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Why are Algebra word problems difficult? Using tutorial log files and the power law of learning to select the best fitting cognitive model
 In J.C. Lester, R.M. Vicari, & F. Parguacu (Eds.) Proceedings of the 7th International Conference on Intelligent Tutoring Systems
, 2004
"... Abstract. Some researchers have argued that algebra word problems are difficult for students because they have difficulty in comprehending English. Others have argued that because algebra is a generalization of arithmetic, and generalization is hard, it’s the use of variables, per se, that cause dif ..."
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Abstract. Some researchers have argued that algebra word problems are difficult for students because they have difficulty in comprehending English. Others have argued that because algebra is a generalization of arithmetic, and generalization is hard, it’s the use of variables, per se, that cause difficulty for students. Heffernan and Koedinger [9] [10] presented evidence against both of these hypotheses. In this paper we present how to use tutorial log files from an intelligent tutoring system to try to contribute to answering such questions. We take advantage of the Power Law of Learning, which predicts that error rates should fit a power function, to try to find the best fitting mathematical model that predicts whether a student will get a question correct. We decompose the question of “Why are Algebra Word Problems Difficult? ” into two pieces. First, is there evidence for the existence of this articulation skill that Heffernan and Koedinger argued for? Secondly, is there evidence for the existence of the skill of “composed articulation ” as the best way to model the “composition effect” that Heffernan and Koedinger discovered? 1
When and Why Does Mastery Learning Work: Instructional Experiments with ACTR
 SimStudents”, Proceedings of 6 th International Conference, ITS 2002
, 2002
"... Abstract. Research in machine learning is making it possible for instructional developers to perform formative evaluations of different curricula using simulated students (VanLehn, Ohlsson & Nason, 1993). Experiments using simulated students can help clarify issues of instructional design, such as w ..."
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Cited by 6 (4 self)
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Abstract. Research in machine learning is making it possible for instructional developers to perform formative evaluations of different curricula using simulated students (VanLehn, Ohlsson & Nason, 1993). Experiments using simulated students can help clarify issues of instructional design, such as when a complex skill can be better learned by being broken into components. This paper describes two formative evaluations using simulated students that shed light on the potential benefits and limitations of mastery learning. Using an ACTR based cognitive model (Anderson & Lebiere, 1998) we show that while mastery learning can contribute to success in some cases (Corbett & Anderson, 1995), it may actually impede learning in others. Mastery learning was crucial to learning success in an experiment comparing a traditional early algebra curriculum to a novel one presenting verbal problems first. However, in a second experiment, an instructional manipulation that contradicts mastery learning led to greater success than one consistent with it. In that experiment learning was better when more difficult problems were inserted earlier in the instructional sequence. Such problems are more difficult not because they have more components but because they cannot be successfully solved using shallow procedures that work on easier problems. 1
Opportunities and Limitations of Computer Algebra in Education
"... Computer algebra systems are frequently used for research. In addition, some instructors have based entire advanced courses around these systems. One benefit is that they allow students to become familiar with the methods of calculus by individual experimentation. However, instructors have general ..."
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Computer algebra systems are frequently used for research. In addition, some instructors have based entire advanced courses around these systems. One benefit is that they allow students to become familiar with the methods of calculus by individual experimentation. However, instructors have generally seen computer algebra systems as unsuitable for introductory algebra since the software is capable of doing all the math for the students. That barrier could be overcome if a suitable user interface were developed. Furthermore, embracing computer algebra could make for a completely different type of introductory course that shifts the focus from calculation to problemsolving. Computer algebra systems could also make life easier for math instructors as problem generators for tests and as automatic online graders, even of nonmultiple choice problems. Computer algebra systems offer many new opportunities for math education. 1
Generalized Learning Factors Analysis: Improving Cognitive Models with Machine Learning
"... Of all the initiatives to improve the math level of U.S. students, vastly improving K12 math education has been a top priority. One major development toward this end is Intelligent Tutoring Systems. The technology that drives intelligent tutoring systems is grounded in research into artificial inte ..."
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Of all the initiatives to improve the math level of U.S. students, vastly improving K12 math education has been a top priority. One major development toward this end is Intelligent Tutoring Systems. The technology that drives intelligent tutoring systems is grounded in research into artificial intelligence and cognitive psychology, which seeks to understand the mechanisms that underlie human thought, including language processing, mathematical reasoning, learning, and memory. As students attempt to solve problems using these tutoring systems, the programs analyze their strengths and weaknesses and on that basis provide individualized instruction. Intelligent tutoring systems do not replace teachers. Rather, they allow teachers to devote more oneonone time to each student, and to work with students of varying abilities simultaneously. They allow teachers to design assignments targeted to individual student needs, thereby increasing student advancement at a better rate. A primary example of Intelligent Tutoring Systems helping U.S. children learn math is Cognitive Tutors, an awardwinning computerbased math program that grows out of the extensive research in human learning and artificial intelligence at Carnegie
CMUML09102 Generalized Learning Factors Analysis: Improving Cognitive Models with Machine Learning
, 2008
"... and the National Science Foundation (PSLC) under contract no. SBE0354420. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government ..."
