novel problem solving

What is creative problem-solving?

Table of Contents

An introduction to creative problem-solving.

Creative problem-solving is an essential skill that goes beyond basic brainstorming . It entails a holistic approach to challenges, melding logical processes with imaginative techniques to conceive innovative solutions. As our world becomes increasingly complex and interconnected, the ability to think creatively and solve problems with fresh perspectives becomes invaluable for individuals, businesses, and communities alike.

Importance of divergent and convergent thinking

At the heart of creative problem-solving lies the balance between divergent and convergent thinking. Divergent thinking encourages free-flowing, unrestricted ideation, leading to a plethora of potential solutions. Convergent thinking, on the other hand, is about narrowing down those options to find the most viable solution. This dual approach ensures both breadth and depth in the problem-solving process.

Emphasis on collaboration and diverse perspectives

No single perspective has a monopoly on insight. Collaborating with individuals from different backgrounds, experiences, and areas of expertise offers a richer tapestry of ideas. Embracing diverse perspectives not only broadens the pool of solutions but also ensures more holistic and well-rounded outcomes.

Nurturing a risk-taking and experimental mindset

The fear of failure can be the most significant barrier to any undertaking. It's essential to foster an environment where risk-taking and experimentation are celebrated. This involves viewing failures not as setbacks but as invaluable learning experiences that pave the way for eventual success.

The role of intuition and lateral thinking

Sometimes, the path to a solution is not linear. Lateral thinking and intuition allow for making connections between seemingly unrelated elements. These 'eureka' moments often lead to breakthrough solutions that conventional methods might overlook.

Stages of the creative problem-solving process

The creative problem-solving process is typically broken down into several stages. Each stage plays a crucial role in understanding, addressing, and resolving challenges in innovative ways.

Clarifying: Understanding the real problem or challenge

Before diving into solutions, one must first understand the problem at its core. This involves asking probing questions, gathering data, and viewing the challenge from various angles. A clear comprehension of the problem ensures that effort and resources are channeled correctly.

Ideating: Generating diverse and multiple solutions

Once the problem is clarified, the focus shifts to generating as many solutions as possible. This stage champions quantity over quality, as the aim is to explore the breadth of possibilities without immediately passing judgment.

Developing: Refining and honing promising solutions

With a list of potential solutions in hand, it's time to refine and develop the most promising ones. This involves evaluating each idea's feasibility, potential impact, and any associated risks, then enhancing or combining solutions to maximize effectiveness.

Implementing: Acting on the best solutions

Once a solution has been honed, it's time to put it into action. This involves planning, allocating resources, and monitoring the results to ensure the solution is effectively addressing the problem.

Techniques for creative problem-solving

Solving complex problems in a fresh way can be a daunting task to start on. Here are a few techniques that can help kickstart the process:

Brainstorming

Brainstorming is a widely-used technique that involves generating as many ideas as possible within a set timeframe. Variants like brainwriting (where ideas are written down rather than spoken) and reverse brainstorming (thinking of ways to cause the problem) can offer fresh perspectives and ensure broader participation.

Mind mapping

Mind mapping is a visual tool that helps structure information, making connections between disparate pieces of data. It is particularly useful in organizing thoughts, visualizing relationships, and ensuring a comprehensive approach to a problem.

SCAMPER technique

SCAMPER stands for Substitute, Combine, Adapt, Modify, Put to another use, Eliminate, and Reverse. This technique prompts individuals to look at existing products, services, or processes in new ways, leading to innovative solutions.

Benefits of creative problem-solving

Creative problem-solving offers numerous benefits, both at the individual and organizational levels. Some of the most prominent advantages include:

Finding novel solutions to old problems

Traditional problems that have resisted conventional solutions often succumb to creative approaches. By looking at challenges from fresh angles and blending different techniques, we can unlock novel solutions previously deemed impossible.

Enhanced adaptability in changing environments

In our rapidly evolving world, the ability to adapt is critical. Creative problem-solving equips individuals and organizations with the agility to pivot and adapt to changing circumstances, ensuring resilience and longevity.

Building collaborative and innovative teams

Teams that embrace creative problem-solving tend to be more collaborative and innovative. They value diversity of thought, are open to experimentation, and are more likely to challenge the status quo, leading to groundbreaking results.

Fostering a culture of continuous learning and improvement

Creative problem-solving is not just about finding solutions; it's also about continuous learning and improvement. By encouraging an environment of curiosity and exploration, organizations can ensure that they are always at the cutting edge, ready to tackle future challenges head-on.

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New Model for Solving Novel Problems Uses Mental Map

• by Andy Fell
• September 02, 2021

How do we make decisions about a situation we have not encountered before? New work from the Center for Mind and Brain at the University of California, Davis, shows that we can solve abstract problems in the same way that we can find a novel route between two known locations — by using an internal cognitive map. The work is published Aug. 31 in the journal Nature Neuroscience .

Humans and animals have a great ability to solve novel problems by generalizing from existing knowledge and inferring new solutions from limited data. This is much harder to achieve with artificial intelligence.

Animals (including humans) navigate by creating a representative map of the outside world in their head as they move around. Once we know two locations are close to each other, we can infer that there is a shortcut between them even if we haven’t been there. These maps make use of a network of “grid cells” and “place cells” in parts of the brain.

In earlier work , Professor Erie Boorman, postdoctoral researcher Seongmin (Alex) Park, Douglas Miller and colleagues showed that human volunteers could construct a similar cognitive map for abstract information. The volunteers were given limited information about people in a two-dimensional social network, ranked by relative competence and popularity. The researchers found that the volunteers could mentally reconstruct this network, represented as a grid, without seeing the original.

The new work takes the research further by testing if people can actually use such a map to find the answers to novel problems.

Matchmaking entrepreneurs

As before, volunteers learned about 16 people they were told were entrepreneurs, ranked on axes of competence and popularity. They never saw the complete grid, only comparisons between pairs.

They were then asked to select business partners for individual entrepreneurs that would maximize growth potential for a business they started together. The assumption was that an entrepreneur scoring high in competence but low on popularity would be complemented by one with a higher popularity score. 

“For example, would Mark Zuckerberg be better off collaborating with Bill Gates or Richard Branson?” Boorman said. (The actual experiment did not use real people.)

While the volunteers were performing the decision task, the researchers scanned their brains with functional magnetic resonance imaging, or fMRI. 

If the volunteers were using the grid cells inside their head to infer the answer, that should be measurable with a tailored analysis approach applied to the fMRI signal, Boorman said.

“It turns out the system in the brain does show the signature of these trajectories being computed on the fly,” he said. “It looks like they are leveraging the cognitive map.”

Computing solutions on the fly

In other words, we can take in loosely connected or fragmentary information, assemble it into a mental map, and use it to infer solutions to new problems.  

Scientists have considered that the brain makes decisions by computing the value of each choice into a common currency, which allows them to be compared in a single dimension, Park said. For example, people might typically choose wine A over wine B based on price, but we know that our preference can be changed by the food you will pair with the wine.

“Our study suggests that the human brain does not have a wine list with fixed values, but locates wines in an abstract multidimensional space, which allows for computing new decision values flexibly according to the current demand,” he said.

The cognitive map allows for computation on the fly with limited information, Boorman said.

“It’s useful when the decisions are novel,” he said. “It’s a totally new framework for understanding decision making.”

The navigational map in rodents is located in the entorhinal cortex, an “early” part of the brain. The cognitive map in humans expands into other parts of the brain including the prefrontal cortex and posterior medial cortex. These brain areas are part of the default mode network, a large “always on” brain network involved in autobiographical memory, imagination, planning and the theory of mind.

The work was supported by the National Science Foundation and National Institutes of Health. 

Media Resources

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Media Contacts:

• Seongmin Park, Center for Mind and Brain, [email protected]
• Erie Boorman, Center for Mind and Brain and Department of Psychology, [email protected]
• Andy Fell, News and Media Relations, 530-304-8888, [email protected]

Primary Category

Secondary categories.

New model for solving novel problems uses mental map

How do we make decisions about a situation we have not encountered before? New work from the Center for Mind and Brain at the University of California, Davis, shows that we can solve abstract problems in the same way that we can find a novel route between two known locations -- by using an internal cognitive map. The work is published Aug. 31 in the journal Nature Neuroscience .

Humans and animals have a great ability to solve novel problems by generalizing from existing knowledge and inferring new solutions from limited data. This is much harder to achieve with artificial intelligence.

Animals (including humans) navigate by creating a representative map of the outside world in their head as they move around. Once we know two locations are close to each other, we can infer that there is a shortcut between them even if we haven't been there. These maps make use of a network of "grid cells" and "place cells" in parts of the brain.

In earlier work, Professor Erie Boorman, postdoctoral researcher Seongmin (Alex) Park, Douglas Miller and colleagues showed that human volunteers could construct a similar cognitive map for abstract information. The volunteers were given limited information about people in a two-dimensional social network, ranked by relative competence and popularity. The researchers found that the volunteers could mentally reconstruct this network, represented as a grid, without seeing the original.

The new work takes the research further by testing if people can actually use such a map to find the answers to novel problems.

Matchmaking entrepreneurs

As before, volunteers learned about 16 people they were told were entrepreneurs, ranked on axes of competence and popularity. They never saw the complete grid, only comparisons between pairs.

They were then asked to select business partners for individual entrepreneurs that would maximize growth potential for a business they started together. The assumption was that an entrepreneur scoring high in competence but low on popularity would be complemented by one with a higher popularity score.

"For example, would Mark Zuckerberg be better off collaborating with Bill Gates or Richard Branson?" Boorman said. (The actual experiment did not use real people.)

While the volunteers were performing the decision task, the researchers scanned their brains with functional magnetic resonance imaging, or fMRI.

If the volunteers were using the grid cells inside their head to infer the answer, that should be measurable with a tailored analysis approach applied to the fMRI signal, Boorman said.

"It turns out the system in the brain does show the signature of these trajectories being computed on the fly," he said. "It looks like they are leveraging the cognitive map."

Computing solutions on the fly

In other words, we can take in loosely connected or fragmentary information, assemble it into a mental map, and use it to infer solutions to new problems.

Scientists have considered that the brain makes decisions by computing the value of each choice into a common currency, which allows them to be compared in a single dimension, Park said. For example, people might typically choose wine A over wine B based on price, but we know that our preference can be changed by the food you will pair with the wine.

"Our study suggests that the human brain does not have a wine list with fixed values, but locates wines in an abstract multidimensional space, which allows for computing new decision values flexibly according to the current demand," he said.

The cognitive map allows for computation on the fly with limited information, Boorman said.

"It's useful when the decisions are novel," he said. "It's a totally new framework for understanding decision making."

The navigational map in rodents is located in the entorhinal cortex, an "early" part of the brain. The cognitive map in humans expands into other parts of the brain including the prefrontal cortex and posterior medial cortex. These brain areas are part of the default mode network, a large "always on" brain network involved in autobiographical memory, imagination, planning and the theory of mind.

The work was supported by the National Science Foundation and National Institutes of Health.

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Story Source:

Materials provided by University of California - Davis . Original written by Andy Fell. Note: Content may be edited for style and length.

Journal Reference :

• Seongmin A. Park, Douglas S. Miller, Erie D. Boorman. Inferences on a multidimensional social hierarchy use a grid-like code . Nature Neuroscience , 2021; 24 (9): 1292 DOI: 10.1038/s41593-021-00916-3

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Knowledge Representations: Individual Differences in Novel Problem Solving

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Data can be accessed by contacting the authors of the study.

The present study investigates how the quality of knowledge representations contributes to rule transfer in a problem-solving context and how working memory capacity (WMC) might contribute to the subsequent failure or success in transferring the relevant information. Participants were trained on individual figural analogy rules and then asked to rate the subjective similarity of the rules to determine how abstract their rule representations were. This rule representation score, along with other measures (WMC and fluid intelligence measures), was used to predict accuracy on a set of novel figural analogy test items, of which half included only the trained rules, and half were comprised of entirely new rules. The results indicated that the training improved performance on the test items and that WMC largely explained the ability to transfer rules. Although the rule representation scores did not predict accuracy on the trained items, rule representation scores did uniquely explain performance on the figural analogies task, even after accounting for WMC and fluid intelligence. These results indicate that WMC plays a large role in knowledge transfer, even when transferring to a more complex problem-solving context, and that rule representations may be important for novel problem solving.

1. Knowledge Representations: Individual Differences in Novel Problem Solving

Research on novel problem solving (i.e., problems with which the solver is not already familiar) is incredibly diverse, with problem solving being studied in the context of intelligence and reasoning (e.g., Bethell-Fox et al. 1984 ; Carpenter et al. 1990 ; Snow 1980 ), analogical transfer (e.g., Cushen and Wiley 2018 ; Gick and Holyoak 1980 ), expertise (e.g., Chi et al. 1981 ; Novick 1988 ; Wiley 1998 ), and even skill acquisition (e.g., Anderson 1987 ; Patsenko and Altmann 2010 ). Separately, these domains have addressed different aspects of problem solving (e.g., learning, transfer, knowledge, individual differences, strategy use, etc.) but the lack of communication across those areas has left a large hole in our understanding of how all of these complex processes interact with each other. In particular, studies involving reasoning tasks primarily focus on the role of stable individual differences, such as working memory capacity (WMC; Ackerman et al. 2005 ; Jarosz and Wiley 2012 ; Unsworth et al. 2014 ). In contrast, studies using classic problem-solving tasks or domain-specific tasks (e.g., physics problems) have focused more on strategies and knowledge ( Chi et al. 1981 ; Holyoak and Koh 1987 ; Novick 1988 ). Although individual differences in reasoning and problem solving ( Kubricht et al. 2017 ; Ricks et al. 2007 ; Sohn and Doane 2003 , 2004 ) have been studied from both a working memory and a knowledge perspective, there remain questions about how the two interact.

2. Working Memory Capacity and Problem Solving

Much of the early work that investigated individual differences in problem solving used tasks designed to measure fluid intelligence (Gf) because they have a great degree of variability and are intended to be novel, and thus they should not be driven by individual differences in knowledge ( Carroll 1993 ). These tasks were helpful in trying to isolate the non-knowledge-based cognitive processes that contributed to reasoning and problem solving. Early work in this area used tasks such as geometric analogies ( Bethell-Fox et al. 1984 ) or Raven’s Advanced Progressive Matrices (RAPM; Raven et al. 1962 ) to assess Gf. In many of these tasks, the stimuli are a series of shapes and patterns that change according to rules. The objective of the task is to extract the rules and apply them. An example of a figural analogy problem is shown in Figure 1 .

Figural analogy example. Note. An example figural analogy item used in the current study. The answer to this item is B.

Specific attributes of the stimuli have been shown to increase the difficulty of the problems. These features include how many rules and transformations are included, as well as the number of objects or elements in the problem ( Bethell-Fox et al. 1984 ; Mulholland et al. 1980 ). The difficulty in the maintenance of these transformations and elements has been ascribed to individual differences in WMC, with storage limits and attentional control being a large barrier to maintaining and manipulating all of the necessary information in memory ( Bethell-Fox et al. 1984 ; Carpenter et al. 1990 ; Jarosz and Wiley 2012 ; Mulholland et al. 1980 ; Primi 2001 ).

Though there are currently many different theories that postulate different structures for working memory ( Baddeley 2000 ; Barrouillet et al. 2004 ; Cowan 2005 ; Oberauer et al. 2003 ; Unsworth 2016 ), Unsworth ( 2016 ) suggested three components that make up WMC: primary memory, attentional control, and retrieval from secondary memory. Each of these subcomponents may play an important role in the relationship between WMC and Gf ( Unsworth et al. 2014 ), and together explain virtually all of their shared variance. For example, the capacity account ( Carpenter et al. 1990 ) of the WMC and Gf relationship would argue that primary memory allows one to hold the various rules, objects, or goals and subgoals required in a temporary storage space during problem solving. According to the distraction account ( Jarosz and Wiley 2012 ; Wiley et al. 2011 ), attentional control provides the necessary resources to focus on desired information while ignoring irrelevant or distracting information coming from within or between problems. Retrieval from secondary memory is the process of retrieving previously learned information and can be useful in retrieving rules and transformations that correctly led to previous solutions, a key process in the learning account ( Verguts and De Boeck 2002 ). Each of these components explains unique variance in WMC ( Unsworth 2016 ; Unsworth et al. 2014 ; Unsworth and Spillers 2010 ) and appear to uniquely contribute to problem solving and reasoning ( Unsworth et al. 2014 ). However, although these subcomponents can fully explain WMC’s relationship with Gf, WMC does not account for all of the variability found in Gf tasks ( Ackerman et al. 2005 ; Kane et al. 2005 ; Oberauer et al. 2005 ).

Although the role of WMC has been heavily studied within reasoning tasks, the mechanistic role it plays during solution and how WMC processes differ from other cognitive processes involved in reasoning tasks remain unclear. Kovacs and Conway ( 2016 ) argue that Gf is unlikely to be a unitary construct and, rather, performance on Gf tasks reflects multiple basic processes that are necessary for most Gf tasks. Given that WMC is also comprised of more basic processes and that the shared processes between WMC and Gf do not account for all of the variability in Gf ( Ackerman et al. 2005 ; Kovacs and Conway 2016 ; Unsworth et al. 2014 ), other mechanisms must be considered to fully understand individual differences in reasoning and problem solving.

3. Individual Differences in Knowledge

The reasoning literature primarily focuses on tasks where the solver has no knowledge, but this is not entirely reflective of problem solving encountered in a real-world setting. In most cases, solvers will have some knowledge pertaining to the problem or will be able to look up information about the problem. Understanding when and how information from memory is used during problem solving is crucial to understanding problem solving as a whole.

3.1. Expertise and Representation

Several studies investigating the effects of expertise on problem solving have shown drastic differences in how experts solve problems in their field when compared to novices. The classic study by Chase and Simon ( 1973 ) demonstrated an extreme form of chunking, with chess experts recalling four times the information that novices did. Additional research indicated that WMC could help to compensate for a lack of knowledge ( Sohn and Doane 2003 , 2004 ), but also that expertise could compensate for a lower WMC, as knowledge becomes more proceduralized and automatic ( Patsenko and Altmann 2010 ).

In addition to having access to more information in memory, experts also show distinct differences from novices in how they represent that information in memory. Experts tend to focus on deeper, more semantically driven representations of problems, whereas novices are more likely to focus on surface features that may not actually be helpful in solving the problem ( Chi et al. 1981 ). Experts’ knowledge representations tend to be highly interconnected, which enables them to move quickly and freely between concepts that are related ( Kohl and Finkelstein 2008 ). Experts are also better at solving problems that are lacking in a clear goal or representation ( Simon 1977 ) by adding in constraints and elaborating on the problem, while novices simply start listing answers or responses to the problem ( Voss et al. 1983 ). Indeed, experts have been shown to create and use goals and subgoals when stuck, whereas novices engage more in exploration ( Hershey et al. 1990 ; Kohl and Finkelstein 2008 ). Experts’ ability to create goals, Hershey et al. ( 1990 ) argue, comes from having plan-like scripts or schemas to help direct them. These highly developed scripts for familiar problems function similarly to productions in the skill acquisition literature, where the scripts and plans increase automaticity. Taken together, this suggests that experts do not simply have more knowledge, but are creating interconnected, abstract representations that benefit them in future problems.

3.2. Analogical Transfer

The analogical transfer literature focuses on the ability to generalize previously learned information given in a source problem to a new target problem. Looking at how participants solve a target problem after being given a structurally identical source, multiple studies have found that participants are unlikely to spontaneously transfer solutions ( Gick and Holyoak 1980 , 1983 ; Holyoak and Koh 1987 ; Novick 1988 ), and even hints do not guarantee transfer ( Gick and Holyoak 1983 ). However, giving participants additional source problems, especially those presented in a spaced-out structure ( Gick and Holyoak 1983 ; Wharton et al. 1994 ), as well as giving participants a delay between problems ( Holyoak and Koh 1987 ), has been shown to be helpful in building better representations. Findings from the skill acquisition literature also replicate the positive effects of including additional source problems and presenting them in a spaced presentation ( Carlson and Yaure 1990 ). Creating a deeper and richer representation appears to be a key element in transferring knowledge.

Although the representation of the source is crucial for understanding structural similarities between analogs, surface similarities (i.e., how much the problems superficially resemble each other) between the source and the analog also contribute to analogical transfer ( Holyoak and Koh 1987 ) by acting as a cue that there is relevant information in memory. In the absence of surface similarities, solvers must rely on their own internal ability to retrieve a previously learned solution. If solvers represent the source based upon its surface features, then it is less likely that solvers will easily retrieve the source when given a dissimilar target. However, if solvers represent the source in a more general and abstract way (i.e., an emphasis on the deep structure of the source problem, and not surface features), then the target should be a sufficient cue to retrieve the source because the structure, rather than the surface features, is the focus point.

