专利汇可以提供Algebraic device and methods of use thereof专利检索,专利查询,专利分析的服务。并且This invention provides a novel affordance to algebra. It combines the immediacy of a manipulative with the power of a computer algebra system. Operations are executed in-place on a mutable expression, by direct manipulation of its terms by the user.Terms, whether simple or complex, can be selected by the user and dragged from one location in the expression to another. The equivalence of the expression is maintained by changes to the moving term and other terms in the expression. A highly interactive interface illuminates to the user the range of potential actions.The beneficiaries of this technology are students, who have a new avenue for exploratory learning and technologists, who have a new tool for symbolic reasoning.,下面是Algebraic device and methods of use thereof专利的具体信息内容。
We claim:
The present application claims priority to and benefit of the prior filed co-pending and commonly owned provisional application, filed in the United States Patent and Trademark Office on Oct. 26, 2009, Application No. 61/254,953 entitled Algebraic Device and Methods of Use Thereof and incorporated herein by reference.
Note that certain terms are defined differently in the two documents. In such cases the scope of the definition is limited to the document in which it appears.
This invention was supported in part by the United States Defense Advanced Research Projects Agency W31P4Q=08-C-0162. The Government has certain rights in the invention.
Our invention relates to a device and method with which a user may perform validated algebraic operations. The primary field of application of the invention may be as a device for mathematics education (434/188). The invention may also be applied to algebraic exercises in fields other than mathematics.
The invention may be used as a working interface of a basic computer algebra system. As such, our invention may be used as a computational device for practical problem-solving. This indicates several fields of art, perhaps fitting best in (708/100/160/171) Electrical Digital Computer with Special Symbolic Output.
Two fundamentally different approaches to education compete for acceptance. The first features the direct transfer of knowledge from master to student—using lecture, reading or demonstration. In the second, the student is guided in an exploratory process during which knowledge is personally discovered. The second method, advocated by Socrates, Maria Montessori and most graduate school programs, claims to achieve deeper understanding and higher retention. But its success depends on the creativity and energy of teachers as they guide individual students. In order to reduce this demand, many tools have been developed to promote the students' discovery of knowledge.
Mathematics education has a history of exploratory learning. A progression of manipulative devices have transformed mathematical abstractions into arrangements of concrete objects made of wood, bone, metal or polymer. Some of these illustrate the properties of numbers, others allow the student to explore operations. Some manipulatives (such as the abacus and the slide rule) demonstrate utility beyond learning and enter the everyday toolbox of those who work with numbers.
Among the aspirations of the present invention is to combine the proven educational value of a math manipulative with the logic engine of a computer algebra systems. These may form a virtual manipulative in which displayed symbols emulate the physicality of material tokens while obeying formal rules.
The roots of the present invention predate the digital computer and the modern patent system. Manipulative devices have been developed to aid math learning and performance since civilization made calculation necessary.
Four millenia ago, the abacus appears. Variations in design support calculation in cultures that use base 60 (Sumerian), base 20 (Aztec) as well as the ubiquitous base 10 number systems. The abacus is a powerful tool for arithmetic processing, but it does little to support symbolic thinking.
An alternative to the abacus was the ancient Chinese system of counting rods. In the 16th century, when the abacus finally displaced them, counting rods were adapted by Japanese algebrist Seki Takakazi as physical tokens that served to introduce his system of symbolic manipulation within the isolated mathematic tradition of wasan.
The slide rule, invented in 18th century Europe, is a familiar manipulative. Unlike the abacus, with its discrete states, the slide rule is an analog computer. It was used for calculation until replaced by the electronic digital calculator in the late 20th century. Although it had a place in schools, the slide rule was principally used for practical engineering. The slide rule offers its user power over numbers, even irrational numbers—but not abstract symbols.
In the twentieth century, classroom manipulatives were developed specifically to expose young children to basic arithmetic and the properties of the base 10 numbers. Most fundamental of these are Cuisenaire rods (1952), colored rods with integer lengths which introduce arithmetic, fractions, and squares.
