A CONCEPTUAL FRAMEWORK FOR ERROR REMEDIATION WITH MULTIPLE EXTERNAL REPRESENTATIONS APPLIED TO LEARNING OBJECTS

In this paper, some of the concepts behind Intelligent Tutoring Systems (ITSs) are used to elaborate a conceptual framework for error remediaton with multple external representatons (MERs) applied to learning objects (LOs). For this purpose, we have developed an LO for teaching the Pythagorean Theorem. This study analyzes the process of error remediaton by classifying mathematcal errors and provides support for the use of MERs in this process. The main objectve of the conceptual framework is to assist the individual learner when rectfying a mistake made during interacton with the LO, due to either carelessness or lack of knowledge. First, we will classify mathematcal errors and explain their relatonship with MERs. Then, the concepts behind the conceptual framework will be presented. Finally, we will discuss an experiment with an LO developed with an authoring tool called FARMA, in which the conceptual framework is used to teach the Pythagorean Theorem.


INTRODUCTION
Mathematcal errors can play an important part in helping learners gain knowledge, but in order for this to happen, it is necessary to analyze them correctly.Classifying mathematcal errors seems to be a good method for understanding the learners' shortcomings.But the variety and complexity of mathematcal errors demand specifc expertse, which complicates the classifcaton task (Peng & Luo, 2009).
Mathematcal errors are considered a natural step during knowledge acquisiton (Fiori & Zuccheri, 2005;Peng & Luo, 2009); they are a common phenomenon, regardless of the student's age and/or level of intelligence.
The use of learning objects as a teaching resource in the classroom improves the ability to analyze the relevant features of educatonal materials in a way that difers from traditonal educaton.It can promote the use of technological resources to acquire concepts during the learning process.
The classifcaton of errors as part of the student's learning process and how this classifcaton can improve the acquisiton of a concept included in a learning object has been discussed in several studies (Marczal & Direne, 2011;Leite, Pimentel & Petruchinski, 2012;Leite, Marczal & Pimentel, 2013).
According to Silva and Fernandez (2007), a learning object must fulfll three characteristcs: • it must stmulate reasoning and critcal thinking (minds-on), • it must make students ask relevant questons (reality-on), and • it must ofer an opportunity to explore (hands-on).
The discussion about the classifcaton of mathematcal errors related to learning objects leads to a necessary discussion about error remediaton in the same context.The present proposal adheres to the ITS concept according to which the learner must be supported when he/she commits an error during problem resoluton.We applied this concept to learning objects using multple external representatons, enabling the learner to review the facts, rules and concepts in order to choose the correct strategy and solve the problem.
Allowing the learner to inspect the resoluton path at each step is one of the key features of remediaton; it is a characteristc of ITSs and can be applied to learning objects.
Error diagnosis and remediaton is thoroughly discussed by Vanlehn (1987), who highlights that some ITSs already include support for error diagnosis and remediaton: a well-known example is Sierra, which interacts with the learner through explanatons of errors, based on concepts presented by means of textual representaton.
Cognitve diagnosis and error remediaton are key steps in the acquisiton of a concept.The mission of cognitve diagnosis is to guide the instructonal plan, in which all the teaching actons depend on a result.
A new element that has been introduced in the error remediaton process is external representatons, which can be understood as a constant process of "taking possession" of stable states to obtain informaton that can then be used in a more fexible way for other purposes.
Evidence of the benefts of error remediaton to support learning has been presented in several studies (Ainsworth, 2006;Ainsworth, 2008;Cleeremans & Jimenez, 2002) that discuss its contributon to improving performance and understanding during the learning process.
A comprehensible, appropriate message or representaton can be crucial to knowledge acquisiton, provided that it is clear and helps communicate the remediaton to the learner.Although Ainsworth (2006) presents a taxonomy of external representatons, it is unclear when or how each representaton is shown to the learner.
The use of MERs in learning environments has shown that students can beneft from the propertes of each representaton and ultmately achieve a broader understanding of the subject being taught (Ainsworth, 2008).The use of error classifcaton in order to choose a type of remediaton allows for a more adequate selecton of external representatons, so that the learner can review facts, rules, concepts and/or fragments that have been forgoten.This aim of this artcle is to create (through the classifcaton of mathematcal errors) a conceptual framework for error remediaton based on MERs and to use it with a learning object to teach the Pythagorean Theorem in primary educaton.

