Suremath

User Study and Related (Re-)Implementation of a Multitouch Application for

Learning Math

Georg J. Schneider and Immanuel Ubl

Dept. of Computer Science, University of Applied Sciences Trier, Schneidershof, Trier, Germany

Keywords: User Study, e-Learning Hardware and Software, Multi-platform Multi-touch Application, e-Learning

Application for Mathematics.

Abstract: Learning of the relation between mathematical effects and the underlying formula is a huge step for learners,

which are used to execute rather basic arithmetic calculations up to this point. This usually happens in high

school and pupils are often overstrained by the amount of abstraction which is required. In order to help

students to overcome this gap, we have developed an application for a multitouch table. The pupils are able

to grasp and move function graphs and see how the parameters of the formula change immediately. We have

claimed that this approach leads to a better and faster understanding of the facts to be learned. In order to

evaluate our approach we have carried through several workshops with a focus group and a small user study

with pupils at the relevant age. In this paper we will describe the findings and we will shortly sketch the

resulting (re-) implementation of our system.

1 INTRODUCTION

Learning mathematical facts is a hard task for many

students. Whereas basic mathematic knowledge more

or less still reflects the daily life. More complex and

abstract features appearing in the pre-calculus

curriculum are hard to match with daily topics.

To help learners to get a better understanding of

the more theoretical concepts we have developed a

multitouch table application, where they can interact

with the mathematical concepts in a natural way,

nevertheless providing a link to the underlying

concept. They can move function graphs using their

fingers by touching and dragging on that table. In

parallel they can observe the changes of the

parameters of the belonging equations in real time

(Blanke and Schneider 2011). Furthermore the

application provides an integrated exercise module.

The system has two application scenarios. First, it

shall add another view and another media to the

regular class. There should be a change between

regular teaching methods with pen and paper and the

multitouch application. We also imagine that pupils

should have time to experiment with the application

on their own for a certain period of time in order to

verify their concepts or to discover correlations.

Second, the pupils shall work independently with the

application at home using the built-in exercise mode

for exam training.

The evaluation of our application in a real world

setting in order to figure out if we can prove that the

application leads to better results in learning these

specific topics was another goal. Therefore we have

carried out several workshops with a focus group and

a small user study with pupils and their teacher.

Having only a relatively small group of participants,

we have focussed on finding hints for improving our

system and its acceptance among learners, which we

wanted to integrate afterwards.

In the following we will describe the study and the

workshop we have carried through as well as the

results.

Finally we have deduced different factors to

support the aspects identified in the studies and

incorporated the results into our application. This has

led to a complete reimplementation of the system

since these features could not be incorporated into the

current system.

Accordingly, we will describe our new approach

at the end of this paper.

198

Schneider, G. and Ubl, I.

Suremath - User Study and Related (Re-)Implementation of a Multitouch Application for Learning Math.

In Proceedings of the 8th International Conference on Computer Supported Education (CSEDU 2016) - Volume 1, pages 198-204

ISBN: 978-989-758-179-3

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

2 RELATED WORK

First of all, we have to ask the question if at all, there

is a hint that computer supported learning leads to

better results. (Means et. Al. 2009) conducted a meta-

analysis of 46 studies in the area of online learning,

incorporating different learning techniques, like

blended learning. One of the main effects they

discovered was, that classes with online learning

(including blended learning) produce stronger student

learning outcomes than classes with solely face-to-

face instruction.

Novel user interfaces based on multitouch tables

in the field of mathematics are presented in (Zeleznik

et al 2010). The application supports the manipulation

of formulas using gestures on the device. A small

prototype evaluation with students of their university

indicated acceptance of the prototype.

Similar applications like our system are

"GeoGebra" (Hohenwarter, Fuchs, 2004) and

"Cinderella" (Kortenkamp, Richter-Gebert, 2002).

However the interaction is more complicated

compared to our system and we strongly believe that

integrated exercises are a very important feature for

learners for self-assessment.

