Patent Publication Number: US-11037463-B2

Title: Liquid flow instructional systems and methods of making and using same

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This Application is a continuation-in-part of U.S. patent application Ser. No. 15/939,153 filed Mar. 28, 2018 and titled “Method for efficiently teaching content using an adaptive engine and a physical input entry device”, which is a continuation-in-part of U.S. patent application Ser. No. 15/369,699 filed Dec. 5, 2016 and titled “Method for Efficiently Teaching Content Using an Adaptive Engine.” The &#39;699 Application is in-turn a continuation-in-part of U.S. patent application Ser. No. 15/044,641 filed Feb. 16, 2016, which is a continuation-in-part of U.S. patent application Ser. No. 14/833,033 filed Aug. 21, 2015, which is a continuation-in-part of U.S. patent application Ser. No. 14/833,037 filed Aug. 21, 2015. The &#39;699 Application also claims priority to U.S. Provisional Application Ser. No. 62/116,707 filed Feb. 16, 2015, and to U.S. Provisional Application Ser. No. 62/040,091 filed Aug. 21, 2014. The disclosure of each these Applications is incorporated by reference herein in its entirety. 
    
    
     BACKGROUND OF THE DISCLOSURE 
     Mathematics is a structured network of cognitive abstractions subject to precise laws, originally presented almost entirely in prose text and numerals. This approach was the norm until symbolic representation was invented around the 15th Century. The introduction of the symbolic representation allowed people to understand and grasp the abstract nature of mathematics easier and quicker, though at a cost in requiring mastery of the notation and its precise grammatical rules. Resultantly, symbolic representation grew in popularity in mathematics and the associated fields, eventually becoming the new norm and standard. Over the years, symbolic representation became ingrained in mathematic problems present in education, research, science, and engineering. In fact, symbol representation has been used for so long that people assume that mathematic problems can be presented and solved only with symbols and resultantly cannot discern the difference between the visual interface, i.e. symbols, and mathematics itself. While extremely beneficial for research and application purposes, symbolic representation does hinder many people in understanding and using mathematics. Numerous research studies going back to the early 1990s have shown that, when ordinary people are repeatedly presented with mathematical problems in a (non-symbolic) meaningful real-world or real-world-like environment, they rapidly achieve a high level of proficiency. This implies that the difficulties many people experience in doing mathematics are primarily of a linguistic nature, also known as the symbol barrier, and do not indicate a lack of mathematical thinking capacity. 
     Modern technology allows for new and novel means for representation of ideas and theories. The present disclosure relates in part to an alternative representation for problems about proportions (including fractions, decimals, percentages, and relative areas/volumes) that eliminates the traditional use of symbols in order to provide an alternative and user-friendly interface for mathematics. More specifically, the present disclosure relates in part to a method of using a liquid flow system, which can be either physical systems or simulated representations thereof, to visually represent and allow for the solving of problems about proportions, thus overcoming the symbol barrier. This alternative approach to representing mathematical problems may have significant potential, both for uses in mathematics and for educational purposes. 
     FIELD OF THE DISCLOSURE 
     The disclosure relates generally to liquid flow instructional systems and methods. More specifically, the disclosure relates to using physical or other tank systems for providing instruction. 
     SUMMARY 
     In an embodiment, a liquid flow instructional system for presenting a proportions problem and for allowing the proportions problem to be solved comprises a refillable input tank. The system includes a plurality of output tanks. The system comprises an adjustable valve selectively and fluidly coupling the refillable input tank to the plurality of output tanks. The adjustable valve has a face comprising a plurality of regions, each of which is configurable by a user. Each of the plurality of configurable regions corresponds to one of the plurality of output tanks. The system includes an activable switch for initiating flow of liquid from the refillable input tank to the plurality of output tanks in accordance with a user configuration of the plurality of configurable regions The proportions problem is solvable by filling each of the plurality of output tanks to capacity without spillage. A volume of the input tank is one of: (a) equal to a collective volume of the plurality of output tanks; and (b) a multiple of the collective volume of the plurality of output tanks. 
     In another embodiment, a liquid flow instructional system for presenting a proportions problem and for allowing the proportions problem to be solved comprises an input container and a plurality of output containers selectively and fluidly coupled to the input container. The system has an adjustable valve having a plurality of configurable regions. Each of the plurality of configurable regions corresponds to one of the plurality of output containers. An activable switch is provided for initiating flow of liquid from the input container to the plurality of output containers in accordance with a user configuration of the plurality of configurable regions. The proportions problem is solvable by filling each of the plurality of output containers to capacity without spillage. The correspondence between the plurality of configurable regions and the plurality of output containers is indicated by a visible indicator. 
     In yet another embodiment, a method for presenting and solving a proportions problem comprises providing an input tank and a plurality of output tanks. The method comprises fluidly coupling the input tank to the plurality of output tanks via an adjustable valve. The adjustable valve has a face comprising a plurality of regions each of which is configurable by a user. Each configurable region corresponds to one of the plurality of output tanks. The method includes the step of configuring the plurality of regions. The method comprises activating a switch for initiating flow of liquid from the input tank to the plurality of output tanks in accordance with the configuration of the plurality of regions to fill each of the plurality of output tanks to capacity without spillage. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
       Illustrative embodiments of the present disclosure are described in detail below with reference to the attached drawing figures and wherein: 
         FIG. 1  is a schematic representation of a problem structure employed by a system (shown below in  FIG. 13 ) for adaptively teaching content to a user, in an embodiment. 
         FIG. 2  is a flowchart depicting a high-level operation of the system of  FIG. 13 , in an embodiment. 
         FIG. 3  is another flowchart depicting a high-level operation of the system of  FIG. 13 , in an embodiment. 
         FIG. 4  is a flowchart outlining a method used by the system of  FIG. 13  to determine whether the user&#39;s answering data matches a solution to a problem presented to the user. 
         FIG. 5  is a flowchart outlining a method used by the system of  FIG. 13  to determine whether the user&#39;s answering data matches an optimal solution to the problem. 
         FIG. 6  is a flowchart depicting a process used by the system of  FIG. 13  to designate a final iteration of the teaching process. 
         FIG. 7  is a flowchart depicting another process used by the system of  FIG. 13  to designate a final iteration of the teaching process. 
         FIG. 8  is a flowchart depicting another process used by the system of  FIG. 13 . 
         FIG. 9  is a flowchart depicting a process used by the system of  FIG. 13  to make an initial assessment of the competencies of the user. 
         FIG. 10  is a flowchart depicting a process used by the system of  FIG. 13  to analyze performance scores of the user so as to place the user within a particular teaching topic from a series of teaching topics. 
         FIG. 11  is a flowchart depicting a process used by the system of  FIG. 13  to ensure adequate curriculum coverage for the user. 
         FIG. 12  is a flowchart depicting the workings of an entry module of the system of  FIG. 13 . 
         FIG. 13  is a schematic representation of the system for adaptively teaching content to the user, in an example embodiment. 
         FIG. 14  is a schematic representation of an example physical input entry device of the system of  FIG. 13 , here a gear system, in an embodiment. 
         FIG. 15  is a schematic representation of another example physical input entry device of the system of  FIG. 13 , here a liquid flow instructional system, in an embodiment. 
         FIGS. 16A-16C  illustrate a puzzle being represented and solved using the liquid flow instructional system of  FIG. 15 . 
         FIGS. 17A-17F  illustrate another puzzle being represented and solved using the liquid flow instructional system of  FIG. 15 . 
         FIGS. 18A-18C  illustrate yet another puzzle being represented and solved using the liquid flow instructional system of  FIG. 15 . 
         FIG. 19  is a schematic representation of yet another example of a physical input entry device of the system of  FIG. 13 , here a tiles instructional system, in an embodiment. 
         FIGS. 20A-20D  illustrates a linear growth problem being represented and solved using the tiles instructional system of  FIG. 19 . 
         FIG. 21  is a table that illustrates example tiles and tile sections of the tile instructional system of  FIG. 19  and growth rules associated therewith. 
     
    
    
     DETAILED DESCRIPTION 
     A major component of digitally implemented learning systems in mathematics (the field used in this application for illustrative purposes) is the regular provision of problems or puzzles that need to be solved to proceed. It is well established in mathematics education that to be most effective, problems or puzzles must be at the upper limit of a user&#39;s ability at that moment—within what is known as the user&#39;s zone of proximal development (ZPD). To achieve this aim, the system must constantly monitor the performance of the user to determine, dynamically, what the user&#39;s current ability level is, and to select problems or puzzles that keep the user in his or her ZPD. Since mathematical problems or puzzles can be developed on a linear scale of difficulty, doing this is straightforward, and has been implemented on many occasions in different systems. Use of such a linear scale of difficulty can work well in a system that focuses on one particular skill or technique. However, for a learning system that covers a range of topics, there is a tension between ensuring curriculum coverage and maintaining the user in his or her ZPD. 
     Some embodiments of the present disclosure relate generally to the field of cognitive testing and adaptive learning. More specifically, some embodiments of the present disclosure include methods and systems for effectively and efficiently teaching educational content using adaptive learning and open-ended problems or puzzles. In embodiments, an individual&#39;s performance is monitored while he or she is solving a problem and the disclosed systems and methods utilize adaptive learning to select subsequent problems or puzzles of the requisite level of difficulty. This ensures that the individual is adequately challenged and is kept in his or her ZPD. At the same time, embodiments of the present disclosure ensure adequate coverage of each offered curriculum by requiring the individual to solve a specific problem from each curriculum; which if solved, demonstrates high degree of proficiency. A variety of problems may be used for the present disclosure in order to suit the education level for each individual. The problems may be represented in the form of a puzzle or may be presented through a variety of mediums. The ideal problem, in embodiments, is an open-ended problem that is presented to the individual in the form of a puzzle, a game essentially. 
     Some embodiments of the present disclosure also relate to systems usable to present such open-ended (or other instructional) problems, and to methods of making and using such systems. These systems, in addition to presenting the instructional problems to the user, may be configured to allow the user to respond to the problems in a step by step fashion such that insight is gleaned into the user&#39;s problem solving thought process. Such insights into the user&#39;s thought process while solving problems may allow the instruction to be better tailored to the user, as compared to, for example, the conventional multiple choice format used in schools today. In an embodiment, the instructional system may be a gear system (such as a physical or virtual gear system). In another embodiment, and as discussed in more detail herein, the instructional system may be a liquid flow system (such as a physical or a virtual liquid flow system). In another embodiment still, the instructional system may be a tiles system (such as a physical or a virtual tiles system). These different instructional systems may be generally configured to present to the user (and educate the user about, determine the user&#39;s mastery in, etc.) different types of problems. For example, the gears system may be generally configured to present algebraic problems. The liquid flow system may be generally configured to present problems regarding proportional reasoning (e.g., fractional quantities, decimals, percentages, relative areas, etc.). The tiles system may be generally configured to present problems relating to linear growth functions, e.g., simultaneous linear equations with a single unknown. And so on. 
