Statistical methods are described for diagnosing and treating three important problems in covariate tests of significance: curvilinearity, covariable effectiveness, and treatment-covariable interaction. Six major assumptions, prerequisites for covariate procedure, are discussed in detail: (1) normal distribution, (2) homogeneity of variances, (3) covariable-group independence, (4) reliability, (5) linearity, and (6) homogeneity of regression. A generalization of the Johnson-Neyman tests of significance (originally developed for two groups and two covariables, but frequently ignored by many critics of analysis of covariance) to cover any number of groups and covariables is presented. The procedure is viewed as a powerful tool for measuring the relationship between learner characteristics and teaching strategies when the regression slopes are not homogeneous. Aided by computer technology, it is proposed as a relatively easy method for classification of students by their individual needs as well as by the characteristics of teaching methods. (CK)