Sellon (U.S. Pat. No. 4,445,865, 1984) improves the Cuisenaire system to better teach base 10 multiplication. Patents for similar grid-like games, based on familiar multiplication tables, stretch back to Verneau (U.S. Pat. No. 381,829.1922).
Rainey (1998) teaches a device for manipulating base 10 numbers. Such manipulatives teach concrete number relationships—but do not help students learn symbolic thinking.
Not all manipulatives involve rods or grids: Lewis (1926) teaches a reconfigurable roulette apparatus for educational games. While it can be applied to mathematics, it assists only in rote memorization and simple drills.
More relevantly, Donecker (1905) produces a tool for learning algebra. Using a system of balance beams and pans, it builds a physical analog for certain linear equations. The device illustrates the laws of proportionality, and provides a concrete sense of the unknown variable. However it offers no insight into symbolic reasoning.
Harder (1980) presents a four-player game of competitive algebraic solution. His board game allows learners to share the experience of solving an equation. Equations (and inequalities) are presented on wooden strips. Using familiar wooden tiles, the players take turns solving the equations. The board rotates 90 degrees at each turn, with each player validating the work of the previous player. Points are scored for correct operations and solving equations.
The Harder game provides limited exercise in symbolic manipulation, but does so without the benefit of technology.
The introduction of the electronic digital computer allows the evolution of logical systems not limited to manipulation of numbers. Symbolic logic is an early concern of computer scientists. An application providing this capability is called a Computer Algebra System (CAS).
The seminal CAS programs predate the patentability of software. Among these are Macsyma, begun in 1968 at MIT, and released commercially in 1982. At CalTech, SMP and Schoonschip (1963) were the antecedents of the commercially successful program Mathematica (1988, Wolfram) which shares market domination with Maple (1980, University of Waterloo).
Like most CAS packages, Maple and Mathematica are solvers—not teaching programs. They do not demonstrate the internal steps required to go from problem to solution.
Handheld calculators have been marketed which include Computer Algebra Systems. Prominent among these is the TI-NSpire CAS (2006) which has an embedded version of Derive (1988), a small-footprint CAS written, like many algebraic solvers, in a form of LISP.
The prior art includes several examples of interfaces built on top of Computer Algebra Systems with the intent of providing a learning experience.
Certain packages, such as the Algebrator (SoftMath, 1990?) explicitly show every step of the transition, and provide textual explanation of each of these steps. This is meant to educate the student, but it offers no exploratory pathways. Marketed to students as an automated tutor, it could also be abused as a homework robot.
Bonadio (1993) teaches a Computer Algebra System in which a graphic user interface (GUI) allows users to enter commands. Bonadio's system, known variously as Theorist, MathView and LiveMath employs a GUI to direct the CAS to derive a new expression from the existing expression, and to display the new one below the old one. A sequence of expressions cascade down the screen. By displaying line after line of expressions, in the manner of a console-driven application, the software does not behave like a manipulative device. Further, Bonadio's device offers no algebraic response to the user's actions until the user had completely specified the operation.
Addressing a different market, Razdow (1994) introduces the concept of a “live” symbolic expression. Using his software, MathSoft 3.1, a user can manipulate an expression in a document and rely on the editor's symbolic algebra engine to alter other expressions so that they remain true. This invention can be seen as combining the ‘live’ nature of a spreadsheet with the graphic sophistication of an equation compositor. The goal is exposition, not education. The actual solutions that MathSoft performs go little beyond altering the initial assignments in a system of equations and performing the consequent substitutions.
Vernon (application 2004) demonstrates concern for the learning process, and introduces an interactive experience that monitors student performance. Given an expression, the student types in a line of text which represents the new expression after applying an algebraic operation, as a single step toward the solution. The software analyzes both initial and subsequent statements, and searches a rulebook to determine the validity of the student's work. Errors, of course, are flagged. The steps are repeated until the solution is arrived at.
