Concept-Oriented Task Design: Making Purposeful Case

Article pubs.acs.org/jchemeduc

Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX

Concept-Oriented Task Design: Making Purposeful Case Comparisons in Organic Chemistry Nicole Graulich*,† and Michael Schween*,‡ †

Justus-Liebig University Giessen, Institute of Chemistry Education, Heinrich-Buff Ring 17, 35392 Giessen, Germany Philipps-University Marburg, Faculty of Chemistry, Hans-Meerwein-Straße 4, 35032 Marburg, Germany

‡

S Supporting Information *

ABSTRACT: Acquiring conceptual understanding seems to be one of the main challenges students face when studying organic chemistry. Traditionally, organic chemistry presents an extensive variety of chemical transformations, which often lead students to recall an organic transformation rather than apply conceptual knowledge. Strong surface level focus and rather weak conceptual knowledge is the consequence. Purposefully designed tasks, which help engaging students to “overlook” the structural features, are proposed here as a means to enhance conceptual understanding and the integration of concepts. Following the idea of contrasting cases, broadly used in science education, and mirroring the epistemic practice of organic chemistry, we illustrate how contrasting cases can be embedded in an inquiry process to highlight the influence of electronic substituent effects on reactive intermediates and rates of organic reactions. KEYWORDS: Second-Year Undergraduate, Upper-Division Undergraduate, Organic Chemistry, Inquiry-Based/Discovery Learning, Kinetics, Reactive Intermediates

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INTRODUCTION Representations, as a semiotic resource, are an inherent part of the scientific discourse in a discipline and are “designed specifically to convey the way of knowing of science”.1 Hence, the ability to “read” domain-specific representations and infer potential reactivities and properties is an integral part of a successful scientific inquiry. In chemistry, this ability is not only essential for drawing and inferring chemical or physical properties appropriately from Lewis structures2 but also for recognizing, for example, an acid-labile ketal function in larger molecules, such as in paranolin.3 However, recognizing a functionality and its property is not enough when estimating favored reaction pathways. Making judgements about the likelihood or rate of reaction steps requires considering the factors which influence the kinetics or thermodynamics of a process. The learner needs to go beyond a static perception of a molecule and to reflect on influential and process-related aspects. A large body of research in chemistry education has documented that undergraduate and graduate students have a rather static perspective on organic reactions, perceiving organic chemistry as a huge collection of single cases. They often memorize or reduce chemical transformations to changes on the surface or symbolic level,4 inferring a similar reactivity based on similar surface features.5 Helpful tools, such as the © XXXX American Chemical Society and Division of Chemical Education, Inc.

electron-pushing formalism, are used more for illustrating the way to the desired and often memorized product than as an explanatory or predictive tool.6−9 Furthermore, student conceptual understanding often seems to be limited to the specific structural context or class of substances in which they encounter a concept and what they learned as the exemplary case of a reaction type. The consideration of electronic substituent effects for novices in organic chemistry, for example, is often restricted to electrophilic aromatic substitution reactions. Lewis structures of substituted aromatic reactants, therefore, usually evoke the association of an electrophilic aromatic substitution mechanism, even in cases where no electrophile is present. Changes at the structural level that differ from their familiar reaction context may, subsequently, distract or hinder the application of the same concept. It seems to be characteristic of students’ difficulties in organic chemistry that they manage, on one hand, to readily reproduce possible products for a given reaction and define fundamental concepts. However, on the other hand, they struggle to combine conceptual understanding and knowledge about reaction types across mechanistic problems.10−12 Received: August 31, 2017 Revised: January 8, 2018

