conceptual problem solving problems

ORIGINAL RESEARCH article

Linking attentional processes and conceptual problem solving: visual cues facilitate the automaticity of extracting relevant information from diagrams.

• 1 Department of Physics, Kansas State University, Manhattan, KS, USA
• 2 Department of Psychology, University of Findlay, Findlay, OH, USA
• 3 Department of Psychological Sciences, Kansas State University, Manhattan, KS, USA

This study investigated links between visual attention processes and conceptual problem solving. This was done by overlaying visual cues on conceptual physics problem diagrams to direct participants’ attention to relevant areas to facilitate problem solving. Participants ( N = 80) individually worked through four problem sets, each containing a diagram, while their eye movements were recorded. Each diagram contained regions that were relevant to solving the problem correctly and separate regions related to common incorrect responses. Problem sets contained an initial problem, six isomorphic training problems, and a transfer problem. The cued condition saw visual cues overlaid on the training problems. Participants’ verbal responses were used to determine their accuracy. This study produced two major findings. First, short duration visual cues which draw attention to solution-relevant information and aid in the organizing and integrating of it, facilitate both immediate problem solving and generalization of that ability to new problems. Thus, visual cues can facilitate re-representing a problem and overcoming impasse, enabling a correct solution. Importantly, these cueing effects on problem solving did not involve the solvers’ attention necessarily embodying the solution to the problem, but were instead caused by solvers attending to and integrating relevant information in the problems into a solution path. Second, this study demonstrates that when such cues are used across multiple problems, solvers can automatize the extraction of problem-relevant information extraction. These results suggest that low-level attentional selection processes provide a necessary gateway for relevant information to be used in problem solving, but are generally not sufficient for correct problem solving. Instead, factors that lead a solver to an impasse and to organize and integrate problem information also greatly facilitate arriving at correct solutions.

Introduction

This study investigated links between visual attention processes and conceptual problem solving. This is challenging, because most of what we know about attention has to do with its lower-level perceptual processes, and most of what we know about problem solving has to do with much higher-level cognitive processes. Thus, forging a link between lower-level perception and higher-level cognition is difficult. A vast literature has developed over the past 40 years explaining the low-level stimulus factors that capture attention and eye movements, and the effects this has on early visual perceptual processes. For example, motion has been shown to reliably capture eye movements (overt attention; Carmi and Itti, 2006 ; Mital et al., 2010 ), as mediated by the superior colliculus in primates ( Kustov and Robinson, 1996 ; Findlay and Walker, 1999 ; Boehnke and Munoz, 2008 ), and the optic tectum in lower animals, including toads ( Borchers and Ewert, 1979 ). In turn, selective attention has been shown to improve perceived brightness, acuity, and contrast sensitivity ( Carrasco et al., 2000 , 2002 ; Cameron et al., 2002 ), as mediated by an increased signal-to-noise ratio of cells as early as the primary visual cortex ( Fischer and Whitney, 2009 ; Pestilli et al., 2011 ). However, despite the tremendous strides that have been made in understanding the low-level causes and effects of visual selective attention, much less is known about high-level cognitive causes and effects of visual selective attention. Admittedly, a sizeable body of research has shown strong relationships between tasks and selective attention, as measured by eye movements ( Foulsham and Underwood, 2007 ; Henderson et al., 2007 ; Einhäuser et al., 2008 ), and between selective attention, as measured by eye movements, and memory ( Hollingworth and Henderson, 2002 ; Zelinsky and Loschky, 2005 ; Pertzov et al., 2009 ). Nevertheless, far less research has investigated such causal relationships between visual selective attention and eye movements on the one hand, and quintessentially higher-level cognitive processes such as those involved in problem solving, on the other.

In the current study, we specifically investigate the relationships between visual selective attention and the cognitive processes involved in solving physics problems, which are among the most intellectually and cognitively demanding that human beings are capable of engaging in. Indeed, one might reasonably ask whether such low-level perceptual functions as those involved in selective attention could really play much of a role in such a high-level cognitive task. However, several studies over the last decade have shown exactly that, namely that cueing people’s attention in specific ways while they solve insight problems can significantly affect their solution accuracy ( Grant and Spivey, 2003 ; Thomas and Lleras, 2007 , 2009 ). In the current study, we have investigated these processes in the context of learning from problem solving. However, evidence of learning, as shown by increased performance on problem solving tasks alone, while clearly implicating memory formation, cannot elucidate the links between online attentional selection and the higher-level cognitive processes involved in physics problem solving. We therefore elucidated the online processes that link attention selection and physics problem solving by using eye movement data in conjunction with increases in problem solving performance.

Selective Attention and Eye Movements

We assume that eye movements are linked to attentional selection as proposed by the rubber band model of eye movements and attention ( Henderson, 1992 , 1993 ). Specifically, at the beginning of each eye fixation, attention is aligned with the point of fixation ( van Diepen and d’Ydewalle, 2003 ; Glaholt et al., 2012 ; Larson et al., 2014 ), but by roughly 80 ms before the next eye movement, covert attention is shifted to the to-be-fixated object ( Kowler et al., 1995 ; Deubel and Schneider, 1996 ; Caspi et al., 2004 ), after which the eyes make a saccade to the newly attended object. Thus, although attention may be at a different location than the point of fixation (especially in the last 80 ms of a fixation, called covert attention; Caspi et al., 2004 ), if the eyes are sent to a location, we know that attention was there at the beginning of the fixation. One can therefore retrospectively measure the location of attentional selection by measuring eye fixation locations, called overt attention.

As noted earlier, research on attentional selection has made tremendous strides in explaining the effects of stimulus characteristics, or bottom-up influences, on overt attention. These studies have shown that stimulus saliency, as measured by contrast along various feature dimensions coded by early visual cortex (e.g., luminance, color, orientation, and motion), plays a moderately strong causal role in determining where the eyes are sent ( Irwin et al., 2000 ; Itti and Koch, 2000 ; Mital et al., 2010 ). Other research has shown non-stimulus-based effects, or top-down influences, on overt attention. These top-down influences can be further divided between those that are involuntary and automatic, based on experience and learning, called mandatory top-down processes, and those that are voluntary and effortful, called volitional top-down processes ( Baluch and Itti, 2011 ). Numerous studies have shown evidence of mandatory top-down effects on overt attentional selection in scenes (e.g., attention to stop signs when they are in expected locations, such as intersections, but not in unexpected locations, such as the middle of a block; Theeuwes and Godthelp, 1995 ; Shinoda et al., 2001 ). A separate body of research has shown effects of volitional top-down processes on overt attention in more laboratory-based tasks (e.g., the anti-saccade task, in which one looks in the opposite direction from a salient visual stimulus; Everling and Fischer, 1998 ). Overall, mandatory top-down processes have been shown to generally have a stronger influence on overt attentional selection than bottom-up visual saliency ( Foulsham and Underwood, 2007 ; Henderson et al., 2007 ; Einhäuser et al., 2008 ). Conversely, because volitional top-down processes require executive attentional and working memory (WM) resources, they generally have weaker effects on overt attentional selection than bottom-up saliency, as shown by the antisaccade task, in which the sudden appearance of a simple stimulus is very difficult to avoid reflexively looking at, while it takes a conscious effort to looking in the opposite direction ( Guitton et al., 1985 ; Mitchell et al., 2002 ). Nevertheless, a far fewer number of studies have investigated the relationships between bottom-up and top-down processes and overt attentional selection in higher-level cognitive tasks such as problem solving.

Cognitive Processes Involved in Problem Solving

In order to discuss the relationship between overt attentional selection and the cognitive processes involved in problem solving, we must first specify what those higher-level cognitive processes might be. We are particularly interested in the cognitive processes involved in conceptual problems requiring insight, in which the solution is not immediately apparent, and solvers cannot simply adopt an algorithmic approach to finding a solution ( Duncker, 1945 ; Ohlsson, 1992 ; Jones, 2003 ). Ohlsson’s (1992) representational change theory provides a framework to understand the cognitive mechanisms involved in solving problems that require conceptual insight, rather than purely algorithmic computation. This framework lends itself to our work on conceptual problem solving. Specifically, the problems we study are conceptual in nature because they require the solvers to recognize the appropriate physics concepts to apply. Recognition of the appropriate concept often comes to the solver in a moment of insight. While encoding the problem, the solver activates (apparently) relevant prior knowledge, which is used to construct a mental representation of the problem. This representation is then used to find a path to the solution. However, in insight problems, solvers commonly make several unsuccessful attempts to solve the problem, which forces them into an impasse, in which they realize that no path to the solution is apparent. In order to break the impasse, the solver must often restructure their mental representation of the problem in order to find a viable solution path. This produces the insight that then rapidly leads to solving the problem. Ohlsson’s (1992) theory provides a good framework for understanding a number of important cognitive processes involved in insight problem solving, but is relatively silent with regard to what roles, if any, attentional selection plays in problem solving.

Prior Research on Overt Attentional Selection and Problem Solving

Prior research on eye movements and problem solving has shown that overt attention can illuminate the cognitive processes involved in problem solving ( Epelboim and Suppes, 2001 ; Knoblich et al., 2001 , 2005 ; Grant and Spivey, 2003 ; Jones, 2003 ; Thomas and Lleras, 2007 , 2009 ; Bilalić et al., 2008 ; Eivazi and Bednarik, 2010 , 2011 ; Madsen et al., 2012 , 2013a , b ; Lin and Lin, 2014 ; Susac et al., 2014 ). However, we are particularly interested in two directions of causal relationships between overt attentional selection and the higher-level cognitive processes involved in problem solving: (1) the causal relationship starting from higher-level cognitive processes involved in problem solving and ending with attentional selection; and (2) the reverse causal relationship starting from attentional selection and ending with the higher-level cognitive processes involved in problem solving. A relatively small number of studies have investigated each of these relationships, with some speaking more to the effect of higher-level cognitive processes in problem solving on attentional selection ( Epelboim and Suppes, 2001 ; Knoblich et al., 2001 ; Madsen et al., 2012 ), and others speaking more to the effect of attentional selection on higher-level cognitive processes in problem solving ( Epelboim and Suppes, 2001 ; Cameron et al., 2002 ; Grant and Spivey, 2003 ; Tai et al., 2006 ; Thomas and Lleras, 2007 , 2009 ; Lin and Lin, 2014 ; Susac et al., 2014 ).

Research on the effect of the cognitive processes involved in problem solving on overt attentional selection has shown that mandatory top-down processes based on prior knowledge can enable solvers to rapidly attend to relevant information when solving a problem ( Epelboim and Suppes, 2001 ; Madsen et al., 2012 ). In the most extreme cases, based on prior knowledge, an expert may attend to the relevant information in a problem within the time frame of a single eye fixation, while a novice may instead take much more time while attending to various sources of irrelevant information ( Charness et al., 2001 ; Reingold et al., 2001 ). Just as importantly, however, even if the solver has previously activated irrelevant knowledge, leading to an impasse, restructuring the problem representation can lead to shifting overt attention away from irrelevant information to relevant but previously ignored information ( Knoblich et al., 2001 ; Jones, 2003 ).

