Improving the Attitude towards Mathematics via an ICT

Rearrangement of the 8

Grade Math Curriculum

Borislav Lazarov

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Blok 8 Georgi Bonchev Str., Sofia, Bulgaria

Keywords: Attitude towards Mathematics, Dynamic Geometry Software, Synthetic Competence, Individual Educational

Trajectory.

Abstract: Under consideration is an experimental teaching in mathematics of an eight-grade high abilities student,

whose attitude towards mathematics initially was negative. The educational goal includes developing

student’s competence of synthetic type, i.e. a package of knowledge, skills and attitudes, which are

multifunctional and transferable. The European framework of key-competences is chosen to design an

individual educational trajectory (IET). An original didactic model is adopted as theoretical base for this IET.

The main issue in the IET was how to change the student’s attitude towards mathematics to positive. For

resolving this issue, a rearrangement of the syllabus is done by incorporating ICT. A large part of the routine

paper-and-pencil drills was replaced with dynamic-geometry-software exercises. Parallel to the compulsory

lessons in math, the IET included two project-oriented initiatives. The goal of these initiatives were to

reinforce the synthesis of the student’s analytical knowledge and skills, which were built separately in math,

ICT and arts. On this stage of the IET, the informal learning served as an accelerator in turning attitude in

positive direction. At the final stage of the experimental teaching, the student covers the general standard for

positive attitude towards mathematics.

1 INTRODUCTION

The concept of competence in European Commission

understanding consist of 3 components: knowledge,

skills and attitude (EU, 2006). The development of

knowledge and skills are parts of any math

curriculum in Bulgaria. There is a long tradition in

teaching any particular math topic in secondary

school and valid didactic methods are approved for

different age groups (Ganchev and Kuchinov, 1996).

The modern textbooks follow these methods and the

teachers are more or less prepared to apply them into

school practice. However, the recent international

assessments pointed on a general drop in performance

of the Bulgarian secondary school students in math

(PISA, 2016). It appears that Bulgarian eighth-

graders are not competent to cover some international

educational standards. Therefore, if the knowledge-

skills part of the education is still in a framework that

assures good results in the past, then maybe the

problem is in the students’ attitude.

We do not know examples of Bulgarian good

practice on improving the attitude toward

mathematics among eighth grade students. Outside

Bulgaria (where the European framework was

adopted) the attitude is related to high achievement,

or motivation, or engagement, which is not the case

we are interested in (Blackweir, 2016, p.16-17).

Therefore, we constituted a teaching experiment to

look for some possible stimuli for an eight-grader to

study math having in mind the Bulgarian educational

context.

The eighth grade mathematics curriculum in

Bulgarian secondary school is crucial in forming

students’ deductive abilities. The main didactical

instrument for this mission is the geometry part that

upgrades the axiomatic approach introduced in

seventh grade by 4 modules: vectors, rigid motions,

circle and geometrical constructions. Every module is

elaborated in a traditional mode lessons: introductory

example, definitions, theorems, drills. The students’

activities are supposed to be performed by hand

drawing and calculating, as well as deductive

reasoning based on a carcass of statements of

Euclidean geometry. We advocate the need of

renovation this traditional approach.

The experimental teaching described below

presents an attempt to reform the teaching method in

eighth grade math by introducing some computer

Lazarov, B.

Improving the Attitude towards Mathematics via an ICT Rearrangement of the 8th Grade Math Curriculum.

DOI: 10.5220/0007744203910397

In Proceedings of the 11th International Conference on Computer Supported Education (CSEDU 2019), pages 391-397

ISBN: 978-989-758-367-4

391

applications, mainly dynamic geometry software

(DGS), but also computer algebra system (CAS). As

we will show, this reform went far beyond the simply

technical issues and caused deep rethinking of the

bases.

