problem solving in experimental psychology

The Process of Problem Solving

• Editor's Choice
• Experimental Psychology
• Problem Solving

In a 2013 article published in the Journal of Cognitive Psychology , Ngar Yin Louis Lee (Chinese University of Hong Kong) and APS William James Fellow Philip N. Johnson-Laird (Princeton University) examined the ways people develop strategies to solve related problems. In a series of three experiments, the researchers asked participants to solve series of matchstick problems.

In matchstick problems, participants are presented with an array of joined squares. Each square in the array is comprised of separate pieces. Participants are asked to remove a certain number of pieces from the array while still maintaining a specific number of intact squares. Matchstick problems are considered to be fairly sophisticated, as there is generally more than one solution, several different tactics can be used to complete the task, and the types of tactics that are appropriate can change depending on the configuration of the array.

Louis Lee and Johnson-Laird began by examining what influences the tactics people use when they are first confronted with the matchstick problem. They found that initial problem-solving tactics were constrained by perceptual features of the array, with participants solving symmetrical problems and problems with salient solutions faster. Participants frequently used tactics that involved symmetry and salience even when other solutions that did not involve these features existed.

To examine how problem solving develops over time, the researchers had participants solve a series of matchstick problems while verbalizing their problem-solving thought process. The findings from this second experiment showed that people tend to go through two different stages when solving a series of problems.

People begin their problem-solving process in a generative manner during which they explore various tactics — some successful and some not. Then they use their experience to narrow down their choices of tactics, focusing on those that are the most successful. The point at which people begin to rely on this newfound tactical knowledge to create their strategic moves indicates a shift into a more evaluative stage of problem solving.

In the third and last experiment, participants completed a set of matchstick problems that could be solved using similar tactics and then solved several problems that required the use of novel tactics.  The researchers found that participants often had trouble leaving their set of successful tactics behind and shifting to new strategies.

From the three studies, the researchers concluded that when people tackle a problem, their initial moves may be constrained by perceptual components of the problem. As they try out different tactics, they hone in and settle on the ones that are most efficient; however, this deduced knowledge can in turn come to constrain players’ generation of moves — something that can make it difficult to switch to new tactics when required.

These findings help expand our understanding of the role of reasoning and deduction in problem solving and of the processes involved in the shift from less to more effective problem-solving strategies.

Reference Louis Lee, N. Y., Johnson-Laird, P. N. (2013). Strategic changes in problem solving. Journal of Cognitive Psychology, 25 , 165–173. doi: 10.1080/20445911.2012.719021

good work for other researcher

APS regularly opens certain online articles for discussion on our website. Effective February 2021, you must be a logged-in APS member to post comments. By posting a comment, you agree to our Community Guidelines and the display of your profile information, including your name and affiliation. Any opinions, findings, conclusions, or recommendations present in article comments are those of the writers and do not necessarily reflect the views of APS or the article’s author. For more information, please see our Community Guidelines .

Please login with your APS account to comment.

Careers Up Close: Joel Anderson on Gender and Sexual Prejudices, the Freedoms of Academic Research, and the Importance of Collaboration

Joel Anderson, a senior research fellow at both Australian Catholic University and La Trobe University, researches group processes, with a specific interest on prejudice, stigma, and stereotypes.

Experimental Methods Are Not Neutral Tools

Ana Sofia Morais and Ralph Hertwig explain how experimental psychologists have painted too negative a picture of human rationality, and how their pessimism is rooted in a seemingly mundane detail: methodological choices. 

APS Fellows Elected to SEP

In addition, an APS Rising Star receives the society’s Early Investigator Award.

Privacy Overview

PERSPECTIVE article

How to detect insight moments in problem solving experiments.

• School of Psychology, The University of Queensland, St. Lucia, QLD, Australia

Arguably, it is not possible to study insight moments during problem solving without being able to accurately detect when they occur ( Bowden and Jung-Beeman, 2007 ). Despite over a century of research on the insight moment, there is surprisingly little consensus on the best way to measure them in real-time experiments. There have also been no attempts to evaluate whether the different ways of measuring insight converge. Indeed, if it turns out that the popular measures of insight diverge , then this may indicate that researchers who have used one method may have been measuring a different phenomenon to those who have used another method. We compare the strengths and weaknesses of the two most commonly cited ways of measuring insight: The feelings-of-warmth measure adapted from Metcalfe and Wiebe (1987) , and the self-report measure adapted from Bowden and Jung-Beeman (2007) . We find little empirical agreement between the two measures, and conclude that the self-report measure of Aha! is superior both methodologically and theoretically, and provides a better representation of what is commonly regarded as insight. We go on to describe and recommend a novel visceral measure of insight using a dynamometer as described in Creswell et al. (2016) .

Introduction

Insight is a multifaceted construct, and is better understood as an umbrella term for more objective features such as: the suddenness and unexpectedness of a solution, a non-linearity in the problem solving process, and the phenomenology of an Aha! experience. A solution to a problem can be anywhere from purely insight (sudden and unexpected), to entirely analytic. When a problem is solved analytically, one proceeds through the problem step-by-step, while conscious of their progress toward a solution. Attempts have been made to understand insight as a feature of certain types of creative problems that elicit insights (e.g., Weisberg, 1996 ; Gilhooly and Murphy, 2005 ), but research shows that even so-called insight problems are often solved without insight, and can be solved through a variety of strategies ( Klein and Jarosz, 2011 ; Fleck and Weisberg, 2013 ; Danek et al., 2014 ). We agree with Danek et al. (2014) who point out that although it is well documented that some problems are more likely to be solved by insight than others, insight problems per se do not exist ( Bowden and Jung-Beeman, 2007 , were also clear in making this distinction). Therefore, a critical challenge for insight researchers is to identify when—case by case—an individual experiences an insight moment. The most popular methods are self-report, and the feelings-of-warmth (warmth) measure developed by Metcalfe (1986) and Metcalfe and Wiebe (1987) . We begin by introducing both measures and our predictions. We then test the two measures for convergent validity. In the discussion, we provide advice about the general usability and conceptual merit of each measure.

The Warmth Measure

During verbal problem solving Metcalfe (1986) asked participants to write down a number between 0 and 10 every 10 s (15 s in experiment two), where 0 is cold (far away from the solution) and 10 is hot, or certain that they had the solution. If a problem-solver exhibits gradual increases in warmth before solving the problem, then they were ostensibly aware of their progress on the problem and therefore the solution was found gradually, or step-by-step. If the problem-solver exhibits a sudden transition from a cold state to a solution, then it appears that the problem was solved through a more sudden and unexpected insight. One year later, Metcalfe and Wiebe (1987) showed that problems that had been previously categorized as insight problems showed more sudden transitions from cold states to solution states, whereas the previously categorized multi-step problems showed gradual warmth ratings preceding the solution. This contribution has had a long-standing impact on insight research and provided some of the first objective evidences that problem solving can occur in a way that resembles the insight construct. It is rare to find research on insight that does not refer to these findings, and variations of the measure are often used (e.g., Chu, 2009 ; Chein et al., 2010 ; Cushen and Wiley, 2012 ; Hedne et al., 2016 ).

The Self-Report Measure

Asking participants to indicate, case by case, whether a problem was solved with an insight moment (i.e., suddenly, unexpectedly, and accompanied by an Aha! experience), or analytically (i.e., gradually, strategically, and step-by-step) is the most common method in recent research. In some cases a rating scale is used (e.g., Bowden and Jung-Beeman, 2003 ), and in other cases a retrospective forced choice paradigm (e.g., Jung-Beeman et al., 2004 ). Some recent research has also measured different features of the Aha! phenomenology on separate scales, which is beginning to provide a more nuanced view of the (often variable) insight experience ( Danek et al., 2014 ; Webb et al., 2016 ).

Predictions

Clearly the ideal situation is to use both the warmth and the self-report measure, and only label insights as those that are corroborated by both (as recommended by Chu and MacGregor, 2011 ). However, there are reasons why this solution may not be appropriate. In particular, insights can occur—at least theoretically—even when the warmth measure indicates gradual progress on the problem, as long as that progress is not related to the content of the insight (more on this in “Discussion”). The self-report measure can also detect the emotional Aha! experience, but the warmth measure can not. If the two measures are not in agreement about whether an insight occurred, at least most of the time, then using the two measures together to identify insights is not going to be productive, since many true insights would go undetected. In further support of a likely divergence between the measures, Hedne et al. (2016) found no differences in warmth ratings between self-reported insight and non-insight solutions in the case of magic tricks. Magic tricks are a relatively new way to elicit insights ( Danek et al., 2014 ), so we should hesitate to generalize this result to the more commonplace stimuli used in insight research—i.e., classical insight problems. If the two measures do not agree, it is also appropriate to discuss which measure is likely to capture what we regard as insight, and which measure is likely to be capturing something else. We don’t have a specific prediction about the degree of convergence, but given our discussion so far, it is quite possible that the two measures do not often agree. We stress that we are not comparing them empirically to find out which measure is better, only to test agreement. Arguments about the merits of each measure must be made on conceptual grounds, since there is no ground truth. We will aim to provide such a perspective in the section “Discussion.”

The participants were eighty undergraduate students (32 males and 48 females) from The University of Queensland who participated in exchange for course credit (mean age = 20.1, SD = 5.1). Each participant was presented with 20 verbal insight problems. We collected the insight problems from either Schooler et al. (1993) , Weisberg (1996) , or online sources (see Appendix A for the list of problems used). We used Weisberg’s (1996) a priori ‘Taxonomy for Identifying Insight Problems,’ which ensures that the problem involves restructuring (a re-interpretation of the problem elements, Ohlsson, 1984 ), and therefore is likely to elicit an insight. We used LiveCode (an open-source programming tool) to create the experiment and presented it to participants on desktop computers. The dependent variables of interest were the self-report insight measure and the feelings-of-warmth measure of insight.

We calculated differential warmth in a similar way to Metcalfe and Wiebe (1987) , and Hedne et al. (2016) . Differential warmth is calculated by finding the difference between the first warmth rating and the last warmth rating prior to a solution. In order to be faithful to the definition of insight as a ‘sudden solution,’ we determined that an insight had occurred when there is no perceived progress on the problem before the solution, as recommended by Kounios and Beeman (2014) . Whereas Metcalfe and Wiebe’s (1987) participants provided a final warmth rating that indicated that they were certain they found the solution, our participants were instructed to provide warmth ratings only before they reached the solution, and the solution itself acted as the final rating. The benefit of using differential warmth in this way, is that only two warmth ratings are required for a problem solution to be categorized as insight or non-insight, whereas the version used by Metcalfe and Wiebe (1987) required a minimum of three. Many problems are solved faster than 30 s (three warmth ratings at 10 s intervals), which means that substantial data are lost. For example, in Metcalfe and Wiebe (1987) , out of 73 subjects, only 39 provided usable data. There is no foreseeable reason why our changes would result in different outcomes than the original formulation of the warmth measure and that used in Hedne et al. (2016) .

We used a self-report measure of insight as recommended by Bowden and Jung-Beeman (2007) . After providing a solution to a problem, participants are asked to indicate whether they experienced an insight moment by providing a rating of 1 (no), 2 (other), or 3 (yes). The 2 (other) option is for participants who guessed, experienced neither insight nor non-insight, were unsure, or did not know the answer (see Appendix B for the instructions script).

The research questions described in this article were assessed as part of another experiment reported elsewhere ( Laukkonen and Tangen, 2017 ). Each participant began by watching pre-recorded instructions, and was provided with examples of insight problems. They were told that throughout problem solving, a warmth scale would appear on the right hand side of the screen every 10 s, at which point they would need to indicate how close they felt they were to solving the problem from 1 (cold/far) to 10 (hot/close). When the warmth bar appeared, the screen was locked so that participants had to immediately make a rating before continuing on the problem. The warmth bar was presented alongside a tone and participants were told not to change their rating once they had solved the problem, and to submit their response as soon as they reached the solution. The warmth bar would no longer appear once the participant started typing their answer. Participants had 1 min to complete each problem, which was presented in the center of the screen in large font, with a text box below it for typing the answer. Once a solution was provided, they completed the self-report measure of insight, and indicated whether the problem was familiar. If the problem was familiar, it was removed from further analysis.

Out of a possible 631 correctly solved insight problems, participants provided two or more warmth ratings in 180 cases. We did not include problems that were left unsolved or solved incorrectly, because omission errors and guesses were likely to add too much noise to the analysis. Initially we found a moderate to strong positive correlation between the total number of self-reported insights ( M = 5.28, SD = 2.74) and the total number of warmth insights ( M = 4.85, SD = 2.7) for each participant ( r = 0.61, n = 51, p < 0.001). This indicates that self-reported insights and sudden warmth ratings are occurring approximately at the same rate, but it does not tell us whether the same problems were categorized as insight. To this end, we ran another Pearson’s correlation analysis across problems case by case (i.e., at the level of the question rather than at the level of participant averages). This analysis showed no significant correlation between the two measures of insight ( r = 0.08, n = 182, p = 0.235). To provide a more nuanced perspective on the low correlation, a contingency matrix of the data is presented in Table 1 . The contingency matrix indicates that when a sudden solution occurred according to warmth ratings, then there was a 75% chance that an insight was also self-reported by participants (i.e., 25% above chance). On the other hand, if no sudden solution was observed according to the warmth measure, then there was a 50% chance that an insight would nevertheless be self-reported.

TABLE 1. A contingency matrix representing the four possible ways that the two measures of insight can converge or diverge.

Our results indicate that agreement between the two most popular measures of insight is low or non-existent. This finding corresponds with Hedne et al. (2016) who found that warmth ratings did not differ for self-reported insights and non-insight solutions when exposed to magic tricks. A closer look at the data using a contingency matrix indicates that the primary source of divergence occurs because gradual warmth ratings have no implication on whether or not an insight is self-reported by the participants. We now consider which measure—self-report or warmth ratings—may be the better option for detecting insight moments.

Aside from the fact that there are difficulties in analyzing and comparing warmth data (see Weisberg, 1992 for a commentary on this point), there are also theoretical limitations to using warmth ratings to measure insight. One problem is that a gradual warmth pattern does not necessarily mean that an insight did not occur. A participant can of course make subjective progress on a problem, and therefore provide increasing warmth ratings, but then have a sudden insight that they were using the incorrect strategy followed by a solution to the problem. If this unexpected shift occurs, then the warmth ratings appear gradual and the solution predictable, when in fact it was sudden and unpredictable. There is no a priori reason why an insight must occur without the feeling of progress, as long as that feeling of progress is illusory or unrelated to the content of the sudden and unexpected solution. We find strong support for this perspective in our data, where participants are just as likely to report insight moments despite gradual warmth patterns.

Insights are in essence a subjective phenomenon—feelings such as pleasure, certainty, relief, drive, and surprise, are key dimensions of the insight experience that cannot be captured by warmth ratings ( Danek and Wiley, 2017 ). Experiencing an Aha! moment is becoming increasingly the core feature of both definitions and measures of insight among researchers in the area ( Bowden and Jung-Beeman, 2007 ; Kounios and Beeman, 2014 ; Webb et al., 2016 ; Danek and Wiley, 2017 ). This also means that, in a hierarchy of measures, the self-report measure of insight will take precedence. If self-reported insights consistently contradict warmth measures, then we would be forced to conclude that the warm measure is not capturing insights. Of course, if the subjective rating of insight fails to map onto anything objective, then it may not be a useful or interesting construct. Fortunately, we now know that self-reported insights map onto different eye-movements ( Salvi et al., 2015 ), different cognitive strategies ( Kounios et al., 2008 ), different neural activity ( Bowden and Jung-Beeman, 2003 ; Jung-Beeman et al. 2004 , 2014; Kounios et al., 2006 , 2008 ; Subramaniam et al., 2009 ), differences in accuracy ( Hedne et al., 2016 ; Salvi et al., 2016 ; Webb et al., 2016 ), and greater positive affect ( Subramaniam et al., 2009 ). This clear mapping onto objective measures for the self-reported insights is not matched by the warmth measure, perhaps partly because it is impractical for neural investigations ( Bowden and Jung-Beeman, 2007 ).