(Iijima 2012) presents a math application based

on HTML5, which hence runs on many output

devices. Therefore popular devices for students as

tablets can be supported as well.

Even though the systems above reflect different

aspects of the field of interactive multitouch math

learning applications there is not yet a clear picture,

which features must be integrated in order to build a

successful application.

3 THE APPLICATION

Our multitouch application consists of a display area,

where the function graph is displayed (see fig. 1 right

part), an equation area, where the equation is

displayed (see fig. 1 left upper part) and an exercise

area, where the exercises are displayed (see fig. 1 left

lower corner).

The students can create mathematical objects by

drawing the shape of the object with their fingers on

the table. When the object is recognized, the

belonging equation is displayed as well. Afterwards

they can move the objects by touching and dragging

and observe how the parameters in the equation

change in real time. Doing so, they can explore the

topics and try to detect the interrelation on their own.

The exercises serve as a means for the students to

check on their own if the concepts have been

understood. The use of a traffic light symbol tells

them if the exercise has been completely, partially or

not at all completed correctly.

Figure 1: The user interface of the multitouch application.

Suremath - User Study and Related (Re-)Implementation of a Multitouch Application for Learning Math

199

4 STUDY DESIGN

In this section we will quickly sketch the design of

our user study and the workshops with a focus group

we have carried through.

4.1 Study Method and Participants

For the study design we have selected 9 pupils from a

local high school for girls at the age between 13 and

14 years. The topics we have selected for the study

were about quadratic equations and parabolas. The

topics have not been taught at school yet. Therefore,

all pupils have were new to the topics that were part

of the study.

Our hypotheses were that:

(A) Pupils will be able to solve the exercises

faster with the multitouch application

compared to pen and paper

(B) Pupils gain a better understanding of the

topic so that they perform better at the pen

and paper exercise compared to the group

that did not work with our application before

the test.

The study started with a short introduction to the

field, explained by one of their teachers.

Then the group has been split up into two groups,

A and B. Both groups had to perform similar tasks.

However in the first round, group A started to work

on an exercise concerning moving a parabola along

the x-axis and determine the effects on the parameters

in the equation and vice versa, using pencil and paper,

whereas group B used the multitouch table

application for this task.

Then the groups switched roles and continued to

work on the same topic. The table group used pencil

and paper and the pencil and paper group used the

application. For the second round, we have focussed

on the movement along the y-axis. This time, group

A started with pencil and paper, whereas group B

used the table. Then they switched their roles again as

in the first round. Each round consisted of 6 exercises,

3 exercises which had to be performed with pencil

and paper and 3 exercises using the multitouch

application.

Exercises looked like this:

The quadratic function f has the form

f(x) = x

2

+ r (1)

Chose a value for r in a way that f has

Two x-intercepts: r= (2)

Only one x-intercept: r= (3)

No x-intercepts: r= (4)

The groups had 20 minutes for the exercises,

independent of the working method.

After completing all exercises, we had handed out

a questionnaire asking their opinion about the

application.

At the end, we had a short discussion with the

whole group.

4.2 Focus Group Workshops

The workshops consisted of a focus group of pupils

from another high school. They all were part of a

STEM special interest course. The number of pupils

varied during the several meetings between 6 and 11.

Their age was between 15 and 17, both male and

female. All of them were familiar with the

mathematical background.

The kick-off of the workshops has been an

introduction into our system. Afterwards the pupils

have met during about 9 months for open discussions.

Altogether, there were approximately 6 general

meetings with all participants. The goal of the

workshops has been to suggest further features, that

the students would find useful for an interactive math

application, having in mind their recent learning

experience.

5 RESULTS OF THE STUDY

Whereas the user study tried to find a hint to support

our hypotheses. Especially we were keen to know if

the pupils perform better, when they use an

interactive application compared to a group with

pencil and paper.

The focus group workshop was targeted at

improving the interactive math application in general

in order to improve its acceptance among pupils in the

classroom.

