WO2015112111A1 - Method of teaching mathematics - Google Patents

Method of teaching mathematics Download PDF

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Publication number
WO2015112111A1
WO2015112111A1 PCT/US2014/012247 US2014012247W WO2015112111A1 WO 2015112111 A1 WO2015112111 A1 WO 2015112111A1 US 2014012247 W US2014012247 W US 2014012247W WO 2015112111 A1 WO2015112111 A1 WO 2015112111A1
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WO
WIPO (PCT)
Prior art keywords
difficulty
operators
candidate
value
equation
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Application number
PCT/US2014/012247
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French (fr)
Inventor
David Jasper
Original Assignee
David Jasper
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Publication date
Application filed by David Jasper filed Critical David Jasper
Priority to PCT/US2014/012247 priority Critical patent/WO2015112111A1/en
Publication of WO2015112111A1 publication Critical patent/WO2015112111A1/en

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Classifications

    • G—PHYSICS
    • G09—EDUCATION; CRYPTOGRAPHY; DISPLAY; ADVERTISING; SEALS
    • G09B—EDUCATIONAL OR DEMONSTRATION APPLIANCES; APPLIANCES FOR TEACHING, OR COMMUNICATING WITH, THE BLIND, DEAF OR MUTE; MODELS; PLANETARIA; GLOBES; MAPS; DIAGRAMS
    • G09B19/00—Teaching not covered by other main groups of this subclass
    • G09B19/02—Counting; Calculating

Definitions

  • the study of mathematics may include, for example, the study of numbers, addition, subtraction, division, and/or order of operations.
  • the study of mathematics may include, for example, the study of algebra, geometry, trigonometry, calculus, finite math, probability and statistics, matrix algebra, linear algebra, differential equations, partial differential equations, multivariable analysis, vector analysis, graph theory, and/or number theory.
  • mathematics relies upon fundamental concepts taught at an early age.
  • a computer-implemented method of teaching mathematics includes generating a plurality of playing numbers, selecting one or more mathematical operators, and displaying a target equation value, the plurality of playing numbers, and the one or more mathematical operators on a display device.
  • a candidate equation is input based on the plurality of playing numbers and at least one of the one or more mathematical operators.
  • the candidate equation is evaluated and the candidate equation value is displayed on the display device.
  • a non-transitory computer-readable medium comprising software instructions that, when executed by a processor, perform a method of teaching mathematics that includes generating a plurality of playing numbers, selecting one or more mathematical operators, and displaying a target equation value, the plurality of playing numbers, and the one or more mathematical operators on a display device.
  • a candidate equation is input based on the plurality of playing numbers and at least one of the one or more mathematical operators.
  • the candidate equation is evaluated and the candidate equation value is displayed on the display device.
  • Figure 1 shows a computing system in accordance with one or more embodiments of the present invention.
  • Figure 2 shows a computer-implemented method of teaching mathematics in accordance with one or more embodiments of the present invention.
  • Figure 3 shows an example of a computer-implemented method of teaching mathematics with two playing numbers in accordance with one or more embodiments of the present invention.
  • Figure 4 shows an example of a computer-implemented method of teaching mathematics with four playing numbers in accordance with one or more embodiments of the present invention.
  • Figure 5 shows an example of a computer-implemented method of teaching mathematics with six playing numbers in accordance with one or more embodiments of the present invention.
  • FIG. 1 shows a computing system 100 in accordance with one or more embodiments of the present invention.
  • Computing system 100 may include one or more computers 105 that each includes one or more printed circuit boards (not shown) or flex circuits (not shown) on which one or more processors (not shown) and system memory (not shown) may be disposed.
  • Each of the one or more processors (not shown) may be a single-core processor (not shown) or a multi-core processor (not shown).
  • Multi-core processors (not shown) typically include a plurality of processor cores (not shown) disposed on the same physical die or a plurality of processor cores (not shown) disposed on multiple die that are disposed in the same mechanical package.
  • Computing system 100 may include one or more input/output devices such as, for example, a display device 1 10, keyboard 1 15, mouse 120, and/or any other human-computer interface device 125.
  • the one or more input/output devices may be integrated into computer 105.
  • Display device 1 10 may be a touch screen that includes a touch sensor (not shown) configured to sense touch.
  • a touch screen enables a user to control various aspects of computing system 100 by touch or gestures. For example, a user may interact directly with objects depicted on display device 110 by touch or gestures that are sensed by the touch sensor and treated as input by computer 105.
  • Computing system 100 may include one or more local storage devices 130.
  • Local storage device 130 may be a solid-state memory device, a solid-state memory device array, a hard disk drive, a hard disk drive array, or any other non-transitory computer readable medium. Local storage device 130 may be integrated into computer 105.
  • Computing system 100 may include one or more network interface devices 140 that provide a network interface to computer 105. The network interface may be Ethernet, Wi-Fi, Bluetooth, WiMAX, Fibre Channel, or any other network interface suitable to facilitate networked communications.
  • Computing system 100 may include one or more network-attached storage devices 140 in addition to, or instead of, one or more local storage devices 130.
  • Network-attached storage device 140 may be a solid-state memory device, a solid-state memory device array, a hard disk drive, a hard disk drive array, or any other non-transitory computer readable medium.
  • Network-attached storage device 150 may not be collocated with computer 105 and may be accessible to computer 105 via one or more network interfaces provided by one or more network interface devices 135.
  • computer 105 may be a server, a workstation, a desktop, a laptop, a netbook, a tablet, a smartphone, a mobile device, and/or any other type of computing system in accordance with one or more embodiments of the present invention.
  • Figure 2 shows a computer-implemented method 200 of teaching mathematics in accordance with one or more embodiments of the present invention.
  • the method 200 of teaching mathematics may be performed by a computing system (e.g., 100 of Figure 1).
  • a playing number may be an integer number randomly selected from a predetermined range of integer values.
  • the predetermined range of integer values may be an inclusive set of integers 1 through x, where x may be any integer larger than 1.
  • the predetermined range of integer values may be selected to correspond to a predetermined level of difficulty. For example, at the lowest level of difficulty, the predetermined range of integer values may include the set of integers 1 through 2 inclusive.
  • the predetermined range of integer values may be extended to higher integer values of x.
  • a high level of difficulty may include the set of integers 1 through 20 inclusive.
  • the predetermined range of integer values may include the set of integers that correspond to the numbered sides of an x-sided die. In this way, the predetermined range of integer values may include all possible outcomes from rolling the x-sided die.
  • the predetermined range of integer values may vary in accordance with one or more embodiments of the present invention.
