Glaser, 1972, p. 7). In searching for measures of aptitude which will have higher predictive value, traditional concepts of aptitude will change (Glaser, 1972) and so must the existing aptitude measures (Cronbach, 1957). At present, "our generally used aptitude constructs are not productive dimensions for measuring those individual differences that interact with different ways of learning" (Glaser, 1972). With respect to ATI, the term, "aptitude", includes all character istics of individuals, rather than being limited to the common singular concept of aptitude. Aptitude then is "a complex of personal character istics that accounts for an individual's end state after a particular educational treatment, i.e., that determines what he learns, how much he learns, or how rapidly he learns" (Cronbach, 1967, p. 23). Cronbach also hypothesized that aptitude "may have as much to do with styles of thought and personality variables as with the abilities covered in con ventional tests" (Cronbach, 1967, p. 24). The topic of concern in this study is mathematical aptitude. As viewed by Cronbach (1967, p. 27) "we haven't the faintest evidence, for example, what constitutes mathematical aptitude, save for the obvious fact that a person who has mastered one mathematical fact or process has an advantage in learning the next process in a hierarchy." The primary purpose of this research study is to define, more precisely, the factors of mathematical aptitude. However, since aptitude information becomes more useful when we know how it interacts with the given treatments (Cronbach, 1967), the secondary purpose is to study the interactions of the variables comprising mathematical aptitude with two treatments, 3 algebra and geometry. The term, "algebra", refers to the first year course in algebra, which generally includes the use of the quadratic formula in factoring; "geometry" refers to the full year course in plane Euclidean geometry. According to Bracht (1970, p. 639), "to be differentially effective for various types of students, the alternative treatments should demand different abilities for successful performance." The function and major differences of these instructional treatments have been described by Salamon (1972, p. 340) in his preferential model for aptitude-treatment interactions. Treatment in his model "call upon and utilize learner's higher aptitudes, neither making up for deficiencies nor compensating for them. Differences may be in content, structure, modality of presen tation, etc." Success can be predicted from this model "when an aptitude in which he [^a student J is proficient is called upon" (Salamon, 1972, p. 340). For example, if having good spatial perception enhances learn ing in geometry, a student's success in geometry could be predicted from his spatial perception ability scores. If the student were found to be deficient in this ability, alternative treatments, which are either compensatory or remedial, could be given to the student in order to max imize his learning and success. The problem in this study ...