Shedding new light on an old problem: The estimation of shadow sizes in children and adults

Two experiments using the 'projection of shadows' paradigm investigated multidimensional reasoning, implicit and explicit knowledge, and the nonlinearity concept in 5-, 9-, and 13-year-olds and adults. Participants estimated the resulting shadow lengths of differently sized objects, placed at varying distances from a light source. Experiment 1 (N=80) revealed that, on the group level, 5-year-olds took both object size and light-object distance into account when estimating shadow size. Moreover, half of the children in this age group even considered the subordinate distance dimension. In addition, we found a large discrepancy between implicit and explicit knowledge about shadows in 5-year-olds that decreased with age. Finally, only a minority of older participants and very few younger children recognized the nonlinear relationship between light-object distance and shadow size, suggesting domain dependence of the nonlinearity concept. Experiment 2 (N=20) predominately replicated the findings for 5-year-olds using slightly different stimuli.     

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.     

Dynamic testing with tangible electronics: Measuring children's change in strategy use with a series completion task

Aims. This study sought to explore the use of a novel approach that incorporates dynamic testing and tangible electronics in the assessment of children's learning potential and strategy use. Sample. A total of 77 children with a mean age 8.9 years participated in the study; half of them were dynamically tested using graduate prompt techniques; the others served as a control group. Method. Children in the experimental group received a series of inputs consisting of a pre‐test, two training sessions, and a post‐test all involving a number of series completion tasks; the controls received only pre‐ and post‐tests. All test sessions were undertaken individually using a specially designed programme incorporating an electronic console and tangible materials equipped internally with sensors. Results. As a consequence of training, children significantly outperformed controls on a number of series completion tasks. Significant individual differences were noted in terms of the children's response to assistance. The study's hypothesis that dynamic testing would increase analytical, and reduce trial‐and‐error, behaviour was supported. While a significant proportion of the children employed strategies that had earlier been identified as optimal, a sizeable minority demonstrated rather more idiosyncratic approaches. Conclusions. Findings from the study demonstrate the potential value of electronic dynamic testing using graduated prompts. However, a number of further refinements to improve the procedure are suggested.     

Spontaneous analogical transfer in 4-year-olds: a microgenetic study

The purpose of the current microgenetic study was to examine children's spontaneous application of analogical problem solving from story problems to physical tasks. Thirty-six 4-year-olds (M=54.7 months), randomly assigned to an experimental or a control condition, participated. The results indicate that 4-year-old children did, with varying success, spontaneously apply analogical solutions to physical problems across sessions. A positive effect over time on children's spontaneous analogical problem solving was found. Additionally, a few children even gave an analogical strategy-related explanation for their own physical behavior. There was much inter- and intra-individual variability, which may indicate that 4 years of age is a period of change in the development of analogical reasoning.     

Young children's organizational strategies on a same-different task: A microgenetic study and a training study.

This study was designed to explore young children's strategy development using a new kind of "same-different" task that minimized memory demands. Four-year-old children were asked to judge whether 2 sets of 7 toys were the same or different. Sets were identical or differed by one toy, with one set presented on a table and the other in a linen bag. Children's spontaneous behavior was highly strategic, and over trials it developed differently for "same" and "different" trials. Most children who did not spontaneously use the most effective strategy, that is, matching, could be trained to do so and were able to transfer this strategy to new materials. Results concerning production and utilization deficiencies are discussed in relation to task characteristics, trial type (same or different), and findings from prior research.     

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.     

Forecasting exponential growth and exponential decline: similarities and differences

Previous research has demonstrated adults' difficulties with explicitly forecasting exponential processes. Exponential growth is usually grossly underestimated, whereas exponential decline is forecast more accurately. By contrast, the present study examined implicit knowledge about exponential processes and how it is affected by function type (growth versus decline) in samples of 7-, 10-, 14-year-olds, and adults (N=80). Different indicators of the quality of forecasts were investigated. As opposed to previous findings, participants of all age groups estimated exponential decline less adequately than exponential growth. This effect could be attributed mainly to the fact that, in relation to fitted exponential functions, the starting value, or intercept, of the function was approximated well for exponential growth but badly with regard to exponential decline. The accuracy of the non-linear component in forecast functions barely differed between function types within the same age group. Furthermore, even 7-year-olds appeared to have a preliminary understanding of exponential processes, while both intercepts and exponents of forecasts became more accurate with age. Theoretical and practical implications are discussed.