One issue pertaining to self-reported Aha! moments is the way that they are described to participants prior to experiments, which may in turn impact which phenomenology the participant classifies as insight. In the literature there are notable inconsistencies, for example Cushen and Wiley (2012) focused on just two dimensions, surprise and suddenness (see also Davidson, 1995 and Bowden, 1997 ), whereas more recent work characterizes insight based on multiple dimensions that often include affective features such as pleasure, certainty, and relief (e.g., Jung-Beeman et al., 2004 ; Webb et al., 2016 ; Danek and Wiley, 2017 ). Danek and Wiley (2017) recently compared experimentally the extent to which different dimensions used in previous research predict participants global Aha! ratings, thus providing a more objective mapping of the insight phenomenology. It is likely that empirically mapping the subjective Aha! experience—as in Danek and Wiley (2017) —will eventually mitigate inconsistencies and ensure more representative descriptions of insight.

A Visceral Alternative

According to Creswell et al. (2016) , “visceral states call for visceral measures.” The authors proposed that the feeling of hunger, like many other non-verbal experiences, is difficult to put into words. It is also known that verbalization can be disruptive to both task performance and subsequent memory (e.g., Schooler and Engstler-Schooler, 1990 ; Schooler, 2002 , 2011 ; Brown et al., 2014 ). To solve this problem, the authors tested whether handgrip pressure over time—as measured by a dynamometer—could be used as a visceral, non-verbal alternative to the commonly used self-report measures of hunger. They found that the visceral measure was a better predictor of subsequent eating behavior than the self-report scale, and was sensitive to a well established food cue exposure paradigm. We propose that the insight experience is also visceral in nature, and may therefore be better captured by a visceral measure that does not interfere with the primary task. To illustrate, a participant can be instructed to begin problem solving with their hand resting on the dynamometer without squeezing, and then be asked to increase grip strength as they make progress on the problem, where a stronger squeeze is equivalent to a higher warmth rating, and a full strength squeeze indicates that an Aha! moment occurred. If the participant solved the problem, but did not experience an Aha! moment, then they can simply release their grip, indicating that the solution was found without the insight phenomenology. With these simple instructions, the dynamometer can provide continuous ratings of progress on a problem (feelings-of-warmth), and can show clearly when an Aha! moment occurs—a light squeeze followed by the sudden onset of a full strength squeeze.

We believe the feelings-of-warmth measure captures only a fraction of the insight solutions that can occur during problem solving, and since the warmth measure does not show agreement with the self-report measure, it may fail to capture some crucial features of the insight experience—namely the Aha! moment. The warmth measure remains an innovative and objective measure of progress during problem solving. We recommend that warmth ratings be used to measure perceived progress on a problem, but that concluding that an insight has or has not occurred without other converging evidence is likely premature. Given the strengths of the self-report measure described as well as the relative ease with which it is administered, it is likely that self-report will continue to be the most popular method for detecting insight moments, and justifiably so. As a promising alternative, we propose that the dynamometer as employed by Creswell et al. (2016) can achieve the best of both worlds by providing an embodied continuous measurement of progress on the problem while also capturing the sudden and ineffable moment of insight.

Ethics Statement

This study was carried out in accordance with the recommendations of the Australian National Statement on Ethical Conduct in Human Research, with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the Human Research Ethics Committee, The University of Queensland, Australia.

Author Contributions

RL contributed to design, data collection, data analysis, and write-up. JT contributed to design and write-up.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fpsyg.2018.00282/full#supplementary-material

Bowden, E., and Jung-Beeman, M. (2007). Methods for investigating the neural components of insight. Methods 42, 87–99. doi: 10.1016/j.ymeth.2006.11.007

PubMed Abstract | CrossRef Full Text | Google Scholar

Bowden, E. M. (1997). The effect of reportable and unreportable hints on anagram solution and the aha! Experience. Conscious. Cogn. 6, 545–573. doi: 10.1006/ccog.1997.0325

Bowden, E. M., and Jung-Beeman, M. (2003). Aha! Insight experience correlates with solution activation in the right hemisphere. Psychon. Bull. Rev. 10, 730–737. doi: 10.3758/BF03196539

Brown, C., Brandimonte, M. A., Wickham, L. H., Bosco, A., and Schooler, J. W. (2014). When do words hurt? A multiprocess view of the effects of verbalization on visual memory. J. Exp. Psychol. Learn. Mem. Cogn. 40, 1244–1256. doi: 10.1037/a0037222

Chein, J., Weisberg, R., Streeter, N., and Kwok, S. (2010). Working memory and insight in the nine- dot problem. Mem. Cogn. 38, 883–892. doi: 10.3758/mc.38.7.883

Chu, Y. (2009). Human Insight Problem Solving: Performance, Processing, and Phenomenology. Berlin: VDM Verlag.

Google Scholar

Chu, Y., and MacGregor, J. (2011). Human performance on insight problem solving: a review. J. Probl. Solving 3, 119–150. doi: 10.7771/1932-6246.1094

CrossRef Full Text | Google Scholar

Creswell, K., Layette, M., Schooler, J., Wright, A., and Pacilio, L. (2016). Visceral states call for visceral measures: verbal overshadowing of hunger ratings across assessment modalities. Assessment 25, 173–182. doi: 10.1177/1073191116645910

Cushen, P., and Wiley, J. (2012). Cues to solution, restructuring patterns, and reports of insight in creative problem solving. Conscious. Cogn. 21, 1166–1175. doi: 10.1016/j.concog.2012.03.013

Danek, A., Fraps, T., von Müller, A., Grothe, B., and Öllinger, M. (2014). Working wonders? Investigating insight with magic tricks. Cognition 130, 174–185. doi: 10.1016/j.cognition.2013.11.003

Danek, A. H., and Wiley, J. (2017). What about false insights? deconstructing the Aha! Experience along its multiple dimensions for correct and incorrect solutions separately. Front. Psychol. 7:2077. doi: 10.3389/fpsyg.2016.02077

Davidson, J. E. (1995). “The suddenness of insight,” in The Nature of Insight , eds R. J. Sternberg and J. E. Davidson (Cambridge, MA: MIT Press), 125–155.

Fleck, J., and Weisberg, R. (2013). Insight versus analysis: evidence for diverse methods in problem solving. J. Cogn. Psychol. 25, 436–463. doi: 10.1080/20445911.2013.779248

Gilhooly, K., and Murphy, P. (2005). Differentiating insight from non-insight problems. Think. Reason. 11, 279–302. doi: 10.1080/13546780442000187

Hedne, M. R., Norman, E., and Metcalfe, J. (2016). Intuitive feelings of warmth and confidence in insight and noninsight problem solving of magic tricks. Front. Psychol. 7:1314. doi: 10.3389/fpsyg.2016.01314

Jung-Beeman, M., Bowden, E., Haberman, J., Frymiare, J., Arambel-Liu, S., Greenblatt, R., et al. (2004). Neural activity when people solve verbal problems with insight. PLoS Biol. 2:e97. doi: 10.1371/journal.pbio.0020097

Klein, G., and Jarosz, A. (2011). A naturalistic study of insight. J. Cogn. Eng. Decis. Mak. 5, 335–351. doi: 10.1177/1555343411427013

Kounios, J., and Beeman, M. (2014). The cognitive neuroscience of insight. Annu. Rev. Psychol. 65, 71–93. doi: 10.1146/annurev-psych-010213-115154

Kounios, J., Fleck, J., Green, D., Payne, L., Stevenson, J., Bowden, E., et al. (2008). The origins of insight in resting-state brain activity. Neuropsychologia 46, 281–291. doi: 10.1016/j.neuropsychologia.2007.07.013

Kounios, J., Frymiare, J., Bowden, E., Fleck, J., Subramaniam, K., Parrish, T., et al. (2006). The prepared mind: neural activity prior to problem presentation predicts subsequent solution by sudden insight. Psychol. Sci. 17, 882–890. doi: 10.1111/j.1467-9280.2006.01798.x

Laukkonen, R., and Tangen, J. (2017). Can observing a Necker cube make you more insightful. Conscious. Cogn. 48, 198–211. doi: 10.1016/j.concog.2016.11.011

Metcalfe, J. (1986). Feeling of knowing in memory and problem solving. J. Exp. Psychol. Learn. Mem. Cogn. 12, 288–294. doi: 10.1037/0278-7393.12.2.288

Metcalfe, J., and Wiebe, D. (1987). Intuition in insight and noninsight problem solving. Mem. Cogn. 15, 238–246. doi: 10.3758/bf03197722

Ohlsson, S. (1984). Restructuring revisited. Scand. J. Psychol. 25, 117–129. doi: 10.1111/j.1467-9450.1984.tb01005.x

Salvi, C., Bricolo, E., Franconeri, S. L., Kounios, J., and Beeman, M. (2015). Sudden insight is associated with shutting out visual inputs. Psychon. Bull. Rev. 22, 1814–1819. doi: 10.3758/s13423-015-0845-0

Salvi, C., Bricolo, E., Kounios, J., Bowden, E., and Beeman, M. (2016). Insight solutions are correct more often than analytic solutions. Think. Reason. 22, 443–460. doi: 10.1080/13546783.2016.1141798

Schooler, J., Ohlsson, S., and Brooks, K. (1993). Thoughts beyond words: when language overshadows insight. J. Exp. Psychol. Gen. 122, 166–183. doi: 10.1037//0096-3445.122.2.166

Schooler, J. W. (2002). Re-representing consciousness: dissociations between consciousness and meta-consciousness. Trends Cogn. Sci. 6, 339–344. doi: 10.1016/S1364-6613(02)01949-6

Schooler, J. W. (2011). Introspecting in the spirit of William James: comment on Fox, Ericsson, and best (2011). Psychol. Bull. 137, 345–350. doi: 10.1037/a0022390

PubMed Abstract | CrossRef Full Text

Schooler, J. W., and Engstler-Schooler, T. Y. (1990). Verbal overshadowing of visual memories: some things are better left unsaid. Cogn. Psychol. 22, 36–71. doi: 10.1016/0010-0285(90)90003-M

Subramaniam, K., Kounios, J., Parrish, T., and Jung-Beeman, M. (2009). A brain mechanism for facilitation of insight by positive affect. J. Cogn. Neurosci. 21, 415–432. doi: 10.1162/jocn.2009.21057

Webb, M. E., Little, D. R., and Cropper, S. J. (2016). Insight is not in the problem: investigating insight in problem solving across task types. Front. Psychol. 7:1424. doi: 10.3389/fpsyg.2016.01424

Weisberg, R. (1992). Metacognition and insight during problem solving: comment on metcalfe. J. Exp. Psychol. Learn. Mem. Cogn. 18, 426–431. doi: 10.1037//0278-7393.18.2.426

Weisberg, R. W. (1996). “Prolegomena to theories of insight in problem solving: a taxonomy of problems,” in The Nature of Insight , eds R. J. Sternberg and J. E. Davidson (Cambridge, MA: The MIT Press), 157–196.

Keywords : insight, problem solving, Aha!, Eureka, creativity, surprise

Citation: Laukkonen RE and Tangen JM (2018) How to Detect Insight Moments in Problem Solving Experiments. Front. Psychol. 9:282. doi: 10.3389/fpsyg.2018.00282

Received: 04 August 2017; Accepted: 20 February 2018; Published: 09 March 2018.

Reviewed by:

Copyright © 2018 Laukkonen and Tangen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Ruben E. Laukkonen, [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

Psychological Research on Insight Problem Solving

Cite this chapter.

• Michael öllinger 3 &
• Günther Knoblich 4  

1245 Accesses

18 Citations

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

• Available as PDF
• Read on any device
• Instant download
• Own it forever
• Compact, lightweight edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info
• Durable hardcover edition

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Unable to display preview.  Download preview PDF.

Ash, M.G. (1998): Gestalt Psychology in German Culture, 1890-1967: Holism and the Quest for Objectivity . Cambridge University Press, Cambridge.

Google Scholar  

Bowden, E.M. and Jung-Beeman, M. (2003): Aha! – Insight experience correlates with solution activation in the right hemisphere. Psychonomic Bulletin and Review 10 , 730–737.

Bowden, E.M., Jung-Beeman, M., Fleck, J. and Kounios, J. (2005): New approaches to demystifying insight. Trends in Cognitive Sciences 9 , 322–328.

Article   Google Scholar  

Chronicle, E.P., MacGregor, J.N. and Ormerod, T.C. (2004): What makes an insight problem ? The roles of heuristics, goal conception, and solution recoding in knowledge-lean problems. Journal of Experimental Psychology: Learning, Memory, and Cognition 30 , 14–27.

Dominowski, R.L. and Dallob, P. (1995): Insight and problem solving. In: R.J. Sternberg and J.E. Davidson (eds.), The Nature of Insight . MIT Press, Cambridge (MA), pp. 33–62.

Duncker, K. (1935): Zur Psychologie des produktiven Denkens . Springer, Berlin.

Duncker, K. (1945): On problem solving. In: J.F. Dashiell (ed.), Psychological Monographs 58 (5). The American Psychological Association, Washington (DC).

Grant, E.R. and Spivey, M.J. (2003): Eye movements and problem solving: Guiding attention guides thought. Psychological Science , 14 , 462–466.

Greeno, J. (1974): Hobbits and Orcs – Acquisition of a sequential concept. Cognitive Psychology 6 , 270–292.

Gruber, H.E. (1995): Insight and affect in the history of science. In: R.J. Sternberg and J.E. Davidson (eds.), The Nature of Insight . MIT Press, Cambridge (MA), pp. 397–431.

Jones, G. (2003): Testing two cognitive theories of insight. Journal of Experimental Psychology: Learning, Memory, and Cognition 29 , 1017–1027.

Jung-Beeman, M., Bowden, E.M., Haberman, J., Frymiare, J.L., Arambel-Liu, S., Greenblatt, R., Reber, P. and Kounios, J. (2004): Neural activity when people solve verbal problems with insight. PLoS Biology 2 , 500–510.

Kaplan, C.A. and Simon, H.A. (1990): In search of insight. Cognitive Psychology 22 , 374–419.

Kershaw, T.C. and Ohlsson, S. (2004): Multiple causes of difficulty in insight: The case of the nine-dot problem. Journal of Experimental Psychology: Learning, Memory, and Cognition 30 , 3–13.

Knoblich, G., Ohlsson, S., Haider, H. and Rhenius, D. (1999): Constraint relaxation and chunk decomposition in insight problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition 25 , 1534–1555.

Knoblich, G., Ohlsson, S. and Raney, G. (1999): Resolving impasses in problem solving. In: M. Hahn and S.C. Stoness (eds.), Proceedings of the 21st Meeting of the Cognitive Science Society . Erlbaum, Mahwah (NJ), pp. 276–281.

Knoblich, G., Ohlsson, S. and Raney, G.E. (2001): An eye movement study of insight problem solving. Memory and Cognition , 29 , 1000–1009.

Knoblich, G. and öllinger, M. (2006): The eureka moment. Scientific American Mind , issue #10, 38–43.

Knoblich, G., öllinger, M. and Spivey, M.J. (2005): Tracking the eyes to obtain insight into insight problem solving. In: G.D.M. Underwood (ed.), Cognitve Processes in Eye Guidance . Oxford University Press, Oxford, pp. 355–375.

Koffka, K. (1935): Principles of Gestalt Psychology . Harcourt, Brace and World, New York.

Köhler, W. (1921): Intelligenzprüfungen an Menschenaffen. Springer, Berlin.

Lavric, A., Forstmeier, S. and Rippon, G. (2000): Differences in working memory involvement in analytical and creative tasks: An ERP study. Neuroreport 11 , 1613–1618.

Lavric, A., Rippon, G. and Forstmeier, S. (1998): ERP studies of working memory in reasoning tasks. International Journal of Psychophysiology 30 , 95–271, see p. 145.

Lindsay, P.H. and Norman, D.A. (1981): Einführung in die Psychologie. Informationsaufnahme und -verarbeitung beim Menschen. Springer, Berlin.

Lovett, M.C. and Anderson, J.R. (1996): History of success and current context in problem solving: Combined influences on operator selection. Cognitive Psychology 31 , 168–217.

Luchins, A.S. (1942): Mechanization in problem solving – the effect of Einstellung. Psychological Monographs 54 (6), 1–95.

Luchins, A.S. and Luchins, E.H. (1959): Rigidity of Behavior: A Variational Approach to the Effect of Einstellung . University of Oregon Books, Eugene (OR).