2 THE INITIAL CONDITIONS

The student V started the eighth grade mathematics

classes declaring she hate math. V was studious and

she was doing regularly the obligatory math

assignments but without any enthusiasm. The most

unpleasant for her were the routine work:

calculations, algebraic transformations etc. During

the first term, the V’s test results were mediocre that

contrasts her high intellectual abilities in general.

Such status quo was unacceptable and the need for

change was obvious. In this connection, the following

questions were top of the agenda:

1) how the attitude towards mathematics of a student

of high potential abilities could be changed to

positive;

2) what part of the curriculum should by reorganized.

Therefor we designed a midterm individual

educational trajectory (IET). By IET we will

understand the organizational framework and plan

for realization of a medium term educational process

that are coherent with the individual specifics of the

learner and provides opportunities for the optimal

development of his/hers creative potential (Lazarov,

2012).

3 THE IET: A BIT OF THEORY

We summarized our previous experience in tutoring

advanced students in (Lazarov 2013A), where we

proposed a didactic model (called DMT) for

designing an IET. The movement along the IET

follows a kind of spiral. However, it is more

convenient to decompose this spiral into two

directions: vertical and horizontal. The vertical

direction is an iterative procedure that includes the

following components.

1) Formation of an individual informational

environment.

2) Individualization of the didactical resources,

including selection of the individual

(re)searching instruments.

3) Individualization of setting the educational goal,

including flexible approach to achieve it.

4) Individualization of the learning temps,

investigation activities, layout style.

5) Taking into account the individual reflexive

abilities and self-organization aptitude in

searching a synergetic effect. [ibid]

The DMT allows visualizing the iterations of an

IET as climbing floors. Here is the list of the steps to

be done in the k-th floor of the IET (the program

maximum).

…(k – 1) →

 A proximal educational goal (of learning,

investigative or research type) is mapped out

with respect to the actual knowledge, skills and

attitude (KSA) of the student (actual

development).

 A (very limited) informational resource is

determined which is focused on the goal.

 All needed activities to extend the actual KSA

to a level required for reaching the stated goal

are performed.

 Student proceeds to the goal in his own manner.

 Analysis of the achievements (tutor

responsibility).

→ (k+1) …

The horizontal direction connects the iterations in

a specific way for which we will give details in the

next section.

4 IMPLEMENTATION OF THE

THEORY

The traditional style math teaching requires just a

white-board and some posters visualizing basic

geometry shapes and math formulae. The students’

classroom activities corresponding to such teaching

are related mainly to problem solving by paper-and-

pencil techniques. Therefore, the first step in our plan

was to expand the classroom equipment by computers

with DGS. We designed for V an IET, based on

implementation of DGS.

4.1 The Ground Floor of V’s IET

Initially the informational environment was limited to

the textbook (Paskaleva et al., 2013). Following the

syllabus, the linear simultaneous equations stood on

agenda. This was the right moment to start our plan

for overcoming V’s negative attitude to math. Among

the didactical tools we selected was learning by

syntax (Lazarov and Karakoleva, 2011). The correct

usage of the DGS requires from V a kind of

mathematics competence related to basic concepts of

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the analytical geometry: Cartesian coordinates, line-

equation correspondence, graphic interpretation of

simultaneous equations etc. Studying the DGS

interface V realized the need to learn the math

concepts deeply. The benefits came quickly: the

paper-and-pencil solution of simultaneous equations

is written on several rows; the DGS solution is limited

to entering the equations and interpretation of the

picture. Let us emphasize that we passed the

obligatory content in full scale to cover knowledge-

and-skills standards. V did just a part of the routine as

a DGS activity.

One can see the V’s uncertain first attempt in

using DGS in Figure 1. The genuine problem was to

solve graphically the simultaneous equations x+2y=3,

y+x/2=0. There are expendable elements as vertical

lines that appear in searching points of intersection

that actually do not exist.

Figure 1: V’s first attempts in applying DGS.

On this floor, V made successfully the first steps

in overcoming the routine calculations by application

of CAS.

