Computer Supported Evolution Inside Van Hiele Levels 1 and 2

Borislav Lazarov

and Rumyana Papancheva

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Soﬁa, Bulgaria

Faculty of Social Sciences, University ”Prof. Dr Asen Zlatarov”, Burgas, Bulgaria

Keywords:

Pre-deductive Phase, Van Hiele Levels, Communicative Abilities.

Abstract:

The goal of the study is to clarify the communicative abilities of 5th grade students related to the computer

supported geometry education. Theoretical frame is the Van Hieles’ model. The experimental teaching gives

an idea of how the language characteristics of Level 1 and 2 could be improved by a short-term game-like math

education. Some coding-decoding activities make students to be more accurate in written communications.

1 A MODERN VECTOR IN MATH

EDUCATION

In the last 30 years ICT enhanced math education be-

came routine practice in secondary school. It was

based mainly on the development of computer al-

gebra systems (CAS) and dynamic geometry soft-

ware (DGS). Key role in this trend plays the so-called

inquiry-based approach (Rocard et al., 2007) which

is a kind of a modern Socratic style (Lazarov, 2014).

But in 2007 there appeared a paper where the role of

the mathematics education in secondary school was

reconsidered (Haapasalo, 2007). It is a matter of fact

that the advanced usage of ICT refers to some speciﬁc

communication skills and applications of mathemat-

ics methods. No other school subject than mathemat-

ics can face better these demands of life, so the vector

of the secondary school math education should con-

tain components that meet the ICT needs of a modern

individual.

Our recent research shows that high school stu-

dents who use dynamic geometry software (DGS) in

studying mathematics developed intuitively speciﬁc

communication skills (Lazarov, 2015). Their writ-

ten math slang includes synthetic symbols and icons

(like dynamic pictures and graphs) parallel to the tra-

ditional formulas and shorthands. This slang evolves

along with the educational process and reﬂects the

level of student’s geometrical reasoning. Our practice

clearly shows that the constructing of DGS applets re-

quires a student to have reached at least Van Hiele

Level 3

. In fact any dynamically stable construction

We are going to make a lot of references to the ﬁrst

(i.e. such that preserves the geometrical properties of

the objects after some transformations) is made fol-

lowing an algorithm reﬂecting the properties of the

ﬁgures. The design of such algorithm requires student

to apply at least short deductive chains using the DGS

syntax. Our experience conﬁrms that the proper us-

age of DGS syntax needs a long training ’within the

following categories concerning what modern tech-

nology can maintain and promote:

(1) Links between conceptual and procedural knowl-

edge,

(2) Metacognitions and problem-solving skills,

(3) Sustainable components of mathematics making,

(4) Interplay between systematic approaches and min-

imalist instruction,

(5) Learning by design’ (Haapasalo, 2007).

But following the Van Hieles’ theory, in order to be

at Level 3 a student should pass consecutively Lev-

els 1 and 2. So the question is what kind of math

activities will contribute to the development of ICT

skills in pre-deductive phase. We started exploring

the mathematics and informatics curriculum to ﬁnd

the most adequate starting point for introducing inte-

grated mathematics and IT approaches.

2 THE STATUS QUO IN

BULGARIA

The change in the teaching style that happens on

the borderline between primary and secondary school

three Van Hiele levels so we give a brief description of them

in an Appendix.

186

Lazarov, B. and Papancheva, R.

Computer Supported Evolution Inside Van Hiele Levels 1 and 2.

In Proceedings of the 8th International Conference on Computer Supported Education (CSEDU 2016) - Volume 2, pages 186-192

ISBN: 978-989-758-179-3

in Bulgaria (4th-5th grade) has several dimensions.

First of all, it considers the subject-oriented approach

which comes to substitute the (more or less) topic-

oriented mode of teaching in primary school. A sec-

ond important moment is the change of the teachers

who are engaged with a particular class. The related

subjects like mathematics and information technol-

ogy are already separate disciplines in the school plan,

sometimes covered by one teacher, but more often by

different persons. The math and IT syllabus are made

by different commissions at the Ministry of Education

which yields to relatively poor interrelated connec-

tions. The Bulgarian math syllabus up to 2015/2016

scholastic year does not provide additional space for

activities of mixed (conventional and IT based) type

in which the necessary skills for inquiry-based educa-

tion to be built. Such status quo makes it hard to take

advantages of IT in traditional math education and

vice versa. So the teachers and educators are search-

ing their own way to achieve integration of IT in math

education in the mode they think is most appropriate

for a particular target group.