There is some evidence to suggest that the ability to generalize previously learned solutions may actually be driven by WMC or reasoning processes. Kubricht et al. ( 2017 ) indicated a positive relationship between analogical transfer and Gf, with additional aids (e.g., providing more source problems) benefitting low-Gf individuals, whereas high-Gf individuals performed well regardless of whether those aids were present. This suggests that high-Gf individuals can more easily form general and abstract representations of the source that can then be transferred more readily. Similar studies have demonstrated the role of diffuse attention in analogical transfer ( Cushen and Wiley 2018 ). Increased attentional resources stemming from WMC may cause fixation on irrelevant details that prevent an individual from noticing that a source and analog are related. In the absence of a more general representation, diffuse attention may help in noticing more remote relationships. Furthermore, the ability to access remote ideas may be helpful in generating more general representations. The results from a study by Cushen and Wiley ( 2018 ) demonstrate that performance on the Remote Associates Test does indeed predict analogical transfer, even after accounting for WMC. Furthermore, Remote Associates Test performance also predicted the completeness of someone’s representation (as measured by a summary of the source problem). However, these results are incongruent with Storm and Bui ( 2016 ), who found that the propensity to mind wander negatively predicted analogical transfer, even if hints were included. It is possible that performance on the Remote Associates Test taps into an ability to reach remote ideas that is not driven by diffuse attention, or that diffuse attention is only helpful for a portion of the analogical transfer process. Cushen and Wiley ( 2018 ) note that once a relationship has been established, the solver must still map the source to the analog. Thus, analogical transfer as a whole may require flexibility in the capacity to use multiple processes.

3.3. Learning on Novel Tasks

Despite the fact that many tasks used to study reasoning are intended to provide novel situations, that does not mean that learning cannot occur throughout the task. Several studies involving Gf tasks have demonstrated that participants learn the rules during the task and use them as they approach new problems ( Bors and Vigneau 2003 ; Harrison et al. 2015 ; Loesche et al. 2015 ; Verguts and De Boeck 2002 ). Although participants may be using this information to help them solve problems, the ability to learn these rules well and apply them is more complicated than simply solving a problem correctly. Loesche et al. ( 2015 ) provided participants with the rules before completing the RAPM, and although performance did improve, it did not increase to such a point that knowing the rules was sufficient for solving all problems on the RAPM. They noted that this finding was especially sensitive to how well participants initially learned the rules. This is consistent with previously discussed findings in both the skill acquisition literature ( Carlson and Yaure 1990 ; Cooper and Sweller 1987 ) as well as studies looking at analogical transfer ( Gick and Holyoak 1983 ; Kubricht et al. 2017 ; Wharton et al. 1994 ). Interestingly, increases in performance due to rule knowledge do not necessarily change the validity of the tests. Schneider et al. ( Schneider et al. 2020 ; Schneider and Sparfeldt 2021a , 2021b ) demonstrated that though learning the rules could increase test performance across several Gf tasks, this increase did not change those tasks’ correlations with other measures of intelligence. This further emphasizes the need to understand how, mechanistically, learning rules influences solution success.

Learning on the RAPM is also sensitive to feedback, as well as exposure to rules throughout the task ( Verguts and De Boeck 2002 ). Additionally, even if participants are given the same problems over multiple sessions, they are inconsistent in solving them. This is true even if they correctly solve the problem the first time ( Bors and Vigneau 2003 ). These results clearly demonstrate that learning and knowledge do play a role in reasoning tasks, but also that previously solving a problem is not a guarantee for solving the same or a similar problem again. An individual’s ability to recognize that previously learned information is relevant, as well as their ability to successfully retrieve the relevant information, are additional processes to consider in conjunction with problem solving or reasoning processes.

Problem solving is clearly a complex task that draws upon many different cognitive processes. Individual differences in WMC are a large contributing factor to problem solving success ( Ash and Wiley 2006 ; Carpenter et al. 1990 ; Sohn and Doane 2003 ; Unsworth et al. 2014 ), but there also appears to be room for other processes. Individual differences in knowledge also play a significant role in problem solving ( Bors and Vigneau 2003 ; Chi et al. 1981 ; Luchins 1942 ; Wiley 1998 ). This knowledge can come in the form of expertise or as information that was learned on a previous problem, depending on the task. There is, however, a more complex relationship between knowledge and problem solving. More knowledge does not necessarily mean improved performance ( Bors and Vigneau 2003 ; Gick and Holyoak 1980 ). Sometimes knowledge can lead to fixation ( Luchins 1942 ; Wiley 1998 ), sometimes knowledge is not necessary if an individual has high levels of other cognitive abilities, such as WMC ( Sohn and Doane 2003 , 2004 ), and sometimes knowledge cannot be used because the individual is unable to transfer it to another problem ( Gick and Holyoak 1983 ; Holyoak and Koh 1987 ; Kubricht et al. 2017 ). What seems to be generally beneficial is the production of a general, abstract representation. In addition, the roles of WMC and knowledge are not necessarily completely distinct, and they likely influence each other.

5. The Present Study

Although the combination of several literatures has provided information on how knowledge and WMC contribute to and interact during problem solving, there are several research questions that have remained unanswered. Of primary concern are the differential problem-solving processes accounted for by WMC and knowledge. WMC clearly contributes to problem solving, but the mechanisms through which it acts remain under debate ( Carpenter et al. 1990 ; Verguts and De Boeck 2002 ; Wiley et al. 2011 ). Because WMC may contribute to knowledge at the encoding stage as well as the retrieval stage, it is important to identify what is breaking down when a solver fails to find or transfer a solution.

To start to isolate knowledge-specific problem-solving processes from general cognitive processes (e.g., attention) in the present study, several changes were made to existing methods in the problem-solving domain. Concerns with analogical transfer stimuli (i.e., using a small sample of fairly difficult problems) were resolved by using a figural analogy task. This ensures that there are multiple target and source problems, as well as making it easier to create problem isomorphs. Additionally, a reasoning task contains more active problem solving than when solely using classical analogical transfer materials. This ensures that there is enough variability in the task that individual differences in problem-solving skill will still be measured.

To keep track of knowledge and transfer throughout the task, stimuli were constructed such that all problems could be decomposed into rules. For example, one problem may require two objects to swap locations, a rotation of those objects, as well as a size change. Another problem may only require a rotation and a size change. Problems were considered unique based upon the combination of rules. To test the effects of increased knowledge on problem solving and transfer, participants first learned a select set of individual rules during a training phase, then were tested on more complex problems involving multiple rules. Figure 2 provides an illustration of the general procedure.

Diagram of figural analogy procedure.

To avoid biasing participants towards a particular representation of the rule, participants were not given the rule explicitly. Rather, rules were learned by solving simple problems that contained only one of those rules, (e.g., a problem where the answer is a single size change). The single-rule problems were designed to be solved fairly easily, such that participants could induce the rule on their own. This was accomplished by only using one rule, reducing the visual complexity of items compared to the standard figural analogies task, and reducing the number of response options from five to four. Furthermore, two variations of each rule were shown during the training phase.

During the test portion, participants solved items that included at least two rules. Half of the test items were comprised of only the rules given in the training portion, and the other half were comprised of entirely new rules. If participants were able to successfully transfer the rules learned in the training portion, even in a context where multiple pieces of information must be used, then participants should have higher accuracy on the test items that were comprised of the trained rules when compared to the items that consist of new rules.

Both the expertise literature and the analogical transfer literature have demonstrated that knowledge representations are crucial to knowledge being used, with an emphasis on structure and less emphasis on surface features being particularly beneficial for transferring that knowledge. As such, the representation quality of the rules may be critically important. To measure this, participants were asked to rate numerically how similar they thought two problems were. Participants were shown only the A:B component of a problem rather than the full version with A:B :: C:? so that they could focus on the rule and not on problem solving. The A:B components they saw came from the rules shown in the initial training phase problems. This ensured that participants were making similarity judgements on items for which they had successfully induced the rule previously. Of primary interest was how closely the individual rated slightly different versions of the same rules compared to completely different rules. For example, did the solver rate a 90-degree rotation rule and a 45-degree rotation rule as equally different when compared to a rotation rule and a size-change rule, or did they treat the two versions of the rotation rule as equally similar as two different 90-degree rotation problems? Where participants fell on this spectrum was used to assess their representation, with participants ranging from more general representations (the 90-degree rotation and 45-degree rotation were very similar) to more item-specific, less general representations (the 90-degree rotation and 45-degree rotation were treated as completely separate rules).

If rule representations do indeed help to facilitate retrieval, then more general representations should correlate with higher accuracy on the trained-rule problems. Ratings on the rule representation measure may also correlate with performance on novel-rule problems because the processes that facilitate the generation of general representations may also play a role in other reasoning processes. However, it is expected that there would be a stronger relationship for the trained-rule problems because the novel-rule problems should not be drawing on retrieval processes as much.

The final set of factors to consider is the role of individual differences in WMC and reasoning. It is unclear whether WMC and/or Gf help to develop more general representations, but it is reasonable to expect that WMC and Gf will interact with the training. Because WMC may play a large role in initially learning the rules and then later facilitating the retrieval of those rules during problem solving, WMC should have a stronger relationship with accuracy for the trained items, and Gf should have a stronger relationship with accuracy for the novel items. Furthermore, relationships between the individual differences measures and the rule representation measures can be investigated. Specifically, one question is whether or not the rule representation measure can explain unique variability on the figural analogies task, or if WMC/Gf determine the quality of the representation. If the rule representation measure does uniquely explain performance on the figural analogies task after accounting for WMC and Gf, then this would provide a novel measure that could potentially explain solution transfer. If, however, the knowledge representation measure can be explained by WMC/Gf, then this would provide some further specificity on why WMC and/or Gf explain performance. Whether rule representations correlate or are predicted by WMC or Gf will also be investigated. This will help to further clarify if rule representations are truly unique, share some processes with WMC and/or Gf, or can be entirely explained by WMC and/or Gf.

6.1. Participants

The target sample size for the study, after accounting for participants who fail to complete the tasks correctly or outliers, was set at 200 participants. This was based on simulated analyses demonstrating that 200 participants would be sufficient for detecting moderate effect sizes within a regression model containing a continuous predictor variable, a binary predictor variable, and an interaction term between the two predictor variables. Furthermore, correlations ranging from .3 to .5 begin to stabilize (remain within a window of ±.1 at 80% confidence) at around 212 and 143 participants, respectively, indicating that 200 participants should be sufficient in detecting moderate correlation coefficients as well ( Schönbrodt and Perugini 2013 ). However, because the study was administered entirely online and previously collected online studies at this university have had attrition rates ranging from 40–50%, a new target of 400 participants was set. A total of 418 participants, collected from a pool of Mississippi State University students, completed all tasks in the study. Of those 418 participants, 173 participants failed to meet the inclusion criteria for one or more of the tasks. Twenty-five additional participants were removed for falling outside of 2.5 standard deviations away from the mean on at least one of the tasks or were multivariate outliers, according to Mahalanobis distance ( Ghorbani 2019 ). This resulted in a final sample of 220 participants.

6.2. Materials

6.2.1. modified figural analogies task.

The modified figural analogies task is based on the figural analogies test by Lohman and Hagen ( 2001 ; Form 6, Level H, Test 8 of the cognitive abilities test). Some of the items from the original task were used in the modified version, but most of the items were created specifically for this experiment. In the standard figural analogies task, participants are shown a set of objects in the form of A:B :: C:? Participants must induce the rule(s) governing the changes from A to B and then apply that rule to the analog, C. Participants select their answer from five response options. An example item is shown in Figure 1 .

The modified figural analogies task was split into three parts: a training portion, the Rule-Similarity Judgement Task (RSJT), and a test portion (see Figure 2 ). For the training portion, participants solved figural analogies items that consisted of only one rule. Thus, participants could learn the rule easily, but it was still through their own induction processes. Then, for the RSJT, participants saw the A:B portion of two figural analogy items and rated how similar they thought the two rules were. The RSJT only used rules that were present in the training portion. Lastly, participants solved a series of figural analogy test items that more closely resembled the items normally shown in the figural analogies task. The test items were comprised of two or three rules, with some items being comprised of rules seen in the training portion and other items being entirely novel. Participants solved both types of problems and the two sets were counterbalanced across participants so that each set of rules was used as the training set and as the novel set at some point.

Figural Analogies Training

The participants solved 14 problems in the training portion; 2 problems for each rule. The two problems for each rule were slightly different versions of the rule. For example, for the “size change” rule, one problem was a size change where the object increased in size and the other was a size change where the object decreased in size. This resulted in a total of 14 single-rule items to be used in the training portion for each set, and 28 different versions in total. Participants only solved one set of 14 problems (7 rules) in the training portion because the other set of 7 rules was used to make up the novel rules condition in the test portion.

Prior to data collection, a larger set of rules were piloted. Some of the rules originated from the original figural analogies task, whereas others were generated specifically for this study. Pilot data indicated that the final set of 14 rules were all treated as distinct by participants and that the expected interpretation of the rule matched with participants’ verbal descriptions of the rules. For example, many participants described a rule wherein an object duplicates as “double”, “duplicating”, “multiply by 2”, “copy shape” or some other variation. These types of responses were consistent for the other rules, wherein there was some variation in the words participants chose, but the overall explanation aligned with the expected interpretation. The full list of rules, and which set they belonged to, can be found in Appendix A . Participants were given an unlimited amount of time to solve the training items. Because the validity of the RSJT and the effect of training both depended on participants successfully inducing the rules, participants were removed from analyses if they missed more than 4 problems in the training portion. Of the 418 participants that completed all tasks, 89 failed to meet the inclusion criteria for the training portion.

Rule-Similarity Judgement Task (RSJT)

After solving the 14 training items, participants made similarity judgements on the rules using a scale that ranged from 0 to 100, in intervals of 5. They were shown all of the items in a random order and had unlimited time to make the judgement.

Participants were shown only the A:B portion of the problem so that they could focus on only the rule. The rules used in the similarity judgement portion were the same rules used in the training portion, so participants had already demonstrated their ability to induce the single rule, which is necessary for making a comparison. Each comparison could be categorized into three conditions: the exact same version of the same rule (“same-version comparison”), two different versions of the same rule (“different-version comparison”), and different versions of different rules (“different-rule comparison”). Examples of the stimuli are provided in Figure 3 . An example of the same-version comparison condition would be a comparison between two different 90-degree rotation items. An example of the different-version condition would be a 90-degree rotation item and a 45-degree rotation item comparison. An example of the different-rule condition would be a 90-degree rotation item being compared to an increase in size; the size-change rule.

RSJT comparison stimuli example. Note. The same version comparison example includes two swap rules, the different version comparison includes two slightly different swap rules, and the different rule comparison includes a swap rule and a size change rule.

Pilot analyses indicated that participants were responsive to the different types of comparisons, with the average rating for same-version comparison being the highest ( M = 82.33, SD = 28.55), and the average rating for different-version comparison ( M = 56.56, SD = 35.95) being between same-version and different-rule comparisons ( M = 18.32, SD = 25.06). Furthermore, there were almost no two rules that were treated as being too similar, except for three different-rule comparisons that had an average rating over 30. These rules, however, were separated into different groups to prevent any issues that might have been caused by their similarities. Finally, subtle differences in the types of rules (quantifiably different rules, such as rotation, vs. qualitatively different rules, such as color change) did not systematically contribute to participants’ similarity judgements.

In order to provide a sufficient number of comparisons, six A:B items were created for each rule, including three for each rule version (i.e., three 90-degree rotation items and three 45-degree rotation items). The six A:B items were drawn with enough variation that there were no isomorphs during the same-version comparisons. However, all items were drawn with most shapes being square-like to encourage participants to look at the rule and not simply make a similarity judgement based upon surface features. An example of the six items for the numeracy rule is shown in Figure 4 . With six items, six same-version comparisons were shown (three per rule), six different-version comparisons were shown (more combinations were technically possible, but six comparisons were randomly selected), and twelve different-rule comparisons per rule were shown. Although more combinations are possible, only two comparisons per rule (one per version) were selected to be matched with other rules. This resulted in a total of 126 comparisons: 42 same-version comparisons, 42 different-version comparisons, and 42 different-rule comparisons. All comparisons were shown in a random order across all participants. Prior to starting the similarity judgement portion, participants were given three practice trials, one per comparison type, so that they could adjust to the procedure and the types of comparisons they would be making.

Numeracy A:B items. Note. Examples of the RSJT items used for the numeracy rule.

To ensure that participants were completing the similarity judgement task appropriately and not simply selecting random responses, a t -test was calculated for each participant to determine if they were treating the same-version rule condition differently than the different-rule condition. Given that these fall on two extremes, participants should rate these differently, and thus this served as a manipulation check. Of the 418 participants that completed all tasks, 57 failed to meet the inclusion criteria for the RSJT.

Figural Analogies Test

For the test portion, participants completed a total of 30 figural analogy problems. Half of the problems were comprised entirely of rules used in the training portion and the other half were comprised of novel rules. However, what constituted trained rules or novel rules depended on which set of rules the participant received in the training portion, with the two set of rules counterbalanced across participants. Pilot analyses from a larger sample of figural analogy problems were used to create the sample of thirty problems to ensure that the two sets of problems were equal in difficulty. The figural analogies test problems consisted of two or three rules (seven two-rule problems for Set A and nine for Set B, with the remainder of the set being three-rule problems), with some problems consisting of entirely distinct rules, whereas other problems used two versions of the same rule in a problem (paired-rule items). The paired-rule items were introduced to increase the number of problems used in the figural analogies test portion while still ensuring that no two problems used the same combination of rules. Furthermore, the practice of using two rules conditional on some other factor (such as the size of the object) is a technique used in the RAPM ( Raven et al. 1962 ) and therefore has precedent for being used in a Gf task. The distinct-rule items could be comprised of two or three rules, but the rules were always unique (e.g., color change, size change, and rotation). The paired-rule items could also consist of two or three rules, but always included two rules that were actually two versions of the same rule. For example, one problem may have a 90-degree rotation for the larger object and a 45-degree rotation for the smaller object. If the paired-rule item consisted of three rules, then the third rule was an additional distinct rule. Pilot analyses indicated that the paired-rule items did not differ in difficulty when compared to the distinct-rule items. Set A did include more paired-rule items than Set B (nine and seven, respectively), but this was a result of prioritizing balanced difficulty across the two sets.

Participants were shown the 30 problems in a random order and were given a maximum of 60 s to solve the problem. If they had not selected a response after 60 s, they were moved to the next problem and that item was marked as incorrect. To encourage accurate performance on the task, a 20 s penalty screen appeared if participants selected a response in less than 5 s and it was incorrect. The screen notified the participant that they selected their response quickly and encouraged them to make sure that they were performing the task correctly. To serve as a manipulation check, the example figural analogies item shown in the instructions (A:a :: R : ?) was drawn to resemble the figural problems more closely (adding triangles and rectangles around the letters) and placed randomly in the figural analogies test portion. Participants that failed to solve the manipulation check item (52/418) were removed from analyses.

6.2.2. Working Memory Capacity

Automated operation span.

The automated operation span ( Unsworth et al. 2005 ) is a complex span working memory task that tests an individual’s ability to remember letters while solving simple math problems in between the presentation of the letters. For the math portion, participants were shown a simple math problem followed by a number. They had to determine if the number was the answer to the previous problem or not. They were then shown a letter that was to be recalled at a later time. After several iterations, participants were asked to recall the letters in the sequence that they saw them in. Participants completed 2 blocks, with each sequence length, ranging from 3 to 7 letters, being presented once per block. The task was scored using the partial scoring method, where a participant’s total is comprised of the total correct letters recalled in the correct position of the sequence. Participants completed a shortened version ( Foster et al. 2015 ) of the task (two blocks instead of three) to try and reduce fatigue because participants completed all tasks online. Participants were removed if they failed to score at least 80% accuracy on the math portion (40/418).

Automated Symmetry Span

The automated symmetry span is very similar to the automated operation span, but it uses visual stimuli rather than verbal stimuli ( Unsworth et al. 2009 ). Instead of math problems, participants determine if an image is symmetrical or not. Then, they are shown a 4 × 4 grid with a single red square filled in. After several iterations, of which range from 2 to 5 to be remembered red squares, participants recall the location of the red squares in the sequence they were presented. Participants completed two blocks ( Foster et al. 2015 ) and the task was scored using the partial scoring method. Participants were removed if they failed to score at least 80% accuracy on the symmetry portion (59/418).

6.2.3. Gf Measures

Paper folding task.

The paper folding task ( Ekstrom et al. 1976 ) shows participants a piece of paper that is then folded several times. A hole is then punched through the folder paper and participants must determine what the unfolded paper would look like. Participants were shown a total of 20 items, with 3 min to complete the first 10 and another 3 min to complete the second set of 10. An example item is shown in Figure 5 . Because it was easy to click through this task and the study was administered online, participants were excluded from all analyses if their average response time across all items was less than two seconds (25/418).

