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An algebraic model uses a set of algebraic equations to describe a situation. Constructing such models is a fundamental skill, but many students still lack the skill, even after taking several algebra courses in high school and college. For such students, we developed instruction that taught students to decompose the to-be-modelled situation into schema applications, where a schema represents a simple relationship such as distance-rate-time or part-whole. However, when a model consists of multiple schema applications, it needs some connection among them, usually representedby letting the same variable appear in the slots of two or more schemas. Students in our studies seemed to have more trouble identifying connections among schema applications than identifying the schema applications themselves. We developed several tutoring systems and evaluated them in university classes. One of them, a step-based tutoring system called OMRaaT (One Mathematical Relationship at a Time), was both reliably superior (p = 0.02, d = 0.67) to baseline and markedly superior (p < 0.001, d = 0.84) to an answer-based tutoring system using only commercially available software (MATLAB Grader). 

An algebraic model uses a set of algebraic equations to describe a situation. Constructing such models is a fundamental skill, but many students still lack the skill, even after taking several algebra courses in high school and college. For underachieving college students, we developed a tutoring system that taught students to decompose the to-be-modelled situation into schema applications, where a schema represents a simple relationship such as distance-rate-time or part-whole. However, when a model consists of multiple schema applications, it needs some connection among them, usually represented by letting the same variable appear in the slots of two or more schemas. Students in our studies seemed to have more trouble identifying connections among schemas than identifying the schema applications themselves. This paper describes a newly designed tutoring system that emphasizes such connections. An evaluation was conducted using a regression discontinuity design. It produced a marginally reliable positive effect of moderate size (d = 0.4).