     The disclosure below discusses the various concepts outlined above in more detail. Specifically,  FIG. 1  and the associated discussion thereof in the disclosure relates to an organizational scheme for storing problems when adaptively and efficiently teaching content to a user.  FIG. 13  shows a system for adaptively and efficiently teaching content to a user, and  FIGS. 2-12  relate to various aspects thereof.  FIG. 14  relates to a physical or virtual gear system which may be a part of or which may be usable with the instructional system of  FIG. 13 .  FIGS. 15, 16A-16C, 17A-17F, and 18A-18C  relate to a physical or virtual liquid flow system which may be a part of or which may be usable with the instructional system of  FIG. 13 .  FIGS. 19, 20A-20D , and  21  relate to a physical or virtual tiles system which may be part of or which may be usable with the instructional system of  FIG. 13 . The disclosure discusses these figures in-turn. 
     Referring to  FIG. 1 , the present disclosure includes a series of teaching topics, wherein each teaching topic includes a lead problem and a plurality of secondary problems (Step A). Each of the teaching topics is associated with a specific curriculum; a curriculum may be focused on a specific concept, puzzle type, theme, or a field of study. For example, one implementation of the present disclosure utilizes different mathematical concepts and problem-solving challenges in order to make up the series of teachings topics. The lead problem and secondary problems for each teaching topic all focus on the same curriculum. Each of the problems is an open-ended problem or puzzle and can be solved in a multitude of ways, with each way being associated with an answer that is satisfactory according to a prescribed measure. More specifically, the lead problem and the secondary problems for each of the teaching topics are associated with an optimal solution and at least one other solution (Step B). The optimal solution may be defined based on the least number of steps used to solve the problem, the highest score attained in solving the problem according to a prescribed scoring system, the exact sequence of steps taken to solve the problem (“solution path”), and/or other similar characteristics. The other solution is any solution other than the optimal solution. The artisan will understand from the disclosure herein that a problem may have two or more optimal solutions (e.g., where the optimal solution is defined based on the least number of steps, two or more solutions may be deemed optimal where they each involve the same (lowest) number of steps). Similarly, the artisan will appreciate that a problem may have two or more solutions other than the optimal solution. Thus, each of the phrases “an optimal solution”, “the optimal solution”, “other solution”, “the other solution”, etc., may but need not refer to a solitary solution. 
     As can be seen in  FIG. 1 , in an embodiment of the present disclosure, the series of teachings topics is organized in a tree-like structure, comprising a central trunk and a multitude of branches. The central trunk comprises the lead problems for each of the teaching topics arranged in a linear fashion. Each of the lead problems is further connected to an emanating branch. Each branch comprises the secondary problems associated with the teaching topic of the lead problem. The secondary problems and lead problem for each of the teaching topics is further associated with a difficulty rank that is used to incrementally increase the problem difficulty for the individual. In an embodiment of the present disclosure, the difficulty rank of the lead problem is greater than the difficulty rank of each secondary problem within each of the teaching topics. Thus, the lead problem is used as a test for the associated teaching topic. If the individual can effectively solve the lead problem for a specific teaching topic, then he or she may skip the secondary problems of the specific teaching topic. This allows an individual that has a high level of proficiency to quickly progress through the series of teaching topics to a curriculum that adequately challenges him or her without having to repeat content which he or she has already mastered. 
     Embodiments of the present disclosure comprise a method and a system. The method delineates the rules and steps necessary to construct a specific path for a user through the series of teaching topics. The specific path is based on the performance of the user and thus is modified after each problem addressed by the user. The system comprises the physical components necessary to execute the method of the present disclosure. The system may comprise a personal computing (PC) device and a physical input entry device discussed further below. The PC device includes a processor and a physical user interface (Step C). As discussed herein, the physical user interface (or the physical input entry device) may be a device not conventionally associated with generic computers. The processor executes the method of the present disclosure in the form of a software application at least in part. The computing device administers the series of teaching topics and the physical input entry device allows the user to interact with the present disclosure to solve and transition through the series of teaching topics. Type of devices that may be used as the PC device include, but are not limited to, desktop computers, laptop computers, smartphones, tablets, and other similar electronic devices. Types of devices usable in the present disclosure as the physical input entry device are discussed further below. As discussed in more detail herein, in some embodiments, the functions of the physical input entry devices may be effectuated by virtual devices, e.g., by means of interactive graphical user interfaces that emulate these physical input entry devices and allow for the solution path of the user to be captured and evaluated. 
     Two important aspects to note for the present disclosure: there are no multiple-choice questions and the user must carry out all key steps of the problem or puzzle with the PC device. This allows the present disclosure to monitor and track every step that the user goes through (“solution path”) in order to solve the problem or puzzle, thus providing raw descriptive information relating to the individual&#39;s cognitive/solving ability. 
     Referring to  FIG. 2  and  FIG. 3 , the overall process of the present disclosure begins with the physical user interface prompting to solve a specific problem within an arbitrary teaching topic, wherein the arbitrary teaching topic can be any topic within the series of teaching topics (Step D). The user then attempts to solve the specific problem through the physical user interface. Answering data for the specific problem is received with the PC device (Step E) to be analyzed. If the answering data is not acceptable, then Steps D and E are repeated until the answering data matches either the optimal solution or the other solution of the specific problem. Once a solution for the specific problem is found, the user&#39;s performance is analyzed based on which solution was found and, resultantly, directed accordingly through the series of teaching topics. 
     If the answering data of the specific problem matches the other solution of the specific problem, then the user is directed to solve a next problem within the arbitrary teaching topic; the computing device prompts to solve the next problem within the arbitrary teaching topic (Step G). The other solution for the specific solution indicates average proficiency in the curriculum of the arbitrary teaching topic. In which case, the user is directed to solve the secondary problems from the arbitrary teaching topic in order to practice, achieve mastery, and ensure curriculum coverage before progressing to the next curriculum, i.e. the next teaching topic following the arbitrary teaching topic. In other words, this conditional moves the user through the branch of the arbitrary teaching topic one problem at a time if any solution besides the optimal solution is entered. Alternatively, if the answering data of the specific problem matches the optimal solution of the specific problem, then the user is prompted to solve the lead problem within a next teaching topic through the physical user interface (Step H). The next teaching topic is defined as the teaching topic following the arbitrary teaching topic within the series of teachings topics. In general, identifying the optimal solution for the specific problem signifies that the user has the required degree of solution proficiency for the curriculum associated to the arbitrary teaching topic. Thus, the user is permitted to skip the rest of the problems within the arbitrary teaching topic and jump to the next point in the trunk. This condition endures that the user is kept within his or her ZPD at each step within the series of teaching topics. 
     Additionally, during Step H, if the specific problem is a last problem within the arbitrary teaching topic, then the user is prompted to solve the lead problem within the next teaching topic, regardless whether the answering data for the specific problem matches the optimal solution or the other solution of the specific problem. Reaching and solving the last problem within the arbitrary teaching topic indicates that the user has reached an acceptable proficiency for the curriculum associated with the arbitrary teaching topic and is thus permitted to move on to the next teaching topic. 
     Finally, the last step in the overall process of the present disclosure is executing the aforementioned steps for the series of teaching topics. In particular, executing a first plurality of iterations for Steps D through H with the processor by using either the next problem within the arbitrary teaching topic of an arbitrary iteration or the lead problem within the next teaching topic of the arbitrary iteration as the specific problem of a subsequent iteration (Step I). This is executed until the arbitrary iteration is circumstantially designated as a last iteration by the processor. The arbitrary iteration and the subsequent iteration are from the first plurality of iterations. Each of the first iterations is Step D through H being executed for a particular problem; the particular problem is dependent on the user&#39;s real-time performance and knowledge/proficiency of the curriculum being addressed. 
     The overall process of the present disclosure is executed until the user demonstrates adequate proficiency in every teaching topic. In relation to the overall process, this is the case when the arbitrary iteration is designated as the last iteration. One such instance is when the user shows adequate proficiency in a final teaching topic by solving one of the problems from the final teaching topic with the optimal solution of said problem; wherein the final teaching topic is the last topic within the series of teaching topics. Referring to  FIG. 6 , the user is finished if the following conditions are met: (1) the teaching topic of the specific problem is the final teaching topic; and (2) the answering data for the specific problem matches the optimal solution of the specific problem. If these conditions are met, then the arbitrary iteration is designated as the last iteration during Step H with the processor. Thus, indicating that the user has mastered the curriculum of the final teaching topic and, resultantly, has finished the series of teaching topics. 
     Another instance is when the user has reached and solved a last problem within the final teaching topic. Referring to  FIG. 7 , in relation to the overall process, the user is finished if the following conditions are met: (1) the teaching topic of the specific problem is the final teaching topic; (2) the answering data for the specific problem matches either the optimal solution or the other solution of the specific problem; and (3) the specific problem is the last problem within the final teaching topic. If these conditions are met, then the arbitrary iteration is designated as the last iteration during Step H by the processor, and the user finishes the series of teaching topics. 
     Referring to  FIG. 4 , if the user finds the other solution for the specific problem, then he or she may be directed onto two different paths. The determining factor is if the specific problem is either the lead problem or one of the secondary problems. Prior to directing the user, the processor first sequentially orders the secondary problems relative to the difficulty rank such that the user is incrementally exposed to harder and harder problems. If the specific problem is the lead problem, then the user is directed to solve the secondary problems within the arbitrary teaching topic. More specifically, a least-difficult secondary problem is chosen and designated as the next problem within the arbitrary teaching topic during Step G, wherein the least-difficult secondary problem is from the plurality of secondary problems within the arbitrary teaching topic. 
     Alternatively, if the specific problem is one of the plurality of secondary problems, then the user is directed to solve the problem after the specific problem within the arbitrary teaching topic. In particular, a next-most-difficult secondary problem is designated as the next problem within the arbitrary teaching topic during Step G. The next-most-difficult secondary problem is from the plurality of secondary problems within the arbitrary teaching topic. Furthermore, it is important to note that the last problem referenced in Step H is the final problem within the arbitrary teaching topic. More specifically, a most-difficult secondary problem is designated as the last problem during Step H; wherein the most-difficult secondary problem is from the plurality of secondary problems within the arbitrary teaching topic. The final problem is the most difficult in order to test the user in the curriculum of the arbitrary teaching topic. 
