ForMath

Intelligent Tutoring System in Mathematics

Piotr Brzoza

, Ewa Łobos

, Janina Macura

, Beata Sikora

and Marek

Zabka

Institute of Informatics, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland

Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland

Keywords:

Inteligent Tutoring System, e-Learning in Mathematics, Learning for Visually Impaired People.

Abstract:

The paper presents main ideas, aims and implementation of ForMath — intelligent tutoring system in mathe-

matics which is a project developed at Silesian University of Technology. Up till now, the project has contained

mathematical problems for engineering students, but in the future it will be adapted to other courses. The pro-

posed platform has both the advantages of e-learning courses and innovative elements such as interactivity

(hints or theory on the user’s request, suggested individual path, remedial sessions when necessary) and the

accessibility of the system for students with disabilities.

1 INTRODUCTION

Modern higher education requires the enrichment of

traditional forms of education by attractive, student-

friendly and custom offers, fusing and structuring

knowledge. Time limited during the course of pro-

gram studies and the growing difﬁculties of assimi-

lation of science, mainly mathematics, increase the

importance of students’ work to which they were not

adequately prepared by their high school. This situa-

tion raises the need for a multi-faceted tool that will

enable the furthest reaching possible contact with the

subject. Education using Web technologies allows for

greater ﬂexibility and the individualization of teach-

ing, which is a chance to break the barriers of time

and social access to university education at the high-

est level.

In the 1990’s, dissemination of the Internet re-

sulted in a rapid development of e-learning, as well as

in the teaching of mathematics. Many e-learning plat-

forms have been created, mainly in the United States,

but also in Europe. In Poland, with some delay, e-

learning is also beginning to play a role in the edu-

cational process. Some universities have e-learning

platforms where academics place their courseware.

Still, not enough attention is given to e-learning in

mathematics, although even here the situation is be-

ginning to improve. The project Development of dis-

tance learning programs at the course - Computer Sci-

ence, funded by the European Social Fund (of the

Sectoral Operational Programme Human Resources

Development 2004 - 2006) exempliﬁes this tendency.

Scientists from leading Polish universities (University

of Warsaw, Jagiellonian University, Poznan Univer-

sity of Technology and Warsaw University of Tech-

nology) are involved in the project. Courseware for e-

learning courses of the main branches of mathematics

have been developed by mathematicians at the Jagiel-

lonian University (Wa˙zniak, 2006). Wroclaw Univer-

sity of Technology has also been using e-learning to

complement traditional educational systems for sev-

eral years and is one of the leading universities in this

area (Eportal, 2011). Until now, theoretical materials

with a set of interactive exercises in algebra and math-

ematical analysis, as well as repertory of mathematics

were implemented.

A very serious problem that touches all techni-

cal universities is a poor background in mathemat-

ics among students beginning the engineering edu-

cation. Our experience of many years of teaching

was also conﬁrmed by PISA (the Program for Inter-

national Student Assessment) investigations (Łobos

and Macura, 2010). In the case of Poland, there is

another factor that intensiﬁes this ‘poor background

problem’. A part of an adjustment of Polish Tertiary

Education System to Bologna Process was a division

of engineering studies into 2 cycles. This change

has resulted in reduction of the number of hours as-

signed for teaching mathematics with unchanged ter-

tiary curricula. In such a situation an intelligent tutor-

ing system in mathematics as a part of blended learn-

ing seems to be the best solution. Although Silesian

118

Brzoza P., Łobos E., Macura J., Sikora B. and Zabka M..

ForMath - Intelligent Tutoring System in Mathematics.

DOI: 10.5220/0003899901180122

In Proceedings of the 4th International Conference on Computer Supported Education (CSEDU-2012), pages 118-122

ISBN: 978-989-8565-06-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Universityof Technology has a remote education plat-

form, it is, as majority of e-learning platforms, not in-

teractive. Therefore we have been working out a sys-

tem of remote interactive education in mathematics

and this paper describes the main ideas of our system.

2 AIMS OF THE PROJECT

The project ForMath - intelligent tutoring system in

mathematics aims to create an interactive platform

for supporting education in mathematics. It can be

used by technical students to self-study selected ﬁelds

of mathematics and expanding knowledge in these

ﬁelds. The platform is primarily a collection of exer-

cises, in which there is a possibility to appeal to the-

ory (e-books), or get hints on how to solve a speciﬁc

problem (at any stage of solving the exercise). It con-

tains exercises of varying difﬁculty within different

branches of mathematics. Errors and mistakes made

by students are registered, analyzed and then become

the basis for preparing individual exercise lists tai-

lored speciﬁcally to the student’s level of knowledge.