Our way started with the target group of high abil-

ity senior secondary school students (Lazarov, 2014).

At this stage all students have reached the Van Hiele

Level 3, and some of them proceeded on Level 4. Stu-

dents’ knowledge, skills and attitude (KSA) are trans-

ferable from the conventional context of math educa-

tion to the DGS environment, therefore we can speak

about students’ competence of synthetic type (syn-

thetic competence). Further, we tried to apply similar

approach to intermediate secondary school students

with average abilities, i.e. incomplete Van Hiele Level

3. The results were far from satisfactory (Shabanova

& Lazarov, 2014). Only a small fraction of the target

group managed to transfer the math KSA into a new

context of DGS exploration. It became clear that the

foundation of the transferability should be established

somewhere in the early secondary school.

3 APPLICATION OF A

CLASSICAL MODEL

The Van Hiele model of learning geometry provides

a convenient base for interpreting and analyzing the

students’ levels of understanding. In parallel, this

model clariﬁes the way they form geometrical rea-

soning. Special role in the model plays the develop-

ment of the language in which students express their

knowledge, as far as each level has its own linguis-

tic symbols and own network of relationships con-

necting those symbols (Usiskin, 1982). For instance,

student’s progress from Level 1 to Level 2 yields a

signiﬁcant structuring of relationships and a reﬁne-

ment of concepts. Teacher should feel how such tran-

sition occurs and should tune his/her language for ad-

equate verbalization of the intuitive knowledge, be-

cause the verbalization goes together with a restruc-

turing of concepts. As we mentioned above, the nec-

essary level for meaningful use of DGS is Level 3,

so the concept restructuring must ﬁrst occur at Level

2 before students can start exploring the logical rela-

tionships needed for creating DGS applets.

The geometry education in Bulgarian 5th grade

is characterized with a signiﬁcant intensiﬁcation. As

evidence we point that the number of new geometri-

cal concepts in 5th grade is about 4 times bigger than

all geometrical concepts introduced in the previous 4

years. Moreover, many problems require a construc-

tion to be done and a solution to be written which

needs more developed language for communications

in both directions - understanding the statements and

composing statements. For instance, let us consider

the following problem:

Draw ∠POQ = 30

◦

and a point A on the ray OQ such

that OA = 10 cm. Find the distance from A to the ray

OP. (Lozanov et al., 2011)

Here students are expected to turn a description

into a picture before starting the solution. They

should draw an arbitrary ray, then measure the an-

gle (using protractor), then ﬁnd a speciﬁc point and

erect a perpendicular from it to a line (to do this, stu-

dents should know that the distance equals the length

of the perpendicular). The solution of the problem

is based on both conceptual and procedural knowl-

edge and skills. This two knowledge types seem to

be developed iteratively (Rittle-Johnson & Koedinger,

2004). In our opinion such activities are accessible for

Level 2 and up and could be built by integrating IT in

math education.

What follows refers to our research in lower sec-

ondary school (Bulgarian 5th grade), which supposes

the Van Hiele Level 2 (sometimes incomplete) has

been reached. We tried to manipulate students’ at-

titude towards elaborating more precise and reliable

communication style in learning mathematics by ap-

plying IT. We hope this approach will guarantee the

foundation of KSA which is necessary for the next

steps in building synthetic competence.

4 FRAMEWORK OF THE STUDY

The goal of our study was to clarify the communica-

tive abilities of 5th grade students related to the com-

puter supported geometry education. Designing our

experimental teaching we took into account the fol-

Computer Supported Evolution Inside Van Hiele Levels 1 and 2

187

lowing two requirements (Kadijevich, 2006):

(1) when utilize mathematics, don’t forget available

tool(s); when make use of tool, don’t forget the un-

derlying mathematics;

(2) to solve the assigned task, use, whenever possi-

ble, a process approach as well as an object approach,

working with different representations .

According to the above requirements, we chose the

MS Paint application with Basic shapes for the IT ac-

tivities. This application provides relevant resources

for Levels 1 and 2: ready-made shapes as isosceles

triangle, right-angled triangle, as well as three types

of transformations: rotation, stretching, dragging.

