In this paper, we describe a new computer-based approach to learning about evolution. In the EACH project (The Evolution of Altruistic and Cooperative Habits) (Centola et al, 2000), students learn about basic processes of evolution through interacting with multi-agent models. Two multi-agent modeling languages, StarLogoT (Wilensky, 1997a) and NetLogo (Wilensky, 1999), serve as the computing substrates for building and running the evolutionary models.

The prevailing approach to teaching evolution in secondary and early post-secondary education is intended to lay the groundwork for students to understand the basic concepts of fitness, natural selection, and evolutionary success. In practice, however, what students retain from this instruction is largely the phrase "survival of the fittest" (Brumby 1984; Centola et al, 2000). Many students who have taken introductory high school and college courses in evolution come away from their studies understanding natural selection in terms of strong individuals dominating over weak individuals. Even students who participate in curricula that do not explicitly teach this view of natural selection will often come out of instruction with these simplified understandings. (Brumby, 1984; Greene, 1990) Recent work suggests that students’ misconceptions about how the evolutionary process works can often be traced beyond the students’ coursework to fundamental intuitions about how natural systems behave (Bishop & Anderson, 1990). Because most students have not had many opportunities to interact reflectively with complex systems, they have not developed the intuitive knowledge that would help them make sense of evolutionary processes. A large body of research has reported the difficulties students have in making intuitive sense of random variation, probabilistic runs, decentralized control — all sub-component processes of natural selection. (Tversky & Kahneman, 1974; Konold, 1991; Resnick, 1996; Wilensky, 1997b; Wilensky & Resnick, 1999). Without further developing and challenging these basic intuitions about how natural systems work, students are unable to internalize an understanding of many of the important advances in evolutionary theory (Greene, 1990).

One important aspect of evolutionary theory that is difficult for introductory students to understand is the multi-level perspective(Sober & Wilson, 1998; Mitteldorf & Wilson, in press; Wilensky & Resnick, 1999). While a complex systems approach to evolution encourages students to think about concepts such as fitness and natural selection from a number of perspectives, students given traditional instruction try to pin down these concepts into singular meanings. In particular, students’ understanding of fitness is often oversimplified by understanding it as an individual’s strength in one-on-one competition for resources (Bishop & Anderson, 1990). Because of this narrow conception of fitness, many interesting new results in evolutionary biology (Axelrod, 1984; Wilson & Sober, 1998; Mitteldorf & Wilson, in press), such as the plausibility of altruistic and cooperative habits being evolutionarily advantageous, are conceptually intractable to students of evolutionary theory (Jacobson, 1996). The EACH project was established to enable students to develop more sophisticated intuitions about evolutionary processes by interacting with multi-agent models of evolutionary systems. The primary focus of the EACH project is helping students to think about the complex dynamics of evolution that would allow for altruistic and cooperative habits to be evolutionarily successful. One important goal of this research is to help students to think more critically about their assumptions about fitness, and how these assumptions play into their expectations for evolutionary scenarios. Specifically, we want students to see the significant differences between understanding fitness at the individual-level , at the gene-level, and at the group level, and how understanding fitness at multiple levels can dramatically change their conceptions of other evolutionary processes, such as the evolution of altruism.

The EACH project is one of several curricular interventions undertaken as part of two NSF funded research projects. The projects, "Making Sense of Complex Phenomena (MSCP)" (Wilensky, 1997).and "Participatory Simulations" (PS) (Wilensky & Stroup, 1999) share the goal of building computer-based tools and curricula to enable students to explore and make sense of complex systems and to study student sense-making using these tools. As part of the MSCP project, we have developed the multi-agent modeling language StarLogoT and as part of the PS project, we have developed NetLogo (Wilensky, 1999). NetLogo is a multi-platform modeling language that is a component of the HubNet (Wilensky & Stroup, 1999) participatory simulation architecture. All activities described herein are written in one of these two multi-agent modeling languages. Both StarLogoT and NetLogo are designed explicitly for exploring systems with multiple interacting "agents". They are instances of a new class of multi-agent (AKA object-based parallel) modeling languages.. Using these languages, students can represent many different types of "agents," such as flashing fireflies, cars in traffic or molecules in a gas They can then build models of the behavior and interactions of thousands of such individual agents. The original EACH models were developed in StarLogoT. More recent project models (including ,many described here) have been developed in NetLogo. We’ve moved to NetLogo because its interaction with HubNet opens new research avenues that will eventually incorporate interactive agents and multi-user scenarios; these possibilities will be discussed briefly in the concluding section.