In the case of repeated difﬁculties the student is di-

rected to remedial exercise sessions.

The platform is aimed at different audiences: stu-

dents, candidates for college, high school students

and active engineers. It also enables competency im-

provement of the young academic teachers. The re-

mote system can be a useful tool in long term learning

programme as well. For example, it would be help-

ful for alumni of other faculties, not necessarily engi-

neering, or any person who wants to refresh or extend

his/her mathematical knowledge.

The project also considers the needs of people

learning disabilities, particularly blind and visually

impaired. One of the research tasks is to interpret

mathematical symbols and formulas allowing their

reading by blind persons. The remote teaching of

blind and visually impaired students is one of the ele-

ments of innovation in the project.

By design the platform is built simultaneously in

two languages: Polish and English. Courses taught in

English are increasingly popular in Poland, where for-

eign students study as well. An option associated with

the English language is meeting the needs of Polish-

speaking students studying in English, who have dif-

ﬁculty with assimilating speciﬁc concepts and terms

(including mathematical), familiarity with which is

already required during the ﬁrst year of studies. This

module enables them to listen to terms, phrases and

mathematical formulas recorded by the teacher. Not

only students will beneﬁt from this offer, but also aca-

demic staff preparing to teach in English (especially

doctoral students and young assistants).

During the project, pedagogical and cognitive re-

search are conducted, which help in choosing the best

learning solutions. This platform exploits elements of

the constructivist theory of learning that focuses on

student activation and motivation and application of

the principle ‘the more we know, the more we can

learn’ by adjusting the difﬁculty level to the level of

knowledgeof the learner, and gradually increasing the

difﬁculty subsequent exercises (ImprovingMathemat-

icsEducation, 2006; Lubina, 2005).

In the long term, the platform will be expanded

and developed to build an interdisciplinary teaching

tool. Additionally, the platform is planned to create

modules from other scientiﬁc disciplines (e.g. math-

ematics for economists, statistics, physics, chemistry

and mechanics, etc.).

3 COMPARISON OF

MATHEMATICAL

EDUCATIONAL PLATFORMS

There are numerous e-learning platforms relative to

(Karczy´nski,2011; Khamsi et al., 2011; Husch, 2001)

that offer well-prepared theoretical content (in the

form of e-books, presentations, animations or video

ﬁlms) and exercises with ready-made solutions, how-

ever we are developing the ForMath system. ForMath

has the advantage of implementing a ‘check, correct

and learn’ as you go system. Essentially teaching

the student step by step as opposed to only recog-

nizing the end results which other popular systems

use currently. There are also systems based on some

type of interactivity — they give hints or explana-

tions at some stages of the process of problem solu-

tion (Wa˙zniak, 2006; ALEKS, 2011; Bogacki, 2009;

Melis, 2011) Additionally some of them provide as-

sessment to the student which motivates him/her to

think and work.

An interesting educational software is ALEKS

(New York University and University of California)

(ALEKS, 2011) which uses ﬂexible and easy to use

answer input tools. ALEKS contains topics in math-

ematics that are lectured at elementary, middle and

high schools. The system individually and continu-

ously assesses each student. At the beginning it as-

sesses the student’s current course knowledge by ask-

ing him/her 20-30 questions. Then the platform offers

a choice of topics that a user is ready to learn. After

the choice of topic student gets practice problems that

teach the topic. All questions are algorithmically gen-

erated (so they are unique) and require a ‘free respon-

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119

se’. Unfortunately, the system does not recognize typ-

ical student mistakes. If a student does not under-

stand a particular problem, he/she can always access

a complete explanation. The explanation offers also

some additional theory if necessary. However, even if

ALEKS recognizes that a part of solution is correct,

it presents the same explanation as for user who does

not know how to start to solve the problem. When

the student gives sufﬁcient amount of correct answers

to a given topic, ALEKS considers that the student

has learned the topic and the student chooses another

topic to learn.

Another example is ActiveMath (DFKI, Saarland

University) (Melis, 2011) — the intelligent and in-

teractive project. It is a web-based, multi-lingual,

learner-centered system for mathematics which con-

tains different topics realized at schools, universities

and it is also a good tool for life-long learning. It helps

the student in self-regulated learning, can adapt to in-

dividual knowledge and personal interests and learn-

ing goals. The system also presents information on

student improvement. The user of this platform has

to ﬁll in the input ﬁeld with the ﬁnal result of his/her

calculations. In the case of a correct answer, the sys-

tem informs the student that the answer is correct and

shows the complete solution. Otherwise it suggests to

try again or to ask for some hints. By clicking several

times on the ‘hint’ button, one may obtain the com-

plete solution.

