Theoretical Underpinnings

This EBTG considers various instructional methods that have been shown to support student learning and problem solving. These methods differ in the nature and timing of instructional guidance and are derived from different theoretical perspectives on learning. This section provides a brief description of the learning theories that support different approaches to teaching problem solving.

  • Constructivism undergirds all of the instructional approaches presented. Constructivism emphasizes the role of the learner in building their knowledge rather than acquiring it and, thus, focuses on learners’ prior knowledge.
  • Some instructional approaches draw on additional theories besides constructivism.
    • Instruction-first approaches derive from cognitive load theory, recognizing that because learners build on prior knowledge the demand on working memory should be minimized in order to maximize learning. The instruction-first approaches reviewed in this guide are worked examples and peer-led team learning, which is a form of guided inquiry.
    • Problem solving-first approaches acknowledge the limits of working memory, yet they argue that initially high levels of cognitive load create benefits that outweigh the costs. They base their approaches on theory about the activation and differentiation of prior knowledge, attention to and encoding of critical features in a problem, desirable difficulties, preparation for future learning, and learner agency and enjoyment. The problem solving-first approaches reviewed in this guide are contrasting cases, productive failure, and process-oriented guided inquiry learning.
  • The term constructivism is applied to a wide range of contexts, pedagogies, and perspectives. For the purpose of this guide, we will focus on two broad categories of constructivism, which are not mutually exclusive:
    • Cognitive constructivism describes how individual learners assimilate new knowledge into existing mental models or accommodate incongruent information by revising those models. It is associated with Jean Piaget, who emphasized the individual learner’s construction of knowledge, particularly during a child’s psychological development.
    • Social constructivism highlights the role of socially-endowed knowledge and tools (e.g., language, the scientific method) in learning. It is linked to Lev Vygotsky’s work emphasizing the social, cultural, and historical drivers of knowledge construction.
Bodner, G.M. (1986). Constructivism: A theory of knowledge. Journal of Chemical Education, 63, 873-878.  This concise review describes the cognitive constructivism theory of knowledge and its relationship to science teaching and learning. The author briefly describes Piaget’s theory of intellectual development, defining the processes of assimilation, or using preexisting mental models to interpret data, and accommodation, or the modification of mental models to account for unexpected results. This model is contrasted with the traditional “realist” view of knowledge, which assumes that knowledge consists of a mental copy of an external reality. The constructivist model asserts instead that each individual constructs mental models that fit the data they perceive from the world. Experiences tend to be interpreted using existing mental understandings, allowing new knowledge to be incorporated into existing mental models. When these existing mental models fail in the face of new experiences/data, however, they can be modified to accommodate these new data. The knowledge that is built through this process is continually tested, in part by new observations and in part by discussions with others. The author describes experiments that support key elements of this constructivist model and explores implications of the model for teaching, noting that the constructivist model requires a shift from framing an instructor’s role as “teaching” to “facilitating learning,” often by posing problems and questioning. Based on this review, instructors should note that teaching involves posing questions and problems that either 1) help students assimilate new knowledge into existing understandings or 2) recognize how their existing understandings fail to fit reality and help them make corresponding modifications to their mental models. 