     Referring to  FIG. 8 , anytime during the overall process of the present disclosure the user is able to return to previously addressed problems and attempt to find a different solution, in particular, the optimal solution. In relation to the overall process, step C and step D may be repeated for a previous iteration during the arbitrary iteration, if the specific problem from the previous iteration and the specific problem from the arbitrary iteration are within the arbitrary teaching topic, wherein the previous iteration is a designated number of iterations back from the arbitrary iteration. The designated number of iterations is set by an administrator account. Any problems further back than the designated number of iterations will not award the user with the ability to skip to the next teaching topic if he or she identifies the optimal solution. In alternative embodiments of the present disclosure, the user may cross to previous topics in order to repeat problems. If the user matches the answering data from the previous iteration to the optimal solution for the specific problem from the previous iteration, then the system executes step H for the previous iteration during the arbitrary iteration. 
     Referring to  FIG. 9 , prior to allowing the user to solve the series of teachings topics, the present disclosure first requires the user to pass through an entry module. The entry module provides a rapid assessment of the user&#39;s ability and proficiency regarding the curriculums within the series of teaching topics. The results from the entry module are used to place the user within the series of teaching topics accordingly. For example, weak users are placed at an initial topic from the series of teaching topics while stronger users may be allowed to skip a number of early topics. 
     The entry module includes a series of assessment problems, wherein each assessment problem is associated with an optimal assessment solution and at least one other assessment solution, similar to the overall process (Step J). The series of assessment problems is populated with questions, problems, or puzzles of different curriculums, thus allowing the system to fully determine the user&#39;s abilities. Additionally, the assessment problems may be easier than the problems from the series of teaching topics. The process for the entry module is similar to the overall process of the present disclosure. First, the user is prompted to solve a specific assessment problem from the series of assessment problems through the physical user interface (Step K). Next, the user solves the specific assessment problem through the physical input entry device. The system receives answering data for the specific assessment problem (Step L). Steps K and L are repeated until the answering data for the specific assessment problem matches either the optimal assessment solution or the other assessment solution of the specific assessment problem. The user&#39;s path through the assessment problems is partially adaptive, i.e. the path is dependent on the user&#39;s performance. 
     If the answering data matches the other assessment solution of the specific assessment problem, then the user is incrementally moved to the next problem within the series of assessment problems. In particular, the user is prompted to solve a first succeeding problem through the physical user interface, wherein the first succeeding problem is sequentially adjacent to the specific assessment problem along the series of assessment problems (Step N). This is similar to the overall process. 
     If the answering data matches the optimal assessment solution of the specific assessment problem, then the user is moved forward through the series of assessment problems a pre-set number of steps. In particular, the user is prompted to solve a second succeeding problem through the physical user interface, wherein the second succeeding problem is sequentially offset from the specific assessment problem along the series of assessment problems (Step O). The offset, the number of steps, may vary depending on the specific assessment problem, the type of educational content, type of problems, or type of puzzles used for the present disclosure. 
     The user is maintained within the entry module until he or she reaches and solves a final problem within the series of assessment problems. More specifically, the processor executes a second plurality of iterations for Steps K through O by using either the first succeeding problem or the second succeeding problem of an arbitrary assessment iteration as the specific assessment problem for a subsequent assessment iteration. The second plurality of iterations is executed until the arbitrary assessment iteration is circumstantially designated as a last assessment iteration by the processor. The arbitrary assessment iteration and the subsequent assessment iteration are any sequential pair of iterations within the second plurality of iterations. 
     Referring to  FIG. 12 , the present disclosure utilizes performance data from the entry module to determine where in the series of teaching topics the user should be placed. In order to achieve this, performance criteria are provided for each of the teaching topics. The performance criteria quantify a minimum proficiency/ability necessary to solve problems within the associated teaching topic. Once the user completes the entry module, the processor assesses a performance score for each of the second plurality of iterations. 
     A variety of scoring methods may be used for determining the performance score. Then, the performance score for each of the second plurality of iterations is compiled into an overall performance score with the processor. The overall performance score is then compared to the performance criteria for each teaching topic with the processor in order to identify a set of matching topics from the series of teaching topics. The set of matching topics is the teaching topics within the series of teaching topics which the user has shown proficiency in and therefore does not need to solve. This ensures that the problems addressed by the user in the overall process of the present disclosure are within his or her ZPD. 
     Once identified, the set of matching topics is then displayed to the user for selection. Referring to  FIG. 10 , the physical user interface prompts the user to select a specific topic from the set of matching topics. Once chosen, the selected topic is designated as the arbitrary teaching topic in Step D of an initial iteration from the first plurality of iterations. This process assesses the user&#39;s ability and places him or her accordingly within the series of teaching topics. 
     In one embodiment, the present disclosure also includes a basics module, essentially a training area (also referred to herein as a tutor module). If at any point the system identifies that the user is struggling to solve a problem, then he or she may be directed towards the basics module. In one embodiment, certain problems within the entry module are dedicated to separating users with strong and weak abilities. The basics module tutors the user through basic elements utilized in the problems within the series of assessment problems and the series of teaching topics. In order for the user to exit the basics module, the user must complete all the problems and tasks within the basics module. Although, there is a one-time exit opportunity, if the user solves the first predetermined number of problems within the basics module by finding the optimal solution in a single try for each one, then the user may exit the basic module. 
     In an embodiment, a system for teaching content using an adaptive engine may include one or more computing devices coupled to one or more input entry devices (also referred to herein as an “interface device”). The input entry device coupled to the computing device(s) may be a physical device other than a conventional computer component, such as a keyboard, mouse, a touchscreen display, etc. For example, in embodiments, the input entry device may be a physical device that includes rotatable gears enmeshed with each other. Or, for instance, the physical input entry device may comprise pieces of a puzzle that can be arranged in predefined patterns. In these embodiments, the user may use the physical input entry device to solve one or more problems (e.g., puzzles or other problems) displayed elsewhere, e.g., on a display of the computing device. The computing device may evaluate the inputs provided by the user via the physical input entry device and, based on this evaluation, adaptively select the next problem to be presented to the user. As discussed above, and depending on the user input, the next problem presented to the user may be a problem within the same teaching topic or a different teaching topic (e.g., a lead problem of a different teaching topic). 
       FIG. 13  shows an example system  100  for teaching content using an adaptive engine and a physical input entry device, as discussed herein. The system  100  may include a structure  102 . The structure  102  may be a computer, a server, a network of computer servers, etc., and is shown with a processor  106  communicatively coupled to a network interface  108 , an input/output device  109 , and a memory  110 . Processor  106  represents one or more digital processors. Network interface  108  may be implemented as one or both of a wired network interface and a wireless network interface, as is known in the art. The input/output device  109  may include any suitable input/output device, such as a display, speakers, a keyboard, a mouse, a touchscreen, etc. Memory  110  represents one or more of volatile memory (e.g., RAM) and non-volatile memory (e.g., ROM, FLASH, magnetic media, optical media, et cetera). Although shown within structure  102 , memory  110  may be, at least in part, implemented as network storage that is external to structure  102  and accessed via network interface  108 . 
     Software  114 , a user database  116 , and a problems database  117  may be stored within a transitory or non-transitory portion of the memory  110 . Software  114  includes machine readable instructions that are executed by processor  106  to perform the functionality of structure  102  as described herein. The user database  116  may include a plurality of records, each pertaining to one of a plurality of users. For example, the user database  116  may include a listing of lead problems attempted and/or solved by each user, a listing of secondary problems attempted and/or solved by each user, and other such user-specific information. The user database  116  may, in embodiments, be omitted. 
     The problems database  117  may include a database of lead problems and associated secondary problems, such as mathematical problems or puzzles, or other problems, arranged for example by teaching topic, concept type, puzzle type, theme, field of study, etc. The problems database  117  may further include each or at least a plurality of solutions for each problem, including the optimal solution thereof, together with a difficulty rank for each problem. The software  102  may be configured to present a user a lead problem, and subsequently, another lead problem or a secondary problem associated with the original lead problem, based on an input provided by the user via the input entry device (as discussed herein). 
     The online structure  102 , using protocol  118  and Application Programming Interface  132 A, may communicate over a wired or wireless network  104  with an input entry device  134  of a user  136 . The user  136  may be any individual (or in embodiments, group of individuals) who are being educated and/or evaluated using the system  100  described herein. 
     Network  104 , which is formed in part by one or more of the Internet, wireless networks, wired networks, local networks, and so on, facilitates communication between the structure  102  and the input entry device. The user  136  views a problem presented by the software  114  on the input/output device  109 , e.g., a display of or associated with the online structure  102 , and in response thereto, utilizes the input entry device  134  to solve the presented problem. The software  114  evaluates the input provided by the user  136  and, based on this evaluation, presents on the output device  109  another lead problem or a secondary problem having a different difficulty rank. The input entry device  134  may include one or more sensors  134 A to allow for relevant interaction of the user  136  with the components of the input entry device  134  to be communicated to the software  114  (e.g., motion and/or rotation detecting sensors such as optical and/or magnetic sensors, pressure detecting sensors, temperature sensors, weight sensors, volume sensors, etc.). In embodiments, the input entry device  134  may also include one or more processors or other such devices to allow for the output of the sensors  134 A to be evaluated. In other embodiments, the input entry device  134  may be devoid of a processor or other comparable device and the adaptive engine  126  may be configured to decipher the output of the sensors  134 A. In other embodiments, the input entry device  134  may be a stand-alone device (e.g., a battery operated or other dedicated device). 
     The input entry device  134 A may further include, in addition to the sensing devices  134 A, responsive devices  134 B. The responsive devices  134 B may be configured to provide a controlled response in reaction to the sensed input. The responsive devices  134 B may be, for example, a pump (e.g., a pump that causes liquid to flow from one location to another based on a user input sensed by a sensor), a light or a speaker that is activated when a puzzle is solved or during the puzzle presentation, a cage that opens when a puzzle is solved by the user, a battery operated lever, a spring activated device, etc. 
     While the structure  102  is shown as having a solitary input entry device  134  coupled thereto, in embodiments, the structure  102  may have a multitude of input entry devices  134  in communication therewith (e.g., the structure  102  may be in communication with a statistically significant number (such as hundreds of thousands) of input entry devices  134 ). In these embodiments, each of the plurality of input entry devices  134  may be associated with a unique user. The user, e.g., the user  136 , may also couple his or her input entry device  134  with the structure  102  indirectly. For example, in embodiments, the structure  102  may be an online structure (e.g., may be a webserver) and each user may interact therewith by coupling their respective input entry device  134  to their personal (or other) computer which is in-turn coupled to the structure  102 . In embodiments, the system  100  may be a dedicated device (e.g., may be configured to effectuate only the purposes described herein). 