THEORETICAL REFERENCE TO CREATE A CLASSIFICATION OF ERRORS
In order to create a classifcaton of mathematcal errors, we have reviewed studies dealing with the analysis of mathematcal errors.Although there is a vast body of literature on the subject, we considered it appropriate to select studies focused only on the mathematcal error itself, rather than other aspects, such as its causes or consequences.Radatz (1979) considers it difcult to clearly categorize the possible causes of an error if there is no close interacton between them.This author carried out a study of the elements of informaton processing theory and suggested fve types of mathematcal errors.
The Conceptual Fields theory is a cognitvist theory by Gerard Vergnaud (1986) that states that the process of fnding a strategy for problem resoluton consists of a relatonal calculaton or a numerical calculaton that appeals to the student's ability to execute mathematcal operatons and algorithms.Therefore, an inconsistency in any of these calculatons generates one incorrect operaton classifed by the author as a relatonal or numerical error.
following categories: inadequate use of informaton, incorrect interpretaton of the language, logically invalid inference, distorton of theorems or defnitons, non-verifed soluton and technical error.Peng and Luo (2009) present a classifcaton based on research into the knowledge of mathematcs teachers in relaton to the analysis of errors commited in mathematcal problems.They identfed four analytcal categories of errors commited by students.Lucas (1974) uses this classifcaton to create a rule-based model for studying how children learn to decode words.This model is based on data collected from errors in word pronunciaton.This author appeals to the concept of sub-generalizaton to classify students' inability to consider an element as a member of a group (e.g.fsh are not animals because they do not have legs), and the concept of generalizaton to classify occurrences of students considering an element as a member of a group when in fact it is not (e.g. a chair is an animal because it has four legs).This classifcaton was adapted by Ramos (2010) for applicaton in inductve mathematcal learning, and three types of conceptual errors were considered: sub-generalizaton, super-generalizaton and miscellaneous.
Research aimed at classifying mathematcal errors has produced difering classifcatons in terms of both the nomenclature and the number of categories.Thus, it is necessary to create a classifcaton that synthesizes this variety.This classifcaton will be presented in Secton 3.

ERROR AND MER FUNCTION CLASSIFICATION
Afer analyzing the research on the classifcaton of mathematcal errors described in Secton 2, it is clear that a unifed classifcaton is needed.We suggest the following more logical error categories: • Misinterpretaton of the language: this type of error refers to the learner's difculty to understand the structure of the problem, i.e. to interpret what is being asked in the problem, as a step prior to formulatng a strategy.
• Directly identfable error: errors in this group can be sub-classifed as errors owing to a domain defciency or a misuse of data; errors due to a defciency in the use of rules, theorems or defnitons; and errors regarding the logical operator.
• Indirectly identfable error: this category includes errors resultng from faulty logic, i.e. incorrect classifcatons, inappropriate problem solving strategies or transformatons with no progress.This type of error demands step-by-step monitoring of the learner.
• Non-categorizable soluton: this category includes errors that cannot be classifed into any other category.For example, the learner is too immature to assimilate a partcular concept and thus proposes random strategies for problem resoluton.
To take this research a step further, a study is needed that links external representatons to the types of errors commited by learners.Such a study is only possible afer elaboratng a broad error classifcaton that allows for a more precise remediaton, since this would enable error mapping.
The Multple External Representatons (MERs) theory (Ainsworth, 2006) is a cognitvist theory that supports the use of techniques to present, organize and transmit knowledge.Ainsworth (2006) classifes MERs into categories according to their functon.Those with a complementary functon support or complement a cognitve process.Those with an interpretaton-constraining functon constrain possible interpretatons that are not relevant to certain concepts.Finally, those with a deeper-understanding constructng functon enable a deeper understanding through the generalizaton of regularites in the presented content.In order to advance the research, it is convenient to create a conceptual framework that will make up the structural part of the study, which will be carried out in the following secton.

CONCEPTUAL FRAMEWORK FOR ERROR REMEDIATION WITH MERS
The conceptual framework of an intelligent tutoring system must dynamically combine informaton about the student model, the domain model and the tutor model, in order to determine what to present to the student.However, emphasis will frst be placed on the modules that make up the specifc conceptual framework for problem resoluton.
The use of error remediaton based on MERs requires a conceptual framework that can help identfy the error commited by the student, classify it into the appropriate category associated with a MER functon, and fnally ofer the learner adequate remediaton.
Figure 1 represents the proposed conceptual framework for remediaton based on MERs, as well as its modules: error classifer, MER functon classifer and MER manager.Other components of the conceptual framework are the rule base for error classifcaton, rule base for MER functon classifcaton and basis for external representatons.

