Students' mental manipulation of a shape at the early educational level *

Vol. 41, 2019, p. 6389

Ewa Swoboda ( Jarosªaw, Poland)

Maªgorzata Zambrowska ( Warsaw, Poland)

Students' mental manipulation of a shape at the early educational level ^*

Abstract: In this article we present results of research aiming at descri- bing the strategies used by 10-year-old children while solving one geo- metric task. The research was lead through three dierent stages. In May 2015 the Educational Research Institute in Poland carried out a su- rvey titled Competences Of Third Grades. One task, related to the do- main the geometric imagination, solved by 199 361 students, achieved a low degree of solvability, also among students achieving good results in other educational domains. To identify the strategy for solving this task, about 3000 submitted solutions were reviewed. One of them was based on imagination of action. We were interested to which extent such dy- namic thinking is present in children's solutions, therefore, in the next stage, individual observations of a child working on this task were carried out. 35 children aged 10 years participated in this stage. The results of all stages are briey presented in this study, with particular attention to the result of the third stage. This last stage supports our opinion that dynamic reasoning is possible to trigger, but requires special teaching methodology and specially designed tasks.

1 Introcuction

In the practice of the Polish school related to teaching early mathematics, the greatest emphasis is placed on arithmetic issues. Tasks proposed to students

*

The article is based on a paper presented during the ICME-13 Conference in Ham- burg. The current text is an extended version of the conference report, supplemented with additional research material.

Key words: square, levels of understanding of geometric concepts, dynamic geometric

reasoning.

expect a correct understanding of arithmetic concepts, use of mathematical relationships within arithmetic. There are still very few didactic proposals tar- geted on teaching geometry (Banasiak et al., 2017; Dobrowolska, Szulc, 2017;

Lankiewicz, Semadeni, 1993). Undoubtedly, this fact is related to the still too narrow understanding of the specicity of learning geometry, to some extent also with the opinion of the place of elementary geometry in modern mathema- tics and the role of geometry in the competences of a modern man (Urba«ska, 1985; D¡browski, Wiatrak, 2009; Kalinowska, 2014; Karpinski, Zambrowska, 2017). There is a conviction, mainly among mathematicians  theorists, that elementary geometry has been exploited as an area of research, which has undoubtedly aected school mathematics for a long and also that the applica- tions of mathematics in real life are related to other elds of mathematics, not to geometry.

However, it turns out that the issue of teaching geometry is gaining more and more popularity both among mathematics educators or didactic specialists dealing with other sciences, as well as developmental psychologists (Kell, 2013;

Harris, 2013; Battista, 2007; Kozhevnikov, 2001; Tumova, 2017a, 2017b; Uttal et al., 2013).

2 Why include geometry in the school mathematics curriculum?

Enthusiasts of teaching geometry in schools, from the youngest educational levels, have no doubts about the benets of such an approach (Alexe, et al., 2015, p. 150). They believe that it supports the development of a child in many

elds, including non-mathematical ones. It is also helpful in understanding other branches of mathematics. Keith Jones (2002) says:

The study of geometry contributes to helping students develop the skills of visualisation, critical thinking, intuition, perspective, problem-solving, conjecturing, deductive reasoning, logical argument and proof. Geometric representations can be used to help students make sense of other areas of mathematics: fractions and multiplication in arithmetic, the relationships between the graphs of functions (of both two and three variables), and graphical representations of data in statistics. Spatial reasoning is impor- tant in other curriculum areas as well as mathematics: science, geography, art, design and technology. Working with practical equipment can also help develop ne motor skills.

(Jones, 2002, p. 125)

It can be assumed that interest in geometry will be more and more visi- ble, which may be associated with even more common use of visual (graphic) computer techniques or while moving in two- and three-dimensional spaces. It is believed that the ability to interpret visual information is fundamental to human existence (Jones, 2002). Acting in the eld of engineering, technology and natural sciences require well-shaped spatial intuitions. There are studies documenting that spatial ability was a signicant predictor of achievement in STEM (Shea, Lubinski, Benbow, 2001). In addition, skills related to under- standing space can be successfully trained (Uttal et al., 2013). On the other hand, results of such surveys like: Czajkowska, 2012; D¡browski, 2013; Kar- pi«ski, Zambrowska, 2015 show that solving geometric problems give students more problems than solving problems from other areas of mathematics.

3 Dierences in learning arithmetic and geometry

In Poland, methodical proposals for shaping mathematical concepts at early educational level have been built based on the Piaget's theory of cognitive de- velopment (Schaer, 2013; Gruszczyk-Kolczy«ska, 2014). According to Piaget, in the case of logical  mathematical concepts, we en-counter the interplay of operations, separated not from the perceived objects, but from the actions taken on them (Piaget, 1966). In Piaget's view, the transformation of reality is of fundamental signicance and action is the tool for that transformation.

Therefore, for teaching arithmetic it is typical to refer to Piaget's theory of interiorization of action.

Uncritical implementation of Piaget's views in the eld of geometry metho- dic raises many objections worldwide (Clements, Battista, 1992; Clements et al., 1999). It is generally believed that there is a discrepancy between theoreti- cal foundations of teaching arithmetic and theoretical foundations of teaching geometry (Gray, 1999; Hejný, 1995). The diversity in approaches to shaping arithmetic and geometric concepts were emphasized by H. Freudenthal:

Viewed developmentally, geometry is the direct opposite of arithmetic.

Space and the bodies around us are early mental objects, the results of

structuring and being structured. To what degree are they mathemati-

cal objects? To what degree are distances viewed as lengths, chunks of

matter as volumes, bodies as gures? Name-giving is a rst step toward

consciousness. But a name for geometric similarity, for instance (which

is one of the earliest geometric experiences), is still far away at the time

it is experienced. It has to be invented. The Greeks did this by lending

the general word for similarity a specic geometrical meaning, but even

nowadays this has hardly inuenced everyday language. One says: copy-

ing, reducing, enlarging. They are the same, uh, this one is big, that one small, uh, the same shape. Geometry, though omnipresent, too often lacks the linguistic means for expression.