Driver, R., Asoko, H., Leach, J., Mortimer, E., & Scott, P. (1994). Constructing scientific knowledge in the classroom. Educational Researcher, 23(7), 5–12. This theoretical paper describes the nature of scientific knowledge in order to demonstrate how such knowledge is socially constructed in a classroom setting. The authors begin by distinguishing observation of the natural world and acquisition of conceptual representations to characterize natural phenomena. Scientific knowledge comprises the latter, i.e., constructs that are imposed on experiences to explain and interpret them in a unifying way. Scientific learning therefore constitutes a process of enculturation more than simple discovery; it entails an introduction to scientific ideas and practices (e.g., hypothesis generation and validation) that are socially constructed and communicated. Next, the authors review Piaget’s theory of individual knowledge construction and Vygotsky’s theory of sociocultural knowledge construction. Scientific learning involves individual students developing new ways of thinking about natural phenomena, which may contradict their common sense understandings acquired through every day culture. Yet individual knowledge acquisition also relies on learners being socialized into the scientific community’s practices for supporting knowledge claims. Thus, a constructivist view portrays scientific learning as inherently social: the cultural tools of science are transmitted through classroom interactions, where the instructor or more advanced learners introduce news ideas and support learners in grappling with inconsistencies between their current understandings and the scientific views they are learning. The authors argue that knowledge construction does not necessarily involve replacing common sense understandings with scientific knowledge; instead, both may ultimately coexist in the learner’s mind if each has utility in different contexts. Nonetheless, learners may struggle to integrate new scientific knowledge when it diverges greatly from their baseline understanding. Based on this article, instructors should note the importance of both individual and social factors in scientific knowledge construction. They should investigate individual learners’ current understandings of phenomena to determine how best to communicate scientific ideas and support students in accommodating this information.
Prince, M. J., & Felder, R. M. (2006). Inductive teaching and learning methods: Definitions, comparisons, and research bases. Journal of Engineering Education, 95(2), 123-138. This review article explores the theoretical foundations and implementations of inductive teaching, where instructors deliver information only after students have identified a need for it through problems, cases, or other active-learning tasks. Importantly, a key theoretical foundation of inductive learning is constructivism, which argues students build knowledge structures to make sense of their experiences, integrating new information into those structures or revising the structures when novel information cannot be accommodated. Based on constructivism, inductive teaching follows several instructional principles: 1) begin with familiar content to facilitate connection-making, 2) do not expect sudden or drastic changes to knowledge structures, 3) ask students to fill in gaps and extrapolate, helping them become more self-directed, and 4) use small-group work to facilitate co-construction of knowledge. Inductive teaching contrasts with traditional science and engineering instruction that relies on deductive teaching, where instructors present information and assert its importance for students’ later study or careers, without students discovering why the content is relevant to themselves.  The article also reviews inductive teaching’s relationships to different learning approaches (e.g, surface, deep, and strategic learning) and learning cycle-based instruction. The second half of the article reviews six inductive teaching methods: inquiry learning, problem-based learning, project-based learning, case-based teaching, discovery learning, and just-in-time teaching. These overviews may help instructors discern the boundaries and overlap among inductive, constructivist techniques; they also explore potential barriers to adoption and evidence for the efficacy of each strategy. Based on this review, instructors should note the potential value of inductive teaching and should consider how learning objectives could guide their choice of a specific inductive technique.
Cognitive Load Theory
Cognitive Load Theory, championed especially by John Sweller and colleagues, is a theory of learning grounded in memory research. It provides the underpinning for the use of instruction-first approaches, such as worked examples and peer-led team learning.