The latest version of StarLogoT and an associated collection of "extensible" models (collectively entitled "Connected Models" (Wilensky, 1998)) are available for download at ccl.northwestern.edu/cm. The models are drawn from a wide range of disciplines including physics, biology, mathematics, computer science, chemistry, materials science, ecology and economics. Similarly, the latest version of NetLogo and associated models can be downloaded from ccl.northwestern.edu/ps.

The EACH project emerged out of the larger scale MSCP project (located at Tufts University’s Center for Connected Learning and Computer-Based Modeling (CCL). The project has continued its work under the umbrella of the Participatory Simulations (PS) project. As part of the MSCP project, we studied students at many different grade levels as they explored and created models of complex phenomena in the language StarLogoT. Students (from middle school to graduate school) came to the CCL and met informally with project staff who mentored them in learning StarLogoT. Students explored models from the project library according to their interests. They were encouraged to vary and extend the underlying StarLogoT code for the project sample models and, when they felt ready, to create new models from "scratch". The topic of evolution, in general, is one that has appealed to many of the student modelers. For the high school students and many of the undergraduates as well, evolution was studied largely in "story" form. That is, they learned stories of how the theory of evolution came into being, how it was resisted and how it is supposed to work. Many of the students found these stories intriguing but unsatisfying. The dissatisfaction consisted of feeling that they did not have an adequate methodology for testing the plausibility of particular evolutionary arguments or for evaluating competing evolutionary claims,

The EACH project is an on-going endeavor to teach students about complexity in evolution. The project began with a focus on getting students to explore models wherein altruism, cooperation, and group behaviors could survive under Darwinian selection. The motivation behind the project was to expand students’ notion of fitness and to get them to take into account the complex field of factors that play into the evolutionary process. The present research concentrates on students’ conceptions of fitness, and is directed at helping them to develop intuitions about evolution as a multi-level process. More recent EACH project research, not included in this paper, addresses student conceptions of randomness in natural selection, and focuses on fostering students’ intuitions about genetic variation in populations and the mechanism of genetic drift..

The students participating in the present study were undergraduates at Tufts University. These four volunteers were drawn from the sciences and social sciences departments. Beverly is a junior biology major; she came to the EACH interested in getting a ‘complex systems’ understanding of evolution. David is a junior biology major who came to the EACH project because he wanted to understand how altruism could be an evolutionary advantage. Amy is a sophomore sociology major who wanted a better understanding of how the social and group dynamics of populations could evolve. Derek is a freshman chemistry major who came to the EACH project because of his interest in combining his interests in evolutionary biology with his interests in mathematics. They had all studied evolutionary theory in either a high school or university class. They came to the study in response to an advertisement offering the opportunity to interact with computer models and simulations that could result in improved understanding of evolution

We began each session by asking the students to discuss their ideas on evolutionary theory. All of the students expressed common intuitions about individual selection, and at some point in the discussion mentioned "survival of the fittest." But when asked to flesh out the idea of fitness, the students had difficulty articulating what ‘fitness’ meant to them beyond the popular notion that fitness was strength: the power of an individual to dominate other individuals. These pre-session interviews gave useful guidance to the research team on how to scaffold student interaction with the models. They also served to set the ground for the project intervention as students started to question their assumptions about the meaning of "fitness" — questions which were elaborated and discussed once they began to interact with the models.

As part of the EACH project, we have developed a series of models that enable students to interact with certain evolutionary scenarios in order to discover what outcomes are plausible, how their assumptions about fitness play into the their expectations, and how different environmental and social considerations can affect the fitness of different populations. These models serve as seeds for developing intuitions about evolutionary processes. Students run these models exploring their behavior under a variety of conditions. They then modify and extend the models to further refine their thinking. Ultimately, our hope is that most students participating in the EACH project will be inspired to develop and discuss their own models of evolutionary scenarios, thereby developing a self-sustaining means for testing their intuitions about evolution, and gaining an increasingly rich understanding of the complexity of the evolutionary process.

