Intelligent tutoring systems are known for providing customized learning opportunities for thousands of users. One feature of many systems is differentiating the amount of practice users receive. To do this, some systems rely on a threshold of consecutive correct responses. For instance, Khan Academy used to use ten correct in a row and now uses five correct in a row as the mastery threshold. The present research uses a series of randomized control trials, conducted in an online learning platform (eg., ASSISTments.org), to explore the effects of different thresholds of consecutive correct responses on learning. Results indicate that despite spending significantly more time practicing there is no significant difference on learning between two, three, four, or five consecutive correct responses. This suggests that systems, and MOOCS, can employ the simple rule of two or three consecutive correct responses when determining the amount of practice provided to users.

Mental models—internal, dynamic, incomplete representations of the external world that people use to guide cognitive processes such as reasoning, decision making, and language comprehension—have practical implications for predicting attitudes and behaviors across various domains. This study examines how mental models of the human–nature relationship predict pro-environmental behavioral intentions directly and indirectly as mediated through anthropocentric and biocentric environmental attitudes. To address these aims, participants were asked about mental model components of the human–nature relationship (human exceptionalism, beliefs about human impact on nature, and beliefs about nature's impact on humans), pro-environmental attitudes (biocentric and anthropocentric), and their pro-environmental behavioral intentions (protection and investment). We found that protection intentions were (1) directly predicted by human exceptionalism beliefs (negatively) and perceived human impact on nature (positively) and (2) indirectly predicted by mental model components via biocentric attitudes. Investment intentions were directly predicted by nature's perceived impact on humans, and were similarly indirectly predicted by mental model components via biocentric attitudes. The results suggest that mental models of the human–nature relationship provide a cognitive foundation for environmental behavioral intentions both directly and through their association with environmental attitudes. These findings have implications for pro-environmental interventions that deal with conceptual and attitudinal change.

Number line estimation tasks are frequently used to learn about numerical thinking, learning, and development. These tasks are often interpreted as though estimates are determined by overall magnitudes of target numerals, rather than specific instantiating digits. Yet estimates are strongly biased by leftmost digits. For example, numbers like "698" are placed too far to the left of numbers like "701" on a 0-1,000 line. This "left digit effect" or "left digit bias" has been investigated little in children, and only on electronic tasks. Here, we ask whether left digit bias appears in paper-and-pencil estimates, and whether it differs for paper-based versus computer-based tasks. In Study 1, 5- to 8-year-old children completed a 0-100 number line task on paper. In Study 2, 7- to 11-year-olds completed a 0-1,000 paper task. In Study 3, adults completed tasks on paper in both ranges. Large left digit effects were observed for children aged 8 years or older and adults, but we did not find evidence for left digit bias in younger children. Study 4 compared paper and computer tasks for adults and children aged 9-12 years. Strong left digit bias was observed in all conditions, with a larger effect for the paper-based task in children. Large left digit effects in number line estimation emerge regardless of task format, with a developmental trajectory broadly consistent with other studies. For children in the age range that reliably exhibits left digit bias (but not adults), paper-and-pencil number line estimation tasks elicit even greater bias than computer-based tasks.

Number line estimation (NLE) tasks are widely employed in research and pedagogical contexts. For both children and adults, leftmost digits disproportionately influence number line placements-a phenomenon known as a left digit effect or left digit bias. Here, we ask whether left digit bias is limited to literal leftmost digits. In Study 1, adults completed a standard 0 to 1000 number line task and a leading zero version in which target numerals began with leftmost zeros (e.g., 0398). Study 2 used the leading zero task (placing numbers like 0398 on a range up to 1000) and a leading one-zero task (placing numbers like 10398 on a 10000-11000 number line). A robust "left digit" bias was observed for the hundreds digits in all cases, showing that left digit effects are deeply context-dependent. Task conditions can lead to strong left-digit-like biases for digits that are not the literal leftmost digits in the target numerals.

A robust left digit effect arises in number line estimation, whereby the leftmost digits of numerals have an undue influence on placements such that, for example, numbers like 298 are placed far to the left of numbers like 302. Past efforts to motivate more accurate performance using trial-by-trial and summary feedback have not led to a reduction in the left digit effect. In two experiments, we asked whether it is possible to reduce or eliminate the left digit effect in number line estimation through an instructional intervention in which one is explicitly taught about the left digit effect. In Experiment 1 (N = 134), participants completed two blocks (60 trials per block) of a self-paced 0-1,000 number line estimation task and were randomly assigned to either an instruction or a control condition. In Experiment 2 (N = 143), the procedure was enhanced with a learning check, and with additional measures to assess changes in behaviour as a result of instruction. In both experiments, a left digit effect was found in each block of each condition. Although there was evidence that instruction changed behaviour, these changes did not result in any reduction in the left digit effect relative to the control condition. These findings demonstrate that the left digit effect cannot be easily reduced by making people aware of it.

