Diagnostic of Students' procedural errors in Addition and subtraction of decimals Using Bayesian Networks
Subject Areas : Educational Psychology
Hashem Hooshyar
1
(Tehran)
Ali Delavar
2
(Tehran)
Noorali Farrokhi
3
(Tehran)
Jalil Younesi
4
(Tehran)
Keywords: Bayesian networks, procedural errors, addition of decimals, subtraction of decimals,
Abstract :
Decimal numbers are one of the most important and useful topics in mathematical education. However, students have trouble with decimal numbers. One way to investigate the reason of difficulty in working decimals for the students is through error analysis. Diagnosis of errors in procedural skills is difficult because of their unstable nature. This study investigates the diagnosis of procedural errors in students’ decimals addition and subtraction performance, by proposing and evaluating a probability-based approach using Bayesian networks. This approach assumes a causal network relating hypothesized decimals addition and subtraction errors to the observed test items. This study is practical and quantitative. The population of this study included all the sixth-grade students of Birjand city in academic year 2017 – 2018. A sample consisted of 407 students was selected through multi-stage sampling method. Network performance was evaluated by two types of testing situations: using binary data (scored as correct or incorrect) and diagnostic use of wrong answers by a multi-choice test. Results showed Bayesian network with binary data had poor performance in errors diagnosis but diagnostic use of students’ wrong answers improved network performance. The Kappa agreement rate for the Bayesian network with wrong answers reached above 90%. Our results suggest that reliable diagnosis of errors can be achieved by using a Bayesian network framework with students’ wrong answers.
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Conati, C., Gertner, A., & VanLehn, K. (2002). Using Bayesian networks to manage uncertainty in student modeling. User Modeling and User-Adapted Interactions, 12, 371-417.
Cox, L. S. (1975). Systematic errors in the four vertical algorithms in normal and handicapped populations. Journal for Research in Mathematics Education, 6, 202-220.
De La Torre, J. (2009). A cognitive diagnosis model for cognitively based multiple-choice options. AppliedPsychological Measurement, 33, 163-183.
Isotani, S., McLaren, B., & Altman, M. (2010). Towards Intelligent Tutoring with Erroneous Examples: A Taxonomy of Decimal Misconceptions. Proc. of the Int. Conference on Intelligent Tutoring Systems. LNCS 6095, 346-348.
Käser, T., Klingler, S., Schwing, A. G., & Gross, M. (2017). Dynamic Bayesian Networks for Student Modeling. IEEE Transactions on Learning Technologies, 10(4), 450-462.
Lauritzen, S. L., & Spiegelhalter, D. J. (1988). Local computations with probabilities on graphical structures and their application to expert systems. Journal of the Royal Statistical Society, Series B, 50,157-224.
Lee, J., & Corter, J. E. (2011). Diagnosis of subtraction bugs using Bayesian Networks. Applied Psychological Measurement, 35(1), 27-47.
Li, X. (2006). Cognitive analysis of students’ errors and misconceptions in variables, equations and functions. Ph.D. Desertation, Texas A&M University.
Mislevy, R. J. (1995). Probability-based inference in cognitive diagnosis. In P. Nichols, S. Chipman, &R. Brennan (Eds.), Cognitively diagnostic assessment (pp. 43-71). Mahwah, NJ: Erlbaum.
Nudelman, Z., Moodley, D., & Berman, S. (2019). Using Bayesian Networks and Machine Learning to Predict Computer Science Success. Paper presented at the ICT Education, Cham.
Pearl, J. (1988). Probabilistic reasoning in intelligent systems. San Mateo, CA: Morgan Kaufmann.
Steinle, V., Stacey, K., & Chambers, D. (2002). Teaching and Learning about Decimals [CD-ROM]: Department of Science and Mathematics Education, The University of Melbourne. Online sample
Templin, J. L., Henson, R. A., Templin, A. E., & Roussos, L. (2008). Robustness of hierarchical modeling of skill association in cognitive diagnosis models. Applied Psychological Measurement, 32, 559-574.
VanLehn, K. (1981). Bugs are not enough: Empirical studies of bugs, impasses and repairs in proceduralskills (Technical Report). Palo Alto, CA: Xerox Palo Alto Research Center.
VanLehn, K. (1990). Minds bugs: The origins of procedural misconceptions. Cambridge, MA: MIT Press.