(Freudenthal, 2006, p. 64) However, it can be assumed that one of the most important reasons for the distinction between ways of shaping arithmetic and geometric concepts is the relationship between action and state. The classical, theoretical basis of cre- ation of geometrical concepts says nothing about the role of action. According to van Hiele's theory (1986), at rst, visual level geometric concepts are reco- gnized globally. It can be considered that at this stage the most important are the perceived qualities, which are static, but not the action. Tall (2001) pre- sents an epistemological model in which creation of arithmetic and geometric concepts are set on opposing elements: action  perception.

Figure 1. Various types of mathematics, from (Tall, 2001).

Milan Hejný and Darina Jirotková (2004) also wrote about the dierences between the world of arithmetic and the world of geometry, emphasizing, among others, the autonomy and independence of geometric entities, which makes this world dicult to order, unlike the world of arithmetic. But this theory, for some time, did not clearly emphasize the role of action, and if so, only in the area of better learning about shapes. By carefully matching and comparing the length of the sides, the child was to learn more about the relationships between the attributes of individual geometric gures.

These paradigms aect how geometry is taught at school. Apart from the

fact that early-school geometry is marginalized, at least in countries of the

former Eastern Bloc, learning through visual representation is still preferred

in shaping geometric concepts. A child learns geometric gures often from drawings presented in the textbook, his activity comes down to, e.g., searching for specic geometric gures in the picture.

With this approach the compounds with some historical sources of geome- try, of which the movement of objects was important (Henderson, Taimina, 2005), are blurred. But competence in this area seems to be important, not just for geometric reasoning. Manipulating images in the head can inspire condence and develop intuitive understanding of spatial situations (Jones, 2000). For solving many geometric problems, it is required to anchor the thin- king in the cause-and-eect relationship. Creation of the understanding of the

gures should be associated with the ability of making mental manipulations of the object. If seeing gures, then through changes and relationships, inc- luding dynamic changes. Visualization of these changes by dynamic computer programs is one of the methods, but probably it requires supplementing with other teaching solutions. The student should be faced with the need to anti- cipate the eects of transformations. This goal should be one of the priorities for teaching design.

4 Reference to related literature

A comprehensive review of psychological research on understanding spatial relationships is provided by the article Uttal et al. (2013). It gives, among others, two fundamental distinctions between types of information related to spatial tasks.

• The rst is between intrinsic and extrinsic information.

• The second is between a static and dynamic task (p. 353).

They write about dynamic tasks following these remarks:

However, objects can also move or be moved. Such movement can change their intrinsic specication, as when they are folded or cut, or rotated in place. In other cases, movement changes an object's position with regard to other objects and overall spatial frameworks. The distinction between static and dynamic skills is supported by a variety of research.

For example, Kozhevnikov, Hegarty, and Mayer (2002) and Kozhevnikov, Kosslyn, and Shephard (2005) found that object visualizers (who excel at intrinsic  static skills in our terminology) are quite distinct from spatial visualizers (who excel at intrinsic  dynamic skills). Artists are very likely to be object visualizers, whereas scientists are very likely to be spatial visualizers.

(Uttal, 2013, p. 3)

Thus, by describing the relationship between these various types of skills, they create a two-dimensional classication, presented in Table 1.

intrinsic (within object) extrinsic (between objects) static the recognition of an object

involves intrinsic, static information

thinking about the relations among locations in the environment, or on a map.

dynamic the mental rotation of the same object

thinking about how one's perception of the relations among the object would change as one moves through the same

environment

Table 1. Classication of situations in the domain spatial skills which require dierent thought processes, according to (Uttal, 2013).

This type of classication can be useful for analyzing issues related to te- aching and learning geometric concepts. Static  intrinsic skills aect the abili- ty to analyze and dene shapes; dynamic-intrinsic are associated with various types of isometric transformations, static-extrinsic relate to the relationship between gures, dynamic-extrinsic support the ability to see objects from dif- ferent points of view. This combination already shows that it does not cover all issues that may be associated with shaping geometric concepts and topics. On the other hand, many didactic problems related to geometry are analyzed by the didactic specialists very selectively, without taking into account all these four areas.

The importance of dynamic images in teaching geometry has gained signi-

cance in early 70s of the last century. It could have been connected with the introduction of computer software to schools. Until today, many researchers analyze the processes running in a dynamic geometry environment, using computer programs (Arzarello, 2002; Markopoulos, 2015; Forsythe, 2011; Gol- denberg, 1998; Hollebrands, 2007; Mariotti, 1995; Pittalis, 2010; Ruthven, 2018;

Jones, 2000). Such works, however, repeatedly focus on the possibility of exa- mining the properties of gures by stretching them (for example, by catching one of its vertices), moving without paying attention to movement planning.

This is evident in many statements, even such as:

(. . . ) visual images, particularly those, which can be manipulated on the computer screen, invite students to observe and conjecture genera- lisations. Proving conjectures requires students to understand how the observed images are related to one another and are linked to fundamen- tal `building blocks'.

(Jones, 2000, p. 9)

Typically, formal deductive reasoning concerning the higher educational levels are tested (Jones, 2000). In this way, tacitly, it is assumed that solving geome- tric problems in a dynamic environment is available only for older students, not only because of technical limitations. In addition, in this type of research, the student has only an opportunity to experiment freely and observe the ef- fects of such experiments. Perhaps it is assumed that the accumulated scope of experience will be interiorized in the future and will allow to act on an imagined level.

So, it is a slightly dierent approach than the one that interests us  that is, where the essence is student's planning, aimed at achieving a specic eect known to him. In addition, it is often emphasized that we still know too little about children's thinking in the area of dynamic transformation of objects:

Although transformational geometry is considered important in suppor- ting children's development of geometric and spatial thinking (Holle- brands, 2003; Jones and Mooney, 2003b), research in the eld seems to have left unanswered questions concerning children's abilities in perfor- ming transformations.

