Kenneth Richard Koedinger

Much problem solving and learning research in math and science has focused on formal representations. Recently researchers have documented the use of unschooled strategies for solving daily problems -- informal strategies which can be as effective, and sometimes as sophisticated, as school-taught formalisms. Our research focuses on how formal and informal strategies interact in the process of doing and learning mathematics. We found that combining informal and formal strategies is more effective than single strategies. We provide a theoretical account of this multiple strategy effect and have begun to formulate this theory in an ACT-R computer model. We show why students may reach common impasses in the use of written algebra, and how subsequent or concurrent use of informal strategies leads to better problem-solving performance. Formal strategies facilitate computation because of their abstract and syntactic nature; however, abstraction can lead to nonsensical interpretations and conceptual errors. Reapplying the formal strategy will not repair such errors; switching to an informal one may. We explain the multiple strategy effect as a complementary relationship between the computational efficiency of formal strategies and the sense-making function of informal strategies.

Despite the accumulation of substantial cognitive science research relevant to education, there remains confusion and controversy in the application of research to educational practice. In support of a more systematic approach, we describe the Knowledge-Learning-Instruction (KLI) framework. KLI promotes the emergence of instructional principles of high potential for generality, while explicitly identifying constraints of and opportunities for detailed analysis of the knowledge students may acquire in courses. Drawing on research across domains of science, math, and language learning, we illustrate the analyses of knowledge, learning, and instructional events that the KLI framework affords. We present a set of three coordinated taxonomies of knowledge, learning, and instruction. For example, we identify three broad classes of learning events (LEs): (a) memory and fluency processes, (b) induction and refinement processes, and (c) understanding and sense-making processes, and we show how these can lead to different knowledge changes and constraints on optimal instructional choices

Dienes & Perner (D&P) argue that nondeclarative knowledge can take multiple forms. We provide empirical support for this from two related lines of research about the development of mathematical reasoning. We then describe how different forms of procedural and declarative knowledge can be effectively modeled in Anderson's ACT-R theory, contrasting this computational approach with D&P's logical approach. The computational approach suggests that the commonly observed developmental progression from more implicit to more explicit knowledge can be viewed as a consequence of accumulating and strengthening mental representations.

This article explores the complementary strengths and weaknesses of grounded and abstract representations in the domain of early algebra. Abstract representations, such as algebraic symbols, are concise and easy to manipulate but are distanced from any physical referents. Grounded representations, such as verbal descriptions of situations, are more concrete and familiar, and they are more similar to physical objects and everyday experience. The complementary computational characteristics of grounded and abstract representations lead to trade‐offs in problem‐solving performance. In prior research with high school students solving relatively simple problems, Koedinger and Nathan (2004) demonstrated performance benefits of grounded representations over abstract representations—students were better at solving simple story problems than the analogous equations. This article extends this prior work to examine both simple and more complex problems in two samples of college students. On complex problems with two references to the unknown, a “symbolic advantage” emerged, such that students were better at solving equations than analogous story problems. Furthermore, the previously observed “verbal advantage” on simple problems was replicated. We thus provide empirical support for a trade‐off between grounded, verbal representations, which show advantages on simpler problems, and abstract, symbolic representations, which show advantages on more complex problems.

Recent studies have shown that self‐explanation is an effective metacognitive strategy, but how can it be leveraged to improve students' learning in actual classrooms? How do instructional treatments that emphasizes self‐explanation affect students' learning, as compared to other instructional treatments? We investigated whether self‐explanation can be scaffolded effectively in a classroom environment using a Cognitive Tutor, which is intelligent instructional software that supports guided learning by doing. In two classroom experiments, we found that students who explained their steps during problem‐solving practice with a Cognitive Tutor learned with greater understanding compared to students who did not explain steps. The explainers better explained their solutions steps and were more successful on transfer problems. We interpret these results as follows: By engaging in explanation, students acquired better‐integrated visual and verbal declarative knowledge and acquired less shallow procedural knowledge. The research demonstrates that the benefits of self‐explanation can be achieved in a relatively simple computer‐based approach that scales well for classroom use.

We analyze naturally occurring datasets from student use of educational technologies to explore a long-standing question of the scope of transfer of learning. We contrast a faculty theory of broad transfer with a component theory of more constrained transfer. To test these theories, we develop statistical models of them. These models use latent variables to represent mental functions that are changed while learning to cause a reduction in error rates for new tasks. Strong versions of these models provide a common explanation for the variance in task difficulty and transfer. Weak versions decouple difficulty and transfer explanations by describing task difficulty with parameters for each unique task. We evaluate these models in terms of both their prediction accuracy on held-out data and their power in explaining task difficulty and learning transfer. In comparisons across eight datasets, we find that the component models provide both better predictions and better explanations than the faculty models. Weak model variations tend to improve generalization across students, but hurt generalization across items and make a sacrifice to explanatory power. More generally, the approach could be used to identify malleable components of cognitive functions, such as spatial reasoning or executive functions.

This study investigated the cognitive processes involved in inductive reasoning. Sixteen undergraduates solved quadratic function–finding problems and provided concurrent verbal protocols. Three fundamental areas of inductive activity were identified: Data Gathering, Pattern Finding, and Hypothesis Generation. These activities are evident in three different strategies that they used to successfully find functions. In all three strategies, Pattern Finding played a critical role not previously identified in the literature. In the most common strategy, called the Pursuit strategy, participants created new quantities from x and y, detected patterns in these quantities, and expressed these patterns in terms of x. These expressions were then built into full hypotheses. The processes involved in this strategy are instantiated in an ACT‐based model that simulates both successful and unsuccessful performance. The protocols and the model suggest that numerical knowledge is essential to the detection of patterns and, therefore, to higher‐order problem solving.

Complex student models often include key parameters critical to their behavior and effectiveness. For example, one meta-cognitive model of student help-seeking in intelligent tutors includes 15 rules and 10 parameters. We explore whether or not this model can be improved both in accuracy and generalization by using a variety of techniques to select and tune parameters.We show that such techniques are important by demonstrating that the normal method of fitting parameters on an initial data set generalizes poorly to new test data sets. We then show that stepwise regression can improve generalization, but at a cost to initial performance. Finally, we show that causal search algorithms can yield simpler models that perform comparably on test data, but without the loss in training set performance. The resulting help-seeking model is easier to understand and classifies a more realistic number of student actions as help-seeking errors