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How does task difficulty schedule affect the rate and efficiency of perceptual learning?

How does task difficulty schedule affect the rate and efficiency of perceptual learning?

In perceptual learning (line length/orientation discrimination, visual target detection, tone frequency discrimination, etc), when training people or animals to perform a task better, is there an optimal schedule for controlling task difficulty, to acheive maximal improvement?

At least 3 reasonable schedules come to (my) mind:

  1. Easy to difficult - In education it is the custom to begin with easy examples or problems, and gradually progress to more difficult ones. Maybe this should work in perceptual learning as well.
  2. Interleave different difficulties
  3. Always the most difficult possible

Has there been a rigorous comparison of such schedules and their effect on the rate of learning, and the maximal performance level achieved at the end of training?


General pedagogical ideas around optimal difficulty

Many theories of instruction suggest that learning is optimal when an appropriate level of challenge is maintained. If a task is too easy, there's little to learn. If a task is too difficult, the learner can be overwhelmed. The implication for practice is that task difficulty should increase in conjunction with the increased skill of the learner. Of course, we can think of exceptions to these ideas, such as the value of over-learning, and the general idea of exposing people to challenging problems to motivate further learning. This also still leaves open the question of precisely how fast difficulty should increase for a given learner.

Academically, I've seen these ideas expressed in several places:

Deliberate practice: This is a concept drawn from the expertise literature. It is posited that amount of deliberate practice (rather than amount of simple experience) is one of the key factors that discriminates experts from non-experts (see Ericsson et al 2006). A key aspect of task difficulty relates to the sequencing of task difficulty with practice. Ericsson et al (2006) writes that

The core assumption of deliberate practice… is that expert performance is acquired gradually and that effective improvement of performance requires the opportunity to find suitable training tasks that the performer can master sequentially… Deliberate practice presents performers with tasks that are initially outside their current realm of reliable performance, yet can be mastered within hours of practice by concentrating on critical aspects and by gradually refining performance through repetitions after feedback.

Challenge point in motor learning: Guadagnoli and Lee present a framework for thinking about optimal difficulty in motor learning. Their framework suggests that optimal learning is achieved by maintaining task difficulty at the point where it provides optimal information (where learning is possible, but it is not too difficult as to be overwhelming).

Research on task difficulty and perceptual learning

In a review article Ahissar and Hochstein (2004) also suggest easy to difficult is the most effective regime for perceptual learning:

As found in our studies, training is more effective if subjects start with easy conditions and gradually move to more difficult con- ditions. The importance of beginning training with easy conditions was first found by Pavlov. When Pavlov reinforced a dog's salivation following its seeing an ellipse but not following its seeing a circle (Figure I, rightmost pair of stimuli), the dog could not avoid generalization and salivated at sight of the circle, as well. Only by using very elongated ellipses, and training along the continuum, from left to right, was it able to achieve good performance for small circle/ellipse differences. This phenomenon was subsequently termed 'transfer along a continuum' (of different degrees of difficulty).

References

  • Ahissar, M. and Hochstein, S. (2004). The reverse hierarchy theory of visual perceptual learning. Trends in cognitive sciences, 8(10):457-464. FREE PDF
  • Ericsson, K. et al. (2006). The influence of experience and deliberate practice on the development of superior expert performance. The Cambridge handbook of expertise and expert performance, 10(3):683-703. FREE PDF
  • Guadagnoli, M. and Lee, T. (2004). Challenge point: a framework for conceptualizing the effects of various practice conditions in motor learning. Journal of Motor Behavior, 36(2):212-224. FREE PDF

Speed-Accuracy Trade-Off

Renée A. Duckworth , . Alexander V. Badyaev , in Advances in the Study of Behavior , 2018

2.2.1 Speed–Accuracy Trade-off

In the speed–accuracy trade-off, decisions are made slowly with high accuracy or fast with high error rate ( Chittka, Skorupski, & Raine, 2009 ). The neurobiological basis of this trade-off is well characterized. In both the prefrontal cortex and subcortical areas of the brain, neurons associated with different perceptual choices gradually increase their firing rate as they integrate inputs from sensory neurons ( Gold & Shadlen, 2007 ). A decision is made when the firing rate of the neurons associated with a particular choice exceeds a critical threshold and individuals told to prioritize speed in a cognitive task showed heightened baseline activation of brain areas involved with decision-making allowing them to reach the decision threshold faster ( Bogacz, Wagenmakers, Forstmann, & Nieuwenhuis, 2009 ). Yet, such flexibility in decision-making processes varies across individuals. Studies have found distinct patterns of brain activity and connectivity among individuals that preferentially prioritize speed and among individuals that vary in their ability to flexibly adjust their level of caution (sometimes prioritizing speed, sometimes accuracy Forstmann et al., 2010 Perri, Berchicci, Spinelli, & Di Russo, 2014 ). In particular, individuals who are better able to flexibly adjust their level of caution have stronger structural connections between the supplementary motor area of the brain and the striatum, a subcortical part of the forebrain and a critical component of the reward system ( Forstmann et al., 2010 ). Moreover, individuals that naturally prioritize speed had higher baseline activation of supplementary motor areas and individuals that naturally prioritize accuracy had higher baseline activity of areas in the prefrontal cortex ( Perri et al., 2014 ). Thus, individual variation in speed vs accuracy of decision-making appears to reflect a trade-off between a greater baseline activity of areas associated with cognitive control (that slow down decision-making processes but increase their accuracy) and greater baseline activity of motor and subcortical areas (that enhance the speed of an action at the expense of accuracy). Finally, variation in how individuals deal with this trade-off has been shown to relate to a variety of personality dimensions such as risk sensitivity ( Nagengast, Braun, & Wolpert, 2011 ), agreeableness ( Bresin, Hilmert, Wilkowski, & Robinson, 2012 ) and neuroticism ( Socan & Bucik, 1998 ) in humans, and alternative proactive and reactive coping styles in other animals ( Sih & Del Giudice, 2012 ).


Training with high perceptual difficulty improves the capacity and fidelity of internal representation in VWM

It has been shown that the capacity of visual working memory (VWM) is a strong predictor of individual intelligence, and researchers have developed various training protocols to improve VWM capacity. However, it seems that whether the fidelity of internal representation in VWM can also be improved by training is largely overlooked in the past literature. Here, we introduced a new training approach that involved increasing the perceptual difficulty of training materials to enhance VWM, and both memory capacity and the fidelity of representation were examined to assess the training efficacy. Participants with normal vision and cognitive abilities received 3-week training on VWM using a change detection task, and the results showed that both the capacity and the fidelity of memory representations were improved for training with perceptually difficult stimuli, while only the fidelity was improved for training with perceptually normal stimuli. In addition, we found that the training effects on memory precision may be subject to capacity constraints. We suggest that long-term adaptive training with perceptually difficult stimuli may facilitate encoding efficiency through familiarizing trainees with an increased baseline of cognitive workload during the encoding process. The present study offers clear evidence that training with high perceptual difficulty is more effective and the improvements in VWM are more stable than training with perceptually normal materials, and the simple manipulation on training stimuli indicates that the method can be generalized to a wider range of training situations and populations.

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Results

Both discriminability and mean RT improved with practice (Fig. 3) and showed highly significant linear and quadratic trends (Table 1). The d’ profiles for easy and difficult discriminations were approximately proportional to each other ( ( d_< m>^prime approx kd_< m>^prime ) with k = 1.65 ± 0.13), in agreement with published data (Petrov et al., 2005). The learning effects were partially specific to the trained reference direction, although the degree of specificity differed significantly for the two dependent measures. The specificity index was Footnote 2 SI = .60 ± .10 for d’ and .37 ± .08 for the mean RT.

Learning profiles for the group-averaged discriminability (a) and mean response times (b) in the raw data, and for various parameters of the diffusion model (cf). The observers practiced motion-direction discrimination for eight blocks (black symbols) and then were tested on the same task with motion in the orthogonal direction (open symbols). The error bars are 90% within-subjects confidence intervals. Shaded areas mark two additional sessions of motion aftereffect measurements.

The DM achieved good fits, evident in the quantile probability plots in Fig. 4 and the scatterplots in Fig. 5. The former show the proportions of correct and error responses (on the x-axis) and the corresponding RT distributions (summarized by the .1, .3, .5, .7, and .9 quantiles on the y-axis). The model (circles) tracks the data (×’s) well. Footnote 3 The scatterplots show that the model can reconstruct the data for each individual on each block to a good approximation. The quality of the fit, coupled with past research (Ratcliff & McKoon, 2008) validating the DM in conditions similar to ours, suggests that the DM parameters offer a concise characterization of the underlying cognitive processes.

Quantile probability plots illustrating the wealth of data and the quality of the fit. Each panel has 22 empirical degrees of freedom: the proportions of errors and correct responses for the easy and difficult discriminations (plotted on the x-axis) and the .1, .3, .5, .7, and .9 quantiles of the corresponding response time distributions (stacked vertically on the y-axis). For example, the x-coordinate of the leftmost, bottommost data point in the top panel indicates the initial error rate (.18) for easy stimuli. The y-coordinate indicates the leading edge (.1 quantile ≈ 530 ms) of the corresponding RT distribution. After 4 days of training (middle panel), the performance improves on both measures (.08 rate and 480 ms, respectively). This illustration is based on group-averaged data the analyses in the text (and the predictions in Fig. 5) are based on model fits to individual data

Scatterplots illustrating the quality of the fit to individual data. The diffusion model was fit separately in each block (297 fits = 27 observers × 11 blocks). Each panel contains 594 points (= 297 × 2 difficulty levels). RT, response time

There were statistically significant learning effects for all DM parameters except the starting point variability s z (Table 1). The twofold improvement in drift rate (Fig. 3c) indicates that the quality of the sensory input to the decision process increases with practice. The learning index for the drift rate v (LI = 0.99 ± 0.23) was significantly Footnote 4 higher than the d’ learning index (.55 ± .08). This is because v reflects learning in both accuracy and speed. The improvement was largely (but not entirely) specific to the trained reference direction (SI = .68 ± .09).

The parameters describing the distribution of nondecision times across trials also improved significantly. The mean nondecision time T er decreased by 20% on average (Fig. 3d). The specificity index for T er (.22 ± .10) was significantly Footnote 5 lower than that for the mean overall RT (.37 ± .08). This is because the improvement in overall RT stems in part from the stimulus-specific increase in drift rate.

The nondecision variability s t is of particular interest. As predicted by the synchronization hypothesis, it was high at first (283 ms during Block 1, Fig. 3f) and decreased steeply to 120 ms by the end of training. Moreover, the improvement transferred fully to the orthogonal direction of motion (SI = .00 ± .08).

There was a small but statistically significant decreasing linear trend in the boundary separation parameter a (Table 1). This suggests a slight adjustment in the speed–accuracy trade-off. The drift-rate increase apparently offset this adjustment and prevented a drop in accuracy. Finally, there was a marginally significant decrease in the across-trial variability in drift rate (η) but no significant changes in the variability in starting point (s z). See the online supplement for details.


EXPERIMENT 1

There is conflicting evidence on the extent to which regions in PPC are driven by the number of spatial locations versus by the number of discrete objects per se (Xu & Chun, 2006, Experiment 4 Xu & Chun, 2007 Cusack, 2005). The aim of this first experiment was to contrast the perception of two objects with one object, in the absence of a difference in the number of spatial locations, and to evaluate a stimulus type that might be developed in the following experiment to distinguish object representation, attention switching, and task difficulty. To do this, we exploited the phenomenon of transparent motion, which has been shown to be an effective substrate for the deployment of object-based attention (Rodriguez, Valdes-Sosa, & Freiwald, 2002 Valdes-Sosa, Cobo, & Pinilla, 1998) and allows matching of spatial location. On some trials, we presented one surface, and on other trials, two overlapping surfaces. The size and the position of the surfaces were the same across conditions.

Methods

Stimuli and Task

On each trial, a set of 1000 white dots was presented on a black background for 1.2 sec, viewed through a circular window of diameter 500 pixels (approximately 11° of visual angle). There were two conditions, schematically illustrated in Figure 1. In Condition 1, all of the dots would oscillate along a single axis chosen at a random angle. The temporal frequency of the oscillatory motion was 4 Hz, and the peak-to-peak amplitude 24 pixels. In Condition 2, half of the dots would move along one axis in a similar manner to Condition 1, and the other half of the dots would move along the orthogonal axis. In this transparent motion condition, two surfaces are clearly perceived, but they are entirely overlapping. The two conditions differ in the number of objects, but not in the number of spatial locations. After each 1.2-sec stimulus, there was a 1.5-sec intertrial interval. To ensure participant vigilance, on each trial, the participant was asked to press one of two buttons with the right hand to indicate whether one or two surfaces were perceived. The button mapping was counterbalanced across subjects. Two blocks of 192 trials were presented. One third of the trials were one surface (Condition 1), one third two surfaces (Condition 2), and one third null trials in which just a fixation cross was presented.

Schematic illustration of the transparent motion stimuli used in Experiment 1. The one-surface (A) and two-surface (B) conditions were distinguished by the oscillatory motion.

Schematic illustration of the transparent motion stimuli used in Experiment 1. The one-surface (A) and two-surface (B) conditions were distinguished by the oscillatory motion.