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DOI: 10.1021/acs.jchemed.7b00672 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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THE EPISTEMIC PRACTICE OF ORGANIC CHEMISTRY Teaching students about the nature and epistemic practice of a discipline is the major goal in all science disciplines.13 The epistemic practice of a discipline is characterized by how members of a “community propose, justify, evaluate, and legitimize knowledge claims within a disciplinary framework”.14 This practice not only includes the “content” but also the “reasoning”, as an enacted epistemic heuristic of a discipline, i.e. heuristics on how to guide and evaluate processes and knowledge.15 Research in organic chemistry education has documented that the inquiry process of a discipline needs to be reflected in the teaching practice to support students’ transition from a reproductive to a reflective approach.16,17 It is common ground that, traditionally, teaching organic chemistry, however, often tends to present the results of this inquiry process instead of making organic chemistry an experience of the process itself. The discipline’s epistemic practice focuses on the identification and evaluation of transformations occurring between molecules on the potential energy surface. This “energy landscape” of chemical transformations is characterized by alternative pathways leading to different products and influenced by a basic set of principles and concepts, which governs the huge variety of seemingly “different looking” reactions.18,19 One major goal in teaching is to engage students in this epistemic practice and let them actively experience how conceptual reasoning is independent of a given molecular structure or reaction type and how chemical concepts help them to derive hypotheses about reaction trajectories and preferences. Nothing else is more characteristic for organic chemistry than formulating and experimentally testing hypotheses about reaction pathways or optimizing synthesis to favor a certain product selectivity. This elaborated view of the interdependence between conceptual understanding and experimental findings in organic chemistry seems to develop only at advanced graduate levels and is missing in undergraduate students’ views of the discipline.16 The key practice in organic chemistry is to investigate correlations and effects through competitive experiments. As such, the conceptual framework in organic chemistry is constantly evolving and refining itself. The pursuit of elucidating reaction pathways offers plenty of examples how knowledge is gathered in organic chemistry. Well-known chemical concepts, which help explain and predict a huge variety of reactions originate from comparing experimental results of only slightly changed substrates. This becomes obvious when looking at the reaction mechanisms, where (unisolable) reactive intermediates play a central role. Evidence for their existence is inferred from kinetic comparison between concurrent reactions.19 Kinetic and other competitive experiments can be modeled to estimate the strength of an influence or determine a favored pathway through subtle changes in the structure, for example, a change of substituents. Hughes and Ingold’s investigations of aliphatic substitutions led to the differentiation between the SN1 and SN2 reactions, one of the most popular examples of how to differentiate kinetically between reactions.20 Bartlett and Knox tested the hypothesis whether carbocations energetically prefer planary structures and whether the Walden inversion is required for nucleophilic substitution by comparing apocamphanyl chloride, substituted at the bridgehead carbon, and tert-butyl chloride (Figure 1). Their competitive experiment created the need for an explanatory concept and thus led to the legitimization of a

Figure 1. tert-butyl chloride undergoes solvolysis in polar solvents to form the tert-butyl cation, while apocamphanyl chloride does not.

well-known concept, i.e. the required planarity of carbocations.21 Basic thermodynamic competition experiments were carried out by Hammett in the 1930s when he investigated electronic substituent effects on the acidities of m- and p-substituted benzoic acids (Figure 2). Representations, as in Figure 2 can be

Figure 2. Comparison of the acidities of p-substituted benzoic acids (in water). The pKA values depend on the nature of the electronic substituent effects.24

found in many organic chemistry textbooks,22,23 but these examples are often descriptive and not embedded in an explicit inquiry process; that is, they often do not create the need for an explanatory concept. Given these historical accounts on how knowledge is acquired in organic chemistry, one may ask how case comparisons and permutation of reaction parameters, as an inherent practice of the discipline, can be used as a design principle to create purposeful, concept-oriented tasks for organic chemistry classes.

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CONTRASTING CASES Research in cognitive psychology has already shown that when cases are examined sequentially, as it is common when following a functional group approach in organic chemistry, learners tend to focus on surface features and encode cases in a situation-specific manner.25 As a result, they have difficulties in retrieving the relevant concepts, underlying various cases.26,27 One way to support students in noticing conceptual similarities or differences, which they might otherwise overlook, is the approach of discriminative or analogy learning.28−31 A meta-analysis on studies using contrasting cases (CC) in various disciplines has shown that purposefully designed comparisons help the learner discriminate more of the possible variables, which are relevant in a problem situation. Students are engaged in discerning critical, differentiating or common features of two or more cases.32 “Near-miss” contrasts, i.e. CCs differing only in one important feature, have especially been shown to improve schema acquisition and subsequent transfer B



