Abstract: A method for representing and solving algebraic equations that allows a user to view and solve algebraic equation with a virtual gear system. The virtual gear system includes a virtual primary cog and virtual secondary cogs. The virtual primary cog represents a range of outcomes for the virtual gear system and contains a number of teeth that is quantitatively greater than a numerical constant of the algebraic equation; amongst the teeth is a target tooth that represents the numerical constant. Each virtual secondary cogs represent a term of the algebraic equation and includes a coefficient and a variable. Each of the virtual secondary cogs contains a number of teeth equal to the coefficient. The equation is solved by rotating the virtual secondary cogs until the target tooth is aligned with a fixed pointer where rotation of the virtual secondary cog represents a value input for the variable of a term.

Abstract: A method for representing and solving algebraic equations that allows a user to view and solve algebraic equations through a novel user-interface. The user-interface includes a primary cog and a plurality of secondary cogs as parts of a gear system. The primary cog represents a range of outcomes for the gear system and contains a plurality of teeth that is quantitatively greater than a numerical constant of the algebraic equation; amongst the teeth is a target tooth that represents the numerical constant. Each of the secondary cogs represent a term of the algebraic equation and includes a coefficient and a variable. Each of the secondary cogs contains a number of teeth equal to the coefficient. The equation is solved by rotating the secondary cogs until the target tooth is aligned with a fixed pointer where rotation of the secondary cog represents a value input for the variable of a term.

Abstract: A teaching method and system which effectively and efficiently teaches content to a user. The method includes administering a series of teaching topics to the user through a personal computing (PC) device, where each topic includes a lead problem and a plurality of secondary problems. As the user solves the problems within each topic, the PC device monitors and analyzes the user's cognitive ability based on which solution was found for each of the problems. Subsequently, the PC device alters which problems are then offered to the user at each incremental step to reflect the user's performance on the previously problems. This is achieved by using open-ended problems with an optimal solution and an at least one other solution. If the optimal solution is identified, then the user can skip certain problems. If the other solution is identified, the user is simply moved to the next problem within the topic.

Abstract: A method for measuring the mathematical fitness of a plurality of user accounts. The method includes prompting an arbitrary account from the user accounts to solve a mathematical problem through a corresponding personal computing (PC) device. After receiving answering data, an external server compares the answering data against each of a plurality of solutions to identify a matching solution. Throughout this method, the external server monitors the PC device and measures a performance score, a persistence score, and records a user solving procedure. The user solving procedure is then compared to a usage frequency data for a plurality of known solving procedures in order to measure a creativity score. The arbitrary account can then view the performance score, the persistence score, and the creativity score in the form of a performance portfolio for the mathematical problem. This is repeated for each user account for a plurality of mathematical problems.

Abstract: A method for representing and solving algebraic equations that allows a user to view and solve algebraic equation with a virtual gear system. The virtual gear system includes a virtual primary cog and virtual secondary cogs. The virtual primary cog represents a range of outcomes for the virtual gear system and contains a number of teeth that is quantitatively greater than a numerical constant of the algebraic equation; amongst the teeth is a target tooth that represents the numerical constant. Each virtual secondary cogs represent a term of the algebraic equation and includes a coefficient and a variable. Each of the virtual secondary cogs contains a number of teeth equal to the coefficient. The equation is solved by rotating the virtual secondary cogs until the target tooth is aligned with a fixed pointer where rotation of the virtual secondary cog represents a value input for the variable of a term.

Abstract: A method for representing and solving algebraic equations that allows a user to view and solve algebraic equations through a novel user-interface. The user-interface includes a primary cog and a plurality of secondary cogs as parts of a gear system. The primary cog represents a range of outcomes for the gear system and contains a plurality of teeth that is quantitatively greater than a numerical constant of the algebraic equation; amongst the teeth is a target tooth that represents the numerical constant. Each of the secondary cogs represent a term of the algebraic equation and includes a coefficient and a variable. Each of the secondary cogs contains a number of teeth equal to the coefficient. The equation is solved by rotating the secondary cogs until the target tooth is aligned with a fixed pointer where rotation of the secondary cog represents a value input for the variable of a term.