In Poland, the platform (Wa˙zniak, 2006) is very

popular. It was built by a few leading Polish univer-

sities and it offers e-learning courses for mathematics

(among other) at the level of secondary and tertiary

education. The systems offers attractive theory (e.g.

with animations), practical tasks, and tests. There are

hints on request or, in case of more complex exercises,

the plans of solution. It is also possible to see the com-

plete solution. The student can solve the exercise on

his/her own, eventually using the hint and then com-

paring his/her solution with the one presented by the

system.

The Inﬁnite Series Tutorial (the part of (Bogacki,

2009) developed by Przemyslaw Bogacki from Old

Dominion University), presents a little different ap-

proach and for this reason it is remarkable. The stu-

dent has to choose the answer (i.e. converges abso-

lutely, converges conditionally, or diverges) and then

give the reason of his answer (the proper convergence

test). Then the system discusses the student’s answer,

eventually suggests the other possible solution paths.

So in the case of an incorrect answer, it is possible to

try again or to give up and see the complete solution.

The tutorial also collects information on numbers of

solved tasks.

Our platform differs from other already exploited

ones. In the described systems (Wa˙zniak, 2006;

ALEKS, 2011; Bogacki, 2009; Melis, 2011) the stu-

dent does not know what kind of error he/she has

done, only comparing his/her solution with the one

given by the system, he/she can ﬁnd the mistake (only

in the case he/she has chosen the same solution path

as is given by the system ), but it is too late to con-

tinue the solution on his/her own. Let us stress the

main differences in the approach to solving exercises

that appear in our and described platforms. In our

system the two main paths are offered. The ﬁrst one

is the path of checking the answer, and the second is

the path of hints. The student that needs more help

can choose this path of hints and then is lead step by

step through the solution of a problem. He/she has to

make some calculations on his/her own and then ﬁll

in the input ﬁelds. The system veriﬁes the correctness

of each step, recognizes typical mistakes and then in-

forms the student about his mistakes, gives more hints

or explanations. It allows the student to correct imme-

diately his/her error and continue the calculations. In

many points of this path of solution there is a possi-

bility to abandon this main path, ﬁnish the exercise on

one’s own and check the answer. In the case of check-

ing the answer, the system examines the correctness

of the solution (if possible) by giving some additional

questions.

4 IMPLEMENTATION

To implement our system we use LaTeX, PHP, C and

JavaScript languages.

4.1 Main Ideas

The main idea of the system is as following: a student

logins into the system and can choose a problem to

learn. After some sessions, the system would be able

to propose some problems according to the difﬁculties

and errors made by the student. To make it possible

the system registers the student’s activity. One distin-

guished action is to solve one problem.

A solution of a problem is divided into pages.

Each page is presented to the student with contents

and some possible actions to be taken. He/she can

make a decision: choose some expression from a list

or write some number/text to an input ﬁeld. After-

wards, the student can chooses an action to be taken

by clicking on a button, although occasionally there

will only be one button present. Each button is related

to one or more pages. If there are more pages then

the choice depends on the previous decisions made

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Figure 1: Screen shot of running system.

by the student. Thusly, we can describe the solution

as a graph, and the student is moving through that

graph choosing a his/her path. Of course, there are

some terminal pages, and after reaching those pages,

the student may move to another problem or log out.

4.2 Preparing of a Problem

Each problem with its correlated solution is written in

one ﬁle in LaTeX. We use a dedicated package ‘jtr’

designed for this system by Authors. Each page is de-

scribed as an environment ‘solutionPage’. The page

contains some text with mathematical formulas and

lists of choices or input ﬁelds. Moreover, this page

has an environment ‘ﬁeldsOfChoice’ with some but-

ton description. On the terminal pages there are no

special button descriptions although there are stan-

dard buttons on each page: ‘log out’ and ‘new prob-

lem’.

Images can be put into a page with ‘img’ envi-

ronment, and compiled with imLaTeX (a new spe-

cial program prepared by M.