Number line estimation tasks are frequently used to study numerical cognition skills. In a typical version, the bounded number line task, target numerals must be placed on a bounded line labeled only at its endpoints (e.g., with 0 and 100). Placements by adults, while highly accurate, reveal a cyclical pattern of over-and underestimation of target numerals. The pattern suggests use of proportion judgment strategies and is well -captured by cyclical power models. Another systematic number line bias that has recently been observed, but has not yet been considered in modeling efforts, is the left digit effect. Numerals with different leftmost digits (e.g., 39 and 41) are placed farther apart on a line than is warranted. In the current study (N = 60), adult estimates were obtained for all numerals on a 0-100 number line estimation task, and fit of the standard cyclical power model was compared with two modified versions of the model. One modified version included a parameter that underweights the rightward digit's place value (e.g., the ones digit here), and the other used the same parameter to underweight all digits' place values. We found that both modifications provided a considerably better fit for individual and median data than the standard model, and we discuss their relative merits and cognitive interpretations. The data and models suggest how a left digit bias might impact estimates across the number line.

Recent work reveals a left digit effect in number line estimation such that adults' and children's estimates for three-digit numbers with different hundreds-place digits but nearly identical magnitudes are systematically different (e.g., 398 is placed too far to the left of 401 on a 0-1000 line, despite their almost indistinguishable magnitudes; Lai et al., 2018, https://doi.org/10.1111/desc.12657). In two preregistered studies (N = 218), we investigate the scope and malleability of the left digit effect. Experiment 1 used a typical forward-oriented 0-1000 number line estimation task and an atypical reverse-oriented 1000-0 number line estimation task. Experiment 2 used the same forward-oriented typical 0-1000 number line estimation task from Experiment 1, but with trial-by-trial corrective feedback. We observed a large left digit effect, regardless of the orientation of the line in Experiment 1 or the presence of corrective feedback in Experiment 2. Further, analyses using combined data showed that the pattern was present across most stimuli and participants. These findings demonstrate a left digit effect that is robust and widely observed, and that cannot be easily corrected with simple feedback. We discuss the implications of the findings for understanding sources of the effect and efforts to reduce it.

A left digit effect has been broadly observed across judgment and decision-making contexts ranging from product evaluation to medical treatment decisions to number line estimation. For example, $3.00 is judged to be a much greater cost than $2.99, and 801 is estimated strikingly too far to the right of 798 on a number line. Although the consequences of the effects for judgment and decision behavior have been documented, the sources of the effects are not well established. The goal of the current work is to extend investigations of the left digit effect to a new complex judgment activity and to assess whether the magnitude of the effect at the individual level can be predicted from performance on a simpler number skills task on which the left digit effect has also recently been observed. In three experiments (N = 434), adults completed a judgment task in which they rated the strength of hypothetical applicants for college admission and a self-paced number line estimation task. In all experiments, a small or medium left digit effect was found in the college admissions task, and a large effect was found in number line estimation. Individual-level variation was observed, but there was no relationship between the magnitudes of the effects in the two tasks. These findings provide evidence of a left digit effect in a novel multiattribute judgment task but offer no evidence that such performance can be predicted from a simple number skills task such as number line estimation.

Number line estimation (NLE) tasks are widely used to investigate numerical cognition, learning, and development, and as an instructional tool. Interpretation of these tasks generally involves an implicit expectation that responses are driven by the overall magnitudes of target numerals, in the sense that the particular digits conveying those magnitudes are unimportant. However, recent evidence shows that numbers with similar magnitudes but different leftmost digits are estimated very differently. For example, 798 is placed systematically much too far to the left of 801 in a 0-1000 NLE task by children aged 7-11 and adults. Here we ask whether this left digit effect generalizes to two-digit numerals in a 0-100 NLE task and whether it emerges in younger children. Children aged 5-8 (Study 1, N = 73), adults (Study 2, N = 44), and children aged 9-11 (Study 3, N = 27) completed a standard 0-100 NLE task on a touchscreen tablet. We observed left digit effects for two-digit numerals in children aged 8-11 and adults, with large effect sizes, demonstrating that these effects generalize to smaller numerical ranges. Left digit effects were not apparent in 5- to 7-year-olds, suggesting that these effects do not emerge at younger ages for smaller, more familiar numerical ranges. We discuss developmental emergence of left digit effects in number line estimation and implications within and beyond the field of cognitive development.