Integrating Rule-Based and Input-Based Approaches for Better Error Diagnosis in Expression Manipulation Tasks

Rein Prank
Institute of Computer Science
University of Tartu
50409 Tartu, Estonia

Abstract. When the student solves an expression manipulation task, he should at each solution step

  1. choose a transformation rule,
  2. select the operands (certain parts of expressions or equations) for this rule,
  3. replace them with the result of the operation.
For proper learning of expression manipulation as well as for diagnosis and assessment, an environment should be available where all the necessary decisions and calculations would be made by the student, and the environment would be able to understand the mistakes and give feedback. The existing software does not address the arising problems in an integrated manner. Some programs (CAS and rule-oriented environments) require from the student only execution of the first and (partially) the second substep. The transformation itself is made by the computer and the student has no possibility to make mistakes. Input-oriented programs (for instance, APLUSIX) use paper-and-pencil-like dialogue design where a transformation step consists purely of entering the next line. As a consequence, practically only syntactical errors and the non-equivalence with the previous line can be diagnosed. In 2004 we launched a project called T-algebra to design and implement an interactive learning environment for expression manipulation tasks in four areas of school algebra: calculation of the values of numerical expressions; operations with fractions; solving of linear equations, inequalities and linear equation systems; operations with monomials and polynomials. T-algebra integrates rule-based and input-based approaches in an Action-Object-Input dialogue scheme, where the student performs all three stages of the step in an explicit manner. At the third stage the student has to enter only the expression replacing the terms marked at the second stage. Information received from the input of the three stages enables reliable diagnosis of the following types of mistakes:
  1. application of the selected operation is impossible,
  2. application of the selected operation does not correspond to the algorithm,
  3. marked term is not a proper subexpression (order of operations misunderstood),
  4. marked term does not have the form required for the operands of the selected rule,
  5. operands do not satisfy the compatibility requirements (are not like terms, etc.),
  6. operands do not satisfy the location requirements (do not belong to the same sum, etc.),
  7. the input does not have the structure required for the result of the applied rule,
  8. the input is not equivalent to the marked part,
  9. the result of substitution of input is not equivalent to the previous line.
In addition, it is possible to formulate error messages pointing exactly to incorrect parts.

The presentation demonstrates our expression manipulation interface and error diagnosis, compares the amount of the student's work with other interfaces, and discusses the problems, which may arise.