  • a plurality of playing numbers may be generated.
  • a number of playing numbers generated may correspond to a predetermined level of difficulty. For example, at the lowest level of difficulty, the number of playing numbers generated may be 2, simulating, for example, two dice. While a single playing number could conceivably be used, it likely would not present a challenge to a user, as they would have no choice but to play the single playing number.
  • the number of playing numbers generated may be increased. For example, at a high level of difficulty, the number of playing numbers generated may be 7, simulating, for example, seven dice, but the number could be higher.
  • the number of playing numbers in the plurality of playing numbers generated may vary in accordance with one or more embodiments of the present invention.
  • each playing number in the plurality of playing numbers may be randomly selected from the predetermined range of integer values. More than one playing number may have the same value. In this way, each playing number generated may correspond to an outcome of a roll of the x-sided die. While computer-based methods of random number generation are, at some level, deterministic, recent trends have focused on the use of seed values derived from random or stochastic processes in an attempt to enhance the randomness of the number generated. Notwithstanding the above, as used herein, random may include numbers generated by software-based random number generators or functions, including those provided by a software development environment, which may not be truly random.
  • a mathematical operator is an operator that performs a function based on one or more operands or numbers.
  • Examples of mathematical operators include, for example, addition (+), subtraction (— ), multiplication (x), division ( ⁇ ), power ( ⁇ ), factorial (!), square root (V), sum ( ⁇ ), and product ( ⁇ ).
  • Examples of mathematical operators may include other more complex operators or functions such as, for example, least common multiple, greatest common multiple, least common factor, greatest common factor, Fibonacci, integration, and derivation. Mathematical operators require one or more operands depending on the type of operator.
  • the factorial operator (!) is a mathematical operator that requires a single operand
  • the addition operator (+) is a mathematical operator that requires two operands
  • the sum operator ( ⁇ ) is a mathematical operator that may operate on more than two operands.
  • One of ordinary skill in the art will recognize that other types of operators may be used in accordance with one or more embodiments of the present invention.
  • one or more mathematical operators may be selected.
  • a number of mathematical operators selected may correspond to the predetermined level of difficulty. For example, at the lowest level of difficulty, the number of mathematical operators selected may be 1. At higher levels of difficulty, the number of mathematical operators selected may be increased. For example, at a high level of difficulty, the number of mathematical operators selected may be 6, but the number could be higher.
  • the number of mathematical operators in the one or more mathematical operators selected may vary in accordance with one or more embodiments of the present invention.
  • Each mathematical operator in the one or more mathematical operators may be randomly selected from a list of operators.
  • the list of operators may correspond to a predetermined level of difficulty. For example, at the lowest level of difficulty, a mathematical operator may be randomly selected from a list of operators that includes a single operator. At a higher level of difficulty, a mathematical operator may be randomly selected from a list of operators that may include, for example, four operators. At an even higher level of difficulty, a mathematical operator may be randomly selected from a list of operators that may include, for example, six operators.
  • the list of operators may vary in accordance with one or more embodiments of the present invention.
  • the list of operators may be a weighted list of operators, where the weight assigned to each operator in the weighted list of operators may correspond to the predetermined level of difficulty.
  • a mathematical operator may be randomly selected from the weighted list of operators, where a higher weight indicates a higher likelihood of random selection. For example, at the lowest level of difficulty, the number of mathematical operators selected may be 1.
  • a mathematical operator may be randomly selected from a list of operators that may include a plurality of operators, where a single operator, such as, for example, addition, has a weight of 100 percent while the remaining operators have a weight of 0 percent, thereby ensuring selection of the addition operator.
  • a mathematical operator may be randomly selected from a list of operators that may include 6 operators, such as, for example, addition, subtraction, multiplication, division, power, and factorial with w r eights of 12.5, 12.5, 12.5, 12.5, 25, and 25 percent respectively.
  • the weights may be distributed to ensure a higher likelihood that more complex operators, such as, for example, power and factorial, are selected.
  • the weights may be distributed among the operators in the list of operators in other ways in accordance with one or more embodiments of the present invention.
  • a target equation value, the plurality of playing numbers, and the one or more mathematical operators selected may be displayed on a display device (1 10 of Figure 1) of a computer ( 105 of Figure 1).
  • the target equation value may have a value in a predetermined range that corresponds to the predetermined level of difficulty. For example, at the lowest level of difficulty, the target equation value may have a value in a predetermined range, such as, between 1 and 9 inclusive. At higher levels of difficulty, the target equation value may have a value in a predetermined range that is larger. For example, at a high level of difficulty, the target equation value may have a value in a predetermined range, such as, for example, between -1000 and 1000 inclusive.
  • the target equation value and the predetermined range of the target equation value may vary in accordance with one or more embodiments of the present invention.
  • the target equation value may be calculated based on a randomly generated target equation.
  • a randomly generated target equation may be constructed by sequentially placing playing numbers from the plurality of playing numbers generated and at least one mathematical operator selected. One or more parentheses may also be placed based on the type of operator selected to ensure a proper order of operations.
  • the target equation value may be evaluated. If the target equation value falls within the predetermined range for a predetermined level of difficulty, the target equation value may be accepted. If the target equation value falls outside the predetermined range for a predetermined level of difficulty, the target equation may be reconstructed.
  • the target equation may be reconstructed by reconstructing the target equation using the same plurality of playing numbers generated and the same at least one mathematical operator selected or the plurality of playing numbers may be regenerated and the at least one mathematical operator may be reselected. Once the target equation is reconstructed, the target equation value may be evaluated and the process may repeat in this manner until a target equation value is calculated that falls within the predetermined range for a predetermined level of difficulty. This ensures there is at least one possible candidate equation that achieves the target equation value. However, there may be more than one possible candidate equation that achieves the target equation value.
  • the process may be reversed such that a suitable target equation value is randomly selected and the plurality of playing numbers and the at least one mathematical operator are generated or selected in a repetitive process until a suitable target equation that achieves the target equation value is identified. This also ensures there is at least one possible candidate equation that achieves the target equation value. However, there may be more than one possible candidate equation that achieves the target equation value.
  • a suitable target equation value is randomly selected and the plurality of playing numbers and the at least one mathematical operator are generated or selected in a repetitive process until a suitable target equation that achieves the target equation value is identified. This also ensures there is at least one possible candidate equation that achieves the target equation value. However, there may be more than one possible candidate equation that achieves the target equation value.
  • the way in which the target equation value is determined may vary in accordance with one or more embodiments of the present invention.
  • a candidate equation may be input by the computer based on the plurality of playing numbers and at least one of the one or more mathematical operators.