Research on the effect of attentional selection on the cognitive processes involved in problem solving suggests that there are at least two qualitatively different types of effects. First, attentional selection can lead either to processing relevant information, which facilitates problem solving by activating relevant domain knowledge, leading to finding a viable solution path, or processing irrelevant information, which impedes problem solving by activating irrelevant knowledge, leading to an incorrect solution path ( Grant and Spivey, 2003 ; Thomas and Lleras, 2007 , 2009 ; Madsen et al., 2012 , 2013a ). This effect of attentional selection on problem solving determines whether or not the solver, in a manner of speaking, gets through the starting gate to finding a viable solution path. Second, if a solver has gotten through the starting gate by attending to relevant information, further attentional selection of aspects of that relevant information appears to be important for not only extracting further relevant information, but also refreshing their WM representations used in finding the solution path. Here, we assume that problem solving occurs in WM ( Ohlsson, 1992 ; Epelboim and Suppes, 2001 ), and that WM has a limited capacity ( Baddeley, 1994 ; Luck and Vogel, 1997 ; Cowan, 2001 ). Thus, if the process of finding a viable solution path involves establishing relationships between numerous conceptual entities, solvers may experience difficulties caused by exceeding their WM capacity ( Epelboim and Suppes, 2001 ). Because maintaining representations in WM requires attention ( Cowan, 2001 ), one can refresh WM representations by attending to them ( Hale et al., 1996 ; Awh et al., 1998 ; D’Esposito et al., 1999 ), for example by repeatedly refixating the eyes on the to-be-processed items ( Zelinsky et al., 2011 ). Thus, during problem solving, attentional selection, as evidenced by refixating relevant information, can facilitate finding a solution path by refreshing the WM representations for the fixated items ( Epelboim and Suppes, 2001 ; Tai et al., 2006 ; Lin and Lin, 2014 ; Susac et al., 2014 ).

A different way in which overt attentional selection can facilitate problem-solving processes in WM is through sustained attention, which involves inhibiting overt and covert attentional shifts. Specifically, when a solver is engaged in complex problem solving processes in WM, longer than normal processing times are sometimes needed in order to attend to the current contents of WM. In those cases, it would be counter-productive to move attention and the eyes to a new location, which automatically triggers extracting the new information there into WM ( Belopolsky et al., 2008 ), potentially displacing some of the current WM contents ( Zelinsky and Loschky, 2005 ). Instead, the solver may inhibit moving the eyes, resulting in a longer eye fixation at the current location ( Findlay and Walker, 1999 ). Thus, during the process of breaking an impasse (i.e., the moment of insight), problem solvers will often produce longer fixation durations, rather than making more fixations on different items ( Knoblich et al., 2001 ; Velichkovsky et al., 2002 ; Jones, 2003 ).

The above discussion sets the stage for discussing our previous work on overt attentional selection and physics problem solving. Our research was inspired by the groundbreaking work of Thomas and Lleras (2007 , 2009 ), which demonstrated that shifting overt or covert attention in ways that embody the solution to Duncker’s (1945) tumor problem improved performance on it, even without solvers being aware of the relevance of the cueing to finding the problem’s solution. The concept of having attentional movement trajectories embody the solution to a problem, while powerful, may not apply to solving a wide array of problems. However, the simpler relationship between what is selected for visual attention and how that affects problem solving cognitive processes can be investigated in most if not all problems involving figures. Our particular approach to investigating this issue has been to use specific physics problems that contain two distinct regions, those associated with well-documented misconceptions and those associated with correctly solving the problems. In this way, a direct connection can potentially be found between overt attentional selection and problem solving cognitive processes. The results of these studies showed that when attempting to solve such problems, solvers’ overt attention was strongly guided by mandatory top-down processes (prior knowledge, either correct or mistaken) to either the relevant or irrelevant regions respectively ( Madsen et al., 2012 , 2013a ). Importantly, those who overtly attended more to the relevant information were more likely to correctly solve the problems, and those who overtly attended to regions associated with well-documented misconceptions more frequently gave incorrect answers in line with those misconceptions. This raised the question of whether guiding solvers’ overt attention to the relevant information would facilitate their correctly solving those or similar problems. In one study, we modified the bottom-up visual saliency (as measured by a computational model) of the relevant vs. irrelevant regions in physics problems (by increasing or decreasing the luminance contrast of the lines in the problem diagrams; Madsen et al., 2013b ). Interestingly, we found that solvers’ mandatory top-down processes (prior knowledge) guided their overt attention, overwhelming any potential effects of stimulus saliency ( Madsen et al., 2013b ). Nevertheless, as before, those who attended more to relevant information were more likely to correctly solve the problems ( Madsen et al., 2012 ). In another study, we used highly salient visual cues (moving colored dots) that mimicked the overt attention shifts that correct solvers often made while solving those problems, and asked the solvers to follow the dots with their eyes (without explaining why) while they solved the problems ( Madsen et al., 2013a ). We found that the moving dot cues often guided solvers’ overt attention to the relevant areas (assumedly based on both bottom-up stimulus saliency and volitional top-down processes). However, the highest percentage of cued participants answering a training problem correctly was 41%, which was not significantly higher than the 32% in the uncued condition. Further, the cued participants significantly outperformed the uncued participants on the training problems in only one of the four problem sets. Likewise, the cued participants significantly outperformed the uncued participants on the transfer problem in only one of the four problem sets ( Madsen et al., 2013a ). Thus, getting solvers through the starting gate, by guiding their overt attention to relevant information, was often insufficient to facilitate correct problem solving.

In sum, our prior work has shown that higher-level cognitive processes involved in physics problem solving very strongly guide solvers’ overt attentional selection. Furthermore, overt attentional selection of relevant (rather than irrelevant) information is associated with a higher probability of correctly solving such problems. However, we have also shown that simply guiding solvers’ overt attention to relevant areas of physics problems is often insufficient to correctly solve those problems, or transfer problems similar to them.

The Current Study

Our prior results described above left important open research questions. Specifically, although previous work clearly showed that higher cognitive processes strongly affect attentional selection during insight problem solving, much less clear is the degree to which attentional selection, as guided by visual cues, can strongly affect higher-level cognitive processes involved in conceptual physics problem solving.

We therefore considered our previous results in terms of their relationship to Ohlsson’s (1992) model of insight problem solving, which suggested that we make several changes to our methodology. These changes were done in order to facilitate both the guidance of overt attention to relevant information, and the use of that information to restructure solvers’ representations of the problems and find correct solution paths. Specifically, although several previous studies had shown that the solvers’ success rate in solving Dunker’s radiation problem could be increased by their cueing attention without explaining why ( Grant and Spivey, 2003 ; Thomas and Lleras, 2007 , 2009 ), we repeatedly found that simply guiding solvers’ attention to the relevant information in a problem was insufficient for them to arrive at a correct solution path ( Madsen et al., 2012 , 2013a ). Thus, we decided to explicitly indicate to solvers that the cues were relevant to solving the problems, by referring to the cues as “hints,” which were meant to help them.

In addition, we previously observed that solvers who were incorrect on the first problem in a set of similar problems tended to repeatedly use the same incorrect solution path for every problem in the set. Thus, in terms of Ohlsson’s (1992) model of insight problem solving, the solvers were apparently not facing an impasse that would force them to restructure their faulty representation of the problem. This points out a difference between our problems and many common insight problems, for example Maier’s two-string problem ( Maier, 1931 ). In our problems the solver may not know that they have failed to reach the goal state, whereas many insight problems are structured such that failure to reach the correct goal state is self-apparent. We therefore decided to provide the solvers with correctness feedback (i.e., saying “correct” or “incorrect” without explaining why) after they gave their answer to each problem. This would facilitate their entering an impasse for those problems they solved incorrectly, with the idea that solvers could then potentially break their impasse by restructuring their representations of those problems. In such cases, the visual cues could direct solvers’ attention to relevant information, which could activate previously dormant relevant knowledge from long-term memory, enabling the solver to create a new representation for the problem that could break the impasse. In order to determine the individual effects of correctness feedback and visual cueing on overt attention and problem solving, we manipulated both factors independently in our experimental design.

We also incorporated a key idea from de Koning et al’s. (2009) model of attentional cueing for learning, specifically that cues can be used not only to facilitate selecting important information for attention, but also to facilitate integrating information across different regions within a problem. For instance, cues can facilitate making comparisons between different elements of a problem, such as comparing the distance traveled at different points in time, or comparing the slopes of two curves on a graph. Such cues still function to direct the solvers’ attention, but go beyond simply directing attention to a location in space by symbolically indicating the types of information to attend to at those locations, and between different locations over time.

In order to measure changes in attentional selection and problem solving over time (i.e., learning), as in our previous studies ( Madsen et al., 2013a ), for each base problem, we created a series of similar problems, which will be discussed in the Methods section. Furthermore, as in our previous studies ( Madsen et al., 2013a ), in order to test for more than just superficial learning, we created transfer problems that used the same underlying reasoning (and solution paths), but had somewhat different surfaces features. In addition, we did not use cues on either the initial problem for each sequence, or on the transfer problem for that sequence, in order to measure both overt attentional selection and problem solving cognitive processes in the absence of cueing.

Given the above discussion, it is worth considering what changes in perceptual and higher level cognitive processing might occur as a consequence of learning engendered by cueing problem solvers on successive trials, each with a similar problem that differs only minimally in its surface features from the previous problem, and then testing on a transfer problem that differs more substantially in its surface features. Changes in solvers’ problem representations could be measured off-line in terms of giving correct answers on the transfer problems by solvers who had given incorrect answers on the initial problem for that problem type. Of particular interest for the current study, we can also measure such changes in the solvers’ problem representations on-line in terms of eye movement data, for example by solvers overtly attending to relevant information on transfer problems that they had previously ignored in the initial problem of that problem type. A more specific hypothesis is that solvers who had previously been cued would have learned to attend to the relevant information, and thus spend more time processing the relevant information on the transfer problem than those solvers who had not been previously cued. We will call this the processing priority hypothesis. Interestingly, however, an alternative competing hypothesis is suggested by considering a further aspect of learning, namely automatization ( Schneider and Shiffrin, 1977 ), which could be measured in terms of increased efficiency of information extraction and integration into a solution path in WM. Assumedly, repeatedly attending to relevant information and using it to create a similar correct solution path would engender greater automaticity (i.e., efficiency) in performing each of these perceptual and cognitive processes. Automatization as shown by eye movements could be measured in terms of fixation durations, which are generally taken as an indication of processing difficulty ( Rayner, 1998 ; Nuthmann et al., 2010 ). Thus, to the degree that relevant information extraction and integration is automatized, it should produce shorter fixation durations. More specifically, an alternative hypothesis is that solvers who had previously been repeatedly cued should process the relevant information in a more automatized manner, and thus have shorter fixation durations on the relevant information on the transfer problem than those solvers who had not been previously cued. We will call this the automatization hypothesis.

Materials and Methods

Participants.

The participants in this study ( N = 80, 44 males, 36 females) were enrolled in a first semester algebra-based physics course and were compensated with course credit. All participants had uncorrected or corrected-to-normal vision.

Four problem sets were investigated in this study and covered the topics of speed and energy conservation. Participants covered the requisite material in their course before being recruited to participate in the study. The problem sets examined in this study all contained diagrams with features consistent with novice-like answers documented in the literature and separate areas relevant to correctly solving the problem ( Madsen et al., 2012 ). Each set consisted of eight problems: an initial problem, six isomorphic training problems, and a transfer problem. The transfer problem assessed the same concept as the other problems in the set, but had different surface features (e.g., Reed, 1993 ). An example of a problem set is provided in Figure 1 .