MRI Acquisition

Data were acquired from 16 subjects using a 3-T Bruker Medspec scanner with a head gradient set at the Wolfson Brain Imaging Centre, Cambridge, UK. Two 8′30″ blocks of 204 EPI acquisitions (TA = 1.1 sec TR = 2.5 sec) were acquired. The first eight scans were discarded to allow for T1 equilibrium. Each volume was 21 slices (Gaussian profile, 4 mm thickness, 1 mm gap) acquired in an ascending interleaved order each of matrix size 64 × 64, giving a resolution of 3.75 × 3.75 mm. The TE was 30 msec and flip angle was 65°. We also acquired fieldmaps using a phase contrast sequence (complex subtraction of a pair GE acquisitions, TE = 7 and 16.1 msec, matrix size 64 × 256 × 64, resolution 4 × 1 × 4 mm), and a whole-brain T1-weighted structural using an SPGR sequence giving approximately 1 mm 3 resolution.

Analysis

The data were analyzed using SPM2 with the automatic analysis (aa) library (http://imaging.mrc-cbu.cam.ac.uk/imaging/AutomaticAnalysisIntroduction) for scripting. Sinc interpolation through time was used to correct for the acquisition of different brain slices at different times (SPM's slice timing correction). Bulk motion of the head through the time series was then estimated. The pattern of inhomogeneity in the magnetic field measured using a fieldmap was used to correct for distortion in the EPIs (Cusack, Brett, & Osswald, 2003). The EPI mean was then coregistered with the structural using a mutual information cost function. Nonlinear warping to MNI space was then accomplished using SPM normalization on the structural image. A single spatial reslicing stage with sinc interpolation applied the transformations of motion correction, undistortion, and normalization to the EPI images. The normalized images were smoothed using a Gaussian kernel of FWHM 10 mm.

Each subject's data were analyzed using a multiple regression model as specified by an SPM design matrix. The model comprised three event-related columns (predictors): one for the fixation trials, one for the trials with a single surface, and one for the trials with two surfaces. Each event began at the onset of the stimuli and lasted for a single scan (2.5 sec). Boxcar functions representing the time course of these events were convolved with the canonical hemodynamic response to form the predictors for the BOLD fMRI data at each voxel. The six parameters derived from motion correction were also included as regressors to partial out the first-order effects of distortion during motion.

Estimation (fitting) of the regression model gave for each voxel a regression coefficient (β) for each of the predictors. The contributions of the events in these predictors to the BOLD signal were then probed with two orthogonal contrasts. One was the main effect of the stimulus and task (βone surface + βtwo surfaces − 2βfixation), and the other the effect of the number of surfaces (βtwo surfaces − βone surface). For each contrast at the first level, group random-effects analyses were then calculated using parametric statistics and reported at a corrected (FDR p < .05) threshold. To visualize the relative strength of each of the contrasts, we calculated the ratio of the t values of the two contrasts (e.g., tnumber of surfaces/tmain effect of stimulus task) for the voxels that survived corrected thresholds in either contrast.

To quantify the effects of task performance and of the number of surfaces across different parietal areas, we conducted a region-of-interest (ROI) analysis. All ROIs were spheres of radius 1 cm, constructed using MarsBar for SPM (http://marsbar.sourceforge.net). Coordinates are given in MNI152 space, converted from Talairach space where necessary using tal2mni http://imaging.mrc-cbu.cam.ac.uk/downloads/MNI2tal/tal2mni.m). Two-tailed statistics were used to calculate p values in t tests. In none of the experiments do we have lateralized stimuli, and we did not expect (or see) lateralized responses, thus all ROI analyses are collapsed across hemispheres. As summarized in the Introduction, a region close to the parietal/occipital boundary in the inferior IPS has been implicated in object representations (Xu, 2007 Xu & Chun, 2006, 2007 Cusack, 2005). This region has also been found to topographically map attention in a paradigm similar to retinotopic mapping, but with attention shifted around the display rather than with a changing stimulus (Silver et al., 2005). To test whether regions previously shown to be topographic are activated to a greater degree by multiple objects even in the absence of any manipulation of the spatial distribution, the peak coordinates in the inferior IPS from Silver et al. (2005) were selected as the center of two ROIs (IPS1: ±23, −80, 38).

To characterize activity in more general, non-task-specific, parietal regions (the MD network), we extended the meta-analysis of Duncan and Owen (2000) to the parietal lobe and used a kernel method to summarize the peak activation coordinates. The peaks of the activations from the studies described by Duncan and Owen appeared symmetric, and so those in the left hemisphere were reflected onto the right. A single point was placed at each coordinate, and the resulting image was smoothed (15 mm FWHM) and then thresholded at 3.5 times the height of the smoothed peak from a single point. The final thresholded regions were then mirrored onto both hemispheres. In addition to the lateral and medial frontal regions reported by Duncan and Owen, inclusion of parietal activations revealed clear foci around the IPS at (±37, −53, 40). As shown in Figure 2, these are more lateral and anterior to the inferior IPS ones, and were used to define another pair of ROIs.

The ROIs from previous studies used to quantify responses. Xu-infIPS, Xu-supIPS, and Xu-LOC are inferior and superior intraparietal sulcus and lateral occipital regions taken from Xu and Chun (2006) MD-IPS and MD-IFS are intraparietal sulcus and inferior frontal sulcus regions in MD network in an extension of the study by Duncan and Owen (2000) Silver-IPS1 is IPS region taken from Silver et al. (2005).

The ROIs from previous studies used to quantify responses. Xu-infIPS, Xu-supIPS, and Xu-LOC are inferior and superior intraparietal sulcus and lateral occipital regions taken from Xu and Chun (2006) MD-IPS and MD-IFS are intraparietal sulcus and inferior frontal sulcus regions in MD network in an extension of the study by Duncan and Owen (2000) Silver-IPS1 is IPS region taken from Silver et al. (2005).

In addition to these key parietal ROIs, we also investigated the response in visual regions (the lateral occipital complex [LOC] taken from Xu & Chun, 2006: −44, −71, 5 and 42, −69, 0) and frontal executive control regions taken from the MD kernel analysis (dorsolateral prefrontal cortex [DLPFC]: ±42, 24, 25). Finally, for direct comparison with Xu and Chun's (2006) studies, we summarized the response in their “inferior IPS” region on the occipital/parietal boundary, and close to Silver's IPS1 (−21, −89, 24 and 26, −85, 28) and the “superior IPS” regions, which are closest to the MD regions (−21, −70, 42 and 23, −56, 46).

Results

The task was easy and performance was excellent (mean trials correct was 97% with a standard error across subjects of 0.7%). Neuroimaging revealed that presentation of the stimulus and performance of the task, when contrasted with fixation, recruited a broad range of regions, including occipital visual areas, left motor cortex, and regions in the fronto-parietal MD network (Figure 3A). The contrast of two surfaces minus one surface revealed several regions in common with this (Figure 3B). Both contrasts activated regions in the posterior parietal lobe to some extent. To visualize the relative strength of the response to the contrast between two surfaces and one surface and the contrast of task versus fixation, we calculated the ratio of the t values of these two contrasts (Figure 3C) for all voxels that were significant in either of the whole-brain corrected contrasts.

Whole-brain results of Experiment 1. The top two panels show whole-brain-corrected SPM-T maps, and the lower panel, their ratio for all voxels where either contrast was significant [warm colors correspond to (B) > (A), cool colors to (A) > (B)].

Whole-brain results of Experiment 1. The top two panels show whole-brain-corrected SPM-T maps, and the lower panel, their ratio for all voxels where either contrast was significant [warm colors correspond to (B) > (A), cool colors to (A) > (B)].

The response in the parietal lobe and other regions was quantified using ROI analyses (Figure 4A). There was a significant effect of the number of surfaces in the inferior IPS but not in the MD-IPS [Silver-IPS1: t(15) = 2.70, p < .02 MD: t(15) = 0.80, ns]. The opposite pattern was seen for the general task demand, with no evidence to reject the null hypothesis in the inferior IPS, but a significant effect in MD-IPS [Silver-IPS1: t(15) = 1.42, ns MD-IPS: t(15) = 3.14, p < .01]. A repeated-measures ANOVA confirmed this Task × Region interaction [F(1, 15) = 8.23, p < .02].

Peak percent signal change in an inferior IPS region (Silver-IPS1) and a more anterior, superior one (MD-IPS), for Experiment 1 (A), Experiment 2B (B), Experiment 3 (C), and Experiment 4 (D). *p < .05 **p < .01, and ***p < .001. The annotation above the horizontal line at the top of the graph indicates the significance of the region by condition interaction (see the text for a description of the contrasts).

Peak percent signal change in an inferior IPS region (Silver-IPS1) and a more anterior, superior one (MD-IPS), for Experiment 1 (A), Experiment 2B (B), Experiment 3 (C), and Experiment 4 (D). *p < .05 **p < .01, and ***p < .001. The annotation above the horizontal line at the top of the graph indicates the significance of the region by condition interaction (see the text for a description of the contrasts).

As shown in Figure 5A and B, the LOC and inferior IPS responded during performance of the task [t(15) = 7.82, p < .001 and t(15) = 5.49, p < .001, respectively] and showed greater recruitment by two surfaces than one [t(15) = 5.09, p < .001 t(15) = 4.47, p < .001]. In contrast, DLPFC was not significantly activated by task [t(15) = 1.65, ns] or by the number of surfaces [t(15) = 0.32, ns]. Finally, the superior IPS region showed a pattern a little like MD-IPS, being significantly activated by performance of any task [t(15) = 2.89, p < .02] but not by the number of surfaces [t(15) = 0.72, ns].

Peak percent signal change across three regions from Xu and Chun (2006) and one region from the MD network (abbreviations as in Figure 2), in a set of contrasts involved in general performance of a task or attention switching (A), and revealing the effect of multiple objects (B).

Peak percent signal change across three regions from Xu and Chun (2006) and one region from the MD network (abbreviations as in Figure 2), in a set of contrasts involved in general performance of a task or attention switching (A), and revealing the effect of multiple objects (B).

Discussion

Regions in PPC responded differentially. The more inferior IPS region (Silver-IPS1) showed selectivity for two surfaces rather than one, whereas the MD-IPS region did not. Conversely, the MD-IPS region responded to the performance of the task in general more strongly than the inferior region. This is consistent with the presence of object-related processing in the inferior IPS (Xu & Chun, 2006 Cusack, 2005), and distinct general processing resources in the MD-IPS (Duncan, 2006). That the object effect was seen even though the two surfaces did not differ in spatial location shows that a difference in the number of locations or spatial extent is not required, as found in the auditory study of Cusack (2005). This is not incompatible with a topographic representation in these regions: They might be topographically mapped (cf. Swisher, Halko, Merabet, McMains, & Somers, 2007 Silver et al., 2005), but over and above this, respond with the number of objects in the display.

Note that this effect of object representation in the absence of spatial differences is at odds with the conclusions drawn from Xu and Chun's (2006) Experiment 4, which used memory for sequentially presented shapes at the same location or at different locations, and concluded that objects at different spatial locations are required to modulate the inferior IPS. However, it is consistent with Xu and Chun (2007), who found that differences in grouping modulate inferior IPS activity without differences in the number of relevant spatial locations. At this stage, it is not clear what the critical aspect is of the difference between Xu and Chun's (2006) Experiment 4 and our experiment.

There was a low-level visual difference in the degree of motion coherence between the two and one surface conditions, which probably directly contributed to the contrast in visual regions. It is also possible that some of these differences in visual regions may have been mediated by the number of objects. It has been shown before that the response in MT can be modified by the number of objects in the presence of only small visual differences (Stoner & Albright, 1992). Conversely, it is also possible, although less likely, that regions in the parietal lobe were modulated directly by the visual differences (see General Discussion also discussed in this section is the possible role of eye movements).


Results

Categorization

To ensure that the counterbalanced variable of set (see Materials and Methods) did not have a significant effect on accuracy or reaction time and to determine whether there was any interaction of set with stimulus type, a repeated-measures ANOVA was performed on the control data only. The ANOVA included a within-subject variable of stimulus (faces vs scenes) and a between-subject variable of set (1 vs 2). Critically, there were no significant main effects of either stimulus type or set, nor was there any interaction. Accordingly, data from the two sets were combined for subsequent analysis.

Figure 4 (a, accuracy b, reaction time) shows the data from both participant groups split according to stimulus type. Although performance on the faces categorization task in the two groups was relatively similar, for both accuracy and reaction time, the patients correctly categorized fewer scenes than controls (67 vs 88%) and took considerably longer (>1 s) when doing so.

Scores on the faces and scenes categorization tasks for the controls (white bars) and hippocampal patients (gray bars). a, Percentage correct b, reaction time (milliseconds). ∗p < 0.05, significant difference between the two groups.

A repeated-measures ANOVA on the accuracy data, with a within-subject variable of stimulus and a between-subject variable of group (control vs hippocampal), confirmed these initial conclusions. Significant main effects of stimulus (F(1,13) = 6.3 p < 0.03) and group (F(1,13) = 9.8 p < 0.01) were evident, as well as a significant interaction between the two (F(1,13) = 7.9 p < 0.02). Additional univariate ANOVAs revealed that there was a significant group effect, with the hippocampal patients categorizing less accurately than controls, for the virtual reality scenes (F(1,13) = 18.8 p < 0.001) but not for faces.

The similar ANOVA investigating reaction time revealed only an interaction between stimulus and group (F(1,13) = 11.8 p < 0.004). Univariate ANOVAs confirmed that the hippocampal patients showed a trend toward slower categorization of virtual reality scenes as predicted (F(1,13) = 4.2 p < 0.06) but not for faces.