5.1 Findings of the User Study

As it concerns the results of the exercises, we could

verify our first hypothesis (A): The multitouch group

was already finished with their exercises mostly in

less than half of the available time, whereas the pen

and paper group needed almost the whole time slot.

Concerning hypothesis (B) there were no clear

findings if the pencil and paper group performed

better or worse than the multitouch group. The pen

and paper groups even performed slightly better in

our study. However, the small number of participants

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200

and the small number of exercises is a weakness of

this study. Hence, we do not see a clear hint that one

of the methods would be better for learning the

concept than the other.

We made several interesting discoveries, when we

met after the exercise to discuss the application.

The teacher made a comment that she could

introduce the concept of the quadratic equation and

the displacement of the function graph in half an hour

compared to a week (about 4 teaching hours) using

the traditional way. This comment points toward a

high potential of the application for the use in a

classroom from the teacher’s point of view.

Afterwards she encouraged the pupils to use the

application for further experimentation and asked

more in depth question. Although the pupils have

performed very well in both exercises, they had not

yet profoundly acquired the underlying concepts. For

several questions, they selected wrong parameters

(especially the wrong sign). Experimenting with the

application, they quickly discovered their mistake and

corrected the parameters. This observation together

with the fact that the pupils executed the exercises

much quicker using the application led us to the belief

that the use of a careful didactic instruction and

especially careful selection of exercises is needed.

Learners may be entrapped by the application to make

faster but not always carefully elaborated decisions.

Further results have been discovered by

evaluating the questionnaires. On a 7 point likert scale

the students had to answer several questions, where

“1” was “strongly agree” and “7” was “strongly

disagree”. In the following, we highlight the most

interesting results. In general, there was a great

acceptance of the multitouch application and the

worst results for individual questions has been an

average of 2.44.

We had asked questions concerning the topics

ease of use, work effort, maturity of the hard- and

software, usage scenarios (basic introduction,

repetition, in-depth studies) and general opinion.

In the following, we mention only some selected

answers that we found the most useful.

The question “The interaction with the table was

obvious” has been rated 1.0. Asking if “The

application motivates to deepen the topics

autonomous” has been rated 2.11. “The critical

examination of the topics is stimulated” scores 2.33

and the question “The examples of the multitouch

application improve my interest in the topic” scores

with 1.66. Most students were convinced that the

application helps them to determine their individual

learning pace (1.55). The pupils wanted to use the

application to get an overview of the related topic

(1.67), acquire basic knowledge (1.62), evaluate

theoretical knowledge (1.56) and repeat and deepen

(1.67) the content. In general the pupils felt well

prepared for the math course at school using the

application (1.72). Summing up they gave a grade of

1.67 for the application and they would recommend

the application to their fellow students with a value of

1.22.

There was an additional possibility to write free-

form text concerning advantages and drawbacks of

the application. As advantages the students

mentioned at first “Having more fun to learn the

topics”, which has been mentioned 5 times. Working

with the multitouch application means spending less

amount of work for the same result has been

mentioned 5 times as well. Three positive comments

relate to the integrated exercises. Two utterances

relate to the positive experience with experimental

learning.

On the other site the pupils mentioned as

drawbacks that the output device is not well suited for

their learning environment (too expensive, too hard to

transport, limited number of users at the same time).

4 comments related to this point. One student pointed

out, that the tool does not replace the explanation of a

teacher.

5.2 Results of the Workshops

The participants of the focus group analysed the

multitouch application and elaborated several topics

in their meetings concerning different parts of the

application and the learning process with the

application in general. The following sections sketch

their proposals.

5.2.1 Support of the Introduction Phase

There should be an introduction into the specific

topics, which is integrated in the application in a way

that the topics are self-contained and do not need an

external explanation.

5.2.2 Different Learning Modi

There should be different ways for working with the

exercises. A learning mode could be used to focus on

specific subjects only.

Assigning difficulty levels to exercises should

help students to deepen their knowledge in a way that

only exercises related to a specific level will be

selected.

An exam training mode should present different

questions of the topic field, possibly also varying in

Suremath - User Study and Related (Re-)Implementation of a Multitouch Application for Learning Math

201

the difficulty.