FIGURE 1. Example of an initial problem (top), one of the six training problems (middle), and a transfer problem (bottom) .

The cues were described to the participants as hints, which were meant to help them solve the problem. When ready to view the cue, the participants pressed a button. All participants in cued conditions were required to view the cue at least once, but there was no limit on the number of times they could replay it. Explanations of the critical information needed to solve each problem, along with examples of the cues for those problems are provided in Figure 2 .

FIGURE 2. Examples of training problems with the cue superimposed from the (A) ball, (B) cart, (C) graph, and (D) skier problem sets, respectively. The cue is represented by the colored shapes (and grayed slopes for the Skier). The cues lasted for 8 s, and the numbers indicate the order in which the shapes appeared on the ball and cart problems. The critical information needed to solve each problem is as follows: in problem (A) the balls will have the same average speed during the time interval in which they travel the same distance. In problem (B) the shapes of the tracks are irrelevant to the carts’ final speeds due to the lack of friction, so it is only necessary to compare the starting and ending heights of the carts. In problem (C) the two objects represented in the graph will travel at the same speed when they have equal slopes, as speed is the rate of change of position. In problem (D) the change in potential energy depends only on the change in vertical height for each segment.

The problems were presented to participants on a computer screen. The screen had a resolution of 1024 × 768 pixels and a refresh rate of 85 Hz. The images subtended 33.3°× 25.5° of visual angle. Participants used a chin and forehead rest that was 24 inches from the screen. Eye movements were recorded with an EyeLink 1000 desktop mounted eye-tracking system which had an accuracy of less than 0.50° of visual angle.

Design and Procedure

This study was part of a larger study in which we investigated the effect on problem correctness due to both feedback and visual cues ( Rouinfar et al., 2014 ). In this paper we focus on the analysis of the eye movement data, though we use accuracy data to show evidence of learning to make arguments linking the eye movements to learning.

Each participant took part in an individual session lasting 50–60 min. At the beginning of the session, participants were given a short explanation of the goal of the interview and given instructions. The eye tracker was calibrated to the individual using a nine-point calibration and validation procedure, with a threshold agreement of 0.5° visual angle required to begin the experiment.

Participants were randomly assigned either a cued condition ( N = 38, 22 males, 16 females) or an uncued condition ( N = 42, 22 males, 20 females). Those in the cued conditions saw colored shapes superimposed on the diagrams of the training problems for 8 s, but not on the transfer problems. All participants worked through four sets of problems. The order of the problem sets and the training problems within each set was randomized. Participants were told to spend as much time as they needed on each question and to give a verbal answer and explanation whenever they were ready. The participants were able to point to areas on the computer screen while explaining their answers if necessary. The experimenter used a pre-defined rubric to determine if the given answer and explanation were correct or incorrect. The experimenter would ask for clarification if the participant provided a vague answer or explanation. To be considered correct, the responses were required to contain both the correct answer and scientifically correct explanation.

The current study reports on eye movement data collected in an experiment reported in more detail in Rouinfar et al. (2014) . That experiment factorially manipulated both cueing and feedback and found significant main effects of both factors, but no interaction between them, on accuracy of physics problem solving. That study did not report on the eye movement data, which is the focus of the current study. The current study analyzed the effects of both cueing and feedback, but found no significant main effects of feedback, nor any interactions of feedback with cueing, on any eye movement measures. Therefore, to streamline our description of our results, we have collapsed across the feedback factor and will not discuss that factor further.

Correctness

We were first interested in the pedagogical effectiveness of the visual cues in helping participants correctly solve and reason about the problems. Figure 3 shows the average percentage of initial and transfer problems solved correctly (correct in terms of both the answer and explanation) by the participants in the cued and uncued conditions. On average, participants in the uncued condition correctly solved 23.4% of initial problems and 35.3% of transfer problems. Participants in the cued condition correctly solved an average of 33.6% of initial problems and 69.7% of transfer problems. To compare the performance of the cued and uncued participants, a repeated measures ANOVA was conducted with the proportion of the initial and transfer problems correctly solved as the within-subjects factor and the condition as the between-subjects factor.

FIGURE 3. The average percentage of initial and transfer problems answered correctly by participants in the cued and uncued conditions. The error bars indicate ± 1 SE of the mean.

The results of the ANOVA indicated that there was a main effect of problem, F (1,78) = 64.55, p < 0.001 and of condition, F (1,78) = 16.45, p < 0.001. These main effects were qualified by a significant interaction, F (1,78) = 16.45, p < 0.001 indicating that participants in the cued and uncued conditions performed differently depending on the problem. Probing the interaction we find that there was no significant difference in the average proportion of initial problems answered correctly by participants in the cued and uncued conditions, F (1,78) = 3.42, p = 0.068. However, those in the cued condition, on average, correctly solved a significantly larger proportion of transfer problems than those in the uncued condition, F (1,78) = 39.38, p < 0.001, d = 1.07. Both those in the cued and uncued conditions showed a significant increase from initial to transfer, F (1,78) = 69.11, p < 0.001, d = 1.23 and F (1,78) = 8.28, p = 0.005, d = 0.45, respectively. After watching cues on the training problems, participants in the cued condition solved nearly twice the proportion of transfer problems correctly as compared to participants in the uncued condition. These results demonstrate that the visual cues significantly improve performance on the transfer problem. More importantly, the results suggest that the visual cues promote higher level cognition as evinced by the improved performance on the transfer problem.

Comparing the Attention of Correct and Incorrect Solvers on the Initial Problem

Madsen et al. (2012) showed that correct and incorrect solvers differ in their allocation of visual attention while solving problems with diagrammatic features consistent with novice-like answers in addition to thematically relevant regions. Specifically, participants who answer the problems correctly spend significantly more time attending to the thematically relevant areas and a significantly smaller proportion of time attending to the features associated with the novice-like answers than participants who answer the problems incorrectly. The novice-like and thematically relevant areas in the problems investigated in this study are depicted in Figure 4 . We performed a similar analysis to determine if the correctness on the initial problem could be attributed to participants’ attention in the thematically relevant and novice-like regions.

FIGURE 4. An example of the thematically relevant area and novice-like area in an initial problem. Respectively, these areas are associated with the correct response (time interval when the balls travel the same distance) and most common incorrect response (time when the balls are at the same position).

To analyze the eye movements, areas of interest (AOI) were drawn around the thematically relevant and novice-like areas associated with each problem with a border of 1.1° of visual angle. The size of the areas was determined by using an error propagation technique ( Preston and Dietz, 1991 ) which took into account both the eye tracker’s accuracy and the spatial extent of the central fovea (0.5° and 1° of visual angle, respectively). When comparing eye movements across several problems, the physical sizes of the thematically relevant and novice-like areas are non-constant and should be normalized. To do this, we divided the percentage of dwell time in the AOI by the percentage of screen that the AOI subtends. This produced a new measure, the percentage of total dwell time divided by the percentage of total area, which is described as the domain relative ratio ( Fletcher-Watson et al., 2008 ).

Figure 5 shows the domain relative ratio spent by correct and incorrect solvers in the thematically relevant and novice-like areas while they solved the initial problem in each set. To compare the proportion of time that correct and incorrect solvers spent attending to the thematically relevant and novice-like areas, we conducted two one-way ANOVAs with the domain relative ratio as the dependent measure and correctness as the between-subjects factor. The results indicate that those who solved the initial problem correctly had a significantly larger domain relative ratio in the thematically relevant area, F (1,318) = 13.20, p < 0.001, d = 0.44 while simultaneously spending a significantly smaller domain relative ratio in the novice-like area, F (1,318) = 14.85, p < 0.001, d = 0.47. These results are consistent with Madsen et al’s. (2012) findings.

FIGURE 5. The domain relative ratio (percentage of dwell time divided by the percentage of area the AOI encompasses) in the thematically relevant and novice-like areas on the initial problem by the correctness of response. Error bars indicate ± 1 standard error of the mean.

Attention While the Cue Played

Participants in the cued condition were required to play the cues on the training problems at least once, but were allowed to replay the cue as many times as desired. The vast majority of the time the participants chose to play the cues just once, accounting for 90.4% of all training problems solved. The cue was played twice 8.1% of the time, 55.4% of which occurred during the first training problem in a set.

We investigated whether participants who most needed to see the cue (namely those who provided an incorrect response to the immediately preceding problem in the set) actually watched the cue while it was on screen. We found that those who switched to a correct response had, on average, a domain relative ratio of 16.5 spent watching the cue while it was on screen, while those who retained an incorrect response had a domain relative ratio of 13.2. To compare these values, a one-way ANOVA was conducted with the domain relative ratio as the dependent measure and correctness pattern as the between-subjects factor. The results indicated that the cued participants who switched to a correct response spent a significantly larger proportion of time per area watching the cue, F (1,277) = 7.71, p = 0.006, d = 0.34. This result demonstrates that watching the cue more closely can be tied to participants switching from an incorrect to correct response.

Changes in Eye Movements Among Participants Who Demonstrated Learning

Thus far, we have demonstrated that cues can be an effective learning tool and that there is a link between the correctness of a student’s response and their allocation of attention while solving the problem. We now consider the subset of participants who we can reasonably assume learned something—that is, those who answered the initial problem incorrectly, but after working through the training problems were successful in correctly solving the transfer problem. Each case in which a participant demonstrated learning was treated as an independent observation in the analyses described later in this section. Across all problem sets, we have 66 cases (34 unique participants) of this occurring in the cued group and 30 cases (21 unique participants) in the uncued group, corresponding to 89.5% and 50.0% of participants in the cued and uncued groups, respectively. There was significantly greater number of participants in the cued condition following this pattern than in the uncued condition, χ 2 (1, N = 320) = 24.83, p < 0.001, V = 0.279. The number of participants demonstrating learning on one or more problems is provided in Table 1 .

TABLE 1. The number of problem sets in which participants demonstrated learning in the cued and uncued conditions.

As indicated by Table 1 , the analyses reported below contained cases in which some participants contributed only a single observation, whereas other participants contributed multiple observations, across all problem sets. Thus, we did not include “problem set” as a within-subjects factor in our analyses due to missing data. Because having different numbers of observations across problem sets as a function of participants could create additional within-subject dependencies in our analyses, we carried out a robustness check. Specifically, we carried out the analyses discussed in this section on a randomly selected subsample of the data in which no participant contributed more than a single observation. The results of these additional analyses showed the same pattern of results reported below—all significant main effects and interactions reported below were also significant with only the randomly chosen subsample. Therefore, for all analyses reported in this section, we have included the full data set shown in Table 1 .

Attention in the thematically relevant area

After finding that correct solvers spent a significantly larger proportion of their time attending to the relevant area, we wanted to see if the participants who demonstrated learning had an increased domain relative ratio in the transfer problem. Figure 6 shows the domain relative ratio that cued and uncued participants spent in the relevant area on the initial and transfer problems.

FIGURE 6. The domain relative ratio (percentage of dwell time divided by the percentage of area the AOI encompasses) in the thematically relevant area on the initial and transfer problems for those who improved from the initial to transfer problem. The error bars indicate ± 1 SE of the mean.