  • Cognitive load theory contrasts the learning of new information, which relies on working memory and is therefore capacity-limited, with the usage of previously learned information, which is stored in long-term memory structures known as schemas and can be accessed freely.
  • Cognitive load theory encourages instructors to:
    • Limit cognitive demands placed on students by the instructional design of an activity (extraneous load), e.g., how much information is presented and what task students are asked to complete.
    • Evaluate the complexity of the information being presented and its relationship to students’ prior knowledge (intrinsic load).
    • Consider the skills and cognitive resources needed to engage in the learning activity (germane load), with active strategies requiring more resources than passive strategies.
  • A drawback of cognitive load theory is the focus on working-memory limitations may lead instructors to shy away from cognitively challenging instructional activities that can benefit learners. Some evidence suggests that very low cognitive load is not beneficial for more advanced students.
  • Specific guidance for managing cognitive load during problem-solving instruction can be found in the Worked Examples section of the guide.


Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257-285. This foundational cognitive load theory article reviews prior experimental studies and provides original computational and experimental evidence that cognitive demands during conventional problem solving can impede learning. The review uses findings from diverse domains (e.g., chess, math, physics) to demonstrate a key difference between novice and expert problem solvers: experts possess problem-solving “schemas” or long-term memory structures that help them automatically categorize and work forward through problems, while novices must rely on cognitively demanding, domain-general strategies that require complex backwards and forwards steps. Sweller proposes that in traditional educational settings where novices are instructed to problem solve for a specific solution or goal, the high cognitive demands of problem solving can prevent learning of underlying structures and expert-like schema acquisition. Instead, students’ attention and mental resources may be consumed by finding the correct answer, to the neglect of building deeper understanding. To support this hypothesis, Sweller presents a simple computational model, simulating how a novice might solve a kinematics problem differently when given a conventional, goal-specific problem vs. a nonspecific problem (e.g., “calculate as many variables as you can”). The simulation confirmed the conventional prompt required more elaborate problem-solving procedures than the open-ended prompt, suggesting higher memory demands and fewer resources available for learning. This interpretation is strengthened by a final experiment, where N = 24 high school students solved conventional or nonspecific geometry problems while holding other information (e.g., prior solutions) in memory. Students in the conventional condition made more recall errors than those in the nonspecific condition, suggesting they had less cognitive capacity available. Based on this evidence, instructors should note that traditional problem-solving activities geared towards finding a specific solution may be cognitively demanding for novice students, causing them to focus on getting the right answer and actually impairing deeper learning. Open-ended problems and worked examples may prove more fruitful for helping learners develop structural knowledge and expert-like schemas.
Sweller, J., Van Merriënboer, J. J., & Paas, F. (2019).  Cognitive architecture and instructional design: 20 years later. Educational Psychology Review, 31(2), 261-292. This article comprehensively reviews the evolution of cognitive load theory, the current state of the literature, and promising lines of future research. Basic cognitive load theory had several key premises: i) working memory for new information is limited, ii) working memory is unbounded when drawing information from long-term memory, and iii) the primary goal of instruction is to help learners transfer new information into long-term memory, thereby increasing their problem-solving resources. Extensive experimental research supported this theory, generating seven instructional effects showing how working memory limitations influence learning and can be manipulated through instructional design (e.g., the goal-free effect from Sweller, 1988, and the worked example effect). The theory later became grounded in evolutionary psychology; new survey, objective, and physiological measures of cognitive load were developed; and additional instructional effects were documented. Importantly for instructors, a four-component instructional design (4C/ID) framework was developed to provide guidance on the design of longer-term educational programs (e.g., courses and curricula) that gradually adjust instruction to students’ growing expertise. The 4C/ID framework assumes a contrast between “recurrent” skills, which apply across tasks and can be routinized, and “non-recurrent” skills, which vary and require reasoning and decision-making. Another key premise is that curricula for complex skill building require 4 key components: 1) learning tasks, 2) supportive information, 3) procedural information, and 4) part-task practice. Going forward, Sweller and colleagues predict cognitive load theory may incorporate research on cognitive resource depletion, self-regulated learning, environmental influences on cognition (i.e., emotions, stress, and uncertainty), and human movement. Based on cognitive load theory, instructors should note the importance of tailoring instruction and learning activities to their students, because a reduced cognitive load is advantageous for novices, but a higher load supports greater learning and transfer among more expert students.


Desirable Difficulties
Desirable difficulties, popularized by Elizabeth and Robert Bjork and colleagues, describe how rigorous learning techniques may elicit poor early performance and feel effortful, yet ultimately produce better long-term retention and transfer outcomes.

  • Desirable difficulties provide a counterpoint to cognitive load theory, emphasizing the value of cognitively demanding learning strategies and the potential benefit of activities where students must actively generate and even struggle to find a solution. This theoretical perspective is an important underpinning for problem solving-first instructional approaches, such as productive failure, contrasting cases, and process-oriented guided inquiry learning.
  • The desirable difficulties framework encourages instructors to:
    • Notice the distinction between performance versus learning, i.e., that student success on early practice tasks does not necessarily predict long-term retention and transfer.
    • Promote evidence-based techniques for improving student learning and transfer, especially retrieval practice and distributed practice.
    • Facilitate transfer-appropriate processing by designing learning and practice activities that involve similar cognitive demands and variability as their eventual transfer tests.
  • A challenge of using desirable difficulties in instruction can be effects on student motivation and attitudes. The increased effort and time required by cognitively demanding strategies may cause students to resist or have negative responses to such techniques, and instructors should consider how to support students as they experience these challenges.