The EACH project methodology is structured around the basic set of "seed" models. The goal of these models is to raise basic questions about fitness, altruism, and group dynamics in natural selection that will encourage students to explore these issues further. The following activities are based on a combination of our "seed" models and a set of models that were developed by students while engaged in the EACH project. The original "seed" models, the Altruism models, were developed during the early stages of the project when the focus of our research was directed more specifically toward helping students to understand how the evolution of altruism is possible. During an early session, we collaborated with a student-learner, who was inspired by the Altruism models to find a more behavioral account of how altruism could evolve, to develop a model called the Cooperation model (Centola et al, 2000). The Cooperation model gave a more embodied feel to the issues surrounding cost, benefit, and altruistic behavior. The Cooperation model shows how an altruistic behavior that would be selected against under normal conditions, could survive under conditions of high population viscosity. Because the Cooperation model is closer to student experience than the original Altruism model, we decided to adapt a version of it to use in our current research on students’ understanding of fitness and how it relates to their understanding of complexity in evolution.

In the research described in this paper, we used two sets of models, the Altruism set and the Cooperation set, as complementary ways of exploring the issues of fitness and complexity in evolution. The activities are organized in two parts. We first presented the Altruism models in order, letting students become familiar with the first model, and its unsurprising consequences, and then gave them the second, more problematic model. After some discussion, we gave them the first Cooperation model, and let them explore the parameter space. Once they became comfortable with this model, we asked them to apply the lessons from the Altruism model to the case of the Cooperation model. Over the course of their involvement with the EACH project, students made surprising advances in their understanding of group behavior and in their conceptions of the differences between individual, group, and gene-level fitness. They came away from the activities with a more critical sense of how distinctions between individual and gene-level fitness determine how they understand group-level phenomena such as the evolution of altruism. Most importantly, they began to see that group behaviors are phenomena that need to be considered from multiple levels of interpretation in order to understand how they can function in evolution.

The first model used in the EACH project was a conservative model of evolutionary behavior based on current theory in evolutionary biology (Hamilton, 1964; Mitteldorf & Wilson, in press). This first model has two types of agents: selfish and altruistic. The premise of the model was that each agent would look around to its neighbors and see whether they were altruistic or selfish. For each altruistic neighbor, the agent increased its fitness by a fixed value. Thus, each agent with altruistic neighbors would have a higher fitness in the reproductive lottery. Each agent then calculated its fitness with one of the following equations:

If I am a selfish individual, my fitness is 1 + my benefit from my altruistic neighbors.

If I am an altruistic individual, my fitness is 1 — the cost of being an altruist + my benefit from my altruistic neighbors.

The values of ‘benefit’ and ‘cost’ were slider-variables that were set at values such that the cost of being an altruist significantly weakened the fitness value of altruists, while the benefit from the neighboring altruists were only significant when there were two or more of them.

Each agent is at the center of a five-agent neighborhood. To enter the reproductive lottery, each agent looks at its neighbors (in the four cardinal directions) and assesses their fitnesses. All of the altruistic agents’ fitnesses are summed (including the central agent, if it is an altruistic agent), and all of the selfish agents’ fitnesses are summed (including the central agent, if it is a selfish agent). These sums constitute the weight of altruistic ‘seeding,’ and the weight of selfish ‘seeding’, respectively, for the center spot in the reproductive lottery. To complete a generational cycle, a random number is picked between 0 and the total of the weights for a spot. The spot is then given to the type of agent whose weighted chance fell on the side of the random number. The higher an agent-type’s weight, the greater the chance that that type of agent will win the spot.

We included control parameters in the model, so that students could set the relative density of the altruistic and selfish populations. The clear outcome of this initial model is that the fitness of the selfish agents is higher than the fitness of the altruistic agents. Depending on the relative population densities, after about two-hundred generations, all of the altruistic agents are extinct; the upper bound on this time, starting with almost no selfish agents, is about five-hundred generations before the altruists are extinct. The outcome of this model fits well with standard adaptationist expectations of evolutionary theory (Hamilton, Williams, 1992) and sociobiology.

Figure 1a below is the control interface for the StarLogoT Altruism-Model 1. Figure 1b shows the model when it starts running. The pink agents are altruistic and the green are selfish. Figure 1c shows the model after it has run for about 100 generations (about thirty seconds running on a typical project iMac computer).