A repeated measures ANOVA with domain relative ratio in the thematically relevant area as the dependent measure and condition as the between-subjects factor was conducted. There was a significant increase in the domain relative ratio in the relevant area from the initial to the transfer problem, F (1,94) = 56.41, p < 0.001 and a significant main effect of condition, F (1,94) = 4.12, p = 0.045. However, these main effects are qualified by a significant interaction, F (1,94) = 10.17, p = 0.002, indicating that the cued and uncued groups performed differently depending on the problem. Probing the interaction we find that the domain relative ratio of both the cued and uncued groups increased significantly from initial to transfer problem, F (1,94) = 14.94, p < 0.001, d = 0.79 and F (1,94) = 41.63, p < 0.001, d = 1.35, respectively. However, while there was no significant difference between the cued and uncued conditions on the initial problem, F (1,94) < 1, the uncued condition had a significantly higher domain relative ratio in the relevant area than the cued condition on the transfer problem, F (1,94) = 14.25, p < 0.001, d = 0.65.

Inconsistent with the processing prioritization hypothesis, among participants who showed evidence of learning (i.e., improved performance on the transfer problem relative to the initial problem), those who saw cues had a significantly smaller domain relative ratio in the relevant area on the transfer problem than those who did not see cues. This is despite the fact that solvers in the cued condition received training to attend the relevant area. This result is surprising, and seems to pose a paradox. Namely, why would those trained to attend to the relevant area spend less time attending to the relevant area than those who were not trained to do so? A possible solution of this paradox is given by the automatization hypothesis, namely that those who were given training with the cues may have developed greater automaticity in extracting the relevant information, and thus spent proportionally less time attending to the relevant area of the transfer problem than those solvers who did not receive the cued training (i.e., the uncued participants).

Automaticity in extracting relevant information

We hypothesized that the reason the cued group had a smaller domain relative ratio in the thematically relevant area on the transfer problem than the uncued group was because the cued group was able to more easily extract the relevant information from the diagram, namely the automatization hypothesis. If so, evidence for the increased efficiency of relevant information extraction should be found by examining their performance on the training problems. Specifically, participants in the cued condition should have had greater success in extracting the relevant information over more trials than participants in the uncued condition, which would then produce greater automaticity of extracting relevant information for the cued group. A test of this hypothesis is shown in Figure 7 , which shows student performance across all problems within the sets.

FIGURE 7. The performance on each problem for the subset of participants who shifted from an incorrect response on the initial problem to a correct response on the transfer problem.

Consistent with the automatization hypothesis, among cued participants who answered the initial problem incorrectly, we find that 73% were able to correctly solve the first training problem, and the proportion increased to 92% by the sixth training problem. In contrast, only 20% of the uncued group answered the first training problem correctly, and by the sixth problem 73% were correct. Because a larger proportion of participants in the cued group were able to answer the training problems correctly, they had more practice doing so, and thus gained more automaticity in extracting the relevant information. In addition, the increase in percentage of correct responses in the two groups from the sixth training problem to the transfer problem was greater for the uncued group – that is, getting the transfer problem correct was a bigger leap for more of those in the uncued condition than those in the cued condition.

To statistically compare the cued and uncued participants’ performance depicted in Figure 7 , a survival analysis was conducted. To do this, the training problem number in which the participant switched to providing only correct responses was considered. Comparing the resulting survival curves using a log-rank test indicates that the participants who saw cues on the training problems switched to a correct response significantly earlier than those in the uncued group. χ 2 (1, N = 96) = 16.17, p < 0.001. Altogether, these conditions likely led to the cued group having greater ease of extracting the relevant information (indicated by the smaller domain relative ratio) on the transfer problem than the uncued group (as shown in Figure 6 ).

Average fixation duration in the thematically relevant area

A further test of the automatization hypothesis is in terms of the successful problem solvers’ average fixation durations. We would expect that increased ease of extracting the relevant information, namely greater automaticity, would be associated with shorter fixation durations in the relevant area. Table 2 shows the average fixation durations of the cued and uncued participants in the relevant area and entire diagram for the transfer problems. (Note that there was no cueing on the transfer problem, even in the cued condition.)

TABLE 2. The average fixation durations (in ms) ± 1 SE of the mean for the cued and uncued groups in the relevant area and entire diagram while viewing the transfer problems.

The average fixation durations of participants while solving the transfer problems were compared using a 2 (cue vs. no cue) × 2 (entire diagram vs. relevant area) ANOVA. The results are summarized in Table 3 .

TABLE 3. Results of a 2 (cue vs. no cue) × 2 (entire diagram vs. relevant area) ANOVA comparing the average fixation duration on the transfer problem.

A significant interaction between the condition and area of interest was found. Probing the significant interaction, we find that within the relevant area, cued participants had significantly shorter mean fixation durations than those in the uncued condition. This is consistent with the hypothesis that the cued participants had indeed developed greater automaticity in extracting information from the relevant area than the uncued participants. We also found that uncued participants had significantly larger fixation durations in the relevant area of the diagram compared to their average fixation durations when considering the entire diagram. However, for those in the cued group, the average fixation duration in the relevant area is not distinguishable from the rest of the problem. The combination of these results indicates that cued participants experience a greater ease of extraction of the relevant information on the transfer problem, as evidenced by their lower fixation durations. This would explain why the cued group spends a smaller proportion of time attending to the relevant area on the transfer problem (as shown in Figure 6 ).

In this study we investigated the relationship between the low-level perceptual processes involved in overt attentional selection by visual cues, on the one hand, and the high-level cognitive processes involved in solving physics problems. Eye movements can be used to elucidate what information within a diagram is being processed and when that information is being processed. This allows for us to investigate how participants’ attention changes over time and relevant cognitive processes associated with problem solving. In the following sections, we revisit our hypotheses and discuss our findings.

Based on the changes we made in the current study in comparison to our previous studies, we were able to show that visual cues did indeed improve problem solving on the transfer problems. The changes we made were those suggested by consideration of both Ohlsson’s (1992) representational change theory and de Koning and colleagues framework of attention cueing ( de Koning et al., 2009 ). In particular, we told solvers that the cues were meant to help them, we provided correctness feedback to induce impasses among those who originally had an incorrect solution path, and we included visual cues that facilitated not only attentional selection of relevant information, but also integration of that information across different regions of the problem. Doing so indeed facilitated solver’s ability to re-represent the problem in a meaningful way allowing for the extraction of the relevant information and thus improved performance.

We found a significantly greater proportion of participants who received training with visual cues were able to subsequently correctly solve the transfer problem without cues than those who received training in the uncued condition. We observed that both the cued and uncued groups performed similarly on the initial problem and both experienced significant increase in performance from the initial to transfer problem. However, nearly twice as many participants in the cued condition were able to correctly solve the transfer problem as compared to participants in the uncued condition (69.7 vs. 35.5%, respectively). This amounted to more than one standard deviation difference between the groups. These results provide evidence that the visual cues facilitated the participants to re-represent the problem enabling them to break an impasse and solve the problem correctly. More importantly, these results provide evidence, consistent with previous studies ( Thomas and Lleras, 2009 ) that manipulation of low-level eye movements can influence high level cognition involved in problem solving. Nevertheless, there is a critically important difference between the results of our studies and those of Grant and Spivey (2003) and Thomas and Lleras (2007 , 2009 ), who proposed the provocative idea that simply having the viewer’s low-level attentional movements embody a problem’s solution is sufficient to facilitate finding the correct solution. Specifically, our research, including both the current and previous studies ( Madsen et al., 2013a , b ) has shown that while attending to relevant information in a problem is a necessary condition for correctly solving the problem, it is generally not sufficient to correctly solving it. The current study has specifically shown that cues, which both draw attention to solution-relevant information, and facilitate organizing and integrating it, facilitate both immediate problem solving and generalization of that ability to new problems. In addition, the current study shows that when such cues are used across multiple problems, solvers can automatize the extraction of problem-relevant information.

Changes in Eye Movements

In the current study, we were particularly interested in the online processes linking overt attentional selection with higher-level cognitive processes involved in problem solving. Thus, we explored how participants’ attention in the relevant area of the diagram changed from the initial problem to the transfer problem. For this set of analyses, we considered the subgroup of participants who demonstrated improvement in their problem solving from the initial to transfer problem. We focused on this subgroup as they were the ones who through the improvement of their responses from the initial to transfer problem, showed evidence that higher order cognitive processes were online.

We presented two competing hypotheses for how cued and uncued participants’ attention in the thematically relevant area of the diagram would compare on the transfer problem. The processing priority hypothesis was that through training of attentional prioritization, solvers in the cued condition would spend a larger percentage of dwell time per percentage of area attending to the relevant features on the transfer problem, namely a higher domain relative ratio in the relevant area of the transfer problem for the cued group compared to the uncued group. Alternatively, the automatization hypothesis was that repeated training in attending to and extracting relevant information from a problem type would increase participants’ efficiency in doing so, and therefore participants in the cued condition would have shorter fixation durations on the relevant features on the transfer problem than those in the uncued group.

We found that successful problem solvers attend to the relevant information in the diagram significantly more than unsuccessful solvers. When provided with cues on the training problems, participants who successfully switch to correct responses overtly attend to the cue significantly more closely. Among the subset of participants who improved their performance from the initial to transfer problem, we found that the cued group nearly doubled their percentage of dwell time per percentage of area in the thematically relevant area while those in the uncued condition more than tripled the domain relative ratio in the relevant area.

While the cued participants had a significantly larger domain relative ratio in the relevant area of the transfer problem than they did while solving the initial problem, it was still significantly less than the domain relative ratio of uncued group on the transfer problem. To investigate if this result could be tied to the cued group having developed an increased ease of extraction of the relevant information, we examined the participants’ performance on the training problems as well as their average fixation durations while solving the initial and transfer problems.

In examining the training problem performance of those who improved from the initial to transfer problem, we found that the cued group showed a significant increase on the first training problem, followed by a more gradual increase on subsequent training problems. By contrast the uncued group showed a slower increase from the first training problem through the sixth training problem with nearly the same proportion of successful solvers on the sixth training problem that the cued group had on the first. This difference in the trajectories of the cued and uncued subgroups going from incorrectly solving the initial problem to correctly solving the transfer problem indicates that participants in the cued group had acquired greater practice than those in the uncued group in extracting information from the relevant area because they correctly solved a larger proportion of training problems. Therefore, the cued group would have achieved greater automaticity in extracting the relevant problem information than the uncued group. This conclusion is consistent with our finding that the cued group showed a lower mean fixation duration in the relevant area on the transfer problem compared to the uncued group.

An open question for further research is the degree to which the cueing effects in the current study were predicated on telling the solvers that the cues were helpful. Based on the previous results of Thomas and Lleras (2007 , 2009 ), in our previous studies we did not inform solvers that the cues would be helpful, but we found only moderate effects of visual cueing on overt attention and successful problem solving. The current study did tell solvers that the cues were “hints” meant to help them, and found strong effects of visual cueing on both overt attention and successful problem solving. Further research can experimentally vary whether solvers are told about the helpfulness of cues and see the degree to which this is important.

A further open question is the degree to which forcing the initially incorrect solvers into an impasse, either explicitly by providing them with correctness feedback, or implicitly by providing them with visual cues that focus on information they have previously ignored, is critical for creating strong effects of cueing on attentional selection and successful insight problem solving. The current study found that both cueing and correctness feedback facilitated solvers to make the transition from incorrect solution paths to correct solution paths. Interestingly, cueing by itself was more effective than feedback by itself. This raises the question of whether both created impasses. Further research will be needed to create on-line measures of impasse in both cueing and feedback conditions to determine the effects of each on entering an impasse during insight problem solving.

