Multiple External Representations in Remediation of Math Errors

Maici Duarte Leite

, Diego Marczal

1,2

and Andrey Ricardo Pimentel

Computer Science Department, Federal University of Paraná, UFPR, Curitiba, PR, Brazil

Technology Systems for Internet Department, Federal Technology University of Paraná,

UTFPR, Guarapuava, PR, Brazil

Keywords: Multiple External Representations, Remediation of Math Errors, Learning Object.

Abstract: The proposition of error remediation is a widely used feature in Intelligent Tutoring Systems, but the use of

Multiple External Representations to assist it, is a research subject. This paper presents (or discuss) the use

of Multiple External Representations contribution in error remediation in Learning Objects. To perform this

study, we present an architectural model, a conceptual framework for mathematical error classification and

Multiple External Representations, using a cognitive remediation for errors. Following is presented the

application of contextual remediation of error based on Multiple External Representations in a Learning

Object. And finally, we present the performance of students during the application of an experiment

consisting of the following steps: pre-test, test and post-test.

1 INTRODUCTION

The analysis of mathematical errors is a great

challenge, once specific knowledge of the content as

well as the factors that originated them are required.

The variety and the complexity of mathematical

errors demand specific knowledge, fact that makes

this task more difficult in regard of error

classification (Peng and Luo, 2009). Nowadays,

mathematical error is considered a natural stage on

knowledge construction (Fiori and Zuccheri, 2005);

(Peng and Luo, 2009), ie, it is a common

phenomenon in the scholar trajectory of the student,

wich is independent of age and/or performance

level.ned the manuscript must be appropriately

modified.

The classification of an error can become an

enhancer agent in the acquisition of a concept by the

student when applied in the proper way. Some

studies applied to Learning Objects (LO) as

(Marczal and Direne, 2011); (Bazzo et al., 2011);

(Leite et al., 2011); (Leite et al., 2012) have already

discussed this issue and presented some approaches,

so that the error can assist the learner in the learning

process.

The discussion presented in this study concerns

the use of mathematical classification errors in order

to provide its remediation through Multiple External

Representations (MER) on LOs. The contribution is

that the error made by the student, can provide a

more appropriate remediation using a MER. This

provides a better direction of learning, since the

study focuses exclusively on the individual needs of

each student.

The error remediation present in Intelligent

Tutoring Systems (ITS) aims to provide students

with the most appropriate feedback, which may be

linked to the student profile or the path that they are

following, interfering even before the learner makes

a mistake. Moreover, the LOs tend to provide

standard feedback to the learner regardless error

itself. Just as educational games has the philosophy

to become learning fun, but in reality usually

provide less care explicit and harsher penalties than

intelligent tutoring systems (Milk et al., 2010;

(Easterday et al., 2011).

The MERs are providing relevant results and

present themselves increasingly involved in

educational materials, as well as the use of MERs

have benefits when incorporated into STI, because

they provide a systematic interaction (Rau et al.,

2012). Other studies conducted in Brazil, also have

shown results in this direction (Leite et al., 2010);

(Leite et al., 2011); (Leite et al., 2012).

The aim of this proposal is to provide support to

remediate student errors through MREs, which can

be tables, lists, pictures, simulations, diagrams,

maps, natural language text, among others. This

519

Duarte Leite M., Marczal D. and Ricardo Pimentel A..

Multiple External Representations in Remediation of Math Errors.

DOI: 10.5220/0004568105190523

In Proceedings of the 15th International Conference on Enterprise Information Systems (ICEIS-2013), pages 519-523

ISBN: 978-989-8565-59-4

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

support is obtained through the identification and

classification of mathematical errors and the

functions of the MREs, allowing the student to

review and create new strategies during the learning

process.

In this paper, we present the results of an

experiment that used the error remediation from the

errors classification and functions of MER applied to

a LO involving mathematical concepts. The

architectural model that supports this study is

discussed as well as the modular composition.

2 METHODS

To discuss classification of errors present in the

literature were chosen authors (Radatz, 1979);

(Vergnaud, 1986); (Movshovitz-hadar and

Zaslavsky, 1987); (Peng and Luo, 2009), which had

in their research subject, the study of the

mathematical error. From the cited authors, an error

categorization was organized and presented by Leite,

Pimentel and Oliveira (2011), with the following

nomenclature: (1) misinterpretation of language, (2)

directly identifiable, (3) indirectly identifiable and

(4) non-categorizable solution. The rationale for the

study of a mathematical classification of errors is the

bond and the complexity required when proposing

an MER as remediation.

The study presented by Ainsworth (2006)

defined a MERs functional taxonomy, emphasizing

the distinct functions of learning (and

communication), used to illustrate the advantages of

MERs. The External Representations (ERs) are

divided in three key functions: complementary roles,

to constrain interpretation and to construct deeper

understanding. This classification was also

incorporated into this study in order to choose an ER

more related to a specific error.

Applying the approach in remediation of errors

in LOs using MERs requires the LO to be

implemented with a functionalist architecture (see

Figure 1). The architecture consists of three main

modules: the error classifier module (1), which aims

to identify and classify the error by comparing the

solution of the learner with the ideal solution, using

production rules; the MER classifier (2), responsible

for binding the proper MER function to the error,

also explores the production rules; and the MRE

Manager Module (3), to identify the most

appropriate MER and present it to the learner in the

learning session.