Discrimination

To ensure that preexposure to the faces and virtual reality scenes actually resulted in perceptual learning in healthy controls, both the accuracy and reaction time data were analyzed in the controls only. Repeated-measures ANOVAs with within-subject variables of stimulus (faces vs scenes) and familiarity (familiar vs novel) and a between-subject variable of set (1 vs 2) revealed a main effect of familiarity only (accuracy, F(1,11) = 10.7, p < 0.007 reaction time, F(1,11) = 6.0, p < 0.04). Categorization of both faces and scenes before undertaking the discrimination task with these stimuli, therefore, resulted in improved accuracy and faster reaction times in control participants for both types of complex stimuli. No other main effects or interactions reached significance. Consequently, as for the categorization task, data from both sets were combined for each stimulus type in additional analyses.

Because perceptual learning was evident for both stimulus types in healthy controls, difference scores (calculated for each subject for both scenes and faces by subtracting the score for novel stimuli from that obtained for familiar items) were used to contrast the level of perceptual learning attained by the hippocampal patients with that of the controls. These difference scores are displayed in Figure 5 (a, accuracy b, reaction times), in which a positive score for accuracy and a negative score for reaction time is evidence of perceptual learning. What is immediately obvious from the two figures is that, although the hippocampal group showed some improvement in accuracy for both faces and scenes, different profiles were evident on the reaction time measure, with a speeding of response for faces but a striking slowing for scenes.

Difference scores (novel score subtracted from familiar score) on the faces and scenes same–different discrimination task for controls (white bars) and hippocampal patients (gray bars). a, Percentage correct b, reaction time (milliseconds) c, inverse efficiency measure. ∗p < 0.05, significant difference between the two groups.

To investigate these findings, repeated-measures ANOVAs were used, with a within-subject variable of stimulus and a between-subjects variable of group, using the difference score data. Although no significant main effects, or an interaction, were revealed in the accuracy ANOVA, a significant interaction between stimulus and group was found for reaction time (F(1,13) = 7.0 p < 0.02). Univariate ANOVAs confirmed that the hippocampal group were performing significantly differently from their controls in the scenes condition (F(1,13) = 10.5 p < 0.006) but not in the faces condition.

These analyses suggest that perceptual learning of scenes is not normal in individuals with hippocampal damage: although the patients accurately discriminated more familiar than novel scenes, they showed a striking increase in reaction time after previous experience with the spatial stimuli. This is a reverse of the improved speed of responding typically seen in perceptual learning and demonstrated by our healthy controls for both faces and scenes. That is to say that the patients appear to be demonstrating a “speed–accuracy trade-off.” To compensate for this, the data can be analyzed using an inverse efficiency measure, which combines reaction time and accuracy (Townsend and Ashby, 1978, 1983). Inverse efficiency was therefore calculated separately for each of the novel and familiar stimulus conditions for each subject and is defined as mean reaction time divided by the proportion of trials correct. These data are displayed as difference scores in Figure 5c, in which more negative scores are evidence of greater perceptual learning.

Figure 5c confirms that the patients and controls show a different pattern of performance on scenes but not on faces. Although both groups show a negative inverse efficiency change for faces (slightly numerically greater in the case of the hippocampal group), the patients showed a large and positive inverse efficiency change for the virtual reality scenes compared with a negative value in the control population. Statistical analyses confirmed a significant main effect of group (F(1,13) = 5.0 p < 0.05), which interacted with stimulus type (F(1,13) = 5.0 p < 0.05). Univariate ANOVAs revealed that the interaction could be accounted for by a difference between the groups for scenes, (F(1,13) = 9.0 p < 0.01) but not for faces.


Thorndike&rsquos Trial and Error Theory | Learning | Psychology

In this article we will discuss about:- 1. Meaning of Thorndike’s Trial and Error Theory 2. Experimental Evidences of Thorndike’s Trial and Error Theory 3. Educational Implications 4. Some Objections.

Meaning of Thorndike’s Trial and Error Theory:

Edward Lee Thorndike (1874-1949) is generally considered to have been the foremost educational psychologist not only of the United States but of the world. He contributed to research and theory in the field of learning and genetic psychology, testing and social psychology, testing and social psychology.

Thorndike first stated the elements of his theory of learning in 1913 that connections are formed in the nervous system between stimuli and response. These connections formed are illustrated by the symbols S-R. Another word used to describe these connections is the word ‘bond’ and hence,’ this theory is sometimes called a ‘Bond Theory of learning’. Thorndike has written- “Learning is connecting. The mind is man’s connection system.”

According to Thorndike learning takes place by trial and error. Some people call it, “Learning by selection of the successful variant,” accordingly when no ready-made solution of a problem is available to the learner, he adopts the method of trial and error. He first, tries one solution. If it does not help him, he rejects it, then, he tries another and so on. In this way he eliminates errors or irrelevant responses which do not serve the purpose and finally discovers the correct solution.

Thus, in trial and error method, the learner makes random activities and finally reaches the goal accidently. Here, one thing should be remembered that in trial and error also, there are often systematic and relevant responses. Activities are not wholly random. All these activities, though apparently random are suggested to him by the situation and the learner proceeds on accordingly. The stages through which the learner has to pass are Goal, Block (hinderances), Random Movements or multiple response, chance success, selection and Fixation.

When and how the connection is accomplished was stated first in the following three laws:

1. Law or Readiness:

First primary law of learning, according to him, is the ‘Law or Readiness’ or the ‘Law of Action Tendency’, which means that learning takes place when an action tendency’ is aroused through preparatory adjustment, set or attitude. Readiness means a preparation for action. If one is not prepared to learn, learning cannot be automatically instilled in him, for example, unless the typist, in order to learn typing prepares himself to start, he would not make much progress in a lethargic and unprepared manner.

2. Law of Exercise:

The second law of learning is the ‘Law of Exercise’, which means that drill, or practice helps in increasing efficiency and durability of learning and according to Thorndike’s S-R Bond Theory, the connections are strengthened with trail or practice and the connections are weakened when trial or practice is discontinued.

The ‘law of exercise’, therefore, is also understood as the ‘law of use and disuse’ in which case connections or bonds made in the brain cortex are weakened or loosened. Many examples of this are found in case of human learning. Learning to drive a motor-car, typewriting, singing or memorizing a poem or a mathematical table, and music etc. need exercise and repetition of various movements and actions May times.

The third law is the ‘Law of Effect’, according to which the trial or steps leading to satisfaction stamps in the bond or connection. Satisfying states lead to consolidation and strengthening of the connection, whereas dis-satisfaction, annoyance or pain leads to the weakening or stamping out of the connections.

In fact, the ‘law or effect’ signifies that if the responses satisfy the subject, they are learnt and selected. While those which are not satisfying are eliminated. Teaching, therefore, must be pleasing. The educator must obey the tastes and interests of his pupils. In other words, greater the satisfaction stronger will be the motive to learn. Thus, intensity is an important condition of the ‘law of effect’.

Besides these three basic laws, Thorndike also refers to five sub-ordinate laws which further help to explain the learning process.

1. Law of Multiple-Response:

According to it the organism varies or changes its responses till an appropriate behaviour is hit upon. Without varying the responses, the correct response for the solution might never be elicited. If the individual wants to solve a puzzle, he is trying in different ways rather than mechanically persisting in the same way. Thorndike’s cat in the puzzle box moved about and tried many ways to come out till finally it hit the latch with her paw which opened the door and it jumped out.

2. The Law of Set or Attitude:

Learning is guided by a total set or attitude of the organism, which determines not only what the person will do but what will satisfy or annoy him. For instance, unless the cricketer sets himself to make a century, he will not be able to score more runs. A student, similarly, unless he sets to get first position and has the attitude of being at the top, would while away the time and would not learn much. Hence, learning is affected more in the individual if he is set to learn more or to excel.

3. Pre-Potency of Elements:

According to this law, the learner reacts selectively to the important or essential element in the situation and neglects the other features or elements which may be irrelevant or non-essential. The ability to deal with the essential or the relevant part of the situation makes analytical and insightful learning possible. In this law of pre-potency of elements, Thorndike is really anticipating insight in learning which was more emphasised by the Gestations.

4. Law of Response by Analogy:

According to this law, the individual makes use of old experiences or acquisitions while learning a new situation. There is a tendency to utilize common elements in the new situation as existed in a similar past situation. The learning of driving a car, for instance, is facilitated by the earlier acquired skill of driving a motor-cycle or even riding a bicycle, because the perspective or maintaining a balance and controlling the handle helps in steering the car.

5. The Law of Associative Shifting:

According to this law we may get any response, of which a learner is capable, associated with any other situation to which he is sensitive. Thorndike illustrated this by the act of teaching a cat to stand up at a command. A fish was dangled before the vat while he said ‘stand up’. After a number of trials by presenting the fish after uttering the command ‘stand up’, he later ousted the fish and the overall command of ‘stand up’ was found sufficient to evoke the response to the cat by standing up on her hind legs.

Experimental Evidences of Thorndike’s Trial and Error Theory:

Various experiments have been performed on men as well as animals to study this method. Thorndike made several experiments on rats and cats. Two important experiments are mentioned here.

Thorndike’s most widely quoted experiment was with the cat placed in a puzzle box. The hungry cat was put in the puzzle box and a fish, as an incentive, was put out-side the cage a little beyond its reach. The box was designed in such a way that the door of the cage can be released by some simple act like depressing a lever inside the cage.

At first, the cat made a great deal of varied attempts to reach the food in a trial and error fashion such as jumping up and down, clawing at the bars, scratching the cage, whaling around trying to push the bars, pawing and shaking movable parts of the cage etc., but all attempts proved to vain.

Ultimately by chance her paw fell on the loop of the rope and the door opened. The cat jumped out immediately and ate the fish. When next day, the cat was put in the box again, this time she took less time in coming out and in the subsequent trials the time decreased further so much so that the stage reached when the cat came out soon after being put inside by directly striking the latch with her paw without any random movement. This is how she learnt to reach its goal.

Expt. 2 (Experiment with Human Subjects):

Gopalaswamy demonstrated trial and error in human beings through Mirror-Drawing Experiment. This is a classical experiment in the psychology of learning. In this experiment the subject is asked to trace a star-shaped drawing, not looking at it directly, but as it is reflected in a mirror, the subject’s hand movements are visible in the mirror only and not directly. The experimenter observes the movements of the hands and thus, records the time of tracing in successive trials and the number of errors committed in each trial.

In first six trials the subject traces the star with the right hand and then in the next six trials he traces it by the left hand. Two graphs-the Time Curve and the Error Curve are then drawn, which show the general characteristics of trial and error learning. In the original experiment Gopalaswamy arranged his apparatus so that a record was automatically made of all the movements of the styles of the subject as it traced out the pattern. In this way the successive times of tracings and a record of errors was obtained.

Gopalaswamy analyzed the errors into two groups-lower level errors and higher level errors. Those errors which do not involve any noble process on the part of the subject in tracing the star are lower-level errors and those which involve higher process of mind on the perceptual and conceptual level are higher-level errors.

He discovered that improvement in the higher-level responses correlated highly with intelligence and that the improvement in the responses of the lower-level errors did not show much correlation with intelligence. This clears the respective share of trial and error and of higher learning.

For Fundulus fishes Thorndike got a glass tub with a dividing wall of glass in the middle. In the dividing wall there was a hole through which the fish could go from one part to another. By nature Fundulus fish like to remain in shade. The glass tub was filled with water and it was put under such a situation that half of its part remained under shade and the other half was in the sunshine. The fishes were kept in the sunny portion.

They began to try to coming over to the shady portion. By trying again and again the fishes succeeded in tracing the hole of the dividing wall and reached the shady portion one by one. But, at first the fishes took more time in reaching the shady portion, then in the second attempt they took less time and in the third attempt they took the least time. Trying it again finally a stage came when the fishes happened to come one after another in a row to the shady portion immediately in the very first attempt i.e., the number of errors of their wandering here and there amounted to a zero.

Educational Implications of Thorndike’s Trial and Error Theory:

Thorndike’s theory of Trial and Error and his three basic laws of learning have direct educational implications. The ‘Law of Readiness’ lays emphasis on motivation while the ‘Law of Exercise’ compels us to accept a well-known fact ‘Practice makes a man perfect’, and the third one i.e., ‘Law of Effect’ opens fairly a large scope to discuss the role of reward and punishment as an incentive in the child’s learning.

Actually, motivation and learning are inter-related concepts. No motivation No learning. Here we can remember a proverb, ‘the one man can take horse to the pool of water but twenty cannot make him drink’. This statement clearly shows the impact of motivation on learning. Clearly speaking motive is a force that compels an individual to act or to behave in a particular direction. And, hence the success of a teacher lies in motivating the roomfuls of energy. His prime duty is to produce ‘thirst’ (a motive to drink water) in the horses. Then and only then he may succeed in making the process of learning easier and interesting.

To quote with the experiment to Tolman and Honzik (1930) which they performed in rats will be of interest and situational here. In this experiment the rats were taught to follow a complex pattern of runs and turns through a maze to reach the food. The rats were divided in three groups. First group of rats was neither hungry nor given any food at the end or trial. The second group was hungry but was not given food. The third one was hungry and given food at the end of a trial.

It was concluded that only the third group learned appreciably i.e., the number of errors went on decreasing in each attempt. The logic is simple. To be motivated and unrewarded leaves to you only frustration instead a notable amount of learning. Also nor is it worthwhile to work for a prize you do not want. Thus, it is the motive that gives the reward its value and the satisfaction of reward that fixes the learning of which it is the effect.

