Concept Model
The model we devised,
benefits from the following properties:
- Supplying rich feedback information to students;
- Being modular in the sense that the exercises – tasks –
that are built can be reused and recombined to create other tasks.
It is based on the following main concepts:
Tasks are the building blocks of our approach.
They represent mathematical challenges that students must overcome.
A task is more than a simple mathematical exercise —
it proposes a problem to the student, and then guides, assesses,
and provides feedback, throughout the resolution process.
In general, a task is a generic module, depending on parameters,
which abstracts some class of problems. Basic tasks
(e.g., expression manipulation, equation solving, logic reasoning)
are initially created by experts.
Subtasks of a given main task are the tasks the student may choose
to perform during the resolution process of the main task.
By default, a task has no subtasks. In this case, the student may have access
to all existing tasks sharing the same Context. This gives the student a great
freedom to choose the resolution path he likes. In other words, no guidance is
provided by the teacher. By enumerating the subtasks of a task, the teacher narrows
the set of possible resolution paths, thus helping the student. The teacher can,
furthermore, define a precedence graph among subtasks, which amounts to even more
guidance through the task resolution process.
By combining existing subtasks, the teacher can build complex multiple step problems,
which automatically benefit from their components rich feedback.
The teacher may create links among the subtasks, forcing the output of one subtask to be the input of another.
Contexts define a grammar specifying the language representing the knowledge
under evaluation, and a meta-language, consisting of a set of meta-functions, for teacher usage.
Meta-Functions are functions designed to evaluate student answers,
providing him/her with error explanations.
Oracles are dynamical objects, created for evaluation of student answers.
Oracles are associated with Contexts, and are able to provide
appropriate feedback to the student's resolution.
Resolution is the sequence of intermediate answers given by the student.
Solution Set comprises the answer type plus a condition
to be fulfilled by the answer.
Answer is a mathematical term, input by the student,
which the system verifies to be in the solution set.
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