Zabka to simplify draw-

ing function graphs in LaTeX ﬁles, now in its’ pre-

release version) concurrently. The ﬁles are loaded

into folders and compiled. The compilation uses La-

TeX, imLaTeX (with MinGW and GD) and speciﬁ-

cally dvipng to translate mathematics to png ﬁles. It is

possible to use MathJax for equations instead dvipng

and get beneﬁts of MathML for visually impaired per-

sons. A system that uses MathML was described in

(Tsonos et al., 2009). Eventually we get some PHP

ﬁles (one for each page) and some graphics with equa-

tions and graphs, as well as the pdf ﬁle with documen-

tation of all the system pages of the problem.

4.3 Runing the System

Let us look closer at how the system runs. Teachers

give the system access code to their students. Thusly

after logging into the system, student’s activities are

saved into data base. The student can choose a group

of problems and a new problem in the chosen group

or the system can propose a new problem to solve.

So a PHP ﬁle with the start page is presented to the

student. He/she can try to solve the problem or can

go to the hints pages which lead to the solution for

the problem. Depending on the page, the student

reads the texts and if necessary, makes choices or puts

a number or letters into the input ﬁeld. Afterwards

he/she should press one of the buttons (or sometimes

only one button). The system checks the choices and

Figure 2: Screen shots, possible steps of a solution.

input ﬁelds and directs the student to an appropri-

ate PHP ﬁle with the new page according to button

descriptions on the source ‘ﬁeldsOfChoice’ environ-

ment. Some JavaScript code controls the behavior of

the page. At the same moment, the choices are saved

ForMath-IntelligentTutoringSysteminMathematics

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to the data base. Fig.1 and Fig.2 show a sequence of

screens, chosen by a student for calculation of an in-

tegral (this shows one of the possible solution paths).

There are some standard buttons on each page:

‘log out’ button, which ends the session, ‘contents’

button, which presents contents of a problem to the

student, as well as a ‘new problem’ button, which

breaks the solution of the current problem and allows

the student to choose another one. On the terminal

page, only text is present and the student can choose

only a standard button.

5 SYSTEM ACCESSIBILITY FOR

PEOPLE WITH DISABILITIES

Currently, there are approximately 20 thousand im-

paired students at Polish universities and over 200

mostly visually impaired at Silesian University of

Technology.

Our system and mathematical e-learning con-

tent are designed conform to web content accessibil-

ity guidelines from the Web Accessibility Initiative

(WAI) of the World Wide Web Consortium (W3C).

The ForMath will be accessible for blind and low vi-

sion people. All graphical user interface (GUI) com-

ponents will be alternativeaccessible by keyboardand

assistive devices. The mathematical content (formu-

las, graphs, etc.) will be described and accessible for

screen readers.

We plan to make system accessibility tests with

impaired students and pupils from the educational

centre for the blind in Laski near Warsaw using main-

stream accessibility tools and software like screen

readers, screen magniﬁers, and Braille displays.

Research with participation of visually impaired

persons will be referred to the understanding of math-

ematical expressions according to applied methods of

interactive reading, expression of verbal descriptions

and synthetic speech parameters.

The research will be carried out with the partici-

pation of both the visually impaired and totally blind

and with the different levels of mathematics knowl-

edge. Test materials will be modiﬁed based on the

conclusions of the analysis previously performed ex-

periments.

Evaluation results will be collected using ques-

tionnaires completed both by students and teachers

assisting in the research. The conclusions of the ac-

cessibility research will be consulted with the experts

in the area of impaired persons education.

Research results will help improve the accessibil-

ity of the developed system for the needs of disabled

people.

6 SUMMARY

The ForMath project is a comprehensive, attrac-

tive and student-friendly platform for remote inter-

active education. Successes of the ALEKS platform

(ALEKS, 2011), although it covers primary and sec-

ondary education, give hope that the proposed tutor-

ing system will be useful for engineering students.

The interactive platform ForMath will cover ter-

tiary education in mathematics and will contain a rep-

etition module at the secondary level. Our years of

experience in teaching students show that such an in-

teractive tool will support the education valuably. The

ForMtah is intended to be more advanced than other

available educational platforms. The continuous anal-

ysis of errors, hints on each stage of solution, different

problem solving solutions (when possible), suggest-

ing a personal path of learning as well as the module

for blind and visually impaired users are the elements

of innovation in the project.

Proofreader: Jerome Keys.

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lus.

ImprovingMathematicsEducation (2006). http://pages.

uoregon.edu/moursund/math/learning-theories.htm.

Karczy´nski, K. (2011). http://www.etrapez.pl/.

Khamsi, M., Knaust, H., and Marcus, N. (2011).

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Łobos, E. and Macura, J. (2010). Mathematical competen-

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