Briefly speaking, without motivation or drive learning is impossible, as firstly, it prods the learner into action and secondly, it introduces light and shadow into an otherwise different field. So, teacher’s concern primarily shall be the motivating of goals and releasing tensions which signalise success. Above all he should have a psychological involvement in reaching and has to be charged with values and therefore, naturally motivated himself. The advice of an old principal of a school is very pertinent here.

“Teachers, you are going to be emulated in your talk and walk by your students, but a little less. If you run, your students will walk. If you walk, your students will stand. If you stand, your students will lie down. If you lie down, your students will sleep. And if you sleep in the class, your students will die”. But, one has to admit here that the organism’s level of performance can’t be beyond a physiological limit, whatever incentive we provide to him. For instance, higher bonus to factory workers, more praise to students may lead to a better performance, but no athlete can jump over the Chinese wall, whatever the intensity of motivation is provided.

Another significant aspect of this theory is that to master a complex situation or to elaborate task, practice is must. It is not possible to handle each difficult situation in a single trial, no matter what the degree of motivation or reward is. One cannot blame the entire constitution of India in one reading even if the reward is a crore of Rupees or the threat is to be shot dead otherwise. Each task initially seems to be difficult and fatiguing but as practice continues, it becomes smoother and requires less effort.

Finally, we say that habit or S-R is established. An expert driver, for instance, goes on driving, listening to the radio and taking to his friend sitting by. In the light of class room teaching blundering is a natural phenomenon associated with student learning. But, the teacher should not regard this as a symptom of inefficient teaching, because this is the way the pupils learn. He should not be at all worried when blundering appears.

Insights will emerge as the blundering progresses from simpler associations to higher units. There is not royal road to success. Kennedy-Fraser, the Psychologist concludes, “The teachers who are responsible for the beginning of any new subject should be the best available, since at the point, the pupils have no defensive system of properly formed habits to protect them from the evil effects of bad teaching.”

Actually, we learn by doing. The teachers’ duty should be to arrange situations in which the student has chance to discover for himself what is significant. The blundering must be directed and methods that are wholly futile must be eliminated. But at the same time the teacher must exercise, constant restraint in his supervision.

Further, both punishment and reward may play a significant role in the process of learning. But, experiments go to show that motivation is successfully handled when it is kept in the positive phase. Drastic forms of inhibition tend to spread their effects over the whole learning situation. Sometimes, the teachers impress upon the negative processes. The false response is effectively inhibited when the correct reaction is fixated and the emphasis should be on the latter process. The fixating rewards are most effective when they afford immediate and complete release.

A delay introduced between the successful performance and the releasing reward has a considerable effect on their rate of learning and co-ordination. In school, the satisfactions should be closely coupled with the activity itself otherwise the likelihood of permanent effects is small. Another aspect of motivating problem is simpler than the manipulations of tensions and releases and can be mastered by all. This is that the learner should be kept informed of his progress and promptly.

Finally, though the theory is not widely accepted for its educational significance, yet, there are certain subjects such as mathematics, tables of mathematics, memorising poetry, rules of grammar etc. in which learning by Trial and Error cannot be avoided. All reasoning subjects afford the greatest opportunity for the application of the Trial and Error method.

In Brief, the implications of the theory are:

1. According to his theory the task can be started from the easier aspect towards its difficult side. This approach will benefit the weaker and backward children.

2. A small child learns some skills through trial and error method only such as sitting, standing, walking, running etc. In teaching also the child rectifies the writing after committing mistakes.

3. In this theory more emphasis has been laid on motivation. Thus, before starting teaching in the classroom the students should be properly motivated.

4. Practice leads a man towards maturity. Practice is the main feature of trial and error method. Practice helps in reducing the errors committed by the child in learning any concept.

5. Habits are formed as a result of repetition. With the help of this theory the wrong habits of the children can be modified and the good habits strengthened.

6. The effects of rewards and punishment also affect the learning of the child. Thus, the theory lays emphasis on the use of reward and punishment in the class by the teacher.

7. The theory may be found quite helpful in changing the behaviour of the delinquent children. The teacher should cure such children making use of this theory.

8. With the help of this theory the teacher can control the negative emotions of the children such as anger, jealousy etc.

9. The teacher can improve his teaching methods making use of this theory. He must observe the effects of his teaching methods on the students and should not hesitate to make necessary changes in them, if required.

10. The theory pays more emphasis on oral drill work. Thus, a teacher should conduct oral drill of the taught contents. This helps in strengthening the learning more.

Some Objections to Thorndike’s Trial and Error Theory:

The theory has been criticised by various psychologists on the following grounds. Firstly, the theory is mechanical, for it leaves no room for an end or purpose in any sense whatsoever. On the contrary psychologist Mc Dougall maintained that even the behaviour of the amoeba or the paramecia consists in learning to face novel conditions to serve some unknown purpose Even repeated trials are of no avail if the tendency to learn is not there.

Again, if the tendency is there, even one trial may be fruitful. Mc Dougall and Woodworth insist on readiness for reaching a goal in learning and Lloyd Morgan lays stress on persistency with varied efforts till the goal of learning is achieved. The hungry cat confined in the puzzle-box with food in front of it goes on persistently trying various means until it gets out of it and has food. So, its trials are not blind and mechanical. In fact, they are guided by perceptual attention and feelings of pleasure and pain. Yet, Thorndike pays no attention to these higher order mental processes.

Secondly, in course or repeated trials the numbers of errors are not corrected of themselves or mechanically. The effects of Trial and Error depend to a great extent upon the psycho-physical state of the animal or man. In the absence of any purpose in view the animal is so puzzled, rather than enlightened by the errors committed that it goes on blindly repeating them without end.

Thirdly, Thorndike assumes that learning consists only in the association of several separate movements. But, learning is a whole process related to a whole situation. The hungry cat confined in a puzzle-box with food placed near it does not perceive the situation in a piece-meal fashion but as a whole of hunger food-puzzle box-confinement.

Finally, the laws of learning formulated by Thorndike appear to be unjustified. For instance, the ‘law of effect’ seems to be in consistent with his mechanical point of view. Satisfaction in or the sense of being rewarded by success and dissatisfaction in or the sense of being punished by failure seen to ascribe higher mental processes to animals like cats and rats than are psychologically ascribable to them. Or, it violates Lloyd Morgans’s law.

Similarly, the ‘Law of Exercise’ has been severely criticised on the grounds that it does not regard other factors like motives, interests, special training etc. Mechanical repetition without motive, interest, significance or understanding does not make anyone learn anything and remember it. One rupee-currency note passes hundred times through the hand of a person, but hardly anyone is able to tell the size, the colour and other details of it.

A boy was asked to write hundred times ‘I have gone’ after school. He wrote it mechanically and correctly all the times. But, when he left the school in the absence of the teacher, he wrote “I have written,” ‘I have gone’ correctly one hundred times and since you are not here “I have went home”. After repeating one correct thing so many times he again committed the same mistake. This shows that repetition without motive, interest or understanding is of no avail.

Thus, learning by Trial and Error is not of very much use and should not be resorted to by the teacher as it lays a stress on cramming. Also, there is much wastage of time and energy by this method.


Discussion

In this article we considered the effect of training accuracy on learning in the case of binary classification tasks and stochastic gradient-descent-based learning rules. We found that the rate of learning is maximized when the difficulty of training is adjusted to keep the training accuracy at around 85%. We showed that training at the optimal accuracy proceeds exponentially faster than training at a fixed difficulty. Finally we demonstrated the efficacy of the Eighty Five Percent Rule in the case of artificial and biologically plausible neural networks.

Our results have implications for a number of fields. Perhaps most directly, our findings move towards a theory for identifying the optimal environmental settings in order to maximize the rate of gradient-based learning. Thus the Eighty Five Percent Rule should hold for a wide range of machine learning algorithms including multilayered feedforward and recurrent neural networks (e.g. including ‘deep learning’ networks using backpropagation 9 , reservoir computing networks 21,22 , as well as Perceptrons). Of course, in these more complex situations, our assumptions may not always be met. For example, as shown in the Methods, relaxing the assumption that the noise is Gaussian leads to changes in the optimal training accuracy: from 85% for Gaussian, to 82% for Laplacian noise, to 75% for Cauchy noise (Eq. (31) in the “Methods”).

More generally, extensions to this work should consider how batch-based training changes the optimal accuracy, and how the Eighty Five Percent Rule changes when there are more than two categories. In batch learning, the optimal difficulty to select for the examples in each batch will likely depend on the rate of learning relative to the size of the batch. If learning is slow, then selecting examples in a batch that satisfy the 85% rule may work, but if learning is fast, then mixing in more difficult examples may be best. For multiple categories, it is likely possible to perform similar analyses, although the mapping between decision variable and categories will be more complex as will be the error rates which could be category specific (e.g., misclassifying category 1 as category 2 instead of category 3).

In Psychology and Cognitive Science, the Eighty Five Percent Rule accords with the informal intuition of many experimentalists that participant engagement is often maximized when tasks are neither too easy nor too hard. Indeed it is notable that staircasing procedures (that aim to titrate task difficulty so that error rate is fixed during learning) are commonly designed to produce about 80–85% accuracy 17 . Similarly, when given a free choice about the difficulty of task they can perform, participants will spontaneously choose tasks of intermediate difficulty levels as they learn 23 . Despite the prevalence of this intuition, to the best of our knowledge no formal theoretical work has addressed the effect of training accuracy on learning, a test of which is an important direction for future work.

More generally, our work closely relates to the Region of Proximal Learning and Desirable Difficulty frameworks in education 24,25,26 and Curriculum Learning and Self-Paced Learning 7,8 in computer science. These related, but distinct, frameworks propose that people and machines should learn best when training tasks involve just the right amount of difficulty. In the Desirable Difficulties framework, the difficulty in the task must be of a ‘desirable’ kind, such as spacing practice over time, that promotes learning as opposed to an undesirable kind that does not. In the Region of Proximal Learning framework, which builds on early work by Piaget 27 and Vygotsky 28 , this optimal difficulty is in a region of difficulty just beyond the person’s current ability. Curriculum and Self-Paced Learning in computer science build on similar intuitions, that machines should learn best when training examples are presented in order from easy to hard. In practice, the optimal difficulty in all of these domains is determined empirically and is often dependent on many factors 29 . In this context, our work offers a way of deriving the desired difficulty and the region of proximal learning in the special case of binary classification tasks for which stochastic gradient-descent learning rules apply. As such our work represents the first step towards a more mathematical instantiation of these theories, although it remains to be generalized to a broader class of circumstances, such as multi-choice tasks and different learning algorithms.

With regard to different learning algorithms, it is important to note that not all models will exhibit a sweet spot of difficulty for learning. As an example, consider how a Bayesian learner with a perfect memory would infer parameters ϕ by computing the posterior distribution given past stimuli, x1:t, and labels, y1:t,

where the last line holds when the label depends only on the current stimulus. Clearly this posterior distribution over parameters is independent of the ordering of the trials meaning that a Bayesian learner (with perfect memory) would learn equally well if hard or easy examples are presented first. This is not to say that Bayesian learners cannot benefit from carefully constructed training sets, but that for a given set of training items the order of presentation has no bearing on what is ultimately learned. This contrasts markedly with gradient-based algorithms, many of which try to approximate the maximum a posteriori solution of a Bayesian model, whose training is order dependent and whose learning is optimized with ER/∂β.

Finally, we note that our analysis for maximizing the gradient, ER/∂β, not only applies to learning but to any process that affects the precision of neural representations, such as attention, engagement, or more generally cognitive control 30,31 . For example, attention is known to improve the precision with which sensory stimuli are represented in the brain, e.g., ref. 32 . If exerting control leads to a change in precision of δβ, then the change in error rate associated with exerting this control is

This predicts that the benefits of engaging cognitive control should be maximized when ER/∂β is maximized, that is at ER * . More generally this relates to the Expected Value of Control theory 30,31,33 which suggests that the learning gradient, ER/∂β, is monitored by control-related areas of the brain such as anterior cingulate cortex.

Along similar lines, our work points to a mathematical theory of the state of ‘Flow’ 34 . This state, ‘in which an individual is completely immersed in an activity without reflective self-consciousness but with a deep sense of control’ [ref. 35 , p. 1], is thought to occur most often when the demands of the task are well matched to the skills of the participant. This idea of balance between skill and challenge was captured originally with a simple conceptual diagram (Fig. 5) with two other states: ‘anxiety’ when challenge exceeds skill and ‘boredom’ when skill exceeds challenge. These three qualitatively different regions (flow, anxiety, and boredom) arise naturally in our model. Identifying the precision, β, with the level of skill and the level challenge with the inverse of true decision variable, 1/Δ, we see that when challenge equals skill, flow is associated with a high learning rate and accuracy, anxiety with low learning rate and accuracy and boredom with high accuracy but low learning rate (Fig. 5b, c). Intriguingly, recent work by Vuorre and Metcalfe, has found that subjective feelings of Flow peaks on tasks that are subjectively rated as being of intermediate difficulty 36 . In addition work on learning to control brain computer interfaces finds that subjective, self-reported measures of ‘optimal difficulty’, peak at a difficulty associated with maximal learning, and not at a difficulty associated with optimal decoding of neural activity 37 . Going forward, it will be interesting to test whether these subjective measures of engagement peak at the point of maximal learning gradient, which for binary classification tasks is 85%.