In summary, the current study has shown two important findings. First, short duration visual cues can improve problem solving performance on a variety of insight physics problems, including transfer problems that do not share the surface features of the training problems, but do share the underlying solution path. In other words, visual cues can facilitate solvers to re-represent a problem and overcome impasse thereby enabling them to correctly solve a problem. These cueing effects on problem solving were not predicated upon the solvers’ overt or covert attentional shifts necessarily embodying the solution to the problem. Instead, the cueing effects were predicated upon having solvers attend to and integrate relevant information in the problems into a solution path. Second, these short duration visual cues when administered repeatedly over multiple training problems resulted in participants becoming more efficient at extracting the relevant information on the transfer problem, showing that such cues can improve the automaticity with which solvers extract relevant information from a problem. These results, when combined with those of our previous studies ( Madsen et al., 2013a , b ) suggest that low-level attentional selection processes provide a necessary gateway for relevant information to be used in problem solving, but are generally not sufficient for correct problem solving. Instead, factors that lead a solver to an impasse (e.g., correctness feedback) and to organize and integrate problem information (e.g., organization and integration cues) also greatly facilitate arriving at correct solutions. We are currently studying the specific effects of these factors on problem solving within the context of a model of the role of visual cueing in conceptual problem solving ( Rouinfar et al., 2014 ). Further research along these lines will enable us to more precisely understand the role of lower-level attentional selection in higher-level problem solving.

Conflict of Interest Statement

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

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant #1138697 was awarded to N. Sanjay Rebello and Lester C. Loschky and Grant #0841414 was awarded to Amy Rouinfar.

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Keywords : overt visual attention, physics education, problem solving, visual cognition, automaticity

Citation: Rouinfar A, Agra E, Larson AM, Rebello NS and Loschky LC (2014) Linking attentional processes and conceptual problem solving: visual cues facilitate the automaticity of extracting relevant information from diagrams. Front. Psychol. 5 :1094. doi: 10.3389/fpsyg.2014.01094

Received: 24 July 2014; Accepted: 10 September 2014; Published online: 29 September 2014.

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Copyright © 2014 Rouinfar, Agra, Larson, Rebello and Loschky. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Lester C. Loschky, Department of Psychological Sciences, Kansas State University, 471 Bluemont Hall, Manhattan, KS 66506, USA e-mail: [email protected]

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

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Conceptual problem solving in high school physics

Jennifer l. docktor, natalie e. strand, josé p. mestre, and brian h. ross, phys. rev. st phys. educ. res. 11 , 020106 – published 1 september 2015.

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Problem solving is a critical element of learning physics. However, traditional instruction often emphasizes the quantitative aspects of problem solving such as equations and mathematical procedures rather than qualitative analysis for selecting appropriate concepts and principles. This study describes the development and evaluation of an instructional approach called Conceptual Problem Solving (CPS) which guides students to identify principles, justify their use, and plan their solution in writing before solving a problem. The CPS approach was implemented by high school physics teachers at three schools for major theorems and conservation laws in mechanics and CPS-taught classes were compared to control classes taught using traditional problem solving methods. Information about the teachers’ implementation of the approach was gathered from classroom observations and interviews, and the effectiveness of the approach was evaluated from a series of written assessments. Results indicated that teachers found CPS easy to integrate into their curricula, students engaged in classroom discussions and produced problem solutions of a higher quality than before, and students scored higher on conceptual and problem solving measures.

• Received 30 April 2015

DOI: https://doi.org/10.1103/PhysRevSTPER.11.020106

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• 1 Department of Physics, University of Wisconsin–La Crosse, La Crosse, Wisconsin 54601, USA
• 2 Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
• 3 Beckman Institute for Advanced Science and Technology, University of Illinois, Urbana, Illinois 61801, USA
• 4 Department of Educational Psychology, University of Illinois, Champaign, Illinois 61820, USA
• 5 Department of Psychology, University of Illinois, Champaign, Illinois 61820, USA
• * [email protected]

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Conceptual Problem Solving in Physics

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Students taking introductory physics courses focus on quantitative manipulations at the expense of learning concepts deeply and understanding how they apply to problem solving. This proclivity toward manipulating equations leads to shallow understanding and poor long-term retention. We discuss an alternative approach to physics problem solving, which we call conceptual problem solving (CPS), that highlights and emphasizes the role of conceptual knowledge in solving problems. We present studies that explored the impact of three different implementations of CPS on conceptual learning and problem solving. One was a lab-based study using a computer tool to scaffold conceptual analyses of problems. Another was a classroom-based study in a large introductory college course in which students wrote conceptual strategies prior to solving problems. The third was an implementation in high school classrooms where students identified the relevant principle, wrote a justification for why the principle could be applied, and provided a plan for executing the application of the principle (which was then used for generating the equations). In all three implementations benefits were found as measured by various conceptual and problem solving assessments. We conclude with a summary of what we have learned from the CPS approach, and offer some views on the current and future states of physics instruction.

• Conceptual assessment
• Conceptual problem solving
• High school
• Introductory physics
• Problem solving
• Science assessment
• Science cognition
• Science education
• Strategy writing

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• Developmental and Educational Psychology

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T1 - Conceptual Problem Solving in Physics

AU - Mestre, Jose P.

AU - Docktor, Jennifer L.

AU - Strand, Natalie E.

AU - Ross, Brian H.

N1 - Funding Information: Work in part supported by the Institute of Education Sciences of the US Department of Education under Award No. DE R305B070085. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the Institute of Education Sciences.

N2 - Students taking introductory physics courses focus on quantitative manipulations at the expense of learning concepts deeply and understanding how they apply to problem solving. This proclivity toward manipulating equations leads to shallow understanding and poor long-term retention. We discuss an alternative approach to physics problem solving, which we call conceptual problem solving (CPS), that highlights and emphasizes the role of conceptual knowledge in solving problems. We present studies that explored the impact of three different implementations of CPS on conceptual learning and problem solving. One was a lab-based study using a computer tool to scaffold conceptual analyses of problems. Another was a classroom-based study in a large introductory college course in which students wrote conceptual strategies prior to solving problems. The third was an implementation in high school classrooms where students identified the relevant principle, wrote a justification for why the principle could be applied, and provided a plan for executing the application of the principle (which was then used for generating the equations). In all three implementations benefits were found as measured by various conceptual and problem solving assessments. We conclude with a summary of what we have learned from the CPS approach, and offer some views on the current and future states of physics instruction.

AB - Students taking introductory physics courses focus on quantitative manipulations at the expense of learning concepts deeply and understanding how they apply to problem solving. This proclivity toward manipulating equations leads to shallow understanding and poor long-term retention. We discuss an alternative approach to physics problem solving, which we call conceptual problem solving (CPS), that highlights and emphasizes the role of conceptual knowledge in solving problems. We present studies that explored the impact of three different implementations of CPS on conceptual learning and problem solving. One was a lab-based study using a computer tool to scaffold conceptual analyses of problems. Another was a classroom-based study in a large introductory college course in which students wrote conceptual strategies prior to solving problems. The third was an implementation in high school classrooms where students identified the relevant principle, wrote a justification for why the principle could be applied, and provided a plan for executing the application of the principle (which was then used for generating the equations). In all three implementations benefits were found as measured by various conceptual and problem solving assessments. We conclude with a summary of what we have learned from the CPS approach, and offer some views on the current and future states of physics instruction.

KW - Assessment

KW - Conceptual

KW - Conceptual assessment

KW - Conceptual problem solving

KW - High school

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KW - Science cognition

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JO - Psychology of Learning and Motivation - Advances in Research and Theory

JF - Psychology of Learning and Motivation - Advances in Research and Theory

Conceptual Model-Based Problem Solving

Teach Students with Learning Difficulties to Solve Math Problems

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Although American students are struggling with many aspects of mathematics, the National Mathematics Advisory Panel has identified “algebra as a central concern” (National Mathematics Advisory Panel, 2008, p. xiii). Interestingly, American students tend to enjoy school mathematics during the early elementary grades. However, they begin to experience difficulty in and come to dislike mathematics after fourth grade when learning becomes more abstract or symbolic and involves more algebraic thinking (Cai, Lew, Morris, Moyer, Ng, & Schmittau, 2004).

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How to master the seven-step problem-solving process

In this episode of the McKinsey Podcast , Simon London speaks with Charles Conn, CEO of venture-capital firm Oxford Sciences Innovation, and McKinsey senior partner Hugo Sarrazin about the complexities of different problem-solving strategies.

Podcast transcript

Simon London: Hello, and welcome to this episode of the McKinsey Podcast , with me, Simon London. What’s the number-one skill you need to succeed professionally? Salesmanship, perhaps? Or a facility with statistics? Or maybe the ability to communicate crisply and clearly? Many would argue that at the very top of the list comes problem solving: that is, the ability to think through and come up with an optimal course of action to address any complex challenge—in business, in public policy, or indeed in life.

Looked at this way, it’s no surprise that McKinsey takes problem solving very seriously, testing for it during the recruiting process and then honing it, in McKinsey consultants, through immersion in a structured seven-step method. To discuss the art of problem solving, I sat down in California with McKinsey senior partner Hugo Sarrazin and also with Charles Conn. Charles is a former McKinsey partner, entrepreneur, executive, and coauthor of the book Bulletproof Problem Solving: The One Skill That Changes Everything [John Wiley & Sons, 2018].

Charles and Hugo, welcome to the podcast. Thank you for being here.

Hugo Sarrazin: Our pleasure.

Charles Conn: It’s terrific to be here.

Simon London: Problem solving is a really interesting piece of terminology. It could mean so many different things. I have a son who’s a teenage climber. They talk about solving problems. Climbing is problem solving. Charles, when you talk about problem solving, what are you talking about?

Charles Conn: For me, problem solving is the answer to the question “What should I do?” It’s interesting when there’s uncertainty and complexity, and when it’s meaningful because there are consequences. Your son’s climbing is a perfect example. There are consequences, and it’s complicated, and there’s uncertainty—can he make that grab? I think we can apply that same frame almost at any level. You can think about questions like “What town would I like to live in?” or “Should I put solar panels on my roof?”

You might think that’s a funny thing to apply problem solving to, but in my mind it’s not fundamentally different from business problem solving, which answers the question “What should my strategy be?” Or problem solving at the policy level: “How do we combat climate change?” “Should I support the local school bond?” I think these are all part and parcel of the same type of question, “What should I do?”

I’m a big fan of structured problem solving. By following steps, we can more clearly understand what problem it is we’re solving, what are the components of the problem that we’re solving, which components are the most important ones for us to pay attention to, which analytic techniques we should apply to those, and how we can synthesize what we’ve learned back into a compelling story. That’s all it is, at its heart.

I think sometimes when people think about seven steps, they assume that there’s a rigidity to this. That’s not it at all. It’s actually to give you the scope for creativity, which often doesn’t exist when your problem solving is muddled.

Simon London: You were just talking about the seven-step process. That’s what’s written down in the book, but it’s a very McKinsey process as well. Without getting too deep into the weeds, let’s go through the steps, one by one. You were just talking about problem definition as being a particularly important thing to get right first. That’s the first step. Hugo, tell us about that.