Schmidt, R. A., & Bjork, R. A. (1992). New conceptualizations of practice: Common principles in three paradigms suggest new concepts for training. Psychological Science, 3(4), 207-218. This paper challenges the assumption that practice conditions conducive to faster improvement and better performance during skill acquisition will lead to better long-term retention and transfer of that skill. Using experimental evidence from various fields, especially motor and verbal skills research, the article distinguishes short-term performance benefits from long-term learning gains. Learning – specifically, durable encoding of information and skills in memory – should be evaluated by retention and transfer tests conducted at a delay, rather than initial outcomes during skill acquisition. To illustrate this point, the authors review three types of training manipulations that impair early practice performance but ultimately enhance long-term retention and transfer. First, randomized practice tasks (i.e., with practice for different skills intermingled) improve retention compared to blocked practice, despite producing worse practice performance. Moreover, an expanding-interval design, where the delay between practice with the same task gradually increases, proves even more effective than uniform-interval random practice. Second, spaced feedback leads to better retention than feedback on every practice trial, although it hinders training performance. Finally, practice that varies along one dimension (e.g., the speed of the task) facilitates better transfer to novel tasks or task conditions than uniform practice, though once again variability hurts early performance. These findings demonstrate the importance of “transfer-appropriate processing” during skill acquisition: training designs that require learners to retrieve information from memory and generalize across variability will produce better outcomes on subsequent tests and real-life challenges that require similar retention and transfer. Instructors should note the important distinction between performance and learning and help students appreciate that challenging practice will lead to better long-term outcomes, even if they struggle early on. Designing practice activities that require students to recall and generalize rather than merely recognizing information will help them gain durable and flexible skills.
Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), 4-58. This article comprises standardized reviews of ten learning techniques, where each receives an overall utility assessment to guide students and instructors towards techniques with broad applicability and strong supporting evidence. Most of the techniques that were found to be more effective at promoting learning are effortful, incorporating desirable difficulties. Each self-contained review examines the potential benefit of a learning technique across four categories of variables: learning materials or content, learning conditions or contextual factors, student characteristics (e.g., age, level of domain knowledge), and types of criterion tasks (i.e., tasks that assess different learning outcomes, such as recall, comprehension, and application). In addition, each review identifies gaps in the research coverage of these variables and considers potential implementation issues for students and instructors. Learning techniques determined to have high utility include practice testing and distributed practice, both of which demonstrate broad applicability, ease of use and efficiency, and robust evidence from authentic educational contexts. Three techniques were ascribed moderate utility: elaborative interrogation, self-explanation, and interleaved practice. Both elaborative interrogation and interleaved practice require further research to establish the generality of their effects; in contrast, self-explanation shows fairly robust applicability, but it can be training and time intensive. Finally, five learning techniques displayed low utility: summarization, highlighting and underlining, keyword mnemonics, imagery use for text learning, and rereading. Notably, the relatively low intensity, student-preferred techniques of highlighting and rereading fall into the low-utility group. This disconnect between student preferences and effective learning techniques indicates an opportunity for instructors to improve student achievement through guidance towards evidence-based strategies. Instructors should note that students may require explicit instruction in effective learning techniques, alongside instruction on content and critical thinking. Instructors can illustrate these techniques, like practice testing and distributed practice, by embedding them into their instruction.
Activating, Differentiating, and Encoding Knowledge
An essential tenet of constructivism is that learning requires accommodation of new information into existing mental schemas. Therefore, learners must activate (that is, recall) what they already know and differentiate between their new and old understandings of phenomena to build understanding. Problem solving-first approaches foster activation of prior knowledge by asking students to try to solve problems before receiving instructions for how to do so. Further, by identifying how their prior knowledge fails, students differentiate between old and new understandings. These processes prepare students to incorporate (that is, encode) new knowledge into their existing schemas during instruction.

