MATHPORT

Web Application to Support Enhancement in Elementary Mathematics Pedagogy

Barbara Koroušić Seljak, Gregor Papa

Computer Systems Department, Jožef Stefan Institute, Jamova c. 39, Ljubljana, Slovenia

Barbara Japelj Pavešić

Educational Research Institute, Gerbičeva 62, Ljubljana, Slovenia

Keywords: Web application, Open-source software, Genetic algorithm, Elementary school mathematics, Pedagogy

enhancement.

Abstract: The paper presents a Web application MatPort that is aimed to support enhancement in elementary

mathematics pedagogy. In Slovenia, there is a lack of e-Learning courses on elementary school mathematics

that would provide teachers and students with verified mathematics problems in Slovene language. The

MatPort offers a databank of verified mathematics problems that may be solved on paper or at a computer,

or combined into a paper and pencil form of tests. In this paper, we describe the structure and the

functionality of the application, providing an insight into the MatPort decision system that supports

automatic search of math problems so that students may reach a certain level of knowledge in a thoughtful

way. The decision support system is based upon the genetic algorithm. The application has been evaluated

by a group of teachers and students, who have been actively involved into the MatPort development.

1 INTRODUCTION

One of modern ways of education delivery is e-

Learning, which is a broad term used to describe

learning done at a computer, usually connected to

the World Wide Web. It is widely accepted that e-

Learning can be as rich and as valuable as the

classroom experience or even more so. With its

unique features e-Learning is an experience that

leads to comprehension and mastery of new skills

and knowledge, just like its traditional counterpart.

Instructional design for e-Learning, which is the

systematic process of translating general principles

of learning and instruction into plans for

instructional materials and learning, has been

perfected and refined over many years using

established teaching methods, with many benefits to

students.

1.1 Learning Styles

There are three predominant learning styles

(Donovan, 2000):

 Visual;

 Auditory;

 Tactile/kinesthetic.

Broken down further, people learn by reading and

seeing (visual style), listening and speaking

(auditory style) and doing (tactile/kinesthetic style).

Reading, listening and seeing are passive types of

learning, while speaking and doing are active types

of learning. How much we tend to remember is a

function of the type of learning we prefer and our

level of involvement in the learning. People often

learn through a combination of the ways described

above. To a lesser degree, environment is a factor

too.

Given a good learning environment, most people

tend to understand and remember (i.e., learn) best

when practicing the real thing. Being actively

engaged in hard and challenging activities strongly

supports them to construct knowledge for

themselves. Next, a combination of observing

people we respect, doing and speaking about what

we learn produces a high retention rate, followed by

speaking alone.

In e-Learning, it is important to provide a variety

of activities supported by human intervention. A lot

Korouši

c Seljak B., Papa G. and Japelj Paveši

c B. (2009).

MATHPORT - Web Application to Support Enhancement in Elementary Mathematics Pedagogy .

In Proceedings of the First International Conference on Computer Supported Education, pages 81-88

DOI: 10.5220/0001974000810088

 SciTePress

of passive learning may be done through reading

text, listening to audio clips, and seeing graphics,

while the active mode should be stimulated through

practising, e.g., solving mathemathics problems.

1.2 Unique e-Learning Features

There are several unique features of e-Learning:

 e-Learning is self-paced and gives students a

chance to speed up or slow down as

necessary;

 e-Learning is self-directed, allowing students to

choose content and tools appropriate to their

differing interests, needs, and skill levels;

 Designed around the learner;

 Geographical barriers are eliminated, opening

up broader education options;

 Enhances computer and internet skills.

1.3 Mathematics e-Learning Support

in Slovenia

Učiteljska.net (http://uciteljska.net/) and e-um

(http://www.e-um.si/) are good examples of Slovene

Web applications aimed to support e-Learning of

elementary school mathematics. While the first one

is intended to support exchange of teacher

experience and information, the second one provides

interactive online courses for students.

