a walk-in clinic of a tertiary care hospital. MAIN OUTCOME MEASURES: Mycoplasma infection, proven by results of antibody testing of paired serum samples (gold standard). We compared the area under the receiver operating characteristic curve (c statistic) of the following 2 prediction models: a scoring system derived from logistic regression analysis and a fast-and-frugal decision tree. RESULTS: Mycoplasma pneumoniae infection was confirmed in 32 (13%) of 253 children. A scoring system based on duration of fever and patient age yielded a c statistic of 0.84 (95% confidence interval [CI], 0.77-0.91), compared with that of the decision tree (c = 0.76 [95% CI, 0.70-0.83]). The scoring system identified 75% of all cases as being at high or very high risk for M pneumoniae infection; the decision tree, 72% at high risk. The scoring system would curtail macrolide prescriptions by 75%; the decision tree, by 68%. CONCLUSIONS: In children with CAP, simple clinical decision rules identify patients at risk for M pneumoniae infection. At present US macrolide resistance rates among invasive S pneumoniae isolates, both rules increase the chance of prescribing effective first-line antibiotics compared with general macrolide administration.

Representation facilitates reasoning: what natural frequencies are and what they are not.

A good representation can be crucial for finding the solution to a problem. Gigerenzer and Hoffrage (Psychol. Rev. 102 (1995) 684; Psychol. Rev. 106 (1999) 425) have shown that representations in terms of natural frequencies, rather than conditional probabilities, facilitate the computation of a cause's probability (or frequency) given an effect--a problem that is usually referred to as Bayesian reasoning. They also have shown that normalized frequencies--which are not natural frequencies--do not lead to computational facilitation, and consequently, do not enhance people's performance. Here, we correct two misconceptions propagated in recent work (Cognition 77 (2000) 197; Cognition 78 (2001) 247; Psychol. Rev. 106 (1999) 62; Organ. Behav. Hum. Decision Process. 82 (2000) 217): normalized frequencies have been mistaken for natural frequencies and, as a consequence, "nested sets" and the "subset principle" have been proposed as new explanations. These new terms, however, are nothing more than vague labels for the basic properties of natural frequencies.