Luo, J. and Niki, K. (2003): Function of hippocampus in “insight” of problem solving. Hippocampus 13 , 316–323.

MacGregor, J.N., Ormerod, T.C. and Chronicle, E.P. (2001): Information processing and insight: A process model of performance on the nine-dot and related problems. Journal of Experimental Psychology: Learning, Memory, and Cognition 27 , 176–201.

Mai, X.-Q., Luo, J., Wu, J.-H. and Luo, Y.-J. (2004). “Aha!” effects in a guessing riddle task: An event-related potential study. Human Brain Mapping 22 , 261–271.

Maier, N.R.F. (1931): Reasoning in humans. II. The solution of a problem and its appearance in consciousness. Journal of Comparative Psychology 12 , 181–194.

McClelland, J.L., McNaughton, B.L. and O’Reilly, R.C. (1995): Why there are complementary learning systems in the hippocampus and neocortex: Insights from the successes and failures of connectionist models of learning and memory. Psychological Review 102 , 419–457.

Metcalfe, J. (1986): Feeling of knowing in memory and problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition 12 , 288–294.

Metcalfe, J. and Wiebe, D. (1987): Intuition in insight and noninsight problem solving. Memory and Cognition 15 , 238–246.

Metzger, W. (1986): Gestalt-Psychologie. Ausgewählte Werke aus den Jahren 1950 bis 1982 . Waldemar Kramer, Frankfurt.

Newell, A. and Simon, H.A. (1972): Human Problem Solving . Prentice Hall, Englewood Cliffs (NJ).

Ohlsson, S. (1992): Information-processing explanations of insight and related phenomena. In: M. Keane and K. Gilhooly (eds.), Advances in the Psychology of Thinking . Harvester-Wheatsheaf, London, pp. 1-44.

öllinger, M., Jones, G. and Knoblich, G. (2006): Heuristics and representational change in two-move matchstick arithmetic tasks. Advances in Cognitive Psychology 2 , 239–253.

öllinger, M., Jones, G. and Knoblich, G. (2008): Investigating the effect of mental set on insight problem solving. Experimental Psychology 55 , 270–282.

Ormerod, T.C., MacGregor, J.N. and Chronicle, E P. (2002): Dynamics and constraints in insight problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition 28 , 791–799.

Reverberi, C., Toraldo, A., D’Agostini, S. and Skrap, M. (2005): Better without (lateral) frontal cortex ? Insight problems solved by frontal patients. Brain 128 , 2882–2890.

Scheerer, M. (1963): Problem-solving. Scientific American , 208 (4), 118–128.

Schooler, J.W., Ohlsson, S. and Brooks, K. (1993): Thoughts beyond words: When language overshadows insight. Journal of Experimental Psychology: General 122 , 166–183.

Thomas, J.C. (1974): An analysis of behavior in the hobbits-orcs problem. Cognitive Psychology 6 , 257-269.

Wagner, U., Gais, S., Haider, H., Verleger, R. and Born, J. (2004): Sleep inspires insight. Nature 427 , 352–355.

Wegner, D.M. (2002): The Illusion of Conscious Will . MIT Press, Cambridge (MA).

Weisberg, R.W. and Alba, J.W. (1982): Problem solving is not like perception: More on Gestalt theory. Journal of Experimental Psychology: General 111 , 326–330.

Weisberg, R.W. (1992): Metacognition and insight during problem solving: Comment on Metcalfe. Journal of Experimental Psychology: Learning, Memory, and Cognition 18 , 426–431.

Weisberg, R W. (1995): Prolegomena to theories of insight in problem solving: A taxonomy of problems. In: R.J. Sternberg and J.E. Davidson (eds.), The Nature of Insight . MIT Press, Cambridge (MA), pp. 157–196.

Wertheimer, M. (1912): Experimentelle Studien über das Sehen von Bewegung. Zeitschrift für Psychologie 61 , 161–265.

Wertheimer, M. (1923): Untersuchungen zur Lehre von der Gestalt. II, Psychologi-sche Forschung 4 , 301–350.

Wertheimer, M. (1925): Drei Abhandlungen zur Gestalttheorie . Verlag der Philosophischen Akademie, Erlangen.

Wertheimer, M. (1959): Productive Thinking . Harper, New York.

Wertheimer, Mich. (1985): A Gestalt perspective on computer simulations of cognitive processes. Computers in Human Behavior 1 , 19–33.

Wickelgren, W.A. (1974). How to Solve Problems: Elements of a Theory of Problems and Problem Solving . Freeman, San Francisco.

Download references

Author information

Authors and affiliations.

Parmenides Center for the Study of Thinking, 80333 München, Germany

Michael öllinger

Psychology Department, Rutgers University, Newark NJ 07102, USA

Günther Knoblich

You can also search for this author in PubMed   Google Scholar

Editor information

Editors and affiliations.

Institut für Grenzgebiete der Psychologie und Psychohygiene e. V. Abt. Theorie und Datenanalyse Wilhelmstr. 3a, 79098 Freiburg, Germany

Harald Atmanspacher ( Dr ) ( Dr )

Kusenstr. 21, 8700 Kuesnacht, Switzerland

Hans Primas ( Prof.Dr ) ( Prof.Dr )

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

öllinger, M., Knoblich, G. (2009). Psychological Research on Insight Problem Solving. In: Atmanspacher, H., Primas, H. (eds) Recasting Reality. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85198-1_14

Download citation

DOI : https://doi.org/10.1007/978-3-540-85198-1_14

Publisher Name : Springer, Berlin, Heidelberg

Print ISBN : 978-3-540-85197-4

Online ISBN : 978-3-540-85198-1

eBook Packages : Humanities, Social Sciences and Law Philosophy and Religion (R0)

Share this chapter

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Publish with us

Policies and ethics

• Find a journal
• Track your research

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

• Publications
• Account settings

Preview improvements coming to the PMC website in October 2024. Learn More or Try it out now .

• Advanced Search
• Journal List
• PMC10219327

Tracing Cognitive Processes in Insight Problem Solving: Using GAMs and Change Point Analysis to Uncover Restructuring

1 Institute of Psychology, University of Klagenfurt, 9020 Klagenfurt, Austria

Amory H. Danek

2 Department of Psychology, Heidelberg University, 69117 Heidelberg, Germany

Nemanja Vaci

3 Department of Psychology, Sheffield University, Sheffield S10 2BP, UK

Merim Bilalić

4 Department of Psychology, University of Northumbria at Newcastle, Newcastle upon Tyne NE1 8ST, UK

Associated Data

Technical details, such as data and code for the analysis, is available at https://osf.io/pwuhs/?view_only=7c52bda4e6fa481e826e5d7570b6ef34 (accessed on 25 April 2023).

Insight problems are likely to trigger an initial, incorrect mental representation, which needs to be restructured in order to find the solution. Despite the widespread theoretical assumption that this restructuring process happens suddenly, leading to the typical “Aha!” experience, the evidence is inconclusive. Among the reasons for this lack of clarity is that many measures of insight rely solely on the solvers’ subjective experience of the solution process. In our previous paper, we used matchstick arithmetic problems to demonstrate that it is possible to objectively trace problem-solving processes by combining eye movements with new analytical and statistical approaches. Specifically, we divided the problem-solving process into ten (relative) temporal phases to better capture possible small changes in problem representation. Here, we go a step further to demonstrate that classical statistical procedures, such as ANOVA, cannot capture sudden representational change processes, which are typical for insight problems. Only nonlinear statistical models, such as generalized additive (mixed) models (GAMs) and change points analysis, correctly identified the abrupt representational change. Additionally, we demonstrate that explicit hints reorient participants’ focus in a qualitatively different manner, changing the dynamics of restructuring in insight problem solving. While insight problems may indeed require a sudden restructuring of the initial mental representation, more sophisticated analytical and statistical approaches are necessary to uncover their true nature.

1. Introduction

In cognitive science, the temporal dynamics of problem-solving processes have always been an important topic of investigation. Most problems are assumed to be solved gradually, by piecing together information in order to arrive at a solution ( Newell and Simon 1972 ). To investigate these problems, several tools have been developed, which allow for the observation of each step of the problem-solving process (e.g., Tower of Hanoi, Hobbits and Orcs problem). In the case of “insight problems”, the solution often comes seemingly out of nowhere ( Duncker 1945 ), despite the problem appearing unsolvable just a moment earlier. To be solved, insight problems are thought to require a fundamental, sudden change in the way the problem is perceived, a process referred to as restructuring or representational change ( Ohlsson 1992 ; Wertheimer 1925 ). The restructuring from the initial, incorrect mental representation to the correct one is the key component in modern theories such as representational change theory (RCT) ( Knoblich et al. 1999 ; Ohlsson 1984 , 1992 , 2011 ).

Although the sudden nature of the underlying restructuring process is a main theoretical assumption about insight, the evidence for this claim is inconclusive. Ohlsson ( 1992 ) even hypothesized that “the sudden appearance of the complete solution in consciousness is an illusion caused by our lack of introspective access to our cognitive processes (...)” (p. 17). To truly understand the temporal nature of insight, the cognitive component of insight (restructuring) must be examined with appropriate tools. Observing changes in solvers’ mental problem representation is a methodological and statistical challenge, which is addressed in the present work. Among the reasons for this lack of clarity is that many measures of insight rely solely on the solvers’ subjective experience of the solution process. Using matchstick arithmetic problems, we demonstrate that it is possible to objectively trace problem-solving processes.

We first review the research on representational change, focusing on the experimental designs. After that, we describe a novel analytical approach that improves upon previous attempts. Finally, and arguably most importantly, we show that this analytical approach needs to be combined with appropriate statistical tools in order to work properly. We demonstrate the feasibility of this approach by re-analyzing eye-tracking data from an already published study ( Bilalić et al. 2019 ). The paper is accompanied by an online supplement , with technical details, such as data and code for the analysis, which is freely available at https://osf.io/pwuhs/?view_only=7c52bda4e6fa481e826e5d7570b6ef3 (accessed on 25 April 2023).

1.1. Temporal Dynamics of the Restructuring Process

In 1994, Durso and colleagues conducted an early study on the temporal dynamics of insight problem solving. They asked participants to rate the relatedness of word pairs in a word puzzle and found that, on average, solution-relevant pairs were rated as increasingly similar as participants approached a solution. The authors concluded that “[l]ike dynamite, the insightful solution explodes on the solver’s cognitive landscape with breathtaking suddenness, but if one looks closely, a long fuse warns of the impending reorganization” ( Durso et al. 1994, p. 98 ). Novick and Sherman ( 2003 ; Experiment 2) provided similar evidence. They asked participants to indicate within a short time window (250 ms after stimulus offset) whether presented anagrams were solvable. They found that, although participants could not find the solution within the allotted time, they were increasingly better at differentiating between solvable and unsolvable anagrams as the presentation time of the anagrams increased. The authors concluded that solvers gradually accumulate information relevant for solving the anagrams.

Several studies have focused on the concept of restructuring in insight problem solving, but have typically not measured the dynamics of the solving process (e.g., Ash et al. 2012 ; Ash and Wiley 2006 , 2008 ; Fleck and Weisberg 2013 ; MacGregor and Cunningham 2009 ). However, a number of studies have attempted to measure the temporal dynamics of restructuring, using different methods to acquire trace data. Some used repeated ratings of problem elements, either regarding their similarity ( Durso et al. 1994 ) or with regard to their relevance for the solution ( Cushen and Wiley 2012 ; Danek et al. 2020 ). Others recorded eye movements ( Ellis et al. 2011 ; Knoblich et al. 2001 ; Bilalić et al. 2019 ; Tseng et al. 2014 ) or employed solvability judgments ( Novick and Sherman 2003 ). In some of these studies, both incremental and sudden solution patterns were found ( Cushen and Wiley 2012 ; Danek et al. 2020 ; Novick and Sherman 2003 ), whereas other studies found only incremental patterns ( Durso et al. 1994 ).

1.2. Eye Movements and Matchstick Arithmetic Problems

Here, we will take a closer look at using eye movement recordings to measure the temporal dynamics of restructuring in insight problems (for a comprehensive overview on eye movements, please see Holmqvist et al. 2011 ). In general, eye movements provide an objective measure of cognitive processes, as they are closely linked to attention (e.g., Just and Carpenter 1976 ; Rayner 1995 ; Reingold et al. 2001 ). Specifically, eye fixations reveal when people pay attention to certain features of a problem and for how long. More importantly, eye tracking is particularly useful when participants might not remember or even concurrently report that they are paying attention to these elements ( Bilalić and McLeod 2014 ; Bilalić et al. 2008 , 2010 ; Kuhn and Land 2006 ; Kuhn et al. 2009 ). This is particularly relevant in the case of insight problems, where it is possible that people are not aware of the dynamics of their solution process.

We use the matchstick arithmetic problems introduced by Knoblich et al. ( 1999 ). Matchstick arithmetic problems are suitable for investigation with eye tracking, as was powerfully demonstrated by the seminal study of Knoblich et al. ( 2001 ). A matchstick arithmetic problem consists of a false arithmetic statement written using Roman numerals, arithmetic operators, and equal signs, all formed using matchsticks ( Knoblich et al. 1999 , 2001 ; see also Figure 1 below). The task is to transform the false arithmetic statement into a true statement by moving only a single stick. Four types of matchstick arithmetic problems have been defined with varying levels of difficulty, depending on the constraints that need to be relaxed and the tightness of the chunks that need to be decomposed. These problem types were theoretically derived from the representational change theory ( Ohlsson 1992 ) and have been empirically confirmed ( Knoblich et al. 1999 ; Öllinger et al. 2006 , 2008 ). The use of matchstick arithmetic problems enables us to build on a well-researched task domain. It is known which problem type should elicit the restructuring process ( Knoblich et al. 1999 ; Öllinger et al. 2006 , 2008 ), and it is possible to contrast it with a type which requires no restructuring. Additionally, based on Knoblich’s study (2001), predictions about eye movement patterns can be made. Furthermore, the matchstick arithmetic domain is well suited for eye tracking because each problem consists of individual matchsticks that do not overlap, allowing for precise differentiation of fixations. In other words, we can determine at any point in time which aspect of the problem is attended to.

Matchstick arithmetic problem. Participants are required to transform the false arithmetic statement to a true statement by moving a single matchstick. This problem requires restructuring, because the initial assumption that only the matchsticks from values can be manipulated needs to be changed. In this case, the operator “+” can be decomposed and its vertical matchstick moved to make another “=” sign (VI = VI = VI). The “+” sign is the critical element that needs to be changed for solution.

Knoblich et al. ( 2001 ) investigated constraint relaxation type problems, which are considered to require restructuring; see Figure 1 for an example. They found that for this problem (constraint relaxation type), both solvers and non-solvers examined the values in the beginning and spent most of their time doing so. This can be seen as an indication that participants were using an initial incorrect problem representation, triggered by previous knowledge, where only values can be changed. Only in the final third of the problem- solving period did later solvers change their mental representation, as demonstrated by their eye movements. Solvers started to pay attention more to the operators and less to the values. In contrast, non-solvers remained stuck in their initial representation, as they continued to attend to values rather than to operators. Similar results for the same problem were found by another eye-tracking study ( Tseng et al. 2014 ).

The Knoblich et al. ( 2001 ) study provides strong evidence for the claim that in problems that require constraint relaxation, a restructuring of the problem representation took place. However, it did not answer the question of whether this change was a sudden or a gradual one. In the final third of the allotted time, solvers paid attention to the important but previously ignored features, which could be interpreted as a result of sudden restructuring. It is nevertheless not that clear, since the final period may have lasted minutes, given that they took around five minutes to solve the problem. Thus, the restructuring might have been a continuous process over time. On the other hand, an eye-tracking study on anagrams by Ellis et al. ( 2011 ; see also Ellis and Reingold 2014 ) found that participants started disregarding the irrelevant problem elements several seconds before they came up with the solution. The viewing times on that problem elements were decreasing gradually. Most intriguingly, both participant groups, those who experienced pop-out insight-like solutions and those who did not, displayed the same gradual accumulation of solution knowledge.