However, there is a lack of online courses in

Slovene language on solving verified mathematics

problems for children under 15 years of age. For this

reason, we have developed a new Web application

for e-Learning of elementary school mathematics in

an active learning mode, called MatPort

(http://sinica.ijs.si/matport/).

The rest of the paper is organized as follows:

Section 2 describes the content and the technology

aspects of the Web application. Section 3 outlines

the concept of the decision support system that may

be used as an automatic search facility. The

experimental work is presented in Section 4.

Conclusions and directions for further work are

given in Section 5.

2 WEB APPLICATION MATPORT

The Web application MatPort is based upon the

national curriculum for elementary school

mathematics and upon verified mathematical

problems.

2.1 Web Content Aspect

For the pilot stage of the project, we decided to use a

collection of verified sets of mathematics problems

in Slovene language, published in the form of flash

cards, which have been used for many years in our

schools. These problem solving items already passed

through many steps of evaluation, therefore, they are

valuable. Some of the real life items needed to be

updated for the present time. By this stage, we have

focused on the grades 6 to 9 or the age group 12 to

15 years old students.

With the help of experienced teachers we

classified the items into subgroups with respect to

the knowledge required for problem solving by

content and three difficulty levels.

Finally, we designed and developed the MatPort

Web application with the following modules for:

1. Entering math items, their solutions and

teaching instructions into a database for

teachers and administrators;

2. Solving math items;

3. Preparing a paper and pencil form of test;

4. Providing other information relevant for

teaching the elementary school mathematics.

The first and the third module are aimed for

teachers, the second one for students and the last one

for teachers and parents.

2.1.1 Entering Math Items

A user registered as a teacher may enter his/her

items into the MatPort database (Fig. 1). Each item

is described with a problem definition, solution(s) in

terms of values and units, content area, difficulty

level, source or author, and teaching instructions.

The user may copy an already validated math item

and adapt it to its needs.

Figure 1: Entering Math Items.

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The math items saved in the database are

accessible only to the user. However, (s)he can

outbid any of his/her items to other MatPort users.

Before publishing this item as a public one, experts

verify it and suggest modifications if required.

Teachers are informed of their activities through

a simple statistics on math items collected in the

MatPort database.

2.1.2 Solving Math Items

A user registered as a teacher or a student may solve

any public or his/her own math item from the

MatPort databank. The application provides manual

and automatic search of math items in content areas

defined by the national curriculum for elementary

school mathematics. The automatic search is

provided by a decision support system that is

described in detail in Section 3.

Ones an item is selected, the user can print it and

solve the problem in a classical paper and pencil

way, or directly provide solution(s) to the MatPort.

The application evaluates the answer(s) and denotes

the item as correctly or incorrectly solved problem

by this user.

Users are given information on the progress in

solving math items through graphycal symbols and

stimulative words (Fig. 2). More specifically, if the

user correctly solves more than 45%, 60%, 75% or

90% of dealt items, respectively, (s)he is rewarded

with one, two, three, four or five stars, respectively.

Figure 2: Solving Math Items – information.

If required, users may get teaching instructions

for math items (Fig. 3). They can also use a forum

integrated into the application to communicate with

other users or experts.

2.1.3 Preparing Tests

The MatPort supports design of paper and pencil

form tests to examine the students’ knowledge in a

classroom. A user may select a set of items, modify

them if needed, define the test’s header and print the

test. The application also offers a printout of

solutions to the selected math items.

Figure 3: Solving Math Items - teaching instructions.

2.1.4 Providing Information

We support teachers and parents with information on

elementary school mathematics, such as (Fig. 4):

 Results on the IEA (International Association

for the Evaluation of Educational

Achievement) TIMSS Advanced Assessment

Frameworks, which – carried out every four

years at the fourth and eighth grades provides

data about trends in mathematics and science

achievement over time;

 The national curriculum for elementary school

mathematics;

 Practical advices to parents on helping students

learning mathematics at home;

 Education debate on motivation methods and

tools for engaging students in the process of

active learning mathematics.

Figure 4: MatPort web application.