(Xistouri, Pitta-Pantazi, 2011) Fortunately, proposals for activities using computers in early education, have increasingly appeared. Using computer programs, children can manipulate blocks in virtual reality, building dierent constructions. Many researchers have maintained that computers have become an essential part of mathematical education (van den Heuvel-Panhuizen and Buys, 2008), where the combination of activities in the real, physical world and the virtual world gives child the ability to collect experiences that help connect 2-D and 3-D representations. On the screen, children can make manipulations: rotations, translations, deletion of blocks from an existing building, approximations, and enlargements. After that, they can see the eects of their actions. Therefore, researchers have focused eorts on developing virtual manipulatives, creating a new trend of designing technology-integrated mathematical instructional materials.

Lately, the group of researchers, gathered around prof. Hejný, began to

emphasize the role of dynamic thinking in geometric reasoning, approaching

this issue in a dierent way. Based on the theory derived from Tall (2001)

according to which the beginnings of geometric cognition are static, and ba-

sed on the perception of geometric phenomena such as the shape and mutual

arrangement of objects in relation to each other, a group of Czech researchers

began to develop a methodology for building children's dynamic thought. It

presented a global approach to problem of geometric education that is focused

on the understanding of plane gure and spatial intuition. In their approach,

the most dicult in learning geometry is the transition from static to dyna-

mic approach (Milan Hejný, Darina Jirotková, et al., 20072011; Milan Hejný, 2012; Jirotková, 2011). Therefore, even in a situation where methodological proposals are focused on examining the shapes of gures (i.e. according to the classication presented by Uttal et al. (2013) static  intrinsic skills), activi- ties related to the possibility of manipulating shapes are proposed. Studies in which children were deprived of the possibility to manipulate objects, has determined that their understanding of the gures is mostly at the visual le- vel. Observations of children who actively investigate shapes (they have the opportunity to manipulate them while solving some problems, including clas- sifying shapes) provide other results. Such an approach clearly improves their recognition, noticing properties such as the size and length of the sides, the number of vertices, convexity, promotes the use of language  which in turn supports the transition to higher levels of understanding of geometric concepts (Coutat-Gousseau, Vendeira-Marechal, 2016).

5 Outlining our own research problem

The issues presented so far, which may have an impact on building the me- thods of teaching early geometry, provoked us to undertake research in this area. One of the most important ones seems to be creating a transition be- tween static reception of visual information and the ability to make mental, dynamic transformations aimed at solving a given problem. In our view, in this approach, rst it is necessary to recognize whether children are able spontane- ously (not guided) to apply dynamic organization of the situation proposed to them within a geometric problem.

Our research problem was thus formulated as follows:

Are children able to spontaneously support their reasoning by dynamic ima- ges, while solving a geometric task?

6 Methodology

The research work was carried out in three stages. The rst stage consisted of analyzing the results of one of the tasks from the survey titled Competences of Third Grades, conducted by the Institute of Educational Research, in May 2015. One of the authors of the article was a member of the research team.

Research tasks were related to the selected compulsory math skills. The mathe-

matical worksheet includes tasks from domains: numerical eciency, geometric

imagination and the ability to solve text problems. Student solutions were te-

sted globally, with no observation of child's work. However, teachers evaluating

the work had to enter the information about students' results and the way of solving the task into a special software program. The task we are interested in, was solved by 199 361 students from primary schools in Poland.

In addition, approx. 3000 working sheets were drawn up. Access to these works gave the opportunity to analyze the solution itself and see the notes created by the children. On this basis it was possible to partially reconstruct the student's way of thinking. The work related to such analysis constituted the second stage of the research.

In the next, third stage of research, we decided to support quantitative results and results obtained through analysis of documents by observing stu- dents solving the same task. To this aim, individual observations were carried out, in which 35 students from various schools of the Podkarpackie region in Poland participated.

At the beginning, an observer was asking the child if she/he knew what a square is. Then he presented the task, which he also read aloud, and explained that one of the given answers should be circled. The child could work as long as they needed, could use various aids or additional sheets of paper. The researcher observed and noted the child's work process (e.g. hand movements performed over the drawing, counting corners, order of marking vertices, how sides were drawn, etc.). The work was anonymous, after nishing the work the student was asked to explain why she/he chose the answer. This oral statement of the child was to verify the observation of the work carried out.

7 Research tool

From 14 tasks listed in the worksheet from the survey Competences of Third Grades, we chose one, related to the domain the geometric imagination.

Figure 2. Research tool.

In the classication (derived from Uttal et al. (2013)), adopted by us and

associated with the purpose of research, this task is an intrinsic and dynamic

one (the object  the square  does not change itself, but is in motion). However,

in the student's perception it could be understood completely dierently. This dierent understanding allows both static and dynamic approaches.

The visual part of this task (the picture) does not impose the movement itself. So it can be of the static  intrinsic type. In the static approach, the student mainly focuses on the presented drawing. From the descriptive content, he takes part of the information saying that the drawing shows the outline of several identical squares.

Dynamic reasoning can be associated with:

a) the oral content of the task (attempt to recreate the situation of stacking napkins, seeing this content through action);

b) assessment of the mutual arrangement of the gures in relation to each other, expressed as the arrangement in a certain rotation around a point lying in the vicinity of the intersection of the diagonals of the square.

Depending on the interpretation of the task, several dierent strategies are available for solving it.

Static strategies could rely on using square properties. Regardless of the

nal position of each napkin, the distance between the vertices and the line- arity of the points lying along one side should remain unchanged. Using a ruler or sketching, the student could try to highlight all existing squares, by measu- ring distances between protruding vertices, or by marking the location of four vertices that belong to one square.

In the dynamic approach, the child could act in various ways:

One of them is to imagine a rigid motion in which the gure itself is not transformed, and all points of the gure change their position. If it is a regular movement, the change of position of all points takes place according to some discovered relationship. The motion that can be associated with this task is rotation on a plane relative to the center located at the intersection of the square's diagonals. In this approach, the child could focus only on the move- ment of one vertex and tracking its subsequent positions. The symmetry of the square, rotation properties and task conditions (how many napkins do you see?) ensured that it is enough to trace the systems distinguished between two consecutive vertices of the rst square (visible in full), including the vertex of this square. So you can see that this type of thinking requires an intuitive understanding of the properties of transformation (in this case  rotation) and referring to these properties while working on a task.

