Understanding linear and exponential growth: Searching for the roots in 6- to 9-year-olds

Previous studies have suggested that children as young as 9 years old have developed an understanding of non-linear growth processes prior to formal education. The present experiment aimed at investigating this competency in even younger samples (i.e., in kindergartners, first, and third graders, ages 6, 7 and 9, respectively). Children (N = 90) solved non-verbal inductive reasoning tasks by forecasting linear and exponential growth. While children of all ages forecasted linear growth adequately, exponential growth was also estimated remarkably well. Surprisingly, kindergartners and third graders showed similar high achievement concerning the magnitude and curve shape of forecasts, whereas first graders performed significantly worse. We concluded that primary knowledge of both linearity and non-linearity exists even in kindergartners. However, children's understanding is quite fragile, as their performance was strongly affected by task sequence: Children underestimated exponential growth when the previous task required a forecast of linear growth, and overestimated linear growth when the previous task required forecasting of exponential growth.     

Implicit and Explicit Knowledge of Linear and Exponential Growth in 5- and 9-Year-Olds

The present study examined children's implicit and explicit knowledge of linear and non-linear processes. Five- and nine-year-olds (N = 60) were asked to forecast linear and exponential growth by providing the corresponding number of beads. Implicit knowledge was assessed via the magnitudes of the forecasts; explicit knowledge was investigated through children's verbal explanations of the growth process. Five year olds demonstrated a primary understanding of both linearity and nonlinearity. These concepts were more stable and more advanced in 9 year olds. Although implicit and explicit knowledge were significantly correlated, results suggested that implicit knowledge develops prior to explicit knowledge in this domain. Furthermore, knowledge of linearity emerged earlier than knowledge of nonlinearity.     

Misperception of exponential growth

Exponential growth in numerical series and graphs is grossly underestimated in an intuitive extrapolation task. Subjects’ extrapolations are well described by a model with two parameters only: one for underestimation of the nonlinear growth, the other for linear compensation. The size of the effect is considerable; it is not unusual that two-thirds of the subjects produce estimates below 10% of the normative value. The effect increases with the exponent of the stimulus series, and with addition of a constant to the stimulus series. Neither special instructions about the nature of exponential growth nor daily experience with growth processes enhanced the extrapolations.

     

Testing the assumptions of exponential,additive reaction time models

Two assumptions commonly made by choice reaction time (RT) models are (1) that certain experimental tasks can be found that cause an extra processing stage to be inserted into the cognitive process and (2) that the duration of one or more processing stages is random with an exponential distribution. Few rigorous tests of either assumption exist. This paper reviews existing tests and presents several new results that can be used to test these assumptions. First, in the case in which the duration of an inserted stage is exponentially distributed, it is shown that the observable RT density functions must always intersect at the mode of the density requiring the extra processing stage. Second, when only the first assumption (Assumption 1) is made, it is shown that the cumulative RT distribution functions and, in many cases, the hazard functions must be ordered. Finally, when only Assumption 2 is assumed, it is shown that, under fairly weak conditions, the taft of the RT density function must be exponential. The first two results are applied to data from a memory scanning experiment, and the two assumptions are found to receive tentative support     

The power law repealed: The case for an exponential law of practice

The power function is treated as the law relating response time to practice trials. However, the evidence for a power law is flawed, because it is based on averaged data. We report a survey that assessed the form of the practice function for individual learners and learning conditions in paradigms that have shaped theories of skill acquisition. We fit power and exponential functions to 40 sets of data representing 7,910 learning series from 475 subjects in 24 experiments. The exponential function fit better than the power function in all the unaveraged data sets. Averaging produced a bias in favor of the power function. A new practice function based on the exponential, the APEX function, fit better than a power function with an extra, preexperimental practice parameter. Clearly, the best candidate for the law of practice is the exponential or APEX function, not the generally accepted power function. The theoretical implications are discussed.     

Beyond curve fitting: a dynamical systems account of exponential learning in a discrete timing task

The authors examined the function for learning a discrete timing task from a dynamical systems perspective rather than solely the traditional curve-fitting viewpoint. Adult participants (N = 8) practiced a single-limb angular movement task of 125 ms over 20 degrees for 200 trials. There was no significant difference in percentage of variance accounted for in 3 parameter exponential and power-law nonlinear fits to the individual and averaged data. The percentage of variance increased in both exponential and power-law equations when the data were averaged over participants and trials. Drawing on a dynamical systems approach to time scales in motor learning and on analysis of the distinctive features of exponential and power-law functions, however, the authors conclude that the exponential is the learning function for that task and that level of practice.     