Fig. 1a

Fig. 1b

Fig. 1c

The results of running this model were consistent with the students’ expectations for evolutionary theory. All of the students predicted that the selfish agents would win. Clearly, the students noted, the selfish agents were the stronger agents, and clearly they had a higher individual fitness. After running this model a few times, students felt comfortable with the parameters, and were confident that they understood how the evolutionary scenario was playing out. By allowing the students to develop a feel for the modeling environment by running a model with a familiar outcome, we hoped to encourage them to develop explanations about how and why the scenario was playing out as it was.

When we asked him why the selfish agents had won, one student, Derek, a freshman chemistry major, said, "the selfish agents are stronger because they have a higher fitness value." But, when we asked Derek to expand on the idea of having a higher fitness value, he said "Having a higher fitness is being strong enough to get more food, which makes it more likely to reproduce." This was a typical response; the students viewed fitness as an individual’s strength in competition for resources. Another student, David, a junior in biology, said, "fitness is the strength of a genotype to out-compete other individuals." David’s notion of fitness demonstrated a conflict between his knowledge that gene-level phenomena exist, and his intuition that fitness must be a property of individuals.

After getting the students’ reactions to the first model, we introduced them to the second model. The second model, based on recent work in the evolutionary biology of cooperation and altruistic behavior (Mitteldorf & Wilson, in press), introduces a new element to the model: adversity. The second model adds a slider-variable called ‘Harshness,’ and a slider variable called ‘Disease.’ The harshness variable gives each empty "patch" (a unit of the screen that is not occupied by a selfish agent or an altruistic agent) a chance of staying empty (resisting population) each turn. Envisioning the model-world as an environment in which individuals need to occupy a space on the grid in order to live, the harshness variable limits population growth by, at each clock "tick" making some of the spaces uninhabitable The disease variable is incorporated into the reproductive lottery for each spot. The value of this variable corresponds to the chance that an agent that occupies a spot on the grid will fail to reproduce, and thus that the spot will become empty. The introduction of these elements into the model alters the relationship between individuals and their environment. The new threats to the well-being of individuals reframe the importance of group behavior on individual success. This second model has a much richer parameter space -- students can explore the effects of the various harshness and disease values on the stability of the altruistic and selfish populations.

Figure 2a below is the control interface for the Altruism-Model 2. Figure 2b shows the model after thirty generations. Figure 2c shows the model after about two-hundred-fifty generations.

 

Fig. 2a

Fig. 2b

Fib. 2c

The Altruism Model 2 in StarLogoT

Students found that when the harshness value is set around .96, and the disease value is set around .2, the model shows the surprising result that the altruistic population fares far better than the selfish population. Why should this be? This counter-intuitive result comes from the fact that when the natural conditions are harsh enough, single agents cannot survive against nature. That is to say, even though the selfish agents would fare better in competition against the altruistic agents, the altruistic agents would fare better against nature than the selfish agents. The reason for this is that altruistic agents survive in groups of altruists. Each altruist adds to the fitness of the other altruists. While a lone altruistic agent can survive no better than a lone selfish agent, because the altruists contribute to the well-being of the agents around them, the community can survive, with its combined altruistic benefit, in the face of harsh conditions.

The students who participated in the project were all very surprised by this result. After running the model a few times, students began to ask questions about how the altruists were winning. It was clear from the outset that the selfish agents were the stronger, or more fit, agents, and yet under harsh conditions the altruists were running the selfish agents to extinction. We asked students to describe what was happening on the screen, and how it was different from the first model. They all recognized that, unlike the first model, the altruists were grouping together.

After discussing this for a few minutes, one student, Beverly, a junior biology major said, "Well, it means that all of the altruistic benefit is going only to altruists; which means that under harsh conditions, either you are an altruist, or you’re on your own." She suggested that altruists were in fact more fit in the second model, but only as a group. We discussed her earlier comments about individual fitness, and asked her how she could make sense of the fact that this group advantage was at odds with her previous focus on individual strength. After some group discussion about how it was possible to have multiple levels of fitness, Beverly resolved this apparent conflict, "so the altruistic gene is more fit under harsh conditions, even though the individuals have a low fitness, because the harshness allows the altruists to clump together, and benefit only one another."