Vernon's technological claims, which—like most CAS engines—focus on the nodal structure of expressions, are interesting in that they amount to a sophisticated equivalent of the “diff” program: they compare two expressions and remove the equivalences. The remaining elements are subjected to a search through a rulebook database.
Vernon's invention compels the student to iteratively enter ASCII text strings with incremental changes to the statement—rather than directly moving the terms of a classical math representation.
It only tests the validity of work after it is performed, rather than supporting the exploratory learner with hints, previews and suggestions.
Our invention relates to a tool for manipulating the terms of an algebraic expression. This device is a computational engine which performs the logical manipulation of symbols as permitted by algebraic rules, with a corresponding display system that exposes representations of these symbols to the user, and a corresponding input system which registers the user's signals of intent.
Advantageously, our invention supports the study and practice of algebraic reasoning with a user experience which is more direct, transparent and facile than any alternative—particularly keyboard entry or pencil and paper methods.
Our invention emulates the physicality of a manipulative device by presenting movable symbols whose behaviors respond to user input even as they reflect the logic of an embedded computer algebra system.
Users perform operations by directly moving terms within a displayed expression while the system alters these terms and/or the remainder of the mutable expression to preserve its validity, consistent with the transformations permitted by the algebra.
Our invention can optionally supply real-time previews indicating which term(s) can be selected, where each can be placed, and which alterations would be performed, if the placement were executed, on the moved term(s) and on the rest of the expression.
Our invention parses the significance of terms in the context from which they are selected and their new significance at each of the various loci where they may be validly placed. When relevant, user intent can be disambiguated by one or more of several information sources which include the path travelled by the moving term, simultaneous movements of other terms, or orthogonal user input. By processing these significances, the invention can compose an expression that conforms to both algebraic rules and user intent.
Our invention optionally offers the unique benefit of recording every step, whether deliberate or tentative, considered by a student. Analysis of the student's process will identify failure points to be remediated by a human teacher, automated mentoring software—or by self-help.
Our invention optionally adapts to the limits and the strengths of various input technologies, such as: Keyboard, Gamepad, Mouse, Touch, Multi-touch, Gesture. Each technology suggests different modalities of selection, drag and release and each presents different feedback opportunity.
Several important features of our invention has been summarized above. Many more are possible; our invention is not to be limited to these examples. Other features and advantages of the inventions may be more clearly understood and appreciated from a review of the following detailed description and by reference to the appended drawings and claims.
All of the drawings display illustrative embodiments of the invention. Other embodiments may be possible.
The invention is described by reference to exemplary embodiments including devices and methods. The invention, however, should not be limited to these embodiments, but may also cover other devices and methods (not specifically described) in accordance with the invention.
Our invention relates to a computational system including a device, and methods of use thereof, that enables the orderly and facile manipulation of one or more symbolic expressions.
Mathematical operators generally include arithmetic, exponential, trigonometric, logarithmic, differential, integral and many other classes of operators. Operators may be represented in any of several accepted forms of notation, some of which vary by domain and context. For example, while addition is almost universally represented by a+b, exponentiation might be shown as YX, Y**X, Y^X or pow(y,x). Conversely, Y^X indicates an exclusive—or operator in other domains. Note that while operators are generally represented explicitly by a symbol, some notations imply a certain operator simply by the graphic relationship of proximate operand symbols. For example, 2X is generally recognized as 2 multiplied by X; while 2X is recognized as 2 raised to the Xth power.
A operand manipulated in this device may be a symbol which represents a value, known or unknown. For example, simple operands are X or θt or 7. A compound operand may be one or more operands combined one or more operators. Examples of compound operands may include: −2 or sin θ or mX or mX+b or πr2. Composited operands such as t0 are accessed by the combined meaning of multiple symbols.