     The software  114  may include an adaptive engine  126 . The adaptive engine  126  may include an evaluator  124 . The adaptive engine  126  may initially present to the user  136  a lead problem associated with a particular topic via the input/output device  109 . The user  136  may use the input entry device  134  in an attempt to solve this lead problem. The user&#39;s input may be communicated to the structure  102  as answering data, and the evaluator  124  thereof may evaluate the answering data to determine if the answering data includes or otherwise corresponds to the optimal solution. If so, the evaluator  124  may subsequently present to the user  136  via the input/output device  109  a suitable problem  127  which is associated with a different teaching topic (see  FIG. 1 ). Alternately, if the input provided by the user  136  via the input entry device  134  includes a non-optimal solution, the evaluator  124  may present to the user  136  the suitable problem  127  which, in this case, may be a secondary problem associated with the same teaching topic. 
     In embodiments, the software  114  may also include a performance module  152 , an entry module  154 , and a tutor module  156 . As is apparent from the disclosure herein, the adaptive engine  126 , together with the performance module  154 , may monitor the user&#39;s performance  136  to ensure that problems are presented to the user  136  so as to adequately challenge the user  136  while keeping the user  136  in his or her ZPD. The entry module  154 , also discussed above, may together with the adaptive engine  126  initially present to the user  136  a series of assessment problems to allow the evaluator  124  to obtain a baseline assessment of the user&#39;s mastery over the teaching curriculum. The tutor module  156 , also referred to as a basics module above, may be configured to tutor the user  136 , e.g., by teaching him or her about the basic elements of a teaching topic, based on a determination that the user  136  is struggling to solve the presented problem. 
     As discussed above, the adaptive engine  126  may adaptively determine the suitable problem  127  based on the input provided by the user  136  via the input entry device  134 . In embodiments, when determining the suitable problem  127  to be presented to the user  136 , the adaptive engine  126  may also take into account inputs provided by other users. For example, where inputs from a multitude of users indicate that a particular problem within a teaching topic is easier to solve than the preceding problem in that topic, the adaptive engine  126  may, based on these inputs, adaptively change the difficulty rank of these problems in the problems database  117 . The artisan will understand that in so doing the system  100  may benefit from a statistically significant number of users  136  (for instance, it may be more beneficial to adaptively change the difficulty rank of a problem based on the input of many thousands of users as compared to changing the difficulty rank of a problem based on the input of two or three users). Thus, use of a statistically significant number of users may facilitate optimal operation of some embodiments of the system  100 . 
     Workings of the disclosure will now be illustrated with an example. The artisan will understand that the example is not intended to be limiting. 
     Focus is directed to  FIG. 14  which shows an input entry device  200 . This input entry device  200  is but one example of the input entry device  134 . The input entry device  200  is modeled after the gear system in U.S. patent application Ser. No. 14/833,037 filed Aug. 21, 2015, which, as noted above, is incorporated by reference herein. The &#39;037 Application illustrates the workings of the physical gear system in detail, but discusses the physical gear system as a stand-alone device. A primary difference between the physical gear system disclosed in the &#39;037 Application and the physical gear system  200  is that the gear system  200  is communicatively coupled to the structure  102 , as illustrated in  FIG. 13  via the input entry device  134 . The physical input entry device (or gear system)  200  is described herein to illustrate use of the system  100  for teaching mathematical content, and particularly, algebraic equations, using the adaptive engine  126 . The artisan will understand that while mathematical content is used as an example to illustrate the workings of the system  100 , that the system  100  may likewise be used to adaptively teach other content to users (e.g., the user  136 ). The disclosure below first details the example input entry device  200 , and then outlines an example use of the input entry device  200  in the system  100  to teach content to the user  136  adaptively. 
     The physical gear system  200  visually represents each entity of an algebraic equation and allows the user  136  to manipulate said entities through the individual gears of the gear system in order to determine a solution to the algebraic equation. Entities of the algebraic equation include a plurality of terms and at least one numerical constant, wherein one side of the equation is the plurality of terms and the other side of the algebraic equation is at least one numerical constant. Each of the plurality of terms includes a coefficient and a variable. The variable is a symbol that represents an undefined value within the algebraic equation, while the coefficient is a constant number which multiples or amplifies the variable. Solving the algebraic equation includes identifying a value for each of the variables, which would balance the two sides of the algebraic equation. 
     The illustrated input entry device  200  includes a primary cog  1 , a plurality of secondary cogs  2 , and a fixed pointer  3 . The primary cog represents a range of solutions for the algebraic equation and includes a plurality of teeth that is quantitatively greater than the numerical constant. For example if the numerical constant is 20, than the number of teeth on the primary cog would need to be greater than 20. The plurality of teeth for the primary cog includes an origin tooth  4  and a target tooth  5 , each marked accordingly. 
     The origin tooth marks a starting point that the user  136  may reference in order to identify the remaining teeth within the plurality of teeth, essentially representing the zero value. The target tooth represents the numerical constant of the algebraic equation. The target tooth is quantitatively offset from the origin tooth by the numerical constant, thus visually displaying the numerical constant as a radial increment on the primary cog. Additional teeth may be marked on the primary cog to indicate their respective offset quantity from the origin tooth. In one embodiment, each tooth on the primary cog is marked with a respective offset quantity from the origin tooth. Alternatively, every incremental tooth may be marked. 
     The plurality of secondary cogs represents the side of the algebraic equation relating to the plurality of terms. Each of the plurality of secondary cogs is associated with a corresponding term from the plurality of terms. This relationship is conveyed to the user by quantitatively matching a plurality of teeth on each secondary cog to the value of the coefficient of its corresponding term. For example, if the corresponding term is “4x”, then the secondary cog representing this particular term would have four teeth. Each of the secondary cogs may be marked with a readable label that depicts the coefficient of the corresponding term, in turn conveying to the user the number of teeth present on said secondary cog. Each of the secondary cogs is designed to mesh with and engage the primary cog such that rotation of each of the plurality of secondary cogs is used to drive the rotation of the primary cog. This includes matching the size and type of the teeth used for each of the plurality of secondary cogs to that of the primary cog; a variety of types of teeth may be used for the primary cog and thus the secondary cogs. As discussed herein, because the number of teeth of each of the three secondary cog  2  is disparate, a full rotation of each secondary cog  2  will cause the primary cog to move by different amounts. 
     The fixed pointer indicates the current output for the input entry device  200 , wherein the output corresponds to the side of the algebraic equation associated with the numerical constant. Additionally, the fixed pointer is used to zero/reset the gear system prior to solving the algebraic equation. The gear system  200  is zeroed by positioning the origin tooth coincident with the fixed pointer. The fixed pointer is preferably shaped similar to an arrowhead and is positioned adjacent to the primary cog, oriented towards the center of the primary cog. 
     In general, the method for solving the algebraic equation involves aligning the target tooth at the fixed pointer, thus setting the current output of the primary cog to be the numerical constant. This is accomplished by first identifying a current tooth at the fixed pointer, wherein the current tooth is any one of the plurality of teeth on the primary cog. If the current tooth is not the origin tooth, then the primary cog is rotated until the origin tooth is set at the fixed pointer, essentially calibrating or resetting the input entry device  200 . Once the device  200  is reset, a plurality of rotations with one or more of the plurality of secondary cogs is then executed in order to rotate the primary cog so that the target tooth aligns with the fixed pointer. This alignment between the target tooth and the fixed pointer yields a possible solution for the algebraic equation. The potential solution lies in the number of rotations executed for each of the secondary cogs. For example, two rotations of the secondary cog that is associated with the term “4x” translates to the variable “x” being two. Once the target tooth is aligned with the fixed pointer, then the plurality of rotations is quantitatively identified for each of the secondary cogs as a potential solution for the variable of the corresponding term. The rotation direction of each of the secondary cogs represents either an increase or decrease in value for the variable of the corresponding term. A clockwise rotation by the secondary cog represents a quantitative increment in the potential solution of the variable for the corresponding term. Similarly, a counterclockwise rotation by the secondary cog represents a quantitative decrement in the potential solution of the variable for the corresponding term. For example, rotating one of the secondary cogs three turns clockwise and two turns counterclockwise means the value for the variable of the corresponding term is one. 
     Positioning the target tooth at the fixed pointer yields a solution for the algebraic equation, wherein the solution includes a potential solution for each of the variables, for each of the terms. However, this solution is only one of many possible solutions for the algebraic equation. The most optimal solution in this example is achieved by minimizing the collective rotations of the secondary cogs  2 . The least amount of rotations for each of the plurality of secondary cogs represents the most efficient and optimal solution for the algebraic equation. 
     The input entry device  200  may also be used to solve the algebraic equation for a plurality of numerical constants, which is also known as a system of equations. Solving for the numerical constants includes repeating the aforementioned method a multitude of times. That is, each of the iterations is executed in order to solve the algebraic equation with a corresponding constant from the numerical constants as one side of the algebraic equation. Similar to solving for one numerical constant, an initial iteration from within the plurality of iterations includes identifying the origin tooth as the current tooth and beginning the plurality of iterations from the origin tooth. An arbitrary iteration from the plurality of iterations is defined as any iteration other than the initial iteration, while the previous iteration is defined as the iteration that is executed prior to the arbitrary iteration. Solving for the numerical constants requires identifying the target tooth of the previous iteration as the current tooth of the arbitrary iteration. Consequently, the primary cog is not zeroed before each iteration. For example, once the target tooth of each numerical constant has been aligned to the fixed pointer, then a solution is identified for the algebraic equation. An optimal solution in this example is achieved when a plurality of collective rotations is minimized during the iterations. The plurality of collective rotations is defined as the summation of the rotations executed by each of the secondary cogs during each iteration. 
     When solving the algebraic equation for more than one numerical constants (e.g. a system of equations), the input entry device  200  allows for constraints in the manner that a user solves for potential solutions. The present disclosure provides a plurality of constraining categories, each of which is associated with a priority rank. The constraining categories are used to guide the steps taken by the user to solve the algebraic equation with the present disclosure. Each numerical constant is assigned to a designated category from the plurality of constraining categories. This allows the system  100  to constraint an execution sequence for the plurality of iterations in accordance to the priority rank of the corresponding constant, and the priority rank is derived from the designated category of the corresponding constant. The execution sequence for the plurality of iterations provides the user with a guide to optimize the manner in which to solve for the potential solutions of the algebraic equation. 