1.3. Metacognitive Processes and Insight Problems

There is evidence that the problem-solving process benefits from hints (e.g., Bowden 1997 ; Bilalić et al. 2019 ; Ammalainen and Moroshkina 2021 ; Becker et al. 2021 ; Korovkin and Savinova 2021 ; Spiridonov et al. 2021 ). This is the case even when hints were unreportable; that is, hints even work when presented briefly below the threshold of consciousness. Ammalainen and Moroshkina ( 2021 ) found evidence that hints can influence the problem-solving ability, which can be both, positive and negative. In a positive way, hints which are helpful to find the solution increase solution rates. On the other hand, misleading hints can negatively affect solution rates by distracting problem solvers and leading to a decrease in their success rate. In our paper ( Bilalić et al. 2019 ), we also provided hints when participants were unable to find the correct solution after a certain time.

These hints serve two purposes: a practical and a theoretical one. On a practical level, they provide an additional check on the main assumption behind the restructuring process. On a theoretical level, they serve as explicit clues that tap into metacognitive processes ( Takeuchi et al. 2019 ; Metcalfe and Shimamura 1994 ). Hints make participants aware of important aspects in the problem, drawing their attention towards elements that may have been neglected. They also change participants’ knowledge about the problem, potentially affecting the way they solve insight problems ( Bowden 1997 ; Bilalić et al. 2019 ; Korovkin and Savinova 2021 ).

The present work is a re-analysis of our paper ( Bilalić et al. 2019 ). In our paper, we also combined solving of insight and non-insight problems with eye tracking. We presented first a non-insight matchstick problem and then the matchstick insight problem depicted here (see Figure 1 ) to 61 participants (5 male; M age = 22.8; SD age = 6.5). The study was designed to take into account the methodological issue discussed in the previous section. It built upon previous attempts that utilized more time periods and sometimes presented the last 5 or 10 s separately (see also Bilalić et al. 2008 , 2010 , 2014 ). In the 2019 study, we provided a more fine-grained temporal analysis of the solution process by using ten time periods of equal length for our eye movement analysis 1 (for more information, please refer to Bilalić et al. 2019 ). We demonstrated that the restructuring is a gradual process on the insight problem as the solvers started paying attention to the important aspects of the problem long before they found the solution. Here, we provide another set of data where the jump is sudden; that is, the solvers started paying attention to the important aspects immediately before they found the solution (as reported by Knoblich et al. 2001 ). This is done to illustrate (1) how classical ways of analyzing data, such as ANOVA, are inappropriate for discovering the sudden changes, and (2) how other non-linear approaches are required.

We expected that all participants would initially focus on the values. Solvers would shift their attention towards the critical element (the “+” operator), while non-solvers would remain fixated on the values. The first question of interest is whether the representational shift in eventual solvers will be sudden or rather incremental. The second question of interest is whether the explicit cue, that is, the hint, will produce a sudden rearrangement of attention towards the critical elements (here “+”, but also “=” because “=” is also an operator). In our design, we included hints for participants who had not solved the problem within five minutes. The hint provided at this point was ‘You can change the operators, too.’ We were interested in whether the hints change the dynamics of problem solving, specifically whether the solution process remains sudden even after receiving an explicit cue.

The problem proved difficult as only 34% found the solution. After the hint was provided, an additional 11% of participants were able to find the solution. We present the eye data analysis below, with a particular focus on the critical element of the problem, the plus sign (+). Additionally, when analyzing the impact of hints, we also focused on the equal sign (=) as the hints should also affect the attention drawn to this operator through metacognitive control. For analysis of other problem elements, please refer to the supplementary materials .

3.1. Is Insight Sudden or Incremental? (Solvers vs. Non-Solvers: First 5 Min Analysis)

In Figure 2 , raw data and means for each bin of the critical element for the first five minutes are presented. 2 The solving pattern follows the typical sudden pattern, where there is not much difference between eventual solvers and non-solvers with regard to the time spent on the critical element (+) until the end of the first five minutes. Solvers suddenly increase their dwell time just before announcing the solution, while non-solvers continue to observe the critical element sporadically until the end of the solving period.

Raw data and means for each bin of the critical element (+). The raw data represents every data point of each participant over the entire problem-solving period. The problem-solving period was divided in 10 proportional bins, each representing 10% of the total problem-solving time. The error bars represent the 68% confidence interval. This figure illustrates a nonlinear increase in the amount of time that solvers spend on the critical element. In the case of solvers, the 100% bin means the participant provided a solution.

The crucial question is how to analyze the temporal changes presented in Figure 2 . The traditional method, which we had chosen in our previous paper ( Bilalić et al. 2019 ), is to use an analysis of variance (ANOVA) where the bins and groups are factors that predict the amount of time spent on the critical element. However, ANOVA not only requires a completely balanced dataset, but it also ignores the clustered nature of data ( van Rij et al. 2020 ). Furthermore, it is based on linear regression, which is not suitable for capturing sudden attentional shifts, which are nonlinear in nature. In order to capture the sudden shift as depicted in Figure 2 (the 100% bin for the solvers), ANOVA would need to adjust the linear trend throughout the whole problem-solving period. In other words, a sudden trend may appear as an incremental one as ANOVA adjusts by increasing previous periods (see Figure 3 , left panel).

Estimated model means based on ( a ) ANOVA with linear term; ( b ) ANOVA with both linear and quadratic terms. Y-Axis: Time on the problem element (%). Please refer to supplementary material for the detailed analysis.

ANOVA can be expressed as linear regression, where an additional quadratic polynomial term is included next to the linear one, in an attempt to capture the shift. However, even in this case, the predicted shift by the ANOVA model would begin earlier, namely at the 80% bin, than it does in the raw data (see Figure 3 , right panel). The general limitation of linear regression, with or without polynomial terms, is that it heavily relies on previous trends. If the change is sudden, the previous time periods will also be adjusted accordingly.

One way around this problem is generalized additive (mixed) modeling (GAM). These models are specifically designed to handle nonlinear relationships, as they are data-driven and use non-linear mixed-effects regression ( van Rij et al. 2020 ). A key benefit of GAMs is that they do not require the user to specify the shape of the nonlinear regression line, as the model determines this based on the data. However, while GAMs have a high level of flexibility in modeling nonlinear changes in time series data, they only allow for the exploration of changes in the function and do not provide parametric estimates such as standard error of estimate or its impact on predictive accuracy of the model. More specifically, GAMs do not provide parametric estimates, which means that they do not give us a set of parameters that describe the shape of the nonlinear function. However, the present work intends to demonstrate the advantages and downsides of the available analysis tools in question, which is why GAMs are included here.

Arguably the most reliable way of checking the assumption of suddenness is the use of change point analysis, which looks for significant deviance from previous trends ( Raftery and Akman 1986 ). Unlike the standard regression analysis (ANOVA) and nonlinear GAMs, change point regression estimates the moment of the function inflection. In other words, it includes the possibility to estimate additional parameters, such as intercept and slope of regression, time point when the function changes, and how the intercept and/or slope of regression changes (see the figures of the MCP analysis for illustrations). This makes the technique particularly valuable in detecting increasing patterns as one would expect several points of change in the attentional pattern on the way towards the solution. In this instance, we use the one implemented in the Multiple Change Points package (MCP; Lindeløv 2020 ).

Below, we address the three main questions using both GAM and MCP analysis. In the supplemental material , we provide the model-estimated values for each case, which include the results and, in the case of the MCPs, how well the model fits the data and which model was used. We begin with the GAM analysis of solvers and non-solvers for the first five minutes to determine whether the insight is sudden or incremental. Figure 4 provides the estimated trend lines for both solvers and non-solvers, as well as the time periods (shaded in orange) where the difference between the two is statistically significant. The model estimates closely follow the raw data (see Figure 2 ), and the difference between solvers and non-solvers is indeed significant at the beginning of the solving phase, as well as at the 90% bin and the 100% bin.

GAM: the difference between two estimated trend lines for solvers and non-solvers of the critical element (+). This figure illustrates that the GAM also found a nonlinear increase in the amount of time that non-solvers spend on the critical element. The orange area determines where the differences between solvers and non-solvers were significant.

Figure 5 shows the results of the MCP analysis for the same data as the GAM above. Similarly to the GAM, the MCP analysis identified a change point around the 90% bin for the solvers, which captures an attentional shift they made. While some non-solvers also shifted their attention towards the “+” sign at the end, it was not as clear as in the case of the solvers.

MCP analysis of the critical element (+) for non-solvers and solvers. This figure illustrates every data point of each participant over the problem-solving period. Lines at the bottom of the figure illustrate the posterior density (estimated likelihood) of the change point for each MCMC chain. There is a nonlinear increase in the amount of time that solvers spend on the critical element.

3.2. Do Explicit Cues Rearrange Attentional Distribution? (An Immediate Change after the Hint)

Figure 6 illustrates the impact of providing an explicit hint to the non-solvers from the first five minutes (presented here as a single group; solvers from the first five minutes are not included in this graph). The attentional shift from values towards operators, “+” and “=”, is substantial immediately after the hint. The operator “=” is attended to twice as much immediately after the hint than before. The change for “+” is slightly less dramatic at first (only 4%), but by the 20% bin, the dwell time has doubled in comparison to before the hint was provided. Note that only non-solvers are shown here, since solvers did not receive any hints.

Raw data and means for each bin of the operators (“+” upper panel; “=” lower panel) in the period before and after the hint. The raw data represents every data point of each participant (non-solvers only) over the problem-solving periods. Each of both problem-solving periods (before and after the hint was provided) were divided in 10 proportional bins, each representing 10% of the total problem-solving time. It is necessary to view the problem-solving periods as distinct periods; therefore, each period is labeled from beginning to end (10% to 100%) to differentiate them. The error bars represent the 68% confidence interval. This figure illustrates the attentional shifts from values (mostly attended to before the hint) towards operators (attended to after the hint was given).

The GAM analysis effectively captures the attentional shift, as depicted in Figure 7 . However, it predicts that the change occurs prior to the hint being provided, starting already at the 90% bin, which is not a correct reflection of the actual data. While GAM is considerably more flexible than regressions with polynomial terms, the same problem of interdependence of neighboring phases remains. The shift caused by the explicit cue is so drastic that the GAM needs to adjust the increase to begin earlier in order to account for it.

GAM: estimated trend line for non-solvers of the critical element (+; upper panel) and the other operator (=; lower panel). This figure illustrates that the GAM also found a nonlinear increase in the amount of time that non-solvers spend on the critical element after receiving a hint. The orange area indicates where there is a significant shift in attention.

In contrast, the switch points of the MCP analysis correctly capture where the change in attention allocation happens (see Figure 8 ).

MCP analysis of the critical element (+; upper panel) and the other operator (=; lower panel). This figure illustrates that the switch points of the MCP analysis correctly captures where the shift in attention happens.

3.3. Does Metacognition Influence Insight Problem Solving? (Solvers vs. Non-Solvers after the Hint)

The final question we aimed to address was whether the explicit cue, and the additional knowledge about the problem associated with it, would alter the way the problem was solved. Figure 9 indicates that both solvers and non-solvers maintain the level of attention on the critical aspects throughout the problem-solving period, which is a direct consequence of the explicit cue. However, this was not sufficient for finding the solution. The eventual solvers initially shifted their attention to “=” around the 30% bin, but starting from the 50% bin, they increasingly focused on “+”. This means that at this point in time, the solvers may have realized that the “+” symbol was the critical element they needed to solve the problem. Consequently, they gathered more information about the symbol by attending to it more closely.

Raw data and means for each bin of the critical element (+; upper panel) and the other operator (=; lower panel) after the hint was provided. The raw data represent every data point of each participant over the remaining problem-solving period after the hint was given. The error bars represent the 68% confidence interval. This figure illustrates a nonlinear increase or decrease in the time solvers spend on the critical element.

This incremental pattern of solving is well captured by GAMs, as Figure 10 illustrates. While the non-solvers attended to the critical “+” operator consistently over the entire problem-solving period, but on a rather low level of 25% of their time, solvers gradually increased their attention towards it. Significant differences were found in the middle and the end of the problem-solving period. This was also the case for the other operator (=). Non-solvers attended to “=” in a consistent manner throughout the problem-solving period, while the solvers attended to “=” more in the middle of the problem-solving period and less at the very end of it, probably because they were then already focusing more on the “+” sign which needs to be changed for a solution.

GAM: the difference between two estimated trend lines for solvers and non-solvers of the critical element (+; upper panel) and the other operator (=; lower panel). This figure illustrates that the GAM also found a nonlinear increase in the time solvers spend on that particular element. The orange area in the figure indicates regions where there are statistically significant differences between the attention patterns of solvers and non-solvers.

Figure 11 illustrates that the attentional shifts after receiving a hint are effectively captured by the MCP analysis. Again, non-solvers attended to the “+” operator on a consistently low level throughout the entire problem-solving process, while solvers attended to the “+” operator more and more. The same trend is observed for the “=” operator. Non-solvers attended to it less, while solvers shifted their attention to it in the middle of the problem-solving process. Towards the end of the problem-solving process, the data suggest that solvers became aware that the “=” operator was not as important for solving the problem and began to focus more on the “+” operator.

MCP analysis of the critical element after the hint (+; upper panel) and the other operator (=; lower panel) for non-solvers and solvers. This figure demonstrates that the switch points of the MCP analysis correctly captures the incremental pattern of solving for the critical element (+). It also demonstrates that after the hint, the non-solvers attended to the noncritical element (=) more in the beginning but not the critical element (+). As the GAMs showed already, the non-solvers attended to the critical element (+) in the same way throughout the whole problem-solving period.

4. Discussion

We have demonstrated that recording eye movements is a valuable method for gaining insight into complex cognitive processes, including mental restructuring in insight problems. It is also an adequate tool for investigating attentional shifts after receiving hints. However, it is important to use eye movement recording with appropriate analytical approaches. Our results show that it is necessary to conduct a more fine-grained analysis of the eye movement data to capture the temporal dynamics of the problem-solving process. This is particularly relevant for insight problems such as the one used here, which are believed to feature a sudden change in eye movement patterns reflecting a change in mental representation.

We were able to identify the point at which solvers and non-solvers start to differ in their attentional patterns by dividing the problem-solving period into ten equal bins. The temporal resolution of the problem-solving period is one aspect, but it is also important to choose an appropriate statistical procedure. We have demonstrated that nonlinear statistical models, such as GAM and MCP, can effectively capture the sudden change that is a hallmark of insight problem solving. The GAM analysis can effectively capture the attentional shift; however, it predicts that the change occurs prior to the correct reflection of the actual data. While GAM is considerably more flexible than regressions with polynomial terms, the same problem of interdependence of neighboring phases remains. The shift caused by the explicit cue is so drastic that the GAM needs to adjust the increase to begin earlier to account for it. In contrast, the change points of the MCP analysis correctly capture where the change in attention allocation happens. A change point is a time point where the statistical properties of a time series change abruptly. However, in contrast to GAMs, one needs a priori knowledge about the number of change points and the form of the segments in between ( Lindeløv 2020 ). Therefore, one might decide from case to case which statistical procedure is appropriate.

Our example illustrates the importance of considering theoretical assumptions when choosing analytical and statistical procedures. The restructuring of mental representations is a key concept in theories of insight ( Knoblich et al. 1999 ; Ohlsson 1984 , 1992 , 2011 ). It is a nonlinear process in essence, which can be operationalized as a sudden burst of attention to the relevant aspects of a problem ( Bilalić et al. 2019 ). The shift inevitably deviates significantly from participants’ previous problem solving. Seen as a part of the overall problem-solving continuum, the sudden shift is difficult to capture with linear statistical procedures. Only truly nonlinear statistical procedures can appropriately capture the sudden nature of representational change.

Providing explicit hints typically alters the dynamics of problem solving. It is obvious that the given hints were effective, as participants’ patterns of attention show a drastic change, which is very well captured by both GAM and MCP. However, it is important to note that the eventual solvers, after receiving the hint, exhibit a gradual, incremental shift, with increasing attention to the main elements during the problem-solving period. In contrast, non-solvers display an immediate burst of refocusing following the hint, but subsequently, their attention to the important aspects diminishes.