Proposed relationship between the Eighty Five Percent Rule and Flow. a Original model of flow as a state that is achieved when skill and challenge are well balanced. Normalized learning rate, ER/∂β, b and accuracy c as a function of skill and challenge suggests that flow corresponds to high learning and accuracy, boredom corresponds to low learning and high accuracy, while anxiety is associated with low learning and low accuracy


Anxiety, Inhibition, Efficiency, and Effectiveness

Effects of anxiety on the antisaccade task were assessed. Performance effectiveness on this task (indexed by error rate) reflects a conflict between volitional and reflexive responses resolved by inhibitory processes (Hutton, S. B., & Ettinger, U. (2006). The antisaccade task as a research tool in psychopathology: A critical review. Psychophysiology, 43, 302–313). However, latency of the first correct saccade reflects processing efficiency (relationship between performance effectiveness and use of resources). In two experiments, high-anxious participants had longer correct antisaccade latencies than low-anxious participants and this effect was greater with threatening cues than positive or neutral ones. The high- and low-anxious groups did not differ in terms of error rate in the antisaccade task. No group differences were found in terms of latency or error rate in the prosaccade task. These results indicate that anxiety affects performance efficiency but not performance effectiveness. The findings are interpreted within the context of attentional control theory (Eysenck, M. W., Derakshan, N., Santos, R., & Calvo, M. G. (2007). Anxiety and cognitive performance: Attentional control theory. Emotion, 7 (2), 336–353).

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Pros and cons of the learning curve theory

Pro of the learning curve theory

Using a learning curve can help a business to improve the performance and productivity of their workforce and reduce costs.

When used to track or predict performance, it can provide psychological motivation and strategic planning:

  • Performance improvement cannot happen on its own and is connected with learning. By incorporating a learning culture within the organization where employees are encouraged and supported to keep learning, performance levels can be expected to increase on the curve.
  • The rate of learning is also considered to be consistent enough that trends can be established using the learning curve, enabling better forecasting and business decisions.

Cons of the learning curve theory

As a con, a learning curve is very dependent on assumptions made about performance. As mentioned earlier, many variables can impact learning and future performance:


Training with high perceptual difficulty improves the capacity and fidelity of internal representation in VWM

It has been shown that the capacity of visual working memory (VWM) is a strong predictor of individual intelligence, and researchers have developed various training protocols to improve VWM capacity. However, it seems that whether the fidelity of internal representation in VWM can also be improved by training is largely overlooked in the past literature. Here, we introduced a new training approach that involved increasing the perceptual difficulty of training materials to enhance VWM, and both memory capacity and the fidelity of representation were examined to assess the training efficacy. Participants with normal vision and cognitive abilities received 3-week training on VWM using a change detection task, and the results showed that both the capacity and the fidelity of memory representations were improved for training with perceptually difficult stimuli, while only the fidelity was improved for training with perceptually normal stimuli. In addition, we found that the training effects on memory precision may be subject to capacity constraints. We suggest that long-term adaptive training with perceptually difficult stimuli may facilitate encoding efficiency through familiarizing trainees with an increased baseline of cognitive workload during the encoding process. The present study offers clear evidence that training with high perceptual difficulty is more effective and the improvements in VWM are more stable than training with perceptually normal materials, and the simple manipulation on training stimuli indicates that the method can be generalized to a wider range of training situations and populations.

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Results

Both discriminability and mean RT improved with practice (Fig. 3) and showed highly significant linear and quadratic trends (Table 1). The d’ profiles for easy and difficult discriminations were approximately proportional to each other ( ( d_< m>^prime approx kd_< m>^prime ) with k = 1.65 ± 0.13), in agreement with published data (Petrov et al., 2005). The learning effects were partially specific to the trained reference direction, although the degree of specificity differed significantly for the two dependent measures. The specificity index was Footnote 2 SI = .60 ± .10 for d’ and .37 ± .08 for the mean RT.

Learning profiles for the group-averaged discriminability (a) and mean response times (b) in the raw data, and for various parameters of the diffusion model (cf). The observers practiced motion-direction discrimination for eight blocks (black symbols) and then were tested on the same task with motion in the orthogonal direction (open symbols). The error bars are 90% within-subjects confidence intervals. Shaded areas mark two additional sessions of motion aftereffect measurements.

The DM achieved good fits, evident in the quantile probability plots in Fig. 4 and the scatterplots in Fig. 5. The former show the proportions of correct and error responses (on the x-axis) and the corresponding RT distributions (summarized by the .1, .3, .5, .7, and .9 quantiles on the y-axis). The model (circles) tracks the data (×’s) well. Footnote 3 The scatterplots show that the model can reconstruct the data for each individual on each block to a good approximation. The quality of the fit, coupled with past research (Ratcliff & McKoon, 2008) validating the DM in conditions similar to ours, suggests that the DM parameters offer a concise characterization of the underlying cognitive processes.

Quantile probability plots illustrating the wealth of data and the quality of the fit. Each panel has 22 empirical degrees of freedom: the proportions of errors and correct responses for the easy and difficult discriminations (plotted on the x-axis) and the .1, .3, .5, .7, and .9 quantiles of the corresponding response time distributions (stacked vertically on the y-axis). For example, the x-coordinate of the leftmost, bottommost data point in the top panel indicates the initial error rate (.18) for easy stimuli. The y-coordinate indicates the leading edge (.1 quantile ≈ 530 ms) of the corresponding RT distribution. After 4 days of training (middle panel), the performance improves on both measures (.08 rate and 480 ms, respectively). This illustration is based on group-averaged data the analyses in the text (and the predictions in Fig. 5) are based on model fits to individual data

Scatterplots illustrating the quality of the fit to individual data. The diffusion model was fit separately in each block (297 fits = 27 observers × 11 blocks). Each panel contains 594 points (= 297 × 2 difficulty levels). RT, response time

There were statistically significant learning effects for all DM parameters except the starting point variability s z (Table 1). The twofold improvement in drift rate (Fig. 3c) indicates that the quality of the sensory input to the decision process increases with practice. The learning index for the drift rate v (LI = 0.99 ± 0.23) was significantly Footnote 4 higher than the d’ learning index (.55 ± .08). This is because v reflects learning in both accuracy and speed. The improvement was largely (but not entirely) specific to the trained reference direction (SI = .68 ± .09).

The parameters describing the distribution of nondecision times across trials also improved significantly. The mean nondecision time T er decreased by 20% on average (Fig. 3d). The specificity index for T er (.22 ± .10) was significantly Footnote 5 lower than that for the mean overall RT (.37 ± .08). This is because the improvement in overall RT stems in part from the stimulus-specific increase in drift rate.

The nondecision variability s t is of particular interest. As predicted by the synchronization hypothesis, it was high at first (283 ms during Block 1, Fig. 3f) and decreased steeply to 120 ms by the end of training. Moreover, the improvement transferred fully to the orthogonal direction of motion (SI = .00 ± .08).

There was a small but statistically significant decreasing linear trend in the boundary separation parameter a (Table 1). This suggests a slight adjustment in the speed–accuracy trade-off. The drift-rate increase apparently offset this adjustment and prevented a drop in accuracy. Finally, there was a marginally significant decrease in the across-trial variability in drift rate (η) but no significant changes in the variability in starting point (s z). See the online supplement for details.


Results

Categorization

To ensure that the counterbalanced variable of set (see Materials and Methods) did not have a significant effect on accuracy or reaction time and to determine whether there was any interaction of set with stimulus type, a repeated-measures ANOVA was performed on the control data only. The ANOVA included a within-subject variable of stimulus (faces vs scenes) and a between-subject variable of set (1 vs 2). Critically, there were no significant main effects of either stimulus type or set, nor was there any interaction. Accordingly, data from the two sets were combined for subsequent analysis.

Figure 4 (a, accuracy b, reaction time) shows the data from both participant groups split according to stimulus type. Although performance on the faces categorization task in the two groups was relatively similar, for both accuracy and reaction time, the patients correctly categorized fewer scenes than controls (67 vs 88%) and took considerably longer (>1 s) when doing so.

Scores on the faces and scenes categorization tasks for the controls (white bars) and hippocampal patients (gray bars). a, Percentage correct b, reaction time (milliseconds). ∗p < 0.05, significant difference between the two groups.

A repeated-measures ANOVA on the accuracy data, with a within-subject variable of stimulus and a between-subject variable of group (control vs hippocampal), confirmed these initial conclusions. Significant main effects of stimulus (F(1,13) = 6.3 p < 0.03) and group (F(1,13) = 9.8 p < 0.01) were evident, as well as a significant interaction between the two (F(1,13) = 7.9 p < 0.02). Additional univariate ANOVAs revealed that there was a significant group effect, with the hippocampal patients categorizing less accurately than controls, for the virtual reality scenes (F(1,13) = 18.8 p < 0.001) but not for faces.

The similar ANOVA investigating reaction time revealed only an interaction between stimulus and group (F(1,13) = 11.8 p < 0.004). Univariate ANOVAs confirmed that the hippocampal patients showed a trend toward slower categorization of virtual reality scenes as predicted (F(1,13) = 4.2 p < 0.06) but not for faces.

Discrimination

To ensure that preexposure to the faces and virtual reality scenes actually resulted in perceptual learning in healthy controls, both the accuracy and reaction time data were analyzed in the controls only. Repeated-measures ANOVAs with within-subject variables of stimulus (faces vs scenes) and familiarity (familiar vs novel) and a between-subject variable of set (1 vs 2) revealed a main effect of familiarity only (accuracy, F(1,11) = 10.7, p < 0.007 reaction time, F(1,11) = 6.0, p < 0.04). Categorization of both faces and scenes before undertaking the discrimination task with these stimuli, therefore, resulted in improved accuracy and faster reaction times in control participants for both types of complex stimuli. No other main effects or interactions reached significance. Consequently, as for the categorization task, data from both sets were combined for each stimulus type in additional analyses.

Because perceptual learning was evident for both stimulus types in healthy controls, difference scores (calculated for each subject for both scenes and faces by subtracting the score for novel stimuli from that obtained for familiar items) were used to contrast the level of perceptual learning attained by the hippocampal patients with that of the controls. These difference scores are displayed in Figure 5 (a, accuracy b, reaction times), in which a positive score for accuracy and a negative score for reaction time is evidence of perceptual learning. What is immediately obvious from the two figures is that, although the hippocampal group showed some improvement in accuracy for both faces and scenes, different profiles were evident on the reaction time measure, with a speeding of response for faces but a striking slowing for scenes.

Difference scores (novel score subtracted from familiar score) on the faces and scenes same–different discrimination task for controls (white bars) and hippocampal patients (gray bars). a, Percentage correct b, reaction time (milliseconds) c, inverse efficiency measure. ∗p < 0.05, significant difference between the two groups.

To investigate these findings, repeated-measures ANOVAs were used, with a within-subject variable of stimulus and a between-subjects variable of group, using the difference score data. Although no significant main effects, or an interaction, were revealed in the accuracy ANOVA, a significant interaction between stimulus and group was found for reaction time (F(1,13) = 7.0 p < 0.02). Univariate ANOVAs confirmed that the hippocampal group were performing significantly differently from their controls in the scenes condition (F(1,13) = 10.5 p < 0.006) but not in the faces condition.

These analyses suggest that perceptual learning of scenes is not normal in individuals with hippocampal damage: although the patients accurately discriminated more familiar than novel scenes, they showed a striking increase in reaction time after previous experience with the spatial stimuli. This is a reverse of the improved speed of responding typically seen in perceptual learning and demonstrated by our healthy controls for both faces and scenes. That is to say that the patients appear to be demonstrating a “speed–accuracy trade-off.” To compensate for this, the data can be analyzed using an inverse efficiency measure, which combines reaction time and accuracy (Townsend and Ashby, 1978, 1983). Inverse efficiency was therefore calculated separately for each of the novel and familiar stimulus conditions for each subject and is defined as mean reaction time divided by the proportion of trials correct. These data are displayed as difference scores in Figure 5c, in which more negative scores are evidence of greater perceptual learning.

Figure 5c confirms that the patients and controls show a different pattern of performance on scenes but not on faces. Although both groups show a negative inverse efficiency change for faces (slightly numerically greater in the case of the hippocampal group), the patients showed a large and positive inverse efficiency change for the virtual reality scenes compared with a negative value in the control population. Statistical analyses confirmed a significant main effect of group (F(1,13) = 5.0 p < 0.05), which interacted with stimulus type (F(1,13) = 5.0 p < 0.05). Univariate ANOVAs revealed that the interaction could be accounted for by a difference between the groups for scenes, (F(1,13) = 9.0 p < 0.01) but not for faces.


EXPERIMENT 1

There is conflicting evidence on the extent to which regions in PPC are driven by the number of spatial locations versus by the number of discrete objects per se (Xu & Chun, 2006, Experiment 4 Xu & Chun, 2007 Cusack, 2005). The aim of this first experiment was to contrast the perception of two objects with one object, in the absence of a difference in the number of spatial locations, and to evaluate a stimulus type that might be developed in the following experiment to distinguish object representation, attention switching, and task difficulty. To do this, we exploited the phenomenon of transparent motion, which has been shown to be an effective substrate for the deployment of object-based attention (Rodriguez, Valdes-Sosa, & Freiwald, 2002 Valdes-Sosa, Cobo, & Pinilla, 1998) and allows matching of spatial location. On some trials, we presented one surface, and on other trials, two overlapping surfaces. The size and the position of the surfaces were the same across conditions.

