Apparatus and method for tools for mathematics instruction

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.

1 . A method for facilitating a user to progress from visual concepts to procedures, said system comprising
 providing at least one processor programmed for:
 presenting one or more models in which at least one of a denominator, a numerator, and a whole number, where present, is unlocked, wherein the value of the at least one of a denominator, a numerator, and a whole number can be directly changed by the user, and in which the user changes one or more unlocked values in a presented model to create a correct answer; 
 presenting one or more models in which at least one of a denominator, a numerator, and a whole number, where present, is unlocked and that includes one or more introduced numbers in which at least one of a denominator, a numerator, and a whole number, where present, is locked, wherein the value cannot be directly changed by the user, and in which at least one of a denominator, a numerator, and a whole number, where present, is linked to corresponding values in the one or more models, wherein a change in the one or more models results in a change in a corresponding value in the one or more numbers, and in which the user changes one or more unlocked values in the one or more models to create a correct answer; 
 presenting one or more models in which at least one of a denominator, a numerator, and a whole number, where present, are locked, and that includes one or more introduced numbers in which at least one of a denominator, a numerator, and a whole number, where present, is unlocked, in which a presented model is not linked to corresponding values in the one or more introduced numbers, wherein a change in the one or more introduced numbers has no effect on the corresponding values in the one or more models, and in which the user changes one or more unlocked values in the one or more introduced numbers to create a correct answer by making one or more unlocked values in the one or more introduced numbers match corresponding values in the one or more models; 
 presenting one or more models in which at least one of a denominator, a numerator, and a whole number, where present, are locked, and that includes one or more introduced numbers in which at least one of a denominator, a numerator, and a whole number, where present, is unlocked and is linked to corresponding values in the one or more presented models, wherein a change in the one or more introduced numbers results in a change in a corresponding value in the one or more models, and in which the user changes one or more unlocked values in the one or more introduced numbers to create a correct answer; and 
   presenting one or more numbers in which at least one of a denominator, a numerator, and a whole number, where present, is unlocked, and in which the user changes one or more unlocked values in a presented number to create a correct answer.   
     
     
         2 . The method of  claim 1 , wherein a correct answer includes any of:
 with regard to all presented models, the numerator, the denominator, and whole number, if present; and   with regard to introduced numbers, a valid value that is neither empty, uninitialized, nor otherwise not a number.   
     
     
         3 . The method of  claim 1 , wherein a correct answer includes:
 an aggregate value of a resulting model that is equivalent to an aggregate value of a corresponding resulting number, when both models and introduced numbers are presented.   
     
     
         4 . The method of  claim 1 , wherein a correct answer includes:
 an aggregate value of a resulting model that is exactly equal to an aggregate value of a corresponding resulting number. when both models and introduced numbers are presented.   
     
     
         5 . The method of  claim 1 , wherein a correct answer includes:
 models in which all of the numerator, the denominator, and the whole number of a resulting model is exactly equal to the corresponding value of a corresponding resulting number, when both models and introduced numbers are presented.   
     
     
         6 . The method of  claim 1 , wherein a correct answer includes:
 all of the numerator, the denominator, and the whole number, if present, in all presented models, and said introduced numbers, are mathematically correct based on a context comprising any of addition, subtraction, multiplication, and division.   
     
     
         7 . The method of  claim 1 , further comprising:
 displaying on a display any of four layers, said four layers comprising:
 a model layer; 
 an equivalence layer; 
 a fractions layer; and 
 a sentence layer; and any of: 
   hiding and showing any layer of said four layers;   linking and unlinking any two layers of said four layers; and   locking and unlocking any portion of any of said four layers.   
     
     
         8 . The method of  claim 7 , wherein said linking and unlinking any two layers of said four layers further comprises linking and unlinking any of:
 said model layer and said fractions layer;   said model layer and said equivalence layer; and   said equivalence layer and said fractions layer.   
     
     
         9 . The method of  claim 1 , wherein said type of model comprises any of:
 a fractions circle;   a vertical fraction bar;   a horizontal fraction bar;   a dot model;   a discrete model;   a number line;   a numeric representation;   a decimal;   a decimal grid;   sets;   rods;   two dimensional arrays;   ten frames;   place value blocks;   number line; and   hundreds chart.   
     
     
         10 . The method of  claim 1 , further comprising:
 determining whether a user input is correct and, when it is determined that said user input is not correct, providing feedback on a display for said user.   
     
     
         11 . The method of  claim 10 , further comprising:
 determining whether a user input is correct based on a particular order of correction.   
     
     
         12 . The method of  claim 10 , further comprising:
 providing corrective feedback incrementally as a user corrects errors on the display.   
     
     
         13 . The method of  claim 10 , wherein said feedback excludes providing answers to problems. 
     
     
         14 . The method of  claim 10 , further comprising:
 generating feedback based on student inputs, which layers are visible, which components are linked, which components are not linked, which components are locked, which components are unlocked, and which model is selected.   
     
     
         15 . The method of  claim 1 , further comprising:
 scaffolding instruction to drive student instruction by providing a particular sequence of steps that are taken by said student, wherein said particular sequence of steps help said student learn a topic, and wherein said particular sequence of steps comprises a particular problem built from a prior problem.   
     
     
         16 . The method of  claim 7 , further comprising:
 linking and unlinking any two layers of said four layers using an observer pattern wherein linked components emit events when any of said components change, wherein   
       corresponding components listen for said emitted events and synchronize themselves to a corresponding changed value, and wherein when any of said components are unlinked, such components stop listening for emitted events or such components ignore said emitted events. 
     
     
         17 . The method of  claim 7 , wherein said locking and unlocking any portion of any of said four layers, further comprises any of:
 causing a locked component to stop listening to user input from a mouse or keyboard;   causing a locked component to ignore user input from a mouse or a keyboard; and   hiding and removing controls which change the value of a locked component.   
     
     
         18 . The method of  claim 7 , wherein said locking and unlocking any portion of any of said four layers prepares fractions for manipulation on a display and wherein said locking and unlocking any portion of any of said four layers has a utilitarian function that allows a user to start interacting with the fractions. 
     
     
         19 . The method of  claim 1 , further comprising:
 providing a multiple models feature allowing a user to interchange models, to select a specific model, and to select a set of models for a given problem.   
     
     
         20 . The method of  claim 19 , further comprising:
 said multiple models feature using an object oriented approach, said object oriented approach creating a template class that declares an application programming interface for the methods and properties common to corresponding fraction models.   
     
     
         21 . The apparatus of  claim 20 , wherein said object oriented approach further comprises a model container that holds shadow copies of corresponding model values and that synchronizes new models with said shadow copies of corresponding model values when any of said new models are selected. 
     
     
         22 . A computer implemented for facilitating a user to progress from visual concepts to procedures, comprising:
 providing at least one processor programmed to perform the steps of:   providing a plurality of visual models; and   allowing a user to change the type of the model presented and interact with a selected model, wherein said model type comprises any of:   a model in which the denominator of a resulting model of the presented models is locked;   a model allowing a user to create an entire fraction in the resulting model;   a model for introducing numbers and requiring the user to provide a numerator in the resulting model;   a model for allowing the user to strategically change the model to promote flexibility;   a model requiring the user provide the numerator and the denominator;   a model presenting procedures only and requiring the user to calculate the numerator; and   a model requiring the user to calculate the numerator and input the denominator.