Both the analytical procedure for capturing the temporal resolution and the nonlinear statistical procedures can be easily extended beyond eye movements to other tracing methods. For example, “importance-to-solution” ratings of individual problem elements that are made repeatedly during the solving process ( Durso et al. 1994 ; Cushen and Wiley 2012 ; Danek et al. 2020 ; Danek and Wiley 2020 ) often reveal patterns of sudden change which could be effectively captured by GAMs and MCPs. Similarly, “Feelings-of-Warmth” that are used to assess metacognitive knowledge about solution progress ( Kizilirmak et al. 2018 ; Hedne et al. 2016 ; Pétervári and Danek 2020 ) are another suitable candidate for nonlinear modeling with GAMs. Other tracing methods, such as mouse-tracing data ( Loesche et al. 2018 ; van Rij et al. 2020 ), think-aloud protocols ( Gilhooly et al. 2010 ; Schooler et al. 1993 ; Blech et al. 2020 ), or even self-reports ( Fedor et al. 2015 ), are also better modeled with GAMs than with commonly applied linear methods, even if they are more appropriate than the classical ANOVA.

5. Conclusions

Our results indicate that for insight problems, the restructuring process leaves a discernible trace of suddenness. Eye movements suggest that just prior to solving the problems, participants shift their focus from elements that constituted the initial problem representation to those crucial for the solution. Our results also demonstrate that receiving hints leads to attentional shifts towards critical aspects, which in turn facilitates the generation of a correct solution. However, in order to accurately capture the sudden shift in attention, a combination of the appropriate methodological approach and statistical procedure is necessary. These nonlinear processes are best captured by nonlinear statistical procedures, such as GAMs and MCPs.

Acknowledgments

The help and cooperation from participants is greatly appreciated, as is Matthew Bladen’s contribution in preparing the text.

Supplementary Materials

The following supporting information can be downloaded at: https://osf.io/pwuhs/?view_only=7c52bda4e6fa481e826e5d7570b6ef34 .

Funding Statement

This research was funded by Talent Austria der OeAD-GmbH, finanziert aus Mitteln des österreichischen Bundesministeriums für Wissenschaft, Forschung und Wirtschaft (BMWFW), grant number ICM-2017-07423 given to the first author.

Author Contributions

Conceptualization, M.B. and M.G.; methodology, M.B. and M.G.; software, M.G.; validation, M.B., N.V., A.H.D., and M.G.; formal analysis, M.G. and N.V.; investigation, M.G.; data curation, M.G.; writing—original draft preparation, M.G.; writing—review and editing, M.B., A.H.D., N.V., and M.G.; visualization, M.G., N.V., and M.B.; project administration, M.G. and M.B.; funding acquisition, M.G. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Since this is a re-analysis of our paper ( Bilalić et al. 2019 ), please refer to the original paper for the Institutional Review Board Statement.

Informed Consent Statement

Informed consent was obtained from all subjects involved in the study.

Data Availability Statement

Conflicts of interest.

The authors declare no conflict of interest.

1 The length of time taken to solve (or not solve) a problem is different from person to person, meaning that one cannot compare the eye tracking data directly between people. For example, some may need only 45 s to solve the problem, whereas others need four minutes to find a solution. In consequence, the data must be transformed in order to be able to compare the data between people properly. While the problem-solving period can be extended by adding more time phases, it is important to note that the duration should not be prolonged beyond a certain point. Utilizing too many time frames may leave too little data (e.g., a 10-second trial should not be divided into 100 bins, as each bin will have the duration of only 100 ms). This can lead to distorted eye movement patterns, masking the underlying effects present before the data were binned. On the other hand, choosing too few bins may not capture the full temporal dynamics of the problem-solving process. In either case, ANOVA is not suitable for analyzing a large number of problem-solving periods, unlike GAM and multiple change point analysis, which can easily accommodate a large number of time frames. MCP analysis is another adequate tool for this type of analysis as it can capture the shift of attention. However, in contrast to GAMs, one needs a priori knowledge about the number of change points and the form of the segments in between ( Lindeløv 2020 ).

2 Please note that the data presented here are simulated to represent a sudden shift, which is difficult to capture by classical analyses. The original data in Bilalić et al. ( 2019 ) indicate a gradual shift.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

• Ammalainen Artur, Moroshkina Nadezhda. The effect of true and false unreportable hints on anagram problem solving, restructuring, and the Aha!-experience. Journal of Cognitive Psychology. 2021; 33 :644–58. doi: 10.1080/20445911.2020.1844722. [ CrossRef ] [ Google Scholar ]
• Ash Ivan K., Wiley Jennifer. The nature of restructuring in insight: An individual-differences approach. Psychonomic Bulletin & Review. 2006; 13 :66–73. [ PubMed ] [ Google Scholar ]
• Ash Ivan K., Wiley Jennifer. Hindsight bias in insight and mathematical problem solving: Evidence of different reconstruction mechanisms for metacognitive versus situational judgments. Memory & Cognition. 2008; 36 :822–37. doi: 10.3758/MC.36.4.822. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Ash Ivan K., Jee Benjamin, Wiley Jennifer. Investigating insight as sudden learning. The Journal of Problem Solving. 2012; 4 :150–76. doi: 10.7771/1932-6246.1123. [ CrossRef ] [ Google Scholar ]
• Becker Maxi, Kühn Simone, Sommer Tobias. Verbal insight revisited—Dissociable neurocognitive processes underlying solutions accompanied by an AHA! experience with and without prior restructuring. Journal of Cognitive Psychology. 2021; 33 :659–84. doi: 10.1080/20445911.2020.1819297. [ CrossRef ] [ Google Scholar ]
• Bilalić Merim, McLeod Peter. Why good thoughts block better ones. Scientific American. 2014; 310 :74–79. doi: 10.1038/scientificamerican0314-74. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Bilalić Merim, McLeod Peter, Gobet Fernand. Why good thoughts block better ones: The mechanism of the pernicious Einstellung (set) effect. Cognition. 2008; 108 :652–61. doi: 10.1016/j.cognition.2008.05.005. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Bilalić Merim, McLeod Peter, Gobet Fernand. The Mechanism of the Einstellung (Set) Effect: A Pervasive Source of Cognitive Bias. Current Directions in Psychological Science. 2010; 19 :111–15. doi: 10.1177/0963721410363571. [ CrossRef ] [ Google Scholar ]
• Bilalić Merim, Graf Mario, Vaci Nemanja, Danek Amory H. The temporal dynamics of insight problem solving–restructuring might not always be sudden. Thinking & Reasoning. 2019; 27 :1–37. [ Google Scholar ]
• Blech Christine, Gaschler Robert, Bilalić Merim. Why do people fail to see simple solutions? Using think-aloud protocols to uncover the mechanism behind the Einstellung (mental set) effect. Thinking & Reasoning. 2020; 26 :552–80. [ Google Scholar ]
• Bowden Edward M. The effect of reportable and unreportable hints on anagram solution and the Aha! Experience. Consciousness and Cognition. 1997; 6 :545–73. doi: 10.1006/ccog.1997.0325. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Cushen Patrick J., Wiley Jennifer. Cues to solution, restructuring patterns, and reports of insight in creative problem solving. Consciousness and Cognition. 2012; 21 :1166–75. doi: 10.1016/j.concog.2012.03.013. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Danek Amory H., Wiley Joshua. What causes the insight memory advantage? Cognition. 2020; 205 :104411. doi: 10.1016/j.cognition.2020.104411. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Danek Amory H., Williams Joshua, Wiley Jennifer. Closing the gap: Connecting sudden representational change to the subjective Aha! experience in insightful problem solving. Psychological Research. 2020; 84 :111–19. doi: 10.1007/s00426-018-0977-8. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Duncker Karl. On problem-solving. Psychological Monographs. 1945; 58 :270. doi: 10.1037/h0093599. [ CrossRef ] [ Google Scholar ]
• Durso Francis T., Rea Cornelia B., Dayton Tom. Graph-theoretic confirmation of restructuring during insight. Psychological Science. 1994; 5 :94–97. doi: 10.1111/j.1467-9280.1994.tb00637.x. [ CrossRef ] [ Google Scholar ]
• Ellis Jessica J., Reingold Eyal M. The Einstellung effect in anagram problem solving: Evidence from eye movements. Frontiers in Psychology. 2014; 5 :679. doi: 10.3389/fpsyg.2014.00679. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Ellis Jessica J., Glaholt Mackenzie G., Reingold Eyal M. Eye movements reveal solution knowledge prior to insight. Consciousness and Cognition. 2011; 20 :768–76. doi: 10.1016/j.concog.2010.12.007. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Fedor Anna, Szathmáry Eörs, Öllinger Michael. Problem solving stages in the five square problem. Frontiers in Psychology. 2015; 6 :1050. doi: 10.3389/fpsyg.2015.01050. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Fleck Jessica I., Weisberg Robert W. Insight versus analysis: Evidence for diverse methods in problem solving. Journal of Cognitive Psychology. 2013; 25 :436–63. doi: 10.1080/20445911.2013.779248. [ CrossRef ] [ Google Scholar ]
• Gilhooly Ken J., Fioratou Evie, Henretty N. Verbalization and problem solving: Insight and spatial factors. British Journal of Psychology. 2010; 101 :81–93. doi: 10.1348/000712609X422656. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Hedne Mikael R., Norman Elisabeth, Metcalfe Janet. Intuitive feelings of warmth and confidence in insight and noninsight problem solving of magic tricks. Frontiers in Psychology. 2016; 7 :1314. doi: 10.3389/fpsyg.2016.01314. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Holmqvist Kenneth, Nyström Marcus, Andersson Richard, Dewhurst Richard, Halszka Jarodzka, Van de Weijer Joost. Eye Tracking: A Comprehensive Guide to Methods and Measures. OUP Oxford; Oxford: 2011. [ Google Scholar ]
• Just Marcel A., Carpenter Patricia A. Eye fixations and cognitive processes. Cognitive Psychology. 1976; 8 :441–80. doi: 10.1016/0010-0285(76)90015-3. [ CrossRef ] [ Google Scholar ]
• Kizilirmak Jasmin M., Serger Violetta, Kehl Judith, Öllinger Michael, Folta-Schoofs Kristian, Richardson-Klavehn Alan. Feelings-of-warmth increase more abruptly for verbal riddles solved with in contrast to without Aha! Experience. Frontiers in Psychology. 2018; 9 :1404. doi: 10.3389/fpsyg.2018.01404. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Knoblich Günther, Ohlsson Stellan, Haider Hilde, Rhenius Detlef. Constraint relaxation and chunk decomposition in insight problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition. 1999; 25 :1534–55. doi: 10.1037/0278-7393.25.6.1534. [ CrossRef ] [ Google Scholar ]
• Knoblich Günther, Ohlsson Stellan, Raney Gary E. An eye movement study of insight problem solving. Memory & Cognition. 2001; 29 :1000–9. doi: 10.3758/BF03195762. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Korovkin Sergey, Savinova Anna. The Effectiveness of Metacognitive Hints in Insight Problem Solving. In: Velichkovsky Boris M., Balaban Pavel M., Ushakov Vadim L., editors. Advances in Cognitive Research, Artificial Intelligence and Neuroinformatics. Vol. 1358. Springer; Cham: 2021. Intercognsci 2020. Advances in Intelligent Systems and Computing. [ CrossRef ] [ Google Scholar ]
• Kuhn Gustav, Land Michael F. There’s more to magic than meets the eye. Current Biology. 2006; 16 :R950–R51. doi: 10.1016/j.cub.2006.10.012. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Kuhn Gustav, Tatler Benjamin W., Cole Geoff. G. You look where I look! Effect of gaze cues on overt and covert attention in misdirection. Visual Cognition. 2009; 17 :925–44. doi: 10.1080/13506280902826775. [ CrossRef ] [ Google Scholar ]
• Lindeløv Jonas K. mcp: An R Package for Regression with Multiple Change Points. OSF Preprints. 2020 doi: 10.31219/osf.io/fzqxv. [ CrossRef ] [ Google Scholar ]
• Loesche Frank, Goslin Jeremy, Bugmann Guido. Paving the way to eureka—Introducing “dira” as an experimental paradigm to observe the process of creative problem solving. Frontiers in Psychology. 2018; 9 :1773. doi: 10.3389/fpsyg.2018.01773. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• MacGregor James N., Cunningham J. Barton. The effects of number and level of restructuring in insight problem solving. The Journal of Problem Solving. 2009; 2 :7. doi: 10.7771/1932-6246.1062. [ CrossRef ] [ Google Scholar ]
• Metcalfe Janet, Shimamura Arthur P., editors. Metacognition: Knowing about Knowing. MIT Press; Cambridge: 1994. [ Google Scholar ]
• Newell Allen, Simon Herbert A. Human Problem Solving. Prentice Hall; Englewood Cliffs: 1972. [ Google Scholar ]
• Novick Laura R., Sherman Steven J. On the nature of insight solutions: Evidence from skill differences in anagram solution. The Quarterly Journal of Experimental Psychology Section A. 2003; 56 :351–82. doi: 10.1080/02724980244000288. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Ohlsson Stellan. Restructuring revisited: II. An information processing theory of restructuring and insight. Scandinavian Journal of Psychology. 1984; 25 :117–29. doi: 10.1111/j.1467-9450.1984.tb01005.x. [ CrossRef ] [ Google Scholar ]
• Ohlsson Stellan. Information-processing explanations of insight and related phenomena. In: Keane M., Gilhooly K. J., editors. Advances in the Psychology of Thinking. Harvester-Wheatsheaf; London: 1992. pp. 1–44. [ Google Scholar ]
• Ohlsson Stellan. Deep Learning: How the Mind Overrides Experience. Cambridge University Press; New York: 2011. [ Google Scholar ]
• Öllinger Michael, Jones Gary, Knoblich Günther. Heuristics and representational change in two-move matchstick arithmetic tasks. Advances in Cognitive Psychology. 2006; 2 :239–53. doi: 10.2478/v10053-008-0059-3. [ CrossRef ] [ Google Scholar ]
• Öllinger Michael, Jones Gary, Knoblich Günther. Investigating the effect of mental set on insight problem solving. Experimental Psychology. 2008; 55 :270–82. doi: 10.1027/1618-3169.55.4.269. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Pétervári Judit, Danek Amory H. Problem solving of magic tricks: Guiding to and through an impasse with solution cues. Thinking & Reasoning. 2020; 26 :502–33. doi: 10.1080/13546783.2019.1668479. [ CrossRef ] [ Google Scholar ]
• Raftery Adrian E., Akman V. E. Bayesian analysis of a Poisson process with a change-point. Biometrika. 1986; 73 :85–89. doi: 10.1093/biomet/73.1.85. [ CrossRef ] [ Google Scholar ]
• Rayner Keith. Studies in Visual Information Processing. Vol. 6. Elsevier; Amsterdam: 1995. Eye movements and cognitive processes in reading, visual search, and scene perception; pp. 3–22. [ Google Scholar ]
• Reingold Eyal M., Charness Neil, Schultetus Richard S., Stampe Dave M. Perceptual automaticity in expert chess players: Parallel encoding of chess relations. Psychonomic Bulletin & Review. 2001; 8 :504–10. [ PubMed ] [ Google Scholar ]
• Schooler Jonathan W., Ohlsson Stellan, Brooks Kevin. Thoughts beyond words: When language overshadows insight. Journal of Experimental Psychology: General. 1993; 122 :166. doi: 10.1037/0096-3445.122.2.166. [ CrossRef ] [ Google Scholar ]
• Spiridonov Vladimir, Loginov Nikita, Ardislamov Vladlen. Dissociation between the subjective experience of insight and performance in the CRA paradigm. Journal of Cognitive Psychology. 2021; 33 :685–99. doi: 10.1080/20445911.2021.1900198. [ CrossRef ] [ Google Scholar ]
• Takeuchi Naoyuki, Mori Takayuki, Suzukamo Yoshimi, Izumi Shin I. Activity of prefrontal cortex in teachers and students during teaching of an insight problem. Mind, Brain, and Education. 2019; 13 :167–75. doi: 10.1111/mbe.12207. [ CrossRef ] [ Google Scholar ]
• Tseng Chien-Chih, Chen Ching-Hui, Chen Hsueh-Chih, Sung Yao-Ting, Chang Kuo-En. Verification of Dual Factors theory with eye movements during a matchstick arithmetic insight problem. Thinking Skills and Creativity. 2014; 13 :129–40. doi: 10.1016/j.tsc.2014.04.004. [ CrossRef ] [ Google Scholar ]
• van Rij Jacolien, Vaci Nemanja, Wurm Lee, Feldman Laurie. Alternative quantitative methods in psycholingistics: Implications for theory and design. In: Pirrelli Vito, Plag Ingo, Dressler Wolfgang., editors. Word Knowledge and Word Usage: A Cross-Disciplinary Guide to the Mental Lexicon. De Gruyter Mouton; Berlin and Boston: 2020. pp. 83–126. [ CrossRef ] [ Google Scholar ]
• Wertheimer Max. Über Schlussprozesse im produktiven Denken. In: Wertheimer Max., editor. Drei Abhandlungen zur Gestalttheorie. Verlag der Philosophischen Akademie; Erlangen: 1925. pp. 164–84. [ Google Scholar ]

7.3 Problem-Solving

Learning objectives.