Methods

Stimuli and Task

On each trial, a set of 1000 white dots was presented on a black background for 1.2 sec, viewed through a circular window of diameter 500 pixels (approximately 11° of visual angle). There were two conditions, schematically illustrated in Figure 1. In Condition 1, all of the dots would oscillate along a single axis chosen at a random angle. The temporal frequency of the oscillatory motion was 4 Hz, and the peak-to-peak amplitude 24 pixels. In Condition 2, half of the dots would move along one axis in a similar manner to Condition 1, and the other half of the dots would move along the orthogonal axis. In this transparent motion condition, two surfaces are clearly perceived, but they are entirely overlapping. The two conditions differ in the number of objects, but not in the number of spatial locations. After each 1.2-sec stimulus, there was a 1.5-sec intertrial interval. To ensure participant vigilance, on each trial, the participant was asked to press one of two buttons with the right hand to indicate whether one or two surfaces were perceived. The button mapping was counterbalanced across subjects. Two blocks of 192 trials were presented. One third of the trials were one surface (Condition 1), one third two surfaces (Condition 2), and one third null trials in which just a fixation cross was presented.

Schematic illustration of the transparent motion stimuli used in Experiment 1. The one-surface (A) and two-surface (B) conditions were distinguished by the oscillatory motion.

Schematic illustration of the transparent motion stimuli used in Experiment 1. The one-surface (A) and two-surface (B) conditions were distinguished by the oscillatory motion.

MRI Acquisition

Data were acquired from 16 subjects using a 3-T Bruker Medspec scanner with a head gradient set at the Wolfson Brain Imaging Centre, Cambridge, UK. Two 8′30″ blocks of 204 EPI acquisitions (TA = 1.1 sec TR = 2.5 sec) were acquired. The first eight scans were discarded to allow for T1 equilibrium. Each volume was 21 slices (Gaussian profile, 4 mm thickness, 1 mm gap) acquired in an ascending interleaved order each of matrix size 64 × 64, giving a resolution of 3.75 × 3.75 mm. The TE was 30 msec and flip angle was 65°. We also acquired fieldmaps using a phase contrast sequence (complex subtraction of a pair GE acquisitions, TE = 7 and 16.1 msec, matrix size 64 × 256 × 64, resolution 4 × 1 × 4 mm), and a whole-brain T1-weighted structural using an SPGR sequence giving approximately 1 mm 3 resolution.

Analysis

The data were analyzed using SPM2 with the automatic analysis (aa) library (http://imaging.mrc-cbu.cam.ac.uk/imaging/AutomaticAnalysisIntroduction) for scripting. Sinc interpolation through time was used to correct for the acquisition of different brain slices at different times (SPM's slice timing correction). Bulk motion of the head through the time series was then estimated. The pattern of inhomogeneity in the magnetic field measured using a fieldmap was used to correct for distortion in the EPIs (Cusack, Brett, & Osswald, 2003). The EPI mean was then coregistered with the structural using a mutual information cost function. Nonlinear warping to MNI space was then accomplished using SPM normalization on the structural image. A single spatial reslicing stage with sinc interpolation applied the transformations of motion correction, undistortion, and normalization to the EPI images. The normalized images were smoothed using a Gaussian kernel of FWHM 10 mm.

Each subject's data were analyzed using a multiple regression model as specified by an SPM design matrix. The model comprised three event-related columns (predictors): one for the fixation trials, one for the trials with a single surface, and one for the trials with two surfaces. Each event began at the onset of the stimuli and lasted for a single scan (2.5 sec). Boxcar functions representing the time course of these events were convolved with the canonical hemodynamic response to form the predictors for the BOLD fMRI data at each voxel. The six parameters derived from motion correction were also included as regressors to partial out the first-order effects of distortion during motion.

Estimation (fitting) of the regression model gave for each voxel a regression coefficient (β) for each of the predictors. The contributions of the events in these predictors to the BOLD signal were then probed with two orthogonal contrasts. One was the main effect of the stimulus and task (βone surface + βtwo surfaces − 2βfixation), and the other the effect of the number of surfaces (βtwo surfaces − βone surface). For each contrast at the first level, group random-effects analyses were then calculated using parametric statistics and reported at a corrected (FDR p < .05) threshold. To visualize the relative strength of each of the contrasts, we calculated the ratio of the t values of the two contrasts (e.g., tnumber of surfaces/tmain effect of stimulus task) for the voxels that survived corrected thresholds in either contrast.

To quantify the effects of task performance and of the number of surfaces across different parietal areas, we conducted a region-of-interest (ROI) analysis. All ROIs were spheres of radius 1 cm, constructed using MarsBar for SPM (http://marsbar.sourceforge.net). Coordinates are given in MNI152 space, converted from Talairach space where necessary using tal2mni http://imaging.mrc-cbu.cam.ac.uk/downloads/MNI2tal/tal2mni.m). Two-tailed statistics were used to calculate p values in t tests. In none of the experiments do we have lateralized stimuli, and we did not expect (or see) lateralized responses, thus all ROI analyses are collapsed across hemispheres. As summarized in the Introduction, a region close to the parietal/occipital boundary in the inferior IPS has been implicated in object representations (Xu, 2007 Xu & Chun, 2006, 2007 Cusack, 2005). This region has also been found to topographically map attention in a paradigm similar to retinotopic mapping, but with attention shifted around the display rather than with a changing stimulus (Silver et al., 2005). To test whether regions previously shown to be topographic are activated to a greater degree by multiple objects even in the absence of any manipulation of the spatial distribution, the peak coordinates in the inferior IPS from Silver et al. (2005) were selected as the center of two ROIs (IPS1: ±23, −80, 38).

To characterize activity in more general, non-task-specific, parietal regions (the MD network), we extended the meta-analysis of Duncan and Owen (2000) to the parietal lobe and used a kernel method to summarize the peak activation coordinates. The peaks of the activations from the studies described by Duncan and Owen appeared symmetric, and so those in the left hemisphere were reflected onto the right. A single point was placed at each coordinate, and the resulting image was smoothed (15 mm FWHM) and then thresholded at 3.5 times the height of the smoothed peak from a single point. The final thresholded regions were then mirrored onto both hemispheres. In addition to the lateral and medial frontal regions reported by Duncan and Owen, inclusion of parietal activations revealed clear foci around the IPS at (±37, −53, 40). As shown in Figure 2, these are more lateral and anterior to the inferior IPS ones, and were used to define another pair of ROIs.

The ROIs from previous studies used to quantify responses. Xu-infIPS, Xu-supIPS, and Xu-LOC are inferior and superior intraparietal sulcus and lateral occipital regions taken from Xu and Chun (2006) MD-IPS and MD-IFS are intraparietal sulcus and inferior frontal sulcus regions in MD network in an extension of the study by Duncan and Owen (2000) Silver-IPS1 is IPS region taken from Silver et al. (2005).

The ROIs from previous studies used to quantify responses. Xu-infIPS, Xu-supIPS, and Xu-LOC are inferior and superior intraparietal sulcus and lateral occipital regions taken from Xu and Chun (2006) MD-IPS and MD-IFS are intraparietal sulcus and inferior frontal sulcus regions in MD network in an extension of the study by Duncan and Owen (2000) Silver-IPS1 is IPS region taken from Silver et al. (2005).

In addition to these key parietal ROIs, we also investigated the response in visual regions (the lateral occipital complex [LOC] taken from Xu & Chun, 2006: −44, −71, 5 and 42, −69, 0) and frontal executive control regions taken from the MD kernel analysis (dorsolateral prefrontal cortex [DLPFC]: ±42, 24, 25). Finally, for direct comparison with Xu and Chun's (2006) studies, we summarized the response in their “inferior IPS” region on the occipital/parietal boundary, and close to Silver's IPS1 (−21, −89, 24 and 26, −85, 28) and the “superior IPS” regions, which are closest to the MD regions (−21, −70, 42 and 23, −56, 46).

Results

The task was easy and performance was excellent (mean trials correct was 97% with a standard error across subjects of 0.7%). Neuroimaging revealed that presentation of the stimulus and performance of the task, when contrasted with fixation, recruited a broad range of regions, including occipital visual areas, left motor cortex, and regions in the fronto-parietal MD network (Figure 3A). The contrast of two surfaces minus one surface revealed several regions in common with this (Figure 3B). Both contrasts activated regions in the posterior parietal lobe to some extent. To visualize the relative strength of the response to the contrast between two surfaces and one surface and the contrast of task versus fixation, we calculated the ratio of the t values of these two contrasts (Figure 3C) for all voxels that were significant in either of the whole-brain corrected contrasts.

Whole-brain results of Experiment 1. The top two panels show whole-brain-corrected SPM-T maps, and the lower panel, their ratio for all voxels where either contrast was significant [warm colors correspond to (B) > (A), cool colors to (A) > (B)].

Whole-brain results of Experiment 1. The top two panels show whole-brain-corrected SPM-T maps, and the lower panel, their ratio for all voxels where either contrast was significant [warm colors correspond to (B) > (A), cool colors to (A) > (B)].

The response in the parietal lobe and other regions was quantified using ROI analyses (Figure 4A). There was a significant effect of the number of surfaces in the inferior IPS but not in the MD-IPS [Silver-IPS1: t(15) = 2.70, p < .02 MD: t(15) = 0.80, ns]. The opposite pattern was seen for the general task demand, with no evidence to reject the null hypothesis in the inferior IPS, but a significant effect in MD-IPS [Silver-IPS1: t(15) = 1.42, ns MD-IPS: t(15) = 3.14, p < .01]. A repeated-measures ANOVA confirmed this Task × Region interaction [F(1, 15) = 8.23, p < .02].

Peak percent signal change in an inferior IPS region (Silver-IPS1) and a more anterior, superior one (MD-IPS), for Experiment 1 (A), Experiment 2B (B), Experiment 3 (C), and Experiment 4 (D). *p < .05 **p < .01, and ***p < .001. The annotation above the horizontal line at the top of the graph indicates the significance of the region by condition interaction (see the text for a description of the contrasts).

Peak percent signal change in an inferior IPS region (Silver-IPS1) and a more anterior, superior one (MD-IPS), for Experiment 1 (A), Experiment 2B (B), Experiment 3 (C), and Experiment 4 (D). *p < .05 **p < .01, and ***p < .001. The annotation above the horizontal line at the top of the graph indicates the significance of the region by condition interaction (see the text for a description of the contrasts).

As shown in Figure 5A and B, the LOC and inferior IPS responded during performance of the task [t(15) = 7.82, p < .001 and t(15) = 5.49, p < .001, respectively] and showed greater recruitment by two surfaces than one [t(15) = 5.09, p < .001 t(15) = 4.47, p < .001]. In contrast, DLPFC was not significantly activated by task [t(15) = 1.65, ns] or by the number of surfaces [t(15) = 0.32, ns]. Finally, the superior IPS region showed a pattern a little like MD-IPS, being significantly activated by performance of any task [t(15) = 2.89, p < .02] but not by the number of surfaces [t(15) = 0.72, ns].

Peak percent signal change across three regions from Xu and Chun (2006) and one region from the MD network (abbreviations as in Figure 2), in a set of contrasts involved in general performance of a task or attention switching (A), and revealing the effect of multiple objects (B).

Peak percent signal change across three regions from Xu and Chun (2006) and one region from the MD network (abbreviations as in Figure 2), in a set of contrasts involved in general performance of a task or attention switching (A), and revealing the effect of multiple objects (B).

Discussion

Regions in PPC responded differentially. The more inferior IPS region (Silver-IPS1) showed selectivity for two surfaces rather than one, whereas the MD-IPS region did not. Conversely, the MD-IPS region responded to the performance of the task in general more strongly than the inferior region. This is consistent with the presence of object-related processing in the inferior IPS (Xu & Chun, 2006 Cusack, 2005), and distinct general processing resources in the MD-IPS (Duncan, 2006). That the object effect was seen even though the two surfaces did not differ in spatial location shows that a difference in the number of locations or spatial extent is not required, as found in the auditory study of Cusack (2005). This is not incompatible with a topographic representation in these regions: They might be topographically mapped (cf. Swisher, Halko, Merabet, McMains, & Somers, 2007 Silver et al., 2005), but over and above this, respond with the number of objects in the display.

Note that this effect of object representation in the absence of spatial differences is at odds with the conclusions drawn from Xu and Chun's (2006) Experiment 4, which used memory for sequentially presented shapes at the same location or at different locations, and concluded that objects at different spatial locations are required to modulate the inferior IPS. However, it is consistent with Xu and Chun (2007), who found that differences in grouping modulate inferior IPS activity without differences in the number of relevant spatial locations. At this stage, it is not clear what the critical aspect is of the difference between Xu and Chun's (2006) Experiment 4 and our experiment.

There was a low-level visual difference in the degree of motion coherence between the two and one surface conditions, which probably directly contributed to the contrast in visual regions. It is also possible that some of these differences in visual regions may have been mediated by the number of objects. It has been shown before that the response in MT can be modified by the number of objects in the presence of only small visual differences (Stoner & Albright, 1992). Conversely, it is also possible, although less likely, that regions in the parietal lobe were modulated directly by the visual differences (see General Discussion also discussed in this section is the possible role of eye movements).