Figure 1. Conceptual framework for using MERs in error remediaton applied to learning objects
Errors detected through interacton with the learner may occur at the beginning, the middle or the end of the resoluton path.The acton refers to the learner's current step along the resoluton path and enables the error to be remediated before the fnal answer is given; moreover, the type of MER depends on this informaton.The number of atempts serves to validate that the inital MER used in the remediaton was appropriate for the learner's advancement.
The aim of the expression classifer module is to establish a connecton between the LO and the system.This module is responsible for the inital communicaton and receives data to determine the degree of correctness of the learner's answer.If the answer is correct, the other modules will not be launched; otherwise, the error classifer module will be launched.
The aim of the error classifer module is to classify the error made by the learner, for which purpose the incorrect data or expression is identfed by the expression identfer module.This module receives an error from the expression classifer module and classifes it using the rules contained in the rule base for error classifcaton, which includes the error classifcaton presented in this study (misinterpretaton, directly identfable error, indirectly identfable error and non-categorizable soluton).
The aim of the rule base for error classifcaton is to lay the basis for classifying the error detected.The acton and the number of atempts are stored as individualized remediaton elements for the learner, in order to track him/her during the resoluton process.The acton will depend on the learner's current resoluton step.Furthermore, the number of atempts will help determine the adequacy of the MER used for remediaton.
At this stage, input is received (error, acton, number of atempts and error type) and the error type is confrmed against the rule base for MER functon classifcaton.This rule base is aimed at determining the MER functon.
The MER manager seems to be one of the most relevant modules, since it determines the type of remediaton needed for the learner to improve his/her problem-solving strategy.This module receives the following inputs: error, acton, number of atempts and MER functon.Note that in this module it is not necessary to contnue storing the error type, as it has already been used for MER classifcaton.
The role of the MER manager is to decide which MER to use to remediate the error once it has been classifed.It should store the last representaton used in order to improve the learner's efectveness.
The MER manager has a sub-module, Object-ER, which is responsible for searching for external representatons according to basic criteria, such as error persistence, success of certain external representatons in previous situatons, and degree of complexity of the situaton faced by the learner.

