Assessing Mathematical Knowledge in a Learning Space

According to Knowledge Space Theory (KST) (cf. Doignon and Falmagne, 1999; Falmagne and Doignon, 2011), a student’s competence in a mathematics or science subject, such as elementary school mathematics or first year college chemistry, can be described by the student’s ‘knowledge state,’ which is the set of ‘problem types’ that the student is capable of solving. (In what follows, we abbreviate ‘problem type’ as ‘problem’ or ‘item.’) As the student masters new problems, she moves to larger and larger states. Some states are closer to the student’s state than others, though, based on the material she must learn in order to master the problems in those states. Thus, there is a structure to the collection of states, and this structure gives rise to a ‘learning space,’ which is a special kind of knowledge space. These concepts have been discussed at length in Chapter 1 of this volume. We recall here that the collection of states forming a learning space always contains the ‘empty state’ (the student knows nothing at all in the scholarly subject considered) and the ‘full state’ (the student knows everything in the subject). The collection of states must also satisfy two pedagogically cogent principles, which we state below in nonmathematical language.