     Essentially, the execution sequence prompts the user to align the fixed pointer to one category of target teeth before aligning the fixed point to another category of target teeth. The plurality of constraint categories places restrictions on the manner on how the present disclosure can be used to solve the algebraic equation, similar to how a system of equations can be solved in multiple ways but is still mathematically constrained. The algebraic equation may but need not contain only whole numbers. Also, in some embodiments, a sequential turn limit may be applied to each of the secondary cogs in order to indicate the number of rotations by a secondary cog has exceeded the most optimal solution by a significant amount. Consequently, the plurality of rotations with each of the secondary cogs  2  may not exceed the sequential turn limit. 
     In the illustrated embodiment, the input entry device  200  is implemented in the form of a physical apparatus. The physical apparatus  200  includes a multitude of gears and a support structure  202 . The primary cog and the secondary cogs are expressed by the gears. The gears are rotatably mounted to the support structure  202 , e.g., on rotatable spindles provided thereon as shown in  FIG. 14 , and are positioned as described herein. The user  236  may rotate the secondary cogs  2  (individually labeled A, B, and C for illustration) in order to identify the solution to the algebraic equation. That is, in this example, to find a solution to an algebraic equation presented to the user  136 , the user  136  must physically rotate the secondary cog(s) A, B, and/or C. And, each rotation of each secondary cog may be a physical action that may be recorded by the structure  102  and evaluated thereby to determine the pros and cons of the solution chosen by the user  136 . The user input may be communicated over the network  104 A to the structure  102 . For example, if the user  136  rotates the secondary cog A once clockwise and the secondary cog C twice counterclockwise, each of these inputs may be communicated to the structure  102  and evaluated by the software  114  as discussed herein. 
     In an embodiment, the adaptive engine  126  may present the problem to the user  136  via the input/output device  109  (e.g., a display). The user  136  may attempt to solve the problem displayed on the display  109  by physically rotating one or more secondary cogs  2  of the input entry device  200 . The adaptive engine  126 , e.g., the evaluator  124  thereof, may evaluate these inputs to determine whether the user  136  provided the optimal solution to the problem. If so, the adaptive engine  126 , using e.g., the performance module  152 , may present to the user  136  via the input/output device  109  a suitable problem  127  from a different teaching topic. Conversely, if the evaluator  124  evaluates the user input and determines that the solution provided by the user  136  is a solution other than the optimal solution, the subsequent suitable problem  127  presented to the user  136  may be from the same teaching topic. The difficulty rank of the problems presented to the user  136  may be increased or decreased by engine  126  in line with the user input. And, as discussed above, the difficulty rank assigned to a particular problem may further be adaptively modified based on the inputs received by a statistically significant number of users. 
     Additional detail is now provided to illustrate how the input entry device  200  may be used to solve a problem—in this case, an algebraic equation—presented to the user  136  by the adaptive engine  126  via the input/output device  109 . 
     As can be seen in  FIG. 14 , the secondary cog A of the example input entry device  200  has three teeth. Secondary cog B has five teeth. And secondary cog C has seven teeth. The primary cog  1  has 40 teeth. The target tooth  5  is seven teeth away from the origin tooth  4  (i.e., counting clockwise from the origin). Based on the configuration of the primary cog and the secondary cogs,  FIG. 1  may be represented by the following equation:
 
3 x+ 5 y+ 7 z =7  [[Eq. 1]]
 
where the 3 in 3x refers to the number of teeth in secondary cog A, the 5 in 5y refers to the number of teeth in secondary cog B, the 7 in 7z refers to the number of teeth in secondary cog C, and 7 at the right hand side of the equation refers to the position of the target tooth of the primary cog relative to the origin tooth. The variable x refers to the number of rotations of cog A (clockwise is positive and counter clockwise is negative), as also discussed herein. The variable y refers to the number of rotations of cog B. And variable z refers to the number of rotations of cog C. The goal in this example is to rotate the primary cog so that the target tooth lands beneath the marker  3 .
 
     The artisan will appreciate that equation 1 has numerous solutions. And each of these solutions helps provide insight into the problem solving prowess of the user  136 . For example, a student Sam can use the input entry device  200  of  FIG. 14  to solve Equation 1 as follows. Sam may physically rotate cog C clockwise once. If cog C is rotated once in the clockwise direction, because it has seven teeth that are enmeshed with the primary cog  1 , the primary cog will move seven teeth counterclockwise. This would leave the target tooth below the marker  3 . In terms of the symbolic equation, since cog A is not rotated, the value of x is zero. Similarly, since cog B is not rotated, the value of y is zero. And because cog C is rotated once, the value of z is 1. This provides one way to solve Equation 1.
         x=0, y=0, z=1; [[Sam&#39;s approach]]
 
i.e., 3(0)+5(0)+7(1)=7.
       

     But, Equation 1 can also be solved in other ways. For example, a student Shelly may rotate cog B clockwise two times, and then rotate cog A counter-clockwise once. That too will result in the target tooth landing beneath the marker  3 . In terms of symbols:
         x=−1, y=2, z=0; [[Shelly&#39;s approach]]
 
i.e., 3(−1)+5(2)+7(0)=7.
       

     Both the solutions above are correct. But, in this example and as noted above, the optimal solution is achieved by minimizing the collective rotation of the secondary cogs. Sam&#39;s solution above required one step whereas Shelly&#39;s solution required two. Therefore, if this data set were the only data set available, the system  100  may determine that Sam is more proficient at solving algebraic equations than Shelly. Therefore, if the suitable problem  127  to be presented to each of Sam and Shelly were an algebraic equation, the adaptive engine  126  may subsequently present an algebraic problem to Sam whose difficulty rank is greater than the difficulty rank of the algebraic problem presented to Shelly. 
     Indeed, the steps that the user  136  takes with the input entry device  200  (and other such input entry devices) may provide much insight into the user&#39;s problem solving abilities with respect to the teaching topic to which the problem belongs. Consider  FIG. 14  again, but now assume that secondary cog C is omitted. As will become clear from the discussion herein, the representative equation would then be:
 
3 x+ 5 y= 7  [[Eq. 2]]
 
     Assume that Sam solves Equation 2 by rotating cog B clockwise two times and cog A counter-clockwise once (i.e., x=−1 and y=2). This would be the most efficient solution to Equation 2. However, to solve Equation 2 in this manner, Sam must know that 2&gt;5=10. That is, if Sam solves Equation 2 in the manner just described, the adaptive engine  126  may determine that Sam understands at least the basics of multiplication operations. The system  100  may therefore chose as a suitable problem (i.e., the problem subsequently presented to Sam) a more complex problem involving multiplication or a problem in a different teaching topic (e.g., division). 
     Assume now that Sam solves Equation 2 a different way. For example, assume Sam solves Equation 2 by rotating cog B clockwise once, rotating cog A counter-clockwise once, and then by rotating cog B clockwise once again. This particular solution indicates that Sam is not proficient at multiplication because he used only addition and subtraction to solve Equation 2. In this case, the adaptive engine  126  may subsequently present to Sam a different problem (e.g., a problem in which the complexity of the addition is increased or a problem in which the complexity of the multiplication is decreased). In this way, thus, the system  100  may continually evaluate the progress of the user  136  and present to him or her problems that challenge the abilities of the user  136  while ensuring that the user  136  is within his ZPD. 
     In embodiments, the physical input entry device  200  may be configurable by the user  136 . For instance, and with respect to the input entry device  200  described as an example herein, the user  136  may be allowed to add or subtract gears from the device  200  (e.g., the support structure  202  may allow for the user  136  to: rotatably couple additional secondary gear(s) to the primary gear; remove one or more secondary gears; add or remove one or more teeth from the primary gear and/or the secondary gear; use a differently sized primary gear, etc.). Such selective configurability of the physical input entry device  200  may further increase the versatility of the system  100 . Other input entry devices (e.g., device  300 , device  700 , etc.) discussed herein may likewise be selectively configurable. 
     The artisan will appreciate from the disclosure herein that the gear system  200  is but one example of the input entry device  134 , and other input entry devices for use with the adaptive system  100  for teaching content are also contemplated.  FIG. 15 , for instance, shows another example  300  of the input entry device  134 . The input entry device  300  may also be referred to herein as a liquid flow instructional device  300 . 
     The disclosure relating to the liquid flow instructional device  300  includes a method for representing a proportions problem and a method for solving the proportions problem. The method for representing the proportions problem utilizes the liquid flow instructional device  300  to express the proportions problem in a non-traditional fashion. The method for solving the proportions problem defines the steps necessary to determine a set of values that solves the proportions problem using the liquid flow system  300 , essentially identifying a solution to the proportions problem. The liquid flow instructional system  300  physically and visually represents each entity of the proportions problem (or purely visually in the case of a digital implementation as described herein) and allows the user to manipulate said entities through an adjustable valve  306  of the liquid flow system  300  to determine a solution to the proportions problem. Entities of the proportions problem may include a number, a plurality of numbers, a geometric shape (circular disk, rectangle, or other regular shape), etc. 
     In more detail, the liquid flow instructional device  300  may include a support structure  301 S onto which a plurality of tanks and/or other containers configured to retain fluid are situated (e.g., mounted). The plurality of tanks may include one or more input tanks and a plurality of output tanks (e.g., two, three, or four or more output tanks, etc.). For example, in the example illustrated in  FIG. 15 , the liquid flow instructional device  300  includes an input tank  302  and output tanks  304 A,  304 B, and  304 N. Each of the input tanks  302  and the output tanks  304 A,  304 B, and  304 N may have a pre-determined capacity which may be displayed thereon or elsewhere. Each of the input tank  302  and the output tanks  304 A,  304 B, and  304 N may be fluidly and selectively coupled to each other. For example, a pipe may connect the input tank  302  to the valve  306 , and a plurality of pipes may connect the valve  306  to the plurality of output tanks. Thus, the adjustable valve  306  may selectively and fluidly couple the input tank  302  to the output tanks  304 A,  304 B, and  304 N. 