An exemplary embodiment of the invention may comprise:
A computer system comprising processor, memory features, communication feature, display device, input device, and appropriate software, which system may have been configured to perform one or more of the following actions, steps or sequence:
(1) The output device may present one or more expressions to the user. (
(2) Using the input device, the user may select a term. (
(3) Continuing, the user may move said term (
(4) In response to the symbol's arrival at a meaningful locus, the software may alter, add or remove one or more various operators to compose a well formed expression that maintains the truth of relationships in the expression, consistent with the rules of the current algebra, while reflecting the intent of the user. (
As it moves from one region to another, the moving term may transform to maintain algebraic integrity. Operands invert as they cross an equation: an addend negates (
These transformations are not limited to movement across a balanced equation, but also appear within a compound operands including exponents (
In some operations one or more new operand(s) may be also introduced. These operations include algebraic distribution (
Solutions requiring systems of equations can be solved when one equation is simplified and becomes the moving term in operations on another equation (
Algebraic operations will vary across different logical disciplines. For example, in the manipulation of statements of propositional logic (
(5) The user may optionally perform a confirmation step. For example, the user might signify acceptance of the altered algebraic expression by release, or deselection, of the symbol. The system's display will reflect this commitment by a change in the appearance of the expression and the relocated term.
The actions, steps, or sequence immediately above may be repeated once or more or may be repeated indefinitely: by the user, automatically, by several users in parallel or series, or any combination thereof.
While an exemplary embodiment of the invention may be implemented with an electronic digital computer system, other embodiments are anticipated in this invention.
Exemplary Details:
One skilled in the art of application development will be familiar with the elements that comprise any of the several embodiments of this invention.
In
The computational resources of the invention include one or more processing units and the instruction store to support algorithmic operations. These resources support three related engines:
The User Interface (UI) (218) translates information between the internal abstractions and the visible, audible and tangible world of the user. Such systems are well known, and are often specialized for a particular computer system.
The Computer Algebra System (CAS) (220) may be of a well known design such as the several systems cited in above. Both User Interface and Computer Algebra Systems are well studied technologies and both are available in open source software.
The Algebra Animation Logic (AAL) (219) is constructed using standard software development tools, to perform the actions described below. It serves as a bridge between the UI and the CAS.
In a simple exemplary embodiment these may share a single central processing unit and the associated instruction store. More common embodiments would employ several central processing units and an array of graphic processing units within a single hand-held, portable or desktop computer. In such an embodiment, much of the User Interface (218) would reside in the graphic processing units white the Algebra Animation Logic (219) and the Computer Algebra System (220) would share the multiple central processing units as directed by the computer operating system. A third class of embodiment could employ a distributed processing architecture in which the three engines are located in multiple computers and communicate across a network.
The memory resources of the computer store the Expression Database (221) and Algebra Rule Set (222). Many embodiments are possible and each has its particular advantages. The Algebra Rule Set can reside in read-only memory such as an optical disk or semiconductor memory. In a simple embodiment, the Rule Set is embedded in the Computer Algebra System itself either as a block of embedded data or as rules distributed in the algorithms themselves. Such a design is viable in a single purpose embodiment. More flexible devices might keep one or more Algebra Rule Sets in the computing system's ready file system which may include optical, magnetic, semiconductor, and remotely located memory resources. This same file system will serve the purpose of Expression Database. Embodiments with no persistent memory devoted to an expression database are perfectly functional as well. The invention needs to only store one expression during actual operation.
Operation of Exemplary Details:
Operation of the embodiment shown in
Employing well known methods of mathematic layout, such as taught by Razdow, the Algebra Animation Logic (ML) (220) will compose representations of available expressions to be rendered by the User Interface (UI) (219).
As the user begins to manipulate the expression, the ML identifies the movable terms from the nodal structure of the expression as maintained by the Computer Algebra System. These movable terms can be indicated to the user by the UI. In a touch screen embodiment, these may be simply highlighted. In a mouse-driven embodiment, these may be identified by their response to cursor rollover.