By the end of this section, you will be able to:

• Describe problem solving strategies
• Define algorithm and heuristic
• Explain some common roadblocks to effective problem solving

   People face problems every day—usually, multiple problems throughout the day. Sometimes these problems are straightforward: To double a recipe for pizza dough, for example, all that is required is that each ingredient in the recipe be doubled. Sometimes, however, the problems we encounter are more complex. For example, say you have a work deadline, and you must mail a printed copy of a report to your supervisor by the end of the business day. The report is time-sensitive and must be sent overnight. You finished the report last night, but your printer will not work today. What should you do? First, you need to identify the problem and then apply a strategy for solving the problem.

The study of human and animal problem solving processes has provided much insight toward the understanding of our conscious experience and led to advancements in computer science and artificial intelligence. Essentially much of cognitive science today represents studies of how we consciously and unconsciously make decisions and solve problems. For instance, when encountered with a large amount of information, how do we go about making decisions about the most efficient way of sorting and analyzing all the information in order to find what you are looking for as in visual search paradigms in cognitive psychology. Or in a situation where a piece of machinery is not working properly, how do we go about organizing how to address the issue and understand what the cause of the problem might be. How do we sort the procedures that will be needed and focus attention on what is important in order to solve problems efficiently. Within this section we will discuss some of these issues and examine processes related to human, animal and computer problem solving.

PROBLEM-SOLVING STRATEGIES

   When people are presented with a problem—whether it is a complex mathematical problem or a broken printer, how do you solve it? Before finding a solution to the problem, the problem must first be clearly identified. After that, one of many problem solving strategies can be applied, hopefully resulting in a solution.

Problems themselves can be classified into two different categories known as ill-defined and well-defined problems (Schacter, 2009). Ill-defined problems represent issues that do not have clear goals, solution paths, or expected solutions whereas well-defined problems have specific goals, clearly defined solutions, and clear expected solutions. Problem solving often incorporates pragmatics (logical reasoning) and semantics (interpretation of meanings behind the problem), and also in many cases require abstract thinking and creativity in order to find novel solutions. Within psychology, problem solving refers to a motivational drive for reading a definite “goal” from a present situation or condition that is either not moving toward that goal, is distant from it, or requires more complex logical analysis for finding a missing description of conditions or steps toward that goal. Processes relating to problem solving include problem finding also known as problem analysis, problem shaping where the organization of the problem occurs, generating alternative strategies, implementation of attempted solutions, and verification of the selected solution. Various methods of studying problem solving exist within the field of psychology including introspection, behavior analysis and behaviorism, simulation, computer modeling, and experimentation.

A problem-solving strategy is a plan of action used to find a solution. Different strategies have different action plans associated with them (table below). For example, a well-known strategy is trial and error. The old adage, “If at first you don’t succeed, try, try again” describes trial and error. In terms of your broken printer, you could try checking the ink levels, and if that doesn’t work, you could check to make sure the paper tray isn’t jammed. Or maybe the printer isn’t actually connected to your laptop. When using trial and error, you would continue to try different solutions until you solved your problem. Although trial and error is not typically one of the most time-efficient strategies, it is a commonly used one.

   Another type of strategy is an algorithm. An algorithm is a problem-solving formula that provides you with step-by-step instructions used to achieve a desired outcome (Kahneman, 2011). You can think of an algorithm as a recipe with highly detailed instructions that produce the same result every time they are performed. Algorithms are used frequently in our everyday lives, especially in computer science. When you run a search on the Internet, search engines like Google use algorithms to decide which entries will appear first in your list of results. Facebook also uses algorithms to decide which posts to display on your newsfeed. Can you identify other situations in which algorithms are used?

A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A “rule of thumb” is an example of a heuristic. Such a rule saves the person time and energy when making a decision, but despite its time-saving characteristics, it is not always the best method for making a rational decision. Different types of heuristics are used in different types of situations, but the impulse to use a heuristic occurs when one of five conditions is met (Pratkanis, 1989):

• When one is faced with too much information
• When the time to make a decision is limited
• When the decision to be made is unimportant
• When there is access to very little information to use in making the decision
• When an appropriate heuristic happens to come to mind in the same moment

Working backwards is a useful heuristic in which you begin solving the problem by focusing on the end result. Consider this example: You live in Washington, D.C. and have been invited to a wedding at 4 PM on Saturday in Philadelphia. Knowing that Interstate 95 tends to back up any day of the week, you need to plan your route and time your departure accordingly. If you want to be at the wedding service by 3:30 PM, and it takes 2.5 hours to get to Philadelphia without traffic, what time should you leave your house? You use the working backwards heuristic to plan the events of your day on a regular basis, probably without even thinking about it.

Another useful heuristic is the practice of accomplishing a large goal or task by breaking it into a series of smaller steps. Students often use this common method to complete a large research project or long essay for school. For example, students typically brainstorm, develop a thesis or main topic, research the chosen topic, organize their information into an outline, write a rough draft, revise and edit the rough draft, develop a final draft, organize the references list, and proofread their work before turning in the project. The large task becomes less overwhelming when it is broken down into a series of small steps.

Further problem solving strategies have been identified (listed below) that incorporate flexible and creative thinking in order to reach solutions efficiently.

Additional Problem Solving Strategies :

• Abstraction – refers to solving the problem within a model of the situation before applying it to reality.
• Analogy – is using a solution that solves a similar problem.
• Brainstorming – refers to collecting an analyzing a large amount of solutions, especially within a group of people, to combine the solutions and developing them until an optimal solution is reached.
• Divide and conquer – breaking down large complex problems into smaller more manageable problems.
• Hypothesis testing – method used in experimentation where an assumption about what would happen in response to manipulating an independent variable is made, and analysis of the affects of the manipulation are made and compared to the original hypothesis.
• Lateral thinking – approaching problems indirectly and creatively by viewing the problem in a new and unusual light.
• Means-ends analysis – choosing and analyzing an action at a series of smaller steps to move closer to the goal.
• Method of focal objects – putting seemingly non-matching characteristics of different procedures together to make something new that will get you closer to the goal.
• Morphological analysis – analyzing the outputs of and interactions of many pieces that together make up a whole system.
• Proof – trying to prove that a problem cannot be solved. Where the proof fails becomes the starting point or solving the problem.
• Reduction – adapting the problem to be as similar problems where a solution exists.
• Research – using existing knowledge or solutions to similar problems to solve the problem.
• Root cause analysis – trying to identify the cause of the problem.

The strategies listed above outline a short summary of methods we use in working toward solutions and also demonstrate how the mind works when being faced with barriers preventing goals to be reached.

One example of means-end analysis can be found by using the Tower of Hanoi paradigm . This paradigm can be modeled as a word problems as demonstrated by the Missionary-Cannibal Problem :

Missionary-Cannibal Problem

Three missionaries and three cannibals are on one side of a river and need to cross to the other side. The only means of crossing is a boat, and the boat can only hold two people at a time. Your goal is to devise a set of moves that will transport all six of the people across the river, being in mind the following constraint: The number of cannibals can never exceed the number of missionaries in any location. Remember that someone will have to also row that boat back across each time.

Hint : At one point in your solution, you will have to send more people back to the original side than you just sent to the destination.

The actual Tower of Hanoi problem consists of three rods sitting vertically on a base with a number of disks of different sizes that can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top making a conical shape. The objective of the puzzle is to move the entire stack to another rod obeying the following rules:

• 1. Only one disk can be moved at a time.
• 2. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod.
• 3. No disc may be placed on top of a smaller disk.

  Figure 7.02. Steps for solving the Tower of Hanoi in the minimum number of moves when there are 3 disks.

Figure 7.03. Graphical representation of nodes (circles) and moves (lines) of Tower of Hanoi.

The Tower of Hanoi is a frequently used psychological technique to study problem solving and procedure analysis. A variation of the Tower of Hanoi known as the Tower of London has been developed which has been an important tool in the neuropsychological diagnosis of executive function disorders and their treatment.

GESTALT PSYCHOLOGY AND PROBLEM SOLVING

As you may recall from the sensation and perception chapter, Gestalt psychology describes whole patterns, forms and configurations of perception and cognition such as closure, good continuation, and figure-ground. In addition to patterns of perception, Wolfgang Kohler, a German Gestalt psychologist traveled to the Spanish island of Tenerife in order to study animals behavior and problem solving in the anthropoid ape.

As an interesting side note to Kohler’s studies of chimp problem solving, Dr. Ronald Ley, professor of psychology at State University of New York provides evidence in his book A Whisper of Espionage  (1990) suggesting that while collecting data for what would later be his book  The Mentality of Apes (1925) on Tenerife in the Canary Islands between 1914 and 1920, Kohler was additionally an active spy for the German government alerting Germany to ships that were sailing around the Canary Islands. Ley suggests his investigations in England, Germany and elsewhere in Europe confirm that Kohler had served in the German military by building, maintaining and operating a concealed radio that contributed to Germany’s war effort acting as a strategic outpost in the Canary Islands that could monitor naval military activity approaching the north African coast.

While trapped on the island over the course of World War 1, Kohler applied Gestalt principles to animal perception in order to understand how they solve problems. He recognized that the apes on the islands also perceive relations between stimuli and the environment in Gestalt patterns and understand these patterns as wholes as opposed to pieces that make up a whole. Kohler based his theories of animal intelligence on the ability to understand relations between stimuli, and spent much of his time while trapped on the island investigation what he described as  insight , the sudden perception of useful or proper relations. In order to study insight in animals, Kohler would present problems to chimpanzee’s by hanging some banana’s or some kind of food so it was suspended higher than the apes could reach. Within the room, Kohler would arrange a variety of boxes, sticks or other tools the chimpanzees could use by combining in patterns or organizing in a way that would allow them to obtain the food (Kohler & Winter, 1925).

While viewing the chimpanzee’s, Kohler noticed one chimp that was more efficient at solving problems than some of the others. The chimp, named Sultan, was able to use long poles to reach through bars and organize objects in specific patterns to obtain food or other desirables that were originally out of reach. In order to study insight within these chimps, Kohler would remove objects from the room to systematically make the food more difficult to obtain. As the story goes, after removing many of the objects Sultan was used to using to obtain the food, he sat down ad sulked for a while, and then suddenly got up going over to two poles lying on the ground. Without hesitation Sultan put one pole inside the end of the other creating a longer pole that he could use to obtain the food demonstrating an ideal example of what Kohler described as insight. In another situation, Sultan discovered how to stand on a box to reach a banana that was suspended from the rafters illustrating Sultan’s perception of relations and the importance of insight in problem solving.

Grande (another chimp in the group studied by Kohler) builds a three-box structure to reach the bananas, while Sultan watches from the ground.  Insight , sometimes referred to as an “Ah-ha” experience, was the term Kohler used for the sudden perception of useful relations among objects during problem solving (Kohler, 1927; Radvansky & Ashcraft, 2013).

Solving puzzles.

   Problem-solving abilities can improve with practice. Many people challenge themselves every day with puzzles and other mental exercises to sharpen their problem-solving skills. Sudoku puzzles appear daily in most newspapers. Typically, a sudoku puzzle is a 9×9 grid. The simple sudoku below (see figure) is a 4×4 grid. To solve the puzzle, fill in the empty boxes with a single digit: 1, 2, 3, or 4. Here are the rules: The numbers must total 10 in each bolded box, each row, and each column; however, each digit can only appear once in a bolded box, row, and column. Time yourself as you solve this puzzle and compare your time with a classmate.

How long did it take you to solve this sudoku puzzle? (You can see the answer at the end of this section.)

   Here is another popular type of puzzle (figure below) that challenges your spatial reasoning skills. Connect all nine dots with four connecting straight lines without lifting your pencil from the paper:

Did you figure it out? (The answer is at the end of this section.) Once you understand how to crack this puzzle, you won’t forget.

   Take a look at the “Puzzling Scales” logic puzzle below (figure below). Sam Loyd, a well-known puzzle master, created and refined countless puzzles throughout his lifetime (Cyclopedia of Puzzles, n.d.).

What steps did you take to solve this puzzle? You can read the solution at the end of this section.

Pitfalls to problem solving.

   Not all problems are successfully solved, however. What challenges stop us from successfully solving a problem? Albert Einstein once said, “Insanity is doing the same thing over and over again and expecting a different result.” Imagine a person in a room that has four doorways. One doorway that has always been open in the past is now locked. The person, accustomed to exiting the room by that particular doorway, keeps trying to get out through the same doorway even though the other three doorways are open. The person is stuck—but she just needs to go to another doorway, instead of trying to get out through the locked doorway. A mental set is where you persist in approaching a problem in a way that has worked in the past but is clearly not working now.

Functional fixedness is a type of mental set where you cannot perceive an object being used for something other than what it was designed for. During the Apollo 13 mission to the moon, NASA engineers at Mission Control had to overcome functional fixedness to save the lives of the astronauts aboard the spacecraft. An explosion in a module of the spacecraft damaged multiple systems. The astronauts were in danger of being poisoned by rising levels of carbon dioxide because of problems with the carbon dioxide filters. The engineers found a way for the astronauts to use spare plastic bags, tape, and air hoses to create a makeshift air filter, which saved the lives of the astronauts.

   Researchers have investigated whether functional fixedness is affected by culture. In one experiment, individuals from the Shuar group in Ecuador were asked to use an object for a purpose other than that for which the object was originally intended. For example, the participants were told a story about a bear and a rabbit that were separated by a river and asked to select among various objects, including a spoon, a cup, erasers, and so on, to help the animals. The spoon was the only object long enough to span the imaginary river, but if the spoon was presented in a way that reflected its normal usage, it took participants longer to choose the spoon to solve the problem. (German & Barrett, 2005). The researchers wanted to know if exposure to highly specialized tools, as occurs with individuals in industrialized nations, affects their ability to transcend functional fixedness. It was determined that functional fixedness is experienced in both industrialized and nonindustrialized cultures (German & Barrett, 2005).

In order to make good decisions, we use our knowledge and our reasoning. Often, this knowledge and reasoning is sound and solid. Sometimes, however, we are swayed by biases or by others manipulating a situation. For example, let’s say you and three friends wanted to rent a house and had a combined target budget of $1,600. The realtor shows you only very run-down houses for $1,600 and then shows you a very nice house for $2,000. Might you ask each person to pay more in rent to get the $2,000 home? Why would the realtor show you the run-down houses and the nice house? The realtor may be challenging your anchoring bias. An anchoring bias occurs when you focus on one piece of information when making a decision or solving a problem. In this case, you’re so focused on the amount of money you are willing to spend that you may not recognize what kinds of houses are available at that price point.

The confirmation bias is the tendency to focus on information that confirms your existing beliefs. For example, if you think that your professor is not very nice, you notice all of the instances of rude behavior exhibited by the professor while ignoring the countless pleasant interactions he is involved in on a daily basis. Hindsight bias leads you to believe that the event you just experienced was predictable, even though it really wasn’t. In other words, you knew all along that things would turn out the way they did. Representative bias describes a faulty way of thinking, in which you unintentionally stereotype someone or something; for example, you may assume that your professors spend their free time reading books and engaging in intellectual conversation, because the idea of them spending their time playing volleyball or visiting an amusement park does not fit in with your stereotypes of professors.

Finally, the availability heuristic is a heuristic in which you make a decision based on an example, information, or recent experience that is that readily available to you, even though it may not be the best example to inform your decision . Biases tend to “preserve that which is already established—to maintain our preexisting knowledge, beliefs, attitudes, and hypotheses” (Aronson, 1995; Kahneman, 2011). These biases are summarized in the table below.