  • a user may input the candidate equation sequentially by selecting one of the plurality of playing numbers and at least one of the one or more mathematical operators until all playing numbers have been played. The user may also use parentheses to ensure proper order of operations. Each playing number from the plurality of playing numbers may be used only once, whereas the one or more mathematical operators may be used repeatedly.
  • a hint may be provided.
  • a hint may help a user that is stuck in a given problem or round of play.
  • the hint may be a portion of a candidate equation that achieves the target equation value.
  • the number of hints available, if any may be controlled. In other embodiments, the number of hints may be unlimited through to construction of candidate equation. This may be particularly helpful at more difficult levels of play.
  • the candidate equation may be evaluated to determine the candidate equation value.
  • the candidate equation value may be displayed on the display device.
  • the candidate equation value may be evaluated and displayed on the display device in real time as the candidate equation is being input to provide feedback to the user.
  • the candidate equation value may be evaluated and displayed on the display device when all playing numbers have been played.
  • the candidate equation value may be compared to the target equation value.
  • the candidate equation value may be compared to the target equation value as the candidate equation is being input.
  • the candidate equation value may be compared to the target equation value only when all playing numbers have been played.
  • an indicator of whether the candidate equation value is equal to the target equation value may be output to the display device. If the candidate equation value is equal to the target equation value, the user has constructed a suitable candidate equation and may receive positive feedback that they solved the problem. In certain embodiments, if a user has given up on a particular problem, the user may resign to exit that round of play.
  • various statistics such as, for example, an amount of time to solve a problem, a number of problems solved (wins), a number of problems solved with hints (wins with hints), average elapsed time per win, a number of problems not solved (resignations), and average elapsed time per resignation may be logged.
  • the various statistics may be provided to the user as metrics of their progress.
  • the various statistics may be provided to an instructor that is monitoring the user's performance.
  • the statistics may be provided to the user or instructor in real time or at a later time upon invocation.
  • the statistics may be provided upon reaching a milestone. For example, the user may be advised when they have exceeded their best time or solved a certain number of problems.
  • the instructor may be advised when they user has exceeded their best time or solved a certain number of problems.
  • the instructor may have the ability to constrain the level of difficulty, clear the logs, and limit the number of hints a user may use in a given round of play.
  • One of ordinary skill in the art will recognize that other statistics may be used in accordance with one or more embodiments of the present invention.
  • a network interface may allow for a networked version of play.
  • the user may compete with other users on various metrics. For example, the user may compete with other users to determine who can solve a given problem first or who can solve a set of problems first.
  • One of ordinary skill in the art will recognize that other modes of competing may be used in accordance with one or more embodiments of the present invention.
  • Figure 3 shows an example 300 of a computer-implemented method of teaching mathematics with two playing numbers in accordance with one or more embodiments of the present invention.
  • the user may select a level of difficulty in advance of play (not shown).
  • the predetermined level of difficulty may determine the number of playing numbers 320 used corresponding, for example, to a number of simulated x-sided dice.
  • the predetermined level of difficulty may also determine the predetermined range of integer values from which each playing number 320 may be generated corresponding, for example, to the possible outcomes from a roll of the x-sided die.
  • the predetermined level of difficulty may also determine the number of mathematical operators 330 selected.
  • the predetermined level of difficulty may also determine the type of mathematical operators 330 available for selection or weight for selection.
  • the predetermined level of difficulty may also determine the predetermined range of the target equation value 340.
  • the user may select from pre-school, elementary school, middle school, high school, or university levels of difficulty.
  • the above- noted levels of difficulty may be partitioned into easy, normal, and difficult sub- levels.
  • the user may select levels of difficulty that correspond to certain areas of mathematics.
  • the user may customize the level of difficulty by selecting one or more of the number of playing numbers 320, the predetermined range of integers from which the playing numbers are generated, the number of mathematical operators 330, the type of mathematical operators, and specifying a target equation value 340.
  • One of ordinary skill in the art will recognize that the manner in which levels of difficulty are defined may vary in accordance with one or more embodiments of the present invention.
  • the user may initiate a round of play by touching, clicking, or otherwise selecting an icon 310 that triggers the generation of a plurality of playing numbers 320 corresponding, for example, to the rolling of the simulated x-sided dice and the selection of one or more mathematical operators 330.
  • a target equation value 340, the plurality of playing numbers generated 320, and the one or more mathematical operators 330 selected may be displayed on a display device 1 10.
  • a user may construct a candidate equation 350 by sequentially selecting a playing number 320 or a mathematical operator 330 until all playing numbers 320 have been played.
  • Each playing number 320 may be used only once, but any one or more of the mathematical operators 330 may be used and a given mathematical operator 330 may be used as many times as a user desires.
  • the candidate equation 350 may be evaluated and the candidate equation value 360 may displayed on the display device 110.
  • the candidate equation 350 may be evaluated and displayed on the display device 110 in real time as the candidate equation 350 is being input. If a user constmcts a candidate equation that fails to achieve the target equation value, the user may edit prior entries and reconstruct the candidate equation. In this way, the running evaluation and display of the running candidate equation value may assist the user in constructing a suitable candidate equation.
  • a hint icon (not shown) and a resign icon (not shown) may be provided.
  • a hint icon may provide a hint to a user that is stuck in a given round of play.
  • the hint may be a portion of a candidate equation that achieves the target equation value.
  • the user may use the hint icon repeatedly until the candidate equation is constructed. This may be particularly helpful at more difficult levels of play.
  • a resign icon may allow a user that has given up on a given round to exit that round of play.
  • the user when a user constmcts a candidate equation that achieves the target equation value, the user may receive positive feedback. The round of play is complete and the user may initiate another round of play by clicking on icon 310.
  • various statistics such as, for example, an amount of time to solve a problem, a number of problems solved (wins), a number of problems solved with hints (wins with hints), average elapsed time per win, a number of problems not solved (resignations), and average elapsed time per resignation may be logged.
  • a timer 370 may show r a running clock that shows the amount of time elapsed since play was initiated and a suitable candidate equation was provided.
  • the various statistics may be provided to the user as metrics of their progress.
  • the various statistics may be provided to an instructor that is monitoring the user's performance.
  • the statistics may be provided to the user or instructor in real time or at a later time upon invocation.
  • the statistics may be provided upon reaching a milestone. For example, the user may be advised when they have exceeded their best time or solved a certain number of problems. Similarly, the instmctor may be advised when they user has exceeded their best time or solved a certain number of problems. The instmctor may have the ability to constrain the level of difficulty, clear the logs, and limit the number of hints a user may use in a given round of play.