Hugo Sarrazin: It is surprising how often people jump past this step and make a bunch of assumptions. The most powerful thing is to step back and ask the basic questions—“What are we trying to solve? What are the constraints that exist? What are the dependencies?” Let’s make those explicit and really push the thinking and defining. At McKinsey, we spend an enormous amount of time in writing that little statement, and the statement, if you’re a logic purist, is great. You debate. “Is it an ‘or’? Is it an ‘and’? What’s the action verb?” Because all these specific words help you get to the heart of what matters.

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Simon London: So this is a concise problem statement.

Hugo Sarrazin: Yeah. It’s not like “Can we grow in Japan?” That’s interesting, but it is “What, specifically, are we trying to uncover in the growth of a product in Japan? Or a segment in Japan? Or a channel in Japan?” When you spend an enormous amount of time, in the first meeting of the different stakeholders, debating this and having different people put forward what they think the problem definition is, you realize that people have completely different views of why they’re here. That, to me, is the most important step.

Charles Conn: I would agree with that. For me, the problem context is critical. When we understand “What are the forces acting upon your decision maker? How quickly is the answer needed? With what precision is the answer needed? Are there areas that are off limits or areas where we would particularly like to find our solution? Is the decision maker open to exploring other areas?” then you not only become more efficient, and move toward what we call the critical path in problem solving, but you also make it so much more likely that you’re not going to waste your time or your decision maker’s time.

How often do especially bright young people run off with half of the idea about what the problem is and start collecting data and start building models—only to discover that they’ve really gone off half-cocked.

Hugo Sarrazin: Yeah.

Charles Conn: And in the wrong direction.

Simon London: OK. So step one—and there is a real art and a structure to it—is define the problem. Step two, Charles?

Charles Conn: My favorite step is step two, which is to use logic trees to disaggregate the problem. Every problem we’re solving has some complexity and some uncertainty in it. The only way that we can really get our team working on the problem is to take the problem apart into logical pieces.

What we find, of course, is that the way to disaggregate the problem often gives you an insight into the answer to the problem quite quickly. I love to do two or three different cuts at it, each one giving a bit of a different insight into what might be going wrong. By doing sensible disaggregations, using logic trees, we can figure out which parts of the problem we should be looking at, and we can assign those different parts to team members.

Simon London: What’s a good example of a logic tree on a sort of ratable problem?

Charles Conn: Maybe the easiest one is the classic profit tree. Almost in every business that I would take a look at, I would start with a profit or return-on-assets tree. In its simplest form, you have the components of revenue, which are price and quantity, and the components of cost, which are cost and quantity. Each of those can be broken out. Cost can be broken into variable cost and fixed cost. The components of price can be broken into what your pricing scheme is. That simple tree often provides insight into what’s going on in a business or what the difference is between that business and the competitors.

If we add the leg, which is “What’s the asset base or investment element?”—so profit divided by assets—then we can ask the question “Is the business using its investments sensibly?” whether that’s in stores or in manufacturing or in transportation assets. I hope we can see just how simple this is, even though we’re describing it in words.

When I went to work with Gordon Moore at the Moore Foundation, the problem that he asked us to look at was “How can we save Pacific salmon?” Now, that sounds like an impossible question, but it was amenable to precisely the same type of disaggregation and allowed us to organize what became a 15-year effort to improve the likelihood of good outcomes for Pacific salmon.

Simon London: Now, is there a danger that your logic tree can be impossibly large? This, I think, brings us onto the third step in the process, which is that you have to prioritize.

Charles Conn: Absolutely. The third step, which we also emphasize, along with good problem definition, is rigorous prioritization—we ask the questions “How important is this lever or this branch of the tree in the overall outcome that we seek to achieve? How much can I move that lever?” Obviously, we try and focus our efforts on ones that have a big impact on the problem and the ones that we have the ability to change. With salmon, ocean conditions turned out to be a big lever, but not one that we could adjust. We focused our attention on fish habitats and fish-harvesting practices, which were big levers that we could affect.

People spend a lot of time arguing about branches that are either not important or that none of us can change. We see it in the public square. When we deal with questions at the policy level—“Should you support the death penalty?” “How do we affect climate change?” “How can we uncover the causes and address homelessness?”—it’s even more important that we’re focusing on levers that are big and movable.

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Simon London: Let’s move swiftly on to step four. You’ve defined your problem, you disaggregate it, you prioritize where you want to analyze—what you want to really look at hard. Then you got to the work plan. Now, what does that mean in practice?

Hugo Sarrazin: Depending on what you’ve prioritized, there are many things you could do. It could be breaking the work among the team members so that people have a clear piece of the work to do. It could be defining the specific analyses that need to get done and executed, and being clear on time lines. There’s always a level-one answer, there’s a level-two answer, there’s a level-three answer. Without being too flippant, I can solve any problem during a good dinner with wine. It won’t have a whole lot of backing.

Simon London: Not going to have a lot of depth to it.

Hugo Sarrazin: No, but it may be useful as a starting point. If the stakes are not that high, that could be OK. If it’s really high stakes, you may need level three and have the whole model validated in three different ways. You need to find a work plan that reflects the level of precision, the time frame you have, and the stakeholders you need to bring along in the exercise.

Charles Conn: I love the way you’ve described that, because, again, some people think of problem solving as a linear thing, but of course what’s critical is that it’s iterative. As you say, you can solve the problem in one day or even one hour.

Charles Conn: We encourage our teams everywhere to do that. We call it the one-day answer or the one-hour answer. In work planning, we’re always iterating. Every time you see a 50-page work plan that stretches out to three months, you know it’s wrong. It will be outmoded very quickly by that learning process that you described. Iterative problem solving is a critical part of this. Sometimes, people think work planning sounds dull, but it isn’t. It’s how we know what’s expected of us and when we need to deliver it and how we’re progressing toward the answer. It’s also the place where we can deal with biases. Bias is a feature of every human decision-making process. If we design our team interactions intelligently, we can avoid the worst sort of biases.

Simon London: Here we’re talking about cognitive biases primarily, right? It’s not that I’m biased against you because of your accent or something. These are the cognitive biases that behavioral sciences have shown we all carry around, things like anchoring, overoptimism—these kinds of things.

Both: Yeah.

Charles Conn: Availability bias is the one that I’m always alert to. You think you’ve seen the problem before, and therefore what’s available is your previous conception of it—and we have to be most careful about that. In any human setting, we also have to be careful about biases that are based on hierarchies, sometimes called sunflower bias. I’m sure, Hugo, with your teams, you make sure that the youngest team members speak first. Not the oldest team members, because it’s easy for people to look at who’s senior and alter their own creative approaches.

Hugo Sarrazin: It’s helpful, at that moment—if someone is asserting a point of view—to ask the question “This was true in what context?” You’re trying to apply something that worked in one context to a different one. That can be deadly if the context has changed, and that’s why organizations struggle to change. You promote all these people because they did something that worked well in the past, and then there’s a disruption in the industry, and they keep doing what got them promoted even though the context has changed.

Simon London: Right. Right.

Hugo Sarrazin: So it’s the same thing in problem solving.

Charles Conn: And it’s why diversity in our teams is so important. It’s one of the best things about the world that we’re in now. We’re likely to have people from different socioeconomic, ethnic, and national backgrounds, each of whom sees problems from a slightly different perspective. It is therefore much more likely that the team will uncover a truly creative and clever approach to problem solving.

Simon London: Let’s move on to step five. You’ve done your work plan. Now you’ve actually got to do the analysis. The thing that strikes me here is that the range of tools that we have at our disposal now, of course, is just huge, particularly with advances in computation, advanced analytics. There’s so many things that you can apply here. Just talk about the analysis stage. How do you pick the right tools?

Charles Conn: For me, the most important thing is that we start with simple heuristics and explanatory statistics before we go off and use the big-gun tools. We need to understand the shape and scope of our problem before we start applying these massive and complex analytical approaches.

Simon London: Would you agree with that?

Hugo Sarrazin: I agree. I think there are so many wonderful heuristics. You need to start there before you go deep into the modeling exercise. There’s an interesting dynamic that’s happening, though. In some cases, for some types of problems, it is even better to set yourself up to maximize your learning. Your problem-solving methodology is test and learn, test and learn, test and learn, and iterate. That is a heuristic in itself, the A/B testing that is used in many parts of the world. So that’s a problem-solving methodology. It’s nothing different. It just uses technology and feedback loops in a fast way. The other one is exploratory data analysis. When you’re dealing with a large-scale problem, and there’s so much data, I can get to the heuristics that Charles was talking about through very clever visualization of data.

You test with your data. You need to set up an environment to do so, but don’t get caught up in neural-network modeling immediately. You’re testing, you’re checking—“Is the data right? Is it sound? Does it make sense?”—before you launch too far.

Simon London: You do hear these ideas—that if you have a big enough data set and enough algorithms, they’re going to find things that you just wouldn’t have spotted, find solutions that maybe you wouldn’t have thought of. Does machine learning sort of revolutionize the problem-solving process? Or are these actually just other tools in the toolbox for structured problem solving?

Charles Conn: It can be revolutionary. There are some areas in which the pattern recognition of large data sets and good algorithms can help us see things that we otherwise couldn’t see. But I do think it’s terribly important we don’t think that this particular technique is a substitute for superb problem solving, starting with good problem definition. Many people use machine learning without understanding algorithms that themselves can have biases built into them. Just as 20 years ago, when we were doing statistical analysis, we knew that we needed good model definition, we still need a good understanding of our algorithms and really good problem definition before we launch off into big data sets and unknown algorithms.

Simon London: Step six. You’ve done your analysis.

Charles Conn: I take six and seven together, and this is the place where young problem solvers often make a mistake. They’ve got their analysis, and they assume that’s the answer, and of course it isn’t the answer. The ability to synthesize the pieces that came out of the analysis and begin to weave those into a story that helps people answer the question “What should I do?” This is back to where we started. If we can’t synthesize, and we can’t tell a story, then our decision maker can’t find the answer to “What should I do?”

Simon London: But, again, these final steps are about motivating people to action, right?

Charles Conn: Yeah.

Simon London: I am slightly torn about the nomenclature of problem solving because it’s on paper, right? Until you motivate people to action, you actually haven’t solved anything.

Charles Conn: I love this question because I think decision-making theory, without a bias to action, is a waste of time. Everything in how I approach this is to help people take action that makes the world better.

Simon London: Hence, these are absolutely critical steps. If you don’t do this well, you’ve just got a bunch of analysis.

Charles Conn: We end up in exactly the same place where we started, which is people speaking across each other, past each other in the public square, rather than actually working together, shoulder to shoulder, to crack these important problems.

Simon London: In the real world, we have a lot of uncertainty—arguably, increasing uncertainty. How do good problem solvers deal with that?

Hugo Sarrazin: At every step of the process. In the problem definition, when you’re defining the context, you need to understand those sources of uncertainty and whether they’re important or not important. It becomes important in the definition of the tree.

You need to think carefully about the branches of the tree that are more certain and less certain as you define them. They don’t have equal weight just because they’ve got equal space on the page. Then, when you’re prioritizing, your prioritization approach may put more emphasis on things that have low probability but huge impact—or, vice versa, may put a lot of priority on things that are very likely and, hopefully, have a reasonable impact. You can introduce that along the way. When you come back to the synthesis, you just need to be nuanced about what you’re understanding, the likelihood.