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abilities in science education.29,30,33 Schwartz et al. could show that students who inferred a general explanation of the concept of density across multiple cases gained a deeper understanding than students who worked with these cases separately.34 Variations of the CC approach have been successfully adopted in mathematics by Rittle-Johnson and Star. Comparing alternative problem-solving strategies to mathematic problems was more beneficial for procedural flexibility than evaluating them sequentially.35 The positive effect of CCs has been observed in domains other than science and mathematics education.26 Alfieri et al. stated that instruction with contrasting cases helps improve the conceptual knowledge of the learner and is additionally useful for the acquisition of procedural or problem-solving strategies, making them a powerful tool for education.32 Typically, in organic chemistry, if asked to formulate the reaction of the deprotonation of phenol or cyclohexanol as single instances, a learner might just be engaged in drawing the deprotonation process simply by describing it based on surface changes. When a student is asked to formulate a justified hypothesis about the acidity of both molecules, he or she has to think about what electronic influences affect the acidity. This may support the learner in building conceptual understanding, which is merely activated when regarding single cases. Despite the reported merits of CCs and their effect on the acquisition of more abstract concepts and procedures, remarkably little has been done in organic chemistry education. This kind of instructional setting might, on one hand, be particularly beneficial for students in this discipline, since they often rely on structural similarity and do not effortlessly perceive conceptual differences.36 On the other hand, case comparisons, as the inherent tool of the epistemic practice in organic chemistry, can be designed such that they serve as the starting point in an inquiry process.

cation strongly through resonance, delocalizing the positive charge. On the other hand, the C−O σ-bond is highly polar and increases the positive charge at the carbocation center. In this case, the resonance effect dominates the electron-withdrawing effect, consequently stabilizing the cation. Confronting learners with this act of weighing effects challenges their case- or rulebased reasoning strategy, offering the opportunity to develop a more differentiated view of chemical concepts. The basic idea of CCs is to combine reactions, slightly different on the structural level and in terms of their kinetics or thermodynamics. By comparing reaction products or intermediates (or even transition states) and asking students to estimate the reaction rate (cf. Figure 3), the search strategy shifts from

Figure 3. SN2 reactions in comparison: three related cases of Williamson’s ether syntheses (solution in blue).

looking at the pure transformation of entities to weighing the favorable and unfavorable effects of the process. As such, the search for similarities is limited not only to what has changed on the Lewis structure level but also to what actually influences the relative reaction rate of the processes and how this could be best rationalized. Figure 3 shows an example of a quite sophisticated CC. Given this case of α-substituted halogenalkanes, estimating the kinetics of these three reactions is not solvable by applying a memorized electron-pushing formalism at the Lewis structure level. The speed of both (b) and (c) reactions is increased considerably compared to (a), due to the carbonyl-group in (b), a -R substituent, and the π double bond in (c), a +R substituent. The comparison of these three reactions requires the learner to retrieve three important factors: (1) The SN2 reaction mechanism as a one-step reaction proceeding through one transition state with a pentacoordinated center; (2) the lower the energy of this transition state, the faster the reaction; and (3) electronic and steric effects influence the energy of this transition state. However, it is not possible to fully explain the reaction speed of (b) and (c) in comparison to (a) without looking beyond the Lewis structure level. Deriving hypotheses about the kinetics requires instead an advanced model to explain how substituents influence the height of the transition state in an SN2 reaction. It is necessary to consider the relevant orbital interactions, as drawn schematically in Figure 4. The π*C−O and the σ*C−Br orbitals in (b) interact and form a new LUMO, which can be attacked much more easily by the HOMO of the nucleophile (= ethanolate). The transition state in (c) is energetically lowered by the interaction of the π-orbital with the p-like orbital of the reactive carbon atom. The primary goal of using CCs is to actively engage the learner in solving mechanistic problems by applying fundamental chemical concepts, for example, regarding the orbital interaction at the transition state. In the case presented, this would lead to the sophisticated assumption that reactions (b)

Contrasting Cases in Organic Chemistry: Electronic Substituent Effects as a Connecting Concept