     A user controlled binary switch, e.g., the fill button  308  communicatively coupled to the valve  306 , may initiate on user command selective flow of the liquid from the input tank to the plurality of output tanks. The initial state of the input tank  302  may be full (i.e., the input tank  302  is full when a puzzle is presented to the user (e.g., user  136 )). The initial state of the output tanks  304 A,  304 B, and  304 N when the puzzle launches may be empty. The initial position of the switch  308  may be off. The user may selectively manipulate the amount of liquid that flows from the input tank  302  to each of the output tanks  304 A,  304 B, and  304 N via the adjustable valve  306 , and more specifically, via movable arms (e.g., rotatable arms, slidable arms, etc.) thereof. When the arms of the valve  306  have been set by the user as discussed herein, the user may set the switch  308  to on, whereupon the entire contents of the input tank  302  may flow first to the valve  306 , and then to the plurality of output tanks  304 A- 304 N as determined by the valve settings. Upon completion of the flow, the switch  308  may automatically re-set to off and the input tank may automatically refill. When the puzzle requires a multi-step solution, as discussed herein, the initial state of the output tanks at the start of any step may be the end-state of the previous step. 
     In embodiments, the input tank  302  (and/or output tanks  304 A,  304 B, and  304 N) may be fluidly coupled to a water source, such as a faucet, a water body, etc. to allow various amounts of fluid to be filled in differently sized input tanks  302 . As discussed herein, the adjustable valve  306  may allow the user to selectively apportion liquid from the input tank  302  into the output tanks  304 A,  304 B, and  304 N. In embodiments, and as discussed below, the adjustable valve  306  may have a plurality of arms (e.g., up to four arms) which the user may use to selectively apportion the liquid from the input tank  302  into the two or more output tanks  304 A,  304 B, and  304 N. The liquid flow instructional device  300  may comprise a fill button  308  or other activation means, which, when employed by the user  136 , may initiate fluid flow from the input tank  302  to the one or more output tanks  304 A,  304 B, and  304 N in line with the adjustable valve  306  settings set by the user. The user&#39;s objective may be to use the adjustable valve  306  to apportion liquid from the input tank  302  to the output tanks  304 A,  304 B, and  304 N so as to exactly fill each of the output tanks  304 A,  304 B, and  304 N without spillage. In puzzles requiring multi-step solutions, as will become clear from the disclosure herein, the fill button  308  may have to be employed two or more times by the user to solve the puzzle. In embodiments, e.g., where the physical input entry device  300  is being used, the input entry device  300  may include means (e.g., pump(s), siphons, gravity fed devices, etc. (i.e., sensing devices  134 A and/or responsive devices  134 B (see  FIG. 13 ))) to cause the liquid to flow from the input tank  302  into the output tanks  304 A,  304 B, and  304 N in line with the adjustable valve settings set by the user. In embodiments, the input entry device  300  may be a modular device such that the size and/or number of input and output tanks may be varied to create new puzzles. The size of the input tank  302  may but need not be the same as the size of the output tanks  304 A,  304 B, and/or  304 N, and the output tanks may likewise have different sizes. In embodiments, the size of the output tanks  304 A,  304 B, and  304 N may be the same but indicia may be provided to indicate that a different amount of liquid is to be filled in each output tank to solve the puzzle. In embodiments, a volume of the input tank  302  may be one of: (a) equal to a collective volume of the plurality of output tanks  304 A,  304 B, and  304 N; and (b) a multiple (i.e., a factor) of the collective volume of the plurality of output tanks  304 A,  304 B, and  304 N. The phrase “collective volume” indicates the actual collective volume of the output tanks and/or the collective volume thereof as indicated thereon or elsewhere. 
       FIG. 16A  shows an example liquid flow instructional device  400 . The liquid flow instructional device  400  is substantially similar to the liquid flow instructional device  300 , except as specifically noted and/or shown, or as would be inherent. Further, those skilled in the art will appreciate that the embodiment  300  (and thus the embodiment  400 ) may be modified in various ways, such as through incorporating all or part of any of the various described embodiments, for example. For uniformity and brevity, corresponding reference numbers may be used to indicate corresponding parts, though with any noted deviations (for example, the input tank is designated  302  in  FIG. 15 and 402  in  FIG. 16A , adjustable valve is designated  306  in  FIG. 15 and 406  in  FIG. 16A , etc.). The artisan will appreciate from the disclosure herein that the configuration of the liquid flow instructional device  400  is one of the many possible configurations of the liquid flow instructional device  300 . In embodiments, the liquid flow instructional device  400  is an example of the liquid flow instructional device  300 . 
     In more detail,  FIG. 16A  shows the liquid flow instructional device  400  in an initial condition (i.e., presenting a puzzle to the user),  FIG. 16B  shows the liquid flow instructional device  400  in an intermediate condition (i.e., where the user has employed the adjustable valve  406  to selectively apportion the liquid from the input tank  402  to the output tanks), and  FIG. 16C  shows the liquid flow instructional device  400  in a final condition (i.e., after the user has interacted with the fill button  408  to cause the liquid to flow from the input tank  402  to the output tanks  404 A and  404 B in line with the settings of the adjustable valve  406 ). The adjustable valve  406  may have arms  406 A and  406 B that may allow the user to selectively define valve regions (or valve face areas)  420 A and  420 B on the valve face. The valve face region  420 A, in this example, may correspond to the output tank  404 A and the valve face region  420 B may correspond to the output tank  404 B. This correspondence may be indicated by color coding (e.g., all or part of each of the valve face region  420 A and the output tank  404 A associated therewith may be red in color) or other means (such as by numerical identification or using another visible indicator). In this example, the input tank  402  has 100 units (e.g., mL, quarts, cups, gallons, etc.) of liquid. The output tanks  404 A and  404 B are currently empty, and the goal here is to use the arms  406 A and  406 B of the adjustable valve  406  to apportion the liquid from the input tank  402  such that 75 units thereof end up in the output tank  404 A and 25 units thereof end up in the output tank  404 B. Specifically, the goal is to selectively configure the valve regions  420 A and  420 B so as to exactly fill the output tanks  404 A and  404 B in a minimum number of attempts. As shown, the output tanks  404 A and  404 B have a capacity of 75 units and 25 units, respectively. 
     The artisan will appreciate from the disclosure herein that the problem represented in  FIG. 16A  is a proportions problem. A typical proportions problem may be: given a number N, and numbers M 1 , . . . , M k , find numbers R 1 , . . . , R k  such that R 1 + . . . +R k =1 and R i ×N=M i , for each i. For example, if N=200, k=3, M 1 =70, M 2 =80, and M 3 =50, then a successful solution is to take R 1 =0.35, R 2 =0.40, and R 3 =0.25, since 0.35+0.40+0.25=1 and 0.35×200=70, 0.4×200=80, and 0.25×200=50. In the example shown in  FIG. 16A , N=100, k=2, M1=75 and M2=25, so a successful solution is to take R1=0.75 and R2=0.25. That is, the input tank  402  represents the given number N, the output tanks  404 A,  404 B, . . .  404   k  represent the numbers M 1 , M k , and R1 and R2 represent the relative size of the valve regions  420 A and  420 B. The user may rotate or otherwise move the moveable valve arms  406 A and  406 B to divide the face of the valve  406  into regions  420 A and  420 B that represent the solution numbers R 1 , . . . , R k . In some instances, the numbers R 1 , . . . , R k  may be restricted to come from a specified collection; for example, fractions in the set {⅕, ⅖, ⅗, ⅘}, decimals in the set {0.20, 0.4, 0.6, 0.8}, etc. 
       FIG. 16B  shows the liquid flow instructional device  400  after the user has properly configured the valve regions  420 A and  420 B to exactly fill up the output tanks  404 A and  404 B. While not expressly shown in  FIG. 16B , and as discussed above, in embodiments the valve regions and the corresponding output tanks may be color coded (e.g., the region  420 A and the output tanks  404 A may each include a red or other identifier to indicate correspondence therebetween and the region  420 B and the output tank  404 B may each include a blue or other identifier to indicate correspondence between the region  420 B and output tank  404 B). In embodiments, indicia may be provided to indicate the ratio of the size of the region  420 A to the size of the region  420 B (e.g., in  FIG. 16B , the fraction ¾ may be provided in region  420 A and the fraction ¼ may be provided in the region  420 B to indicate that the regions  420 A and  420 B respectively take up ¾ th  and ¼ th  of the total valve region). Alternately, the indicia may include percentages, decimals between zero and one, and/or other such indicators to indicate the relationship between the valve regions  420 A and  420 B and/or the size thereof. Indicia may also be provided to indicate the ratio of the volume of the output tank  404 A to the volume of the output tank  404 B. 
     Once the valve arms  406 A and  406 B are set up as shown in  FIG. 16B , the user may depress or otherwise interact with the fill button  408 . The flow button  408  may be communicatively coupled to the input tank  402  and/or the valve  406  (e.g., over a network, via a mechanical connection, etc.), such that interacting with the fill button  408  may initiate flow from the input tank  402  to the output tanks  404 A and  404 B. Thus, when the fill button  408  is depressed or otherwise interacted with, the liquid to flow from the input tank  402  to the output tanks  404 A and  404 B in line with the valve regions  420 A and  420 B set by the user. For instance, interacting with the fill button  408  may cause a pump or other liquid flow means to cause liquid to be pushed from the input tank  402  into the output tanks  404 A and  404 B. 
       FIG. 16C  shows the liquid flow instructional device  400  in its final condition, after the valve regions  420 A and  420 B have been set by the user and the fill button  408  has been depressed. As can be seen in  FIG. 16C , the liquid has flown from the input tank  402  to the output tanks  404 A and  404 B such that the output tanks are exactly filled (the phrase “exactly filled”, as used herein, means that a tank is filled to capacity without spillage). Had the user configured the valve regions  420 A and  420 B differently, e.g., if the user had configured the valve regions such that the region  420 A had an area equal to that of region  420 B, output tank  404 B would have overflown once the fill button  408  was depressed. As noted, the input tank  402  may be configured to be refilled immediately (or at another time) for the presentation of the next puzzle. 
       FIG. 17A  shows an example liquid flow instructional device  500  to illustrate a more complex problem than that shown in  FIG. 16A . The liquid flow instructional device  500  is substantially similar to the liquid flow instructional device  300  and  400 , except as specifically noted and/or shown, or as would be inherent. Further, those skilled in the art will appreciate that the embodiment  500  may be modified in various ways, such as through incorporating all or part of any of the various described embodiments, for example. For uniformity and brevity, corresponding reference numbers may be used to indicate corresponding parts, though with any noted deviations (for example, the input tank is designated  302  in  FIG. 15, 402  in  FIGS. 16A-16C , and  502  in  FIG. 17A ; the adjustable valve is designated  306  in  FIG. 15, 406  in  FIG. 16A-16C , and  506  in  FIG. 17A , etc.). In use, and in line with the disclosure herein, the problem in  FIG. 17A  may be presented to the user after the user has solved the comparatively easier problem in  FIG. 16A . 