Uncertainty forecasts improve weather-related decisions and attenuate the effects of forecast error

Although uncertainty is inherent in weather forecasts, explicit numeric uncertainty estimates are rarely included in public forecasts for fear that they will be misunderstood. Of particular concern are situations in which precautionary action is required at low probabilities, often the case with severe events. At present, a categorical weather warning system is used. The work reported here tested the relative benefits of several forecast formats, comparing decisions made with and without uncertainty forecasts. In three experiments, participants assumed the role of a manager of a road maintenance company in charge of deciding whether to pay to salt the roads and avoid a potential penalty associated with icy conditions. Participants used overnight low temperature forecasts accompanied in some conditions by uncertainty estimates and in others by decision advice comparable to categorical warnings. Results suggested that uncertainty information improved decision quality overall and increased trust in the forecast. Participants with uncertainty forecasts took appropriate precautionary action and withheld unnecessary action more often than did participants using deterministic forecasts. When error in the forecast increased, participants with conventional forecasts were reluctant to act. However, this effect was attenuated by uncertainty forecasts. Providing categorical decision advice alone did not improve decisions. However, combining decision advice with uncertainty estimates resulted in the best performance overall. The results reported here have important implications for the development of forecast formats to increase compliance with severe weather warnings as well as other domains in which one must act in the face of uncertainty.     

On weighted exponential entropies

As an alternative to Shannon's classical entropy measure of information, an exponential entropy function was proposed by Pal and Pal in 1989 and 1991. To generalize Pal's entropy further, this author introduced two different families of exponential entropies that are one-parameter generalizations of Pal's entropy. The purpose of the present paper is to define weighted entropies corresponding to those one-parameter generalizations. Some properties and examples of such weighted exponential entropies are discussed.     

Bias in exponential and power function fits due to noise: Comment on Myung,Kim, and Pitt

Myung, Kim, and Pitt (2000) demonstrated that simple power functions almost always provide a better fit to purely random data than do simple exponential functions. This result has important implications, because it suggests that high noise levels, which are common in psychological experiments, may cause a bias favoring power functions. We replicate their result and extend it by showing strong bias for more realistic sample sizes. We also show that biases occur for data that contain both random and systematic components, as may be expected in real data. We then demonstrate that these biases disappear for two- or three-parameter functions that include linear parameters (in at least one parameterization). Our results suggest that one should exercise caution when proposing simple power and exponential functions as models of learning. More generally, our results suggest that linear parameters should be estimated rather than fixed when one is comparing the fit of nonlinear models to noisy data.     

Concave utility,transaction costs,and risk in measuring discounting of delayed rewards

Research has consistently found that the decline in the present values of delayed rewards as delay increases is better fit by hyperbolic than by exponential delay-discounting functions. However, concave utility, transaction costs, and risk each could produce hyperbolic-looking data, even when the underlying discounting function is exponential. In Experiments 1 (N = 45) and 2 (N = 103), participants placed bids indicating their present values of real future monetary rewards in computer-based 2nd-price auctions. Both experiments suggest that utility is not sufficiently concave to account for the superior fit of hyperbolic functions. Experiment 2 provided no evidence that the effects of transaction costs and risk are large enough to account for the superior fit of hyperbolic functions.     

The relationship between Luce's Choice Axiom,Thurstone's Theory of Comparative Judgment,and the double exponential distribution

Holman and Marley have shown that Thurstone's Case V model becomes equivalent to the Choice Axiom if its discriminal processes are assumed to be independent double exponential random variables instead of normal ones. It is shown here that for pair comparisons, this representation is not unique; other discriminal process distributions (specifiable only in terms of their characteristic functions) also yield a model equivalent to the Choice Axiom. However, none of these models is equivalent to the Choice Axiom for triple comparisons: There the double exponential representation is unique. It is also shown that within the framework of Thurstone's theory, the double exponential distribution, and hence the Choice Axiom, is implied by a weaker assumption, called “invariance under uniform expansions of the choice set.”     