Another strategy could be a process-based view of the task itself, described

in words. Since we are talking about stacking napkins into one pile, you could

try to track the eects of this stacking and relate this process to the presented

gure, assuming that all napkins are at least partially visible. Which one of these napkins was placed rst? How were the next ones, up to the last one, seen in its entirety?

By analyzing the available solutions, we tried to identify the strategies used. Access to these sheets gave the opportunity to analyze the solution's record itself and see the notes created by the children. We were interested in the extent to which we will be able to see traces of dynamic thinking in the solutions presented to us.

It was obvious, however, that information collected in the manner described above may give an incomplete picture of the strategies used  and of the way of students' reasoning. Therefore, it became important to observe the actual course of the child's work on this task.

8 The results of the rst stage of the research

Global results obtained on a large (approx. 200,000) group of 10-year-old stu- dents have shown that the task was not easy at all, even though the concept of the square is known for children since preschool age.

Task results are presented in the table and the graph (gure 3). To be able to compare results of all tasks, they were normalized, so that the average was 100. The dots on the charts measure the next deciles of students. Numbers on the horizontal axis indicates the scores for the whole test. The vertical axis is marked with the percentage of students providing the answers of the task.

Figure 3. Responses in the task Napkins.

Freq. Percent

A 3,428 1.72

B 14,961 7.50

C 5,705 2.86

∗ D 101,150 50.74

E 72,616 36.42

Lack or n > 1 answ. 1,501 0.75

Total 199,361 100

Table 2. The results of the rst stage of the research (in numbers and in percentages).

As it is visible, for the entire group of students the task proved to be quite dicult. The correct answer D was given by 51% of students, mostly from the

nal deciles (table 2).

Among the wrong answers, the answer E (4 napkins) was chosen most frequently. 34% of students chose it. Those students were not always the weak

students. Even among the top 10% of students (tenth decile) up to 18% chose the answer E. This result is quite dicult to interpret. Did the students count only the covered napkins? It is possible that focusing on this procedure turned their attention away from the most obvious napkin, the fth one.

Answer B (16 napkins) was the most popular among the weakest students (20% of the students from this decile). But also approx. 10% of average stu- dents indicated that response, which is related to the number of all corners protruding outside the napkin visible as the whole. Other answers were chosen exceptionally, even among the weakest students.

Such results provoked a closer look at the ways students solve this task.

For now, we have focused on strategies used in the tasks solved correctly.

9 Analysis of the correct solutions, based on the writ- ten work of students, collected in the second stage of research

Students used dierent strategies to choose their answers. We can conclude this on the basis of sketches that they made. By analyzing their work you can inquire what properties they took into consideration.

9.1 Size as the invariant

In the rst approach, the student has counted all protruding corners (answer

A-20). Such a response indicates that the student was not referring to the drawn

square, the student he did not think about this property, which is invariant in this transformation  that was about the size. At this stage the student has proceeded as did almost 2% of the surveyed students. Apparently, however, this answer did not match the student with the visual assessment of the fully visible gure, as at the next attempt the student took into account the size and shape of the gures. In the next approach the student did it by shading the protruding portions tried to match the parts of the individual squares. In this perspective, it is dicult to say that the student was thinking dynamically.

Rather, the student received the situation statically (as a fact  the napkins are already lying), trying only to count them.

Figure 4. Size as the invariant in the student's solution.

9.2 Focus on the globally perceived gures

In most sketches done by the students we can see that they were trying to

extract all the squares by matching the protruding corners. They mark pieces

related to one square, sometimes using dierent symbols for successive squares

(circles, crosses  as shown in gure 5b). In some sketches you can see the

corrections  from E to D (gure 5a); perhaps the idea to introduce the separate

signs for each square appeared only at a later stage of the work, that is after

the step of assigning vertices to covered squares. Such an interpretation would

explain the high score for E in the whole study group (37%).

Figure 5. Dierent strategies presented in one work.

Another example, more advanced, in which the student was able to recon- struct the shape and size of the square, to refer it to the square visible in its entirety, is a work in which the student marked only one vertex of each co- unted square. This vertex was somehow a representative of each gure. This approach has not always been easy, as can be seen from the deletions on the answer card.

Figure 6. Focus on the globally perceived gures in the student's solution  only one vertex of each counted square is marked.

9.3 Dynamic approach

In this approach, the student tried to apply those elements which resulted from a dynamic interpretation of the situation described in the task. These elements were dierent.

9.3.1 Laying napkins as a process

This situation is illustrated in the sketch presen- ted in gure 7. The student numbered the tops of the particular squares. Number 1 is located on the square, which, according to the student, was placed

rst.

Figure 7.

9.3.2 The transition from static to dynamic interpretation

A student began to solve the problem from creating a comprehensive picture of individual squares. After nishing three full objects the student resigned from the drawing.

The degree of certainty (showing by the accuracy of line) indicates that the

rst square drawn was the one that is presented in its entirety, the second one

 having diagonals parallel to sides of the rst one. It is possible that while drawing subsequent square the student noticed that it is enough just to focus on one vertex that changes its position within a certain angle. The answer could be read from the picture, where 3 vertices were in bold and 2 were not in bold. A student didn't x the whole outlines for following squares and strongly painted over the answer D: 5 (gure 8)

Figure 8.

Carrying out these analyzes, we were constantly aware that our interpreta- tions may be erroneous or incomplete. The inability to talk with learning, the lack of observation of the actual course of work, were important restrictions for making real conclusions. Therefore, it was necessary to continue research.

This took place in the next stage, described below.

10 Results of observing the students' behavior while solving the napkins task

Observation conducted in the last stage of these studies, has brought many interesting results.

One is the fact of observing dierent ways of working, those that were

reected both in the design of the task itself and during the second stage

of research, by the distinction into a static-dynamic approach. The students

presented dierent strategies. They calculated all the vertices (suggestion of

answer A), or calculated only the protruding vertices (suggestion B)  without

grouping them by 4. There were also behaviors leading to answer D and an-

swer E. We did not nd a student who circled answer A and C, but this should not come as a surprise; the observed group was not numerous, and during the

rst stage of research, done on a national scale, such answers were marked by a very low percentage of students.