       FIG. 17A  shows the liquid flow instructional device  500  in an initial condition (i.e., presenting a puzzle to the user),  FIG. 17B  shows the liquid flow instructional device  500  in an intermediate condition (i.e., where the user has employed the adjustable valve  506  to apportion the liquid from the input tank  502  to the output tanks),  FIG. 17C  shows the liquid flow instructional device  500  in another intermediate condition (i.e., after the user has interacted with the fill button  508  to cause the liquid to flow from the input tank  502  to the output tanks  504 A and  504 B in line with the settings of the adjustable valve  506  but before the output tanks  504 A and  504 B are exactly filled,  FIG. 17D  shows the liquid flow instructional device  500  in another intermediate condition (i.e., after the input tank  502  is refilled),  FIG. 17E  shows the liquid flow instructional device  500  in yet another intermediate condition (i.e., after the valve areas  520 A and  520 B have been set by the user  136  for a second time), and  FIG. 17F  shows the liquid flow instructional device  500  in a final condition (i.e., after the fill button  508  has been depressed for the second time to complete the puzzle). In this example, the valve region  520 A corresponds to the output tank  504 A and the valve region  520 B corresponds to the output tank  504 B. The input tank  502  has 90 units of liquid, and can be refilled. The output tanks  504 A and  504 B are currently empty, and the goal here is to use the arms  506 A and  506 B of the adjustable valve  506  to apportion the liquid from the input tank  502  to exactly fill the output tanks  504 A and  504 B in a minimum number of attempts. Each of the output tanks  504 A and  504 B has a capacity of 90 units. 
     In the event that there are no numbers R 1 , . . . , R k  such R 1 + . . . +R k =1 and R i ×N=M i , for each i, completion of the puzzle may require two or more applications of the settings. This may be referred to as multi-step problem herein, which is solved by obtaining a series of partial solutions, all but the final solution being a partially complete configuration. The artisan will appreciate from the disclosure herein that the problem disclosed in  FIG. 17A  is a multi-step (specifically, a two-step) problem (i.e., the input tank having 90 units will need to be refilled once after the 90 units therein are transferred to the output tanks, because the output tanks  504 A and  504 B collectively require 180 units). Puzzles may likewise take a minimum of three or more steps to resolve. 
       FIG. 17B  shows the first setting of the control valve  506 , and  FIG. 17C  shows the results once the fill button  508  is first depressed.  FIG. 17D  shows the input tank  502  being refilled (which may, in embodiments, happen automatically),  FIG. 17E  shows the second setting of the control valve  506 , and  FIG. 17F  shows the final result once the fill button  508  is depressed by the user the second time. In this way, the user may fill each of two 90 unit output tanks with a refillable 90 unit input tank in two tries (i.e., using the fill button  508  twice). In essence,  FIGS. 17A-17F  show the following: 
     Turn 1: Valve areas  520 A and  520 B take up ⅔ rd  and ⅓ rd  of the valve face, respectively; and 
     Turn 2: Valve areas  520 A and  520 B take up ⅓ rd  and ⅔ rd  of the valve face, respectively. 
     The artisan will appreciate that the puzzle may likewise be solved in other ways, including in two (or a different number of) attempts. For example, the puzzle may be been solved as follows: 
     Turn 1: Valve areas  520 A and  520 B take up ½ and ½ of the valve face; and 
     Turn 2: Valve areas  520 A and  520 B take up ½ and ½ of the valve face. 
     Had the user taken more than two tries to solve this puzzle, the system  100  may have gleaned that the user does not have mastery over proportions problems and may have presented to him additional (e.g., easier) proportions puzzles to solve. 
       FIG. 18A  shows an example liquid flow instructional device  600  to illustrate another problem, wherein the numbers R 1 , . . . , R k  are indicated as percentages. The liquid flow instructional device  600  is substantially similar to the liquid flow instructional device  300 ,  400 , and  500 , except as specifically noted and/or shown, or as would be inherent. Further, those skilled in the art will appreciate that the embodiment  600  may be modified in various ways, such as through incorporating all or part of any of the various described embodiments, for example. For uniformity and brevity, corresponding reference numbers may be used to indicate corresponding parts, though with any noted deviations. 
       FIG. 18A  shows the liquid flow instructional device  600  in an initial condition (i.e., presenting a puzzle to the user),  FIG. 18B  shows the liquid flow instructional device  600  in an intermediate condition (i.e., where the user has employed the adjustable valve  606  to apportion the liquid from the input tank  602  to the output tanks  604 A,  604 B, and  604 C), and  FIG. 17C  shows the liquid flow instructional device  600  in a final condition (i.e., after the user has interacted with the fill button  608  to cause the liquid to flow from the input tank  602  to the output tanks  604 A,  604 B, and  604 C in line with the settings of the adjustable valve  606 ). In this example, the valve region  620 A corresponds to the output tank  604 A, the valve region  620 B corresponds to the output tank  604 B, and the valve region  620 C corresponds to the output tank  604 C. The input tank  602  has 200 units of liquid, and can be refilled. The output tanks  604 A,  604 B, and  604 C are currently empty, and the goal here is to use the arms  606 A,  606 B, and  606 C of the adjustable valve  606  to apportion the liquid from the input tank  602  to exactly fill the output tanks  604 A,  604 B, and  604 C in a minimum number of attempts. The output tanks  604 A,  604 B, and  604 C have a capacity of 70 units, 80 units, and 50 units, respectively. 
       FIG. 18B  shows the device  600  in the intermediate condition, after the regions  620 A,  620 A, and  620 C have been set to cause the output tanks  604 A,  604 B, and  604 C to be filled exactly.  FIG. 18C  shows the final result once the fill button  608  is depressed by the user. While in this example the numbers R 1 , . . . , R k  are expressed as percentages, the artisan will appreciate that these numbers may likewise be expressed as fractions, decimals, or without any symbols (e.g., the user may simply evaluate the valve  606  to discern the relative size of the valve regions). While not expressly shown in the figures, the correspondence between the valve regions  620 A,  620 B, and  620 C to the output tanks  604 A,  604 B, and  604 C, respectively, may be indicated by color coding, numeric indicia, or other appropriate means. 
     Proportions problems of the general nature discussed herein may provide a proven and effective way to develop an understanding of, and a facility to manipulate, fractions, and proportions in a variety of settings. Thus, by playing such games, e.g., the ones shown in  FIGS. 16A-16C and 17A-17F, and 18A-18C , the user may get a deeper understanding of fractions and proportions. Applicant&#39;s research has confirmed that this is indeed the case. The particular proportions problems may be formulated specifically to facilitate the construction of a physical or digital device that provides a learning experience that breaks the symbol barrier. In some embodiments, instead of exactly filling the output tanks, the puzzle may be solved by spilling the one or more output tanks by a particular amount (or by no more than a specified amount). Such may help develop the understanding of proportions to include estimation skills. The artisan understands that mathematics educators may consider skills related to approximate reasoning to be valuable in their own right. 
     While  FIGS. 16A-16C, 17A-17F and 18A-18B  show that the adjustable valve is generally circular, such is merely exemplary, and the valve (and the regions thereof formed by the arms) may be rectangular, square, or take on other regular or irregular shapes. Similarly, the valve arms may extend radially as shown (akin to the hands of a clock), but may extend vertically, horizontally, or in other directions in other embodiments. 
       FIG. 19  shows yet another embodiment  700  of the input entry device  134 . The input entry device  700  may, like the other input entry devices  134  disclosed herein, be a physical device that is communicatively coupled to the structure  102 . Alternately, the input entry device  700  (and the other input entry devices) may be implemented digitally, e.g., via a graphical user interface and machine readable instructions. The input entry device  700  may be directed to presenting and solving simultaneous linear equations in a single unknown, with the intention of developing a deep and productive understanding of linear growth functions. The input entry device  700  may also provide exercise in spatial reasoning. The artisan will understand from the disclosure herein that linear growth is a ubiquitous phenomenon in many walks of life, and that assisting people in developing an understanding of linear growth can accordingly play a major role in mathematics education. 
     The disclosure relating to the input entry device  700  includes a method for representing and solving a problem involving a linear growth function and simultaneous linear equations in a single unknown. The primary objective of the input entry device  700  is not to demonstrate to users how systems of linear equations may be solved by hand. Rather, in embodiments, a primary objective of the input entry device  700  may be to cultivate in users a meaningful understanding of linear growth and to allow them to reason successfully about linear growth situations. Linear growth lends itself to instantiation in a simple mechanical device. The disclosure incorporates an element of engaging, challenging spatial reasoning that may provide a visualization of the growth. 
     Linear equations may normally be written in the symbolic algebraic form: y=ax+b, where x is an input variable, y an output variable, and a, b are constants. They can be viewed both statically and dynamically. 
     Viewed statically, linear equations may capture a specific relation between two numbers. For example, for the equation y=3x+4, the equation says what when x=7, then y=25, so the equation outlines a relationship between 7 and 25. In other words, the equation specifies an algorithm that, given a number x, produces a number y. 
     Viewed dynamically, a linear equation may specify a function. One common way to represent the function defined by the equation y=ax+b is by drawing its graph. While effective, the graphical representation may obscure the inherently dynamic, procedural aspect of a function. The input entry device  700  may provide an alternative representation of such a function that brings out the dynamic feature, drawing the user&#39;s attention to the growth-pattern of the function. 
     Focus on the growth pattern of linear functions may be achieved by representing the function in terms of small tiles or blocks that have pre-specified linear growth patterns. Of course, the tiles may take on other regular or irregular shapes. The input entry device  700  may also be referred to herein as a tile or block instructional system  700 . 
     In more detail, the tile instructional system  700  may include a support structure  701 S onto which one or more remaining components of the device  700  may be situated. The device  700  may include an input tray  702 . The input tray  702  may comprise one or more individual movable tiles (or blocks)  704  and/or movable tile sections  706 . Each tile section  706  may comprise a plurality of individual tiles  704  that are grouped together. 
     One or more of the tiles  704  (i.e., one or more of the individual tiles  704  and/or one or more of the tiles  704  forming the tile sections  706 ) may include a growth rule. The growth rule may be indicated by, e.g., Chevron markings or other indicia. For example, as shown in  FIG. 19 , a tile may have a left chevron marking  708  or a right chevron marking  710 . The term “chevron marking”, as used herein, includes any marking that indicates a direction, such as a left arrow, a right arrowhead, etc. The chevron markings  708 ,  710 , when provided on tile sections  706 , may be provided on the leftmost and/or the rightmost tile  704  of that section  706 . As shown in  FIG. 19 , some tiles  704  may be devoid of any growth functions (indicated here by chevron markings) and other tiles  704  may include a plurality of chevron markings. 