Example of paper folding item. Note. A paperfolding example from Ekstrom et al. ( 1976 ).

Letter Series

The letter series task ( Kotovsky and Simon 1973 ) requires participants to identify a pattern in a sequence of letters and provide what the next letter in the sequence would be according to that pattern. For example, participants might be given ‘ababababa’, with the correct answer being ‘b’. Participants solved 15 items with no time limit for the task. Similar to the figural analogies test portion, however, participants were shown a 20 s time penalty screen if they selected their response in less than 5 s and it was incorrect. To find participants that were clicking through the task, the problem “abcde” was added as the tenth item (changing the total to 16 items). However, after data collection was completed, it was realized that “abcde” could be solved with multiple answers, so instead participants were excluded from analyses if their accuracy was less than three. Three was chosen because the first two items of the letter series task are extremely easy and most participants did solve the “abcde” item correctly, suggesting that most participants genuinely doing the task should be able to solve at least three problems correctly. Of the 418 participants that completed all data, 66 failed to meet the inclusion criteria for the letter series task.

6.3. Procedure

Participants completed all tasks in a single session, entirely online. Participants were told prior to starting the study that they must complete the study in one sitting so they should only begin when they are ready. Participants were given a final warning screen before starting the study to not begin unless they were ready. Participants completed the modified figural analogies task first, followed by the automated operation span task, the symmetry span task, the paper folding task, and the letter series task. All tasks were completed in this order, with a short one-to-two-minute break provided between each task.

Composite scores were generated for the WMC and Gf measures by taking performance on the two tasks (operation span and symmetry span for WMC and paper folding and letter series for gf), z -transforming the scores, and then taking the average of the two tasks 1 . For the modified figural analogies task, three rule-representations scores were generated from the RSJT, one score for each type of comparison (i.e., the same rule and the same version of the rule, or “same version”; the same rule but a different version of the rule, or “different version”; or a different rule and different version, or “different rule”). These were calculated by first recentering all the scores for each participant such that their minimum score became −1 and their highest score became 1. The recentering was performed across all of their judgement scores and was not split by type of comparison. This was carried out to account for participants using the scale differently 2 . Next, the average score was taken for each type of comparison for each participant, producing three scores: their average same-version comparison score, different-version comparison score, and different-rule comparison score. The primary measure of focus was their average score for the different-version comparisons. If participants have a positive value for their different-version score from the RSJT, then this would indicate that they are treating the different-version items similar to the same-version items, suggesting their representation of that rule is more general and that they are not treating different versions as entirely different rules. Conversely, if their different-version score is negative, then this would indicate that they are treating the different-version rules as separate rules.

7.1. Summary Statistics and Correlations

All summary statistics are included in Table 1 and the correlations between all tasks are shown in Table 2 . The RSJT scores were split by the two stimulus sets in Table 1 so that reliability could be calculated for each set. Reliability was calculated using Cronbach’s alpha at the item level for the figural analogies test items, letter series, and the RSJT same-version and different-version scores (averaged across each rule, treating it as an item). Split-half reliability was used for paper folding and the RSJT different-rule scores. Split-half reliability was used for the RSJT different-rule scores because they could not be averaged by rule, as two rules were shown in the different-rule comparisons. Parallel-forms reliability was used for operation span and symmetry span.

Descriptive statistics.

Note . FA refers to figural analogies, SV refers to same version, DV refers to different version, and DR refers to different rule.

Task correlations.

Note. * p < .05. FA refers to figural analogies, SV refers to same version, DV refers to different version, and DR refers to different rule.

7.2. Predicting Figural Analogies Accuracy with Training, RSJT Scores, WMC, and Gf

The first set of analyses investigated whether accuracy on the figural analogies test items changed as a function of training, the participant’s rule representation as measured by the RSJT different-version scores, and the WMC and Gf measures. To determine this, generalized linear mixed-effects models (GLMM) were used to predict item accuracy on the test items with the fixed effects hierarchically introduced. A logit link function was used to predict the binary accuracy data. The maximum random effects structure included random intercepts for subject and item, with random slopes added for training for both subject and item. The general procedure for the models was to include all fixed effects and the maximum random effects structure initially and to then gradually reduce the complexity of the random effects structure if the model failed to converge or was singular ( Bates et al. 2015 ; Matuschek et al. 2017 ). The structure could be reduced by removing correlation parameters or removing random effects that failed to explain a significant portion of variance (i.e., if variance was not greater than zero at α = 0.20). All binary predictors were coded using sum contrast coding (−1, 1) and all continuous predictors were z -transformed.

The first model predicted figural analogy accuracy with fixed effects for the training (items comprised of previously seen or novel rules) and the participants’ RSJT different-version score with a test for an interaction between RSJT different-version scores and training. The results of the model, Model 1, are shown in Table 3 . There was a significant interaction between the training manipulation and RSJT different-version scores, shown in Figure 6 ; however, the relationship was in the opposite direction of what was predicted. The interaction illustrates a small relationship between RSJT different-version scores and accuracy, but only when participants are solving problems comprised of novel rules. The RSJT scores did not predict accuracy at all for the trained items. It is worth noting that only three participants fell outside of −1.7 standard deviations away from the mean for the RSJT different-version scores but were within −2.5 standard deviations. The removal of these participants did not change the results of the model.

Interaction between RSJT different-version scores and training. Note. The plot was generated using the predict function in R to generate log odds ratio accuracy data based upon Model 1 ( Table 3 ). The data points represent item level data for all participants and the linear slopes were generated with the geom_smooth function in ggplot. The circles represent trained-rule items and the squares represent novel-rule items.

Predicting figural analogy accuracy with RSJT different-version scores, WMC, and Gf.

Note. Training was coded with −1 for the novel-rules condition and 1 for the trained-rules condition. DV refers to different version.

The next analysis introduced the WMC and Gf composite measures into the GLMM. WMC and Gf were added into the models to determine if rule representations accounted for unique variance or if they could be explained by other processes. Both WMC and Gf measures were included because prior work investigating analogical transfer has shown that Gf may contribute to building representations ( Kubricht et al. 2017 ), but WMC was not accounted for in that study.

WMC and Gf were added on top of the RSJT different-version scores, with WMC and Gf also tested for interactions with training. Because the inclusion of interaction terms can make it difficult to interpret main effects, nonsignificant interactions were removed for the final model, Model 4. However, the WMC and Gf models with all interaction terms are also included. All models’ results are shown in Table 3 . For WMC, shown in Model 2, there was no significant interaction with training, though WMC did predict accuracy on the figural analogies task. For Gf, in Model 3, there was also no significant interaction, but Gf did predict accuracy on the figural analogies task. The interaction between RSJT different-version scores and training did remain significant after including both Gf and WMC, indicating that the RSJT scores were measuring something unique from Gf and WMC.

7.2.1. Interim Discussion: Task Learning

The interaction between RSJT different-version scores and training was in an unexpected direction. These results may indicate that processes that would normally be beneficial in solving figural analogies problems become less important or less likely to be used when participants have been trained on the rules. This is similar to findings in the expertise literature, where dependence on other cognitive processes, such as working memory, decreases as expertise increases. The lack of an interaction between WMC and training was also unexpected, given prior work showing that the relationship between performance on Gf tasks and WMC changes depending on whether items are comprised of novel or previously seen rules ( Harrison et al. 2015 ; Loesche et al. 2015 ; Wiley et al. 2011 ). However, it has also been noted that participants learn throughout a task, even if all of the rules are novel, and that the frequency of rules used can bias participants’ expectations and use of rules ( Verguts and De Boeck 2002 ). Therefore, a post hoc analysis was conducted to investigate the relationship of WMC and training on accuracy while also accounting for learning throughout the task.

7.2.2. Task Learning Post Hoc Analysis

Learning throughout the task was assessed using trial number, so a separate analysis tested for a three-way interaction between WMC, training, and trial number, while predicting accuracy on the figural analogies task. The results of the GLMM are shown in Table 4 . The three-way interaction was significant ( p = .003), as shown in Figure 7 . Post hoc comparisons demonstrated that WMC and trial continued to interact for the novel-rule items ( p = .021) but not for the trained-rule items ( p = .07). However, there was still a main effect of WMC ( p < .001) and trial ( p = .02) for the trained-rule items (even with the interaction term removed). The models for the post hoc comparisons can be found in Appendix A . The results of the three-way interaction and the post hoc comparisons suggest that high-WMC individuals improve on the novel-rule items over time whereas the low-WMC individuals do not. Although the interaction term between WMC and trial was not significant for the trained-rule items, the main effects of WMC and trial indicate that high-WMC individuals were more likely to solve trained-rule items, but both high and low-WMC individuals improved on the trained-rule items over time. Notably, there were no significant interactions with Gf for training and trial number, so these results appear to be unique to WMC.

Interaction between WMC, training, and trial. Note. The plot was generated using the predict function in R to generate log odds ratio accuracy data based upon the model in Table 4 . The data points represent item-level data for all participants and the linear slopes were generated with the geom_smooth function in ggplot. The model was analyzed with WMC as a continuous predictor but a tertiary split was used to plot the data. The circles represent trained-rule items and the squares represent novel-rule items.

Predicting figural analogy accuracy with WMC, training, and trial.

Note. Training was coded with −1 for the novel-rules condition and 1 for the trained-rules condition.

7.3. Predicting RSJT Different-Version Scores

After investigating the relationship between memory retrieval and accuracy, a closer look at the rule representation measure was conducted to determine if WMC and/or Gf could explain the ability to generate more general rule representations. The second analysis used a linear mixed-effects model (LMM) to predict the RSJT different-version scores for each rule, with the WMC and Gf measures used as predictors. The random effects structure included random intercepts for subject and rule. The p -values were generated using the lmertest package in R. The results of the model can be found in Table 5 . Neither Gf nor WMC predicted the RSJT different-version scores.

Predicting RSJT different-version scores.

7.3.1. Interim Discussion: Further Investigating the RSJT Scores

The lack of a relationship between RSJT different-version scores and figural analogies accuracy on trained items, as well as the fact that RSJT same-version scores appeared to better correlate with figural analogy accuracy, prompted a deeper inspection of the RSJT scores and what they may actually measure. The original intention with the RSJT was to use the different version comparison scores only, because it was assumed that most participants would rate the same-version comparisons as very close to 1 and the different-rule comparisons as very close to −1, reducing those measures to be at the ceiling or at the floor. Although same-version comparisons were rated the highest on average and the different-rule comparisons were rated the lowest on average, there were no ceiling or floor effects, thus making these other RSJT scores potentially meaningful.

It is also possible that different processes are beneficial when solving an item that has two very similar rules included in a single item. The two types of figural analogy items, distinct-rule items (items comprised of all unique rules) and paired-rule items (items where two versions of the same rule appear in the same item), were originally developed to help control for difficulty across the two sets and to allow for a larger set of items by repeating rules. This requires the solver to be open to different variations of a rule and to not be overly fixated on a single element of the rule. Yet, it also requires that the solver does not over-generalize such that they cannot distinguish between the two highly similar rules. As such, the following analyses explore both the other two RSJT measures and their impact on different types of items.

7.3.2. Further Investigating the RSJT Scores Post Hoc Analyses

The first analysis sought to determine if each of the three RSJT scores measure different constructs. The three measures were used as simultaneous predictors in a GLMM predicting figural analogies accuracy. The results are shown in Table 6 . The RSJT same-version and RSJT different-rule measures did uniquely predict accuracy, but the RSJT different-version did not predict accuracy, congruent with prior analyses.

Predicting figural analogy accuracy with all three RSJT measures.

Note. SV refers to same version, DV refers to different version, and DR refers to different rule.

The next analysis explored whether the type of figural analogy item being solved contributed to the predictiveness of the RSJT measures. To test if the type of problem mattered, three mixed-effects models were run, each with a different RSJT measure. The models tested for a three-way interaction between the RSJT measure, training, and the type of figural analogy item (distinct or paired) with random intercepts for subject and item, and random slopes for training for both subject and item. The final models are shown in Table 7 . The type of figural analogies item only interacted with the RSJT different-version measurements. Figure 8 shows the three-way interaction between RSJT different version, training, and type of figural analogy item. Post hoc comparisons specified that the significant differences in slopes were between the novel paired-rule items and the trained paired-rule items, as well as the novel paired-rule items and the novel distinct items. All the models for the post hoc comparisons can be found in Appendix A . The results indicate that how an individual rates different versions of the same rule does predict accuracy on the figural analogies task, but only for novel paired-rule items. Furthermore, this interaction was not significant with the RSJT same-version or different-rule measures, suggesting that this is particular to the different version measure. Notably, the results found with the three-way interaction likely explain the effect found in the first analysis, with RSJT different version interacting with training. Finally, the lack of a main effect of item type, as well as a two-way interaction between training (namely in the same version and different rule models that do not include the three-way interaction term), indicate that the type of item did not contribute to overall accuracy and is also not responsible for any of the training effects.

Interaction between RSJT different version scores, training, and item type. Note. The plot was generated using the predict function in R to generate log odds ratio accuracy data based upon the different-version model in Table 7 . The data points represent item-level data for all participants and the linear slopes were generated with the geom_smooth function in ggplot. The circles represent trained-rule items and the squares represent novel-rule items.

Predicting figural analogy accuracy with RSJT measures, training, and item type.

Note. Training was coded with −1 for the novel-rules condition and 1 for the trained-rules condition. Item type was coded with −1 for the paired-rule items and 1 for the distinct-rule items. SV refers to the same-version score, DV refers to the different-version score, and DR refers to the different-rule score.

7.3.3. Interim Discussion: Measuring Different Version Relative to Same Version and Different Rule

The original different-version measure represented an individual’s average score for the different-version comparisons after rescaling all of their scores from −1 to 1. Although this method also provides measures for same-version and different rule, it fails to account for participant and item effects, it does not take into account the spread of an individual’s ratings (only the average), and it does not provide any information on how an individual rates different-version comparisons relative to the same version or different rule. For example, some participants may rate different-version rule comparisons as similar, but they may also rate different-rule comparisons as similar, suggesting that they have a general propensity toward calling all rules similar. Alternatively, participants may view same-version rules as being equally as distinct as different-version rules. Although all participants included in the sample needed to have a significant difference between the same-version and different-rule conditions to be included, the current measures do not provide information on how large of a difference there exists between the same-version and different-version comparisons, as well as the different-version and different-rule comparisons.

7.3.4. Relative Rule Post Hoc Analysis

To better address ratings of different-version rules relative to the other two rule comparison types, new measures were generated using a mixed-effects model. The linear mixed- effect model predicted raw similarity judgement ratings with type of comparison (same version, different version, different rule) as a fixed effect. The random effects included random intercepts for item and participant, with random slopes for type of comparison for each participant. Type of comparison was dummy-coded as two variables, same version–different version and different rule–different version, with same version coded as 1 for the same version–different version measure and different rule coded as 1 for the different rule–different version measure, and everything else coded as 0. The random slopes for the same version–different version and different rule–different version measures for each participant were extracted and used as predictors in a series of analyses. The same version–different version and different rule–different version measures represented the difference between the two types of comparisons. Individuals with higher (positive) values for the same version–different version measure on average rated same-version comparisons as higher than different-version comparisons, whereas individuals with lower (negative) values rated different-version comparisons as being closer to same-version comparisons. For the different rule–different version measure, individuals with larger (positive) values rated different-version comparisons as closer to different-rule comparisons, whereas individuals with lower (negative) values rated different-version comparisons as different from different-rule comparisons. Because the new measures produced some outliers for those measures (beyond 2.5 standard deviations away from the mean), some participants were removed for subsequent analyses. This resulted in a final sample of 215 participants. The correlations between the new RSJT measures, the old RJST measures, figural analogies accuracy, WMC, and Gf are in included in Table 8 .

Correlations for new RSJT measures.

Note. * p < .05 FA refers to figural analogies, SV refers to same version, DV refers to different version, and DR refers to different rule. SV-DV refers to the same version–different version score and DR-DV refers to the different rule–different version score.

The first analysis used a GLMM to predict figural analogy accuracy as a function of the two new measures, the training manipulation, WMC, and Gf, with a test for an interaction between the two new measures and the training manipulation. The random effects structure was identical to previous analyses, with random intercepts for subject and item, with random slopes added for training for both subject and item. The same version–different version and different rule–different version measures were z -transformed prior to including them in the model. The final model is shown in Table 9 . There was a significant interaction between the same version–different version measure and training, but not for the different rule–different version measure. The interaction is shown in Figure 9 . The interaction indicated a positive relationship between the same version–different version measure and figural analogies accuracy for the trained items only. Thus, individuals that tend to rate different-version comparisons as different from same-version comparisons show improved accuracy for the trained-rule items. These results are the opposite of what was found previously when just using the RSJT different version measure. Although there was no significant interaction with the different rule–different version measure, it did predict figural analogy accuracy as a main effect, with a larger difference between different version and different rule scores corresponding with higher accuracy on the figural analogies test items.

Interaction between different rule–different version and the training manipulation. Note. The plot was generated using the predict function in R to generate log odds ratio accuracy data based upon the model in Table 9 . The data points represent item-level data for all participants and the linear slopes were generated with the geom_smooth function in ggplot. The circles represent trained-rule items and the squares represent novel-rule items.

Predicting figural analogy accuracy with same version–different version and different rule–different version measures and training.

Note. Training was coded with −1 for the novel-rules condition and 1 for the trained-rules condition. SV-DV refers to same version–different version and DR-DV refers to different rule–different version.

The next analysis tested for interactions between the two new RSJT measures, the training manipulation, and item type (distinct vs. paired rules). The final models are shown in Table 10 . There was a significant interaction between all three measures when different rule–different version was included, but not when same version–different version was included. The results of the interaction are shown in Figure 10 . Post hoc comparisons indicated that the only significant difference in slopes was between the novel paired-rule items and the trained paired-rule items. The full models for the post hoc comparisons are located in Appendix A . These results are largely congruent with previous results showing a significant three-way interaction between different-version scores, training, and item type. Compared to the previous analyses, the current model specifies that not only is it higher different-version scores that predict accuracy on novel paired-rule items, but that higher different-version scores relative to different-rule scores predict accuracy on novel paired-rule items.

Interaction between different rule–different version, training, and item type. Note. The plot was generated using the predict function in R to generate log odds ratio accuracy data based upon the different rule–different version model in Table 10 . The data points represent item-level data for all participants and the linear slopes were generated with the geom_smooth function in ggplot. The circles represent trained-rule items and the squares represent novel-rule items.

Predicting figural analogy accuracy with same version–different version and different rule–different version measures, training, and item type.

Note. Training was coded with −1 for the novel-rules condition and 1 for the trained-rules condition. Item type was coded with −1 for the paired-rule items and 1 for the distinct-rule items. SV-DV refers to same version–different version and DR-DV refers to different rule–different version.

8. Discussion

8.1. knowledge representations and transfer.

The present work examined the unique and interacting impact of processes driven by WMC and knowledge representations on success during reasoning. The potential for more thorough knowledge representations to better facilitate transfer is not a novel concept ( Anderson 1987 ; Gick and Holyoak 1983 ; Holyoak and Koh 1987 ; Wharton et al. 1994 ) and neither is the contribution of knowledge to performance on Gf tasks ( Bors and Vigneau 2003 ; Harrison et al. 2015 ; Loesche et al. 2015 ; Verguts and De Boeck 2002 ). Similarly, the role of WMC during reasoning has been explored previously ( Carpenter et al. 1990 ; Verguts and De Boeck 2002 ; Wiley et al. 2011 ). Where the present study breaks new ground is in looking at these mechanistically, during a Gf task.

Although both knowledge representations and WMC did predict success during the figural analogies task, the results did not align with a priori predictions. It was assumed that the presence of general rule representations would better facilitate transfer, as measured by the RSJT different-version scores. However, although RSJT different-version scores did interact with the training manipulation, they only predicted accuracy on novel items, not the trained items. This was opposite of what was expected. The original reasoning was that generalized rule representations would facilitate retrieval, but it is possible that the training or even the RSJT itself helped most participants conceptualize the rules in a way that was useful, thus leaving retrieval of the trained rules to depend on other processes, such as WMC. That being said, the analyses that included same version–different version measure did provide some support for rule representations facilitating retrieval. The same version–different version measure only predicted accuracy for the trained-rule items, not the novel-rule items. However, it also showed a positive relationship, suggesting that individuals who treat different-version comparisons as being less similar than the same-version comparisons had improved accuracy on the figural-analogy-trained items. This is also contrary to what was expected. Although these interactions differed from expectations, they did remain significant once WMC and Gf were introduced into the models. This indicates that the RSJT can uniquely explain variance on novel and trained figural analogies items beyond what Gf and WMC can explain.