2.2 Web Technology Aspect

The MatPort was designed and developed using

state-of-the-art technologies. We applied a UML

(Unified Modelling Language) based model-driven

MATHPORT - Web Application to Support Enhancement in Elementary Mathematics Pedagogy

methodology (http://www.uml.org/) to cover the

life-cycle of the Web application development.

2.2.1 Design

To design the application sympathetically with the

way students, teachers and parents actually use the

Web, not how we think they should, we directly

involved experienced teachers into the project.

The MatPort database was designed as a

rational database, consisting of tables that store data

on:

 User profiles;

 Math items;

 Knowledge required for solving items;

 Relations between the content areas;

 History of items’ solving.

The database structure is given in Fig. 5.

Figure 5: Rough draft of the MatPort database structure.

The application consists of several dynamic and

static modules, such as:

 Module for providing math items;

 Math item solver module;

 Test generator module;

 Informer module;

 Forum;

 Download center.

The informer module has a static content that is

managed by a content management system, while

the others may change their content in a dynamic

way. The Web application structure is shown in Fig.

2.2.2 Development

As the project’s budget was low and we needed to

minimize the cost, we decided to make good use of

open-source and freely available software. We

applied:

Figure 6: Rough draft of the MatPort use case.

 Apache (http://www.apache.org/) as a Web

server that replies to Web clients’ requests via

HTTP (HyperText Transfer Protocol);

 MySQL (http://www.mysql.com/) as a

database management system that is based

on the relational model;

 PHP5 (HyperText Preprocessor)

(http://www.php.net/) as a server-side

technology. This is also a scripting language

that has evolved to include a command line

interface capability and can be used in

standalone graphical applications. PHP5 offers

standardized means for specifying the variety

of property scopes typically offered by object-

oriented languages (Hayder, 2007);

 JavaScript (Darie et al, 2006) as a client-side

technology. This is another scripting

language, whose code is written in plain text

and can be embedded into HTML (HyperText

Markup Language) pages to empower them.

The MatPort is aimed for doing mathematics,

which means that mathematical symbols need to be

supported. For this reason, we integrated TinyMCE

(http://tinymce.moxiecode.com/) as a platform

independent JavaScript WYSIWYG editor. It has

the ability to convert HTML TEXTAREA fields or

other HTML elements to editor instances. Because

CSEDU 2009 - International Conference on Computer Supported Education

TinyMCE supports only standard HTML math

symbols, we extended its library with a module for

handling mathematical expressions. Once entered,

these are rendered into images using the

LatexRender scripts

(http://www.mayer.dial.pipex.com/tex.htm).

All MatPort printouts have the standard PDF

(Portable Document Format) format.

The MatPort forum is based upon the PHPBB

(http://www.phpbb.com/) open-source forum

solution.

Last but not least, we set formatting MatPort

visual options in a centralized document that is

referenced from PHP files by using CSS (Cascading

Style Sheets).

3 DECISION SUPPORT SYSTEM

We are aware of the fact that only providing a

dataset of math items is not enough. We need to

incorporate an extrinsic motivation system to

“bribe” the 6

to 9

grade students to practice

mathematics. There are rare children who are

intrinsically motivated to do repetitive, boring tasks.

3.1 How to Motivate Children?

A student who solves a MatPort math item receives

information on the progress through graphical

symbols and stimulative words (Fig. 2).

In addition, the application provides information

on math items that need to be further solved to

receive a higher score. These can be supplementary

or additional items to help strengthen or increase

knowledge, respectively. The information is

provided by the MatPort decision support system,

when the automatic search facility is used.

3.2 Genetic Algorithm

The MatPort decision support system that motivates

students to continue solving exercises is based upon

an evolutionary computation method, i.e., the

genetic algorithm (GA) (Goldberg, 1989; Bäck,

1996).

The GA is based on a heuristic method, which

requires little information to search effectively in a

large search space. The algorithm employs an initial

population of chromosomes, which evolve to the

next generation by probabilistic transition rules

(randomized genetic operators) such as selection,

crossover and mutation. The objective function

evaluates the quality (fitness) of solutions coded as

chromosomes. This information is used to perform

an effective search for better solutions. There is no

need of other auxiliary knowledge. The GA tends to

take advantage of the fittest solutions by giving them

greater weight and by concentrating the search in the

regions of the search space with likely improvement.