Most of the students made similar inferences about the relevance of grouping to gene-level fitness. Many of them remarked that the selfish agents were still more ‘fit’, even though the altruists could survive better. But, they were puzzled about this apparent equivocation in the term ‘fitness.’ We encouraged them to address this confusion by concluding this part of the sessions with a discussion of the idea of the "survival of the fittest", and what that meant. Amy, a sophomore sociology major, had earlier been confident that this catch-phrase summed up natural selection as a process that favored the strongest individuals, but was now unsure of how to think of ‘fitness’: "The altruistic group has a higher fitness, as a group, than the selfish group, but I don’t think that they are any stronger as individuals. They [the altruists] are stronger now [under harsh conditions] because being in a group makes gene-level selection matter." Amy was beginning to see that thinking about fitness at the gene-level could change her perspective on group-level selection. Ultimately, she suggested that the phrase "survival of the fittest" was misleading. Instead of "survival of the fittest", Amy and the other students preferred to talk about how a population could survive, or ‘be more fit’ under certain circumstances, due to the benefits that a trait gave to the individuals that carried it. Indeed, most students were talking about traits as having fitness, and individuals as ‘carrying’ this fitness. Another EACH project participant, Derek, suggested that in this case, "survival of the groupiest" was a better phrase to describe the selection process.

After the students had explored the Altruism models, and had reflected on how the complexity of environmental and social factors affect an agent’s fitness, we gave them the first Cooperation model. The Cooperation model consists of two parts: cows and grass. The grass was designed to grow in such a way that above a certain height, it would have a high percentage chance of growing back to its full height each turn, but below a certain height, it would grow back very slowly. The high grass is thus considered to be healthy, or sustainable grass, and the low grass is considered to be unhealthy grass. In terms of the model, these variables can be expressed in terms of the following parameter settings: the maximum grass height is 10; the threshold for healthy grass is 5; the chance for growing back for healthy grass is 75%; and, the chance for unhealthy grass growing back is 30%. These parameters are all slider variables that have been put into the Cooperation model.

There are two types of cows. The first type of cow, the "greedy", eats the grass as far down as possible. The second type of cow, the "cooperative", only eats the grass if it is above the fast-growth threshold. All the cows have the same metabolism, and require a fixed amount of food to live. Each turn they lose a percentage of their energy, and if their energy runs out, they die. Eating grass restores a cow’s energy by a fixed amount. Finally, if a cow’s energy reaches a certain threshold, it reproduces. In terms of the model, these variables can be expressed in terms of the following parameter settings: all the cows get 51 energy units for eating from a patch of grass; they all lose 10 energy units each turn for moving (regardless of how far they move); they all reproduce when their energy level reaches 101; and, they all lose 40 energy units for reproducing. These parameters, too, have been made into slider variables in the Cooperation model.

Thus, all things being equal, the only difference between the greedy cows and the cooperative cows is that the former eat all the grass and over-harvest the land, while the latter will go hungry rather than damage the well-being of the food supply. Supposing that these habits are not intentional states, but manifestations of different genetic traits, we asked students to discuss the scenario, and predict the outcomes of the models.

All of the students said that the evolutionary advantage goes to the cows who can get as much energy as quickly as possible. David said, "The greedy cows will have the clear advantage because they will be able to eat more food." He suggested that the higher an agent’s energy got, the more it would reproduce, and the more it would reproduce the more resources it would use. Another student, Beverly, said, "once the greedy cows begin to reproduce, they will spread out and eat the grass below the level where the cooperative cows can eat." The cooperative cows would be keeping the grass high, so that they and the other cooperative cows would have plenty of food; however, the greedy cows would eat the grass below the healthy threshold, and the cooperative cows would have no food. Amy said, "the cooperative cows will not be able to eat once the greedy cows start to eat the grass; and they [the cooperative cows] will go to extinction."

Below is the NetLogo Cooperation model. Figure 3a is the model when it is starting. The sliders determine the values of the following variables: metabolism, energy from grass, reproduction- cost, level of energy needed to reproduce, the maximum grass height, the healthy grass threshold, the percentage chance to grow back for healthy grass, the percentage chance to grow back for unhealthy grass, and the numbers of the cooperative and greedy populations. Students explored changing these variables, but found the result to be the same in most every scenario where there was a substantial environmental difference between healthy and unhealthy grass. Figure 3b is the Cooperation model after about 30 generations. The greedy cows out-grazed and out-reproduced the cooperative cows; eventually, they ran the cooperative population to extinction.