The interaction between training and RSJT different-version scores also increased when accounting for the type of figural analogy item (distinct vs. paired rules). It is likely that the initial two-way interaction between training and RSJT different-version scores is driven by the three-way interaction with figural analogy items, and that in truth RSJT different-version scores really only predict novel paired-rule items. However, the relationship was positive, congruent with the initial hypotheses of the study. Mentally representing two versions of the same rule as closely similar, as opposed to treating them as distinct rules, corresponded with higher accuracy on novel problems where two versions of the same rule were included. Furthermore, the different rule–different version analyses indicated that it may be more important that individuals treat different-version comparisons as different from different rules than it is for different-version comparisons to be treated as similar to same-version comparisons. Notably, the fact that the RSJT different-version scores, as well the different rule–different version scores, only predicted novel items suggests that individuals with a propensity to treat different versions of the same rule as more similar, but still as separate rules, may have an easier time inducing related, but slightly different rules when they are exposed to them for the first time. It is likely that the training or the RSJT helped participants induce and develop representations to a point where individual differences in RSJT different-version scores no longer mattered for solving the trained-rule problems.

Unlike the RSJT different-version scores, WMC did appear to explain transfer, showing interactions with the training manipulation when accounting for learning across the task. Both high- and low-WMC individuals improved on trained rules over time, but high-WMC individuals were more likely to solve trained items correctly than low-WMC individuals. Furthermore, high-WMC individuals improved on the novel-rules items as they progressed, with performance increasing as they solved more items. Low-WMC individuals did not show this benefit, appearing to struggle on novel items throughout the task. It is possible that high-WMC individuals improved on the novel items by learning the rules and using them later in the task ( Harrison et al. 2015 ; Loesche et al. 2015 ; Verguts and De Boeck 2002 ), or they may have simply improved in solving novel problems, learning new strategies or becoming more comfortable with the task ( Klauer et al. 2002 ; Tomic 1995 ). The low-WMC individuals did not improve on the novel problems over time, suggesting that they struggled to induce the rules at all, or that they may need substantially more practice before they begin to benefit from learned rules.

These results are congruent with prior work in multiple ways. First, the fact that high-WMC individuals were able to benefit from trained rules more than low-WMC individuals indicates that WMC is important for retrieving previously learned information ( Loesche et al. 2015 ; Unsworth 2016 ). Second, only the high-WMC individuals improved on novel items over time, whereas both low- and high-WMC individuals improved on the trained items over time, indicating that WMC also contributes to learning throughout the task ( Harrison et al. 2015 ; Unsworth 2016 ). Finally, because the high-WMC individuals were generally better at solving novel items and low-Gf individuals did not improve at the novel items over time, WMC seems important for initially inducing the rules ( Carpenter et al. 1990 ). Although these results support several explanations for the role of WMC in Gf tasks, it is worth noting that the findings only became apparent after testing for the three-way interaction with trial number. Furthermore, these results fail to replicate studies showing a stronger relationship between WMC and repeated rules (items wherein the combination of rules has been seen previously) on the RAPM ( Loesche et al. 2015 ; Harrison et al. 2015 ) or studies showing the opposite, where WMC better predicts novel (unique combinations) RAPM items ( Wiley et al. 2011 ). Taking into consideration learning across the task, as well as accounting for individual rules rather than just unique combinations of rules, may explain the mixed findings on the role of WMC in Gf tasks.

Finally, it is worth noting that WMC may play a large role in how well the rules are initially learned. Previous work with analogical transfer has found that Gf may explain the ability to develop more general representations that are more easily transferred ( Kubricht et al. 2017 ). However, WMC and Gf are highly correlated and so it is possible that the high-Gf individuals in Kubricht et al.’s study did not benefit from the training manipulation because they were also high in WMC, thus learning the source problem better. This would explain why WMC is shown to explain transfer in the current study and not Gf. Yet, it is worth noting that Cushen and Wiley ( 2018 ) found that WMC only explained the completeness of an individual’s summary of the source problem when performance on the Remote Associates Test was not accounted for. Thus, although WMC may be important for retrieving learned rules, other processes may still be important in helping to generate more general representations, especially in cases of more complex learned information.

8.2. The Rule-Similarity Judgement Task

The original intention with the RSJT was to only use the different-version scores, but the same-version and different-rule scores ultimately did provide meaningful information, as did the difference-score measures, with each predicting figural analogies accuracy differently and uniquely. As many of these analyses were post hoc considerations, it is thus worth considering what processes these measures are tapping into. The RSJT same-version measure was the best predictor of figural analogies accuracy overall, even though it did not interact with any of the training manipulations. Because it was comparing two identical rules, it is possible that the measure is tapping into an individual’s propensity to let surface features of the items factor into their similarity judgements. For example, consider Figure 4 , which utilizes a numeracy rule. In this case, all of the items look slightly different. Some are just lines, some make a whole shape, and some include both aspects. Thus, even though they all increase an edge by two or lose an edge by one, some participants may have been accounting for surface features, with this potentially affecting their ability to solve figural analogy problems. Given that the same-version measure did predict accuracy and did also correlate with paperfolding and the Gf composite measure, it may be that a reliance on surface features is something that generally correlates with accuracy on Gf tasks. Furthermore, because the same version–different version measure did not correlate with figural analogy accuracy, Gf, WMC, or even the RSJT same version measure, but did correlate with the different-version and different-rule scores, it does appear that the RSJT same-version measure is tapping into something unique that happens when same-version comparisons are made.

The same version–different version and different rule–different version difference scores showed complementary as well as distinct findings from the RSJT same-version, different-version, and different-rule scores. The same version–different version measure was the only measure to predict trained-rule items and it predicted positively, indicating that representing different-version comparisons as less similar to same-version was beneficial, rather than what was originally hypothesized. The different rule–different version measure did support the initial findings with the RSJT different-version scores, but further specified that it was beneficial for accuracy on novel paired-rule items to represent different-version comparisons as more similar than different-rule comparisons. The same version–different version and different rule–different version measures indicate that generally treating the three comparisons as different is beneficial for accuracy. Although the same version–different version measures and the different rule–different version measures better target the difference between the three types of comparisons, it appears that all five measures produced by the RSJT measure slightly different propensities and further work will need to be conducted to better understand the differences and the similarities between all of the measures.

Although the RSJT did explain some transfer when using the same version–different version measure, it is possible that the RSJT did not explain transfer in the expected way because the measure was designed to look at the abstractness of the rule representations. Prior work has primarily looked at whether the representations include all the necessary information that is needed for transfer or mapping ( Cushen and Wiley 2018 ), rather than looking at the quality of a representation that includes all the necessary information. Lacking information may produce a more salient effect with transfer, whereas the quality of a whole representation may matter less. Ultimately, more information is needed on the three RSJT measures to ascertain what they are actually measuring and why they are distinct from WMC and Gf, in order to determine the role of rule representations in novel problem solving and transfer.

8.3. Conclusions

In conclusion, individual differences in knowledge representations and WMC play independent roles in reasoning performance. WMC appears to be important for both learning and retrieving rules, as well as contributing to novel problem solving. However, beyond WMC, individual differences in knowledge representations were able to explain some aspects of performance via retrieval as well as for novel problems. The results indicate that there may be a happy medium with knowledge representations, wherein an individual will want to recognize similarities when present while also understanding distinctions between rules. Furthermore, the propensity to develop a more general or stimuli-specific knowledge representation was not explained by WMC and Gf. Thus, understanding what prompts individuals to build better representations may be key to not only understanding novel reasoning but problem solving as a whole.

Figural analogy rules.

Post hoc comparisons: RSJT different-version × training × item type.

Note. Training was coded with −1 for the novel-rules condition and 1 for the trained-rules condition. Item type was coded with −1 for the paired-rule items and 1 for the distinct-rule items. DV refers to different version.

Post hoc comparisons: WMC × training × trial.

Post hoc comparisons: RSJT different-rule × Gf × training.

Note. Training was coded with −1 for the novel-rules condition and 1 for the trained-rules condition. DR refers to different rule.

Post hoc comparisons: different-rule–different-version × training × item type.

Note. Training was coded with −1 for the novel-rules condition and 1 for the trained-rules condition. Item type was coded with −1 for the paired-rule items and 1 for the distinct-rule items. DR-DV refers to different rule–different version.

Funding Statement

This research received no external funding.

Author Contributions

Conceptualization, M.J.R. and A.F.J.; methodology, M.J.R. and A.F.J.; software, M.J.R.; validation, M.J.R.; formal analysis, M.J.R.; investigation, M.J.R.; resources, A.F.J.; data curation, M.J.R.; writing—original draft preparation, M.J.R.; writing—review and editing, M.J.R. and A.F.J.; visualization, M.J.R.; supervision, M.J.R. and A.F.J.; project administration, M.J.R. and A.F.J. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Review Board (or Ethics Committee) of Mississippi State University (IRB #21-342; approved 9/1/21).

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 To address concerns that composite variables for WMC and Gf may not be appropriate, several models were checked with each of the individual indicators for these variables, rather than the composite variables. In the WMC x Novel/Learned x Trial interaction, using Operation Span instead of the composite moved the p -value to p = .07, rather than being significant, while the interaction remained significant using only Symmetry Span. In the Representation Scores x Novel/Learning + WMC + Gf analysis, using the individual Gf tasks moved the main effect of DV representation scores from non-significant to significant. When predicting SV-DV training, changes from composites to individual measures changed nothing. As these are minor differences, composites were used throughout the paper for analyses.

2 Only 3 participants failed to use the highest marking on the scale (100) for any of their scores; all 3 instead used 95 as their top score. A total of 29 participants put their lowest raw score as something other than 0. For 2, the lowest score was 25, for 2 others it was 20, and for 4 it was 15. All of the rest used 5 or 10 as their lowest number. Because 86% of the sample used the full scale, it is not anticipated that this caused substantial changes to the analyses.

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• Ackerman Phillip L., Beier Margaret E., Boyle Mary O. Working memory and intelligence: The same or different constructs? Psychological Bulletin. 2005; 131 :30–60. doi: 10.1037/0033-2909.131.1.30. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Anderson John R. Skill acquisition: Compilation of weak-method problem situations. Psychological Review. 1987; 94 :192–210. doi: 10.1037/0033-295X.94.2.192. [ 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. doi: 10.3758/bf03193814. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Baddeley Alan. The episodic buffer: A new component of working memory? Trends in Cognitive Sciences. 2000; 4 :417–23. doi: 10.1016/S1364-6613(00)01538-2. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Barrouillet Pierre, Bernardin Sophie, Camos Valerie. Time constraints and resource sharing in adults’ working memory spans. Journal of Experimental Psychology General. 2004; 133 :83–100. doi: 10.1037/0096-3445.133.1.83. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Bates Douglas, Kliegl Reinhold, Vasishth Shravan, Baayen Harald. Parsimonious mixed models. arXiv. 2015 1506.04967 [ Google Scholar ]
• Bethell-Fox Charles E., Lohman David F., Snow Richard E. Adaptive reasoning: Componential and eye movement analysis of geometric analogy performance. Intelligence. 1984; 8 :205–38. doi: 10.1016/0160-2896(84)90009-6. [ CrossRef ] [ Google Scholar ]
• Bors Douglas A., Vigneau François. The effect of practice on Raven’s Advanced Progressive Matrices. Learning and Individual Differences. 2003; 13 :291–312. doi: 10.1016/S1041-6080(03)00015-3. [ CrossRef ] [ Google Scholar ]
• Carlson Richard A., Yaure Robin G. Practice schedules and the use of component skills in problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition. 1990; 16 :484–96. doi: 10.1037/0278-7393.16.3.484. [ CrossRef ] [ Google Scholar ]
• Carpenter Patricia A., Just Marcel A., Shell Peter. What one intelligence test measures: A theoretical account of the processing in the Raven Progressive Matrices Test. Psychological Review. 1990; 97 :404–31. doi: 10.1037/0033-295X.97.3.404. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Carroll John B. Human Cognitive Abilities: A Survey of Factor-Analytic Studies. Cambridge University Press; Cambridge: 1993. [ Google Scholar ]
• Chase William G., Simon Herbert A. Perception in chess. Cognitive Psychology. 1973; 4 :55–81. doi: 10.1016/0010-0285(73)90004-2. [ CrossRef ] [ Google Scholar ]
• Chi Michelene T., Feltovich Paul J., Glaser Robert. Categorization and representation of physics problems by experts and novices. Cognitive Science. 1981; 5 :121–52. doi: 10.1207/s15516709cog0502_2. [ CrossRef ] [ Google Scholar ]
• Cooper Graham, Sweller John. Effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology. 1987; 79 :347. doi: 10.1037/0022-0663.79.4.347. [ CrossRef ] [ Google Scholar ]
• Cowan Nelson. Essays in Cognitive Psychology. Working Memory Capacity. Psychology Press; New York: 2005. [ Google Scholar ]
• Cushen Patrick. J., Wiley Jennifer. Both attentional control and the ability to make remote associations aid spontaneous analogical transfer. Memory & Cognition. 2018; 46 :1398–412. doi: 10.3758/s13421-018-0845-1. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Ekstrom Ruth B., French John W., Harman Harry, Dermen Diran. Kit of Factor-Referenced Cognitive Tests. Educational Testing Service; Princeton: 1976. [ Google Scholar ]
• Foster Jeffrey L., Shipstead Zach, Harrison Tyler L., Hicks Kenny L., Redick Thomas S., Engle Randall W. Shortened complex span tasks can reliably measure working memory capacity. Memory & Cognition. 2015; 43 :226–36. doi: 10.3758/s13421-014-0461-7. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Ghorbani Hamid. Mahalanobis distance and its application for detecting multivariate outliers. Facta Universitatis, Series: Mathematics and Informatics. 2019; 2019 :583–95. doi: 10.22190/FUMI1903583G. [ CrossRef ] [ Google Scholar ]
• Gick Mary L., Holyoak Keith J. Analogical Problem Solving. Cognitive Psychology. 1980; 12 :306–55. doi: 10.1016/0010-0285(80)90013-4. [ CrossRef ] [ Google Scholar ]
• Gick Mary L., Holyoak Keith J. Schema induction and analogical transfer. Cognitive Psychology. 1983; 15 :1–38. doi: 10.1016/0010-0285(83)90002-6. [ CrossRef ] [ Google Scholar ]
• Harrison Tyler L., Shipstead Zach, Engle Randall W. Why is working memory capacity related to matrix reasoning tasks? Memory & Cognition. 2015; 43 :389–96. doi: 10.3758/s13421-014-0473-3. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Hershey Douglas A., Walsh David A., Read Stephen J., Chulef Ada S. The effects of expertise on financial problem solving: Evidence for goal-directed, problem-solving scripts. Organizational Behavior and Human Decision Processes. 1990; 46 :77–101. doi: 10.1016/0749-5978(90)90023-3. [ CrossRef ] [ Google Scholar ]
• Holyoak Keith J., Koh Kyunghee. Surface and structural similarity in analogical transfer. Memory & Cognition. 1987; 15 :332–40. doi: 10.3758/bf03197035. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Jarosz Andrew F., Wiley Jennifer. Why does working memory capacity predict RAPM performance? A possible role of distraction. Intelligence. 2012; 40 :427–38. doi: 10.1016/j.intell.2012.06.001. [ CrossRef ] [ Google Scholar ]
• Kane Michael J., Hambrick David Z., Conway Andrew R. Working memory capacity and fluid intelligence are strongly related constructs: Comment on Ackerman, Beier, and Boyle (2005) Psychological Bulletin. 2005; 131 :66–71. doi: 10.1037/0033-2909.131.1.66. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Klauer Karl J., Willmes Klaus, Phye Gary D. Inducing inductive reasoning: Does it transfer to fluid intelligence? Contemporary Educational Psychology. 2002; 27 :1–25. doi: 10.1006/ceps.2001.1079. [ CrossRef ] [ Google Scholar ]
• Kohl Patrick B., Finkelstein Noah D. Patterns of multiple representation use by experts and novices during physics problem solving. Physical Review Special Topics-Physics Education Research. 2008; 4 :010111. doi: 10.1103/PhysRevSTPER.4.010111. [ CrossRef ] [ Google Scholar ]
• Kotovsky Kenneth, Simon Herbert A. Empirical tests of a theory of human acquisition of concepts for sequential patterns. Cognitive Psychology. 1973; 4 :399–424. doi: 10.1016/0010-0285(73)90020-0. [ CrossRef ] [ Google Scholar ]
• Kovacs Kristof, Conway Andrew R. Process overlap theory: A unified account of the general factor of intelligence. Psychological Inquiry. 2016; 27 :151–77. doi: 10.1080/1047840X.2016.1153946. [ CrossRef ] [ Google Scholar ]
• Kubricht James R., Lu Hongjing, Holyoak Keith J. Individual differences in spontaneous analogical transfer. Memory & Cognition. 2017; 45 :576–88. doi: 10.3758/s13421-016-0687-7. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Loesche Patrick, Wiley Jennifer, Hasselhorn Marcus. How knowing the rules affects solving the Raven Advanced Progressive Matrices Test. Intelligence. 2015; 48 :58–75. doi: 10.1016/j.intell.2014.10.004. [ CrossRef ] [ Google Scholar ]
• Lohman David F., Hagen Elizabeth P. Cognitive Abilities Test: Form 6, Level A. Riverside Pub; Rolling Meadows: 2001. [ Google Scholar ]
• Luchins Abraham S. Mechanization in problem solving: The effect of einstellung. Psychological Monographs. 1942; 54 :i-95. doi: 10.1037/h0093502. [ CrossRef ] [ Google Scholar ]
• Matuschek Hannes, Kliegl Reinhold, Vasishth Shravan, Baayen Harald, Bates Douglas. Balancing Type I error and power in linear mixed models. Journal of Memory and Language. 2017; 94 :305–15. doi: 10.1016/j.jml.2017.01.001. [ CrossRef ] [ Google Scholar ]
• Mulholland Timothy M., Pellegrino James W., Glaser Robert. Components of geometric analogy solution. Cognitive Psychology. 1980; 12 :252–84. doi: 10.1016/0010-0285(80)90011-0. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Novick Laura R. Analogical transfer, problem similarity, and expertise. Journal of Experimental Psychology: Learning, Memory, and Cognition. 1988; 14 :510–20. doi: 10.1037/0278-7393.14.3.510. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Oberauer Klaus, Süß Heinz-Martin, Wilhelm Oliver, Wittman Werner W. The multiple faces of working memory: Storage, processing, supervision, and coordination. Intelligence. 2003; 31 :167–93. doi: 10.1016/S0160-2896(02)00115-0. [ CrossRef ] [ Google Scholar ]
• Oberauer Klaus, Schulze Ralf, Wilhelm Oliver, Süß Heinz-Martin. Working memory and intelligence–their correlation and their relation: Comment on Ackerman, Beier, and Boyle (2005) Psychological Bulletin. 2005; 131 :61–65. doi: 10.1037/0033-2909.131.1.61. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Patsenko Elena G., Altmann Erik M. How planful is routine behavior? A selective-attention model of performance in the Tower of Hanoi. Journal of Experimental Psychology: General. 2010; 139 :95–116. doi: 10.1037/a0018268. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Primi Ricardo. Complexity of geometric inductive reasoning tasks: Contribution to the understanding of fluid intelligence. Intelligence. 2001; 30 :41–70. doi: 10.1016/S0160-2896(01)00067-8. [ CrossRef ] [ Google Scholar ]
• Raven John. C., Raven John C., Court John H. Advanced Progressive Matrices. HK Lewis London; London: 1962. [ Google Scholar ]
• Ricks Travis R., Turley-Ames Kandi Jo, Wiley Jennifer. Effects of working memory capacity on mental set due to domain knowledge. Memory & Cognition. 2007; 35 :1456–62. doi: 10.3758/bf03193615. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Schneider Benedikt, Sparfeldt Jörn R. How to solve number series items: Can watching video tutorials increase test scores? Intelligence. 2021a; 87 :101547. doi: 10.1016/j.intell.2021.101547. [ CrossRef ] [ Google Scholar ]
• Schneider Benedikt, Sparfeldt Jörn R. How to Get Better: Taking Notes Mediates the Effect of a Video Tutorial on Number Series. Journal of Intelligence. 2021b; 9 :55. doi: 10.3390/jintelligence9040055. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Schneider Benedikt, Becker Nicolas, Krieger Florian, Spinath Frank M., Sparfeldt Jörn R. Teaching the underlying rules of figural matrices in a short video increases test scores. Intelligence. 2020; 82 :101473. doi: 10.1016/j.intell.2020.101473. [ CrossRef ] [ Google Scholar ]
• Schönbrodt Felix D., Perugini Marco. At what sample size do correlations stabilize? Journal of Research in Personality. 2013; 47 :609–12. doi: 10.1016/j.jrp.2013.05.009. [ CrossRef ] [ Google Scholar ]
• Simon Herbert A. Models of Discovery. Springer; Berlin and Heidelberg: 1977. The structure of ill-structured problems; pp. 304–25. [ Google Scholar ]
• Snow Richard E. Aptitude, Learning, and Instruction. Vol. 1. Routledge; Philadelphia: 1980. Aptitude processes; pp. 27–63. [ Google Scholar ]
• Sohn Young W., Doane Stephanie M. Roles of working memory capacity and long-term working memory skill in complex task performance. Memory & Cognition. 2003; 31 :458–66. doi: 10.3758/bf03194403. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Sohn Young W., Doane Stephanie M. Memory processes of flight situation awareness: Interactive roles of working memory capacity, long-term working memory, and expertise. Human Factors. 2004; 46 :461–75. doi: 10.1518/hfes.46.3.461.50392. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Storm Benjamin C., Bui Dung C. Individual differences in mind wandering while reading predict lower rates of analogical transfer. Learning and Individual Differences. 2016; 51 :427–32. doi: 10.1016/j.lindif.2015.02.001. [ CrossRef ] [ Google Scholar ]
• Tomic Welco. Training in inductive reasoning and problem solving. Contemporary Educational Psychology. 1995; 20 :483–90. doi: 10.1006/ceps.1995.1036. [ CrossRef ] [ Google Scholar ]
• Unsworth Nash. Psychology of Learning and Motivation. vol. 65. Elsevier; Amsterdam: 2016. The Many Facets of Individual Differences in Working Memory Capacity; pp. 1–46. [ CrossRef ] [ Google Scholar ]
• Unsworth Nash, Spillers Gregory J. Working memory capacity: Attention control, secondary memory, or both? A direct test of the dual-component model. Journal of Memory and Language. 2010; 62 :392–406. doi: 10.1016/j.jml.2010.02.001. [ CrossRef ] [ Google Scholar ]
• Unsworth Nash, Fukuda Keisuke, Awh Edward, Vogel Edward K. Working memory and fluid intelligence: Capacity, attention control, and secondary memory retrieval. Cognitive Psychology. 2014; 71 :1–26. doi: 10.1016/j.cogpsych.2014.01.003. [ PMC free article ] [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Unsworth Nash, Heitz Richard P., Schrock Josef C., Engle Randall W. An automated version of the operation span task. Behavior Research Methods. 2005; 37 :498–505. doi: 10.3758/BF03192720. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Unsworth Nash, Redick Thomas S., Heitz Richard P., Broadway James M., Engle Randall W. Complex working memory span tasks and higher-order cognition: A latent-variable analysis of the relationship between processing and storage. Memory. 2009; 17 :635–54. doi: 10.1080/09658210902998047. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Verguts Tom, De Boeck Paul. The induction of solution rules in Raven’s Progressive Matrices Test. European Journal of Cognitive Psychology. 2002; 14 :521–47. doi: 10.1080/09541440143000230. [ CrossRef ] [ Google Scholar ]
• Voss James F., Greene Terry R., Post Timothy A., Penner Barbara C. Psychology of Learning and Motivation. Vol. 17. Elsevier; Amsterdam: 1983. Problem-solving skill in the social sciences; pp. 165–213. [ Google Scholar ]
• Wharton Charles M., Holyoak Keith J., Downing Paul E., Lange Trent E., Wickens Thomas D., Melz Eric R. Below the surface: Analogical similarity and retrieval competition in reminding. Cognitive Psychology. 1994; 26 :64–101. doi: 10.1006/cogp.1994.1003. [ CrossRef ] [ Google Scholar ]
• Wiley Jennifer. Expertise as mental set: The effects of domain knowledge in creative problem solving. Memory & Cognition. 1998; 26 :716–30. doi: 10.3758/BF03211392. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Wiley Jennifer, Jarosz Andrew F., Cushen Patrick J., Colflesh Gregory J. New rule use drives the relation between working memory capacity and Raven’s Advanced Progressive Matrices. Journal of Experimental Psychology: Learning, Memory, and Cognition. 2011; 37 :256. doi: 10.1037/a0021613. [ PubMed ] [ CrossRef ] [ Google Scholar ]