About the attitude component of the competence,

our proximal educational goal was related with the

first two points of the Mathematical literacy key-

competence, as well as points ii. and iii. of the

Learning to learn key-competence (see the

Appendix). So, our program maximum at this stage

was fulfilled: V started dealing math with no bad

feelings.

4.2 The Second Floor of V’s IET

The next big step that V made was related to geometry

lessons: the rigid motions. The chapter in the textbook

started with a discussion about how to imagine a

plane that can move onto itself. Here the advantages

of the DGS versus paper-and-pencil style are

indisputable. No imagination was needed in general

but one can see the matter of a particular rigid motion

just dealing with DGS instruments to perform

translation, rotation, or symmetry.

V and a classmate of her started a project about

congruent shapes in real life environment. They

captured images of buildings, electric lighting poles,

street tessellation etc. and classified the repeating

shapes according to the type of the rigid motion that

could be recognized. For instance, the front view of

some buildings repeat the same form of the floors;

hence, they could be obtained by translation about a

vertical vector. Our educational goal in its attitude

part referred to points i., iv., and vi. of the key-

competence Learning to learn (see the Appendix).

We will skip more details of that stage of V’s IET

just mentioning that V’s attitude towards math ceased

being negative. Such change was probably caused by

the decontextualization of the math theory connecting

it with the everyday student’s experience.

4.3 The Third Floor of V’s IET

The project-oriented education was a part of the

school strategy at that time (Lazarov, 2018).

Moreover, students like working on educational

projects and we catch this opportunity to apply

integrated approach. V was no exception. The project

about the rigid motion in the city environment lifted

the V’s enthusiasm.

‘Rigid motion’ in Bulgarian translation is

‘ednakvost’ (pl. ednakvosti). Ednakvost has an

everyday meaning, which is slightly different from

the mathematical concept. It is close to ‘mapping a

figure onto congruent figure’ (figuratively, copy-

pasting). Here the ‘rigid’ part is not obligatory that

allows extending the math concept beyond the

geometry. In the following, we will use the word

ednakvost for expanding the meaning of the concept

of rigid motion beyond geometry.

So, we started another project on ednakvosti, this

time about figures in art. Our idea was to integrate

with math as much art as possible. After consulting

the music teacher, who suggested to consider the

music of Ravel’s bolero, we decided to go further and

to analyze the Maya Plisetskaya’s bolero dance

(YouTube, 2016). In our opinion, this movie is a pure

synthesis of music, dance, and filming. The project

idea was to observe how the show is composed of

figures in music and dance that are multiplied through

different ednakvosti.

The rhythmic part of Ravel’s bolero repeated 168

times the initial figure. V recognized here translation

(Figure 2). Another translation V observed in the two

Improving the Attitude towards Mathematics via an ICT Rearrangement of the 8th Grade Math Curriculum

393

Figure 2: Translation in the rhythmic section (a slide from

V’s presentation).

melodies, which are on top of this rhythm (Figure 3).

The second translation moves the musical figure from

flute to clarinet, then to bassoon, and so on. This was

the first coordinate of the translation vector. The

second coordinate is in the loudness, constantly rising

from pianissimo to fortissimo.

Figure 3: Translation in the melody (a slide from V’s

presentation).

Another ednakvoti V saw in the Maya

Plisetskaya’s dance. She recognized symmetry and

rotation. V managed to cut the YouTube clip into

short parts, which illustrated the corresponding

ednakvost. For this purpose, V found the free

software WEVIDEO in the www, learned by herself

how it works and applied it to a presentation. Let us

point that the filming of the dance emphasized the

dance figures in a very suitable manner for the project

goals. V caught this opportunity. E.g. she splitted one

of the movements into symmetrical origin and image,

which are shown together in the next movement that

completes the symmetry (ibid., 0:22-1:27); similarly,

V extracted rotation (ibid., 8:04-8:15).

V presented the project to her teachers and

schoolmates. At the end of the presentation, after the

final accords of bolero, she declared ‘ednakvosti are

everywhere’. This we accepted as the turning point

in change V’s attitude towards math to positive.

