Abstract: In a computer-implemented educational environment, four particular features are provided, which may be used separately or in combination, to drive the instruction of a particular curriculum, e.g. the instruction of fractions. In another implementation, educators and authors may use permutations and combinations of such features to construct in a graphical user interface a progression of interactive elements which help guide a student from visual concepts to procedures, such as equations and formulas. In a particular implementation, by using these features carefully, a curriculum author may design instructional sequences that provide careful, step-by-step sequences that help students progress from understanding the visual concepts, e.g. of fractions, to mastering procedures, e.g. fractions equations.

Abstract: An apparatus and method for computer-implemented tools for mathematics instruction are provided. These tools enable providing a context for a given problem, e.g. a mathematical problem. Such tools enable demonstrating understanding of the problem by enabling paraphrasing the context. The tools enable further understanding of the problem by enabling using models to depict the paraphrase. The tools further enable solving the problem. For example, a tool enables students to learn what it means to multiply fractions, to represent multiplication of fractions using visual models, and to use equations to compute answers. For example, students may be given a story problem as context. They paraphrase this context, choosing between two types of multiplication problems: groups of and part of. Students use one of two models to depict the paraphrase: the Two Number Line Model or the Double Area Model. Students solve the equation.