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September 3, 2021

New model for solving novel problems uses mental map

by UC Davis

How do we make decisions about a situation we have not encountered before? New work from the Center for Mind and Brain at the University of California, Davis, shows that we can solve abstract problems in the same way that we can find a novel route between two known locations—by using an internal cognitive map. The work is published Aug. 31 in the journal Nature Neuroscience .

Humans and animals have a great ability to solve novel problems by generalizing from existing knowledge and inferring new solutions from limited data. This is much harder to achieve with artificial intelligence.

Animals (including humans) navigate by creating a representative map of the outside world in their head as they move around. Once we know two locations are close to each other, we can infer that there is a shortcut between them even if we haven't been there. These maps make use of a network of "grid cells" and "place cells" in parts of the brain.

In earlier work, Professor Erie Boorman, postdoctoral researcher Seongmin (Alex) Park, Douglas Miller and colleagues showed that human volunteers could construct a similar cognitive map for abstract information. The volunteers were given limited information about people in a two-dimensional social network, ranked by relative competence and popularity. The researchers found that the volunteers could mentally reconstruct this network, represented as a grid, without seeing the original.

The new work takes the research further by testing if people can actually use such a map to find the answers to novel problems.

Matchmaking entrepreneurs

As before, volunteers learned about 16 people they were told were entrepreneurs, ranked on axes of competence and popularity. They never saw the complete grid, only comparisons between pairs.

They were then asked to select business partners for individual entrepreneurs that would maximize growth potential for a business they started together. The assumption was that an entrepreneur scoring high in competence but low on popularity would be complemented by one with a higher popularity score.

"For example, would Mark Zuckerberg be better off collaborating with Bill Gates or Richard Branson?" Boorman said. (The actual experiment did not use real people.)

While the volunteers were performing the decision task, the researchers scanned their brains with functional magnetic resonance imaging, or fMRI.

If the volunteers were using the grid cells inside their head to infer the answer, that should be measurable with a tailored analysis approach applied to the fMRI signal, Boorman said.

"It turns out the system in the brain does show the signature of these trajectories being computed on the fly," he said. "It looks like they are leveraging the cognitive map."

Computing solutions on the fly

In other words, we can take in loosely connected or fragmentary information, assemble it into a mental map, and use it to infer solutions to new problems.

Scientists have considered that the brain makes decisions by computing the value of each choice into a common currency, which allows them to be compared in a single dimension, Park said. For example, people might typically choose wine A over wine B based on price, but we know that our preference can be changed by the food you will pair with the wine.

"Our study suggests that the human brain does not have a wine list with fixed values, but locates wines in an abstract multidimensional space, which allows for computing new decision values flexibly according to the current demand," he said.

The cognitive map allows for computation on the fly with limited information, Boorman said.

"It's useful when the decisions are novel," he said. "It's a totally new framework for understanding decision making."

The navigational map in rodents is located in the entorhinal cortex , an "early" part of the brain. The cognitive map in humans expands into other parts of the brain including the prefrontal cortex and posterior medial cortex. These brain areas are part of the default mode network, a large "always on" brain network involved in autobiographical memory, imagination, planning and the theory of mind.

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How humans use objects in novel ways to solve problems

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Human beings are naturally creative tool users. When we need to drive in a nail but don’t have a hammer, we easily realize that we can use a heavy, flat object like a rock in its place. When our table is shaky, we quickly find that we can put a stack of paper under the table leg to stabilize it. But while these actions seem so natural to us, they are believed to be a hallmark of great intelligence — only a few other species use objects in novel ways to solve their problems, and none can do so as flexibly as people. What provides us with these powerful capabilities for using objects in this way?

In a new paper published in the Proceedings of the National Academy of Sciences describing work conducted at MIT’s Center for Brains, Minds and Machines , researchers Kelsey Allen, Kevin Smith, and Joshua Tenenbaum study the cognitive components that underlie this sort of improvised tool use. They designed a novel task, the Virtual Tools game , that taps into tool-use abilities: People must select one object from a set of “tools” that they can place in a two-dimensional, computerized scene to accomplish a goal, such as getting a ball into a certain container. Solving the puzzles in this game requires reasoning about a number of physical principles, including launching, blocking, or supporting objects.

The team hypothesized that there are three capabilities that people rely on to solve these puzzles: a prior belief that guides people’s actions toward those that will make a difference in the scene, the ability to imagine the effect of their actions, and a mechanism to quickly update their beliefs about what actions are likely to provide a solution. They built a model that instantiated these principles, called the “Sample, Simulate, Update,” or “SSUP,” model, and had it play the same game as people. They found that SSUP solved each puzzle at similar rates and in similar ways as people did. On the other hand, a popular deep learning model that could play Atari games well but did not have the same object and physical structures was unable to generalize its knowledge to puzzles it was not directly trained on.

This research provides a new framework for studying and formalizing the cognition that supports human tool use. The team hopes to extend this framework to not just study tool use, but also how people can create innovative new tools for new problems, and how humans transmit this information to build from simple physical tools to complex objects like computers or airplanes that are now part of our daily lives.

Kelsey Allen, a PhD student in the Computational Cognitive Science Lab at MIT, is excited about how the Virtual Tools game might support other cognitive scientists interested in tool use: “There is just so much more to explore in this domain. We have already started collaborating with researchers across multiple different institutions on projects ranging from studying what it means for games to be fun, to studying how embodiment affects disembodied physical reasoning. I hope that others in the cognitive science community will use the game as a tool to better understand how physical models interact with decision-making and planning.”

Joshua Tenenbaum, professor of computational cognitive science at MIT, sees this work as a step toward understanding not only an important aspect of human cognition and culture, but also how to build more human-like forms of intelligence in machines. “Artificial Intelligence researchers have been very excited about the potential for reinforcement learning (RL) algorithms to learn from trial-and-error experience, as humans do, but the real trial-and-error learning that humans benefit from unfolds over just a handful of trials — not millions or billions of experiences, as in today’s RL systems,” Tenenbaum says. “The Virtual Tools game allows us to study this very rapid and much more natural form of trial-and-error learning in humans, and the fact that the SSUP model is able to capture the fast learning dynamics we see in humans suggests it may also point the way towards new AI approaches to RL that can learn from their successes, their failures, and their near misses as quickly and as flexibly as people do.” 

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The 11 Best Problem Solving Books

Adventurer, Tech Geek and Lover of Productivity Hacks. 

Learn how you can improve your problem solving skills with this curated list of the 11 Best Problem Solving Books on the market.

Looking for new insights and best practices when it comes to coming up with proven, quality solutions to the problems we face both at home and in the workplace?

Fortunately, there are a variety of problem solving books out there that are filled from front to back with new and exciting ways to conquer the issues that we deal with on a daily basis.

Whether we like to admit it or not, problem solving skills are high in demand these days whether it’s in the workplace or in the comforts of your own home.

One thing that is for sure is that life is definitely easier when you have the skills to solve problems with ease.

The best part is that problem solving is that it’s a skill that anyone can learn.

Below, you’ll find a list of the best problem solving books that should be helpful for those interested in really diving into the art of problem solving.

Table of Contents

Best problem solving books, sprint, how to solve big problems and test new ideas in just five days, the innovator’s dilemma, switch, how to change things when change is hard, problem solving 101, seeking wisdom: from darwin to munger.

• The Art of Thinking Clearly 

Mastermind: How to Think Like Sherlock Holmes

How to solve it: a new aspect of mathematical method, what do you do with a problem.

• The Art and Craft of Problem Solving 

The Back of the Napkin: Solving Problems and Selling Ideas with Pictures

Author Dr. Jason Selk and Tom Bartow

Sprint offers a transformative formula for testing ideas that work whether it is for yourself or for a large corporation. The ideas that Sprint provides you are already tested and successful ones therefore you have nothing to lose giving them a try. Whenever you are feeling stuck and don’t know how to solve an issue, check out these ideas and test them out to see which one works best in your favor.

Author Clayton M. Christensen

Named one of 100 Leadership & Success Books to Read in a Lifetime by Amazon Editors, The Innovator’s Dilemma offers a different approach to problem-solving. This book helps you look at your problem from an outsider point of view. Whenever you don’t know where to go next and how to solve an issue, the best thing is stepping out of the box and seeing whatever is that you are missing to identify in order to solve it. 

Authors Chip Heath and Dan Heath

Psychologists have discovered that our minds are ruled by two different systems: the rational mind and the emotional mind that compete for control. The concept of this book is using our internal “switch” and learning when to use each type of mind. Mainly, allowing ourselves to have full control on when we decide to use the emotional side and the rational side. These factors will allow you to make decisions in a more concise manner and therefore have a smarter mentality when it comes to problem-solving. 

Author Ken Watanabe

Originally written to help Japanese schoolchildren learn how to be better problem solvers, this book ended in every businessman’s desk as the information was just too valuable. This book is filled with simple-to-follow case studies to illustrate different solutions to problem-solving. 

Author Peter Bevelin

This book covers everything from the exact moment we come up with an idea, to the point where we are stuck and don’t know how to move on past the issue. Through a psychological point of view, the author helps us understand the way our minds evolve. He essentially leads out a misjudgment point of view to one of a better and wiser thinker. 

The Art of Thinking Clearly

Author Rolf Dobelli

The Art of Thinking Clearly isn’t just another one of the best problem-solving books, but it is a guide to living a more convenient life, where every step that we take has its own action and consequence. Understanding that problem-solving comes from having an organized mind is the first place to get started when we are capable of thinking clearly, the solutions come to us in a clear manner as well. 

Author  Maria Konnikova

Who wouldn’t want to have the mind and the skills of problem-solving that Sherlock Holmes has? Well with this book you are able to acquire some of those astounding skills to use into your daily life. Holmes is one of the world’s most proficient problem solvers and Konnikova highlights the key characteristics that make him so effective in order for the reader to apply them. 

Author George Pólya

George Pólaya uses this mathematical method to help people to think straight. Through his brilliant method he has helped a lot of people tackle their problems only by changing the way that they think. Our mind is more powerful than we know, and therefore knowing how to work our way around it might help people deal with daily life struggles. 

Author by Kobi Yamada

What Do You Do with a Problem? Especially one that you can’t get rid of and can’t find a way to fix? Kobi Yamada tackles this exact scenario and offers the reader multiple ideas to deal with that one problem that seems to not go away. The key in the book is to never avoid a problem, the more we avoid dealing with one issue, the bigger it will become. 

The Art and Craft of Problem Solving

Author  Paul Zeitz

This text offers unique skills and solutions to approach a problem. Not only it helps to identify how to fix the problem but also to understand the problem itself. Understanding how the problem developed and when it started to become a problem for us, is important in order to avoid future conflicts. Tackling the problem is one thing, learning how to stop problems for developing is another great quality. 

Author Dan Roam

Herb Kelleher was brainstorming about the traditional method we deal with problem solving and it was in this exact moment where he grabbed a bar napkin and a pen and decided to scribble what problem solving would look like. He believed that people could understand something better by looking at it, and for that reason he decided to incorporate this lesson into his book. 

Used properly, a simple drawing was more demonstrative than a simple PowerPoint, but it can help crystallize ideas, think outside the box.

Did you find this list of problem solving books to be helpful? If I missed one that you recommend, please leave a comment below.

You Might Also Enjoy: Top 25 Books About Productivity & The Top 25 Productivity Blogs for 2020

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12 Best Problem Solving Books to Read in 2024

You found our list of top problem solving books .

Problem solving books are guides that improve critical thinking capability and the ability to resolve issues in the workplace. These works cover topics like bias and logical fallacies, problem prevention, and prioritizing. The purpose of these books is to help workers remain calm under pressure and come up with solutions more quickly.

These guides are similar to decision making books , negotiation books , and conflict resolution books . To improve competency in this area, one can also play problem solving games .

This list includes:

• problem solving books for adults
• creative problem solving books
• business problem solving books
• problem solving books for programmers

Here we go!

List of problem solving books

Here is a list of books to improve problem solving skills in the workplace.

1. Fixed: How to Perfect the Fine Art of Problem Solving by Amy E Herman

Fixed is one of the most useful new books on problem solving. The book calls for problem solvers to look beyond instinctual and obvious answers and provides a framework for more creative thinking. While most folks think about problem solving in terms of logic, reason, and disciplines like math and science, this book shows the role that art and imagination play in the process. Amy Herman consulted on leadership training with Silicon Valley companies and military organizations and brings this expertise into the text to train readers on how to adopt a more innovative critical thinking approach.

Notable Quote: “Working through problems is critical for productivity, profit, and peace. Our problem-solving skills, however, have been short-circuited by our complicated, technology-reliant world.”

Read Fixed .

2. Cracked it!: How to solve big problems and sell solutions like top strategy consultants by Bernard Garrette, Corey Phelps, and Olivier Sibony

Cracked it! is one of the best creative problem solving books. Drawing inspiration from the tactics of consultants, this guide is a practical playbook for approaching business problems. The authors outline a “4S” method– State – Structure – Solve – Sell– to tackle obstacles and get support from stakeholders. While many problem solving books simply focus on how to think through issues, this guide also demonstrates how to gain approval for ideas and get others onboard with the solution. The book explains how to best use these techniques, and presents case studies that show the theories in action. Cracked it! is a handy reference for any professional that faces tough challenges on the regular.

Notable Quote: “If you want to know how a lion hunts, don’t go to a zoo. Go to the jungle.”

Read Cracked it!

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3. Upstream: The Quest to Solve Problems Before They Happen by Dan Heath

Upstream takes a proactive approach to problem solving. The book urges readers to not only be responsive to issues, but also try to prevent obstacles from occurring. The guide opens with an exploration of “problem blindness,” and the psychological factors that cause folks to be oblivious to issues, along with a reminder that many problems are more controllable and avoidable than first assumed. The pages that follow outline a series of questions leaders can ask to fine-tune the system and steer clear of major headaches, for instance, “How Will You Unite the Right People?” and “How Will You Avoid Doing Harm?” Upstream is full of real world examples of how minor tweaks achieved major results and allowed organizations to sidestep serious holdups.

Notable Quote: “The postmortem for a problem can be the preamble to a solution.”

Read Upstream .

4. Problem Solving 101: A Simple Book for Smart People by Ken Watanabe

Problem Solving 101 is one of the most fun problem solving books for adults. Written by Ken Watanabe, the guide draws on Japanese philosophy as well as the author’s experience as a consultant at McKinsey to help readers understand and approach problems in productive ways. The pages provide blueprints for problem-solving methods such as logic trees and matrixes, and include scenarios and illustrations that help readers visualize the process more clearly. Problem Solving 101 breaks down the problem solving procedure into the most basic parts and lays out step-by-step instructions for choosing the best action in any situation.

Notable Quote: “When you do take action, every result is an opportunity to reflect and learn valuable lessons. Even if what you take away from your assessment seems to be of small consequence, all of these small improvements taken together make a huge difference in the long term.”

Read Problem Solving 101 .

5. What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve by Thomas Wedell-Wedellsborg

What’s Your Problem? insists that the most important step in the problem solving process is to start by honing in on the correct problem. The root of much frustration and wasted efforts is that professionals often pick the wrong points to focus on. This book teaches readers how to reframe and approach issues from a different perspective. The guide outlines a repeatable three step process “Frame, Reframe, and Move Forward” to ensure that workers prioritize effectively and stay on track to achieve desired results. What’s Your Problem? teaches professionals of all levels how to be less rigid and more results-focused and adopt a more agile approach to fixing issues.

Notable Quote: “The problems we’re trained on in school are often quite different from the ones we encounter in real life.”

Read What’s Your Problem?

6. Sprint: How to Solve Big Problems and Test New Ideas in Just Five Days by Jake Knapp, John Zeratsky, et al

Sprint is one of the best problem solving books for programmers. The authors are the creators of the five-day-process at Google. This guide describes best practices for conducting sprints and solving problems in limited timeframes. The book provides a day-by-day breakdown of tasks for each day of the workweek, with the final steps being designing a prototype and a plan for implementation. Though this idea originated in the tech world and is most widely used in the software industry, this problem-solving and product design approach can be useful for any position that needs to find fixes in a time crunch.

Notable Quote: “We’ve found that magic happens when we use big whiteboards to solve problems. As humans, our short-term memory is not all that good, but our spatial memory is awesome. A sprint room, plastered with notes, diagrams, printouts, and more, takes advantage of that spatial memory. The room itself becomes a sort of shared brain for the team.”

Read Sprint , and check out this guide to virtual hackathons and this list of product design books .

7. Think Like a Rocket Scientist: Simple Strategies You Can Use to Make Giant Leaps in Work and Life by Ozan Varol

Think Like a Rocket Scientist lays out formulas and instructions for thinking more strategically. The guide reveals common problem solving approaches used by rocket scientists when exploring the unknown and testing new technology. The book is split into three sections– launch, accelerate, and achieve– with deep dives into concepts such as moonshot thinking and overcoming failure. The anecdotes revolve around space exploration and rocket science yet the methods can be applied to more commonplace and less complex problems as well. Think Like a Rocket Scientist proves that one does not need to be a genius to be a genius problem solver and lets readers learn tricks from one of the most complex professions on the planet.