However, we consider the attitude in a synthesis with

some attitude components from other key-

competences, e.g. Cultural awareness and ICT ones

(Appendix).

4.4 The Top Floor of V’s IET

The time to harvest the fruits from the V’s

metamorphosis came with the curriculum unit about

Geometrical constructions. However, the application

of DGS in standard tasks requires deep revision of the

method. The ruler-and-compass axiomatic is not

coherent with DGS instruments. The conflict points

are discussed in details by Lazarov (2011).

Another issue relates the reasoning. Here two

points stand on agenda:

1) the traditional proof that the construction algorithm

produces the proper figure can be replaced by a check

for dynamical steadfastness of the DGS construction;

2) the conditions under which a figure exists can be

replaced by examination of the initial data.

4.4.1 The First Point

We illustrate the reasoning instead the proof by the

example in Figure 4. The theorem states that the

symmetric point of the orthocenter of a triangle about

any side lies on the circumcircle.

Figure 4: V’s applet illustrates the property of the

symmetric point of the orthocenter of a triangle.

The V’s applet allows checking different type of

triangles: acute, obtuse, right. The symmetric point

D’ of the orthocenter D of triangle ABC about the line

BC lies on the circumcircle, every time when any of

the vertices A, B, C changes its location. Let us point

that the rigor proof is not easy even for some math

teachers.

4.4.2 The Second Point

The analysis of existence of a figure we illustrate by

Figure 5 where V constructed isosceles triangle

having given base a and the circumradius r.

She introduces two sliders: one for the base a,

another for the radius r. The dynamic construction

allows examining the existence of a triangle with the

given properties.

Let us explain that the change of the method does

not change the educational goal of the topic Geomet-

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Figure 5: Construction of an isosceles triangle with given

base and circumradius.

rical constructions. The replacement of the axiomatic

of ruler-and-compass by the DGS operators develops

the same kind of algorithmic thinking. Moreover, the

computer-based application of the theory is a solid

ground to cultivate some designer abilities. E.g., V

made just for fun an applet for topological

transformation of a dog’s face into a cat’s face and

vice versa.

At this stage, we emphasize the last three points

of math key-competence, again in synthesis with key-

competence Learning to learn (Appendix).

5 DEVELOPING COMPETENCE

ALONG THE IET

The DMT describes in general a hypothetic

movement inside the floors of an IET (Lazarov,

2013). However, the details are rather specific for any

particular IET. Bulgarian educational legislation

operates with the category of key competences

(MON, 2015), while PISA framework takes

mathematical literacy as the basic construct (PISA,

2018). Our standing point differs from both cases:

building up a kind of synthetic competence is what

should be the educational goal in the secondary

school (Lazarov, 2013B).

The synthetic competence is a construct for the

methodology when designing IET, but it becomes a

personal attribute for any individual as an outcome of

the IET. Let us emphasize that the synthetic

competence is context sensitive. This is why the

educational context should be challenging and to

include stimuli of synthetic type. In our case, some of

the stimuli were the two project-oriented initiatives

on ednakvosti (remember 4.2. and 4.3.). Other stimuli

were set during the discussions on extracurricular

topics as the nine-point circle of a triangle, Euler’s

line, etc. Any time the DGS illustrations were done

by V. As result of her activity an archive of DGS

applets were created.

The knowledge and skills built on any floor of V’s

IET were tested by implementation in DGS activities

that assured a kind of transferability and

multifunktionality of knowledge and skills. Parallel

to this the V’s attitude towards mathematics lifted on

any floor (in our opinion). Therefore, we can speak

for a development of V’s knowledge-skills-attitude

package in a context that is different from the one, in

which the KSA package was elaborated. In other

words, the V’s synthetic competence was upgraded at

any floor of her IET.