The pond-and-duckweed problem; Three experiments on the misperception of exponential growth

The representation of duckweed multiplying itself in a pond is used as a research paradigm to study underestimation of exponential growth. The advantage of this paradigm is that the growth process is presented in a direct non-numerical way. The results show that the underestimation observed in earlier studies where growth was presented by means of tables or graphs, occurs in the pond-and-duckweed situation as well. By manipulating the way the process is presented it is possible to obtain some insight into the sampling strategies used by the subjects when they subjectively extrapolate the perceived processes. These experiments lead to the conclusion that subjects base their extrapolations on three or four samples only.     

Global and domain-specific changes in cognition throughout adulthood

Normative adult age-related decrements are well documented for many diverse forms of effortful cognitive processing. However, it is currently unclear whether each of these decrements reflects a distinct and independent developmental phenomenon, or, in part, a more global phenomenon. A number of studies have recently been published that show moderate to large magnitudes of positive relations among individual differences in rates of changes in different cognitive variables during adulthood. This suggests that a small number of common dimensions or even a single common dimension may underlie substantial proportions of individual differences in aging-related cognitive declines. This possibility was directly examined using data from 1,281 adults 18-95 years of age who were followed longitudinally over up to 7 years on 12 different measures of effortful processing. Multivariate growth curve models were applied to examine the dimensionality of individual differences in longitudinal changes. Results supported a hierarchical structure of aging-related changes, with an average of 39% of individual differences in change in a given variable attributable to global (domain-general) developmental processes, 33% attributable to domain-specific developmental processes (abstract reasoning, spatial visualization, episodic memory, and processing speed), and 28% attributable to test-specific developmental processes. Although it is often assumed that systematic and pervasive sources of cognitive decline only emerge in later adulthood, domain-general influences on change were apparent for younger (18-49 years), middle aged (50-69 years), and older (70-95 years) adults.     

Executive function, intellectual decline and daily living skills

Understanding the cognitive changes associated with compromised daily living skills in elderly individuals is important for making appropriate recommendations about the capacity for independent functioning. To this end, we retrospectively examined data from 92 elderly individuals presenting with cognitive decline who were administered measures of executive functioning, general intelligence, and daily living skills. Multiple regression analyses were used to examine the relationship between executive functioning and daily living skills, while controlling for age, depression, and either IQ decline or current IQ. Executive functioning accounted for additional variance in a broad range of daily living skills after controlling for IQ decline. When FSIQ was used in the regression model rather than IQ decline, executive functioning was no longer uniquely associated with daily living skills. Executive functions appear to be important for daily living skills until a critical threshold of low intellectual functioning is reached, reflecting the combined influence of premorbid ability and the extent of intellectual decline. Our results suggest that understanding the relative contribution of different cognitive domains to functional decline in elderly individuals should take into account general intellectual functioning and estimated decline, and that the initiation and/or persistence of self-directed cognitive processes may be important for adaptive daily functioning. These findings have implications for making more evidence-based recommendations about the capacity for independent living.     

Effects of weather and weather forecasts on driver behaviour

Winter-time slipperiness is a considerable source of elevated road accident risk, especially in northern countries such as Canada, Finland and Sweden. The national road administrations often offer a service to inform drivers of forthcoming weather and driving conditions in different regions. This study addressed the effects of adverse weather and traffic weather forecasts on driver behaviour in Finland. Drivers (n = 1437) answered a questionnaire on perceptions of weather, self-reported driving behaviour, pre-trip acquisition of weather information, and possible travel plan changes. The questionnaires were distributed and instantaneously collected in rural service stations in different weather and driving conditions. Data from traffic weather forecasts, automatic traffic counters and weather measurement stations concerning the same area (and road) were also collected. Acquisition of weather information for the trip was associated with low recent driving experience, increasing age, female gender, long trip in question and very poor (local) conditions perceived by the driver. Drivers who had acquired information had also made more changes to travel plans, but information acquisition did not have an effect on their on-road driving behaviour. However, they estimated prevailing risks higher than those who did not acquire weather information. Drivers generally considered the driving conditions better than the forecast, but significantly less so in darkness than in daylight or civil twilight. Leisure trips were clearly underrepresented during very poor driving conditions forecasts, suggesting that some trips are postponed as a result of adverse weather conditions or forecasts thereof. Drivers reported various kinds of compensatory behaviour during adverse conditions, including a 6–7 km/h target speed decrement. This corresponded to traffic flow speed measurements. The results suggest that the on-road driving behaviour is predominantly affected by the prevailing observable conditions, rather than traffic weather forecasts. It is suggested that if administrators wish to help drivers in adjusting their on-road behaviour in adverse weather conditions, the methods should be more local and technical by nature.     