  • Problem-solving activities that ask learners to invent general principles, e.g., to explain several contrasting cases, can help learners attend to sources of meaningful variation and encode the critical features of the problem structure.
  • Research shows that problem solving-first approaches can benefit the learning and transfer of students with both high and low prior achievement.


DeCaro, M. S., & Rittle-Johnson, B. (2012). Exploring mathematics problems prepares children to learn from instruction. Journal of Experimental Child Psychology, 113(4), 552-568. This experimental study examines how exploration benefits learning, comparing a guided exploration approach where problem solving precedes instruction (“solve-instruct”) to the traditional approach where direct instruction precedes practice problem solving (“instruct-solve”). 159 elementary school students were random assigned to each order-of-instruction condition: solve-instruct or instruct-solve. They were instructed and tested (pretest, midtest, immediate posttest, retention test after two weeks) on the concept of mathematical equivalence, which states that expressions on two sides of the equal sign represent the same quantity. The learning assessment measured both 1) procedural knowledge of the action sequences required to solve the target problems and 2) abstract conceptual knowledge, including the relational definition of the equal sign and knowledge of the problem structures. Results showed order of instruction had a clear impact on learning: while procedural knowledge was comparable across conditions, students had higher conceptual knowledge scores in the solve-instruct condition compared to the instruct-solve condition. The midtest administered between the problem-solving and instruction blocks revealed that halfway through training, instruct-solve participants had superior knowledge of the relational definition of the equal sign, whereas solve-instruct participants had superior knowledge of problem structures. In addition, solve-instruct participants tried a wider range of (incorrect) strategies during training, and they gave more accurate assessments of their own understanding after a two week delay. The authors concluded that exploration through problem solving prior to direct instruction improves students’ conceptual learning by enhancing attention to the structural features of problems, facilitating usage of diverse strategies, and prompting more accurate self-assessment. Instructors should note that problem solving before instruction provides an opportunity for students to activate and apply prior knowledge to a novel context. While challenging, exploration may produce more effective conceptual learning and knowledge differentiation, illuminating students’ current knowledge and skills and how those relate to new concepts.
Schwartz, D. L., Chase, C. C., Oppezzo, M. A., & Chin, D. B. (2011). Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer. Journal of Educational Psychology, 103(4), 759. This paper describes two experimental studies manipulating order of instruction, comparing problem solving- and instruction-first approaches that use contrasting cases. In both studies (Ns = 128 & 120), eighth-grade students learned how the principle of ratios underlies various physics phenomena (e.g., density, speed). They were randomly assigned an “inventing with contrasting cases” activity, which presents multiple cases varying along one dimension and prompts students to determine the underlying structure, or a “tell-and-practice” activity, where the instructor explicitly explains the underlying structure before providing practice problems with the same contrasting cases. Study 1 revealed no difference between inventing with contrasting cases and tell-and-practice participants on surface feature recall or word problem performance, but inventing with contrasting cases learners exhibited better recall of the deep ratio structure and more frequent transfer of that structure to a novel context than tell-and-practice learners. Study 2 replicated and extended these findings, showing the inventing with contrasting cases advantage for deep learning and transfer applied to both high- and low-achieving students. These findings indicate that (guided) problem solving can facilitate better attention to and encoding of deep structural knowledge than instruction-first approaches, particularly when contrasting cases prompt students to search for underlying structure and identify general explanatory principles. Instructors should note that learners, regardless of prior achievement, can benefit from problem solving-first approaches. If problem solving guides students to engage in abstract processes like comparing and contrasting, explanation, and induction of general principles (e.g., from contrasting cases), then it may induce more robust conceptual learning and transfer than instruction-first approaches. 
Agency and Preparation for Future Learning
While desirable difficulties may challenge student motivation, activities where students are engaged in actively exploring and inventing solutions to novel problems also have the potential to excite motivation by inviting learner agency.