Speed-Accuracy Trade-Off

Renée A. Duckworth , . Alexander V. Badyaev , in Advances in the Study of Behavior , 2018

2.2.1 Speed–Accuracy Trade-off

In the speed–accuracy trade-off, decisions are made slowly with high accuracy or fast with high error rate ( Chittka, Skorupski, & Raine, 2009 ). The neurobiological basis of this trade-off is well characterized. In both the prefrontal cortex and subcortical areas of the brain, neurons associated with different perceptual choices gradually increase their firing rate as they integrate inputs from sensory neurons ( Gold & Shadlen, 2007 ). A decision is made when the firing rate of the neurons associated with a particular choice exceeds a critical threshold and individuals told to prioritize speed in a cognitive task showed heightened baseline activation of brain areas involved with decision-making allowing them to reach the decision threshold faster ( Bogacz, Wagenmakers, Forstmann, & Nieuwenhuis, 2009 ). Yet, such flexibility in decision-making processes varies across individuals. Studies have found distinct patterns of brain activity and connectivity among individuals that preferentially prioritize speed and among individuals that vary in their ability to flexibly adjust their level of caution (sometimes prioritizing speed, sometimes accuracy Forstmann et al., 2010 Perri, Berchicci, Spinelli, & Di Russo, 2014 ). In particular, individuals who are better able to flexibly adjust their level of caution have stronger structural connections between the supplementary motor area of the brain and the striatum, a subcortical part of the forebrain and a critical component of the reward system ( Forstmann et al., 2010 ). Moreover, individuals that naturally prioritize speed had higher baseline activation of supplementary motor areas and individuals that naturally prioritize accuracy had higher baseline activity of areas in the prefrontal cortex ( Perri et al., 2014 ). Thus, individual variation in speed vs accuracy of decision-making appears to reflect a trade-off between a greater baseline activity of areas associated with cognitive control (that slow down decision-making processes but increase their accuracy) and greater baseline activity of motor and subcortical areas (that enhance the speed of an action at the expense of accuracy). Finally, variation in how individuals deal with this trade-off has been shown to relate to a variety of personality dimensions such as risk sensitivity ( Nagengast, Braun, & Wolpert, 2011 ), agreeableness ( Bresin, Hilmert, Wilkowski, & Robinson, 2012 ) and neuroticism ( Socan & Bucik, 1998 ) in humans, and alternative proactive and reactive coping styles in other animals ( Sih & Del Giudice, 2012 ).


Thorndike&rsquos Trial and Error Theory | Learning | Psychology

In this article we will discuss about:- 1. Meaning of Thorndike’s Trial and Error Theory 2. Experimental Evidences of Thorndike’s Trial and Error Theory 3. Educational Implications 4. Some Objections.

Meaning of Thorndike’s Trial and Error Theory:

Edward Lee Thorndike (1874-1949) is generally considered to have been the foremost educational psychologist not only of the United States but of the world. He contributed to research and theory in the field of learning and genetic psychology, testing and social psychology, testing and social psychology.

Thorndike first stated the elements of his theory of learning in 1913 that connections are formed in the nervous system between stimuli and response. These connections formed are illustrated by the symbols S-R. Another word used to describe these connections is the word ‘bond’ and hence,’ this theory is sometimes called a ‘Bond Theory of learning’. Thorndike has written- “Learning is connecting. The mind is man’s connection system.”

According to Thorndike learning takes place by trial and error. Some people call it, “Learning by selection of the successful variant,” accordingly when no ready-made solution of a problem is available to the learner, he adopts the method of trial and error. He first, tries one solution. If it does not help him, he rejects it, then, he tries another and so on. In this way he eliminates errors or irrelevant responses which do not serve the purpose and finally discovers the correct solution.

Thus, in trial and error method, the learner makes random activities and finally reaches the goal accidently. Here, one thing should be remembered that in trial and error also, there are often systematic and relevant responses. Activities are not wholly random. All these activities, though apparently random are suggested to him by the situation and the learner proceeds on accordingly. The stages through which the learner has to pass are Goal, Block (hinderances), Random Movements or multiple response, chance success, selection and Fixation.

When and how the connection is accomplished was stated first in the following three laws:

1. Law or Readiness:

First primary law of learning, according to him, is the ‘Law or Readiness’ or the ‘Law of Action Tendency’, which means that learning takes place when an action tendency’ is aroused through preparatory adjustment, set or attitude. Readiness means a preparation for action. If one is not prepared to learn, learning cannot be automatically instilled in him, for example, unless the typist, in order to learn typing prepares himself to start, he would not make much progress in a lethargic and unprepared manner.

2. Law of Exercise:

The second law of learning is the ‘Law of Exercise’, which means that drill, or practice helps in increasing efficiency and durability of learning and according to Thorndike’s S-R Bond Theory, the connections are strengthened with trail or practice and the connections are weakened when trial or practice is discontinued.

The ‘law of exercise’, therefore, is also understood as the ‘law of use and disuse’ in which case connections or bonds made in the brain cortex are weakened or loosened. Many examples of this are found in case of human learning. Learning to drive a motor-car, typewriting, singing or memorizing a poem or a mathematical table, and music etc. need exercise and repetition of various movements and actions May times.

The third law is the ‘Law of Effect’, according to which the trial or steps leading to satisfaction stamps in the bond or connection. Satisfying states lead to consolidation and strengthening of the connection, whereas dis-satisfaction, annoyance or pain leads to the weakening or stamping out of the connections.

In fact, the ‘law or effect’ signifies that if the responses satisfy the subject, they are learnt and selected. While those which are not satisfying are eliminated. Teaching, therefore, must be pleasing. The educator must obey the tastes and interests of his pupils. In other words, greater the satisfaction stronger will be the motive to learn. Thus, intensity is an important condition of the ‘law of effect’.

Besides these three basic laws, Thorndike also refers to five sub-ordinate laws which further help to explain the learning process.

1. Law of Multiple-Response:

According to it the organism varies or changes its responses till an appropriate behaviour is hit upon. Without varying the responses, the correct response for the solution might never be elicited. If the individual wants to solve a puzzle, he is trying in different ways rather than mechanically persisting in the same way. Thorndike’s cat in the puzzle box moved about and tried many ways to come out till finally it hit the latch with her paw which opened the door and it jumped out.

2. The Law of Set or Attitude:

Learning is guided by a total set or attitude of the organism, which determines not only what the person will do but what will satisfy or annoy him. For instance, unless the cricketer sets himself to make a century, he will not be able to score more runs. A student, similarly, unless he sets to get first position and has the attitude of being at the top, would while away the time and would not learn much. Hence, learning is affected more in the individual if he is set to learn more or to excel.

3. Pre-Potency of Elements:

According to this law, the learner reacts selectively to the important or essential element in the situation and neglects the other features or elements which may be irrelevant or non-essential. The ability to deal with the essential or the relevant part of the situation makes analytical and insightful learning possible. In this law of pre-potency of elements, Thorndike is really anticipating insight in learning which was more emphasised by the Gestations.

4. Law of Response by Analogy:

According to this law, the individual makes use of old experiences or acquisitions while learning a new situation. There is a tendency to utilize common elements in the new situation as existed in a similar past situation. The learning of driving a car, for instance, is facilitated by the earlier acquired skill of driving a motor-cycle or even riding a bicycle, because the perspective or maintaining a balance and controlling the handle helps in steering the car.

5. The Law of Associative Shifting:

According to this law we may get any response, of which a learner is capable, associated with any other situation to which he is sensitive. Thorndike illustrated this by the act of teaching a cat to stand up at a command. A fish was dangled before the vat while he said ‘stand up’. After a number of trials by presenting the fish after uttering the command ‘stand up’, he later ousted the fish and the overall command of ‘stand up’ was found sufficient to evoke the response to the cat by standing up on her hind legs.

Experimental Evidences of Thorndike’s Trial and Error Theory:

Various experiments have been performed on men as well as animals to study this method. Thorndike made several experiments on rats and cats. Two important experiments are mentioned here.

Thorndike’s most widely quoted experiment was with the cat placed in a puzzle box. The hungry cat was put in the puzzle box and a fish, as an incentive, was put out-side the cage a little beyond its reach. The box was designed in such a way that the door of the cage can be released by some simple act like depressing a lever inside the cage.

At first, the cat made a great deal of varied attempts to reach the food in a trial and error fashion such as jumping up and down, clawing at the bars, scratching the cage, whaling around trying to push the bars, pawing and shaking movable parts of the cage etc., but all attempts proved to vain.

Ultimately by chance her paw fell on the loop of the rope and the door opened. The cat jumped out immediately and ate the fish. When next day, the cat was put in the box again, this time she took less time in coming out and in the subsequent trials the time decreased further so much so that the stage reached when the cat came out soon after being put inside by directly striking the latch with her paw without any random movement. This is how she learnt to reach its goal.

Expt. 2 (Experiment with Human Subjects):

Gopalaswamy demonstrated trial and error in human beings through Mirror-Drawing Experiment. This is a classical experiment in the psychology of learning. In this experiment the subject is asked to trace a star-shaped drawing, not looking at it directly, but as it is reflected in a mirror, the subject’s hand movements are visible in the mirror only and not directly. The experimenter observes the movements of the hands and thus, records the time of tracing in successive trials and the number of errors committed in each trial.

In first six trials the subject traces the star with the right hand and then in the next six trials he traces it by the left hand. Two graphs-the Time Curve and the Error Curve are then drawn, which show the general characteristics of trial and error learning. In the original experiment Gopalaswamy arranged his apparatus so that a record was automatically made of all the movements of the styles of the subject as it traced out the pattern. In this way the successive times of tracings and a record of errors was obtained.

Gopalaswamy analyzed the errors into two groups-lower level errors and higher level errors. Those errors which do not involve any noble process on the part of the subject in tracing the star are lower-level errors and those which involve higher process of mind on the perceptual and conceptual level are higher-level errors.

He discovered that improvement in the higher-level responses correlated highly with intelligence and that the improvement in the responses of the lower-level errors did not show much correlation with intelligence. This clears the respective share of trial and error and of higher learning.

For Fundulus fishes Thorndike got a glass tub with a dividing wall of glass in the middle. In the dividing wall there was a hole through which the fish could go from one part to another. By nature Fundulus fish like to remain in shade. The glass tub was filled with water and it was put under such a situation that half of its part remained under shade and the other half was in the sunshine. The fishes were kept in the sunny portion.

They began to try to coming over to the shady portion. By trying again and again the fishes succeeded in tracing the hole of the dividing wall and reached the shady portion one by one. But, at first the fishes took more time in reaching the shady portion, then in the second attempt they took less time and in the third attempt they took the least time. Trying it again finally a stage came when the fishes happened to come one after another in a row to the shady portion immediately in the very first attempt i.e., the number of errors of their wandering here and there amounted to a zero.

Educational Implications of Thorndike’s Trial and Error Theory:

Thorndike’s theory of Trial and Error and his three basic laws of learning have direct educational implications. The ‘Law of Readiness’ lays emphasis on motivation while the ‘Law of Exercise’ compels us to accept a well-known fact ‘Practice makes a man perfect’, and the third one i.e., ‘Law of Effect’ opens fairly a large scope to discuss the role of reward and punishment as an incentive in the child’s learning.

Actually, motivation and learning are inter-related concepts. No motivation No learning. Here we can remember a proverb, ‘the one man can take horse to the pool of water but twenty cannot make him drink’. This statement clearly shows the impact of motivation on learning. Clearly speaking motive is a force that compels an individual to act or to behave in a particular direction. And, hence the success of a teacher lies in motivating the roomfuls of energy. His prime duty is to produce ‘thirst’ (a motive to drink water) in the horses. Then and only then he may succeed in making the process of learning easier and interesting.

To quote with the experiment to Tolman and Honzik (1930) which they performed in rats will be of interest and situational here. In this experiment the rats were taught to follow a complex pattern of runs and turns through a maze to reach the food. The rats were divided in three groups. First group of rats was neither hungry nor given any food at the end or trial. The second group was hungry but was not given food. The third one was hungry and given food at the end of a trial.

It was concluded that only the third group learned appreciably i.e., the number of errors went on decreasing in each attempt. The logic is simple. To be motivated and unrewarded leaves to you only frustration instead a notable amount of learning. Also nor is it worthwhile to work for a prize you do not want. Thus, it is the motive that gives the reward its value and the satisfaction of reward that fixes the learning of which it is the effect.

Briefly speaking, without motivation or drive learning is impossible, as firstly, it prods the learner into action and secondly, it introduces light and shadow into an otherwise different field. So, teacher’s concern primarily shall be the motivating of goals and releasing tensions which signalise success. Above all he should have a psychological involvement in reaching and has to be charged with values and therefore, naturally motivated himself. The advice of an old principal of a school is very pertinent here.

“Teachers, you are going to be emulated in your talk and walk by your students, but a little less. If you run, your students will walk. If you walk, your students will stand. If you stand, your students will lie down. If you lie down, your students will sleep. And if you sleep in the class, your students will die”. But, one has to admit here that the organism’s level of performance can’t be beyond a physiological limit, whatever incentive we provide to him. For instance, higher bonus to factory workers, more praise to students may lead to a better performance, but no athlete can jump over the Chinese wall, whatever the intensity of motivation is provided.

Another significant aspect of this theory is that to master a complex situation or to elaborate task, practice is must. It is not possible to handle each difficult situation in a single trial, no matter what the degree of motivation or reward is. One cannot blame the entire constitution of India in one reading even if the reward is a crore of Rupees or the threat is to be shot dead otherwise. Each task initially seems to be difficult and fatiguing but as practice continues, it becomes smoother and requires less effort.

Finally, we say that habit or S-R is established. An expert driver, for instance, goes on driving, listening to the radio and taking to his friend sitting by. In the light of class room teaching blundering is a natural phenomenon associated with student learning. But, the teacher should not regard this as a symptom of inefficient teaching, because this is the way the pupils learn. He should not be at all worried when blundering appears.

Insights will emerge as the blundering progresses from simpler associations to higher units. There is not royal road to success. Kennedy-Fraser, the Psychologist concludes, “The teachers who are responsible for the beginning of any new subject should be the best available, since at the point, the pupils have no defensive system of properly formed habits to protect them from the evil effects of bad teaching.”