Another moment in the research design was the

choice of the topic. We recognize the unavoidable

usage of familiar everyday concepts on the ﬁrst two

levels, so we expected students’ descriptions to be in

the form of a meta-language (mixture of geometrical

and everyday life language, expanded with pictures

and shorthands). Students were assigned to sketch a

monster using some geometrical shapes. Among the

other educational reasons, this topic was selected be-

cause of the anthropomorphic and zoomorphic terms

that potentially could help the composite ﬁgures to be

properly depicted. Let us point the anthropomorphic

origin of some concepts in geometry like legs of a tri-

angle and trapezium in Bulgarian math language. The

idea of the topic came from a creative writing project,

proposed by Linda Yollis in her blog (Yollis, 2014).

The accent in her project work was on developing cre-

ative thinking and writing skills.

5 PARAMETERS OF THE STUDY

The target group was composed of 5th grade students

(no indication for additional interest in mathematics

among them). The experimental teaching lasted 6

academic hours distributed in the following manner:

– Diagnostic test.

– Math class. Revision of geometrical students’

knowledge.

– ICT class. Students created on computer their mon-

sters, constructed by different geometrical ﬁgures, us-

ing a simple computer graphic program - MS Paint.

– Language class. Writing a description of a picture.

Each student wrote a short description of his/her own

monster.

– ICT class. Students exchanged their descriptions

and based on the written texts they created a copy of

the original monsters on computer.

– Control test.

Students worked individually but grouped by pairs on

the next assignments:

(A1) A monster to be drawn using the following basic

shapes: rectangle (including square), triangle (isosce-

les, right), and circle.

(A2) The own monster to be described by words and

to be sent to a classmate for depicting.

(A3) The descriptions to be interchanged in the class-

mate pair, the other monster to be depicted following

only the description and to be compared with the orig-

inal.

During the lessons some comments and remarks on

the assignment were done. Students were pointed out

that the ﬁgure type is invariant when applying any of

the three transformations.

6 INDICATORS AND DATA

COLLECTION

We observed several indicators but some of them were

covered by all students (like classifying the general

type of a polygon or reconstructing an abstract ﬁgure

following a verbal description), so we took them away

from the analysis. The following indicators were used

to determine the initial Van Hiele level including the

degree of completeness inside the level:

(i1) recognizing the square and the rectangle no mat-

ter how it is oriented;

(i2) the square is considered as rectangle;

(i3) recognizing the type of a triangle (isosceles,

right);

(i4) combining two properties of triangle (isosceles

and right);

(i5) usage of geometrical properties of the ﬁgure in

description;

(i6) usage of elements of the ﬁgure like side and ver-

tex in depiction;

(i7) applying elements of the ﬁgure like side and ver-

tex in reconstruction;

(i8) reconstruction of abstract ﬁgure following verbal

description.

Indicators for determining the individual Van Hiele

level are based on the Burger-Shaughnessy opera-

tionalization (Burger & Shaughnessy, 1986), given in

the Appendix. Most of them refer to Level 2, but some

indicate Level 1, like (i1). Burger-Shaughnessy fea-

tures suppose a direct communications with students

to analyze the geometrical reasoning. Our approach

is oriented mostly to analyze the features of the stu-

dent’s written language at Levels 1 and 2. So we col-

lected data from the tests, students’ computer pictures

and written descriptions of their pictures. There were

22 pairs of students who took part in the experimental

teaching. Below we are going to present the details

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188

about 4 pairs that are representative for the most typ-

ical cases. The observed students are coded as PgA,

PpA, RsA, RkA, KoG, KrG, RaG, SaG.

7 STATISTICS

Table 1 shows the coverage of the indicators: 0 means

that an indicator is not covered and 1 stands for a cov-

ered indicator. Some indicators were partially cov-

ered, e.g. the corresponding test item is correct but in

the written material there were mistakes or gaps – we

scored these cases with 1.