  • the user may be advised when they have exceeded their best time or solved a certain number of problems.
  • the instmctor may have the ability to constrain the level of difficulty, clear the logs, and limit the number of hints a user may use in a given round of play.
  • One of ordinary skill in the art will recognize that other statistics may be used in accordance with one or more embodiments of the present invention.
  • Figure 4 shows an example 400 of a computer- implemented method of teaching mathematics with four playing numbers in accordance with one or more embodiments of the present invention.
  • the example depicted in Figure 4 is substantially similar to that depicted in Figure 3 with a few notable exceptions.
  • the example in Figure 4 depicts a higher level of difficulty as evidenced by the fact that the number of playing numbers generated is higher, the predetermined range of integer values from which the playing numbers are generated is larger, the number of mathematical operators selected is higher, the type of mathematical operators selected are more complex, and the predetermined range of the target equation value is larger.
  • each playing number 320 must be used, but any one or more of the mathematical operators 330 may be used and a given mathematical operator 330 may be used as many times as a user desires.
  • the candidate equation 350 depicted in this example all playing numbers 320 are used and the mathematical operator 330 of addition is used more than once.
  • Figure 5 shows an example 500 of a computer- implemented method of teaching mathematics with six playing numbers in accordance with one or more embodiments of the present invention.
  • the example depicted in Figure 5 is substantially similar to that depicted in Figure 4 with a few notable exceptions.
  • the example in Figure 5 depicts a higher level of difficulty as evidenced by the fact that the number of playing numbers generated is higher, the predetermined range of integer values from which the playing numbers are generated is larger, the number of mathematical operators selected is higher, the type of mathematical operators selected are more complex, and the predetermined range of the target equation value is larger.
  • each playing number 320 must be used, but any one or more of the mathematical operators 330 may be used and a given mathematical operator 330 may be used as many times as a user desires, hi the candidate equation 350 depicted in this example, all playing numbers 320 are used and the mathematical operator 330 of multiplication is used more than once.
  • the candidate equation 350 depicted in this example demonstrates the use of parentheses to ensure a desired order of operations in the candidate equation 350. In the candidate equation 350 depicted, the parentheses are necessary to achieve the target equation value 340.
  • Advantages of one or more embodiments of the present invention may include one or more of the following:
  • a method of teaching mathematics teaches or reinforces mathematics skills in a simple, fun, and interactive manner.
  • a method of teaching mathematics teaches or reinforces mathematics skills in an interactive game-like format.
  • a method of teaching mathematics teaches or reinforces mathematics skills suitable for use in an intuitive touch screen application.
  • a method of teaching mathematics challenges a user to construct a candidate equation that evaluates to a candidate equation value that is equal to a target equation value.
  • a method of teaching mathematics challenges a user to construct a candidate equation from a plurality of playing numbers, at least one of one or more mathematical operators, and zero or more parentheses that evaluate to a candidate equation value that is equal to a target equation value.
  • a method of teaching mathematics generates a plurality of playing numbers.
  • a number of playing numbers in the plurality of playing numbers generated may correspond to a predetermined level of difficulty.
  • Each playing number in the plurality of playing numbers may be randomly selected from a predetermined range of integer values.
  • the predetermined range of integer values may correspond to a predetermined level of difficulty.
  • a method of teaching mathematics selects one or more mathematical operators.
  • a number of mathematical operators in the one or more mathematical operators selected may correspond to a predetermined level of difficulty.
  • Each mathematical operator in the one or more mathematical operators may be randomly selected from a list of operators.
  • the list of operators may be a weighted list of operators where a weight assigned to each operator in the weighted list of operators may correspond to a predetermined level of difficulty. The weights may vary to ensure a higher likelihood of selection of more advanced operators at higher levels of difficulty.
  • a method of teaching mathematics displays a target equation value, a plurality of playing numbers, and one or more mathematical operators on a display device of a computer.
  • the user may be challenged to construct a candidate equation that evaluates to a candidate equation value that is equal to the target equation value.
  • the target equation value may be in a predetermined range corresponding to a predetermined level of difficulty.
  • a method of teaching mathematics inputs a candidate equation based on a plurality of playing numbers and at least one of one or more mathematical operators.
  • the candidate equation value may be evaluated as the candidate equation is being input and displayed on a display device to assist the user in constructing the equation.

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Abstract

A computer-implemented method includes generating a plurality of playing numbers, selecting one or more mathematical operators, and displaying a target equation value, the plurality of playing numbers, and the one or more mathematical operators on a display device. A candidate equation is input based on the plurality of playing numbers and at least one of the one or more mathematical operators. The candidate equation is evaluated and the candidate equation value is displayed on the display device.

Description

METHOD OF TEACHING MATHEMATICS
BACKGROUND OF THE INVENTION
[0001] Mathematics is the science of quantities, shapes, and their relationships and has evolved from efforts to count, measure, describe, or otherwise quantify the physical world. Li this way, mathematics has become the universal language of the sciences and the foundation of many other fields of interest. Because of its pervasive application, mathematics is typically taught from early childhood through to university-level academic studies.
[0002] At lower levels, the study of mathematics may include, for example, the study of numbers, addition, subtraction, division, and/or order of operations. At higher levels, the study of mathematics may include, for example, the study of algebra, geometry, trigonometry, calculus, finite math, probability and statistics, matrix algebra, linear algebra, differential equations, partial differential equations, multivariable analysis, vector analysis, graph theory, and/or number theory. However, even in advanced areas of study, mathematics relies upon fundamental concepts taught at an early age.
[0003] Because of the building-block nature of the study of mathematics, students that fail to master fundamental concepts in one area tend to have difficulty in more advanced areas that build upon them. As a consequence, many students lose interest in mathematics at some point in their study of the subject and do not pursue it further, forgoing potential careers in science, technology, engineering, and mathematics. Others, who previously developed proficiency in mathematics, including those who work in science, technology, engineering, and mathematics, often lose mathematics skills from periods of lack of use. As a consequence, those once proficient require remediation to refresh what they had previously learned.
BRIEF SUMMARY OF THE INVENTION
[0004] According to one aspect of one or more embodiments of the present invention, a computer-implemented method of teaching mathematics includes generating a plurality of playing numbers, selecting one or more mathematical operators, and displaying a target equation value, the plurality of playing numbers, and the one or more mathematical operators on a display device. A candidate equation is input based on the plurality of playing numbers and at least one of the one or more mathematical operators. The candidate equation is evaluated and the candidate equation value is displayed on the display device.