The need for contextual and technical knowledge in judgmental forecasting

The purpose of this research is to assess the extent to which judgmental forecasts are improved by having more contextual and technical knowledge. Contextual information is knowledge gained by practitioners through experience on the job, consisting of general forecasting experience in the industry as well as specific product knowledge. Technical knowledge is knowledge about data analysis and formal forecasting procedures, including information on how to analyze data judgmentally. We directly compared judgmental forecasts of business practitioners with those generated by students, using 22 real-world time series. The practitioners had considerable contextual but no technical knowledge. The students had no contextual but two different levels of technical knowledge. We also generated forecasts with statistical methods to benchmark performance. Results show that contextual knowledge is particularly important in making good judgmental forecasts, while technical knowledge has little value. Practitioner forecasts are better than student forecasts in almost all comparisons. A decisive factor affecting forecast performance appears to be data variability, measured by the coefficient of variation of the time-series data. As the variability of a time series increases, the performance of all forecasts deteriorates, but judgmental forecasts by practitioners become more preferable. Statistical methods have difficulty achieving reasonable forecasts when the data are more variable, whereas judgemental forecasts reinforced by contextual information do relatively well. Data variability is one explanation for the mixed findings of past studies, relative to how well statistical techniques compare with judgment as a forecasting method.     

The pessimistic induction and the exponential growth of science reassessed

My aim is to evaluate a new realist strategy for addressing the pessimistic induction, Ludwig Fahrbach’s (Synthese 180:139–155, 2011) appeal to the exponential growth of science. Fahrbach aims to show that, given the exponential growth of science, the history of science supports realism. I argue that Fahrbach is mistaken. I aim to show that earlier generations of scientists could construct a similar argument, but one that aims to show that the theories that they accepted are likely true. The problem with this is that from our perspective on the history of science we know their argument is flawed. Consequently, we should not be impressed or persuaded by Fahrbach’s argument. Fahrbach has failed to identify a difference that matters between today’s theories and past theories. But realists need to find such a difference if they are to undermine the pessimistic induction.     

6～11岁儿童执行功能发展研究

以168名6~11岁儿童为被试,使用9种执行功能任务研究了儿童三种执行功能成分的发展,三种执行功能是,抑制(Inhibition)、转换(Shifting)和刷新(Updating),结果发现:整个儿童阶段,三种执行功能一直表现出随年龄递增而增长的趋势;不同的执行功能表现出不同的发展速率,其中抑制控制大约在6~7岁表现出一个快速的增长期,记忆刷新从7岁到10岁一直呈线性增长,转换表现出两个快速增长期,第一个快速增长则发生在7~8岁之间,第二个快速增长则发生在9-10岁之间,三种执行功能的发展趋势到大约10岁后趋于平缓,10岁和11岁儿童的表现没有显著性差异;所有执行功能任务测量中没有发现性别主效应,年龄和性别的交互作用不显著。     

Age-Related Changes in Auditory Abilities and in a Competing Task

This study explores the relationship between age and performance on auditory tasks. The primary auditory factors which define broad auditory functions (Ga) show a clear decline with age. This finding supports a claim that Ga is an independent function, separate from crystallized intelligence (Gc). This study also explores the assumption, derived from several theories about the causes of intellectual decline, that competing (dual) tasks would show greater decline in performance than single tests. Present data do not provide evidence to support this assumption.     

Executive dysfunctions across adulthood: measurement properties and correlates of the DEX self-report questionnaire

Difficulties in executive processes can disturb daily life functioning. Using data obtained from two independent community-dwelling samples (n(1) = 468, n(2) = 669, 18-97 years), we examined the factor structure of the Dysexecutive Functioning Questionnaire (DEX) and explored the frequencies and potential correlates of self-reported executive difficulties. Our results revealed that executive problems are parsimoniously described with one underlying factor. Everyday executive dysfunctions were moderately frequent throughout adulthood. Reports of executive problems were associated with individual difference characteristics including age, subjective health, personality, affect, and cognition. We also found that, although executive functions are known to decline with advancing adult age, younger age groups reported more problems than older groups, an effect that was partially mediated by a negative affect factor. We discuss implications for the validity of the instrument as well as directions for future research on executive functioning difficulties in everyday life.