Table 1: Coverage of the indicators.

i1 i2 i3 i4 i5 i6 i7 i8 Σ

PgA 1 1 1 1 1 0 0 1 6

PpA 1 1 1 0 0 0 0 0 3

RsA 1 1 1 1 1 1 1 1 8

RkA 1 0 1 1 1 0 0 0 4

KoG 1 0 1 1 1 0 0 1 5

KrG 0 0 1 0 0 0 0 0 1

RaG 0 0 0 0 0 0 0 0 0

SaG 1 0 1 1 1 0 0 1 5

We consider a total score Σ ≥ 4 as reaching Level

2, and Σ ≤ 3 as reaching Level 1. Initially, we in-

troduced more indicators, but these indicators were

either covered by all students or there was no student

who covered them. For instance:

usage of non geometrical concepts in description –

all;

application of elements of the ﬁgure like side and ver-

tex in description – none.

8 EXAMPLES AND COMMENTS

In this section we are going to consider some particu-

lar cases which are emblematic for the different stages

of language forming inside Levels 1 and 2. Let us

highlight that we did not register any usage of pure

geometrical concepts. All students’ descriptions of

their monsters were based on anthropomorphic fea-

tures; all geometrical shapes were colored and usually

the color stands before the shape in description.

Case 1

The monster created by PgA and its replica recon-

structed by KrG are shown in Figure 1.

PgA (Level 2, Σ = 6) uses simple sentences to de-

scribe his monster, e.g. My body is a gray rectangle

Figure 1: Monster created by PgA and its decoded replica

by KrG.

as a trafﬁc light. My neck is an orange rectangle.

There is no detailed information about sizes, direc-

tions, triangle types and so on. Just shapes, colors

and relations over and bellow.

Nevertheless, even with this simple and insufﬁ-

cient information, the replica created by KrG (Level

1, Σ = 1) and based on the text description, is quite

accurate. We could explain the poor description with

limited students’ knowledge of geometry at this level.

But this limited geometrical resource was sufﬁcient

enough for the students to communicate with each

other.

Case 2

The monster created by PpA (Level 1, Σ = 3) is shown

in Figure 2 (left). The monster created by PpA is

shown in Figure 2. The student, from the position

of his monster, wrote: My head is a blue square with

rounded edges.

Figure 2: Monster created by PpA and a picture from his

test.

From mathematical point of view, this description

is completely wrong, because a polygon cannot have

rounded edges. But in computer graphic software ap-

plications (including MS Paint) we could see an icon

called rounded rectangle. Some students expand their

language including icon-labels on an equal level with

geometrical concepts. The teacher could use such

contradictions to clarify the terms and to upgrade stu-

dents’ knowledge.

The right image on Figure 2 is taken from the

PpA’s control test. The test task was to draw a ﬁgure

Computer Supported Evolution Inside Van Hiele Levels 1 and 2

189

where the arms are triangles and legs are rectangles;

the arms and the legs should be connected with the

circle body at only one vertex. PpA follows mainly

the anthropomorphic context, without paying enough

attention to geometrical details.

Case 3

The monster created by KoG (Level 2, Σ = 5) and its

replica reconstructed by PpA are shown in Figure 3.

Figure 3: Monster created by KoG and its replica decoded

by PpA.

KoG’s description is the richest one in geometrical

concepts. He described the arms as two orange equi-

lateral triangles, the horns as right triangles. But no

details about the position of these triangles was given.

So the reconstructor PpA has put the arms connected

to the body by side, not by vertex. Such problem solv-

ing examples, connected with coding and decoding

processes, could be used to develop students’ criti-

cal thinking. Here one can see how anthropomorphic

context dominates over geometrical knowledge.

Case 4

The monster, created by KrG and its replica by PgA

(Level 2, Σ = 6) are shown in Figure 4.

Figure 4: Monster created by KrG and the reconstruction

by PgA.

In her description KrG wrote that the arms are

formed by four rectangles: the straight are green and

the down are yellow. Such multimodality of the spo-

ken language (Ginsberg, 2015) applied to geometri-

cal purposes is an evidence for incomplete Level 2 of

formation of mathematical concepts and terminology.

However, KrG showed signiﬁcant progress during the

experimental teaching. Her starting point was recog-

nized as Level 1, but further in her description lan-

guage appeared properly used geometrical concepts

as right triangle and rhombus. KrG covered (i6) and

(i7) and approached Level 2.

