Consider diagnosing the errors school students make when adding multi-digit numbers. Consider adding two multi-digit numbers to form a third multi-digit number, as in A1 A0

+ B1 B0 C2 C1 C0 where Ai, Bi, and Ci are all digits.

(a) Suppose you want to model whether students know the skills of single-digit addition and carrying. If students know how, they usually get the correct answer, but sometimes make mistakes. Students who do not know simply guess. Draw a belief network for a single student and problem involving adding two 2-digit numbers to get a 3-digit number. [Hint: Each of the Ai, Bi, and Ci, and the carries, are variables, and there is a variable for each skill of the student.]

(b) Suppose there are multiple students and multiple problems. Give a plate representation, either drawing plates or specifying the parametrized random variables. There should be plates for students and problems. Think about what variables depend on the student, the problem, or both.

(c) Suppose now you want to also represent multiple digits and multiple times.
Assume that the students’ knowledge can change through time. Draw the plate model. What challenge arises in plate models for times and digits that did not arise for students and problems?