[0005] According to one aspect of one or more embodiments of the present invention, a non-transitory computer-readable medium comprising software instructions that, when executed by a processor, perform a method of teaching mathematics that includes generating a plurality of playing numbers, selecting one or more mathematical operators, and displaying a target equation value, the plurality of playing numbers, and the one or more mathematical operators on a display device. A candidate equation is input based on the plurality of playing numbers and at least one of the one or more mathematical operators. The candidate equation is evaluated and the candidate equation value is displayed on the display device.
[0006] Other aspects of the present invention will be apparent from the following description and claims.
BRIEF DESCRIPTION OF THE DRAWINGS
[0007] Figure 1 shows a computing system in accordance with one or more embodiments of the present invention.
[0008] Figure 2 shows a computer-implemented method of teaching mathematics in accordance with one or more embodiments of the present invention.
[0009] Figure 3 shows an example of a computer-implemented method of teaching mathematics with two playing numbers in accordance with one or more embodiments of the present invention.
[0010] Figure 4 shows an example of a computer-implemented method of teaching mathematics with four playing numbers in accordance with one or more embodiments of the present invention.
[001 1] Figure 5 shows an example of a computer-implemented method of teaching mathematics with six playing numbers in accordance with one or more embodiments of the present invention.
DETAILED DESCRIPTION OF THE INVENTION
[0012] One or more embodiments of the present invention are described in detail with reference to the accompanying figures. For consistency, like elements in the various figures are denoted by like reference numerals, hi the following detailed description of the present invention, specific details are set forth in order to provide a thorough understanding of the present invention. In other instances, well-known features to one of ordinary skill in the art are not described to avoid obscuring the description of the present invention.
[0013] Figure 1 shows a computing system 100 in accordance with one or more embodiments of the present invention. Computing system 100 may include one or more computers 105 that each includes one or more printed circuit boards (not shown) or flex circuits (not shown) on which one or more processors (not shown) and system memory (not shown) may be disposed. Each of the one or more processors (not shown) may be a single-core processor (not shown) or a multi-core processor (not shown). Multi-core processors (not shown) typically include a plurality of processor cores (not shown) disposed on the same physical die or a plurality of processor cores (not shown) disposed on multiple die that are disposed in the same mechanical package. Computing system 100 may include one or more input/output devices such as, for example, a display device 1 10, keyboard 1 15, mouse 120, and/or any other human-computer interface device 125. The one or more input/output devices may be integrated into computer 105. Display device 1 10 may be a touch screen that includes a touch sensor (not shown) configured to sense touch. A touch screen enables a user to control various aspects of computing system 100 by touch or gestures. For example, a user may interact directly with objects depicted on display device 110 by touch or gestures that are sensed by the touch sensor and treated as input by computer 105.
[0014] Computing system 100 may include one or more local storage devices 130.
Local storage device 130 may be a solid-state memory device, a solid-state memory device array, a hard disk drive, a hard disk drive array, or any other non-transitory computer readable medium. Local storage device 130 may be integrated into computer 105. Computing system 100 may include one or more network interface devices 140 that provide a network interface to computer 105. The network interface may be Ethernet, Wi-Fi, Bluetooth, WiMAX, Fibre Channel, or any other network interface suitable to facilitate networked communications. Computing system 100 may include one or more network-attached storage devices 140 in addition to, or instead of, one or more local storage devices 130. Network-attached storage device 140 may be a solid-state memory device, a solid-state memory device array, a hard disk drive, a hard disk drive array, or any other non-transitory computer readable medium. Network-attached storage device 150 may not be collocated with computer 105 and may be accessible to computer 105 via one or more network interfaces provided by one or more network interface devices 135. One of ordinary skill in the art will recognize that computer 105 may be a server, a workstation, a desktop, a laptop, a netbook, a tablet, a smartphone, a mobile device, and/or any other type of computing system in accordance with one or more embodiments of the present invention.
[0015] Figure 2 shows a computer-implemented method 200 of teaching mathematics in accordance with one or more embodiments of the present invention. In one or more embodiments of the present invention, the method 200 of teaching mathematics may be performed by a computing system (e.g., 100 of Figure 1). A playing number may be an integer number randomly selected from a predetermined range of integer values. The predetermined range of integer values may be an inclusive set of integers 1 through x, where x may be any integer larger than 1. In certain embodiments, the predetermined range of integer values may be selected to correspond to a predetermined level of difficulty. For example, at the lowest level of difficulty, the predetermined range of integer values may include the set of integers 1 through 2 inclusive. At higher levels of difficulty, the predetermined range of integer values may be extended to higher integer values of x. For example, a high level of difficulty may include the set of integers 1 through 20 inclusive. In certain embodiments, the predetermined range of integer values may include the set of integers that correspond to the numbered sides of an x-sided die. In this way, the predetermined range of integer values may include all possible outcomes from rolling the x-sided die. One of ordinary skill in the art will recognize that the predetermined range of integer values may vary in accordance with one or more embodiments of the present invention.
[0016] In step 210, a plurality of playing numbers may be generated. In certain embodiments, a number of playing numbers generated may correspond to a predetermined level of difficulty. For example, at the lowest level of difficulty, the number of playing numbers generated may be 2, simulating, for example, two dice. While a single playing number could conceivably be used, it likely would not present a challenge to a user, as they would have no choice but to play the single playing number. At higher levels of difficulty, the number of playing numbers generated may be increased. For example, at a high level of difficulty, the number of playing numbers generated may be 7, simulating, for example, seven dice, but the number could be higher. One of ordinary skill in the art will recognize that the number of playing numbers in the plurality of playing numbers generated may vary in accordance with one or more embodiments of the present invention.
[0017] In certain embodiments, each playing number in the plurality of playing numbers may be randomly selected from the predetermined range of integer values. More than one playing number may have the same value. In this way, each playing number generated may correspond to an outcome of a roll of the x-sided die. While computer-based methods of random number generation are, at some level, deterministic, recent trends have focused on the use of seed values derived from random or stochastic processes in an attempt to enhance the randomness of the number generated. Notwithstanding the above, as used herein, random may include numbers generated by software-based random number generators or functions, including those provided by a software development environment, which may not be truly random.
