Disclosed is a digital pen that tracks the user's writing and provides useful feedback based on the user's writing. In one embodiment, the pen may provide feedback when the user has written a misspelled word, invalid mathematical expression, or any noncompliant expression. The pen may also provide feedback relating to the user's handwriting. The feedback may be visual, auditory, or tactile, and may be realtime or delayed. Statistics relating to the user's performance may be tracked, uploaded to external devices, and shared with others. This allows the user and interested parties to track the user's progress over time. The disclosed pen will be useful in educational settings.

A tile instructional system for presenting a linear growth problem and for allowing the linear growth problem to be solved comprises an input tray having at least a first tile and a second tile. The first tile has a first marking and the second tile has a second marking. Each of the first tile and the second tile is configured to selectively expand in accordance with its respective marking. The system includes a tile bed comprising a plurality of tile slots. Each of the plurality of tile slots is configured to receive a tile. The linear growth problem is solvable by situating in the tile bed at least one of the first tile and the second tile and by causing each tile situated in the tile bed to expand in accordance with its respective marking to fill the tile bed.

A tile instructional system for presenting a linear growth problem and for allowing the linear growth problem to be solved comprises an input tray having at least a first tile and a second tile. The first tile has a first marking and the second tile has a second marking. Each of the first tile and the second tile is configured to selectively expand in accordance with its respective marking. The system includes a tile bed comprising a plurality of tile slots. Each of the plurality of tile slots is configured to receive a tile. The linear growth problem is solvable by situating in the tile bed at least one of the first tile and the second tile and by causing each tile situated in the tile bed to expand in accordance with its respective marking to fill the tile bed.

An apparatus, method, and/or system for teaching the value of monetary coins including first, second, third, and fourth bases, each having a plurality of recesses, and first, second, third and fourth inserts, each having at least one protrusion for inserting into a corresponding recess of one or more of the first, second, third, and fourth bases. The first, second, third, and fourth bases may be provided in one kit or as part of one overall apparatus, wherein the first base has only one hundred recesses, the second base has only twenty-five recesses, the third base has only ten recesses, and the fourth base has only five recesses. The first, second, third, and fourth inserts may be configured to cover twenty-five, ten, five, and one recesses, respectively to represent twenty five cents, ten cents, five cents, and one cent of a U.S. dollar.

An apparatus, method, and/or system for teaching the value of monetary coins including first, second, third, and fourth bases, each having a plurality of recesses, and first, second, third and fourth inserts, each having at least one protrusion for inserting into a corresponding recess of one or more of the first, second, third, and fourth bases. The first, second, third, and fourth bases may be provided in one kit or as part of one overall apparatus, wherein the first base has only one hundred recesses, the second base has only twenty-five recesses, the third base has only ten recesses, and the fourth base has only five recesses. The first, second, third, and fourth inserts may be configured to cover twenty-five, ten, five, and one recesses, respectively to represent twenty five cents, ten cents, five cents, and one cent of a U.S. dollar.

An educational kit for teaching mathematics includes easily manipulated elements, which serve as cognitive reinforcement during the learning process. These physical elements are used in conjunction with a set of grouping rules. The educational kit and corresponding grouping rules determine a model and process to represent both an algebraic linear equation and its algebraic solution. A simple element, such as an item/figure including a variable “X”, is used to denote the unknown quantity. An item/figure including a number is used to represent a numerical value. These items/figures contain the exact elements used to form the expressions of the linear equation and are not items/figures that simulate the elements that form the expressions of the equation. By the use of this educational kit and associated grouping, students learn to simplify a given linear algebraic equation to the point where the solution is obvious.

An educational kit for teaching mathematics includes easily manipulated elements, which serve as cognitive reinforcement during the learning process. These physical elements are used in conjunction with a set of grouping rules. The educational kit and corresponding grouping rules determine a model and process to represent both an algebraic linear equation and its algebraic solution. A simple element, such as an item/figure including a variable “X”, is used to denote the unknown quantity. An item/figure including a number is used to represent a numerical value. These items/figures contain the exact elements used to form the expressions of the linear equation and are not items/figures that simulate the elements that form the expressions of the equation. By the use of this educational kit and associated grouping, students learn to simplify a given linear algebraic equation to the point where the solution is obvious.

Shape-matrix geometric instruments having numerous applications including, but not limited to, anti-counterfeiting, graphical passwording, games, and geometry education. A shape-matrix geometric instrument is a manufacture and/or a method whose design is based on a shape-matrix that, in turn comprises a set of building blocks that are N-dimensional polytopes. Corner shapes are positioned in or near the interior corner spaces of at least ones of the shape-matrix building blocks. At least ones of the corner shapes differ from others in at least one property or aspect including, for example, geometric shape, orientation within the building block, and one or more surface finishes, such as color, shading, cross-hatching or real or apparent texture.

Shape-matrix geometric instruments having numerous applications including, but not limited to, anti-counterfeiting, graphical passwording, games, and geometry education. A shape-matrix geometric instrument is a manufacture and/or a method whose design is based on a shape-matrix that, in turn comprises a set of building blocks that are N-dimensional polytopes. Corner shapes are positioned in or near the interior corner spaces of at least ones of the shape-matrix building blocks. At least ones of the corner shapes differ from others in at least one property or aspect including, for example, geometric shape, orientation within the building block, and one or more surface finishes, such as color, shading, cross-hatching or real or apparent texture.

An apparatus for teaching statistics concepts includes a simulation module that generates a simulation of subject elements moving from a first portion of a simulated space to a second portion of the simulated space, where the simulated space has a treatment path and a non-treatment path. A propensity module utilizes one or more propensity inputs to the simulation module to affect a propensity of the subject elements to move on the treatment path to have treatment functions applied. A treatment module applies the treatment functions to the subject elements moving on the treatment path, where the treatment functions are configured to affect at least one output measure of the subject elements. A display module displays the simulation of the subject elements having the propensity inputs applied, application of the treatment functions, and the at least one output measure.