The main goal of using CCs in organic chemistry is to engage students to “see” beyond the Lewis structure level and apply conceptual knowledge while problem-solving. Suitable CCs, in this sense, should not be easily solvable by applying correlative shortcuts, e.g. “primary alkyl halides always make SN2reactions” or “the nitro group directs substituents in the meta-position”, but require the learner to search for differences and effects on the conceptual level. We have chosen electronic substituent effects as a conceptual subject to create CCs for organic chemistry, because they span across various reaction types and problem situations, such as directing effects in substitution reactions, stabilizing intermediates, and estimating the strength of acids. Although some substituents may not be apparently involved in a reaction, they might change the rate of organic reactions, nevertheless, or favor one of various possible pathways. Estimating the indirect influence of substituents on the reaction outcome can be difficult, as one has to consider more than a depicted functional group and needs to reflect on the underlying reaction process: Do I have more favorable resonance structures or additional electron-deficit compensation through orbital interaction? Estimating the electronic effects of functional groups is not always straightforward. Depending on the structural context, they can be conflicting and need to be balanced. A methoxy group adjacent to a carbocation, for example, stabilizes the C



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suggest embedding CCs in a Compare−Predict−Observe− Explain (CPOE) cycle (Figure 5). This refers to the Predict− Observe−Explain cycle commonly used in science education and which has been revealed to be effective for conceptual change.37 These four steps are not limited to lab classes or lectures, and we do not aim to prescribe how CCs should be used, but rather we want to outline how CCs can be meaningfully embedded in an inquiry process.38 The first step while working with CCs generally starts with Prompting to compare and a subsequent focused and effortful search for commonalities and differences. Richland et al.showed that using comparisons in direct instruction requires appropriate supportive cues to help learners to see the differing features.39 This primary step of comparison is common in the use of contrasting cases. Alfieri et al. could show that the instruction is equally beneficial for students with varying levels of experience.32 Based on this first search for commonalities, the learner is asked to formulate a hypothesis, for example, which of the reactions proceeds faster or slower. This second step in the CPOE cycle is different compared to the steps in CC instructions. Within the steps of a CC instruction, the “alignment of target features” would be the next step. As the CPOE cycle follows the inquiry process, the generation of hypotheses is the next logical step. The hypotheses, generated by the learner can then, in the OBSERVE step, be subsequently tested in an experimental or theoretical setting. The latter decision is dependent on the learning goal, the type of class, and the time available for experimental testing (cf. following chapter). If used in a lab class, the students should be able to test their hypotheses in an experimental setting. There are various experiments, which we have already designed for these cases and which can be used in the CPOE cycle as experimental setting.40,41 If the time and context of a course do not allow the testing of the hypotheses experimentally, theoretical data, for example, diagrams of kinetic measurements or stating the result qualitatively (fast/slow), can be used. For the purpose of this paper, we chose to give the relative speed or reaction outcome (given in blue in the figures). Depending on the outcome of this OBSERVE step, the learner should rationalize the result with his or her prior hypotheses, confirm or refute them explicitly, and derive an explanatory concept for the case given. This step is important to reduce the case-specific details and

Figure 4. Orbital interactions in SN2 reactions of (a) alkyl-substituted, (b) α-acylated (-R substituent), and (c) α-vinylated haloalkanes (+R substituent).

and (c) are proceeding faster than (a). Furthermore, the conceptual difference between (b) and (c) requires a differentiation of the concept (e.g., evaluating or calculating the orbital coefficients to estimate the most favorable orbital interactions). How To Implement These Cases in a Learning Environment?

Contrasting cases can be embedded in a scientific inquiry process, i.e. to derive or advance an explanatory concept from concrete experimental settings if previous concepts are not sufficiently explanatory. This process of deriving explanatory concepts is an integral part of how organic chemistry works, and it needs to be made explicit to the learner. Therefore, we

Figure 5. Illustration of the proposed CPOE cycle for use in contrasting cases. D



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Figure 6. Illustration of the CPOE cycle for a lab experiment.43

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build an abstract representation of the cases.30,42 It should result in a concept which is transferable to future cases. In some cases, learners might benefit from guidance in the EXPLAIN step. Then the teacher should support the learner to abstract from the concrete example. Following this cycle in a subsequent round, further CCs can be used for differentiating the concept.

LEARNING OBJECTIVES Three types of learning objectives can be imagined with the CCs in the CPOE cycle. Depending on the cases and the learner, the corresponding concept can be either developed, applied, or expanded. The prior knowledge is essential for all cases illustrated here and should be taken into account. The classification of CCs, used either for developing, applying, or expanding a concept, is a suggestion on how to build sequences of CCs, but is not mandatory. A variety of additional contrasting cases for typical substituent effects is provided in the Supporting Information.