Fig. 3a

Fig. 3b

In discussing the outcome of this model with the students, we asked them to explore ways in which the cooperative population could possibly survive. Reflecting on their work with the Altruism models, a number of the students suggested that if we could get the cooperative population to group together, then they might be able to survive. We encouraged this line of thought, and asked students to consider how this change would affect the fitness of cooperative individuals. A few students immediately responded: Derek said, "if we got the cooperators to group together, they would all have higher grass to eat, and would all benefit from their cooperative behavior…The cooperative gene would benefit only the cooperative gene, and it would be more fit."

For the second Cooperation model, we introduced the variable of population viscosity into the model. We explained population viscosity as the limitation in a cow’s range of movement due to environmental barriers, and then asked students, individually, to predict what would happen when different values were entered in the Viscosity slider. Many of them suggested that if the Viscosity value was high enough, that the greedy population couldn’t reach the cooperative population, and that the cooperative population would therefore survive. Figure 3c, below, is the second Cooperation model with the Viscosity set at eleven (which effectively limited the cows to one-eleventh their normal travelling distance) after about 300 generations.

Fig. 3c

After running the model, students found that, given the above values for the other variables, at a Viscosity value of around about eleven the greedy population was kept to about a twentieth of the cooperating population. The students were surprised to find that at higher values of Viscosity, the greedy population was so localized that it over-grazed and killed itself off. The students found a number of interesting metaphors for explaining the success of the cooperative population under high population viscosity. But, more interestingly, they all understood that the localization of the populations was key to the success of cooperative behavior because the benefit of cooperation fell on the cooperators alone. Beverly said, "the greedy cows were stronger when they could spread out because it takes a larger portion of land to support them. They are more fit when there is no population viscosity. But, the cooperative cows are more fit when there is no competition with greedy cows, because all of their cooperative benefit goes to other cooperative cows." After a few more minutes of discussion, Derek said, "So, the cooperators succeed because of gene-level fitness; the cooperation gene is benefited by itself." Amy concluded, "When the cooperative gene helps itself, there is no way to beat it."

After their work with the cooperative models, many of the students began, in discussion, to explore important issues in the evolution of altruism and cooperation. The level at which they had envisioned agents being altruistic or cooperative had implicitly assumed individual fitness as the standard for cost and benefit. Their new conceptions of multi-level fitness raised many questions about whether the altruistic habits were really altruistic, or whether they were only genes helping themselves. Of course, these are precisely the questions that occupy contemporary thought in the field, and we were encouraged to see students beginning to understand the different levels at which these problems can be addressed: what is altruistic at the individual level, is selfishness at the gene level.

The focus of the EACH project was to allow students to explore their understanding of fitness and its relation to their conceptions of natural selection and the evolution of altruism. All the students who participated in this study initially believed that altruism could not survive because it was not a benefit to individual fitness. After students interacted with the models, they began to think of fitness as a multi-level phenomenon, and to think of the possible ways in which group and environmental factors can make altruism an advantageous trait. Through experimenting with the models, students came to a new understanding of fitness, and of the role that conceptions of fitness play in their understanding of evolutionarily plausible scenarios.

We encouraged students participating in the EACH project to explore the NetLogo environment, and to build models that would explore their ideas of how different evolutionary scenarios would work. Eventually, we also hope to involve students in participating in on-line interactive simulations of evolutionary systems. Such participatory simulations are a new and emerging technology in which individual students act out the role of individual agents in a simulation. As this technology advances, we expect that participatory simulations will be a powerful new tool for exploring the roles of individual and group-level phenomena in evolutionary systems. Ultimately, we hope to develop on-going evolutionary scenarios in which each user is an agent that develops a survival strategy that is either passed on, if successful, or dies out, if unsuccessful. We anticipate that allowing users to participate in these simulations as first-person agents will add new dimensions of understanding to their conceptions of evolutionary processes and the multiple levels of fitness.

The preparation of this paper was supported by the National Science Foundation (Grants RED-9552950, REC-9632612). The ideas expressed here do not necessarily reflect the positions of the supporting agency. We are indebted to Josh Mitteldorf for the ideas in the first EACH model and for reviewing EACH project models. We thank Dan Dennett for stimulating discussions and critique.

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