[0018] A mathematical operator is an operator that performs a function based on one or more operands or numbers. Examples of mathematical operators include, for example, addition (+), subtraction (— ), multiplication (x), division (÷), power (Λ), factorial (!), square root (V), sum (∑), and product (Π). Examples of mathematical operators may include other more complex operators or functions such as, for example, least common multiple, greatest common multiple, least common factor, greatest common factor, Fibonacci, integration, and derivation. Mathematical operators require one or more operands depending on the type of operator. For example, the factorial operator (!) is a mathematical operator that requires a single operand, the addition operator (+) is a mathematical operator that requires two operands, and the sum operator (∑) is a mathematical operator that may operate on more than two operands. One of ordinary skill in the art will recognize that other types of operators may be used in accordance with one or more embodiments of the present invention.
[0019] In step 220, one or more mathematical operators may be selected. In certain embodiments, a number of mathematical operators selected may correspond to the predetermined level of difficulty. For example, at the lowest level of difficulty, the number of mathematical operators selected may be 1. At higher levels of difficulty, the number of mathematical operators selected may be increased. For example, at a high level of difficulty, the number of mathematical operators selected may be 6, but the number could be higher. One of ordinary skill in the art will recognize that the number of mathematical operators in the one or more mathematical operators selected may vary in accordance with one or more embodiments of the present invention.
[0020] Each mathematical operator in the one or more mathematical operators may be randomly selected from a list of operators. In certain embodiments, the list of operators may correspond to a predetermined level of difficulty. For example, at the lowest level of difficulty, a mathematical operator may be randomly selected from a list of operators that includes a single operator. At a higher level of difficulty, a mathematical operator may be randomly selected from a list of operators that may include, for example, four operators. At an even higher level of difficulty, a mathematical operator may be randomly selected from a list of operators that may include, for example, six operators. One of ordinary skill in the art will recognize that the list of operators may vary in accordance with one or more embodiments of the present invention.
[0021] In other embodiments, the list of operators may be a weighted list of operators, where the weight assigned to each operator in the weighted list of operators may correspond to the predetermined level of difficulty. A mathematical operator may be randomly selected from the weighted list of operators, where a higher weight indicates a higher likelihood of random selection. For example, at the lowest level of difficulty, the number of mathematical operators selected may be 1. As such, a mathematical operator may be randomly selected from a list of operators that may include a plurality of operators, where a single operator, such as, for example, addition, has a weight of 100 percent while the remaining operators have a weight of 0 percent, thereby ensuring selection of the addition operator. At a higher level of difficulty, a mathematical operator may be randomly selected from a list of operators that may include 6 operators, such as, for example, addition, subtraction, multiplication, division, power, and factorial with wreights of 12.5, 12.5, 12.5, 12.5, 25, and 25 percent respectively. In this way, at a higher level of difficulty, the weights may be distributed to ensure a higher likelihood that more complex operators, such as, for example, power and factorial, are selected. One of ordinary skill in the art will recognize that the weights may be distributed among the operators in the list of operators in other ways in accordance with one or more embodiments of the present invention.
[0022] In step 230, a target equation value, the plurality of playing numbers, and the one or more mathematical operators selected may be displayed on a display device (1 10 of Figure 1) of a computer ( 105 of Figure 1). The target equation value may have a value in a predetermined range that corresponds to the predetermined level of difficulty. For example, at the lowest level of difficulty, the target equation value may have a value in a predetermined range, such as, between 1 and 9 inclusive. At higher levels of difficulty, the target equation value may have a value in a predetermined range that is larger. For example, at a high level of difficulty, the target equation value may have a value in a predetermined range, such as, for example, between -1000 and 1000 inclusive. One of ordinary skill in the art will recognize that the target equation value and the predetermined range of the target equation value may vary in accordance with one or more embodiments of the present invention.
[0023] In certain embodiments, the target equation value may be calculated based on a randomly generated target equation. A randomly generated target equation may be constructed by sequentially placing playing numbers from the plurality of playing numbers generated and at least one mathematical operator selected. One or more parentheses may also be placed based on the type of operator selected to ensure a proper order of operations. Once the target equation is constructed, the target equation value may be evaluated. If the target equation value falls within the predetermined range for a predetermined level of difficulty, the target equation value may be accepted. If the target equation value falls outside the predetermined range for a predetermined level of difficulty, the target equation may be reconstructed. The target equation may be reconstructed by reconstructing the target equation using the same plurality of playing numbers generated and the same at least one mathematical operator selected or the plurality of playing numbers may be regenerated and the at least one mathematical operator may be reselected. Once the target equation is reconstructed, the target equation value may be evaluated and the process may repeat in this manner until a target equation value is calculated that falls within the predetermined range for a predetermined level of difficulty. This ensures there is at least one possible candidate equation that achieves the target equation value. However, there may be more than one possible candidate equation that achieves the target equation value. In other embodiments, the process may be reversed such that a suitable target equation value is randomly selected and the plurality of playing numbers and the at least one mathematical operator are generated or selected in a repetitive process until a suitable target equation that achieves the target equation value is identified. This also ensures there is at least one possible candidate equation that achieves the target equation value. However, there may be more than one possible candidate equation that achieves the target equation value. One of ordinary skill in the art will recognize that the way in which the target equation value is determined may vary in accordance with one or more embodiments of the present invention.
[0024] In step 240, a candidate equation may be input by the computer based on the plurality of playing numbers and at least one of the one or more mathematical operators. A user may input the candidate equation sequentially by selecting one of the plurality of playing numbers and at least one of the one or more mathematical operators until all playing numbers have been played. The user may also use parentheses to ensure proper order of operations. Each playing number from the plurality of playing numbers may be used only once, whereas the one or more mathematical operators may be used repeatedly. In certain embodiments, a hint may be provided. A hint may help a user that is stuck in a given problem or round of play. The hint may be a portion of a candidate equation that achieves the target equation value. In certain embodiments, the number of hints available, if any, may be controlled. In other embodiments, the number of hints may be unlimited through to construction of candidate equation. This may be particularly helpful at more difficult levels of play.
[0025] In step 250, the candidate equation may be evaluated to determine the candidate equation value. In step 260, the candidate equation value may be displayed on the display device. In certain embodiments, the candidate equation value may be evaluated and displayed on the display device in real time as the candidate equation is being input to provide feedback to the user. In other embodiments, the candidate equation value may be evaluated and displayed on the display device when all playing numbers have been played. In step 270, the candidate equation value may be compared to the target equation value. In certain embodiments, the candidate equation value may be compared to the target equation value as the candidate equation is being input. In other embodiments, the candidate equation value may be compared to the target equation value only when all playing numbers have been played. In step 280, an indicator of whether the candidate equation value is equal to the target equation value may be output to the display device. If the candidate equation value is equal to the target equation value, the user has constructed a suitable candidate equation and may receive positive feedback that they solved the problem. In certain embodiments, if a user has given up on a particular problem, the user may resign to exit that round of play.