Below we present three dierent protocols, done by students who partici- pated in the third stage of research (gure 9)

a) In the rst phase of work, the student sketches the in- visible shape of the square lying directly below the rst one, then represents the squ- are through one highlighted vertex.

b) The student recreates the

overall shape of the square. c) The student counts only one vertex of the protruding napkin, marks E.

Figure 9. Three dierent protocols, done by children, from the third stage of our research.

Another interesting fact is the time that students devoted to solving this task. This time was very dierent for dierent students. Sometimes students very quickly decided to mark the answer. After reading the task, they briey analyzed the drawing, quickly calculated or circled something, and chose one of the answers. Often, they did not really want to talk about their strategies for solving the task. The time devoted to the solution ranged from 1 to 2 mi- nutes. Therefore, for these students, the time planned by the organizers of the nationwide study was appropriate.

However, there was a group of students who spent up to 20 minutes on the

task. These students analyzed the drawing in great detail, sometimes measuring

the sides of the squares, marking out the highlighted squares with dierent

colors. Repeatedly, they used dierent strategies to solve the task, treating the

result achieved with the second strategy as a verication of what they achieved

the rst. Here is an example:

Figure 10. Dierent strategies presented in one work

The student described their behavior as follows:

At rst I looked at the rst square and marked its corners. Then I took the next corner, dragged my nger and looked for the another corner to connect with and marked the next color there. And so I did with every square. At the end I counted how many colors I have and hence how many napkins there were. Just to make sure I counted all the corners, it turned out 20, so I divided it by 4, because each square has 4 vertices and it came out to me 5. I marked the answer 5.

This girl knew that to solve the task she should use the properties of the square.

She decided to use the property the square has four vertices, but additionally she used the relation of mutual arrangement of vertices, equal length of the sides, mutual position of the sides. Using these relationships, she supported herself with gestures, sliding her nger on the drawing and circling the shapes of subsequent squares on it. Only then she used the numerical description, counting all drawn vertices and dividing the obtained number by 4.

These observations suggest that the poor results of this task in the nation- wide test could be, for example, related to the insuciently long time that children could devote to solving this task.

However, our main research problem was related to dynamic representation

of the situation presented in the task. It has been observed that this strategy

is not as rare as it seemed to us. In this approach, the children tried to identify

the squares in an orderly manner, starting from the potentially rst placed

napkin, or  starting from the rst (most clearly visible) but then trying to

determine their order going back to the rst one. There were 7 such children

in the observations  starting from the bottom square, and 6  starting from

the top square. Interestingly, these children have not always been able to nd

the right answer. Here are examples of their work:

Figure 11. Dierent strategies presented in one work.

Example 1 The child focused only on the protruding vertices. He indicated which vertices he counted and in which order. He started with the potentially

rst square (i.e. from the bottom), lost one protruding vertex, but took into account the one from the whole square. He marked the answer E.

Figure 12.

Example 2 The student counted 4 vertices protruding in the lower left cor- ner, starting from the potentially rst located square. He missed counting of the rst visible square. He marked the answer E.

Figure 13.

Example 3 He studied the drawing for a long time. Then he began to count

the protruding vertices. He counted from each side separately, constantly star-

ting with the potentially rst napkin  that is, four times he took into account

the order in which napkins were laid. He counted all protruding vertices, but did not include those of the whole square. He marked answer B.

Figure 14.

Example 4 The student said to himself in a low voice that the square has 4 vertices. He began to number the vertices of the squares, but not consecutive, only the ones corresponding to one square. He started with the rst fully visible, then he chose the second one directly below him, i.e. acting in the reverse order than those which were laid out. So he numbered to 20. He stated that the correct answer is 5. When asked how he knows this he wrote the calculation 20: 4 = 5. Why? Because if there are 20 vertices and a square has 4 vertices, there must be 5 napkins.

Among observed students there was one boy who indicated a dierent ap- proach  by presenting dynamically the relationship of napkins in relation to each other. This child's work lasted very shortly. The boy focused on the top square. Then he made a characteristic movement circling the squares. He stated that it is simply visible that there are 5, he marked the answer D.

11 Conclusions from conducted research

Conducting didactic research in one area, using many dierent methods, can allow you to look globally at the studied problem. It seems that this was also the case this time, because each stage of the research allowed to see new, interesting phenomena or gave a chance for their better interpretation.

In the third stage of research, we were not interested in the distribution of solutions divided into quadrilles, assigning them to gifted and less able

students. It was also irrelevant whether the child would circle the correct an-

swer. We were looking for an answer to the question of whether children would

spontaneously interpret the task dynamically and whether this approach would

help them nd the right answer. The very fact that this type of approach (dy-

namic thinking) may appear was suggested by an analysis of solutions carried

out in the second stage of the research.

Straightway observations of the child's work have conrmed that dynamic understanding is not uncommon. It was present in 14 out of 35 observed chil- dren's work. In the vast majority it was caused by the story layer of the task, which turned out to be important for children. The very course of child's ap- proach to the solution repeatedly began similarly. It was the case that children started their work by analyzing the shape visible in its entirety. Such starting point is obvious, because it was the rst basic information that enabled further solving the task. Sometimes, for a better understanding of this shape, the child additionally outlined it, or repeated its shape with a movement of the hand.

However, later it could be seen that the children tried to reproduce the order of laying napkins, from the rst one or from the last one. In addition, one stu- dent recreated the relationship of arranging subsequent napkins, by suggesting rotation.