LEARNING OBJECT FOR TEACHING THE PYTHAGOREAN THEOREM
An ITS is based on the assumpton that a student's thinking process can be modeled, tracked and corrected (Self, 1999).This enables not only teaching, but also using diferent teaching methods, as well as discovering the paths that the learner follows to acquire the desired knowledge.
A learning object for mastering the Pythagorean Theorem was developed for applicaton in a public school in Curitba, State of Paraná, Brazil, during the 9th grade of primary educaton.
In order to develop this LO, we used a Web-based authoring tool called FARMA (an authoring tool for error remediaton with learning disabled students), which allowed for the applicaton of the conceptual framework presented in Secton 4.
FARMA's characteristcs made it suitable for implementng the conceptual framework; these characteristcs are: • less efort required to create educatonal materials, whose main feature is their intuitveness, • fewer skills required to manage content outside the author's feld of expertse, and • easy and quick prototyping, i.e. it is possible to validate the fnal draf of the LO in real tme.
These aspects contributed to the completon and validaton of the present study, as they facilitated the implementaton of the architecture for mathematcal error remediaton based on MERs (Marczal & Direne, 2011).
The Adaptve Control of Thought (ACT) theory by John Anderson ( 1983) is a unifed theory on informaton processing that states that learning mechanisms are closely related to the way the content is transmited to the learner, partcularly the way in which it is presented.
In general, proposals based on the ACT theory consist of four premises: a model, in other words, a model of producton rules for basic skills, in this case, the ideal-student model; actons on the right path, i.e. correct actons by the student, which are part of a set of solutons included in the model; actons on the wrong path, which enable verifying whether the student contnues on the right path towards the soluton; and answers about errors and assistance systems, which refer to the system's interacton instructon (Anderson, 2005).
Another relevant aspect is that the ACT theory explores the decompositon of a goal into sub-goals, allowing the student to proceed in stages, which is one of the characteristcs of the use of producton rules (Anderson, 2005).
This theory has been used as a mechanism for submitng representatons for error remediaton, in order to optmize the student's knowledge acquisiton during the resoluton process.
In order to exemplify the applicaton of the ACT theory concepts, the error remediaton process and the use of MERs, we frst presented the students with a traditonal exercise on the Pythagorean Theorem and then we redesigned it according to the aforementoned approaches.The descripton of the exercise is: An acrobatc bike rider wishes to cross from one building to another along a rigid steel cable with a special bike.The height of the building from which he will depart is 75 meters and the height of the building to which he will arrive is 25 meters.The distance between the two buildings is 120 meters.
Queston: What is the minimum length required for the cable?(The correct answer is 130 meters.)Such an exercise requires a great capacity for abstracton on the part of the learner, especially to understand the problem, retrieve the data and then perform the necessary calculatons.It must be noted that the learner's ability to do mathematcal calculatons is not enough to solve the problem: he/she must refect on the descripton of the problem, retrieve the correct data, identfy the concepts involved and fnally calculate the answer.Another downside of this approach is the inability to ofer relevant error-based feedback to the student, as it is only possible to validate the fnal answer and the tutor cannot determine at which point the learner made a mistake (he/she may have misunderstood the problem, misused the data or even chosen an answer at random).
The ACT theory supports the monitoring and analysis of the entre interacton between the student and the LO (the ACT theory-based LO developed in this research is called Pythagoras Max) as a mechanism for defning when to use an external representaton for remediatng an error commited by the student.The objectve is to optmize the path the student takes when acquiring a concept, for which purpose a tracing model is used to enable the exploraton of each step in the student's resoluton process in an individualized manner.In this case, it was necessary to divide the problem into steps in order to fnd out the point at which the student deviated from the ideal path, and then establish an adequate error remediaton plan based on an external representaton.
In the case of the exercise above, which was designed to compare two teaching approaches, a traditonal approach (Pythagoras Mix learning object) and an ACT theory-based approach that includes error remediaton using MERs (Pythagoras Max learning object), the student would have to abstract all informaton in order to answer the queston.However, in the Pythagoras Max LO, the exercise was divided into sub-steps in order to track the student's resoluton path.Moreover, this second approach allows for a gradual development of the learner's capacity for abstracton.
In the exercise included in the Pythagoras Mix LO, typical feedback would be "try again", "wrong answer" or signaling in diferent colors, which would simply draw atenton to the student's misconcepton.The Pythagoras Max LO allows for a very diferent situaton, as the division into stages makes it possible to monitor the students' progress during problem resoluton, identfy the mistake, classify it in associaton with a MER functon, and fnally use a MER for error remediaton.
Therefore, the advantage of the Pythagoras Max LO is that it enables gradual learning by breaking down the problem-resoluton process into steps, which allows for the applicaton of ACT concepts such as tracing models.
Below, the exercise is divided into steps and the MERs to be used (which depend on the learner's number of atempts) are presented.
An acrobatc bike rider wishes to cross from one building to another along a rigid steel cable with a special bike.The height of the building from which he will depart is 75 meters and the height of the building to which he will arrive is 25 meters.The distance between the two buildings is 120 meters.The MERs for error remediaton at this stage of the exercise are presented in Figure 4.The MERs for error remediaton at this stage of the exercise are presented in Figure 5.The MERs for error remediaton at this stage of the exercise are presented in Figure 6.Therefore, the main advantages of this approach are: • Providing immediate feedback on errors, giving the learner guidance throughout the entre process, not just in the fnal step • Minimizing the memory efort, since the problem can be analyzed in stages • Enabling the student to advance towards the soluton one step at a tme • Monitoring the student's acton against the expected correct path and determining through MERs whether he/she needs to review his/her strategy • Possibility of reducing the cognitve load by using MERs together with the error classifcaton as relevant error-remediatng tools In order to validate this approach, we carried out an experiment with 20 ninth-grade students from a public school in Brazil.They were divided into two groups: the experimental group and the control group.The experimental group used the Pythagoras Max LO, which includes error remediaton based on MERs.The control group used the Pythagoras Mix LO, with identcal exercises, but without employing the ACT concepts or error remediaton based on MERs.
The students were distributed into the groups based on the results of a pre-test which consisted of 6 questons about the Pythagorean Theorem.A t-test was used to analyze the data in order to identfy any signifcant gain in learning.
The experiment was aimed at confrming the adequacy of MERs for error remediaton.Thus, we expected to obtain signifcant results from the use of the Pythagoras Max LO (which included error remediaton based on MERs).
The results confrmed that the use of error remediaton based on MERs related to error categories contributes to increase student's knowledge.The hypothesis of the experiment was that the Pythagoras Max LO helps the learner acquire concepts (Leite et al., 2013).
The performance of partcipants using the Pythagoras Max LO led us to reject the null hypothesis (level of signifcance: 0.05%) and confrmed with a confdence of 95% that this LO facilitated the acquisiton of mathematcal concepts (Leite et al., 2013).
The null hypothesis for the Pythagoras Max LO stated that the post-test average results would be inferior or equal to the pre-test average results.However, they were signifcantly higher, which demonstrates a gain in learning.To confrm this, we used a t-test, since the sample included less than 30 subjects.With a confdence level of 95% (α = 0.05), we obtained p = 0.000412178 (t = 4.9202, df = 9).Thus, as p < α, we can reject the null hypothesis in terms of concept acquisiton.
The statstcal results and the details of the experiment, as well as other evidence on the efectveness of the Pythagoras Max LO as compared to the Pythagoras Mix LO can be found in the studies already validated by the scientfc community (Leite et al., 2013).

Queston 1 :
What are the measurements shown in the fgure?(The correct answers are 75, 25 and 120 meters.)TheMERs for error remediaton at this stage of the exercise are presented in Figure2.

Table 1
summarizes the use of MERs with diferent functons for error remediaton, in order to contextualize this study.This table includes error types and sub-types, associated MER functons and proposed remediaton mechanisms.

Table 1 .
Summarizes the use of MERs with diferent functons for error remediaton