In order to validate this study two LOs with

distinct profiles were consolidated. These LO were

developed to be applied in Brazil, and this is the

reason why their screenshots are presented in

portuguese. The Pythagoras Max (sess Figure 2) and

Pythagoras Mix (see Figure 3). The former had all

the assumptions of the study to apply the MER-

based remediation, and the latter had only the

mathematical problem in structured form of

statement, signaling that the answer was incorrect or

not.

Figure 1: Functionalist architecture of the LO developed.

Figure 2: Pythagoras Max with MER Error Remedion.

Figure 3: Pythagoras Mix without MER Error Remedion.

ICEIS2013-15thInternationalConferenceonEnterpriseInformationSystems

520

3 EXPERIMENTAL DESIGN

AND PROCEDURE

For the experiment 20 students from 9th grade of

elementary school of a public school were taken as

subjects, they were divided into two groups, the

experimental group and the control group. The

experimental group used the Pythagorean Max LO,

which explore the process of remediation of error

based on MER. The control group used the

Pythagoras Mix LO, with identical questions, but

without remediation of error process with MER.

The students were distributed according to the

pre-test results, this test consisted of 6 questions

involving concepts related to the Pythagorean

Theorem, which were defined by the success

percentage, and thus group they belong to. In order

to make more precise and formal, pre-test grades

were turned into hit percentage.

The experiment was conducted in 4 lessons, with

50 minutes long each. In the first lesson, we

performed a pre-test in order to identify the prior

knowledge about the content covered. In another

day, participants were invited to interact with the LO

in the proposed activities, to take the post-test,

followed by the application of a satisfaction

questionnaire. The questionnaire consisted of 12

questions regarding aspects related to the use of

Learning Objects, 6 questions about ease of use,

three questions regarding the feedback provided to

the student and, finally, 3 issues covering the impact

of using LO to learn a concept.

Figure 4: Pythagoras Max with MER Error Remediation:

First question.

The paired t-test was used for data analysis in order

to identify any significant gain in learning.

Additionally, we applied a satisfaction questionnaire

to the learners. Figure 4 shows an example of the

Pythagoras Max problem and the respective MERs

presented to remedy the error. Thus, for each type of

error made (misinterpretation of language; directly

identifiable; indirectly identifiable and non-solution

categorizable), it is identified and classified by the

Error Classification Module. Following the MRE

classifier module selected the MRE function

compatible with the classification error

(complementary roles, barring in understanding

construction and deep knowledge). Subsequent to

the manager module MRE selected the most

compatible representation.

4 EXPERIMENTAL DESIGN

AND PROCEDURE

For the experiment 20 students from 9th grade of

elementary school of a public school from Brazil

were taken as subjects, they were divided into two

groups, the experimental group and the control

group. The experimental group used the Pythagorean

Max LO, which explore the process of remediation

of error based on MER. The control group used the

Pythagoras Mix LO, with identical questions, but

without remediation of error process with MER.

The students were distributed according to the

pre-test results, this test consisted of 6 questions

involving concepts related to the Pythagorean

Theorem, which were defined by the success

percentage, and thus group they belong to. In order

to make more precise and formal, pre-test grades

were turned into hit percentage.

The experiment was conducted during 4 (four)

lessons, with 50 minutes long each. In the first

lesson, we performed a pre-test in order to identify

the prior knowledge about the content covered. In

another day, participants were invited to interact

with the LO in the proposed activities, to take the

post-test, followed by the application of a

satisfaction questionnaire. The questionnaire

consisted of 12 questions regarding aspects related

to the use of Learning Objects, 6 questions about

ease of use, three questions regarding the feedback

provided to the student and, finally, 3 issues

covering the impact of using LO to learn a concept.

The paired t-test was used for data analysis in

order to identify any significant gain in learning.

Additionally, we applied a satisfaction questionnaire

to the learners. Figure 4 shows an example of the

Pythagoras Max problem and the respective MERs

presented to remedy the error. Thus, for each type of

MultipleExternalRepresentationsinRemediationofMathErrors

521

error made (misinterpretation of language; directly

identifiable; indirectly identifiable and non-solution

categorizable), it is identified and classified by the

Error Classification Module. Following the MRE

classifier module selected the MRE function

compatible with the classification error

(complementary roles, barring in understanding

construction and deep knowledge). Subsequent to

the manager module MRE selected the most

compatible representation.

The experiment aimed to find a positive

confirmation on the use of MERs in the remediation

of error. Thus, we expected to find significant results

in the use of Max LO, by proposing remediation of

error with MERs. Students' grades, the average and

standard deviation of pre-test and post-test are

shown in Tables 1.

The results confirm the hypothesis, the use of

remediation of error supported in MERs from the

classification error contributed to increased the

student knowledge. The hypothesis of the

experiment is that the Max LO helps the learner

learn concepts providing a significant gain.