A self- assessment mode should only go from one

topic to another if a certain number of exercises has

been solved correctly.

Furthermore the participants suggested to

incorporate some kind of gamification e.g. using high

scores so that pupils of a class could compete against

each other.

5.2.3 Misconception Detection

The application should be able to discover a

misconception and to actively help the learner to

overcome the error. E.g. if a student drags the

parabola in the wrong direction, the application

should detect this (common) mistake and remind the

user to rethink her approach and possibly give her

relevant information that is needed in this situation.

6 OPERATIONALIZATION OF

THE RESULTS

The results of the study and the workshops require to

integrate certain concepts that are not part of the

system yet.

As it concerns the exercises, right now they are

“hard-wired” into our application. In order to

guarantee a high amount of exercises to make the

application attractive to use, it is necessary to provide

an authoring tool which can be easily used by teachers

and pupils. Hence there must be a possibility to load

the exercises into the system at run-time.

We have decided to integrate a web based

authoring tool and to store the exercises on a server in

order to build-up a pool of exercises that everybody

can use.

A further requirement was the dynamic

compilation of exercises.

Therefore each exercise can be associated with

metadata, which will be used form the system to

select exercises at run-time. Accordingly different

selection criteria have to be integrated into the

application for supporting the different learning modi

(Section 5.2.2).

Concerning the remarks about the poor

availability of the system in schools, we have decided

to port the implementation to a platform that supports

several clients, which are more commonly available

at schools, like whiteboards, PCs or tablets (Section

5.1).

We are about to make it possible to display text,

sound and animation. This enables possibilities for

the introduction phase (Section 5.2.1), like explaining

mathematical concepts and instructions how to work

with the system. Especially concerning the

animations, we will offer the possibility to control the

display of graphs, formula text and sound using a

simple scripting language.

Finally, the detection of mistakes (Section 5.2.3)

is supported by the system through attaching events

to the mathematical objects. If a user has to move a

parabola to the right in order to correctly solve the

exercise, it is possible to attach an event to the

parabola so that moving the parabola to the left will

trigger an event displaying hints or pointing again to

the theoretical foundations.

7 SYSTEM DESIGN

The program is implemented in the C# language,

using the Monogame framework (Monogame 2016).

This framework evolved from Microsofts XNA

Game Studio (Microsoft 2016) and can support

multiple operating systems like Microsoft Windows,

Linux, OS X, Android, iOS and more. For

implementing applications for Android and iOS, the

Mono implementation from Xamarin (Xamarin 2016)

is necessary. The programming is mostly done loop

based like a video game rather than event based. This

allows easier programming and smoother handling

like moving the function graphs on the drawing area.

Each frame, the input is read and the program reacts

on changes.

7.1 Input

Suremath has input handling for mouse, touch and

pixelsense (SUR40 (Samsung 2016). Input from most

interactive whiteboards is possible as mouse input

simulation. As long as the input is not the mouse

device (or its emulation) the program can handle

multitouch.

7.2 Gesture Recognition

The recognition of the mathematical functions drawn

on the grid is realized using the Protractor algorithm

(Li 2010). This algorithm is a geometric template

matcher that is based on the angular distance between

the template and the drawn input. This allows very

easy and fast recognition, but some additional work

on the templates is needed. For example a normal

parabola (5) and one with a negative leading

coefficient (6) have different angular sums, and hence

need two separate templates.

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202

Figure 2: Formula creation using the authoring tool.

y = x

2

(5)

y = -x

2

(6)

However the algorithm is so flexible that new

functions can be recognized by simply adding the

corresponding templates. The templates are stored in

an XML file. This guarantees a user friendly way to

easily extend the set of gestures.

7.3 UI Elements

Since XNA/Monogame does not have any GUI

elements included, all UI elements like windows,

buttons, slider or textboxes have to be implemented

separately. In our implementation the required images

are generated programmatically on the fly.