It was not surprising that most children did not see any process, they tre- ated the situation statically. They were focused on the analysis of the picture, trying to identify the squares marked there. This was probably the expecta- tion of the authors of the task, which resulted from the assumptions of the school curriculum. Gestures also helped in this analysis  simulating the shape of a square by trying to connect visible vertices. Sometimes such analysis was purely arithmetic  by calculating all vertices and dividing the obtained num- ber by 4. In this strategy, the student referred to his knowledge about gure, did not try to extract the shape of the square. However, it seems that the children's activities predominantly took place at a level dened by van Hiele (1986) and Milian Hejný (1993) as a visual level  the identication of the shape was either global or focused on the most characteristic elements of the shape  i.e. vertices. Only one of the observed children reached for a ruler to measure the length of the side of the square, and this knowledge was used in further proceedings, when this distance was the main tool for assigning visi- ble vertices to one square. And just such behavior may indicate a conscious reference to the properties of the square, i.e. to actions from higher levels of understanding geometric objects.

And one nal note  the overall results of the research show that geometric tasks are still dicult for children. This may be inter alia, due to the fact that both global and analytical views are needed to solve them; ability to choose the right strategy for solving and consistency in implementing this strategy.

Observation of child's work showed, among other things, that choosing a good

strategy did not have to lead to a good solution, because, for example, a child

who focused on the most dicult part of the task (nding hidden squares)

forgot about the obvious part (adding the square fully visible).

Geometric tasks are typical tasks having too much data, and the selection of the information useful for solving the problem and consistency combined with exibility are skills that must be developed from the lowest stages of education.

12 Final conclusion

The task that the students were solving could trigger the strategy of imagining the action. In Polish schools students are not accustomed to such approach.

The results conrm that dynamic thinking is not a natural intuitive strategy for children, or  perhaps  it is a strategy for which they are not prepared through the existing school practice. But on the other hand, the students showed that they are able to think dynamically in the geometric environment. It follows that an attempt for shaping a dynamic intuition can give a good results, because such strategies appear in the children's work.

Visualization of changes in dynamic software is one of methods of work in this direction. But it is not the only one. The student need to anticipate the eects of carried out transformations, not only have the possibility to do their testing. This objective should be one of the priorities for designing the teaching. A child can work in the familiar environment, using bricks, sticks, badges, models, gures and solids. Then manipulate real objects should be associated with solving a specic task, and the child will subordinate its actions with expected results of the transformation.

The proposed tasks should include various implementation of the assump- tions of dynamic understanding of space. One way of suggesting imagination of action can be the appropriately selected content of the task  as was the case in the reported research. Another way may be to recommend a specic trans- formation, supported by manipulation. Such tasks are already appearing (e.g.

in didactic proposals developed by Hejný and the team: Milan Hejný, Darina Jirotková, et al. (20072011); Kloboucková, Darina Jirotková, and Slezáková (2013)). It is also still worth conducting research to check how they inuence the raising of competences and skills related to dynamic organization of space.

References

A l e x e, S., A l e x e, G., V o i c a, C., V o i c a, C.: 2015, X-colony

educational activities enhance spatial reasoning in middle school students, Pro-

ceedings of the 9th mathematical creativity and giftedness International Confe-

rence, in: Florence Mihaela Singer, Florentina Toader, Cristian Voica, (Eds.),

Sinaia, Romania, 146153.

A r z a r e l l o, F., O l i v e r o, F., P a o l a, D., R o b u t t i, O.:

2002, A cognitive analysis of dragging practises in Cabri environments, ZDM, 34(3), 6672.

B a n a s i a k, A., B u r d z i « s k a, A., D y m a r s ka, J., K a m i « - s k a, A., K o ªa c z y « s k a, M., N a d a r z y « s k a, B., D a n i e - l e w i c z - M a l i n o w s k a, A.: 2017, Nowi Tropiciele, Podr¦cznik Klasa 1, Wydawnictwa Szkolne i Pedagogiczne, Warszawa.

B a t t i s t a, M. T.: 2007, The development of geometric and spatial thin- king, in: F. K. Lester Jr. (ed.), Second handbook of research on mathematics teaching and learning, 843908. Charlotte, NC: Information Age Publishing Inc.

C l e m e n t s, D. H., B a t t i s t a, M. T.: 1992, Geometry and spatial re- asoning, in: D. A. Grouws (ed.), Handbook of research on mathematics teaching and learning, New York, Mcmillan, 420464.

C l e m e n t s, D. H., S w a m i n a t h a n, S., H a n n i b a l, M. A. Z., S a r a m a, J.: 1999, Young Children's Concepts of Shape, Journal for Research in Mathematics Education, 30(2), 192212.

C o u t a t, S., V e n d e i r a, C.: 2016, Shape recognition in early school.

Paper present during the 13th international congress on mathematical educa- tion, ICME-13, Hamburg, Germany.

C z a j k o w s k a, M.: 2012, Umiej¦tno±ci matematyczne przyszªych polskich nauczycieli edukacji wczesnoszkolnej w ±wietle wyników badania TEDS, Pro- blemy Wczesnej Edukacji, 1(16), 67.

D ¡ b r o w s k i, M., W i a t r a k, E.: 2009, Nauczyciele nauczania po- cz¡tkowego w ±wietle ankiet, in: Trzecioklasista i jego nauczyciel. Raport z ba- da« ilo±ciowych 2008, (ed.) M. D¡browski, Centralna Komisja Egzaminacyjna, Warszawa.

D ¡ b r o w s k i, M.: 2013, O matematycznych wynikach polskich trzeciokla- sistów w badaniach TIMSS, Problemy Wczesnej Edukacji 9, 4(23), 3238.

D o b r o w o l s k a, M., S z u l c, A.: 2017, Lokomotywa 3. Matematyka, Gda«skie Wydawnictwo O±wiatowe.

F o r s y t h e, S. K.: 2011, Developing perceptions of symmetry in a Dynamic Geometry Environment, Research in Mathematics Education 13(2), 225226.

F r e u d e n t h a l, H.: 2002, Revisiting Mathematics Education, China Lec- tures, Kluwer Academic Publishers Dordrecht/Boston/London.

G o l d e n b e r g, P., C u o c o, A.: 1998, What is Dynamic Geometry?

in: Lehrer, R. and Chazan, D. (Eds.) Designing Learning Environments for Developing Understanding of Geometry and Space, Hilldale, NJ: LEA, 351

367.

G r a y, E., P i n t o, M., P i t t a, D., T a l l, D.: 1999, Knowledge Con- struction and Diverging Thinking in Elementary and Advanced Mathematics, Educational Studies in Mathematics, 38, 111113.

G r u s z c z y k - K o l c z y « s k a, E.: (2015). Edukacja matematyczna w klasie I. Ksi¡»ka dla nauczycieli i rodziców, CEBP 24.12, Kraków.

H a r r i s, J., H i r s h - P a s e k, K., N e w c o m b e, N. S.: 2013, Under- standing spatial transformations: similarities and dierences between mental rotation and mental folding. Cognitive processing, 14(2), 105115.

H e j n ý, M.: 2007, Budování matematických schémat. [Building mathemati- cal schemes], in: A. Ho²pesová, N. Stehlíková, M. Tichá (Eds.), Cesty z doko- nalování kultury v yu£ování matematice (81122). ƒeské Bud¥jovice: Jiho£eská univerzita v ƒeských Bud¥jovicích.

H e j n ý, M.: 1993, The understanding of geometrical concepts, in: Proce- edings of the 3rd Bratislava international symposium on mathematical educa- tion, BISME3, Bratislava: Comenius University.

H e j n ý, M.: 1995, Development of Geometrical Concepts, Proceedings of SEMT 95, Prague, 1318.

H e j n ý, M.: (2012). Exploring the cognitive dimension of teaching mathe- matics through scheme-oriented approach to education, Orbis Scholae, 6(2), 4155. http://www. orbisscholae.cz/archiv/2012/2012_2_03.pdf

H e j n ý, M., J i r o t k o v á, D., S l e z á k o v á, J., B o m e r ová, E., M i c h n o v á, J.: 20072011, Matematika pro 1.-5. ro£ník Z’. U£ebnice a P°íru£ka u£itele, Plze¬: Fraus.

H e j n ý, M., J i r o t k o v á, D.: 2004, Sv¥t aritmetiky a sv¥t geometrie, in: Milan Hejny, Jarmila Novotnae Nadá Stehlíková (eds.) Dvacet p¥t kapitol z didaktiky matematiky, Univerzita Karlova v Praze, Pedagogicka Fakulta, 125

136.

H e n d e r s o n, D. W., T a i m i n a, D.: 2005, Experiencing Geometry, Euclidean and Non-Euclidean with History, third Edition, Pearson Prentice Hall, Upper Saddle River, New Jersey.

V a n H e u v e l - P a n h u i z e n, M., B u y s, K. (Eds.).: 2004, Young Children Learn Measurements and Geometry, TAL Project, Freudenthal Insti- tute, Utrecht University, National Institute for Curriculum Developmnet (SLO) in collaboration with CED Rotterdam.

V a n H i e l e, P. M.: 1986, Structure and Insight, Academic press Orlando Florida.

H o l l e b r a n d s, K.: 2007, The role of a dynamic software program for

geometry in the strategies high school mathematics students employ, Journal

for Research in Mathematics Education, 38(2), 164192.

H o l l e b r a n d s, K.: 2003, High school students' understandings of geo- metric transformations in the context of a technological environment, Journal of Mathematical Behavior, 22(1), 5572.

J i r o t k o v á, D.: 2011, Generic models in geometry. in: J. Novotná, H. Moraová (eds.), Proceedings of the SEMT `11-International symposium, ele- mentary maths teaching, Praha: UK v Praze, PedF, 174181.

J o n e s, K.: 2000, Providing a Foundation for Deductive Reasoning: Stu- dents' Interpretations When Using Dynamic Geometry Software and Their Evolving Mathematical Explanations, Educational Studies in Mathematics, 44, 5585.

J o n e s, K.: 2002, Issues in the Teaching and Learning of Geometry, in: Lin- da Haggarty (ed.), Aspects of Teaching Secondary Mathematics: perspectives on practice, London: RoutledgeFalmer, Chapter 8, 121139. Retrieved from http://eprints.soton.ac.uk/13588

J o n e s, K., M o o n e y, C.: 2003, Making space for geometry in primary mathematics, in: I. Thomson (ed.), Enhancing primary mathematics teaching 31,5 London: Open University Press.

K a l i n o w s k a, A.: 2014, Poznawczy i kulturowy wymiar dezintegracji wczesnoszkolnych poj¦¢ matematycznych, (Anty)edukacja wczesnoszkolna, (ed.) D. Klus-Sta«ska, Ocyna Wydawnicza Impuls, Kraków, 373.

K a r p i « s k i, M., Z a m b r o w s k a, M.: 2015, Raport z badania nauczania matematyki w szkole podstawowej [Report of the survey ndings on Teaching Mathematics in Grammar School], Warsaw: IBE.

K e l l, H. J., L u b i n s k i, D., B e n b o w, C. P., S t e i g e r, J.

H.: 2013, Creativity and Technical Innovation Spatial Ability's Unique Role, Psychological science, 24(9), 18311836.

K l o b o u £ k o v á, J., J i r o t k o v á, D., S l e z á k o v á, J.: 2013, Enhancement of 3D imagination in the 1st and 2nd grades, Procedia  Social and Behavioral Sciences 93, Elsevier Ltd, 984989.

K o z h e v n i k o v, M., H e g a r t y, M.: 2001, A dissociation betwe- en object manipulation spatial ability and spatial orientation ability, Memory

& Cognition, 29(5), 745756.

K o z h e v n i k o v, M., H e g a r t y, M., M a y e r, R. E.: 2002, Revi- sing the visualizerverbalizer dimension: Evidence for two types of visualizers, Cognition and Instruction, 20(1), 4777.

K o z h e v n i k o v, M., K o s s l y n, S., S h e p h a r d, J. 2005, Spatial versus object visualizers: A new characterization of visual cognitive style, Memory & Cognition, 33(4), 710726.

L a n k i e w i c z, B., S e m a d e n i, Z.: 1993, Matematyka 2. Podr¦cznik,

WSiP, Warszawa.

L o p e z - R e a l, F., L e u n g, A.: 2006, Dragging as a conceptual to- ol in dynamic geometry environments, International journal of mathematical education in science and technology 37(6), 665679.

