The acquisition of basic mathematical skills and competence in systematic sequential problem-solving techniques is important to success in a variety of fields of study, including numerous "non-mathematical" college disciplines such as economics, psychology, and philosophy. However many students, particularly many of the non-traditional students, do not have a firm foundation in these skills. They have no schema for sequential, step-by-step analysis of problems, and thus cannot solve mathematical problems involving even very low levels of abstraction. They tend to believe, particularly when dealing with word problems, that they either know the answer to a problem or do not, and they lack the skill to break a problem into steps, and then proceed in an orderly fashion toward a solution. Or they may fail to identify relevant information and important variables.
At Bowling Green State University we have been experimenting with devices for teaching analytical reasoning in primary mathematics to educationally disadvantaged students. Evidence of the need for such instruction comes from a recent analysis of the mathematics placement test scores earned by 200 freshmen from low-income families. At least 63 percent of these students were in need of basic pre-algebra math. Other evidence comes from the steady increase in enrollment in a noncredit algebra course offered at the University. Thirty-one percent of the students who take the University math placement test are advised to enroll in this course because they are seriously deficient in math skills.
The experimental program at Bowling Green State University is designed to teach analytical reasoning to educationally disadvantaged students. The program is based on principles derived from Piaget and other investigators of human thought processes. This research recognizes that effective thinkers in any area engage in mental activities which are different from the activities of novices. The cognitive-skills approach to teaching which we employed begins by identifying the mental activities used by successful thinkers as they solve problems and master ideas. Other students who have not yet demonstrated these competencies are then taught the techniques used by high aptitude thinkers. Mastery of such skills is facilitated initially by providing guidelines that lead the student through all the necessary steps or operations and then by providing enough drill and variety to facilitate refinement and generalization of the new skills. This training process is not haphazard or random. Rather, all the