Cognitive Tutor: Applied research in mathematics education

For 25 years, we have been working to build cognitive models of mathematics, which have become a basis for middle- and high-school curricula. We discuss the theoretical background of this approach and evidence that the resulting curricula are more effective than other approaches to instruction. We also discuss how embedding a well specified theory in our instructional software allows us to dynamically evaluate the effectiveness of our instruction at a more detailed level than was previously possible. The current widespread use of the software is allowing us to test hypotheses across large numbers of students. We believe that this will lead to new approaches both to understanding mathematical cognition and to improving instruction.     

Iterating between lessons on concepts and procedures can improve mathematics knowledge

Background Knowledge of concepts and procedures seems to develop in an iterative fashion, with increases in one type of knowledge leading to increases in the other type of knowledge. This suggests that iterating between lessons on concepts and procedures may improve learning. Aims The purpose of the current study was to evaluate the instructional benefits of an iterative lesson sequence compared to a concepts‐before‐procedures sequence for students learning decimal place‐value concepts and arithmetic procedures. Samples In two classroom experiments, sixth‐grade students from two schools participated (N = 77 and 26). Method Students completed six decimal lessons on an intelligent‐tutoring systems. In the iterative condition, lessons cycled between concept and procedure lessons. In the concepts‐first condition, all concept lessons were presented before introducing the procedure lessons. Results In both experiments, students in the iterative condition gained more knowledge of arithmetic procedures, including ability to transfer the procedures to problems with novel features. Knowledge of concepts was fairly comparable across conditions. Finally, pre‐test knowledge of one type predicted gains in knowledge of the other type across experiments. Conclusions An iterative sequencing of lessons seems to facilitate learning and transfer, particularly of mathematical procedures. The findings support an iterative perspective for the development of knowledge of concepts and procedures.     

The knowledge-learning-instruction framework: bridging the science-practice chasm to enhance robust student learning

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.     

Experiencing malevolent voices is associated with attentional dysfunction in psychotic patients

Inattention in people with schizophrenia is common. However, there has been little research on the association between inattention and auditory hallucinations. The aim of the study was to investigate how inattention is affected by beliefs about voices as benevolent and malevolent and perceived control of voices. A total of 31 patients who experienced auditory hallucinations and who met the criteria for schizophrenia or other psychosis completed the attention subscale of the Scale for the Assessment of Negative Symptoms (SANS) and the Connors’ Continuous Performance Test II (CCPT‐II). The revised Beliefs About Voices Questionnaire (BAVQ‐R) was used to assess malevolent and benevolent beliefs about voices, and severity of auditory hallucinations (the Psychotic Symptom Rating Scales; PSYRATS) was used to assess perceived control of voices and frequency of voices. Levels of depression (the Beck Depression Inventory; BDI), anxiety (the Beck Anxiety Inventory; BAI), severity of overall psychiatric symptoms (the Brief Psychiatric Rating Scale; BPRS), and severity of negative symptoms (SANS) were assessed to control for their potential confounding effects. The relations between the variables were explored with correlations and multiple hierarchical regression analyses. The results indicated that more malevolent, but not more benevolent, beliefs about voices predicted lower levels of attention, independently of general psychiatric symptoms and various other psychotic symptoms such as frequency of and perceived control of voices. These findings suggest an important relationship between malevolent beliefs about voices and levels of inattention. The possible impact of changing beliefs about voices to improve attentional functioning is discussed.     

Are diagrams always helpful tools? Developmental and individual differences in the effect of presentation format on student problem solving

Background. High school and college students demonstrate a verbal, or textual, advantage whereby beginning algebra problems in story format are easier to solve than matched equations ( Koedinger & Nathan, 2004 ). Adding diagrams to the stories may further facilitate solution ( Hembree, 1992 ; Koedinger & Terao, 2002 ). However, diagrams may not be universally beneficial ( Ainsworth, 2006 ; Larkin & Simon, 1987 ). Aims. To identify developmental and individual differences in the use of diagrams, story, and equation representations in problem solving. When do diagrams begin to aid problem‐solving performance? Does the verbal advantage replicate for younger students? Sample. Three hundred and seventy‐three students (121 sixth, 117 seventh, 135 eighth grade) from an ethnically diverse middle school in the American Midwest participated in Experiment 1. In Experiment 2, 84 sixth graders who had participated in Experiment 1 were followed up in seventh and eighth grades. Method. In both experiments, students solved algebra problems in three matched presentation formats (equation, story, story + diagram). Results. The textual advantage was replicated for all groups. While diagrams enhance performance of older and higher ability students, younger and lower‐ability students do not benefit, and may even be hindered by a diagram's presence. Conclusions. The textual advantage is in place by sixth grade. Diagrams are not inherently helpful aids to student understanding and should be used cautiously in the middle school years, as students are developing competency for diagram comprehension during this time.     

Goals and Learning in Microworlds

We explored the consequences for learning through interaction with an educational microworld called Electric Field Hockey (EFH). Like many microworlds, EFH is intended to help students develop a qualitative understanding of the target domain, in this case, the physics of electrical interactions. Through the development and use of a computer model that learns to play EFH, we analyzed the knowledge the model acquired as it applied the game-oriented strategies we observed physics students using. Through learning-by-doing on the standard sequence of tasks, the model substantially improved its EFH playing ability; however, it did so without acquiring any new qualitative physics knowledge. This surprising result led to an experiment that compared students' use of EFH with standard-goal tasks against two alternative instructional conditions, specific-path and no-goal, each justified from a different learning theory. Students in the standard-goal condition learned less qualitative physics than did those in the two alternative conditions, which was consistent with the model. The implication for instructional practice is that careful selection and analysis of the tasks that frame microworld use is essential if these programs are to lead to the learning outcomes imagined for them. Theoretically, these results suggest a new interpretation for numerous empirical findings on the effectiveness of no-goal instructional tasks. The standing “reduced cognitive load” interpretation is contradicted by the success of the specific-path condition, and we offer an alternative knowledge-dependent interpretation.     

Abstract Planning and Perceptual Chunks: Elements of Expertise in Geometry

We present a new model of skilled performance in geometry proof problem solving called the Diagram Configuration model (DC). While previous models plan proofs in a step-by-step fashion, we observed that experts plan at a more abstract level: They focus on the key steps and skip the less important ones. DC models this abstract planning behavior by parsing geometry problem diagrams into perceptual chunks, called diagram configurations, which cue relevant schematic knowledge. We provide verbal protocol evidence that DC's schemas correspond with the step-skipping inferences experts make in their initial planning. We compare DC with other models of geometry expertise and then, in the final section, we discuss more general implications of our research. DC's reasoning has important similarities with Larkin's (1988) display-based reasoning approach and Johnson-Laird's (1983) mental model approach. DC's perceptually based schemas are a step towards a unified explanation of (1) experts' superior problem-solving effectiveness, (2) experts' superior problem-state memory, and (3) experts' ability, in certain domains, to solve relatively simple problems by pure forward inferencing. We also argue that the particular and efficient knowledge organization of DC challenges current theories of skill acquisition as it presents an end-state of learning that is difficult to explain within such theories. Finally, we discuss the implications of DC for geometry instruction.     

Solving Inductive Reasoning Problems in Mathematics: Not‐so‐Trivial Pursuit

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.