Were you able to determine how many marbles are needed to balance the scales in the figure below? You need nine. Were you able to solve the problems in the figures above? Here are the answers.

   Many different strategies exist for solving problems. Typical strategies include trial and error, applying algorithms, and using heuristics. To solve a large, complicated problem, it often helps to break the problem into smaller steps that can be accomplished individually, leading to an overall solution. Roadblocks to problem solving include a mental set, functional fixedness, and various biases that can cloud decision making skills.

References:

Openstax Psychology text by Kathryn Dumper, William Jenkins, Arlene Lacombe, Marilyn Lovett and Marion Perlmutter licensed under CC BY v4.0. https://openstax.org/details/books/psychology

Review Questions:

1. A specific formula for solving a problem is called ________.

a. an algorithm

b. a heuristic

c. a mental set

d. trial and error

2. Solving the Tower of Hanoi problem tends to utilize a  ________ strategy of problem solving.

a. divide and conquer

b. means-end analysis

d. experiment

3. A mental shortcut in the form of a general problem-solving framework is called ________.

4. Which type of bias involves becoming fixated on a single trait of a problem?

a. anchoring bias

b. confirmation bias

c. representative bias

d. availability bias

5. Which type of bias involves relying on a false stereotype to make a decision?

6. Wolfgang Kohler analyzed behavior of chimpanzees by applying Gestalt principles to describe ________.

a. social adjustment

b. student load payment options

c. emotional learning

d. insight learning

7. ________ is a type of mental set where you cannot perceive an object being used for something other than what it was designed for.

a. functional fixedness

c. working memory

Critical Thinking Questions:

1. What is functional fixedness and how can overcoming it help you solve problems?

2. How does an algorithm save you time and energy when solving a problem?

Personal Application Question:

1. Which type of bias do you recognize in your own decision making processes? How has this bias affected how you’ve made decisions in the past and how can you use your awareness of it to improve your decisions making skills in the future?

anchoring bias

availability heuristic

confirmation bias

functional fixedness

hindsight bias

problem-solving strategy

representative bias

trial and error

working backwards

Answers to Exercises

algorithm:  problem-solving strategy characterized by a specific set of instructions

anchoring bias:  faulty heuristic in which you fixate on a single aspect of a problem to find a solution

availability heuristic:  faulty heuristic in which you make a decision based on information readily available to you

confirmation bias:  faulty heuristic in which you focus on information that confirms your beliefs

functional fixedness:  inability to see an object as useful for any other use other than the one for which it was intended

heuristic:  mental shortcut that saves time when solving a problem

hindsight bias:  belief that the event just experienced was predictable, even though it really wasn’t

mental set:  continually using an old solution to a problem without results

problem-solving strategy:  method for solving problems

representative bias:  faulty heuristic in which you stereotype someone or something without a valid basis for your judgment

trial and error:  problem-solving strategy in which multiple solutions are attempted until the correct one is found

working backwards:  heuristic in which you begin to solve a problem by focusing on the end result

Share This Book

• Increase Font Size

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

• Publications
• Account settings
• My Bibliography
• Collections
• Citation manager

Save citation to file

Email citation, add to collections.

• Create a new collection
• Add to an existing collection

Add to My Bibliography

Your saved search, create a file for external citation management software, your rss feed.

• Search in PubMed
• Search in NLM Catalog
• Add to Search

Investigating the effect of mental set on insight problem solving

Affiliation.

• 1 Parmenides Center for the Study of Thinking, Munich, Germany. [email protected]
• PMID: 18683624
• DOI: 10.1027/1618-3169.55.4.269

Mental set is the tendency to solve certain problems in a fixed way based on previous solutions to similar problems. The moment of insight occurs when a problem cannot be solved using solution methods suggested by prior experience and the problem solver suddenly realizes that the solution requires different solution methods. Mental set and insight have often been linked together and yet no attempt thus far has systematically examined the interplay between the two. Three experiments are presented that examine the extent to which sets of noninsight and insight problems affect the subsequent solutions of insight test problems. The results indicate a subtle interplay between mental set and insight: when the set involves noninsight problems, no mental set effects are shown for the insight test problems, yet when the set involves insight problems, both facilitation and inhibition can be seen depending on the type of insight problem presented in the set. A two process model is detailed to explain these findings that combines the representational change mechanism with that of proceduralization.

PubMed Disclaimer

Similar articles

• Once more with feeling: Normative data for the aha experience in insight and noninsight problems. Webb ME, Little DR, Cropper SJ. Webb ME, et al. Behav Res Methods. 2018 Oct;50(5):2035-2056. doi: 10.3758/s13428-017-0972-9. Behav Res Methods. 2018. PMID: 29052169 Review.
• In search of the 'Aha!' experience: Elucidating the emotionality of insight problem-solving. Shen W, Yuan Y, Liu C, Luo J. Shen W, et al. Br J Psychol. 2016 May;107(2):281-98. doi: 10.1111/bjop.12142. Epub 2015 Jul 17. Br J Psychol. 2016. PMID: 26184903
• The cognitive neuroscience of insight. Kounios J, Beeman M. Kounios J, et al. Annu Rev Psychol. 2014;65:71-93. doi: 10.1146/annurev-psych-010213-115154. Annu Rev Psychol. 2014. PMID: 24405359 Review.
• Working wonders? investigating insight with magic tricks. Danek AH, Fraps T, von Müller A, Grothe B, Ollinger M. Danek AH, et al. Cognition. 2014 Feb;130(2):174-85. doi: 10.1016/j.cognition.2013.11.003. Epub 2013 Dec 1. Cognition. 2014. PMID: 24300080
• Neural activity when people solve verbal problems with insight. Jung-Beeman M, Bowden EM, Haberman J, Frymiare JL, Arambel-Liu S, Greenblatt R, Reber PJ, Kounios J. Jung-Beeman M, et al. PLoS Biol. 2004 Apr;2(4):E97. doi: 10.1371/journal.pbio.0020097. Epub 2004 Apr 13. PLoS Biol. 2004. PMID: 15094802 Free PMC article.
• Variability and harshness shape flexible strategy-use in support of the constrained flexibility framework. Pope-Caldwell S, Deffner D, Maurits L, Neumann T, Haun D. Pope-Caldwell S, et al. Sci Rep. 2024 Mar 27;14(1):7236. doi: 10.1038/s41598-024-57800-w. Sci Rep. 2024. PMID: 38538731 Free PMC article.
• Tracing Cognitive Processes in Insight Problem Solving: Using GAMs and Change Point Analysis to Uncover Restructuring. Graf M, Danek AH, Vaci N, Bilalić M. Graf M, et al. J Intell. 2023 May 3;11(5):86. doi: 10.3390/jintelligence11050086. J Intell. 2023. PMID: 37233335 Free PMC article.
• Search and insight processes in card sorting games. Öllinger M, Szathmáry E, Fedor A. Öllinger M, et al. Front Psychol. 2023 May 5;14:1118976. doi: 10.3389/fpsyg.2023.1118976. eCollection 2023. Front Psychol. 2023. PMID: 37213381 Free PMC article.
• Transcranial direct current stimulation (tDCS) targeting the postcentral gyrus reduces malevolent creative ideation. Gao Z, Lu K, Hao N. Gao Z, et al. Soc Cogn Affect Neurosci. 2023 Mar 22;18(1):nsad019. doi: 10.1093/scan/nsad019. Soc Cogn Affect Neurosci. 2023. PMID: 36961729 Free PMC article.
• Surprise! Why Insightful Solution Is Pleasurable. Savinova A, Korovkin S. Savinova A, et al. J Intell. 2022 Nov 7;10(4):98. doi: 10.3390/jintelligence10040098. J Intell. 2022. PMID: 36412778 Free PMC article.

Publication types

• Search in MeSH

LinkOut - more resources

Full text sources.

• Ovid Technologies, Inc.

• Citation Manager

NCBI Literature Resources

MeSH PMC Bookshelf Disclaimer

The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). Unauthorized use of these marks is strictly prohibited.

American Psychological Association

Younger workers feel stressed, lonely, and undervalued

Nearly half of workers aged 18–25 say they feel lonely at work, according to APA’s 2024 Work in America survey

U.S. workers adjust to changing nature of employment

Survey highlights include remote work, four-day workweeks, and AI adoption

5 ways to improve employee mental health

Supportive workplace practices can boost employee well-being, company morale

Psychological safety in the changing workplace

Survey shows link with job satisfaction, including creativity and innovation

Membership in APA

APA Community

A new exclusive destination tailored for APA members

Membership benefits

Unlock the tools, discounts, and services included with your membership

Renew your membership

Keep your benefits and access to leading psychological information

Psychology topics spotlight

Misinformation and disinformation

Resources to navigate trauma

Tips to foster a healthy workplace

Science and practice of psychology

Ethics Code

Continuing Education

Grants, Awards, and Funding

Standards and Guidelines

Networks and communities

Network with peers, enhance your professional development, expand your personal growth, and more

APA Divisions

High school teachers

Undergraduate educators

Graduate students

Early career psychologists

Managing your career

Resources to help you throughout your career in psychology, including finding a job, salary data, finances and money management, mentoring and supervision, and training and professional development

Explore career paths

Psychologist profiles

How did you get that job?

Events and training

Featured jobs

Apa publications and products.

Write with clarity, precision, and inclusion

Children’s books

Monitor on Psychology

Newsletters

Reports and surveys

Continuing education

Merchandise

Real Siblings

Jacob's Missing Book

Harper Becomes a Big Sister

Attachment-Based Family Therapy for Sexual and Gender Minority Young Adults and Their Non-Accepting Parents

Dismantling Everyday Discrimination

APA Services

Learn how you can help APA advocate for psychology-informed federal policy and legislation, and support psychological research

APA Services, Inc.

A companion professional organization to APA, serving all members and advocating for psychology

Help | Advanced Search

Computer Science > Computation and Language

Title: chain of preference optimization: improving chain-of-thought reasoning in llms.

Abstract: The recent development of chain-of-thought (CoT) decoding has enabled large language models (LLMs) to generate explicit logical reasoning paths for complex problem-solving. However, research indicates that these paths are not always deliberate and optimal. The tree-of-thought (ToT) method employs tree-searching to extensively explore the reasoning space and find better reasoning paths that CoT decoding might overlook. This deliberation, however, comes at the cost of significantly increased inference complexity. In this work, we demonstrate that fine-tuning LLMs leveraging the search tree constructed by ToT allows CoT to achieve similar or better performance, thereby avoiding the substantial inference burden. This is achieved through Chain of Preference Optimization (CPO), where LLMs are fine-tuned to align each step of the CoT reasoning paths with those of ToT using the inherent preference information in the tree-search process. Extensive experimental results show that CPO significantly improves LLM performance in solving a variety of complex problems, including question answering, fact verification, and arithmetic reasoning, demonstrating its effectiveness. Our code is available at this https URL .

Submission history

Access paper:.

• HTML (experimental)
• Other Formats

References & Citations

• Google Scholar
• Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

• Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

Online ordering is currently unavailable due to technical issues. We apologise for any delays responding to customers while we resolve this. For further updates please visit our website: https://www.cambridge.org/news-and-insights/technical-incident

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings .

Login Alert

• > The Psychology of Problem Solving
• > The Acquisition of Expert Performance as Problem Solving: Construction and Modification of Mediating Mechanisms through Deliberate Practice

Book contents

• Frontmatter
• Contributors
• PART I INTRODUCTION
• 1 Recognizing, Defining, and Representing Problems
• 2 The Acquisition of Expert Performance as Problem Solving: Construction and Modification of Mediating Mechanisms through Deliberate Practice
• PART II RELEVANT ABILITIES AND SKILLS
• PART III STATES AND STRATEGIES
• PART IV CONCLUSION AND INTEGRATION

2 - The Acquisition of Expert Performance as Problem Solving: Construction and Modification of Mediating Mechanisms through Deliberate Practice

Published online by Cambridge University Press:  05 June 2012

How do experts reach their high level of performance? Recent reviews (Ericsson, 1996, 1998b, 2001; Ericsson & Lehmann, 1996) dispel the common belief that “talented” expert performers attain very high levels of performance virtually automatically through cumulative domain-related experience. Instead, empirical evidence strongly implies that even the most “talented” individuals in a domain must spend over ten years actively engaging in particular practice activities (deliberate practice) that lead to gradual improvements in skill and adaptations that increase performance.

In this chapter I argue that the acquisition of expert performance can be described as a sequence of mastered challenges with increasing levels of difficulty, such as playing pieces of music, performing challenging gymnastic routines, and solving complex mathematical problems. Different levels of mastery present the learner with different kinds of problems that must be solved for the skill to develop further. And each individual's path toward skilled performance is distinct; it depends on when technical challenges were encountered and the specific methods used to help the individuals continue their development.

When beginners are first introduced to a domain of expertise they can successfully perform only the most simple tasks and activities. With the aid of instruction and training many individuals are able to master increasingly difficult tasks, thus gradually improving and slowly approaching the level of expert performers. The incremental nature of gaining mastery means that tasks that were initially impossible to perform can be executed effortlessly as increased skill is attained.

Access options

Save book to kindle.

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle .

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service .

• The Acquisition of Expert Performance as Problem Solving: Construction and Modification of Mediating Mechanisms through Deliberate Practice
• By K. Anders Ericsson , Florida State University
• Edited by Janet E. Davidson , Lewis and Clark College, Portland , Robert J. Sternberg , Yale University, Connecticut
• Book: The Psychology of Problem Solving
• Online publication: 05 June 2012
• Chapter DOI: https://doi.org/10.1017/CBO9780511615771.003

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox .

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive .

share this!

June 12, 2024

This article has been reviewed according to Science X's editorial process and policies . Editors have highlighted the following attributes while ensuring the content's credibility:

fact-checked

peer-reviewed publication

trusted source

Psychology researchers find collaborative imagination increases social connection

by University at Albany

The ability to imagine is pivotal for human development, driving creativity and problem-solving. It may also influence our relationship with others, according to new research.

A team of researchers, led by Brendan O'Connor of the University at Albany, has developed a new framework for investigating imagination as a process in which individuals co-create shared representations of hypothetical events —what they're calling "collaborative imagination."

Using the framework, O'Connor and colleagues from the University of British Columbia in Canada and University of Nottingham in the UK, believe they are making important strides toward understanding the ways that imagination fosters social connection .

Findings were published in the Proceedings of the National Academy of Sciences .

"The research community has learned a great deal over the last few decades about the science of imagination," said O'Connor, an associate professor in UAlbany's Department of Psychology and director of the Imagination and Moral Cognition Lab.

"However, it has been viewed as an individual process, focusing on how people imagine personal events independently. Our research instead explores imagination as a collaborative process, in a way that has never been done before."

Imagining a future together

To observe collaborative imagination, the researchers conducted two observational studies . The first was in-person with 120 undergraduate students , and the second via Zoom with 124 participants from the general public.

Participants were assigned to pairs and introduced to each other. They were then asked to imagine future events in as much detail as possible, either collaboratively with the other participant, or independently in a separate room. Another experiment involved either collaborating on a game with the other participant or collaboratively discussing an image depicting an event with people and objects in a specific location.

The researchers used a social connection rating scale and natural language processing tools to examine the impact of collaborative imagination across the different experiments.

"Our research has strong potential to change the way we view imagination by providing a novel theoretical framework and evidence that imagination itself is a socially creative process," said Zoë Fowler, a graduate researcher in O'Connor's lab, who helped develop the study concept and experiment design.

In both studies, results revealed that collaboratively imagining a shared future increased social connection among participants, compared to imagining a shared future independently or collaborating on non-imaginative tasks.

Furthermore, collaborative imagination increased participants' engagement in mentalizing (for example, considering their partner's thoughts and feelings) and heightened the vividness of the imagined event.

"These findings shed new light on the nature and structure of imagination with implications for better understanding interpersonal relationships, future thinking and the formation of collective beliefs across social networks," said Daniela Palombo, an assistant professor in the Department of Psychology at the University of British Columbia and study co-author. "We believe the findings will be relevant to researchers in a variety of disciplines."

Using imagination for social good

The new findings are the latest in a series of research articles published by O'Connor investigating the impacts of imagination, including a study from 2019.

His lab has a particular interest in understanding how imagining the future and remembering past events can influence the moral decisions we make in the present.