Notable Quote: “Critical thinking and creativity don’t come naturally to us. We’re hesitant to think big, reluctant to dance with uncertainty, and afraid of failure. These were necessary during the Paleolithic Period, keeping us safe from poisonous foods and predators. But here in the information age, they’re bugs.”

Read Think Like a Rocket Scientist .

8. Bulletproof Problem Solving: The One Skill That Changes Everything by Charles Conn and Robert McLean

Bulletproof Problem Solving is one of the best business problem solving books. This workbook-style-guide breaks down a “bulletproof” method of problem solving favored by consultants at McKinsey. The authors distill the process into seven simple steps–define the problem, disaggregate, prioritize, workplan, analyze, synthesize, and communicate– and give numerous examples of how to follow this cycle with different dilemmas. The chapters explore each stage in depth and outline the importance and finer points of each phase. The book also provides practical tools for readers to build skills, including an appendix with exercise worksheets.

Notable Quote: “Problem solving doesn’t stop at the point of reaching conclusions from individual analyses. Findings have to be assembled into a logical structure to test validity and then synthesized in a way that convinces others that you have a good solution. Great team processes are also important at this stage.”

Read Bulletproof Problem Solving .

9. Think Like a Programmer: An Introduction to Creative Problem Solving by by V. Anton Spraul

Think Like a Programmer is one of the top problem solving books for programmers. The guide lays out methods for finding and fixing bugs and creating clean, workable code. The text emphasizes that programming is not merely a matter of being competent in the language, but also knowing how to troubleshoot and respond to unexpected occurrences. The chapters present examples of problems and puzzles and work through the answers to help strengthen professional competencies. The book provides an introductory crash course and practical toolkit for beginning coders, with a focus on C++. Yet since the text outlines general theory and approach, the book is also helpful for dealing with other programming languages, or for solving problems in non-tech industries as well. The point of the text is to provide a proper mindset and attitude for reacting to these developments, and the book can be a benefit for folks in any field.

Notable Quote: “Don’t Get Frustrated The final technique isn’t so much a technique, but a maxim: Don’t get frustrated. When you are frustrated, you won’t think as clearly, you won’t work as efficiently, and everything will take longer and seem harder. Even worse, frustration tends to feed on itself, so that what begins as mild irritation ends as outright anger.”

Read Think Like a Programmer .

10. The Founder’s Dilemmas: Anticipating and Avoiding the Pitfalls That Can Sink a Startup by by Noam Wasserman

The Founder’s Dilemmas lays out the most common problems entrepreneurs face and gives advice on how to avoid or solve these issues. The book tackles topics such as managing relationships, hiring, and rewarding or correcting employees. The chapters outline the mistakes inexperienced leaders often make and offer strategies for handling these tough situations with more smarts and skill. By reading this book, founders can learn from predecessors and avoid making obvious and avoidable errors in judgment. The Founder’s Dilemmas is a problem-solving resource for startup leaders and team members who lack more traditional guidance.

Notable Quote: “Ideas are cheap; execution is dear.”

Read The Founder’s Dilemmas , and check out more entrepreneurial books .

11. The Scout Mindset: Why Some People See Things Clearly and Others Don’t by Julia Galef

The Scout Mindset challenges readers to move beyond gut reactions and preconceptions and rethink problems. The book offers instructions for overcoming bias and central beliefs to gather more objective data. Julia Galef encourages readers to act more like scouts than soldiers and gather information without judging to make more informed decisions. The text outlines the common reasons folks jump to conclusions and offers advice on how to avoid incorrect assumptions and conduct level-headed analyses. The Scout Mindset is a call to action for objectivity and an instruction manual for breaking away from unhelpful mental patterns that can lead to poor choices.

Notable Quote: “Discovering you were wrong is an update, not a failure, and your worldview is a living document meant to be revised.”

Read The Scout Mindset .

12. Super Thinking: The Big Book of Mental Models by Gabriel Weinberg and Lauren McCann

Super Thinking is a comprehensive resource that explains various mental models for problem solving. The book identifies logical fallacies and shows readers how to avoid these pitfalls. The pages also lay out appropriate strategies, tools, techniques to use in different situations, such as matrices, pointed questions, and philosophies. The point of the guide is to teach readers how to evaluate information and make quick yet accurate judgements. The guide helps readers decide the best approach to use for each circumstance. Though packed with information, the pages also contain images and humor that prevent the material from getting too dry. Super Thinking is the ultimate cheat sheet for thinking rationally and acting with intention.

Notable Quote: “Unfortunately, people often make the mistake of doing way too much work before testing assumptions in the real world.”

Read Super Thinking .

Final Thoughts

Problem solving is one of the most essential skills for modern industry. With the breakneck pace at which the current business world changes, there is no shortage of new developments that professionals must contend with on a daily basis. Operating the same way for years at a time is impossible, and it is almost guaranteed that workers at every level will have issues to unravel at some point in their careers.

Books about problem solving help professionals predict, prevent, and overcome issues and find more viable and sustainable solutions. These guides not only provide skills, but also methods for survival in a highly competitive business landscape. These texts show workers that they are more capable than may first appear and that sometimes, seemingly insurmountable obstacles are beatable with a combination of creativity, teamwork, and proper process.

For more ways to beat the odds, check out this list of books on innovation and this list of books on business strategy .

We also have a list of the best communication books .

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FAQ: Problem solving books

Here are answers to common questions about problem solving books.

What are problem solving books?

Problem solving books are guides that teach critical thinking skills and strategies for resolving issues. The purpose of these works is to help professionals be more creative and strategic in problem solving approaches.

What are some good problem solving books for work?

Some good problem solving books for work include Sprint by Jake Knapp, John Zeratsky, et al, Upstream by Dan Heath, and Think Like a Rocket Scientist by Ozan Varol.

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Do You Understand the Problem You’re Trying to Solve?

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Problem solving skills are invaluable in any job. But all too often, we jump to find solutions to a problem without taking time to really understand the dilemma we face, according to Thomas Wedell-Wedellsborg , an expert in innovation and the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

In this episode, you’ll learn how to reframe tough problems by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for just one root cause can be misleading.

Key episode topics include: leadership, decision making and problem solving, power and influence, business management.

HBR On Leadership curates the best case studies and conversations with the world’s top business and management experts, to help you unlock the best in those around you. New episodes every week.

• Listen to the original HBR IdeaCast episode: The Secret to Better Problem Solving (2016)
• Find more episodes of HBR IdeaCast
• Discover 100 years of Harvard Business Review articles, case studies, podcasts, and more at HBR.org .

HANNAH BATES: Welcome to HBR on Leadership , case studies and conversations with the world’s top business and management experts, hand-selected to help you unlock the best in those around you.

Problem solving skills are invaluable in any job. But even the most experienced among us can fall into the trap of solving the wrong problem.

Thomas Wedell-Wedellsborg says that all too often, we jump to find solutions to a problem – without taking time to really understand what we’re facing.

He’s an expert in innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

  In this episode, you’ll learn how to reframe tough problems, by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for one root cause can be misleading. And you’ll learn how to use experimentation and rapid prototyping as problem-solving tools.

This episode originally aired on HBR IdeaCast in December 2016. Here it is.

SARAH GREEN CARMICHAEL: Welcome to the HBR IdeaCast from Harvard Business Review. I’m Sarah Green Carmichael.

Problem solving is popular. People put it on their resumes. Managers believe they excel at it. Companies count it as a key proficiency. We solve customers’ problems.

The problem is we often solve the wrong problems. Albert Einstein and Peter Drucker alike have discussed the difficulty of effective diagnosis. There are great frameworks for getting teams to attack true problems, but they’re often hard to do daily and on the fly. That’s where our guest comes in.

Thomas Wedell-Wedellsborg is a consultant who helps companies and managers reframe their problems so they can come up with an effective solution faster. He asks the question “Are You Solving The Right Problems?” in the January-February 2017 issue of Harvard Business Review. Thomas, thank you so much for coming on the HBR IdeaCast .

THOMAS WEDELL-WEDELLSBORG: Thanks for inviting me.

SARAH GREEN CARMICHAEL: So, I thought maybe we could start by talking about the problem of talking about problem reframing. What is that exactly?

THOMAS WEDELL-WEDELLSBORG: Basically, when people face a problem, they tend to jump into solution mode to rapidly, and very often that means that they don’t really understand, necessarily, the problem they’re trying to solve. And so, reframing is really a– at heart, it’s a method that helps you avoid that by taking a second to go in and ask two questions, basically saying, first of all, wait. What is the problem we’re trying to solve? And then crucially asking, is there a different way to think about what the problem actually is?

SARAH GREEN CARMICHAEL: So, I feel like so often when this comes up in meetings, you know, someone says that, and maybe they throw out the Einstein quote about you spend an hour of problem solving, you spend 55 minutes to find the problem. And then everyone else in the room kind of gets irritated. So, maybe just give us an example of maybe how this would work in practice in a way that would not, sort of, set people’s teeth on edge, like oh, here Sarah goes again, reframing the whole problem instead of just solving it.

THOMAS WEDELL-WEDELLSBORG: I mean, you’re bringing up something that’s, I think is crucial, which is to create legitimacy for the method. So, one of the reasons why I put out the article is to give people a tool to say actually, this thing is still important, and we need to do it. But I think the really critical thing in order to make this work in a meeting is actually to learn how to do it fast, because if you have the idea that you need to spend 30 minutes in a meeting delving deeply into the problem, I mean, that’s going to be uphill for most problems. So, the critical thing here is really to try to make it a practice you can implement very, very rapidly.

There’s an example that I would suggest memorizing. This is the example that I use to explain very rapidly what it is. And it’s basically, I call it the slow elevator problem. You imagine that you are the owner of an office building, and that your tenants are complaining that the elevator’s slow.

Now, if you take that problem framing for granted, you’re going to start thinking creatively around how do we make the elevator faster. Do we install a new motor? Do we have to buy a new lift somewhere?

The thing is, though, if you ask people who actually work with facilities management, well, they’re going to have a different solution for you, which is put up a mirror next to the elevator. That’s what happens is, of course, that people go oh, I’m busy. I’m busy. I’m– oh, a mirror. Oh, that’s beautiful.

And then they forget time. What’s interesting about that example is that the idea with a mirror is actually a solution to a different problem than the one you first proposed. And so, the whole idea here is once you get good at using reframing, you can quickly identify other aspects of the problem that might be much better to try to solve than the original one you found. It’s not necessarily that the first one is wrong. It’s just that there might be better problems out there to attack that we can, means we can do things much faster, cheaper, or better.

SARAH GREEN CARMICHAEL: So, in that example, I can understand how A, it’s probably expensive to make the elevator faster, so it’s much cheaper just to put up a mirror. And B, maybe the real problem people are actually feeling, even though they’re not articulating it right, is like, I hate waiting for the elevator. But if you let them sort of fix their hair or check their teeth, they’re suddenly distracted and don’t notice.

But if you have, this is sort of a pedestrian example, but say you have a roommate or a spouse who doesn’t clean up the kitchen. Facing that problem and not having your elegant solution already there to highlight the contrast between the perceived problem and the real problem, how would you take a problem like that and attack it using this method so that you can see what some of the other options might be?

THOMAS WEDELL-WEDELLSBORG: Right. So, I mean, let’s say it’s you who have that problem. I would go in and say, first of all, what would you say the problem is? Like, if you were to describe your view of the problem, what would that be?

SARAH GREEN CARMICHAEL: I hate cleaning the kitchen, and I want someone else to clean it up.

THOMAS WEDELL-WEDELLSBORG: OK. So, my first observation, you know, that somebody else might not necessarily be your spouse. So, already there, there’s an inbuilt assumption in your question around oh, it has to be my husband who does the cleaning. So, it might actually be worth, already there to say, is that really the only problem you have? That you hate cleaning the kitchen, and you want to avoid it? Or might there be something around, as well, getting a better relationship in terms of how you solve problems in general or establishing a better way to handle small problems when dealing with your spouse?

SARAH GREEN CARMICHAEL: Or maybe, now that I’m thinking that, maybe the problem is that you just can’t find the stuff in the kitchen when you need to find it.

THOMAS WEDELL-WEDELLSBORG: Right, and so that’s an example of a reframing, that actually why is it a problem that the kitchen is not clean? Is it only because you hate the act of cleaning, or does it actually mean that it just takes you a lot longer and gets a lot messier to actually use the kitchen, which is a different problem. The way you describe this problem now, is there anything that’s missing from that description?

SARAH GREEN CARMICHAEL: That is a really good question.

THOMAS WEDELL-WEDELLSBORG: Other, basically asking other factors that we are not talking about right now, and I say those because people tend to, when given a problem, they tend to delve deeper into the detail. What often is missing is actually an element outside of the initial description of the problem that might be really relevant to what’s going on. Like, why does the kitchen get messy in the first place? Is it something about the way you use it or your cooking habits? Is it because the neighbor’s kids, kind of, use it all the time?

There might, very often, there might be issues that you’re not really thinking about when you first describe the problem that actually has a big effect on it.

SARAH GREEN CARMICHAEL: I think at this point it would be helpful to maybe get another business example, and I’m wondering if you could tell us the story of the dog adoption problem.

THOMAS WEDELL-WEDELLSBORG: Yeah. This is a big problem in the US. If you work in the shelter industry, basically because dogs are so popular, more than 3 million dogs every year enter a shelter, and currently only about half of those actually find a new home and get adopted. And so, this is a problem that has persisted. It’s been, like, a structural problem for decades in this space. In the last three years, where people found new ways to address it.

So a woman called Lori Weise who runs a rescue organization in South LA, and she actually went in and challenged the very idea of what we were trying to do. She said, no, no. The problem we’re trying to solve is not about how to get more people to adopt dogs. It is about keeping the dogs with their first family so they never enter the shelter system in the first place.

In 2013, she started what’s called a Shelter Intervention Program that basically works like this. If a family comes and wants to hand over their dog, these are called owner surrenders. It’s about 30% of all dogs that come into a shelter. All they would do is go up and ask, if you could, would you like to keep your animal? And if they said yes, they would try to fix whatever helped them fix the problem, but that made them turn over this.

And sometimes that might be that they moved into a new building. The landlord required a deposit, and they simply didn’t have the money to put down a deposit. Or the dog might need a $10 rabies shot, but they didn’t know how to get access to a vet.

And so, by instigating that program, just in the first year, she took her, basically the amount of dollars they spent per animal they helped went from something like $85 down to around $60. Just an immediate impact, and her program now is being rolled out, is being supported by the ASPCA, which is one of the big animal welfare stations, and it’s being rolled out to various other places.

And I think what really struck me with that example was this was not dependent on having the internet. This was not, oh, we needed to have everybody mobile before we could come up with this. This, conceivably, we could have done 20 years ago. Only, it only happened when somebody, like in this case Lori, went in and actually rethought what the problem they were trying to solve was in the first place.

SARAH GREEN CARMICHAEL: So, what I also think is so interesting about that example is that when you talk about it, it doesn’t sound like the kind of thing that would have been thought of through other kinds of problem solving methods. There wasn’t necessarily an After Action Review or a 5 Whys exercise or a Six Sigma type intervention. I don’t want to throw those other methods under the bus, but how can you get such powerful results with such a very simple way of thinking about something?

THOMAS WEDELL-WEDELLSBORG: That was something that struck me as well. This, in a way, reframing and the idea of the problem diagnosis is important is something we’ve known for a long, long time. And we’ve actually have built some tools to help out. If you worked with us professionally, you are familiar with, like, Six Sigma, TRIZ, and so on. You mentioned 5 Whys. A root cause analysis is another one that a lot of people are familiar with.

Those are our good tools, and they’re definitely better than nothing. But what I notice when I work with the companies applying those was those tools tend to make you dig deeper into the first understanding of the problem we have. If it’s the elevator example, people start asking, well, is that the cable strength, or is the capacity of the elevator? That they kind of get caught by the details.

That, in a way, is a bad way to work on problems because it really assumes that there’s like a, you can almost hear it, a root cause. That you have to dig down and find the one true problem, and everything else was just symptoms. That’s a bad way to think about problems because problems tend to be multicausal.

There tend to be lots of causes or levers you can potentially press to address a problem. And if you think there’s only one, if that’s the right problem, that’s actually a dangerous way. And so I think that’s why, that this is a method I’ve worked with over the last five years, trying to basically refine how to make people better at this, and the key tends to be this thing about shifting out and saying, is there a totally different way of thinking about the problem versus getting too caught up in the mechanistic details of what happens.

SARAH GREEN CARMICHAEL: What about experimentation? Because that’s another method that’s become really popular with the rise of Lean Startup and lots of other innovation methodologies. Why wouldn’t it have worked to, say, experiment with many different types of fixing the dog adoption problem, and then just pick the one that works the best?

THOMAS WEDELL-WEDELLSBORG: You could say in the dog space, that’s what’s been going on. I mean, there is, in this industry and a lot of, it’s largely volunteer driven. People have experimented, and they found different ways of trying to cope. And that has definitely made the problem better. So, I wouldn’t say that experimentation is bad, quite the contrary. Rapid prototyping, quickly putting something out into the world and learning from it, that’s a fantastic way to learn more and to move forward.

My point is, though, that I feel we’ve come to rely too much on that. There’s like, if you look at the start up space, the wisdom is now just to put something quickly into the market, and then if it doesn’t work, pivot and just do more stuff. What reframing really is, I think of it as the cognitive counterpoint to prototyping. So, this is really a way of seeing very quickly, like not just working on the solution, but also working on our understanding of the problem and trying to see is there a different way to think about that.

If you only stick with experimentation, again, you tend to sometimes stay too much in the same space trying minute variations of something instead of taking a step back and saying, wait a minute. What is this telling us about what the real issue is?

SARAH GREEN CARMICHAEL: So, to go back to something that we touched on earlier, when we were talking about the completely hypothetical example of a spouse who does not clean the kitchen–

THOMAS WEDELL-WEDELLSBORG: Completely, completely hypothetical.

SARAH GREEN CARMICHAEL: Yes. For the record, my husband is a great kitchen cleaner.

You started asking me some questions that I could see immediately were helping me rethink that problem. Is that kind of the key, just having a checklist of questions to ask yourself? How do you really start to put this into practice?

THOMAS WEDELL-WEDELLSBORG: I think there are two steps in that. The first one is just to make yourself better at the method. Yes, you should kind of work with a checklist. In the article, I kind of outlined seven practices that you can use to do this.

But importantly, I would say you have to consider that as, basically, a set of training wheels. I think there’s a big, big danger in getting caught in a checklist. This is something I work with.

My co-author Paddy Miller, it’s one of his insights. That if you start giving people a checklist for things like this, they start following it. And that’s actually a problem, because what you really want them to do is start challenging their thinking.

So the way to handle this is to get some practice using it. Do use the checklist initially, but then try to step away from it and try to see if you can organically make– it’s almost a habit of mind. When you run into a colleague in the hallway and she has a problem and you have five minutes, like, delving in and just starting asking some of those questions and using your intuition to say, wait, how is she talking about this problem? And is there a question or two I can ask her about the problem that can help her rethink it?

SARAH GREEN CARMICHAEL: Well, that is also just a very different approach, because I think in that situation, most of us can’t go 30 seconds without jumping in and offering solutions.

THOMAS WEDELL-WEDELLSBORG: Very true. The drive toward solutions is very strong. And to be clear, I mean, there’s nothing wrong with that if the solutions work. So, many problems are just solved by oh, you know, oh, here’s the way to do that. Great.

But this is really a powerful method for those problems where either it’s something we’ve been banging our heads against tons of times without making progress, or when you need to come up with a really creative solution. When you’re facing a competitor with a much bigger budget, and you know, if you solve the same problem later, you’re not going to win. So, that basic idea of taking that approach to problems can often help you move forward in a different way than just like, oh, I have a solution.

I would say there’s also, there’s some interesting psychological stuff going on, right? Where you may have tried this, but if somebody tries to serve up a solution to a problem I have, I’m often resistant towards them. Kind if like, no, no, no, no, no, no. That solution is not going to work in my world. Whereas if you get them to discuss and analyze what the problem really is, you might actually dig something up.

Let’s go back to the kitchen example. One powerful question is just to say, what’s your own part in creating this problem? It’s very often, like, people, they describe problems as if it’s something that’s inflicted upon them from the external world, and they are innocent bystanders in that.

SARAH GREEN CARMICHAEL: Right, or crazy customers with unreasonable demands.

THOMAS WEDELL-WEDELLSBORG: Exactly, right. I don’t think I’ve ever met an agency or consultancy that didn’t, like, gossip about their customers. Oh, my god, they’re horrible. That, you know, classic thing, why don’t they want to take more risk? Well, risk is bad.

It’s their business that’s on the line, not the consultancy’s, right? So, absolutely, that’s one of the things when you step into a different mindset and kind of, wait. Oh yeah, maybe I actually am part of creating this problem in a sense, as well. That tends to open some new doors for you to move forward, in a way, with stuff that you may have been struggling with for years.

