Marcus Lindskog

Measuring acuity of the approximate number system reliably and validly: the evaluation of an adaptive test procedure.

Two studies investigated the reliability and predictive validity of commonly used measures and models of Approximate Number System acuity (ANS). Study 1 investigated reliability by both an empirical approach and a simulation of maximum obtainable reliability under ideal conditions. Results showed that common measures of the Weber fraction (w) are reliable only when using a substantial number of trials, even under ideal conditions. Study 2 compared different purported measures of ANS acuity as for convergent and predictive validity in a within-subjects design and evaluated an adaptive test using the ZEST algorithm. Results showed that the adaptive measure can reduce the number of trials needed to reach acceptable reliability. Only direct tests with non-symbolic numerosity discriminations of stimuli presented simultaneously were related to arithmetic fluency. This correlation remained when controlling for general cognitive ability and perceptual speed. Further, the purported indirect measure of ANS acuity in terms of the Numeric Distance Effect (NDE) was not reliable and showed no sign of predictive validity. The non-symbolic NDE for reaction time was significantly related to direct w estimates in a direction contrary to the expected. Easier stimuli were found to be more reliable, but only harder (7:8 ratio) stimuli contributed to predictive validity.

The association between higher education and approximate number system acuity

Humans are equipped with an approximate number system (ANS) supporting non-symbolic numerosity representation. Studies indicate a relationship between ANS-precision (acuity) and math achievement. Whether the ANS is a prerequisite for learning mathematics or if mathematics education enhances the ANS remains an open question. We investigated the association between higher education and ANS acuity with university students majoring in subjects with varying amounts of mathematics (mathematics, business, and humanities), measured either early (First year) or late (Third year) in their studies. The results suggested a non-significant trend where students taking more mathematics had better ANS acuity and a significant improvement in ANS acuity as a function of study length that was mainly confined to the business students. The results provide partial support for the hypothesis that education in mathematics can enhance the ANS acuity.

The role of ANS acuity and numeracy for the calibration and the coherence of subjective probability judgments

The purpose of the study was to investigate how numeracy and acuity of the approximate number system (ANS) relate to the calibration and coherence of probability judgments. Based on the literature on number cognition, a first hypothesis was that those with lower numeracy would maintain a less linear use of the probability scale, contributing to overconfidence and nonlinear calibration curves. A second hypothesis was that also poorer acuity of the ANS would be associated with overconfidence and non-linearity. A third hypothesis, in line with dual-systems theory (e.g., Kahneman and Frederick, 2002) was that people higher in numeracy should have better access to the normative probability rules, allowing them to decrease the rate of conjunction fallacies. Data from 213 participants sampled from the Swedish population showed that: (i) in line with the first hypothesis, overconfidence and the linearity of the calibration curves were related to numeracy, where people higher in numeracy were well calibrated with zero overconfidence. (ii) ANS was not associated with overconfidence and non-linearity, disconfirming the second hypothesis. (iii) The rate of conjunction fallacies was slightly, but to a statistically significant degree decreased by numeracy, but still high at all numeracy levels. An unexpected finding was that participants with better ANS acuity gave more realistic estimates of their performance relative to others.

Are there rapid feedback effects on Approximate Number System acuity

Humans are believed to be equipped with an Approximate Number System (ANS) that supports non-symbolic representations of numerical magnitude. Correlations between individual measures of the precision of the ANS and mathematical ability have raised the question of whether the precision can be improved by feedback training. A study (DeWind and Brannon, 2012) reported improvement in discrimination precision occurring within 600–700 trials of feedback, suggesting ANS malleability with rapidly improving acuity in response to feedback. We tried to replicate the rapid improvement in a control group design, while controlling for the use of perceptual cues. The results indicate no learning effects, but a minor constant advantage for the feedback group. The measures of motivation suggest that feedback has a positive effect on motivation and that the difference in discrimination is due to the greater motivation of participants with feedback. These results suggest that at least for adults the number sense may not respond to feedback in the short-term.

Naïve Point Estimation

The capacity of short-term memory is a key constraint when people make online judgments requiring them to rely on samples retrieved from memory (e.g., Dougherty & Hunter, 2003). In this article, the authors compare 2 accounts of how people use knowledge of statistical distributions to make point estimates: either by retrieving precomputed large-sample representations or by retrieving small samples of similar observations post hoc at the time of judgment, as constrained by short-term memory capacity (the naïve sampling model: Juslin, Winman, & Hansson, 2007). Results from four experiments support the predictions by the naïve sampling model, including that participants sometimes guess values that they, when probed, demonstrably know have the lowest probability of occurring. Experiment 1 also demonstrated the operations of an unpredicted recognition-based inference. Computational modeling also incorporating this process demonstrated that the data from all 4 experiments were better predicted by assuming a post hoc sampling process constrained by short-term memory capacity than by assuming abstraction of large-sample representations of the distribution.