A Contrasting Case for Lab Classes

The CC in Figure 6, developed for educational purposes, shows exemplarily how the CPOE cycle can be used for the concept of hyperconjugation and resonance effects.43 The initial task is to COMPARE the protonation of the double bond in (1) 1,1diphenylethene and (2) trans-1,2-diphenylethene and derive a hypothesis about the reaction speed (PREDICT). The learner should be familiar with the concept of charge stabilization through inductive or resonance effects. Depending on prior knowledge, the learner can derive two hypotheses leading to the same assumption. Just by comparing the possible sites for hyperconjugation or resonance for the process of forming the positive charge, reaction 1 has two sites of resonance effects and one site of hyperconjugative effects, whereas reaction 2 only has one of each effect. These structurally derived hypotheses may lead to the hypothesis that the formation of the carbocation in reaction 1 is stabilized through additional effects and is thus formed faster than reaction 2. Another way to hypothesize about these two reactions can proceed by counting the resonance forms for the carbocation in each reaction. Reaction 1 can form seven resonance forms, whereas reaction 2 can only form four resonance forms. As charge delocalization is a key concept for stability in organic chemistry, the same hypothesis about the reaction speed can be drawn. However, this hypothesis needs to be tested. In this specific case of carbocations, the resulting conductivity of the solution can be taken as a qualitative indicator to compare the rate of formation. Quantitative nuclear magnetic resonance analysis can also be used in upper-level classes. Both compounds react with trifluoromethanesulfonic acid, and a change in color and conductivity is observed and, respectively, measured (OBSERVE). The reaction time measured in the experiment can be qualitatively compared, and the initial hypotheses can be confirmed or refuted. In order to link the concept of carbocation stabilization to the reaction rate, one can relate the stabilization of the intermediates to Hammond’s postulate that more stable intermediates are formed faster as they proceed through lower transition states (EXPLAIN).

Learning Objective: Developing a Concept

Depending on the prior knowledge, the CCs in this setting are meant to initiate hypotheses about the outcome or speed of contrasted cases and derive initial ideas about basic concepts. As students tend to focus on differences on the structural level, the COMPARE step is important to engage the learner to brainstorm about all differences, which are not apparent, for example, electronegativity differences, orbital interactions, and structural constraints, and derive a hypothesis. When using CCs to develop a concept, for example, hyperconjugation, the PREDICT step can be challenging, as the appropriate prior knowledge has not yet been learned. As such, it is necessary to either help students with additional information or provide the learners with the kinetic results (given in blue in Figure 7 and in the other cases). In this case, the students are not left in the dark concerning their hypothesis.

Figure 7. Developing the concept of hyperconjugation (solution in blue).

The CCs in Figure 7 may lead the learner to recognize that for the dissociation of an alkyl iodide to a carbocation and an iodide anion, the presence of three instead of two methyl groups obviously makes the difference in the rate of reaction. One step in proposing a hypothesis for this result is to realize that the sextet center at the carbocation profits from adjacent E



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group of Figure 8 into a trifluoromethyl group in Figure 9 increases the challenge. Building upon the concept learned

methyl groups (+I effect). Given the kinetic results, the learner can infer that methyl groups act as electron-donor groups. If required, this electron-donating property can be discussed further on the orbital level, as a donating effect from the σC−Horbital of the adjacent CH bonds into the empty pz-orbital, leading to the concept of hyperconjugation. The last, but also the most complicated step for the learner for this CC could be to rationalize the kinetics of the reactions by using Hammond’s postulate in a qualitative way: The more stabilized carbocation forms through a lower transition state and, therefore, it forms faster.

Figure 9. Disturbing the application of Markovnikov’s rule: addressing the effect of negative hyperconjugation.