SARAH GREEN CARMICHAEL: So, we’ve surfaced a couple of questions that are useful. I’m curious to know, what are some of the other questions that you find yourself asking in these situations, given that you have made this sort of mental habit that you do? What are the questions that people seem to find really useful?

THOMAS WEDELL-WEDELLSBORG: One easy one is just to ask if there are any positive exceptions to the problem. So, was there day where your kitchen was actually spotlessly clean? And then asking, what was different about that day? Like, what happened there that didn’t happen the other days? That can very often point people towards a factor that they hadn’t considered previously.

SARAH GREEN CARMICHAEL: We got take-out.

THOMAS WEDELL-WEDELLSBORG: S,o that is your solution. Take-out from [INAUDIBLE]. That might have other problems.

Another good question, and this is a little bit more high level. It’s actually more making an observation about labeling how that person thinks about the problem. And what I mean with that is, we have problem categories in our head. So, if I say, let’s say that you describe a problem to me and say, well, we have a really great product and are, it’s much better than our previous product, but people aren’t buying it. I think we need to put more marketing dollars into this.

Now you can go in and say, that’s interesting. This sounds like you’re thinking of this as a communications problem. Is there a different way of thinking about that? Because you can almost tell how, when the second you say communications, there are some ideas about how do you solve a communications problem. Typically with more communication.

And what you might do is go in and suggest, well, have you considered that it might be, say, an incentive problem? Are there incentives on behalf of the purchasing manager at your clients that are obstructing you? Might there be incentive issues with your own sales force that makes them want to sell the old product instead of the new one?

So literally, just identifying what type of problem does this person think about, and is there different potential way of thinking about it? Might it be an emotional problem, a timing problem, an expectations management problem? Thinking about what label of what type of problem that person is kind of thinking as it of.

SARAH GREEN CARMICHAEL: That’s really interesting, too, because I think so many of us get requests for advice that we’re really not qualified to give. So, maybe the next time that happens, instead of muddying my way through, I will just ask some of those questions that we talked about instead.

THOMAS WEDELL-WEDELLSBORG: That sounds like a good idea.

SARAH GREEN CARMICHAEL: So, Thomas, this has really helped me reframe the way I think about a couple of problems in my own life, and I’m just wondering. I know you do this professionally, but is there a problem in your life that thinking this way has helped you solve?

THOMAS WEDELL-WEDELLSBORG: I’ve, of course, I’ve been swallowing my own medicine on this, too, and I think I have, well, maybe two different examples, and in one case somebody else did the reframing for me. But in one case, when I was younger, I often kind of struggled a little bit. I mean, this is my teenage years, kind of hanging out with my parents. I thought they were pretty annoying people. That’s not really fair, because they’re quite wonderful, but that’s what life is when you’re a teenager.

And one of the things that struck me, suddenly, and this was kind of the positive exception was, there was actually an evening where we really had a good time, and there wasn’t a conflict. And the core thing was, I wasn’t just seeing them in their old house where I grew up. It was, actually, we were at a restaurant. And it suddenly struck me that so much of the sometimes, kind of, a little bit, you love them but they’re annoying kind of dynamic, is tied to the place, is tied to the setting you are in.

And of course, if– you know, I live abroad now, if I visit my parents and I stay in my old bedroom, you know, my mother comes in and wants to wake me up in the morning. Stuff like that, right? And it just struck me so, so clearly that it’s– when I change this setting, if I go out and have dinner with them at a different place, that the dynamic, just that dynamic disappears.

SARAH GREEN CARMICHAEL: Well, Thomas, this has been really, really helpful. Thank you for talking with me today.

THOMAS WEDELL-WEDELLSBORG: Thank you, Sarah.  

HANNAH BATES: That was Thomas Wedell-Wedellsborg in conversation with Sarah Green Carmichael on the HBR IdeaCast. He’s an expert in problem solving and innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

We’ll be back next Wednesday with another hand-picked conversation about leadership from the Harvard Business Review. If you found this episode helpful, share it with your friends and colleagues, and follow our show on Apple Podcasts, Spotify, or wherever you get your podcasts. While you’re there, be sure to leave us a review.

We’re a production of Harvard Business Review. If you want more podcasts, articles, case studies, books, and videos like this, find it all at HBR dot org.

This episode was produced by Anne Saini, and me, Hannah Bates. Ian Fox is our editor. Music by Coma Media. Special thanks to Maureen Hoch, Adi Ignatius, Karen Player, Ramsey Khabbaz, Nicole Smith, Anne Bartholomew, and you – our listener.

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Polytomous Effectiveness Indicators in Complex Problem-Solving Tasks and Their Applications in Developing Measurement Model

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• Pujue Wang   ORCID: orcid.org/0000-0001-6931-6829 1 , 2 &
• Hongyun Liu   ORCID: orcid.org/0000-0002-3472-9102 1 , 2  

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Recent years have witnessed the emergence of measurement models for analyzing action sequences in computer-based problem-solving interactive tasks. The cutting-edge psychometrics process models require pre-specification of the effectiveness of state transitions often simplifying them into dichotomous indicators. However, the dichotomous effectiveness becomes impractical when dealing with complex tasks that involve multiple optimal paths and numerous state transitions. Building on the concept of problem-solving, we introduce polytomous indicators to assess the effectiveness of problem states \(d_{s}\) and state-to-state transitions \({\mathrm {\Delta }d}_{\mathrm {s\rightarrow s'}}\) . The three-step evaluation method for these two types of indicators is proposed and illustrated across two real problem-solving tasks. We further present a novel psychometrics process model, the sequential response model with polytomous effectiveness indicators (SRM-PEI), which is tailored to encompass a broader range of problem-solving tasks. Monte Carlo simulations indicated that SRM-PEI performed well in the estimation of latent ability and transition tendency parameters across different conditions. Empirical studies conducted on two real tasks supported the better fit of SRM-PEI over previous models such as SRM and SRMM, providing rational and interpretable estimates of latent abilities and transition tendencies through effectiveness indicators. The paper concludes by outlining potential avenues for the further application and enhancement of polytomous effectiveness indicators and SRM-PEI.

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Data Availability

The data analyzed in the empirical example of this study are available on this project’s Open Science Framework (OSF) page: https://osf.io/fw82q/ .

Anderson, J. R., Funke, J., & Plata, G. (Eds.). (2007). Cognitive psychologic (6 Aufl.). Spektrum Akademischer Verlag http://www.gbv.de/dms/bs/toc/529836963.pdf

Arieli-Attali, M., Ou, L., & Simmering, V. R. (2019). Understanding test takers’ choices in a self-adapted test: A hidden Markov modeling of process data. Frontiers in Psychology, 10 , 83. https://doi.org/10.3389/fpsyg.2019.00083

Article   PubMed   PubMed Central   Google Scholar  

Bergner, Y., & von Davier, A. A. (2019). Process data in NAEP: Past, present, and future. Journal of Educational and Behavioral Statistics, 44 (6), 706–732.

Article   Google Scholar  

Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37 (1), 29–51. https://doi.org/10.1007/BF02291411

Brooks, S. P., & Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7 , 434–455.

Google Scholar  

Buchner, A., & Funke, J. (1993). Finite-state automata: Dynamic task environments in problem-solving research. The Quarterly Journal of Experimental Psychology, 46 (1), 83–118.

Chen, Y. (2020). A continuous-time dynamic choice measurement model for problem-solving process data. Psychometrika, 85 (4), 1052–1075. https://doi.org/10.1007/s11336-020-09734-1

Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2022). Introduction to algorithms (4th ed., pp. 563–572). Cambridge: MIT Press.

Fu, Y., Zhan, P., Chen, Q., & Jiao, H. (2023). Joint modeling of action sequences and action time in computer-based interactive tasks. Behav Res Methods . https://doi.org/10.3758/s13428-023-02178-2

Article   PubMed   Google Scholar  

Funke, J. (2001). Dynamic systems as tools for analysing human judgement. Think Reason, 7 , 69–89.

Geisser, S., & Eddy, W. F. (1979). A predictive approach to model selection. J Am Stat Assoc, 74 , 153–160.

Gelfand, A. E., & Dey, D. K. (1994). Bayesian model choice: Asymptotics and exact calculations. Journal of the Royal Statistical Society Series B, 56 , 501–514.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2014). Bayesian data analysis (3rd ed.). Boca Raton: Chapman & Hall/CRC Press.

Gelman, A., Meng, X.-L., & Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statistica Sinica, 6 , 733–760.

Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7 (4), 457–472. https://doi.org/10.1214/ss/1177011136

Griffin, P., & Care, E. (2014). Assessment and teaching of 21st century skills: Methods and approach . New York, NY: Springer.

Guttman, I. (1967). The use of the concept of a future observation in goodness-of-fit problems. Journal of the Royal Statistical Society: Series B (Methodological), 29 (1), 83–100. https://doi.org/10.1111/j.2517-6161.1967.tb00676.x

Han, Y., Liu, H., & Ji, F. (2022). A sequential response model for analyzing process data on technology-based problem-solving tasks. Multivariate Behavioral Research, 57 (6), 960–977. https://doi.org/10.1080/00273171.2021.1932403

Han, Y., & Wilson, M. (2022). Analyzing student response processes to evaluate success on a technology-based problem-solving task. Applied Measurement in Education, 35 (1), 33–45.

He, Q., & von Davier, M. (2015). Identifying feature sequences from process data in problem-solving items with n-grams. In L. A. van der Ark, D. M. Bolt, W.-C. Wang, J. A. Douglas, & S.-M. Chow (Eds.), Quantitative psychology research (pp. 173–190). Berlin: Springer. https://doi.org/10.1007/978-3-319-19977-1_13

Chapter   Google Scholar  

He, Q., & von Davier, M. (2016). Analyzing process data from problem-solving items with N-Grams: Insights from a computer-based large-scale assessment. In Y. Rosen, S. Ferrara, & M. Mosharraf (Eds.), Handbook of research on technology tools for real-world skill development (pp. 750–777). Pennsylvania: IGI Global.

Koehler, E., Brown, E., & Haneuse, J. P. A. (2009). On the assessment of monte carlo error in simulation-based statistical analyses. The American Statistician, 63 (2), 155–162. https://doi.org/10.1198/tast.2009.0030

LaMar, M. M. (2018). Markov decision process measurement model. Psychometrika, 83 (1), 67–88. https://doi.org/10.1007/s11336-017-9570-0

Levy, R., & Mislevy, R. J. (2016). Bayesian psychometric modeling . Cambridge: CRC Press.

Li, M., Liu, H., Cai, M., & Yuan, J. (2023). Estimation of individuals’ collaborative problem solving ability in computer-based assessment. Education and Information Technologies . https://doi.org/10.1007/s10639-023-12271-w

Liu, H., Liu, Y., & Li, M. (2018). Analysis of process data of PISA 2012 computer-based problem solving: Application of the modified multilevel mixture IRT model. Frontiers in Psychology, 9 , 1372.

Mayer, R. E., & Wittrock, M. C. (2006). Problem solving. In P. A. Alexander & P. H. Winne (Eds.), Handbook of educational psychology (2nd ed., pp. 287–304). Mahwah: Erlbaum.

Newell, A., & Simon, H. A. (1972). Human problem solving (Vol. 104). Englewood Cliffs: Prentice-Hall.

OECD (2014). PISA 2012 results: Creative problem solving: Students ’ skills in tackling real-life problems (Vol. V). OECD.

OECD. (2016). PISA 2015 Assessment and analytical framework: Science . Reading, mathematic and financial literacy: PISA. OECD Publishing. https://doi.org/10.1787/9789264255425-en

OECD. (2018). The future of education and skills: Education 2030 . Paris: OECD Publishing.

Patz, R. J., & Junker, B. W. (1999). Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses. Journal of Educational and Behavioral Statistics, 24 (4), 342–366. https://doi.org/10.3102/10769986024004342

Patz, R. J., & Junker, B. W. (1999). A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, 24 (2), 146–178. https://doi.org/10.3102/10769986024002146

Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applies statistician. The Annals of Statistics, 12 , 1151–1172. https://doi.org/10.2307/2240995

Shu, Z., Bergner, Y., Zhu, M., Hao, J., & von Davier, A. A. (2017). An item response theory analysis of problem-solving processes in scenario-based tasks. Psychological Test and Assessment Modeling, 59 (1), 109–131.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van der Linde, A. (1998). Bayesian deviance, the effective number of parameters, and the comparison of arbitrarily complex models . MRC Biostatistics Unit: Technical report.

Tang, X. (2023). A latent hidden Markov model for process data. Psychometrika . https://doi.org/10.1007/s11336-023-09938-1

Tang, X., Wang, Z., He, Q., Liu, J., & Ying, Z. (2020). Latent feature extraction for process data via multidimensional scaling. Psychometrika, 85 (2), 378–397.

von Davier, A. A. (2017). Computational psychometrics in support of collaborative educational assessments. Journal of Educational Measurement, 54 (1), 3–11.

Xiao, Y., He, Q., Veldkamp, B., & Liu, H. (2021). Exploring latent states of problem-solving competence using hidden Markov model on process data. Journal of Computer Assisted Learning, 37 (5), 1232–1247.

Xiao, Y., & Liu, H. (2023). A state response measurement model for problem-solving process data. Behavior Research Methods, 1–20 . https://doi.org/10.3758/s13428-022-02042-9

Yuan, J., Xiao, Y., & Liu, H. (2019). Assessment of collaborative problem solving based on process stream data: A new paradigm for extracting indicators and modeling dyad data. Frontiers in Psychology, 10 , 369. https://doi.org/10.3389/fpsyg.2019.0036

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Appendix a. algorithm for automatically calculating state effectiveness in the balance beam task.

The interface of the initial state in the Chinese version of the Balance Beam task.

The diagram for the four types of transitions that can occur when a weight moves among ten possible positions in the Balance Beam task.

In the Balance Beam task, the ten potential positions for each weight are categorized into four groups: (1) Positions 1–4: Positioned on side A of the beam; (2) Position 5: Not suspended on side A; (3) Position 6: Not suspended on side B; (4) Positions 7–10: Positioned on side B of the beam. Figure 5 illustrates the transition of each weight among ten positions through four types of operations: (1) removing a weight from the beam; (2) hanging an unhung weight; (3) passing a weight to the other student; and (4) shifting the position of a weight on the same side. Each arrow represents an operation that can lead to a transition. Through this figure, we can easily find the minimum number of transitions between any two positions for one weight. Since an operation can only alter the position of one weight once, the shortest distance between states s and \(s'\) equals the sum of the minimum number of operations required for each of the four weights to change its position from state s to \(s'\) . Then, we can quickly and accurately calculate the shortest distance \(d_{s}^{(k)}\) between a state s and the target state \(s_{target}^{(k)}\) using the state code and rules to change the position according to Fig. 6 . Finally, we select the minimum distance \(d_{s}=\min \left( d_{s}^{(1)}, d_{s}^{(2)}, \ldots , d_{s}^{(k)}\right) \) as the effectiveness indicator \(d_{s}\) of the state s

During the process of programming the calculations mentioned above, the position of each weight can be assigned a unique number from one to ten. Therefore, any given state in the Balance Beam task can be encoded by a sequence of four numbers, a representation we refer to as the state code. For one weight, calculating the shortest distance between any two positions can be simplified by several rules. The R code for evaluating the effectiveness of states for the Balance Beam task that requires the use of two weights to achieve balance is available at https://osf.io/fw82q/ .

In the example of the code, the four positions for hanging weights on the balance beam on student A’s side are coded as 1 to 4, and the four positions on student B’s side are coded as \(-1\) to \(-4\) . The unhung weights are coded as 0.5 when in student A’s hand and \(-\) 0.5 when in student B’s hand. In the initial state, all four weights are in the hand of A, and the state code is (0.5, 0.5, 0.5, 0.5). The effectiveness of the initial state is equal to 3, which means that the balance state using two weights can be achieved after a minimum of three transitions. Another example is that Student B holds the 50 g and 100 g weights and Student A has hung the 300 g weight at position 1 and the 500 g weight at position 2. This state is at a minimum distance of 2 from the balance state.

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Wang, P., Liu, H. Polytomous Effectiveness Indicators in Complex Problem-Solving Tasks and Their Applications in Developing Measurement Model. Psychometrika (2024). https://doi.org/10.1007/s11336-024-09963-8

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A novel and effective method for solving the router nodes placement in wireless mesh networks using reinforcement learning

Contributed equally to this work with: Le Huu Binh, Thuy-Van T. Duong

Roles Investigation, Methodology, Software, Writing – original draft, Writing – review & editing

Affiliation Faculty of Information Technology, University of Sciences, Hue University, Hue City, Vietnam

Roles Conceptualization, Investigation, Methodology, Writing – review & editing

* E-mail: [email protected]

Affiliation Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

• Le Huu Binh, 
• Thuy-Van T. Duong

• Published: April 10, 2024
• https://doi.org/10.1371/journal.pone.0301073
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Router nodes placement (RNP) is an important issue in the design and implementation of wireless mesh networks (WMN). This is known as an P-hard problem, which cannot be solved using conventional algorithms. Consequently, approximate optimization strategies are commonly used to solve this problem. With heavy node density and wide-area WMNs, solving the RNP problem using approximation algorithms often faces many difficulties, therefore, a more effective solution is necessary. This motivated us to conduct this work. We propose a new method for solving the RNP problem using reinforcement learning (RL). The RNP problem is modeled as an RL model with environment, agent, action, and reward are equivalent to the network system, routers, coordinate adjustment, and connectivity of the RNP problem, respectively. To the best of our knowledge, this is the first study that applies RL to solve the RNP problem. The experimental results showed that the proposed method increased the network connectivity by up to 22.73% compared to the most recent methods.

Citation: Binh LH, Duong T-VT (2024) A novel and effective method for solving the router nodes placement in wireless mesh networks using reinforcement learning. PLoS ONE 19(4): e0301073. https://doi.org/10.1371/journal.pone.0301073

Editor: Mohammed Balfaqih, University of Jeddah, SAUDI ARABIA

Received: August 9, 2023; Accepted: March 9, 2024; Published: April 10, 2024

Copyright: © 2024 Binh, Duong. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting information files.

Funding: The authors received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Wireless communication is growing and being widely applied in many fields. In the local area network of agencies, businesses, schools, and so on, wireless mesh networks (WMN) [ 1 , 2 ] are the best choice today because of their significant advantages compared to wireless networks using traditional access points. The most notable benefit of the WMN is that it reduces congestion owing to its ability to balance the loads. In addition, the installation of a WMN is very convenient because there is no need to construct wired connections from the gateway to all routers. Fig 1 illustrates an example of a WMN consisting of six mesh routers (represented by r 1 to r 6 ) and eleven mesh clients (represented by c 1 to c 11 ). In addition, at least one the router of the Internet service provider serves as a gateway for clients to access the Internet. If two mesh routers are within range of each other, a wireless link is established between them. A mesh topology consists of of all the mesh routers and wireless links. For a WMN to deliver Internet services, several mesh routers must be connected to the gateway router via wireless or cable links. As shown in Fig 1 , the mesh routers r 1 and r 2 are connected to the gateway router (GPON or FTTh router) via wireless links. Mesh clients are terminal devices that are users of network services. When a mesh client enters the network region, it can be covered by one or more mesh routers; the mesh client connects to the nearest mesh router to access network services.

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https://doi.org/10.1371/journal.pone.0301073.g001

With the rapid development of wireless and mobile communication technologies, network services are becoming more diverse and rich, especially those on fifth-generation (5G) and sixth-generation (6G) wireless network platforms. To effectively provide these services, WMNs must be designed and installed in the most efficient manner possible, allowing network resources to be fully utilized. This is the motivation for researchers to focus on WMN. Some of the most prevalent subjects that have been implemented include network topology control [ 3 – 7 ], router node placement (RNP) [ 8 – 24 ], optimum routing protocols [ 25 – 29 ], and access point allocation [ 30 – 33 ], with the RNP challenge being the most fascinating. Because the RNP problem is known to be NP-hard, it cannot be solved using conventional algorithms. Recently, approximate optimization methods have become useful for solving this problem [ 8 – 12 ]. The authors of [ 8 ] have used the coyote optimization algorithm (COA) to solve the RNP problem. Their proposed method optimizes both network connectivity and user coverage, which are two critical performance criteria. Using MATLAB simulations, the authors demonstrated that the COA algorithm outperformed other well-known optimization algorithms. In [ 10 ], the authors suggested an optimal method called the Chemical Reaction Optimization (CRO) algorithm to solve this problem. The CRO algorithm was inspired by how molecules interact to achieve a low, stable energy state in chemical reactions. In terms of client coverage and network connection, the simulation findings reveal that their suggested approach outperforms the Genetic approach (GA) and Simulated Annealing (SA). Another study employed a genetic algorithm and simulated annealing to discover a low-cost WMN configuration while satisfying restrictions and identifying the number of gateways needed [ 34 ]. Experiments showed that the evolutionary algorithm and simulated annealing were successful in lowering WMN network expenses while maintaining QoS. The new models significantly outperformed the conventional solutions. QoS was also considered in the RNP problem in [ 23 ]. The authors described a unique particle swarm optimization method for improving network connectivity and client coverage. The QoS restrictions for this study are the delay, relay load, and Internet gateway capacity. In [ 35 ], the authors suggested an improved version of the Moth Flame Optimization (MFO) algorithm, namely, Enhanced Chaotic Lévy Opposition-based MFO (ECLO-MFO), for solving the RNP problem. To improve the optimization performance of MFO, the proposed method integrates three strategies: the chaotic map concept, Lévy flying strategy, and Opposition-Based Learning (OBL) technique. The simulation results showed that the proposed algorithm was more efficient than the method of applying popular optimization algorithms.

