Abstract not found

There is a significant gap between theories of general psychological functions on one hand (e.g., memory) and theories of mathematical content knowledge on the other (e.g., content of algebra). To better guide the design of ground breaking and demonstrably better mathematics instruction, we need instructional principles and associated design methods to fill this gap in a way that is not only consistent with psychological and content theories but prompts and guides us beyond what those theories can do. Toward this goal, I reflect on lessons from past and current Cognitive Tutor mathematics projects. From this experience, I have abstracted four instructional bridging principles, Situation-Abstraction, Action-Generalization, Visual-Verbal, and Conceptual-Procedural, and associated methods for applying them. I illustrate these in the context of the design of the successful Cognitive Tutor Algebra course (now in more than 800 schools) and the on-going research and development of a Cognitive Tutor course for 6 th grade mathematics.

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. For 25 years, we have been working to understand mathematical cognition through the use of cognitive modeling and applying that knowledge to constructing curricula (both text and software) that are more educationally effective than preexisting approaches. This work has been successful on many levels. It has advanced knowledge of cognition in general and of mathematical cognition in particular; the resulting curricula have proven to be educationally effective in school settings; and the curricula, as commercial products, have found a strong following in the school marketplace. We believe that our development model, which involves a close and continuing relationship among basic research, applied research, and field testing, can serve as a model for other efforts to apply cognitive psychology to education. In this article, we describe some of the history of our efforts, our view of the relationship between basic research and development, and some directions for further research.

This study examined a corpus of 10 widely used pre-algebra and algebra textbooks, with the goal of investigating whether they exhibited a symbol precedence view of mathematical development as is found among high school teachers. The textbook analysis focused on the sequence in which problem-solving activities were presented to students. As predicted, textbooks showed the symbol precedence view, presenting symbolic problems prior to verbal problems.

In a series of 5 experiments in 2000 and 2001, several hundred students at two different universities with three different professors and six different teaching assistants took a semester long course on causal and statistical reasoning in either traditional lecture/recitation or online/recitation format. In this article we compare the pre-post test gains of these students, we identify features of the online experience that were helpful and features that were not, and we identify student learning strategies that were effective and those that were not. Students who entirely replaced going to lecture with doing online modules did as well and usually better than those who went to lecture. Simple strategies like incorporating frequent interactive comprehension checks into the online material (something that is difficult to do in lecture) proved effective, but online students attended face-to-face recitations less often than

recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Keywords: computational modeling, cognitive modeling, instructional theory, machine learning, learning science, second language learning, mathematics learning, science learning, robust learning, learning theory, knowledge componentsExecutive Summary The volume of research on learning and instruction is enormous. Yet progress in improving educational outcomes has been slow at best. Many learning science results have not been translated into general practice and it appears that most that have been fielded have not yielded significant results in randomized control trials. Addressing the chasm between learning science and educational practice will require massive efforts from many constituencies, but one of these efforts is to develop a theoretical framework that permits a more systematic accumulation of the relevant research base. A key piece in such a theoretical framework is the development of levels of analyses that are fine enough to be supported by cognitive science and cognitive neuroscience, but also at levels appropriate to guide the design of effective educational practices. An ideal scientific solution would be a small set of universal instructional principles that can be applied to produce efficient

Animations are a versatile media for displaying changes over time. They can show cellular processes, a billion years of continental drift, the assembly of a desk, and even the invisible shifting of political tides. Most animations depict changes to a situation, such as a desk being assembled. In this chapter, we describe a series of software environments, called Teachable Agents (TAs) that use animations in another way. Rather than displaying a situation, the TAs animate the thoughts an individual might use to reason about that situation. For example, using the same well-structured representations as experts, TAs can visually model how to reason through the causal chains of an ecosystem. This is worthwhile, because the goal of learning is often to emulate an expert’s reasoning processes, and animations of thought make that reasoning visible. For novices, learning to reason with an expert’s knowledge organization is as important as learning the bare facts themselves. We build TA systems to capitalize on the adage that an effective way to learn something is to teach it, and this framework has allowed us to introduce some uncommon uses of animation. One novelty is that students help build the animation rather than just

The examination of user data as a basis for developing production models of user behavior has been a major focus in the PAT Algebra I Tutor's development. In recent work, we have investigated relationships between related tasks and the solution strategies displayed by students. To solve a PAT Algebra I problem, students must complete several related arithmetic and algebraic tasks. The sequences in which these tasks are completed suggest problem-solving strategies of students. We have observed a characteristic pattern of students' success rates on related tasks. We have also observed that students' success on specific skills (e.g. constructing a symbolic representation) may differ depending on whether students previously carried out related tasks in the same problem (e.g. solving an analogous arithmetic question). This information has important implications for our user model and our modeling approach.

The potential to use mathematics so ftware to enhance student thinking and development is discussed and a taxonomy of software categories is outlined in this paper. Briefly, there are five categories of tool-based mathematics software that can be used fruitfully in a mathematics curriculum: (a) review and practice, (b) general, (c) specific, (d) environment, and (e) communication. A description of the affordances and constraints of the five types of software and how each facilitates different aspects of student learning clarifies the ways in which diverse off-the-shelf offerings can be used to address the goals of mathematics instruction, from building basic skills to exploring mathematical applications in the real world. Money spent on computers in public schools has increased at a steady rate over the last 20 years. According to the President’s Committee of Advisors on Science and Technology (1997), over $3.5 billion was spent on computers in 1997 alone (Hooper & Hokanson, 2000). Even with this increase in computers in the classroom, most instruction utilizes

Mathematics teachers and mathematics educational researchers were asked to rank order arithmetic and algebra problems for their predicted problem-solving difficulty for students.