ABSTRACT
Suppose you wanted to improve how subtraction is taught. Clearly, in designing an appropriate curriculum, you may want to understand how children go about solving the task and why they make the mistakes they do. One approach to developing such an understanding would be to build, test, and refine computational models of subtraction behavior. Of course, different children approach subtraction in different ways, so your model would need to include parameters that allow its behavior to be tailored to that of individual students. For example, some students fail to borrow and always subtract the smaller digit in a column from the larger even when the smaller digit appears in the upper number: https://www.w3.org/1998/Math/MathML"> 42 − 18 ¯ 36 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203052969/ccbfad87-d853-4478-b73d-ab51292f55c3/content/math_29_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Thus, you would probably want to include a parameter that controls whether your model makes this error: Parameter
Smaller-from-larger
Values
makes this error
does not make this error
Similarly, many children treat blanks as if they were Is and make errors like the following: https://www.w3.org/1998/Math/MathML"> 35 − 2 ¯ 23 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203052969/ccbfad87-d853-4478-b73d-ab51292f55c3/content/math_30_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> So you might include another parameter, Sub-one-over-blank, that controls whether or not your model behaves this way. Some mistakes may be due to more than one error. For example, the following behavior could be explained by assuming that the student made both errors described previously: https://www.w3.org/1998/Math/MathML"> 671 − 28 ¯ 557 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203052969/ccbfad87-d853-4478-b73d-ab51292f55c3/content/math_31_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> In addition to these two, Brown and Burton (1978) identified a large number of other common subtraction errors, which they called “bugs” (also see Burton, 1982; VanLehn, 1990). For a subtraction model to predict the behavior of different students with any accuracy, it would seem to require a fairly large number of parameters.