VanLehn, K., & Martin, J. (1998). Evaluation of an assessment system based on Bayesian student modeling. International Journal of Artificial Intelligence in Education, 8, 179-221.
Watson, S. M. R., Lopes, J., Oliveira, C., & Judge, S. (2018). Error patterns in Portuguese students’ addition and subtraction calculation tasks: implications for teaching. Journal for Multicultural Education, 12(1), 67-82.
_||_Almond, R. G., & Mislevy, R. J. (1999). Graphical models and computerized adaptive testing. Applied Psychological Measurement, 23, 223-237.
Almond, R. G., Mislevy, R. J., Steinberg, S. L., Yan, D., & Williamson, D. M. (2015). Bayesian Networks in Educational Assessment. New York, NY: Springer.
Bakhshalizadeh, Sh., & Broojerdian, N. (2014). Identifying common misconceptions in fourth-grade students in the field of mathematical content. Organization for Research and Planning. Institute for Education Studies. [In Persian]
Brown, J. S., & Burton, R. B. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2, 155-192.
Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4, 379-426.
Conati, C., Gertner, A., & VanLehn, K. (2002). Using Bayesian networks to manage uncertainty in student modeling. User Modeling and User-Adapted Interactions, 12, 371-417.
Cox, L. S. (1975). Systematic errors in the four vertical algorithms in normal and handicapped populations. Journal for Research in Mathematics Education, 6, 202-220.
De La Torre, J. (2009). A cognitive diagnosis model for cognitively based multiple-choice options. AppliedPsychological Measurement, 33, 163-183.
Isotani, S., McLaren, B., & Altman, M. (2010). Towards Intelligent Tutoring with Erroneous Examples: A Taxonomy of Decimal Misconceptions. Proc. of the Int. Conference on Intelligent Tutoring Systems. LNCS 6095, 346-348.
Käser, T., Klingler, S., Schwing, A. G., & Gross, M. (2017). Dynamic Bayesian Networks for Student Modeling. IEEE Transactions on Learning Technologies, 10(4), 450-462.
Lauritzen, S. L., & Spiegelhalter, D. J. (1988). Local computations with probabilities on graphical structures and their application to expert systems. Journal of the Royal Statistical Society, Series B, 50,157-224.
Lee, J., & Corter, J. E. (2011). Diagnosis of subtraction bugs using Bayesian Networks. Applied Psychological Measurement, 35(1), 27-47.
Li, X. (2006). Cognitive analysis of students’ errors and misconceptions in variables, equations and functions. Ph.D. Desertation, Texas A&M University.
Mislevy, R. J. (1995). Probability-based inference in cognitive diagnosis. In P. Nichols, S. Chipman, &R. Brennan (Eds.), Cognitively diagnostic assessment (pp. 43-71). Mahwah, NJ: Erlbaum.
Nudelman, Z., Moodley, D., & Berman, S. (2019). Using Bayesian Networks and Machine Learning to Predict Computer Science Success. Paper presented at the ICT Education, Cham.
Pearl, J. (1988). Probabilistic reasoning in intelligent systems. San Mateo, CA: Morgan Kaufmann.
Steinle, V., Stacey, K., & Chambers, D. (2002). Teaching and Learning about Decimals [CD-ROM]: Department of Science and Mathematics Education, The University of Melbourne. Online sample
Templin, J. L., Henson, R. A., Templin, A. E., & Roussos, L. (2008). Robustness of hierarchical modeling of skill association in cognitive diagnosis models. Applied Psychological Measurement, 32, 559-574.
VanLehn, K. (1981). Bugs are not enough: Empirical studies of bugs, impasses and repairs in proceduralskills (Technical Report). Palo Alto, CA: Xerox Palo Alto Research Center.
VanLehn, K. (1990). Minds bugs: The origins of procedural misconceptions. Cambridge, MA: MIT Press.
VanLehn, K., & Martin, J. (1998). Evaluation of an assessment system based on Bayesian student modeling. International Journal of Artificial Intelligence in Education, 8, 179-221.
Watson, S. M. R., Lopes, J., Oliveira, C., & Judge, S. (2018). Error patterns in Portuguese students’ addition and subtraction calculation tasks: implications for teaching. Journal for Multicultural Education, 12(1), 67-82.