Learning Objective: Applying a Concept

Furthermore, a chemical concept will only be meaningful to a learner if it is beneficial in a new problem-solving context; that is, if the concept is applicable for explaining new reactions or a different reaction outcome. Therefore, each concept needs to be challenged in new CCs, as concepts such as electronic substituent effects are often learned in a specific context and neglected in another one.44 If a concept has been developed through a prior CC, the CPOE cycle can be used in a subsequent step to train the application of the concept in a new structural or mechanistic setting. If possible, the CC to apply the concept should “look” different from the prior one, to prevent students from focusing again on structural similarities. In contrast to the scenario in which students develop a concept, CCs requiring the application of a concept learned previously can follow the full CPOE cycle. As the learners already have the prior knowledge, they are asked to hypothesize about the rate of reaction or the outcome of the CC based on their learned concept. In the example in Figure 8, the concept

previously, a learner might derive the hypothesis that the speed of formation of the second carbocation is faster, as it forms a tertiary cation. The experimental finding in the OBSERVE step generates a cognitive conflict, as the formation of the secondary carbocation in the upper pathway seems to be favored over the tertiary one. The concept of hyperconjugation still applies in this case; however, the trifluoromethyl group seems to destabilize the cation center. In contrast to a C−H bond, a C−F bond adjacent to a cation center increases the charge and destabilizes the cation, increasing the energy of the transition state. The −I effect of the fluoro substituents reverses the effect, compared to a typical σ-donor effect of a methyl group. With the CC in Figure 10, the concept of stabilization of carbocations is expanded to the important phenomenon that n-

Figure 10. Expanding the stabilization of carbocations to carboxonium ions.

Figure 8. Applying the concept of hyperconjugation in another mechanistic context (solution in blue).

donor atoms, such as oxygen, stabilize adjacent sextet centers perfectly by resonance, with the consequence that the carboxonium ion in the first case is much better stabilized than the carbocation in the second case (Figure 10). Carboxonium ions occur in all cases of Brønsted or Lewis acid-catalyzed reactions of carbonyl compounds. The selection of examples presented here show that the CCs can be used to aim at different learning objectives. Each learning environment, however, also depends on the teacher’s engagement to discuss the CCs on a conceptual basis, shifting the focus to the reaction process. This is especially important when reactive intermediates are involved and the discussion of concurrent reaction processes is meaningful.

of hyperconjugation is taken into the context of electrophilic addition reactions. Applying the effects of hyperconjugation on the rate of both reactions should lead the learner to hypothesize that the carbocation of the upper pathway is formed faster after protonation of the double bond, as it profits from the hyperconjugative effect of an additional methyl group. It is important for learners to experience that, independent of the structural reaction context, a concept, which helps to derive hypotheses about the reaction rate, is also valuable in other reaction contexts: in the SN1 reaction (Figure 7) and in electrophilic addition reactions of Brønsted acids to alkenes (Figure 8).

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CONCLUSION Electronic substituent effects play a major role in all cases of organic reactions, proceeding through reactive intermediates, and they are, thus, a connecting concept in organic chemistry. Following the research results on using CCs in education, we illustrated here that this design idea can help to purposefully contrast the substituent effects of organic reactions. Using these tasks as an instructional tool in organic chemistry classes might help to shift the focus toward the actual process of a mechanism

Learning Objective: Expanding a Concept

Expanding a concept is meant to differentiate the concept learned based on more complex cases where additional effects play a role. As such, a learner needs to combine various effects. In this case, the learner should experience that, depending on experimental findings, which may contradict the initial hypotheses, additional prevailing effects need to be considered. Consequently, the concept learned previously needs to be adjusted or differentiated accordingly. Changing the methyl F



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by always comparing two slightly differing reactions or reaction steps. If learners are prompted to derive hypotheses about the kinetics and thermodynamics of the process and experience that they can infer preferences by applying concepts in various contexts, then it may become a valuable practice. Rolf Huisgen mentioned in the 1970s that the term “mechanism” should not lead students to think only about structural changes but to look on the interplay of the molecules on the “energy landscape”.19 The CCs and the CPOE cycle illustrated in this paper focused explicitly on recurring concepts, such as stabilization of reactive intermediates through resonance, hyperconjugation, n-donors, or other orbital interactions. These concepts are discussed herein in various “structural” contexts to span across multiple reaction types. Whether CCs help students sustainably to acquire conceptual knowledge is currently under investigation in laboratory classes for high school chemistry,45 in student teacher education, and in our ongoing research efforts to elicit students’ reasoning at the university level.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.7b00672. Contrasting cases for each reactive intermediate (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: schweenm@staff.uni-marburg.de. ORCID

Nicole Graulich: 0000-0002-0444-8609 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank Tobias Fredlund for permitting us to use his analogy of screws and bolts for showing the variation theory of learning, which we transferred in the graphical abstract to the idea of contrasting cases.

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REFERENCES

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