The GA mechanism is presented in Fig. 7.

Initialize the population of

chromosomes;

While stop condition not met do:

 Calculate the fitness for each

member in the population using the

fitness function

;

 Select and reproduce individuals

according to their fitness;

 Perform genetic operators

(crossover and mutation) on the

population.

Figure 7: The GA’s pseudocode.

The GA is a population-based evolutionary

approach that allows searching within a broad set of

solutions from the search space simultaneously.

Namely, because there are many math items (few

hundreds or even more than thousand math items per

a grade) and interrelated content areas (more than

100 content areas per a grade), the student may

continue solving items in many possible ways that

may or may not lead to a higher score. Moreover,

math items are dynamically generated by teachers

(i.e., the item dataset expands with time) and the

student may start solving them anywhere in the

dataset. In the GA, there is a risk of converging to a

local optimum, but good results of various research

work obtained in other optimization problem areas

(Papa and Koroušić Seljak, 2005; Koroušić Seljak,

2006; Tušar et al, 2007; Korošec and Šilc, 2008)

encouraged us to consider the GA approach as a

promising approach to the decision making problem.

The idea is to find a set of math items within

different content areas that, when solved

correctly, improve the user’s knowledge and

increase his/her score as much as possible. The set

of items should consist of math problems from all

poorly scored content areas and the areas that

precede these areas. Therefore, before start

searching, the system identifies all the feasible

items, i.e., math problems from the poorly scored

content areas. These items form some kind of a pool

of relevant items P for current score improvement.

3.2.1 Encoding

The suggested list of math items needed to improve

the score is encoded into a chromosome, where each

MATHPORT - Web Application to Support Enhancement in Elementary Mathematics Pedagogy

gene represents the identification (ID) number of the

item in the MatPort database. The chromosome

length has been fixed to 15, while this number

represents a reasonable number of items to perform,

in order to significantly improve the score. Fig. 8

presents the chromosome with such number of genes

and the IDs of the items to be performed in the order

as encoded in the chromosome.

1 2 3 4 5 14

45 124 33 79 186 … 247

Figure 8: Chromosome of length 15 that represents a set of

items to be solved to increase the current score.

3.2.2 Population Initialization

The initial population consists of n chromosomes.

Each chromosome is initialized with randomly

chosen items from the pool of relevant math items.

3.2.3 Genetic Operators

In the selection process, the elitism strategy is

applied through the substitution of the least-fit

chromosomes with the equal number of the best-

ranked chromosomes.

With the one-point crossover scheme,

chromosome mates are chosen randomly and with a

probability p

all values after randomly chosen

position are swapped, which leads to two new

solutions that replace their original sources. Fig. 9

shows a crossover example.

45 124 33 79 186 … 247

12 14 233 56 228 … 269

⇓

45 124 33 56 228 … 269

12 14 233 79 186 … 247

Figure 9: One-point crossover.

In the mutation process each value of the

chromosome mutates with a probability p

. If the

value of the chromosome needs to be changed, than

some new value from the pool of relevant math

items is chosen. However, since a high mutation rate

resulted in a random walk through the GA search

space, p

has to be chosen to be somewhat low. Fig.

10 shows a mutation example.

124

186 … 247 31

⇓

137

12 … 269 41

Figure 10: Mutation.

3.2.4 Fitness Evaluation

After the recombination operators modify the

solutions, the whole new population of

chromosomes is ready to be evaluated. In the

evaluation process the set of math items is assumed

to be solved correctly and the score improvement is

calculated. The calculated score improvement is

used as a fitness value of each chromosome. Here all

the items are weighted with their difficulty grade;

the order of items is relevant when the problems

belong to different content areas that derive from

each other.