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and the National Science Foundation (PSLC) under contract no. SBE0354420. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. Keywords: cognitive models, intelligent tutoring systems, machine learning, educational data mining, learning factors, psychometrics, additive factor models, latent variable models, exponential principal component analysis, logistic regression, combinatorial search ii To my parents and to my wife iii CONTENTS Generalized Learning Factors Analysis: Improving Cognitive Models with Machine Learning........................................................................................................................ i
Likelihoodbased estimation and model selection for ACTR cognitive models
, 2004
"... ACTR cognitive models are examples of complex hidden Markov processes with a large state space and a sparse transition matrix with absorbing states. While these models are designed to be theoretically interpretable as the thought processes generating observable data, mathematical model estimation a ..."
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ACTR cognitive models are examples of complex hidden Markov processes with a large state space and a sparse transition matrix with absorbing states. While these models are designed to be theoretically interpretable as the thought processes generating observable data, mathematical model estimation and evaluation methods are based on simulation and prediction. There is little methodology applied to ACTR that addresses questions of model complexity, identifiability and generalizability – all topics that can be addressed through study of the likelihood. In this paper I develop likelihoodbased estimation using MCMC methods for a class of ACTR models. I compare selection of ACTR models for scatterplot generation using current predictive methods and using likelihoodbased estimation and Bayes Factors. I propose extending this methodology to include a larger range of ACTR models and to incorporate other model selection criteria (BIC, MDL, etc.) and decision theoretic approaches, with applications to ACTR models including individual differences, and general hidden Markov processes. 1 1
How Abstract Is Symbolic Thought?
"... In 4 experiments, the authors explored the role of visual layout in rulebased syntactic judgments. Participants judged the validity of a set of algebraic equations that tested their ability to apply the order of operations. In each experiment, a nonmathematical grouping pressure was manipulated to ..."
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In 4 experiments, the authors explored the role of visual layout in rulebased syntactic judgments. Participants judged the validity of a set of algebraic equations that tested their ability to apply the order of operations. In each experiment, a nonmathematical grouping pressure was manipulated to support or interfere with the mathematical convention. Despite the formal irrelevance of these grouping manipulations, accuracy in all experiments was highest when the nonmathematical pressure supported the mathematical grouping. The increase was significantly greater when the correct judgment depended on the order of operator precedence. The result that visual perception impacts rule application in mathematics has broad implications for relational reasoning in general. The authors conclude that formally symbolic reasoning is more visual than is usually proposed.
AND EDUCATIONAL TESTING SERVICE
, 2002
"... The paper surveys 15 years of progress in three psychometric research areas: latent dimensionality structure, test fairness, and skills diagnosis of educational tests. It is proposed that one effective model for selecting and carrying out research is to chose one’s research questions from practical ..."
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The paper surveys 15 years of progress in three psychometric research areas: latent dimensionality structure, test fairness, and skills diagnosis of educational tests. It is proposed that one effective model for selecting and carrying out research is to chose one’s research questions from practical challenges facing educational testing, then bring to bear sophisticated probability modeling and statistical analyses to solve these questions, and finally to make effectiveness of the research answers in meeting the educational testing challenges be the ultimate criterion for judging the value of the research. The problemsolving power and the joy of working with a dedicated, focused, and collegial group of colleagues is emphasized. Finally, it is suggested that the summative assessment testing paradigm that has driven test measurement research for over half a century is giving way to a new paradigm that in addition embraces skills level formative assessment, opening up a plethora of challenging, exciting, and societally important research problems for psychometricians.
Bethany RittleJohnson
, 2010
"... We present a methodology for designing better learning environments. In Phase 1, 6thgrade students ’ (n = 223) prior knowledge was assessed using a difficulty factors assessment (DFA). The assessment revealed that scaffolds designed to elicit contextual, conceptual, or procedural knowledge each imp ..."
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We present a methodology for designing better learning environments. In Phase 1, 6thgrade students ’ (n = 223) prior knowledge was assessed using a difficulty factors assessment (DFA). The assessment revealed that scaffolds designed to elicit contextual, conceptual, or procedural knowledge each improved students’ability to add and subtract fractions. Analyses of errors and strategies along with cognitive modeling suggested potential mechanisms underlying these effects. In Phase 2, we designed an intervention based on scaffolding this prior knowledge and implemented the computerbased lessons in mathematics classes. In Phase 3, we used the DFA and supporting analyses to assess student learning from the intervention. The posttest results suggest that scaffolding conceptual, contextual, and procedural knowledge are promising tools for improving student learning. Wellstructured, organized knowledge allows people to solve novel problems and to remember more information than do memorized facts or procedures (see Bransford, Brown, & Cocking, 2001, for a review). Such wellstructured knowledge requires that people integrate their contextual, conceptual and procedural knowledge in a domain. Unfortunately, U.S. students rarely have such integrated and robust knowledge in mathematics or science (Beaton et al., 1996; Reese, Miller, Mazzeo, & Dossey, 1997). Designing learning environments that support integrated knowledge is a key challenge for the field, especially given the low number of established tools for guiding this design process (Lesh, Lovitts, & Kelly, 2000). In this article, we use our design of a lesson within a larger design exRequests for reprints should be sent to Bethany RittleJohnson, Vanderbilt University, 230