No evidence of learning in non-symbolic numerical tasks – A comment on Park and Brannon (2014)

Two recent studies - one of which was published in this journal - claimed to have found that learning on a non-symbolic arithmetic task improved performance on a symbolic arithmetic task (Park & Brannon, 2013, 2014). This finding has potentially far-reaching implications, because it would constitute evidence for a causal link between the Approximate Number System (ANS) and symbolic-math ability. Here, we argue that, due to the methodology used in both studies, the interpretation of data in terms of an improvement in ANS performance is problematic. We provide arguments and simulations showing that the trends in the data are similar to what one would expect for a non-learning observer. We discuss the implications for the original interpretation in terms of causality between non-symbolic and symbolic arithmetic performance.

An Embodied Account of Early Executive-Function Development: Prospective Motor Control in Infancy Is Related to Inhibition and Working Memory

The importance of executive functioning for later life outcomes, along with its potential to be positively affected by intervention programs, motivates the need to find early markers of executive functioning. In this study, 18-month-olds performed three executive-function tasks—involving simple inhibition, working memory, and more complex inhibition—and a motion-capture task assessing prospective motor control during reaching. We demonstrated that prospective motor control, as measured by the peak velocity of the first movement unit, is related to infants’ performance on simple-inhibition and working memory tasks. The current study provides evidence that motor control and executive functioning are intertwined early in life, which suggests an embodied perspective on executive-functioning development. We argue that executive functions and prospective motor control develop from a common source and a single motive: to control action. This is the first demonstration that low-level movement planning is related to higher-order executive control early in life.

Individual Differences in Nonverbal Number Skills Predict Math Anxiety

Math anxiety (MA) involves negative affect and tension when solving mathematical problems, with potentially life-long consequences. MA has been hypothesized to be a consequence of negative learning experiences and cognitive predispositions. Recent research indicates genetic and neurophysiological links, suggesting that MA stems from a basic level deficiency in symbolic numerical processing. However, the contribution of evolutionary ancient purely nonverbal processes is not fully understood. Here we show that the roots of MA may go beyond symbolic numbers. We demonstrate that MA is correlated with precision of the Approximate Number System (ANS). Individuals high in MA have poorer ANS functioning than those low in MA. This correlation remains significant when controlling for other forms of anxiety and for cognitive variables. We show that MA mediates the documented correlation between ANS precision and math performance, both with ANS and with math performance as independent variable in the mediation model. In light of our results, we discuss the possibility that MA has deep roots, stemming from a non-verbal number processing deficiency. The findings provide new evidence advancing the theoretical understanding of the developmental etiology of MA.

Is there something special with probabilities? – Insight vs. computational ability in multiple risk combination

While a wealth of evidence suggests that humans tend to rely on additive cue combination to make controlled judgments, many of the normative rules for probability combination require multiplicative combination. In this article, the authors combine the experimental paradigms on probability reasoning and multiple-cue judgment to allow a comparison between formally identical tasks that involve probability vs. other task contents. The purpose was to investigate if people have cognitive algorithms for the combination, specifically, of probability, affording multiplicative combination in the context of probability. Three experiments suggest that, although people show some signs of a qualitative understanding of the combination rules that are specific to probability, in all but the simplest cases they lack the cognitive algorithms needed for multiplication, but instead use a variety of additive heuristics to approximate the normative combination. Although these heuristics are surprisingly accurate, normative combination is not consistently achieved until the problems are framed in an additive way.

Calculate or wait: Is man an eager or a lazy intuitive statistician?

Research on peoples ability to act as intuitive statisticians has mainly focused on the accuracy of estimates of central tendency and variability. In this paper, we investigate two hypothesised cognitive processes by which people make judgements of distribution shape. The first claims that people spontaneously induce abstract representations of distribution properties from experience, including about distribution shape. The second process claims that people construct beliefs about distribution properties post hoc by retrieval from long-term memory of small samples from the distribution, implying format dependence with accuracy that differs depending on judgement format. Results from two experiments confirm the predicted format dependence, suggesting that people are often constrained by the post hoc assessment of distribution properties by sampling from long-term memory. The results, however, also suggest that, although post hoc sampling from memory seems to be the default process, under certain predictable circumstances people do induce abstract representations of distribution shape.