Developing proficiency in math: cause and effect in a cognitive process

Symbolic mathematical software has become inexpensive enough that it is almost practical to require each first-year calculus student to purchase it. But having a program do math for you might not be learning, and does not teach you how to apply it. This paper examines research issues related to learning calculus including cognitive processes, the role of proficiency and a proposed causal model. Current trends in calculus curriculum reform are reviewed including the need for better measurements of learning effects. John R. Anderson´s ACT-R theory is proposed as a comprehensive cognitive learning theory and basis for the causal model. The primary theoretical contribution is advocating proficiency as a construct for measuring learning in both individual and collaborative environments. Proficiencies (such as subject mastery, missing elements, and faulty elements) are both causes and effects, and are therefore modeled as control feedback mechanisms (more similar to an instrument panel than a direct outcome). Based on the causal model and precise measurements of proficiency, an experiment is proposed to study the impact of symbolic math software and collaborative processes in the teaching of calculus to university pre-business students