[0026] In certain embodiments, various statistics, such as, for example, an amount of time to solve a problem, a number of problems solved (wins), a number of problems solved with hints (wins with hints), average elapsed time per win, a number of problems not solved (resignations), and average elapsed time per resignation may be logged. The various statistics may be provided to the user as metrics of their progress. The various statistics may be provided to an instructor that is monitoring the user's performance. The statistics may be provided to the user or instructor in real time or at a later time upon invocation. The statistics may be provided upon reaching a milestone. For example, the user may be advised when they have exceeded their best time or solved a certain number of problems. Similarly, the instructor may be advised when they user has exceeded their best time or solved a certain number of problems. The instructor may have the ability to constrain the level of difficulty, clear the logs, and limit the number of hints a user may use in a given round of play. One of ordinary skill in the art will recognize that other statistics may be used in accordance with one or more embodiments of the present invention.
[0027] In certain embodiments, a network interface may allow for a networked version of play. In such an embodiment, the user may compete with other users on various metrics. For example, the user may compete with other users to determine who can solve a given problem first or who can solve a set of problems first. One of ordinary skill in the art will recognize that other modes of competing may be used in accordance with one or more embodiments of the present invention.
[0028] Figure 3 shows an example 300 of a computer-implemented method of teaching mathematics with two playing numbers in accordance with one or more embodiments of the present invention. In one or more embodiments of the present invention, the user may select a level of difficulty in advance of play (not shown). The predetermined level of difficulty may determine the number of playing numbers 320 used corresponding, for example, to a number of simulated x-sided dice. The predetermined level of difficulty may also determine the predetermined range of integer values from which each playing number 320 may be generated corresponding, for example, to the possible outcomes from a roll of the x-sided die. The predetermined level of difficulty may also determine the number of mathematical operators 330 selected. The predetermined level of difficulty may also determine the type of mathematical operators 330 available for selection or weight for selection. The predetermined level of difficulty may also determine the predetermined range of the target equation value 340.
[0029] In certain embodiments, the user may select from pre-school, elementary school, middle school, high school, or university levels of difficulty. The above- noted levels of difficulty may be partitioned into easy, normal, and difficult sub- levels. In other embodiments, the user may select levels of difficulty that correspond to certain areas of mathematics. In still other embodiments, the user may customize the level of difficulty by selecting one or more of the number of playing numbers 320, the predetermined range of integers from which the playing numbers are generated, the number of mathematical operators 330, the type of mathematical operators, and specifying a target equation value 340. One of ordinary skill in the art will recognize that the manner in which levels of difficulty are defined may vary in accordance with one or more embodiments of the present invention.
[0030] In certain embodiments, the user may initiate a round of play by touching, clicking, or otherwise selecting an icon 310 that triggers the generation of a plurality of playing numbers 320 corresponding, for example, to the rolling of the simulated x-sided dice and the selection of one or more mathematical operators 330. A target equation value 340, the plurality of playing numbers generated 320, and the one or more mathematical operators 330 selected may be displayed on a display device 1 10. A user may construct a candidate equation 350 by sequentially selecting a playing number 320 or a mathematical operator 330 until all playing numbers 320 have been played. Each playing number 320 may be used only once, but any one or more of the mathematical operators 330 may be used and a given mathematical operator 330 may be used as many times as a user desires. The candidate equation 350 may be evaluated and the candidate equation value 360 may displayed on the display device 110. [0031] In certain embodiments, the candidate equation 350 may be evaluated and displayed on the display device 110 in real time as the candidate equation 350 is being input. If a user constmcts a candidate equation that fails to achieve the target equation value, the user may edit prior entries and reconstruct the candidate equation. In this way, the running evaluation and display of the running candidate equation value may assist the user in constructing a suitable candidate equation.
[0032] In certain embodiments, a hint icon (not shown) and a resign icon (not shown) may be provided. A hint icon may provide a hint to a user that is stuck in a given round of play. The hint may be a portion of a candidate equation that achieves the target equation value. The user may use the hint icon repeatedly until the candidate equation is constructed. This may be particularly helpful at more difficult levels of play. A resign icon may allow a user that has given up on a given round to exit that round of play.
[0033] In certain embodiments, when a user constmcts a candidate equation that achieves the target equation value, the user may receive positive feedback. The round of play is complete and the user may initiate another round of play by clicking on icon 310.
[0034] In certain embodiments, various statistics, such as, for example, an amount of time to solve a problem, a number of problems solved (wins), a number of problems solved with hints (wins with hints), average elapsed time per win, a number of problems not solved (resignations), and average elapsed time per resignation may be logged. For example, a timer 370 may showr a running clock that shows the amount of time elapsed since play was initiated and a suitable candidate equation was provided. The various statistics may be provided to the user as metrics of their progress. The various statistics may be provided to an instructor that is monitoring the user's performance. The statistics may be provided to the user or instructor in real time or at a later time upon invocation. The statistics may be provided upon reaching a milestone. For example, the user may be advised when they have exceeded their best time or solved a certain number of problems. Similarly, the instmctor may be advised when they user has exceeded their best time or solved a certain number of problems. The instmctor may have the ability to constrain the level of difficulty, clear the logs, and limit the number of hints a user may use in a given round of play. One of ordinary skill in the art will recognize that other statistics may be used in accordance with one or more embodiments of the present invention.
[0035] Figure 4 shows an example 400 of a computer- implemented method of teaching mathematics with four playing numbers in accordance with one or more embodiments of the present invention. The example depicted in Figure 4 is substantially similar to that depicted in Figure 3 with a few notable exceptions. The example in Figure 4 depicts a higher level of difficulty as evidenced by the fact that the number of playing numbers generated is higher, the predetermined range of integer values from which the playing numbers are generated is larger, the number of mathematical operators selected is higher, the type of mathematical operators selected are more complex, and the predetermined range of the target equation value is larger. As noted above, each playing number 320 must be used, but any one or more of the mathematical operators 330 may be used and a given mathematical operator 330 may be used as many times as a user desires. In the candidate equation 350 depicted in this example, all playing numbers 320 are used and the mathematical operator 330 of addition is used more than once.
[0036] Figure 5 shows an example 500 of a computer- implemented method of teaching mathematics with six playing numbers in accordance with one or more embodiments of the present invention. The example depicted in Figure 5 is substantially similar to that depicted in Figure 4 with a few notable exceptions. The example in Figure 5 depicts a higher level of difficulty as evidenced by the fact that the number of playing numbers generated is higher, the predetermined range of integer values from which the playing numbers are generated is larger, the number of mathematical operators selected is higher, the type of mathematical operators selected are more complex, and the predetermined range of the target equation value is larger.