Often, people lack humility in the way they make their recommendations: “This is the answer.” They’re very precise, and I think we would all be well-served to say, “This is a likely answer under the following sets of conditions” and then make the level of uncertainty clearer, if that is appropriate. It doesn’t mean you’re always in the gray zone; it doesn’t mean you don’t have a point of view. It just means that you can be explicit about the certainty of your answer when you make that recommendation.

Simon London: So it sounds like there is an underlying principle: “Acknowledge and embrace the uncertainty. Don’t pretend that it isn’t there. Be very clear about what the uncertainties are up front, and then build that into every step of the process.”

Hugo Sarrazin: Every step of the process.

Simon London: Yeah. We have just walked through a particular structured methodology for problem solving. But, of course, this is not the only structured methodology for problem solving. One that is also very well-known is design thinking, which comes at things very differently. So, Hugo, I know you have worked with a lot of designers. Just give us a very quick summary. Design thinking—what is it, and how does it relate?

Hugo Sarrazin: It starts with an incredible amount of empathy for the user and uses that to define the problem. It does pause and go out in the wild and spend an enormous amount of time seeing how people interact with objects, seeing the experience they’re getting, seeing the pain points or joy—and uses that to infer and define the problem.

Simon London: Problem definition, but out in the world.

Hugo Sarrazin: With an enormous amount of empathy. There’s a huge emphasis on empathy. Traditional, more classic problem solving is you define the problem based on an understanding of the situation. This one almost presupposes that we don’t know the problem until we go see it. The second thing is you need to come up with multiple scenarios or answers or ideas or concepts, and there’s a lot of divergent thinking initially. That’s slightly different, versus the prioritization, but not for long. Eventually, you need to kind of say, “OK, I’m going to converge again.” Then you go and you bring things back to the customer and get feedback and iterate. Then you rinse and repeat, rinse and repeat. There’s a lot of tactile building, along the way, of prototypes and things like that. It’s very iterative.

Simon London: So, Charles, are these complements or are these alternatives?

Charles Conn: I think they’re entirely complementary, and I think Hugo’s description is perfect. When we do problem definition well in classic problem solving, we are demonstrating the kind of empathy, at the very beginning of our problem, that design thinking asks us to approach. When we ideate—and that’s very similar to the disaggregation, prioritization, and work-planning steps—we do precisely the same thing, and often we use contrasting teams, so that we do have divergent thinking. The best teams allow divergent thinking to bump them off whatever their initial biases in problem solving are. For me, design thinking gives us a constant reminder of creativity, empathy, and the tactile nature of problem solving, but it’s absolutely complementary, not alternative.

Simon London: I think, in a world of cross-functional teams, an interesting question is do people with design-thinking backgrounds really work well together with classical problem solvers? How do you make that chemistry happen?

Hugo Sarrazin: Yeah, it is not easy when people have spent an enormous amount of time seeped in design thinking or user-centric design, whichever word you want to use. If the person who’s applying classic problem-solving methodology is very rigid and mechanical in the way they’re doing it, there could be an enormous amount of tension. If there’s not clarity in the role and not clarity in the process, I think having the two together can be, sometimes, problematic.

The second thing that happens often is that the artifacts the two methodologies try to gravitate toward can be different. Classic problem solving often gravitates toward a model; design thinking migrates toward a prototype. Rather than writing a big deck with all my supporting evidence, they’ll bring an example, a thing, and that feels different. Then you spend your time differently to achieve those two end products, so that’s another source of friction.

Now, I still think it can be an incredibly powerful thing to have the two—if there are the right people with the right mind-set, if there is a team that is explicit about the roles, if we’re clear about the kind of outcomes we are attempting to bring forward. There’s an enormous amount of collaborativeness and respect.

Simon London: But they have to respect each other’s methodology and be prepared to flex, maybe, a little bit, in how this process is going to work.

Hugo Sarrazin: Absolutely.

Simon London: The other area where, it strikes me, there could be a little bit of a different sort of friction is this whole concept of the day-one answer, which is what we were just talking about in classical problem solving. Now, you know that this is probably not going to be your final answer, but that’s how you begin to structure the problem. Whereas I would imagine your design thinkers—no, they’re going off to do their ethnographic research and get out into the field, potentially for a long time, before they come back with at least an initial hypothesis.

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Hugo Sarrazin: That is a great callout, and that’s another difference. Designers typically will like to soak into the situation and avoid converging too quickly. There’s optionality and exploring different options. There’s a strong belief that keeps the solution space wide enough that you can come up with more radical ideas. If there’s a large design team or many designers on the team, and you come on Friday and say, “What’s our week-one answer?” they’re going to struggle. They’re not going to be comfortable, naturally, to give that answer. It doesn’t mean they don’t have an answer; it’s just not where they are in their thinking process.

Simon London: I think we are, sadly, out of time for today. But Charles and Hugo, thank you so much.

Charles Conn: It was a pleasure to be here, Simon.

Hugo Sarrazin: It was a pleasure. Thank you.

Simon London: And thanks, as always, to you, our listeners, for tuning into this episode of the McKinsey Podcast . If you want to learn more about problem solving, you can find the book, Bulletproof Problem Solving: The One Skill That Changes Everything , online or order it through your local bookstore. To learn more about McKinsey, you can of course find us at McKinsey.com.

Charles Conn is CEO of Oxford Sciences Innovation and an alumnus of McKinsey’s Sydney office. Hugo Sarrazin is a senior partner in the Silicon Valley office, where Simon London, a member of McKinsey Publishing, is also based.

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Framing a Conceptual Problem

At-home/in-class exercise.

This handout (inspired by the Little Red Schoolhouse  approach ) explains how to frame a conceptual problem in a paper’s introduction. Students may use this handout to consider the discrete rhetorical moves an introduction involves, especially when creating research problems of their own in WR 15x. 

To help students reflect on the key elements of framing a problem engagingly in order to motivate readers to care about what they’re arguing; to review together an example (in this case, drawn loosely from conversations about the purpose and identity of higher education) and prompt students to reflect on how this example might connect to their own projects.

introduction; problem; question

Example/Analysis

Step 1: Begin by establishing an element of common ground for your readers.

• Example:  Contemporary high schools emphasize college attendance as the goal for most students. They demonstrate this emphasis through the development of an increasing number of college prep programs and through the consideration of statistics on how many graduates go on to college as a metric for a high school’s “success.”
• Analysis: This statement serves as common ground because it is likely that very few people would disagree with this statement, especially not the parents, students, or educators who might make up our audience.

Step 2: Continue by offering a problem or complication of which readers may may not already be aware. This problem serves as a destabilizing moment, making readers no longer certain about the common ground.

• Example:  But, in 2006, the Wall Street Journal   published an article that suggested that “skilled [manual] labor is becoming one of the sure paths to a good living.”
• Analysis:  This new piece of information complicates what our audience understands about the contemporary experience of education leading to work.

Step 3: To continue to convince our readers that our conceptual problem is important, we must present the potential consequences if this problem is not resolved, or the rationale for why this problem matters.

• Example: If we don’t further examine this conflict, our nation’s educational system could be preparing a generation of students for jobs that will  be extremely scarce when these students enter the workforce.
• Analysis:  This explanation of the potential consequences gives our audience a chance to see why it is vital that we explore the problem, and how the problem might relate to themselves or students they know.

Step 4: Lastly, we need to propose a solution that demonstrates that there is still something that can be done to forestall the potential consequences–a potential claim, in other words.

• Example : In addition to rigorous college prep, high schools must   reinvigorate trade studies to better prepare students for a wider variety of employment possibilities.
• Analysis:  Proposing a possible solution completes our problem by offering one way to solve it. The argument now has a direction from our perspective, but it still has room for others to propose their own solutions.

What are the steps for solving conceptual problems?

List and describe the two steps for solving conceptual problems. Analyze and Solve. Analust the conceptual problem and then solve it.

How do you solve a conceptual physics problem?

• Focus on the Problem. Establish a clear mental image of the problem. A.
• Describe the Physics. Refine and quantify your mental image of the problem. A.
• Plan a Solution. Turn the concepts into math. A.
• Execute the Plan. This is the easiest step – it’s just the algebra/calculus/etc. A.
• Evaluate the Answer. Be skeptical.

What is conceptual problem solving?

This study describes the development and evaluation of an instructional approach called Conceptual Problem Solving (CPS) which guides students to identify principles, justify their use, and plan their solution in writing before solving a problem.

What is an example of a conceptual problem?

When we are starting out, what we really have to worry about are conceptual problems, for example: “This population of people live in a dense, urban setting where everything they would possibly need is within walking distance.”

What problems can be solved using physics?

• Global warming. Global warming is a real threat that needs no further explanation.
• Food production.
• Medical applications.
• Energy crisis.

What are the four steps for solving physics problems?

There may be more than one way to solve the problem so group the equations by the type of possible solution. Solve the equation(s). Solve algebraically for the unknown(s). Substitute known values into the solved equation.

What are the five steps to solving a physics problem?

The strategy we would like you to learn has five major steps: Focus the Problem, Physics Description, Plan a Solution, Execute the Plan, and Evaluate the Solution. Let’s take a detailed look at each of these steps and then do an sample problem following the strategy.

What is an example of conceptual thinking?

Conceptual thinking means that when a new project lands on your plate, you’re not one to roll up your sleeves and jump into tasks or start delegating responsibilities. You prefer to step back and conceptualize or theorize the project before getting into action.

What are some examples of conceptual skills?

• Able to ignore extraneous information.
• Broad thinking.
• Critical thinking.
• Breaking down a project into manageable pieces.
• Decision making.
• Executing solutions.
• Formulating effective courses of action.

What is a conceptual question in physics?

Physics concept questions or concept checking questions are questions prepared to examine learners’ understanding of core physics topics. Asking questions is one of the fundamental ways of understanding any physics topic. If students can answer all such questions, they have understood the concept well.

What is the difference between practical and conceptual problems?

The main difference is that the practical claim explains the causes of the problem to later propose a feasible solution to the problem. However, the conceptual claim focuses on understanding the situation and decoding it in some way to use as information.

What are conceptual answers?

Conceptual questions or conceptual problems in science, technology, engineering, and mathematics (STEM) education are questions that can be answered based only on the knowledge of relevant concepts, rather than performing extensive calculations.

What is the last thing you should do when solving a problem physics?

Answer and Explanation: The last thing that we do is rechecking of the answer, our answer should be correct and full fill all the requirements. Also, at last, recheck the unit and if there is not the unit, then provide the sign for the answer, checking all these things, at last, improve the accuracy of the answer.

How is physics used in everyday life?

• Alarm Clock. Physics gets involved in your daily life right after you wake up in the morning.
• Steam Iron.
• Ball Point Pen.
• Headphones/Earphones.
• Car Seat-Belts.
• Camera Lens.
• Cell Phones.

What is the importance of physics in our daily life?

Physics improves our quality of life by providing the basic understanding necessary for developing new instrumentation and techniques for medical applications, such as computer tomography, magnetic resonance imaging, positron emission tomography, ultrasonic imaging, and laser surgery.

Who is the father of problem solving method?

George Polya, known as the father of modern problem solving, did extensive studies and wrote numerous mathematical papers and three books about problem solving.