Formation of the V’s individual informational

environment included web-based resources, e.g. the

paper about Ravel’s bolero (WikipediA, 2018). V was

not restricted in copy-pasting some paragraphs from

internet papers, but the citation ethics was strictly

under control. We agree with Petrovic (2018) about

the benefits of informal learning even in such

reproductive form. Moreover, we consider this type

of informal learning as an opportunity to

decontextualize the classroom knowledge. However,

the key-competence ICT attitude points (Appendix)

must be taken into account in any copy-paste

activities.

6 CONCLUDING WORDS

The Bulgarian secondary school theoretical

framework adopts explicitly the key-competences

from the European one (EU, 2006). However, the

responsibility for development of any key-

competence is spread all over the subjects and vice

versa: any subject covers several key-competences.

E.g. any of the themes Plane figures and solid bodies,

Probability and Functions covers 7 of 8 key-

competences (MON, 2015) (in fact there are 9 key-

competences in Bulgarian educational legislation).

Implicitly this means that a kind of synthetic

competence is in the scope of the compulsory

education, which is coherent with our standing point

about one competence of synthetic type.

Further, the attitude as a component of the

synthetic competence inherits the synthetic nature of

the competence. This was clearly seen during our

experimental teaching.

The concept of attitude is operationalized more

comprehensive in Bulgarian legislation (MON, 2015)

than the points we cite in the Appendix. However, the

existing instruments to measure the student’s attitude

in quantitative way, which we know, do not

correspond to the individual approach we applied.

E.g., take the attitude point 3.1.iii and think about

what it means to register 30% higher respect for truth.

The common instrument for this purpose is an inquiry

of the type given by Zakariya (2017, p. 82).

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On the other hand, a qualitative evaluation is

possible and we observed (indirectly) rather positive

lift of V’s attitude towards math. Taking into account

that the V’s math test results are related mainly to the

knowledge and skills side of her learning process, we

would like to think that the higher score she achieved

at the end of the school year is also influenced by the

positive attitude she get during the experimental

teaching presented here.

ACKNOWLEDGEMENTS

The study is supported by the Educational and

Research Program Cherorizec Hrabar of the Institute

of Mathematica and Informatics, Bulgarian Academy

of Sciences. The author thanks the reviewers for the

comments.

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APPENDIX

The attitude component of Mathematics key-

competence in EUROPEAN COMMISSION, 2018.

ANNEX to the Proposal for a Council

Recommendation on Key Competences for Lifelong

Learning (page C 189/9) is defined as A positive

attitude in mathematics is based on the respect for

truth and a willingness to look for reasons and to

assess their validity. (https://eur-lex.europa.eu/legal-

content/EN/TXT/PDF/?uri=CELEX:32018H0604(0

1)&rid=7)

Below is an extraction from the communication Key

Competences in the Knowledge Based Society – A

framework of eight key competences. European

Commission (Sept. 2004). The quotations are taken

from the column ATTITUDES of the table on pages

5-13. Here the key-competence number corresponds

the source numbering.

3.1. Key-competence Mathematical literacy

i. overcoming ‘fear of numbers’;

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ii. willingness to use numerical computation in

order to solve problems in the course of

day-to-day work and domestic life;

iii. respect for truth;

iv. willingness to look for reasons to support one’s

assertions;

v. willingness to accept or reject the opinions of

others on the basis of valid (or invalid) reasons

or proofs.

4. Key-competence ICT (digital competence)

i. desire critically to assess information available;

ii. awareness that the lower threshold to access

information may need to be balanced by

higher standards of ethics and taste –

ability to distinguish what is ‘accessible’

from what is ‘acceptable’.

5. Key-competence Learning to learn

i. adaptability and flexibility;

ii. self-motivation and confidence in one’s

capability to succeed;

iii. a self-concept that upholds one’s willingness to

change and further develop competences;

iv. sense of initiative (to learn);

v. positive appreciation of learning as a life-

enriching activity.

8. Key-competence Cultural awareness

The term ‘cultural awareness’ comprises an

appreciation of popular culture and general social

mores, as well as the ability to appreciate

literature, art, music and other forms of creative

expression.

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