  • Problem solving-first approaches provide students with a sense of ownership over their learning and instill the goal on growing their understanding (mastery) rather than just performing well.
  • Such approaches also empower students to learn independently: research shows that problem solving-first approaches help students prepare to learn alone from novel future resources.


Belenky, D. M., & Nokes-Malach, T. J. (2012). Motivation and transfer: The role of mastery-approach goals in preparation for future learning. Journal of the Learning Sciences, 21(3), 399-432. This study explores the impact of motivation, specifically achievement goals, and instruction on transfer. Achievement goals describe the reasons why and the manner in which students engage in academic activities, and they are defined along two dimensions: mastery vs. performance and approach vs. avoidance orientations. This study focuses primarily on mastery-approach orientations, where students aim and use adaptive strategies to gain understanding, because prior work suggests mastery-approach goals support conceptual learning, engagement, and therefore transfer. Transfer is construed as the ability to effectively learn from a new resource (e.g., instruction or activity), rather than the ability to apply knowledge to a new problem or task. This conceptualization of transfer relies on students’ interpretive knowledge rather than replicative or application knowledge. In a 2 (learning activity: invention vs. tell-and-practice) x 2 (learning resource: present vs. not) experimental design, the authors assessed how students’ intrinsic goal orientations (measured pre- and post-study) influence transfer. They also examined whether instruction via invention versus tell-and-practice activities induced different short-term goal adoption (measured immediately after the learning activity), with potential downstream effects on transfer. Results from N = 104 undergraduates learning statistics in introductory psychology confirmed the authors’ three hypotheses: 1) students who began the experiment with stronger mastery-approach orientations were more likely to transfer, regardless of instruction, 2) invention (vs. tell-and-practice) activities induced more mastery-approach goal adoption, and 3) instruction moderated the effect of motivation on transfer; all students with strong mastery-approach orientations were likely to transfer, but for students with weak mastery-approach orientations, likelihood of transfer depended on instruction (invention led to more transfer). Based on this study, instructors should note that student motivation can influence learning and transfer. Moreover, instruction that prioritizes learner agency (e.g., invention activities, problem solving-first approaches) can enhance motivation, focusing student attention on conceptual understanding and supporting transfer.
Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22(2), 129-184. This study tests the hypothesis that invention activities better prepare students to learn from future resources (e.g., instruction or activity) than tell-and-practice instruction, despite the fact that students may not invent canonical solutions. The goal of “inventing-to-prepare-for-learning” is to help students develop prior knowledge that will allow them to interpret future lessons. In statistics, students must appreciate the quantitative properties of problems and understand what quantitative work is accomplished by different mathematical procedures or tools available to them. Two ninth-grade classroom experiments (N = 95 & 102) used a 2 x 2 design crossing instructional method (inventing-to-prepare-for-learning vs. tell-and-practice) with transfer test format (with vs. without additional learning resources). This design is described as a “double-transfer” paradigm, because when the transfer test is preceded by an additional learning resource, students must (a) “transfer in” knowledge from the initial instruction to learn from the extra resource and then (b) “transfer out” learning from that extra resource to the final transfer test. The same results emerged regardless of whether the authors (study 1) or classroom instructors (study 2) administered the experiments, which were embedded in a multi-part learning cycle about descriptive statistics. Without an additional learning resource, inventing-to-prepare-for-learning and tell-and-practice participants performed similarly on the transfer test; however, when the additional resource was provided, invention participants outperformed tell-and-practice participants. Based on this study, instructors should note that while student solutions during invention activities may be incorrect, the activities prepare students for future learning by allowing them to notice problem features, discern sources of variability, and perform hands-on work with representations. Invention instruction may seem less efficient, but it can make learners more adaptive and ready to transfer knowledge in an authentic way (i.e., from formal instruction to independent learning with other resources to eventual application).

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Cite this guide: Frey RF, Brame CJ, Fink A, and Lemons PP. (2022) Evidence Based Teaching Guide: Problem Solving. CBE Life Science Education. Retrieved from
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