Actually, we learn by doing. The teachers’ duty should be to arrange situations in which the student has chance to discover for himself what is significant. The blundering must be directed and methods that are wholly futile must be eliminated. But at the same time the teacher must exercise, constant restraint in his supervision.

Further, both punishment and reward may play a significant role in the process of learning. But, experiments go to show that motivation is successfully handled when it is kept in the positive phase. Drastic forms of inhibition tend to spread their effects over the whole learning situation. Sometimes, the teachers impress upon the negative processes. The false response is effectively inhibited when the correct reaction is fixated and the emphasis should be on the latter process. The fixating rewards are most effective when they afford immediate and complete release.

A delay introduced between the successful performance and the releasing reward has a considerable effect on their rate of learning and co-ordination. In school, the satisfactions should be closely coupled with the activity itself otherwise the likelihood of permanent effects is small. Another aspect of motivating problem is simpler than the manipulations of tensions and releases and can be mastered by all. This is that the learner should be kept informed of his progress and promptly.

Finally, though the theory is not widely accepted for its educational significance, yet, there are certain subjects such as mathematics, tables of mathematics, memorising poetry, rules of grammar etc. in which learning by Trial and Error cannot be avoided. All reasoning subjects afford the greatest opportunity for the application of the Trial and Error method.

In Brief, the implications of the theory are:

1. According to his theory the task can be started from the easier aspect towards its difficult side. This approach will benefit the weaker and backward children.

2. A small child learns some skills through trial and error method only such as sitting, standing, walking, running etc. In teaching also the child rectifies the writing after committing mistakes.

3. In this theory more emphasis has been laid on motivation. Thus, before starting teaching in the classroom the students should be properly motivated.

4. Practice leads a man towards maturity. Practice is the main feature of trial and error method. Practice helps in reducing the errors committed by the child in learning any concept.

5. Habits are formed as a result of repetition. With the help of this theory the wrong habits of the children can be modified and the good habits strengthened.

6. The effects of rewards and punishment also affect the learning of the child. Thus, the theory lays emphasis on the use of reward and punishment in the class by the teacher.

7. The theory may be found quite helpful in changing the behaviour of the delinquent children. The teacher should cure such children making use of this theory.

8. With the help of this theory the teacher can control the negative emotions of the children such as anger, jealousy etc.

9. The teacher can improve his teaching methods making use of this theory. He must observe the effects of his teaching methods on the students and should not hesitate to make necessary changes in them, if required.

10. The theory pays more emphasis on oral drill work. Thus, a teacher should conduct oral drill of the taught contents. This helps in strengthening the learning more.

Some Objections to Thorndike’s Trial and Error Theory:

The theory has been criticised by various psychologists on the following grounds. Firstly, the theory is mechanical, for it leaves no room for an end or purpose in any sense whatsoever. On the contrary psychologist Mc Dougall maintained that even the behaviour of the amoeba or the paramecia consists in learning to face novel conditions to serve some unknown purpose Even repeated trials are of no avail if the tendency to learn is not there.

Again, if the tendency is there, even one trial may be fruitful. Mc Dougall and Woodworth insist on readiness for reaching a goal in learning and Lloyd Morgan lays stress on persistency with varied efforts till the goal of learning is achieved. The hungry cat confined in the puzzle-box with food in front of it goes on persistently trying various means until it gets out of it and has food. So, its trials are not blind and mechanical. In fact, they are guided by perceptual attention and feelings of pleasure and pain. Yet, Thorndike pays no attention to these higher order mental processes.

Secondly, in course or repeated trials the numbers of errors are not corrected of themselves or mechanically. The effects of Trial and Error depend to a great extent upon the psycho-physical state of the animal or man. In the absence of any purpose in view the animal is so puzzled, rather than enlightened by the errors committed that it goes on blindly repeating them without end.

Thirdly, Thorndike assumes that learning consists only in the association of several separate movements. But, learning is a whole process related to a whole situation. The hungry cat confined in a puzzle-box with food placed near it does not perceive the situation in a piece-meal fashion but as a whole of hunger food-puzzle box-confinement.

Finally, the laws of learning formulated by Thorndike appear to be unjustified. For instance, the ‘law of effect’ seems to be in consistent with his mechanical point of view. Satisfaction in or the sense of being rewarded by success and dissatisfaction in or the sense of being punished by failure seen to ascribe higher mental processes to animals like cats and rats than are psychologically ascribable to them. Or, it violates Lloyd Morgans’s law.

Similarly, the ‘Law of Exercise’ has been severely criticised on the grounds that it does not regard other factors like motives, interests, special training etc. Mechanical repetition without motive, interest, significance or understanding does not make anyone learn anything and remember it. One rupee-currency note passes hundred times through the hand of a person, but hardly anyone is able to tell the size, the colour and other details of it.

A boy was asked to write hundred times ‘I have gone’ after school. He wrote it mechanically and correctly all the times. But, when he left the school in the absence of the teacher, he wrote “I have written,” ‘I have gone’ correctly one hundred times and since you are not here “I have went home”. After repeating one correct thing so many times he again committed the same mistake. This shows that repetition without motive, interest or understanding is of no avail.

Thus, learning by Trial and Error is not of very much use and should not be resorted to by the teacher as it lays a stress on cramming. Also, there is much wastage of time and energy by this method.


Discussion

In this article we considered the effect of training accuracy on learning in the case of binary classification tasks and stochastic gradient-descent-based learning rules. We found that the rate of learning is maximized when the difficulty of training is adjusted to keep the training accuracy at around 85%. We showed that training at the optimal accuracy proceeds exponentially faster than training at a fixed difficulty. Finally we demonstrated the efficacy of the Eighty Five Percent Rule in the case of artificial and biologically plausible neural networks.

Our results have implications for a number of fields. Perhaps most directly, our findings move towards a theory for identifying the optimal environmental settings in order to maximize the rate of gradient-based learning. Thus the Eighty Five Percent Rule should hold for a wide range of machine learning algorithms including multilayered feedforward and recurrent neural networks (e.g. including ‘deep learning’ networks using backpropagation 9 , reservoir computing networks 21,22 , as well as Perceptrons). Of course, in these more complex situations, our assumptions may not always be met. For example, as shown in the Methods, relaxing the assumption that the noise is Gaussian leads to changes in the optimal training accuracy: from 85% for Gaussian, to 82% for Laplacian noise, to 75% for Cauchy noise (Eq. (31) in the “Methods”).

More generally, extensions to this work should consider how batch-based training changes the optimal accuracy, and how the Eighty Five Percent Rule changes when there are more than two categories. In batch learning, the optimal difficulty to select for the examples in each batch will likely depend on the rate of learning relative to the size of the batch. If learning is slow, then selecting examples in a batch that satisfy the 85% rule may work, but if learning is fast, then mixing in more difficult examples may be best. For multiple categories, it is likely possible to perform similar analyses, although the mapping between decision variable and categories will be more complex as will be the error rates which could be category specific (e.g., misclassifying category 1 as category 2 instead of category 3).

In Psychology and Cognitive Science, the Eighty Five Percent Rule accords with the informal intuition of many experimentalists that participant engagement is often maximized when tasks are neither too easy nor too hard. Indeed it is notable that staircasing procedures (that aim to titrate task difficulty so that error rate is fixed during learning) are commonly designed to produce about 80–85% accuracy 17 . Similarly, when given a free choice about the difficulty of task they can perform, participants will spontaneously choose tasks of intermediate difficulty levels as they learn 23 . Despite the prevalence of this intuition, to the best of our knowledge no formal theoretical work has addressed the effect of training accuracy on learning, a test of which is an important direction for future work.

More generally, our work closely relates to the Region of Proximal Learning and Desirable Difficulty frameworks in education 24,25,26 and Curriculum Learning and Self-Paced Learning 7,8 in computer science. These related, but distinct, frameworks propose that people and machines should learn best when training tasks involve just the right amount of difficulty. In the Desirable Difficulties framework, the difficulty in the task must be of a ‘desirable’ kind, such as spacing practice over time, that promotes learning as opposed to an undesirable kind that does not. In the Region of Proximal Learning framework, which builds on early work by Piaget 27 and Vygotsky 28 , this optimal difficulty is in a region of difficulty just beyond the person’s current ability. Curriculum and Self-Paced Learning in computer science build on similar intuitions, that machines should learn best when training examples are presented in order from easy to hard. In practice, the optimal difficulty in all of these domains is determined empirically and is often dependent on many factors 29 . In this context, our work offers a way of deriving the desired difficulty and the region of proximal learning in the special case of binary classification tasks for which stochastic gradient-descent learning rules apply. As such our work represents the first step towards a more mathematical instantiation of these theories, although it remains to be generalized to a broader class of circumstances, such as multi-choice tasks and different learning algorithms.

With regard to different learning algorithms, it is important to note that not all models will exhibit a sweet spot of difficulty for learning. As an example, consider how a Bayesian learner with a perfect memory would infer parameters ϕ by computing the posterior distribution given past stimuli, x1:t, and labels, y1:t,

where the last line holds when the label depends only on the current stimulus. Clearly this posterior distribution over parameters is independent of the ordering of the trials meaning that a Bayesian learner (with perfect memory) would learn equally well if hard or easy examples are presented first. This is not to say that Bayesian learners cannot benefit from carefully constructed training sets, but that for a given set of training items the order of presentation has no bearing on what is ultimately learned. This contrasts markedly with gradient-based algorithms, many of which try to approximate the maximum a posteriori solution of a Bayesian model, whose training is order dependent and whose learning is optimized with ER/∂β.

Finally, we note that our analysis for maximizing the gradient, ER/∂β, not only applies to learning but to any process that affects the precision of neural representations, such as attention, engagement, or more generally cognitive control 30,31 . For example, attention is known to improve the precision with which sensory stimuli are represented in the brain, e.g., ref. 32 . If exerting control leads to a change in precision of δβ, then the change in error rate associated with exerting this control is

This predicts that the benefits of engaging cognitive control should be maximized when ER/∂β is maximized, that is at ER * . More generally this relates to the Expected Value of Control theory 30,31,33 which suggests that the learning gradient, ER/∂β, is monitored by control-related areas of the brain such as anterior cingulate cortex.

Along similar lines, our work points to a mathematical theory of the state of ‘Flow’ 34 . This state, ‘in which an individual is completely immersed in an activity without reflective self-consciousness but with a deep sense of control’ [ref. 35 , p. 1], is thought to occur most often when the demands of the task are well matched to the skills of the participant. This idea of balance between skill and challenge was captured originally with a simple conceptual diagram (Fig. 5) with two other states: ‘anxiety’ when challenge exceeds skill and ‘boredom’ when skill exceeds challenge. These three qualitatively different regions (flow, anxiety, and boredom) arise naturally in our model. Identifying the precision, β, with the level of skill and the level challenge with the inverse of true decision variable, 1/Δ, we see that when challenge equals skill, flow is associated with a high learning rate and accuracy, anxiety with low learning rate and accuracy and boredom with high accuracy but low learning rate (Fig. 5b, c). Intriguingly, recent work by Vuorre and Metcalfe, has found that subjective feelings of Flow peaks on tasks that are subjectively rated as being of intermediate difficulty 36 . In addition work on learning to control brain computer interfaces finds that subjective, self-reported measures of ‘optimal difficulty’, peak at a difficulty associated with maximal learning, and not at a difficulty associated with optimal decoding of neural activity 37 . Going forward, it will be interesting to test whether these subjective measures of engagement peak at the point of maximal learning gradient, which for binary classification tasks is 85%.

Proposed relationship between the Eighty Five Percent Rule and Flow. a Original model of flow as a state that is achieved when skill and challenge are well balanced. Normalized learning rate, ER/∂β, b and accuracy c as a function of skill and challenge suggests that flow corresponds to high learning and accuracy, boredom corresponds to low learning and high accuracy, while anxiety is associated with low learning and low accuracy


Anxiety, Inhibition, Efficiency, and Effectiveness

Effects of anxiety on the antisaccade task were assessed. Performance effectiveness on this task (indexed by error rate) reflects a conflict between volitional and reflexive responses resolved by inhibitory processes (Hutton, S. B., & Ettinger, U. (2006). The antisaccade task as a research tool in psychopathology: A critical review. Psychophysiology, 43, 302–313). However, latency of the first correct saccade reflects processing efficiency (relationship between performance effectiveness and use of resources). In two experiments, high-anxious participants had longer correct antisaccade latencies than low-anxious participants and this effect was greater with threatening cues than positive or neutral ones. The high- and low-anxious groups did not differ in terms of error rate in the antisaccade task. No group differences were found in terms of latency or error rate in the prosaccade task. These results indicate that anxiety affects performance efficiency but not performance effectiveness. The findings are interpreted within the context of attentional control theory (Eysenck, M. W., Derakshan, N., Santos, R., & Calvo, M. G. (2007). Anxiety and cognitive performance: Attentional control theory. Emotion, 7 (2), 336–353).

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Pros and cons of the learning curve theory

Pro of the learning curve theory

Using a learning curve can help a business to improve the performance and productivity of their workforce and reduce costs.

When used to track or predict performance, it can provide psychological motivation and strategic planning:

  • Performance improvement cannot happen on its own and is connected with learning. By incorporating a learning culture within the organization where employees are encouraged and supported to keep learning, performance levels can be expected to increase on the curve.
  • The rate of learning is also considered to be consistent enough that trends can be established using the learning curve, enabling better forecasting and business decisions.

Cons of the learning curve theory

As a con, a learning curve is very dependent on assumptions made about performance. As mentioned earlier, many variables can impact learning and future performance: