9.3 Theory

For those interested in looking beneath the surface, these questions concern the principles on which ALEKS is designed and constructed.

What is the theory behind ALEKS?
[Chapter 8] ALEKS is based on a field of Cognitive Science (Mathematical Psychology) called "Knowledge Spaces" (or "Learning Spaces"). The purpose of research in Knowledge Spaces is to model human knowledge in any subject, using mathematical tools such as Set Theory, Combinatorics, and Markovian Processes, so as to make possible fast and accurate assessment through interactive computer applications. There are numerous scientific publications in the field of Knowledge Spaces dating back to the early 1980's. A recent, authoritative treatment (with Bibliography) is Doignon and Falmagne, Learning Spaces (Berlin, Heidelberg: Springer-Verlag, 2011).

What is an "item"?
[Sec. 8.2.1] In Knowledge Space theory, an "item" is a concept or skill to be learned, the mastery of which is captured by a "problem type" serving as the basis for specific assessment and practice problems. Thus the item "Addition of two-digit numbers without carry" might produce the problem (instance) "What is 25 plus 11?"

What is a "domain"?
[Sec. 8.2.1] In Knowledge Space theory, a "domain" is the set of all items making up a particular subject matter, such as Behavioral Science Statistics. A learner is considered to have mastered the domain when that learner can solve problems corresponding to all the items in the domain.

What is a "knowledge state"?
[Sec. 8.2.2] In Knowledge Space theory, a "knowledge state" is the set of items belonging to a domain that a learner has mastered at some point in time. We speak of knowledge states in relation to a particular learner and a particular domain. Obviously, a learner's knowledge changes in time, and the goal of learning is that the knowledge state should eventually include (correspond to) the entire domain.

What is the "outer fringe" of a knowledge state?
[Sec. 8.2.4] In Knowledge Space theory, a learner's "outer fringe" is the set of items, any one of which can be added to the current knowledge state without others to make a new, feasible knowledge state. These are the items that the student is considered most "ready to learn." Progress is made from one state to another through one of the items in the first state's "outer fringe."

What is the "inner fringe" of a knowledge state?
[Sec. 8.2.4] In Knowledge Space theory, a learner's "inner fringe" is the set of items, any one of which can be taken away without any others from the current knowledge state to make a new, feasible knowledge state. These are the items that the student may have learned recently, and thus whose knowledge might need reinforcement.

What is a "knowledge structure"? What is a "knowledge space"?
[Sec. 8.2.3] In Knowledge Space theory, "knowledge structure" or "knowledge space" (the two concepts differ in a technical way) refers to the collection of feasible knowledge states for a particular domain. It is a key point that not all sets of items from the domain (subsets of the domain) are feasible knowledge states. For instance, in mathematics there can be no knowledge state containing the item "finding the square root of an integer" that does not contain the item "addition of two-digit numbers without carry," since no one will master the first without having mastered the second.

How was the structure created?
The knowledge structures (or, briefly, "structures") used by ALEKS are created by analysis of the subject matter and refined on the basis of data obtained from students' learning experiences. When ALEKS assesses a student, it is actually searching the structure for knowledge states that match the student's present competence.

What is the educational philosophy behind ALEKS?
The educational use of ALEKS is not tied to any particular theory of education or knowledge acquisition. A key insight underlying ALEKS is the existence of a vast multiplicity of diverse "learning paths" or sequences of topics by which a field can be mastered. Based on an inventory of knowledge states that numbers in the tens of thousands (for the subjects currently covered by ALEKS), the specialized tools of Knowledge Space theory make it possible for the system to accommodate literally billions of possible individual learning paths implied by the relations among states. ALEKS does not embody a particular philosophy of teaching mathematics or statistics; it is compatible with any pedagogical approach.