"Individuals imagining future events together can lead to the formation of collective beliefs," said O'Connor. "We hope to continue building on our findings in a way that helps people harness the power of imagination to make positive changes in their everyday lives and society at large."

Journal information: Proceedings of the National Academy of Sciences

Provided by University at Albany

Explore further

Feedback to editors

Saturday Citations: Bacterial warfare, a self-programming language model, passive cooling in the big city

15 hours ago

Some CRISPR screens may be missing cancer drug targets

21 hours ago

Novel photocatalyst enables efficient ester reduction with blue light

Jun 15, 2024

Physicists confirm quantum entanglement persists between top quarks, the heaviest known fundamental particles

Jun 14, 2024

25 years of massive fusion energy experiment data open on the 'cloud' and available to everyone

Quantum entangled photons react to Earth's spin

Q&A: Barrier islands and dunes protect coastlines, but how are environmental changes affecting them and adjacent land?

A new weapon in the battle against antibiotic resistance: Temperature

Sharks have depleted functional diversity compared to the last 66 million years, study finds

Quebec lake meteorite impact yields rare rocks and evidence of extreme heat

Relevant physicsforums posts, what is your favorite drawing.

7 hours ago

Another Word I Got Wrong : Vile

14 hours ago

Cover songs versus the original track, which ones are better?

22 hours ago

Biographies, history, personal accounts

Tell us about left-right hand coordination when playing musical instruments, especially for piano, today's fusion music: t square, cassiopeia, rei & kanade sato.

Jun 12, 2024

More from Art, Music, History, and Linguistics

Related Stories

Infant-toddlers' play and imagination sparked by adult involvement

Jun 5, 2024

How humans struggle to differentiate imagination from reality

Apr 21, 2023

Why creative experts may be better at imagining the future

May 8, 2019

Researchers find connection between autobiographical memory and aphantasia

Feb 27, 2024

Imagination can reduce pain

Feb 6, 2018

The secret to creativity – according to science

Jan 3, 2018

Recommended for you

Fans of long-running TV show experienced grief similar to losing a close friend when show ended, study finds

Jun 13, 2024

Greater gender equality associated with men eating meat more frequently than women, study finds

Study shows facially expressive people to be more likable and socially successful

Study suggests ambivalence and polarized views can promote political violence

Are men dissatisfied with their penis size more likely to own a gun? Researchers find out

Jun 11, 2024

Study shows the power of social connections to predict hit songs

Let us know if there is a problem with our content.

Use this form if you have come across a typo, inaccuracy or would like to send an edit request for the content on this page. For general inquiries, please use our contact form . For general feedback, use the public comments section below (please adhere to guidelines ).

Please select the most appropriate category to facilitate processing of your request

Thank you for taking time to provide your feedback to the editors.

Your feedback is important to us. However, we do not guarantee individual replies due to the high volume of messages.

E-mail the story

Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. The information you enter will appear in your e-mail message and is not retained by Phys.org in any form.

Newsletter sign up

Get weekly and/or daily updates delivered to your inbox. You can unsubscribe at any time and we'll never share your details to third parties.

More information Privacy policy

Donate and enjoy an ad-free experience

We keep our content available to everyone. Consider supporting Science X's mission by getting a premium account.

E-mail newsletter

Information

• Author Services

Initiatives

You are accessing a machine-readable page. In order to be human-readable, please install an RSS reader.

All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited. For more information, please refer to https://www.mdpi.com/openaccess .

Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications.

Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive positive feedback from the reviewers.

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

Original Submission Date Received: .

• Active Journals
• Find a Journal
• Proceedings Series
• For Authors
• For Reviewers
• For Editors
• For Librarians
• For Publishers
• For Societies
• For Conference Organizers
• Open Access Policy
• Institutional Open Access Program
• Special Issues Guidelines
• Editorial Process
• Research and Publication Ethics
• Article Processing Charges
• Testimonials
• Preprints.org
• SciProfiles
• Encyclopedia

Article Menu

• Subscribe SciFeed
• Recommended Articles
• Author Biographies
• Google Scholar
• on Google Scholar
• Table of Contents

Find support for a specific problem in the support section of our website.

Please let us know what you think of our products and services.

Visit our dedicated information section to learn more about MDPI.

JSmol Viewer

Parameter identification of flexible link manipulators using evolutionary algorithms.

1. Introduction

2. materials and methods, 2.1. flexible-link manipulator dynamics, 2.2. parameter identification approach, 3. case study: one-link flexible manipulator, 3.1. testbed, 3.2. numerical model, 4. results and discussion.

• The parameters used by the DE algorithm [ 35 ] are the following: population size N P = 100, weighting factor F = 0.5, crossover probability C R = 0.8, 100 generations, and D E / r a n d / 1 / e x p strategy for the generation of candidates.
• The parameters used by the GA algorithm [ 36 ] are the following: N P = 100, selection rate S R = 0.5, crossover rate C R = 0.8, mutation rate M R = 0.2, and 100 generations.
• The parameters used by the PSO algorithm [ 32 ] are the following: number of particles N P = 100, inertia weigth w = 1.4, c 1 = 1.5, c 2 = 2.5, and 100 iterations.
• The stopping criteria considered was the maximum number of generations/iterations.
• The study cases were run 10 times, and the average values were obtained.
• To establish a fair comparison among the evolutionary algorithms, the seeds 0, 1, 2, …, 9 were used to initialize the random generator for each simulation.
• The aforementioned case studies, using DE, the GA, and PSO, were run 10 times to obtain the upcoming average values.

Model Validation

• Three different test inputs were considered for the torque applied by the servomotor that permits assessment of the numerical and experimental dynamic response: triangular (see Figure 8 a), pulse (see Figure 9 a), and sinusoidal that considers the positive part (see Figure 10 a). These inputs are three different signal profiles of torque that produce different angular accelerations at the flexible link. These torques were applied from 0 (seg) to 0.3 (seg) to move the joint angle ϕ to a maximum angular displacement of 80 (deg). Figure 8 a, Figure 9 a and Figure 10 a show the torque applied torque that was measured using the current sensor of the servomotor.
• The identified parameters p ^ considered in the numerical model were obtained from the best case of DE, and these parameters are presented in Table 2 .
• The numerical and experimental outputs of the joint angle ϕ for the corresponding test inputs are presented in Figure 8 b, Figure 9 b and Figure 10 b.
• The numerical and experimental outputs of link’s tip acceleration u ¨ for the corresponding test inputs are presented in Figure 8 c, Figure 9 c and Figure 10 c. Moreover, the frequency response functions (toque input/link’s tip acceleration) for the numerical and experimental outputs are also computed in Figure 8 d, Figure 9 d and Figure 10 d.
• For the error analysis, the error between the numerical model and experimental outputs in terms of the joint angle ϕ and the F R F s were estimated based on the Normalized Root Mean Square Error ( R M S E ) according to the expressions of Equation ( 7 ).

5. Conclusions

Data availability statement, acknowledgments, conflicts of interest.

• Cordier, J.; Friconneau, J.; Gargiulo, L.; Grisolia, C.; Palmer, J.; Perrot, Y.; Samaille, F. Articulated inspection arm for ITER, a demonstration in the Tore Supra tokamak. In Proceedings of the 20th IEEE/NPSS Symposium on Fusion Engineering, San Diego, CA, USA, 14–17 October 2003; IEEE: Piscataway, NJ, USA, 2003; pp. 197–200. [ Google Scholar ]
• Zhang, Y.; Lu, M. A review of recent advancements in soft and flexible robots for medical applications. Int. J. Med. Robot. Comput. Assist. Surg. 2020 , 16 , e2096. [ Google Scholar ] [ CrossRef ]
• Sąsiadek, J. Space robotics and its challenges. In Aerospace Robotics: Selected Papers from I Conference on Robotics in Aeronautics and Astronautics ; Springer: Berlin/Heidelberg, Germany, 2013; pp. 1–8. [ Google Scholar ]
• Li, B.; Li, X.; Gao, H.; Wang, F.Y. Advances in Flexible Robotic Manipulator Systems—Part I: Overview and Dynamics Modeling Methods. IEEE/ASME Trans. Mechatron. 2024 , 29 , 1100–1110. [ Google Scholar ] [ CrossRef ]
• Lara-Molina, F.A.; Gonçalves, R.S. Reliability-based optimization of flexible manipulators. J. Vib. Eng. Technol. 2023 , 11 , 3147–3162. [ Google Scholar ] [ CrossRef ]
• Lismonde, A.; Sonneville, V.; Brüls, O. A geometric optimization method for the trajectory planning of flexible manipulators. Multibody Syst. Dyn. 2019 , 47 , 347–362. [ Google Scholar ] [ CrossRef ]
• Lara-Molina, F.A.; Dumur, D.; Assolari Takano, K. Multi-objective optimal design of flexible-joint parallel robot. Eng. Comput. 2018 , 35 , 2775–2801. [ Google Scholar ] [ CrossRef ]
• Sayahkarajy, M.; Mohamed, Z.; Mohd Faudzi, A.A. Review of modelling and control of flexible-link manipulators. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 2016 , 230 , 861–873. [ Google Scholar ] [ CrossRef ]
• Wu, J.; Wang, J.; You, Z. An overview of dynamic parameter identification of robots. Robot. Comput.-Integr. Manuf. 2010 , 26 , 414–419. [ Google Scholar ] [ CrossRef ]
• Cammarata, A.; Sinatra, R.; Rigano, A.; Lombardo, M.; Maddio, P.D. Design of a large deployable reflector opening system. Machines 2020 , 8 , 7. [ Google Scholar ] [ CrossRef ]
• Urrea, C.; Pascal, J. Design, simulation, comparison and evaluation of parameter identification methods for an industrial robot. Comput. Electr. Eng. 2018 , 67 , 791–806. [ Google Scholar ] [ CrossRef ]
• Pires, I.; Ayala, H.V.H.; Weber, H.I. Nonlinear ensemble gray and black-box system identification of friction induced vibrations in slender rotating structures. Mech. Syst. Signal Process. 2023 , 186 , 109815. [ Google Scholar ] [ CrossRef ]
• Yazdizadeh, A.; Khorasani, K.; Patel, R.V. Identification of a two-link flexible manipulator using adaptive time delay neural networks. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 2000 , 30 , 165–172. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• Liu, K.; Sun, X. System identification and model reduction for a single-link flexible manipulator. J. Sound Vib. 2001 , 242 , 867–891. [ Google Scholar ] [ CrossRef ]
• Ziaei, K.; Wang, D.W. Application of orthonormal basis functions for identification of flexible-link manipulators. Control Eng. Pract. 2006 , 14 , 99–106. [ Google Scholar ] [ CrossRef ]
• Zhu, C.; Wang, J.; Chen, Z.; Liu, B. Dynamic characteristic parameters identification analysis of a parallel manipulator with flexible links. J. Mech. Sci. Technol. 2014 , 28 , 4833–4840. [ Google Scholar ] [ CrossRef ]
• Pappalardo, C.M.; Lök, Ş.İ.; Malgaca, L.; Guida, D. Experimental modal analysis of a single-link flexible robotic manipulator with curved geometry using applied system identification methods. Mech. Syst. Signal Process. 2023 , 200 , 110629. [ Google Scholar ] [ CrossRef ]
• Lara-Molina, F.A. Modeling of Flexible-Link Manipulators Under Uncertain Parameters Based on Stochastic Finite Element Method. J. Mech. Robot. 2022 , 14 , 061002. [ Google Scholar ] [ CrossRef ]
• Meng, D.; She, Y.; Xu, W.; Lu, W.; Liang, B. Dynamic modeling and vibration characteristics analysis of flexible-link and flexible-joint space manipulator. Multibody Syst. Dyn. 2018 , 43 , 321–347. [ Google Scholar ] [ CrossRef ]
• Mehrjooee, O.; Fathollahi Dehkordi, S.; Habibnejad Korayem, M. Dynamic modeling and extended bifurcation analysis of flexible-link manipulator. Mech. Based Des. Struct. Mach. 2020 , 48 , 87–110. [ Google Scholar ] [ CrossRef ]
• Perry, M.; Koh, C.; Choo, Y. Modified genetic algorithm strategy for structural identification. Comput. Struct. 2006 , 84 , 529–540. [ Google Scholar ] [ CrossRef ]
• Trinh, T.N.; Koh, C.G. An improved substructural identification strategy for large structural systems. Struct. Control Health Monit. 2012 , 19 , 686–700. [ Google Scholar ] [ CrossRef ]
• Tam, J.H.; Ong, Z.C.; Ismail, Z.; Ang, B.C.; Khoo, S.Y.; Li, W.L. Inverse identification of elastic properties of composite materials using hybrid GA-ACO-PSO algorithm. Inverse Probl. Sci. Eng. 2018 , 26 , 1432–1463. [ Google Scholar ] [ CrossRef ]
• Wang, X.; Zhang, G.; Wang, X.; Ni, P. Output-only structural parameter identification with evolutionary algorithms and correlation functions. Smart Mater. Struct. 2020 , 29 , 035018. [ Google Scholar ] [ CrossRef ]
• Zhou, H.; Zhang, G.; Wang, X.; Ni, P.; Zhang, J. A hybrid identification method on butterfly optimization and differential evolution algorithm. Smart Struct. Syst. Int. J. 2020 , 26 , 345–360. [ Google Scholar ]
• Xu, X.; Lin, P. Parameter identification of sound absorption model of porous materials based on modified particle swarm optimization algorithm. PLoS ONE 2021 , 16 , e0250950. [ Google Scholar ] [ CrossRef ] [ PubMed ]
• Yoshikawa, T.; Hosoda, K. Modeling of flexible manipulators using virtual rigid links and passive joints. Int. J. Robot. Res. 1996 , 15 , 290–299. [ Google Scholar ] [ CrossRef ]
• Theodore, R.J.; Ghosal, A. Comparison of the assumed modes and finite element models for flexible multilink manipulators. Int. J. Robot. Res. 1995 , 14 , 91–111. [ Google Scholar ] [ CrossRef ]
• Jonker, J.B.; Aarts, R.G. A perturbation method for dynamic analysis and simulation of flexible manipulators. Multibody Syst. Dyn. 2001 , 6 , 245–266. [ Google Scholar ] [ CrossRef ]
• Usoro, P.B.; Nadira, R.; Mahil, S.S. A Finite Element/Lagrange Approach to Modeling Lightweight Flexible Manipulators. J. Dyn. Syst. Meas. Control 1986 , 108 , 198–205. [ Google Scholar ] [ CrossRef ]
• Inman, D.J.; Singh, R.C. Engineering Vibration ; Prentice Hall: Englewood Cliffs, NJ, USA, 1994; Volume 3. [ Google Scholar ]
• Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the ICNN’95—International Conference on Neural Networks, Perth, WA, Australia, 27 November–1 December 1995; IEEE: Piscataway, NJ, USA, 1995; Volume 4, pp. 1942–1948. [ Google Scholar ]
• Lara-Molina, F.A.; Dumur, D. A fuzzy approach for the kinematic reliability assessment of robotic manipulators. Robotica 2021 , 39 , 2095–2109. [ Google Scholar ] [ CrossRef ]
• Lara-Molina, F.A.; Dumur, D. Robust multi-objective optimization of parallel manipulators. Meccanica 2021 , 56 , 2843–2860. [ Google Scholar ] [ CrossRef ]
• Storn, R.; Price, K. Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 1997 , 11 , 341–359. [ Google Scholar ] [ CrossRef ]
• Holland, J.H. Genetic algorithms. Sci. Am. 1992 , 267 , 66–73. [ Google Scholar ] [ CrossRef ]

Share and Cite

Lara-Molina, F.A. Parameter Identification of Flexible Link Manipulators Using Evolutionary Algorithms. Machines 2024 , 12 , 409. https://doi.org/10.3390/machines12060409

Lara-Molina FA. Parameter Identification of Flexible Link Manipulators Using Evolutionary Algorithms. Machines . 2024; 12(6):409. https://doi.org/10.3390/machines12060409

Lara-Molina, Fabian Andres. 2024. "Parameter Identification of Flexible Link Manipulators Using Evolutionary Algorithms" Machines 12, no. 6: 409. https://doi.org/10.3390/machines12060409

Article Metrics

Article access statistics, further information, mdpi initiatives, follow mdpi.

Subscribe to receive issue release notifications and newsletters from MDPI journals