The UI, when it detects the selection of a term, highlights the term and moves its location as indicated by the user. The UI informs the ML of the selection. The ML traverses the nodal representation of the expression. Querying the CAS, which in turn consults the active Algebra Rule Set, the ML lists the potential loci to which the term can be moved. The ML constructs a target area around each potential locus. These targets are frequently nested to allow a term to be placed at different loci at different depths of a compound expression.
When crossing between certain nodes of the expression structure, the moving term will itself be altered. For example, when crossing between the two sides of an equation, a multiplicand will become its reciprocal. The ML establishes target regions around the graphic representation of these nodes as well. (
The UI may indicate the potential loci as it did the potential terms, such as by highlighting, or by response to proximity of the cursor. When the moving term enters a target area, the UI informs the ML. The ML employs the CAS to determine the new expression. (
New Utility
An important feature of this invention is the simplification of certain operations. Algebra students frequently execute each step of an algebra problem by performing the same operation to both sides of an equation. With this invention, they can execute the same step more quickly, by directly moving a term. The moving term may invert its value when it crosses an equal sign, an inequality or a division bar.
For example, in elementary polynomial algebra, moving an addend across the equal sign will generally invert the sign of the addend, changing addition to subtraction, or the reverse. Similarly, moving a multiplicand or an exponent across the equal sign will effectively change the term to its reciprocal. These operations may have the same result as the traditional methods of balancing an equation, but do so in a single step in direct response to the translocation of the term by the user.
Automatic introduction, substitution and deletion of operators by this device and methods of use thereof may be governed by the laws of the symbolic system being represented and driven by the actions of the user.
The symbolic grammars over which this device and methods of use may function include (but are not limited to) the basic algebra of polynomial equations, as well as the entire spectrum of symbolic mathematics, including calculus (
Fe+Cu2+→Fe2++Cu
In such use, operands may include representations of atoms and subatomic particles, and operators may represent ionization. A relational operator may include additional information, such as an asymmetric thermal transformation which may or may not be reversible.
Many input devices are appropriate to this system, including ones not yet known. The various sensors which now are used to acquire signals from a user include mouse, joystick, touch screen, game controller, tablet, keypad, touchpad, joypad, footpad, motion sensor, force board, body sensor, camera, brain monitor, speech control, and many others.
Many output devices are appropriate to the system, including ones not yet known. Such devices include, without limitation, visual displays based on cathode rays, light emitting diodes, liquid crystals, plasma, micro-mirror projection, electronic ink, holographic projection, reconfigurable keyboards, tactile screens based on Braille, audio interfaces including formant, concatenative or synthetic speech, and direct neural stimulation.
An embodiment of this device may consist purely of software that directs the performance of a user's general computing hardware.
An embodiment of the device may implement the operational rules of a single symbolic system, such as simple polynomial algebra. Another embodiment might contain the logic to perform operations in a plurality of systems, such as elementary algebra and inorganic chemistry. A third embodiment might include a feature that reads a formal rule set, and can perform operations in any system which is described using this formal grammar. Students may use a device or method according to the invention to learn algebra and other disciplines. Engineers and other technicians may use a device or method according to the invention to solve practical symbolic problems, just as a calculator is used to solve arithmetic problems. Other uses are myriad.
Users of the device or method according to the invention can employ common algebraic strategies including (but not limited to) simplification, extraction of roots, solving for one quantity in terms of another, and/or substitution of terms through a system of equations as well as other methods and processes. The device or method according to the invention may allow such symbolic operations to be performed more quickly, and/or with fewer errors, and/or less training than manual methods.
The exemplary embodiments of the present inventions were chosen and described above in order to explain the principles of the invention and their practical applications so as to enable others skilled in the art to utilize the inventions including various embodiments and various modifications as are suited to the particular uses contemplated. The examples provided herein are not intended as limitations of the present invention. Other embodiments will suggest themselves to those skilled in the art. Therefore, the scope of the present invention is to be limited only by the claims below.
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