[0037] As noted above, each playing number 320 must be used, but any one or more of the mathematical operators 330 may be used and a given mathematical operator 330 may be used as many times as a user desires, hi the candidate equation 350 depicted in this example, all playing numbers 320 are used and the mathematical operator 330 of multiplication is used more than once. The candidate equation 350 depicted in this example demonstrates the use of parentheses to ensure a desired order of operations in the candidate equation 350. In the candidate equation 350 depicted, the parentheses are necessary to achieve the target equation value 340. [0038] Advantages of one or more embodiments of the present invention may include one or more of the following:
[0039] In one or more embodiments of the present invention, a method of teaching mathematics teaches or reinforces mathematics skills in a simple, fun, and interactive manner.
[0040] In one or more embodiments of the present invention, a method of teaching mathematics teaches or reinforces mathematics skills in an interactive game-like format.
[0041] In one or more embodiments of the present invention, a method of teaching mathematics teaches or reinforces mathematics skills suitable for use in an intuitive touch screen application.
[0042] In one or more embodiments of the present invention, a method of teaching mathematics challenges a user to construct a candidate equation that evaluates to a candidate equation value that is equal to a target equation value.
[0043] In one or more embodiments of the present invention, a method of teaching mathematics challenges a user to construct a candidate equation from a plurality of playing numbers, at least one of one or more mathematical operators, and zero or more parentheses that evaluate to a candidate equation value that is equal to a target equation value.
[0044] In one or more embodiments of the present invention, a method of teaching mathematics generates a plurality of playing numbers. A number of playing numbers in the plurality of playing numbers generated may correspond to a predetermined level of difficulty. Each playing number in the plurality of playing numbers may be randomly selected from a predetermined range of integer values. The predetermined range of integer values may correspond to a predetermined level of difficulty.
[0045] In one or more embodiments of the present invention, a method of teaching mathematics selects one or more mathematical operators. A number of mathematical operators in the one or more mathematical operators selected may correspond to a predetermined level of difficulty. Each mathematical operator in the one or more mathematical operators may be randomly selected from a list of operators. The list of operators may be a weighted list of operators where a weight assigned to each operator in the weighted list of operators may correspond to a predetermined level of difficulty. The weights may vary to ensure a higher likelihood of selection of more advanced operators at higher levels of difficulty.
[0046] In one or more embodiments of the present invention, a method of teaching mathematics displays a target equation value, a plurality of playing numbers, and one or more mathematical operators on a display device of a computer. The user may be challenged to construct a candidate equation that evaluates to a candidate equation value that is equal to the target equation value. The target equation value may be in a predetermined range corresponding to a predetermined level of difficulty.
[0047] In one or more embodiments of the present invention, a method of teaching mathematics inputs a candidate equation based on a plurality of playing numbers and at least one of one or more mathematical operators. The candidate equation value may be evaluated as the candidate equation is being input and displayed on a display device to assist the user in constructing the equation.
[0048] While the present invention has been described with respect to the above- noted embodiments, those skilled in the art, having the benefit of this disclosure, will recognize that other embodiments may be devised that are within the scope of the invention as disclosed herein. Accordingly, the scope of the invention should be limited only by the appended claims.

Claims

CLAIMS What is claimed is:
1. A computer-implemented method comprising:
generating a plurality of playing numbers;
selecting one or more mathematical operators;
displaying a target equation value, the plurality of playing numbers, and the one or more mathematical operators on a display device;
inputting a candidate equation based on the plurality of playing numbers and at least one of the one or more mathematical operators;
evaluating the candidate equation; and
displaying the candidate equation value on the display device.
2. The computer-implemented method of claim 1 , further comprising comparing the candidate equation value to the target equation value.
3. The computer-implemented method of claim 1, further comprising outputting an indicator of whether the candidate equation value equals the target equation value on the display device.
4. The computer-implemented method of claim 1, wherein a number of playing numbers generated corresponds to a predetermined level of difficulty.
5. The computer-implemented method of claim 1 , wherein each playing number generated is randomly selected from a predetermined range of integer values.
6. The computer-implemented method of claim 5, wherein the predetermined range of integer values corresponds to a predetermined level of difficulty.
7. The computer-implemented method of claim 1 , wherein a number of mathematical operators selected corresponds to a predetermined level of difficulty.
8. The computer-implemented method of claim 1 , wherein each mathematical operator selected is randomly selected from a weighted list of operators and wherein a weight assigned to each operator in the weighted list of operators corresponds to a predetermined level of difficulty.
9. The computer-implemented method of claim 1, wherein the target equation value has a value in a predetermined range corresponding to a predetermined level of difficulty.
10. The computer-implemented method of claim 1, wherein the candidate equation value is evaluated and displayed on the display device as the candidate equation is being input.
1 1. A non-transitory computer-readable medium comprising software instructions that, when executed by a processor, perform a method comprising:
generating a plurality of playing numbers;
selecting one or more mathematical operators;
displaying a target equation value, the plurality of playing numbers, and the one or more mathematical operators on a display device;
inputting a candidate equation based on the plurality of playing numbers and at least one of the one or more mathematical operators;
evaluating the candidate equation; and
displaying the candidate equation value on the display device.
12. The non-transitory computer-readable medium of claim 11, further comprising comparing the candidate equation value to the target equation value.
13. The non- transitory computer-readable medium of claim 1 1, further comprising outputting an indicator of whether the candidate equation value equals the target equation value on the display device.
14. The non-transitory computer-readable medium of claim 11, wherein a number of playing numbers generated corresponds to a predetermined level of difficulty.
15. The non-transitory computer-readable medium of claim 11, wherein each playing number generated is randomly selected from a predetermined range of integer values.
16. The non-transitory computer-readable medium of claim 15, wherein the predetermined range of integer values corresponds to a predetermined level of difficulty.
17. The non-transitory computer-readable medium of claim 11, wherein a number of mathematical operators selected corresponds to a predetermined level of difficulty.
18. The non-transitory computer-readable medium of claim 1 1 , wherein each mathematical operator selected is randomly selected from a weighted list of operators and wherein a weight assigned to each operator in the weighted list of operators corresponds to a predetermined level of difficulty.
19. The non-transitory computer-readable medium of claim 11, wherein the target equation value has a value in a predetermined range corresponding to a predetermined level of difficulty.
20. The non-transitory computer-readable medium of claim 1 1 , wherein the candidate equation value is evaluated and displayed on the display device as the candidate equation is being input.
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