3.2.5 Parameter Settings

In order to ensure optimal solutions in a reasonable

response time robust parameter settings need to be

found for the population size, number of

generations, selection criteria and genetic operator

probabilities:

 If the population size and the number of

generations are too small, the GA converges

too quickly to a local optimal solution and

may not find the best solution. On the other

hand, a large population and too much

iteration require long time to converge to a

region of the search space with significant

improvement. In our case, we have used the

population size n=15 and number of

generations n

=30;

 Applying the elitism strategy, fitter solutions

have greater chance to be reproduced. But

when the number of worse solutions to be

exchanged with better ones (the selection

criteria) is too high, the GA is trapped too

quickly in a local optimum solution. Our

selection rate has been 20%;

 Too low crossover probability preserves

solutions to be interchanged and longer time is

required to converge. This probability should

be large enough to crossover almost all mated

solutions. In our case, efficient setting for p

has been 70%;

 Too high mutation probability may introduce

too much diversity and takes longer time to

reach an optimal solution. Too low mutation

CSEDU 2009 - International Conference on Computer Supported Education

probability tends to miss some near-optimal

solutions. Again, the efficient setting for p

has been 5%.

3.2.6 Termination

When a certain number of populations are generated

and evaluated, the system is assumed to be in a non-

converging state. A chromosome with the highest

score improvement is chosen as a final result.

On average, the GA finds an optimal selection of

math items that need to be further solved by the user

to receive a higher score in order of few seconds. As

it is implemented as a background process, it does

not slow down the application.

4 EXPERIMENTAL WORK

During the current pilot stage of the MatPort

application, we have involved a group of elementary

school teachers and a group of the 8

grade students

to test the application facilities.

We have examined the MatPort effectiveness for

delivery of elementary mathematics e-Learning. In

the group of teachers, we have been interested in:

1. The number of new math items provided by

the teachers;

2. The number of copied and modified already

validated math items;

3. The number of written tests designed by the

MatPort.

In the group of students, we have been interested

in:

1. The time spent in each content area;

2. The number of correctly and incorrectly

solved exercises in each knowledge field;

3. The correlation between the MatPort scores

and school marks before and after the period

of testing the MatPort;

4. Frequency of using the MatPort automatic

search (decision support system) facility.

Both groups have been asked to complete a

survey that includes an assessment of likeability,

intention to use exercises and information, and

demographic indicators.

The main aim of the study is to prove a predicted

hypothesis that participants keep or even increase

their level of knowledge in elementary mathematics

by practising math problems provided by the

MatPort. In this way, we could conclude that the

MatPort actually supports enhancement in

elementary mathematics pedagogy.

5 CONCLUSIONS

In this paper, the instructional design of the MatPort

Web application, which is aimed for elementary

mathematics e-Learning in the active way, were

presented. We upgraded this design by incorporating

a high-performance evolutionary computation

method to support automatic search of relevant math

items. In this way, the MatPort may lead its users

toward higher scores in a thoughtful way. Finally,

we described the method for evaluation of the

application’s effectiveness.

After the pilot stage of the project, we are

planning to expand the dataset of math items to other

elementary school grades. We will increase its

efficiency through additional motivation tools, such

as winner lists or computer games, which will be

activated as soon as a student will gain a certain

score. Much work needs to be done to find an

adequate level of human intervention. In cooperation

with teachers, we will try to improve the way of

providing teaching instructions and intermediate

solutions. Last but not least, we will discuss the

problem of cheating.

In addition, we will do some experimental work

on the application of the efficient parameter-less

evolutionary search method (Papa, 2008) as a

substitution for the currently implemented genetic

algorithm.

ACKNOWLEDGEMENTS

This research project was supported by a joint action

of the Jožef Stefan Institute and the Educational

Research Institute Ljubljana under financial support

to education and training of the European Social

Fund.

The authors are most grateful to Breda Koroušič

for her guidance in the MatPort development. As an

experienced teacher of mathematics for many years

she has significantly contributed to the project,

providing a large collection of math items and all the

knowledge and experience gained through many

years of successful work with children.

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MATHPORT - Web Application to Support Enhancement in Elementary Mathematics Pedagogy

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