     Each chevron marking may indicate a growth rule. For example, a solitary left chevron marking  708  on a tile  704  may indicate that the particular tile  704  can grow a tile to the left. For example, when the growth function is invoked on a solitary tile  704  having a left chevron marking, that tile  704  may grow a tile to the left and become a two tile section. A solitary right chevron marking  710  on a tile  704  may indicate that the particular tile  704  can grow a tile to the right. For example, when the growth function is invoked on a solitary tile  704  having a right chevron marking, that tile  704  may grow a tile to the right and become a two tile section. 
     As shown in  FIG. 19 , some tiles sections  706  may have each of a left chevron marking and a right chevron marking; for example, the right most tile of a tile section  706  may have a right chevron marking, and the left most tile of a tile section  706  may have a left chevron marking. When the growth function is invoked, the right most tile may grow a tile to its right and the left most tile may grow a tile to its left. That is, on initiating a move (e.g., a Grow move), each tile  704  with a chevron marking may expand in the direction of the chevron and the expansion thereof may correspond to the number of chevrons associated therewith. For example, the particular tile section  712  which is shown as having four tiles  704  may, upon invocation of the grow move, initially expand one unit square to the left and two unit squares to the right to form a tile section having an overall length of 7 unit squares, with a single left chevron pointed at its leftmost end and a double right-pointed chevron at its right most end. This expanded tile, once expanded, may be expanded yet again via a second Grow move that will cause this tile section to have an overall length of 10 unit squares. And so on. A tile having no chevron marking, conversely, may not grow as it does not have growth markings thereon. 
     Invocation of the growth function may be effectuated in the physical domain (e.g., mechanically) or virtually. For example, in physical embodiments, each tile with a chevron marking may have one or more additional tiles stacked atop the lowermost tile, and one or more of these tiles may be configured to extend outward (i.e., to the left or to the right depending on the chevron marking) when the Grow move (or growth function) is invoked. In one embodiment, one or more of the tiles stacked above the lowermost tile may be spring loaded and may be configured to extend to the left or the right by the force of a spring when the growth function is invoked. In another embodiment, powered (e.g., battery operated) arms may be used to cause a tile to extend to the left or the right of another tile. In another embodiment still, the tiles may be nested (akin to Russian dolls) within each other and may be pushed out using mechanical means when the growth function is invoked. 
     The tile instructional system  700  may further include one or more tile beds, such as tile beds  720 A,  720 B. Each tile bed  720 A and  720 B may include a label indicating the total number of unit tile receiving slots in that bed (e.g.,  18  and  4  in beds  720 A and  720 B, respectively). In another embodiment, each tile receiving slot in each tile bed may be numbered to assist the user  136  in the determination of a solution. 
     The goal of the user  136  solving the linear growth problem presented by the device  700  may be to take the appropriate tile(s) and/or tile beds from the input tray  702  and position them in the tile beds  720 A and  720 B in such a way that, by invoking a minimum number of Grow moves, all tile beds are filled exactly (with no overlapping). Activation of a Grow move may cause all tiles that have been placed in a bed to grow according to their specified growth rule. There may be any number of trays and/or tile beds. The device  700  may provide a mechanism for solving simultaneous linear equations, with a focus on mathematical growth rules (functions), as discussed herein. In this example, the optimal solution may be of two forms: fewest number of applications of the Grow move and fewest number of tiles used. If the user  136  determines an optimal solution to a problem, a different (more complex problem from the same or a different topic) may be presented to the user  136 . Alternately, if the user  136  is unable to determine the optimal solution, a different (e.g., a less complex problem from the same topic) may be presented to the user  136 . In this way, thus, the input entry device  134  may allow the system  100  to educate the user  136  while ensuring that the user  136  remains within his or her ZPD. 
     In embodiments, the device  700  may include an activator or binary switch, such as a grow button  722 , which, when activated, may cause the tiles and tile sections to expand in accordance with their respective growth functions. In physical embodiments, the grow button  722  may be communicatively coupled to the tile sections (e.g., using RF or other network). In other embodiments, the growth function of a tile and/or tile section may be activated by interacting with (e.g., depressing) the chevron marking thereon. In some embodiments, a deactivator (such as an ‘ungrow’ or undo button  724 ) may be provided to reverse a grow move. The tile instructional system  700  may, in embodiments, be modular to allow for different puzzles to be presented to the user  136  (e.g., a tile bed may be replaced with a differently sized tile bed, a tile section may be replaced with a differently sized and/or marked tile section, etc.). As discussed above, the puzzles may increase or decrease in complexity depending on user progress. In embodiments, a guide having a plurality of linear growth problems of various difficulties may be provided so that the user  136  and/or another (e.g., an educator) may configure the device  700  to present and solve different problems. A similar guide may likewise be provided in association with the other input devices discussed herein. 
       FIG. 20A  shows an example tile instructional system  800  for educating users about linear growth functions. The tile instructional system  800  is substantially similar to the tile instructional system  700 , except as specifically noted and/or shown, or as would be inherent. Further, those skilled in the art will appreciate that the embodiment  800  (and thus the embodiment  700 ) may be modified in various ways, such as through incorporating all or part of any of the various described embodiments, for example. For uniformity and brevity, corresponding reference numbers may be used to indicate corresponding parts, though with any noted deviations (for example, the input tray is designated  702  in  FIG. 19 and 802  in  FIG. 20A ). The artisan will appreciate from the disclosure herein that the configuration of the tile instructional system  800  is one of the many possible configurations of the tile instructional system  700 . In embodiments, the tile instructional system  800  is an example of the tile instructional system  700 . 
     As can be seen, the device  800  has in its input tray  802  three tile sections  806 A,  806 B, and  806 C each comprising individual tiles  804 . In this example, each of the tile sections  806 A,  806 B, and  806 C have two individual tiles  804 . In the tile section  806 A, one tile has a left chevron marking and the other tile has a right chevron marking. In the tile section  806 B, the left tile has a left chevron marking and the right tile has two right chevron markings. In the tile section  806 C, the left tile has two left chevron markings and the right tile has two right chevron markings. As discussed above, when the Grow move is initiated (e.g., using the grow button  822 ), each tile may expand in accordance with the chevron markings thereon. The tile instructional system  800  further includes two tile beds  820 A and  820 B. As shown, each tile of each tile bed may be numbered, though such is not required in all embodiments. The tile bed  820 A has 6 tiles and the tile bed  820 B has 10 tiles. Of course, a different number and configuration of tiles and/or tile sections may be provided in the input tray and/or a different number and configuration of tile beds may likewise be provided. 
     It will be appreciated that  FIG. 20A  shows the tile instructional device  800  in an initial condition (i.e., presenting a linear growth problem to the user  136 ).  FIG. 20B  shows the tile instructional device  800  in an intermediate condition (i.e., where the user has placed at least one of the tile sections  806 A,  806 B, and  806 C into each of the tile beds  820 A and  820 B). Specifically, as can be seen, the user  136  has placed the tile section  806 A in the tile bed  820 A such that it covers the third and the fourth tile receiving slot of the tile bed  820 A. The user  136  has further placed the tile section  806 C in the tile bed  820 B such that it covers the fifth and the sixth tile receiving slot of the tile bed  820 B. 
       FIG. 20C  shows the tile instructional device  800  in another intermediate condition (i.e., after the user has activated one Grow move). As can be seen, in response to the Grow move, the tile section  806 A has grown from two tiles to four tiles, and more specifically, the tile section  806 A has expanded one tile to the left and one tile to the right such that it now covers slots  2 ,  3 ,  4 , and  5  of the bed  820 A. The chevron markings continue to be on the outermost tiles, but the left chevron marking is now on the tile of the tile section  806 A that corresponds to slot number  2  of the bed  820 A, and the right chevron marking is now on the tile of the tile section  806 A that corresponds to slot number  5  of the bed  820 A. The tile bed  820 B has likewise expanded in response to the activation of the Grow move. However, as can be seen, because originally each of the two tiles of the tile section  806 C has two chevron markings each, the tile section  806 C has expanded from two tiles to six tiles. 
       FIG. 20D  shows the tile instructional device  800  in the final condition (i.e., after the user has initiated the Grow move a second time to fill each tile bed  820 A and  820 B exactly). Once the user initiates another Grow move, the tile section  806 A expands from four tiles (as shown in  FIG. 20A ) to six tiles to fill the tile bed  820 A. Similarly, once the user initiates another Grow move, the tile section  806 B expands from six tiles (as shown in  FIG. 20A ) to ten tiles to fill the tile bed  820 B. 
     The artisan will understand from the disclosure herein that the tiles are representations of linear equations. Unlike the gears system disclosed above, where the focus is on solving equations in several variables, with tiles the focus is on linear functions as a way to describe growth processes. An element of spatial reasoning is also incorporated in the positioning of the tiles and tile sections.  FIG. 21  shows some example growth rules, i.e., example tiles and tile sections and linear functions corresponding thereto. The examples in  FIG. 21  are not exhaustive and are not meant to be independently limiting. 
     The artisan will understand that while the disclosure focuses on a physical input entry device  134  usable by the user  136  to provide inputs that are then captured and evaluated by the structure  102 , that in other embodiments, the input entry device  134  may be provided as software with which the user  136  may interact via conventional means (e.g., via a keyboard and mouse, etc.). For example, an interactive graphical user interface may comprise the gears system shown in  FIG. 14 , the liquid flow system shown in  FIGS. 15, 16A-16C, 17A-17F and 18A-18B , the tiles system shown in  FIGS. 19 and 20A-20C , etc., and the user may interact with these systems using conventional computer means (e.g., via a keyboard, a mouse, or other controller). However, in some embodiments, it may be preferable to use the physical input entry device  134  at least because the real-world experience provided thereby may be more memorable for the user  136  (as compared to pressing the keys of a keyboard and/or moving the mouse to cause virtual gears on the screen to rotate in like fashion). 
     Many different arrangements of the various components depicted, as well as components not shown, are possible without departing from the spirit and scope of the present disclosure. Embodiments of the present disclosure have been described with the intent to be illustrative rather than restrictive. Alternative embodiments will become apparent to those skilled in the art that do not depart from its scope. A skilled artisan may develop alternative means of implementing the aforementioned improvements without departing from the scope of the present disclosure. 
     It will be understood that certain features and subcombinations are of utility and may be employed without reference to other features and subcombinations and are contemplated within the scope of the claims. Not all steps described herein and/or listed in the various figures need be carried out or need to be carried out in the specific order described.