9 CONCLUSIONS

Ginsberg (ibid., pp 4-5) gives very interesting ex-

ample of misunderstanding in communication be-

tween teacher and students. Similar misunderstand-

ing appears every time when the teacher’s expecta-

tions about the geometry reasoning of the students are

not coherent with their actual Van Hiele level. There

were also quite different standing points between the

authors of this article before the analysis of the exper-

imental data was done. One of us was quite sure that

students operate at least at Level 2, but the other was

more skeptic. Our experimental teaching was held in

the beginning of 5th grade before the new geometry

topics from the school plan started. Based on the out-

comes of our study, we recommend Bulgarian teach-

ers to be very careful when using professional math

slang in their instructions.

Indeed, our observed students used mainly every-

day life expressions and images instead of geometri-

cal concepts. Some of them recognized the impor-

tance of clarity in communications after getting some

coding-decoding experience during the experimental

teaching. However, we consider problems like the one

quoted in section 3 to be still beyond the average 5th

graders’ zone of proximal development.

De Villiars stated the following open question:

could hierarchical thinking be developed earlier at

Van Hiele Levels 1 and 2 through various strategies

and using tools such as dynamic geometry software?

(De Villiers, 2010). We claim that the ability to use

DGS is equally related to mathematics and ICT, but

also it needs speciﬁc communication skills to express

the KSA. Evolution along the Van Hiele levels causes

development of a meta-language that reﬂects the de-

gree of geometrical reasoning but also accelerates the

evolution itself. However, DGS is not a relevant ed-

ucational tool for construction activities at Levels 1

and 2 – there is not enough mathematical KSA accu-

mulated for proper use of DGS. Even the basic under-

standing of a DGS interface requires signiﬁcant math

knowledge (compare with the Level 3 features in Ap-

pendix). Thus some preparatory training should be

done for some connections between elements of the

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190

ﬁgures at Level 1 and 2. Short deductive chains ap-

pear naturally during such training and computer ap-

plications of lower class than DGS allow to achieve

clarity and precision of the expression of these chains.

ACKNOWLEDGEMENTS

The authors thank to the reviewers for the suggestions

which are taken into account in the ﬁnal version of

this paper. The authors are very thankful to Albena

Vassileva for the improvement of the text.

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APPENDIX

Operationalization of Van Hiele Levels 1-3

Burger & Shaughnessy characterized pupils’ geomet-

rical reasoning at the ﬁrst three Van Hiele levels as

follows:

Level 1 (Recognition)

(1) Often use irrelevant visual properties to identify

ﬁgures, to compare, to classify and to describe.

(2) Usually refer to visual prototypes of ﬁgures, and

is easily misled by the orientation of ﬁgures.

(3) An inability to think of an inﬁnite variation of a

particular type of ﬁgure (e,g. in terms of orientation

and shape).

(4) Inconsistent classiﬁcations of ﬁgures; for exam-

ple, using non-common or irrelevant properties to sort

ﬁgures.

(5) Incomplete descriptions (deﬁnitions) of ﬁgures by

viewing necessary (often visual) conditions as sufﬁ-

cient conditions.

Level 2 (Analysis)

(1) An explicit comparison of ﬁgures in terms of their

underlying properties.

(2) Avoidance of class inclusions between different

classes of ﬁgures, eg. squares and rectangles are con-

sidered to be disjoint.

(3) Sorting of ﬁgures only in terms of one property,

for example, properties of sides,

while other properties like symmetries, angles and di-

agonals are ignored.

(4) Exhibit an uneconomical use of the properties of

ﬁgures to describe (deﬁne) them, instead of just using

sufﬁcient properties.

(5) An explicit rejection of deﬁnitions supplied by

other people, e.g. a teacher or textbook, in favour of

their own personal deﬁnitions.

Computer Supported Evolution Inside Van Hiele Levels 1 and 2

191

(6) An empirical approach to the establishment of the

truth of a statement; e.g. the use of observation and

measurement on the basis of several sketches.

Level 3 (Ordering)

(1) The formulation of economically, correct deﬁni-

tions for ﬁgures.

(2) An ability to transform incomplete deﬁnitions into

complete deﬁnitions and a more spontaneous accep-

tance and use of deﬁnitions for new concepts.

(3) The acceptance of different equivalent deﬁnitions

for the same concept.

(4) The hierarchical classiﬁcation of ﬁgures, e.g.

quadrilaterals.

(5) The explicit use of the logical form ”if ... then’ in

the formulation and handling of conjectures, as well

as the implicit use of logical rules such as modus po-

nens.

(6) An uncertainty and lack of clarity regarding the

respective functions of axioms, deﬁnitions and proof.

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