M a r i o t t i, M. A.: 1995, Images and concepts in geometrical reasoning, in:

R. Sutherland and J. Mason (Eds), Exploiting mental imagery with computers in mathematics education, Berlin: Springer.

M a r k o p o u l o s, C., P o t a r i, D., B o y d, W., P e t t a, K., C h a s e l i n g, M.: 2015, The Development of Primary School Students' 3D Geometrical Thinking within a Dynamic Transformation Context, Creative Education, 6, 15081522.

P i a g e t, J., I n h e l d e r, B.: 1966, La psychologie de l`enfant, Presses Universitaires de France. Paris.

P i t t a l i s, M., C h r i s t o u, C.: 2010, Types of reasoning in 3D geo- metry thinking and their relation with spatial ability, Educational Studies in Mathematics, 75(2), 191212.

R u t h v e n, K.: 2018, Constructing Dynamic Geometry: Insights from a Stu- dy of Teaching Practices in English Schools, in: G. Kaiser et al. (eds.), In- vited Lectures from the 13th International Congress on Mathematical Edu- cation, ICME-13 Monographs, https://doi.org/10.1007/978-3-319-72170-5_29.

521540

S c h a f f e r, R. H.: 2013, Psychologia dziecka, Wydawnictwo Naukowe PWN, Warszawa.

S h e a, D. L., L u b i n s k i, D., B e n b o w, C. P.: 2001, Importance of assessing spatial ability in intellectually talented young adolescents: A 20-year longitudinal study, Journal of Educational Psychology, 93(3), 604614.

T a l l, D.: 2001, What mathematics is needed by teachers of young children?, in: J. Novotná & H. Moraová (Eds.), Proceedings of the international sympo- sium elementary maths. teaching SEMT 01, Czech State: Prague.

T · m o v á, V.: 2017, How do pupils of the 5th and 6th grade structure spa- ce, in:] J. Novotná, H. Moraová (Eds.), International Symposium Elementary Maths Teaching SEMT '17, Proceedings. Equity and diversity in elementa- ry mathematics education, Prague: Charles University, Faculty of Education, 411421.

T · m o v á, V., V o n d r o v á, N.: 2017, Links between Success in Non- measurement and Calculation Tasks in Area and Volume Measurement and Pupils' Problems. Scientia in educatione, 8(2), 100129.

U r b a « s k a, E.: 1985, O stosunku kandydatów na nauczycieli klas pocz¡t- kowych do matematyki jako przedmiotu ich studiów, Dydaktyka matematyki, Seria V, Roczniki Polskiego Towarzystwa Matematycznego, 5, 97122.

U t t a l, D. H., M e a d o w, G., N. G., T i p t o n, E., H a n d, L. J.,

A l d e n, A. R., N e w c o m b e, N. S., W a r r en, C h.: 2013, The Malleability of Spatial Skills: A Meta-Analysis of Training Studies, Psychological Bulletin, American Psychological Association, 139(2), 352402.

X i s t o u r i, X., P i t t a - P a n t a z i, D.: 2011, Elementary students' transformational geometry abilities and cognitive style, Proceedings of Seventh European Research Conference in Mathematics Education, Rzeszów.

Umiej¦tno±¢ prowadzenia my±lowych manipulacji gurami geometrycznymi przez uczniów edukacji wczesnoszkolnej

S t r e s z c z e n i e

W artykule prezentujemy przebiegaj¡ce na trzech ró»nych etapach badania, dotycz¡ce strategii rozwi¡zywania jednego geometrycznego zadania. W maju 2015, Instytut Bada« Edukacyjnych w Polsce przeprowadziª badanie zatytuªo- wane Kompetencje trzecioklasistów. Analiza wyników jednego z zada« testu, rozwi¡zywanego przez 199 361 uczniów, dotycz¡cego obszaru wyobra¹nia geo- metryczna, stanowiªa pierwszy etap opisywanych bada«. W wyniku tej anali- zy okazaªo si¦, »e osi¡gn¦ªo ono niski stopie« rozwi¡zywalno±ci, równie» w±ród uczniów plasuj¡cych si¦ wysoko w caªym badaniu. Pojawiªa si¦ wi¦c natural- na potrzeba lepszego zdiagnozowania tego zjawiska. Dla rozpoznania strategii rozwi¡zywania tego zadania zostaªo przegl¡dni¦tych ok. 300 przesªanych roz- wi¡za«, co umo»liwiªo zidentykowanie ró»nych uczniowskich strategii. Jedn¡

z nich byªo wyobra»enie sztywnego ruchu (obrotu). Odkrycie tej strategii zmo- tywowaªo do postawienia kolejnego pytania: w jakim stopniu takie dynamiczne rozumowanie jest obecne w dzieci¦cych strategiach dotycz¡cych rozwi¡zywania wybranych zada« geometrycznych? Dlatego w kolejnym etapie zostaªy przepro- wadzone indywidualne obserwacje pracy dzieci nad tym zadaniem, w których uczestniczyªo 35 dzieci w wieku 10 lat. Wyniki wszystkich etapów zostaªy po- krótce przedstawione w tym opracowaniu, ze szczególnym zwróceniem uwagi na przebieg etapu trzeciego. Pomógª on bowiem ustali¢, »e rozumowanie dy- namiczne jest mo»liwe do wywoªania, ale wymaga specjalnych zabiegów dy- daktycznych. Taki wynik uwa»amy za wa»ny, gdy» mo»e on wytycza¢ drog¦

dla projektowania dydaktycznego, zgodnego z sugestiami zarówno teorii psy-

chologicznych zwi¡zanych z rozumowaniami przestrzennymi, jak i zaªo»eniami

dydaktycznymi teorii Milana Hejný'ego ksztaªtowania poj¦¢ geometrycznych.

Ewa Swoboda

The State Higher School of Technology and Economics in Jarosªaw Jarosªaw

Poland

e-mail: eswoboda@ur.edu.pl Maªgorzata Zambrowska

Maria Grzegorzewska University Warsaw

Poland

e-mail: mzambrowska@aps.edu.pl