Based on the results of published works, we find that the method of using approximate optimal algorithms provide good solutions. However, because randomness is used in several steps of the algorithm, the results often differ for different executions. For accurate results, each script must be executed multiple times, and then the average of all executions is obtained. For example, the authors of [ 8 , 11 ] executed each simulation scenario 50 times. Furthermore, with heavy node density and wide-area WMNs, solving the RNP problem with approximation algorithms often presents many difficulties, necessitating a more effective solution. In this paper, we propose a new and effective algorithm to solve this problem. The main contributions of this study are summarized as follows:

• (i) We proposed a novel and effective method for solving the RNP problem using RL. The RNP problem is modeled as an RL model, with the environment, agent, action, and reward representing the network system, routers, coordinate adjustment, and connectivity respectively, of the RNP problem. To the best of our knowledge, this is the first study to apply reinforcement learning to the RNP problem.
• (ii) We compared and evaluated the performance of the RNP problem solving method using the heuristic algorithms and the RL method.

The remainder of this paper is organized as follows. The next section describes the formulation of the RNP problem in the WMN. The following sections present our proposed solution and experimental results. Finally, concluding remarks and promising future studies are presented in the last section.

RNP problem

In this section, we formulate the RNP problem in a WMN. First, graph theory was used to describe the WMN. We then define some metrics to use for the objective function of the RNP problem, similar to [ 11 ]. Finally, the RNP problem was formulated as a nonlinear programming problem. For convenience, we define the mathematical symbols shown in Table 1 .

https://doi.org/10.1371/journal.pone.0301073.t001

Mathematical model of a WMN using graph theory

Consider a WMN comprising m mesh routers, n mesh clients, and k gateway routers. Mathematically, this WMN can be represented as an undirected graph, denoted by G = ( V , E ), where V and E are the vertex and edge sets, respectively. V is equivalent to the set of all nodes in the WMN and is determined by V = R ∪ C ∪ W , where R , C and W are the sets of mesh routers, mesh clients, and gateway routers, respectively. E is equivalent to the set of all wireless links in the WMN and consists of three types: links between mesh routers, links between mesh client and mesh router, and links between gateway and mesh router.

RNP problem formulation

In this section, we formulate the RNP problem using some concepts and metrics from [ 11 ], including the connected router, connected client, connected router ratio, and connected client ratio.

Connected router.

The mesh router r i is a connected router if and only if at least one path exists between it and the gateway router. If we return to the WMN example in Fig 1 , we can see that mesh routers r 1 , r 2 , r 3 , r 4 and r 6 are the connected routers but r 5 is not because no path exists from this mesh router to the gateway router.

Connected router ratio (CRR).

Connected client.

Connected client ratio (CCR).

Formulate the RNP into a nonlinear programming problem.

The RNP problem in the WMN is stated as follows: Consider a case where it is necessary to design and install a WMN with the following assumptions:

• The network system is located in an area of W × H meters.

• The number of mesh routers was m , and the coverage radius of each mesh router was d r .

RL-based mesh router nodes placement

Fundamentals of rl.

https://doi.org/10.1371/journal.pone.0301073.g002

RL has been successfully applied to control protocols in wireless networks, typically routing in WMN [ 25 , 27 , 29 ], topology control in wireless sensor networks [ 37 ], improving the performance of energy-harvesting wireless body area networks [ 38 , 39 ]. In this paper, we apply RL to solve the RNP in WMN. Details of this new proposal are presented in the following sections.

Solving the RNP in WMN using RL

The RL has recently been successfully employed to solve technical challenges in wireless communication such as routing [ 27 , 36 ], topology management [ 37 ], and resource allocation. In this study, we use RL to solve the RNP problem. To the best of our knowledge, this is the first study to use RL to address the RNP problem. To do this, the RNP problem must be modeled as a reinforcement learning model with five characteristic factors: agent, environment, state, action, and reward.

An agent is a mesh router that regularly adjusts its coordinates to obtain an optimal topology.

Environment.

In a RL model, the environment is everything that exists around the agent, and it is where the agent acts and interacts. The environment for the RNP problem using RL is the network system, which includes a set of mesh routers, clients, gateway routers, and network area.

Each state is determined by a triple { P c , P r , P w }, where P c , P r and P w are the sets of coordinates for the mesh clients, mesh router, and gateway routers, respectively. The sets are listed in Table 1 .

Action is the way in which the agent interacts with the environment to change its state. For the RNP problem using reinforcement learning, the agents are the mesh routers. Each action was defined by a mesh router that adjusted its coordinates. The set of actions at a specific state s t for each mesh router r i is defined as A t = { mn1s, me1s, ms1s, mw1s, mn2s, me2s, ms2s, mw2s }, where the actions are described in Table 2 , step is a given distance.

https://doi.org/10.1371/journal.pone.0301073.t002

RL algorithm for solving RNP problem.

Algorithm 1 The pseudo-code of the reinforcement learning algorithm for solving RNP problem

• Network area ( W × H );

• The set of mesh routers ( R = { r i | i = 1.. m }), and the coverage radius of each mesh router ( d r );

2: while ( learn ≤ numLearn ) do

3:  Randomly choose mesh router r i ∈ R ;

4:   for ( each action a j ∈ A ) do

5:   Update Q ( r i , s t , a t , j ) using ( 11 );

6:   end for

7:  Choose the action a t , k ∈ A t using policy derived from Q-values (e.g., ε -greedy ) according to ( 12 );

8:  Take action a t , k , observe reward R ( r i , s t , a t , k ) and next state s t +1 ;

9:  Update next state ( s t +1 ) to current state ( s t );

10:   learn ← learn + 1;

11: end while

12: P r ← P r in state s t ;

Analyze the computational complexity.

The computational complexity of Algorithm 1 depends mainly on the iteration in Step (2), the number of possible actions in Step (4), and the algorithm for updating the Q value in Step (5). Q ( r i , s t , a t , j ) is updated using Eq. ( 11 ), where the greatest complexity is the calculation of RW ( r i , s t , a t ) according to ( 10 ). RW ( r i , s t , a t ) contains two metrics, CRR and CCR , which are defined by ( 1 ) and ( 5 ), respectively. To determine CRR , we employed a breadth-first search algorithm on a network of m vertices, which is the number of mesh routers. Therefore, the computational complexity was O ( m 2 ). The CCR is calculated using two nested loops of sizes m and n , where n is the number of mesh clients. Therefore, the complexity was O ( m × n ). Because n is always greater than m in a WMN, the computational complexity of RW ( r i , s t , a t ) is O ( m × n ). Consequently, the computational complexity of Algorithm 1 is O ( I × | A | × m × n ), where I is the number of iterations and | A | is the number of possible actions.

The computational complexity of Algorithm 1 is greater than that of the algorithms solving the RNP problem using GA [ 40 ], PSO [ 24 ], and WOA [ 41 ], which we compare in the following section. However, because its computing complexity is a polynomial function, it can be implemented in practice. Furthermore, because the algorithms for solving the RNP problem are run offline, the polynomial complexity is acceptable.

Simulation results and discussion

Simulation scenarios.

The performance of the proposed method was evaluated through a simulation using Python. Our proposed method is compared with the most recent methods that use approximate optimization algorithms to address the RNP problem, including GA [ 40 ], PSO [ 24 ], WOA [ 41 ], and MVO [ 11 ]. All experiments were run on a 3.6 GHz Core i7 CPU computer. The surveyed network instances (NI) are presented in Table 3 . NI-1 and NI-2 were used to investigate the effect of the number of mesh routers on the network performance, with the number of mesh routers ranging from 20 to 45 covering 150 mesh clients (NI-1) and 350 mesh clients (NI-2). NI-3 and NI-4 ware used to study the effect of client density, varying from 100 to 400. In NI-5 and N-6, the effect of the coverage radius of each mesh router was thoroughly examined. The final two NIs were used to investigate the influence of the network area. The parameters of the simulation scenarios and algorithms are presented in Table 4 , where th parameters of the GA, PSO, WOA, and MVO are set as in [ 11 ].

https://doi.org/10.1371/journal.pone.0301073.t003

https://doi.org/10.1371/journal.pone.0301073.t004

Simulation results

Topology evaluation..

First, we evaluate the topology obtained when solving the RNP problem using the GA, PSO, WOA, MVO, and our proposed method, which employs reinforcement learning. The results obtained in Fig 3 clearly show topological differences between the methods. These findings were obtained using NI-2 with 30 mesh routers covering 350 mesh clients in an area of 2000 × 2000 [ m 2 ] and a coverage radius of 200 [ m ] for each mesh router. We can observe that the method using RL provides the most optimal topology compared with the methods using approximate optimization algorithms, GA, WOA, PSO, and MVO. Specifically, for the method using reinforcement learning, there are 334 mesh clients covered by at least one mesh router, corresponding to a rate of 95.43%. These values were 292 (83.43%), 309 (88.29%), 313 (89.43%), and 313 (88.86%) for the WOA, GA, PSO, and MVO algorithms, respectively. In addition, the topology of the reinforcement learning method has a wider coverage area than the other methods, which can increase the percentage of clients covered in the case of denser clients.

(a) WOA, (b) GA, (c) PSO, (d) MVO, and (e) reinforcement learning.

https://doi.org/10.1371/journal.pone.0301073.g003

Impact of mesh router density.

The results obtained in Fig 4 clearly show the difference in network connectivity between the proposed method and the method using approximate optimization algorithms. These findings were obtained using NI-1, in which the number of mesh routers varieed from 20 to 45, covering 150 mesh clients in an area of 2000 × 2000 [ m 2 ] and a coverage radius of 200 [ m ] for each mesh router. We can observe that the NC increases proportionally with the number of mesh routers for all methods. This is evident because as the number of mesh routers increases, the coverage area expands, increasing the probability of mesh clients being covered. Comparing the methods of solving RNP problems, the method using RL (legend namely RL-based RNP) gives the highest NC. For example, considering the case of 35 mesh routers, The NC values of the methods using the WOA, PSO, GA, MVO, and RL are 85.64, 87.42, 90.67, 93.42, and 95.68%, respectively. Thus, compared with the method using algorithms WOA, PSO, GA, and MVO, the proposed method improved NC by 10.03, 8.25, 5.01%, and 2.25%, respectively. This is a significant result in improving WMN performance.

https://doi.org/10.1371/journal.pone.0301073.g004

The results obtained were quite similar for the implementation on NI-2, as shown in Fig 5 . The assumptions of this simulation scenario are the same as those in NI-1, except that the number of mesh clients increases to 350. We can see that the proposed method is highly effective in terms of NC. We can observe that the proposed solution provides high efficiency in terms of NC for most values of the number of mesh routers. The NC of the method using RL increases by an average of 4 to 20% compared with the cases where approximate optimization algorithms are used. As is the case with 35 mesh routers, the NC of the RL is 98.71%. These values of the WOA, PSO, GA, and MVO algorithms were 81.66%, 86.89%, 88.59%, and 94.44% respectively. Thus, the method using RL improved the NC from 4.26% to 17.04%.

https://doi.org/10.1371/journal.pone.0301073.g005

Based on the findings in Figs 4 and 5 , we can conclude that changing the number of mesh routers affects on network performance in terms of NC. The larger the number of routers, the higher the NC for all investigated RNP problem solving methods. In particular, the method based on RL is the most efficient.

Impact of mesh client density.

In this section, we investigate the effect of client density on network performance. In a WMN, the denser the clients, the greater is the number of connection requests to the routers. As a result, network performance was affected. This is more evident in Fig 6 , where we plot NC as a function of the number of mesh clients. These results are obtained by executing NI-3, where the number of mesh routers is 30, covering 150 to 300 mesh clients. We can easily observe that the method using RL always yields the highest NC regardless of whether the client density is sparse or dense. The NC value of this method from 90.43% to 95.79%. Meanwhile, the NC values for the cases of algorithm WOA, PSO, GA, and MVO are fom 74.59% to 84.08%, from 77.00% to 85.63%, from 82.32% to 91.46%, and from 88.27% to 90.83%, respectively. When 45 mesh routers were used (NI-4), the NC value increased for all methods. This is clearly shown in Fig 7 , where we represent NC versus the number of mesh clients. Comparing the methods, we find that the method using RL outperforms the method using approximate optimal algorithms in terms of NC.

https://doi.org/10.1371/journal.pone.0301073.g006

https://doi.org/10.1371/journal.pone.0301073.g007

Impart of the coverage radius of mesh routers.

The coverage radius of the mesh routers is another technological parameter that has a considerable impact on the WMN performance. In this section, we investigate the effect of this technological parameter on the NC metric. The results obtained in Fig 8 clearly show the change in NC with respect to the coverage radius of the mesh routers. These results were implemented using NI-5, which has 30 mesh routers and 150 mesh clients. The coverage radius of each router ranged from 150 to 300 [ m ]. The plots in Fig 8 indicate that the NC increases proportionally to the coverage radius of the mesh routers. This is because expanding the coverage radius increases the likelihood that clients will be covered. As a result, NC increases. In particular, the method using RL yielded the highest NC, reaching close to 100% when the coverage radius was 250 [m] or more. The results are also similar for NI-6, as shown in Fig 9 . The NC value of this NI is greater than that of the NI-5 because this uses more mesh routers. As in the previous scenarios, the method using RL always yields the highest NC.

https://doi.org/10.1371/journal.pone.0301073.g008

https://doi.org/10.1371/journal.pone.0301073.g009

Impact of network area.

In the last section, we investigate the effect of network area on the efficiency of RNP problem solving methods. Figs 10 and 11 show the results obtained by executing NI-7 and NI-8, respectively. In these NIs, the network area varies from 2000 × 2000 [ m 2 ] to 3000 × 3000 [ m 2 ]. The NC value decreased according to the network area for all the algorithms. This is because, for a given number of mesh routers, the larger the network area, the lower the percentage of area covered, leading to a decrease in the NC value. However, the NC value of the method using RL is always the largest.

https://doi.org/10.1371/journal.pone.0301073.g010

https://doi.org/10.1371/journal.pone.0301073.g011

Based on the above findings, we can conclude that the proposed method, which uses reinforcement learning to solve the RNP problem, is more efficient than a method that uses approximate optimal algorithms. This is a crucial result in the design and implementation of a WMN, which helps find an optimal network topology to exploit network resources more efficiently.

The placement of router nodes in wireless mesh networks is a significant problem that has recently attracted the interest of several research groups. This problem is recognized as NP-hard, and cannot be resolved using conventional algorithms. In this study, we proposed a new and effective method for solving this problem using RL. The process of finding the optimal coordinates for placing mesh routers is modeled as an RL with the main components being environment, agent, action, and reward, which are equivalent to the network system, routers, coordinate adjustment, and network connectivity of the RNP problem, respectively. Simulation results show that our proposed method outperforms the most recent methods in terms of coverage and network connectivity.

In future work, we will continue to develop this method by considering additional constraints on the quality of transmission and load balancing to improve network performance. In addition, the deep reinforcement learning method can also be applied to static and dynamic RNP problems to further improve the performance of the WMN.

Supporting information

S1 dataset..

https://doi.org/10.1371/journal.pone.0301073.s001

• 1. Zhang Y, Luo J, Hu H. Wireless Mesh Networking—Architectures, Protocols and Standards. Taylor & Francis Group, LLC; 2007.
• 2. Akyildiz IF, Xudong Wang. Wireless Mesh Networks. John Wiley & Sons Ltd; 2009.
• View Article
• Google Scholar
• 4. Aron FO, Olwal TO, Kurien A, Odhiambo MO. Energy Efficient Topology Control Algorithm for Wireless Mesh Networks. In: 2008 International Wireless Communications and Mobile Computing Conference; 2008. p. 135–140.
• 6. Yang L, Quan L. A Topology Control Algorithm Using Power Control for Wireless Mesh Network. In: 2011 Third International Conference on Multimedia Information Networking and Security; 2011. p. 141–145.
• PubMed/NCBI
• 15. Bello OM, Taiwe KD. Mesh Node Placement in Wireless Mesh Network Based on Multiobjective Evolutionary Metaheuristic. In: Proceedings of the International Conference on Internet of Things and Cloud Computing. ICC’16. New York, NY, USA: Association for Computing Machinery; 2016.Available from: https://doi.org/10.1145/2896387.2896444 .
• 18. Hamdi M, Mhiri S. Dynamic mesh router placement for connectivity maximization in wireless mesh networks; 2015. p. 1–6.
• 20. Sayad L. Optimal placement of mesh routers in a wireless mesh network with mobile mesh clients using simulated annealing. In: 2017 5th International Symposium on Computational and Business Intelligence (ISCBI); 2017. p. 45–49.
• 21. Rezaei M, Sarram M, Derhami V, Sarvestani H. Novel Placement Mesh Router Approach for Wireless Mesh Network. 2012;.
• 22. Seetha S, Anand John Francis S, Grace Mary Kanaga E. Optimal Placement Techniques of Mesh Router Nodes in Wireless Mesh Networks. In: Haldorai A, Ramu A, Mohanram S, Chen MY, editors. 2nd EAI International Conference on Big Data Innovation for Sustainable Cognitive Computing. Cham: Springer International Publishing; 2021. p. 217–226.
• 23. Lin CC, Chen TH, Jhong SY. Wireless mesh router placement with constraints of gateway positions and QoS. In: 2015 11th International Conference on Heterogeneous Networking for Quality, Reliability, Security and Robustness (QSHINE); 2015. p. 72–74.
• 35. Mekhmoukh Taleb S, Meraihi Y, Mirjalili S, Acheli D, Ramdane-Cherif A, Benmessaoud Gabis A. Mesh Router Nodes Placement for Wireless Mesh Networks Based on an Enhanced Moth–Flame Optimization Algorithm. Mobile Networks and Applications. 2023. https://doi.org/10.1007/s11036-022-02059-6

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A novel hippo swarm optimization: for solving high-dimensional problems and engineering design problems.

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Guoyuan Zhou, Jiaxuan Du, Jia Guo, Guoliang Li, A Novel Hippo Swarm Optimization: For solving high-dimensional problems and engineering design problems, Journal of Computational Design and Engineering , 2024;, qwae035, https://doi.org/10.1093/jcde/qwae035

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In recent years, scholars have developed and enhanced optimization algorithms to tackle high-dimensional optimization and engineering challenges. The primary challenge of high-dimensional optimization lies in striking a balance between exploring a wide search space and focusing on specific regions. Meanwhile, engineering design problems are intricate and come with various constraints. This research introduces a novel approach called Hippo Swarm Optimization (HSO), inspired by the behavior of hippos, designed to address high-dimensional optimization problems and real-world engineering challenges. HSO encompasses four distinct search strategies based on the behavior of hippos in different scenarios: starvation search, alpha search, margination, and competition. To assess the effectiveness of HSO, we conducted experiments using the CEC2017 test set, featuring the highest-dimensional problems, CEC2022 and four constrained engineering problems. In parallel, we employed 14 established optimization algorithms as a control group. The experimental outcomes reveal that HSO outperforms the 14 well-known optimization algorithms, achieving first average ranking out of them in CEC2017 and CEC2022. Across the four classical engineering design problems, HSO consistently delivers the best results. These results substantiate HSO as a highly effective optimization algorithm for both high-dimensional optimization and engineering challenges.

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Title: saas: solving ability amplification strategy for enhanced mathematical reasoning in large language models.

Abstract: This study presents a novel learning approach designed to enhance both mathematical reasoning and problem-solving abilities of Large Language Models (LLMs). We focus on integrating the Chain-of-Thought (CoT) and the Program-of-Thought (PoT) learning, hypothesizing that prioritizing the learning of mathematical reasoning ability is helpful for the amplification of problem-solving ability. Thus, the initial learning with CoT is essential for solving challenging mathematical problems. To this end, we propose a sequential learning approach, named SAAS (Solving Ability Amplification Strategy), which strategically transitions from CoT learning to PoT learning. Our empirical study, involving an extensive performance comparison using several benchmarks, demonstrates that our SAAS achieves state-of-the-art (SOTA) performance. The results underscore the effectiveness of our sequential learning approach, marking a significant advancement in the field of mathematical reasoning in LLMs.

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