How do you think logically in physics?

The best way to deal with this is to “start with the basics” of any subject you are studying. In physics, go back to main principles. Acceleration is velocity/time because acceleration is the rate at which velocity changes. Just like that, take a basic principle that you do understand and move forward from there.

Is physics easy or hard?

Students and researchers alike have long understood that physics is challenging. But only now have scientists managed to prove it. It turns out that one of the most common goals in physics—finding an equation that describes how a system changes over time—is defined as “hard” by computer theory.

How many steps of problem solving are there?

All six steps are followed in order – as a cycle, beginning with “1. Identify the Problem.” Each step must be completed before moving on to the next step. redefine the problem.

How can I improve my physics?

• Master the Basics.
• Learn How to Basic Equations Came About.
• Always Account For Small Details.
• Work on Improving Your Math Skills.
• Simplify the Situations.
• Use Drawings.
• Always Double-Check Your Answers.
• Use Every Source of Physics Help Available.

How do you solve NEET physics Numericals?

• Study and practice Physics every day.
• Don’t miss your classes and make class notes.
• Read/ Preview the topic before the class.
• Revise everything after the class.
• Follow NEET study material to understand concepts well.
• Solve problems from NCERT and coaching modules.

What are the 7 steps of problem-solving?

• 7 Steps for Effective Problem Solving.
• Step 1: Identifying the Problem.
• Step 2: Defining Goals.
• Step 3: Brainstorming.
• Step 4: Assessing Alternatives.
• Step 5: Choosing the Solution.
• Step 6: Active Execution of the Chosen Solution.
• Step 7: Evaluation.

What are the three problem-solving techniques?

• Trial and Error.
• Difference Reduction.
• Means-End Analysis.
• Working Backwards.

What are the 3 steps of problem-solving?

Stop 1: Problem (Define the problems in the case.) Stop 2: Cause of the Problem (Identify the OB concepts or theories to use to solve the problem.) Stop 3: Recommendation (Explain what you would do to correct the situation.)

What is conceptual example?

Conceptual definition An example of conceptual is when you formulate an abstract philosophy to explain the world which cannot be proven or seen. Of, or relating to concepts or mental conception; existing in the imagination. We defined a conceptual model before designing the real thing.

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How to solve ai’s roi problem.

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Chief Product Officer at Nasuni .

We're 18 months into the generative AI (GenAI) revolution and we still haven't begun the inexorable slide down from peak expectations. Every time the GenAI movement appears to plateau, a new model with staggering capabilities such as OpenAI's Sora is released, reinvigorating the hype.

Yet, enterprises are beginning to question exactly how and when they'll start to see the ROI in AI technologies. This question is coming up more often in my conversations with executives. The cost of both publicly available and privately maintained AI solutions is proving to be higher than anticipated. Token, GPU and energy costs are high as demand exceeds supply.

As executives, we want the time, cash and human capital sunk into these technologies to produce measurable results. The problems is that with AI in the enterprise still being an immature technology, it isn't clear what the correct use cases are or what the results should be. Even though we're at or close to the peak of GenAI hype, it's still too early to measure AI initiatives in terms of the hard ROI benefits they're providing to the enterprise.

This isn't a subversive attempt on my part to extend the runway; my company isn't an AI solution provider. But we're deeply entrenched in the business of data, so we're always engaged in conversations about how our customers can use their datasets to see AI results.

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Nvidia are splitting 10 for 1 here s what it means and how to profit, massive dota 2 7 36 patch notes add innate abilities and facets.

When considering an AI project, the first question that should be asked is: Is this really a use case that needs or can leverage AI? Not all AI use cases are necessarily going to generate hard ROI (i.e., cost saving or revenue increases). The key is to focus on soft ROI for initial AI projects, as this will steer your organization toward a sustainable, hard-ROI-oriented deployment of GenAI in the years ahead. Why? Let's discuss.

You'll get more out of AI.

People often make the mistake of looking at AI as a like-for-like replacement for an existing role. I'd advise against this. The introduction of large language models (LLMs) doesn't signal the end of the technical writing industry. Sora doesn't mean filmmakers will never work again. Code-generating models haven't decimated the programming world, either. The like-for-like approach suggests that it's possible to trade an LLM for a certain number of employees performing a given job or task, but AI is far more efficient when used to augment our capabilities as people. Instead of looking at which roles within your organization you might replace with AI, consider where and how these tools might be able to help your people work more productively.

AI has hard-to-quantify benefits.

The 2024 Artificial Intelligence Index from Stanford University reports that AI helps workers accomplish tasks faster and produce higher-quality work. This isn't as easy to measure, but it's the right way to initially think about putting AI technologies to work within your organization.

There are additional advantages to GenAI that you simply can't quantify. Creatives can use LLMs as a sounding board for innovative ideas or digital brainstorming partners. Salespeople or technical engineers who need to generate an email or slide deck can enlist the help of an AI tool to produce a first draft. An LLM integrated with a retrieval-augmented generation (RAG) tool can help your employees search and access internal company knowledge faster. Translating these types of benefits into hard results is almost impossible. Yet, the tools are directly impacting the productivity of your employees.

You'll ease the cultural shift.

According to the results of a 2023 Pew Research Center study , "52% of Americans are more concerned than excited about AI," up from 38% the prior year. Yet, a 2023 PwC survey of about 54,000 global workers revealed that nearly one-third of employees believe AI will help them increase their productivity. More than one-fifth reported believing that AI will create new job opportunities. There's indubitably a real divide in the world.

By emphasizing the augmentative aspects of AI, you can counter the fear and uncertainty. You'll be equipping your employees with the tools to make them better at their jobs, not attempting to replace them with AI solutions. Educating and advertising this shift internally will be critical. You'll need to demonstrate how AI is helping people within your organization do their job at a higher level.

You can improve the customer experience.

Any effective executive understands the value of listening to your customers. Deeply engaged and passionate customers can help you steer the evolution of your product or service. A better customer experience can't necessarily be measured in a hard ROI sense, but as you integrate AI tools to gather more intelligence regarding customer sentiment and preferences, you can really grow and sharpen your understanding of what your customers want and need. This kind of information will not only help you improve the customer experience but also shape the development and quality of your products and services.

You'll learn more about AI.

Finally, by focusing on soft ROI benefits, you can gain valuable AI experience and start to understand how and where AI solutions could be best deployed within your organization. Working with your employees, you can pinpoint the tasks within your workflow that might be better suited to AI-led automation. Then, you can identify viable hard ROI use cases. In manufacturing, automation can drive a more efficient production cycle, increase yields and reduce time to market when a single repetitive task is turned over to a machine. We should expect similar patterns in the office to what we've witnessed in the factory as specific tasks are replaced or augmented—not an entire department or role—and the organization becomes more efficient and productive.

This is the sort of ROI we should all be aiming for as enterprise executives and technology providers: more productive and efficient users contributing to a stronger organization by leveraging data and AI to automate routine tasks so individuals can focus on higher-value contributions.

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Plans to spend billions on a flood-prone East Texas highway may not solve the problem

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LUFKIN, Texas (AP) — U.S. Highway 59, a major evacuation route from Houston, has been a problem for East Texas for decades. And as flooding rivaling that of Hurricane Harvey inundated the region in April, the highway closed in several places, cutting off a major evacuation route for countless people seeking shelter from the floods.

Plans to upgrade the highway , which stretches more than 600 miles through Texas from Laredo to Texarkana, to interstate standards have been on the books for decades. But the Texas Department of Transportation says it cannot guarantee that the billions of dollars being poured into the project will fix the flooding problem.

“U.S. 59 was one of the issues during (hurricanes) Rita, Ike and Katrina,” Polk County Judge Sydney Murphy said. “So you think by now we would be committed to expanding that roadway.”

Texas has poured millions of dollars over the past 30 years into upgrading parts of the highway to interstate standards — an effort known as the I-69 project — with the goal of relieving traffic congestion, supporting economic development, improving safety for travelers and upgrading a major evacuation route for the state’s most populous city.

Thus far, only the part of U.S. 59 that runs through Houston has been upgraded to interstate standards, with a minimum of two travel lanes in each direction, 12-foot lane widths and paved shoulders of a specific width on both sides.

Steps to upgrade the highway in other areas have focused on larger population centers, such as Lufkin and Nacogdoches.

Portions of U.S. 59 between Cleveland and Shepherd as well as between Shepherd and Livingston saw significant flooding in April. Those stretches of highway were closed multiple times between April 29 and May 4 — then again when more heavy rain came the weekend of May 16 — and are supposed to receive upgrades in the next four years.

Those sections are part of nearly $6 billion the state plans to pour into the highway over the next decade or more to upgrade the highway to interstate standards, address safety issues and cover basic maintenance. TxDOT says it has allocated $1.5 billion for projects already underway or that begin soon on U.S. 59. The agency has another $4.3 billion allocated for future projects scheduled to begin in the next four to 10 years.

But it’s unclear whether those upgrades will prevent the kind of flooding that submerged parts of the highway this spring and during Hurricane Rita in 2005 and Hurricane Ike in 2008.

TxDOT said the upgraded highway will be engineered to avoid flooding during a 100-year flood event. However, 100-year floods — which have a 1% chance of happening in any given year — have become more commonplace, as have 500-year floods — which are more severe and have a .2% chance of occurring in any given year.

For example, Hurricane Harvey in 2017 was the third 500-year flood to hit Houston in three years. Memorial Day floods in 2015 and 2016 were also classified as 500-year floods.

John Nielsen-Gammon , the state’s climatologist at Texas A&M University, warned that floods are becoming more extreme in Texas.

“East Texas in general has experienced a large increase in extreme rainfall compared to last century,” Nielsen-Gammon said. “Part of that is due to climate change. Climate change has increased the intensity of very heavy rainfall across the southern U.S. by nearly 20%.”

TxDOT would not say whether current improvement plans take into account warnings from climatologists of even more severe flooding to come due to climate change.

“The projects being developed along the future I-69 corridor are designed to be serviceable for a 100-year flood event, however TxDOT cannot predict the amount of rain or potential flooding our lakes, rivers and streams could see in the future,” said Rhonda Oaks, the public information officer for TxDOT’s Lufkin District, where plans are currently underway to upgrade around a dozen miles of U.S. 59 to interstate standards.

Laura Butterbrodt, another TxDOT spokesperson, said the agency is currently developing the Statewide Resiliency Plan , “which will specifically target critical routes for the most appropriate design, maintenance and operations to foster resilience.”

The first draft will be available for review by the Resilience Steering Committee in June.

When the federal government authorized building 41,000 miles of interstate highways crisscrossing the nation in the 1950s, the federal government paid 90% of the cost, leaving the remaining 10% to the states.

But the I-69 project was not included in the original plans and didn’t receive federal designation until the early 2000s, state Sen. Robert Nichols said. When completed, the interstate will stretch more than 2,600 miles across multiple states from the Texas-Mexico border to the Michigan-Canada border.

But each state along the proposed interstate highway is expected to cover the cost — not the federal government.

“At present, there is no dedicated federal funding for the entire conversion of U.S. 59 to a future I-69 route through Texas,” the Federal Highway Administration said in an email statement. “It is up to the State (Texas Department of Transportation), to move projects forward.”

This story was originally published by The Texas Tribune and distributed through a partnership with The Associated Press.