The performance of participants in the

Pythagoras Max LO can say that it is possible to rule

out the Null Hypothesis, which reached 0.05% of

significance, concluding with 95% confidence that

the LO brought gains to the acquisition of

mathematical concepts.

Table 1: Results of Pythagoras MAX and MIX.

MAX Pythagoras MIX Pythagoras

Student Pretest(%) Postest(%) Student Pretest(%)

Postest

(%)

A1 66,7 100,0 A1 93,3 100,0

A2 50,0 66,7 A2 56,7 96,7

A3 80,0 83,3 A3 66,7 66,7

A4 96,7 96,7 A4 93,3 100,0

A5 50,0 83,3 A5 96,7 100,0

A6 83,3 100,0 A6 93,3 100,0

A7 66,7 83,3 A7 83,3 83,3

A8 67,5 90,6 A8 100,0 66,7

A9 75,5 85,9 A9 96,7 100,0

A10 68,5 86,3 A10 79,2 93,7

Average 70,5 87,6 Average 85,9 90,7

Standard

Deviation

14,3 10,0

Standard

Deviation

14,4 13,7

The null hypothesis of Pythagoras Max LO is the

average of the post-test is less than or equal to the

average of the pretest. Furthermore, the claim

whether the post-test average was significantly

higher than the average pretest identifying a gain in

student learning. For this purpose, we used a paired

t-test, since the sample size is smaller than 30. With

a confidence level of 95% (α = 0.05), we obtain p =

0.000412178 (t = 4.9202, df = 9). Thus, as p < α, we

can deny the null hypothet hypothepants s in the

acquisition of concepts.

The null hypothesis of Pythagoras Max LO is the

average of the post-test is less than or equal to the

average of the pretest. Furthermore, the claim

whether the post-test average was significantly

higher than the average pretest identifying a gain in

student learning. For this purpose, we used a paired

t-test, since the sample size is smaller than 30. With

a confidence level of 95% (α = 0.05), we obtain p =

0.000412178 (t = 4.9202, df = 9). Thus, as p < α, we

can deny the null hypothesis in the acquisition of

concepts.

The null hypothesis of Pythagoras Mix LO is the

average post-test is less than or equal to the average

of the pre-test. With a confidence level, concluding

that there is evidence to say with 95% confidence

the Pythagoras Max LO helped the particiof 95% (α

= 0.05), we obtain p = 0.20834 (t = 0.8527, df = 9).

Thus, as p > α, it can not rule out the null

hypothesis, concluding that there is no evidence to

say that Pythagoras Mix LO assisted the participants

in the acquisition of concepts. This perhaps is the

fact it is a reproduction model of didactic classroom,

composed solely of the problems statements.

This highlights the importance of paradigm shift

when migrating from the traditional approach to

computer-mediated. If LOs are not built with proper

care can not help the learner and even more may end

up hindering their learning.

As for satisfaction questionnaire applied to the

end of the interaction with the LOs, Pythagoras Max

and Pythagoras Mix: 48% of the group of questions

regarding the ease of use found fully satisfactory

aspects regarding the ease of use of the LO. In the

other group, independently, analysing aspects related

feedback, 42% manifested in a fully satisfactory as a

form of feedback displayed. While the group solving

tasks using the LO also analysed independently,

54% considered fully relevant using an LO for the

acquisition of a concept.

5 DISCUSSION

AND CONCLUSIONS

There are many advantages in using a diagnosis

followed by an intervention, may be mentioned

detection and remediation of errors in the same

context, also is possible, in ITS, analyse partial

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522

solutions of the learner.

The MRE offered in LOs through the remedies

are intended to provide further insight apprentice

path to which it is following, from diagnostic error.

The aim of the proposal was to provide support to

students through an MRE, which can be a sentence

of natural language, tables, lists, pictures,

simulations, diagrams, maps, etc.

The use of remediation enables an intervention

on the learner before progression on a particular

error, thereby avoiding a complete solution, but

misguided. As a consequence there is a reduction in

the number of errors that can occur, this aspect was

considered in the relevant proposal of the study.

The average found in MIX LO pre-test is bigger

than the MAX LO pre-test, but in post-test we can

see this difference between averages was decreased.

This, is more one signal that indicating that the LO

MAX positively influenced the learner acquisition of

knowledge.

The architecture presented allowed facilitate the

remediation of errors made by the learner, through a

more specific categorization error, which is to be

split in more categories, providing opportunities for

a range of varieties that allows the learner to acquire

mathematical knowledge. As a way to meet a higher

level of granularity regarding appropriate

presentation of MER, we used the classification of

MERs functions.

In conclusion, the present study extends the

concepts involving ITS and concept acquisition,

classification of errors linked to MERs. Still want to

do more experiments with a larger sample of

students, expanding the validation study. This

experiment was aimed to validate assumptions.

Future works, we intend to apply the architecture

of remediation of errors based on a classification of

errors, in the other areas of knowledge, in order to

validate the modularity of the architecture and the

use of MREs in remediation of errors in LO.

ACKNOWLEDGEMENTS

We thank CAPES (Coordination of Improvement of

Higher Education Personnel), the Federal University

of Paraná and School Good Shepherd for their

support for the development of this work.

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