7.4 Exercises

Our authoring tool allows to create exercises and to

store them on a server. The exercises can be retrieved

over the internet by entering a code in an input field

on the exercises menu. A set of several exercises is

stored in a ZIP file that contains a SQLite database

file together with the metadata of the exercises and

the images that contain the instructions. The use of

images instead of text is necessary since complex

formulas have to be displayed.

Exercises can have attached events, text or sound

hints and animations. Giving an example, it is

possible to add a text hint that is shown when a

parabola is moved to the left instead of to the right.

This allows the creation of self-explaining exercises.

Furthermore the integration of sound and

animation can be used for an introduction into the

subject. Using evented math objects it is even

possible to integrate a mixed initiative introduction

where the user is required to take some action before

an animation continues.

7.5 Authoring Tool

The authoring tool is a web-based tool, which is

integrated in a web page. It allows creating exercises

as a static and/or dynamic set of exercises. Static sets

are a fixed list of exercises, created by the author.

These sets are presented to the learner in the sequence

as they have been created. Dynamic exercise sets can

be compiled on the fly, based on metadata like

difficulty, or topic.

The creation of the individual exercises in the web

page is done using a very simple WYSIWIG LaTeX

editor, based on tokens. Therefore it is possible to

create professionally looking instructions for

exercises without the need of knowing LaTeX or any

other language to render equations.

Every token has two icons on the right hand side

(see fig 2.).

A little “x” in a circle, used to delete this token.

A “+” in a circle adds a new token right to this

token.

This tool uses conventional web technologies like

Javascript and jQuery on the client side. For the user

interface jQueryUI is used. The rendering of the

equations is done with MathJax.

On the server-side asp.net and LaTeX are used.

The application runs on a Linux system to render the

text into images and create ZIP files and SQLite

databases for deploying exercise-sets to the clients.

8 SUMMARY

In this paper, we have presented the evaluation of our

multitouch application for learning math and the

improved implementation based on the results we

have gathered through our evaluation with pupils and

workshops with a focus group.

First we have described the setup of our study and

the approach for the workshops.

In the user study, we could show that learners can

work faster with our application, compared to

working with pen and paper.

Based on the findings of the comments from the

user study and the remarks from the focus group, we

have formulated the belonging features and changes

of our application. These changes had to be integrated

in order to support the learner and to improve the

Suremath - User Study and Related (Re-)Implementation of a Multitouch Application for Learning Math

203

acceptance and usability of our system. These were

concepts like providing a kind of scripting to create

an introduction to the domain. We have introduced

meta data, e.g. in form of difficulty levels and

different learning modes like competition, exam

training etc.. Additionally we have introduced events

to react flexibly on the user interaction.

Finally, we have shortly sketched our system (re)

implementation, based on a video game based

concept with an independent authoring tool, which is

browser based.

In the future, we want to extend the use of our

system. We have it already installed in one of the high

schools running on a whiteboard for being used in the

classroom. We also plan to make the Android version

available in the app store soon. We hope to receive

much more input from pupils and teachers over this

channel. Additionally we want to carry through a

more profound user study with our improved system.

As a further step, we plan to extend the

functionality of our application to fields with similar

challenges, e.g. trigonometric functions. More

specifically, we want to extend the system to

exercises of the type: “At a point 15 feet from the base

of a church, the angle of elevation of the top of the

church is 43°. Find the height of the church to the

nearest foot.” We imagine that moving objects back

and forth and changing the angle of lines interactively

while displaying the values of angles, lengths and

trigonometric functions can help to grasp the concept

behind relatively abstract functions like sine or

cosine.

ACKNOWLEDGEMENTS

We especially want to express our gratitude to the

Angela Merici Gymnasium Trier, Germany, for

supporting our research, especially Miss Daniela

Kiefer and her pupils who participated in the user

study. Furthermore we want to thank the Auguste-

Viktoria-Gymnasium, Trier, Germany especially

Miss Karin Brezina and Miss Anne Bläsius and her

pupils who supported our work by participating in our

workshops and still continue working with us on this

topic.

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