Perception of Symmetry and Spatial Reasoning in 11–12-Year-Old Pupils

Abstract

Visual perception and visualization are crucial in mathematical thinking and, more importantly, in geometric thinking. Our research study aimed to follow the link of geometric transformations, mental rotation, spatial ability, and visualisation in geometric thinking. During a longitudinal study, we examined the spatial ability-related geometric competencies of 11–12-year-old students from three lower secondary schools. We analysed their test results from axial symmetry and its application in practical problems, orientation in spatial labyrinths, and building spatial constructions with three views. During a detailed analysis of the written results, a grounded theory hypothesis arose that there was a possible relationship between symmetry perception and spatial thinking. This hypothesis was tested with CHIC statistical analysis. The analysis, however, did not show any deeper connection between symmetry perception and spatial thinking from the given data. We can conclude that this connection occurs in individual cases but not in general.

1. Introduction

1.1. Visual Thinking, Visualisation, and Spatial Reasoning

Visual perception is a fundamental cognitive process important in mathematical thinking because it allows students to visualise abstract concepts and relationships. Some authors do not distinguish between visualisation and visual thinking, or when we focus on the study of three-dimensional geometry, the concepts visualisation and spatial visualisation are considered equivalent [1]. Visualisation is obviously assumed in solving geometric and solid geometry problems, but not exclusively. Concept of mental imagery can be found also in studies [2,3]. According to Lean and Clements [2] (pp. 267–268), mental imagery is the “occurrence of mental activity corresponding to the perception of an object, but when the object is not presented to the sense organ”. Cook in Clements [3] (p. 34), on the other hand, understands mental imagery as the “formation and retention of an image that involves no mental movement of the image once formed”.

The concept of visual mental imagery can be found in Kosslyn’s publications [4,5]. He states that although visual mental imagery and visual perception are not identical, they are functionally equivalent. It means that while visual perception occurs if “a stimulus is being viewed and includes functions such as visual recognition … and identification, visual mental imagery is a set of representations that gives rise to the experience of viewing a stimulus in the absence of appropriate sensory input” [5] (p. 334).

According to a broad analysis of the relevant literature carried out by Vágová [1], we can state that visualisation has a different meaning for many authors. This concept can be understood as a kind of reasoning activity and a way of thinking, including a process of a mental or physical action, an ability to represent and transform mental images, as well as a set of external representations and processes.

As stated by Duval [6], visualisation and intuition are the key elements in the study of geometric mathematical reasoning. Battista defines spatial reasoning as follows: “Underlying most geometric thought is spatial reasoning, which is the ability to ‘see’, inspect, and reflect on spatial objects, images, relationships, and transformations. Spatial reasoning includes generating images, inspecting images to answer questions about them, transforming and operating on images, and maintaining images in the service of other mental operations.” [7] (p. 843).

1.2. Perception of Symmetry

Perception of symmetry is fundamental in revealing spatial relations beginning from early childhood. The child identifies the symmetry of left and right side on his own body and many various objects carrying the sign of symmetry also. The name symmetry originates from the words together (son) and measure (metron). The lefthand–righthand connection is easily transferable from 3D space to 2D plane as an axial symmetry around a vertical axis. In Kuchemann [8], it was shown that children aged 6–10 clearly distinguish between axial symmetry around a vertical axis, a horizontal axis, and other types of axial symmetries. For them, these are distinct categories of symmetry. The opportunity to generalise and group these symmetries into one category comes later and varies significantly with age. “Despite the importance of the concept in so many areas and at so many levels of complexity, and despite the evident interest it holds for students of all ages, symmetry remains a relatively underdeveloped area of study in elementary school curricula. Once established, it tends to be a mechanical and static concept” [9] (p. 1) rather than a richly generative mathematical principle. Real mirroring in 3D space is tightly connected with imagining of motion in space. To distinguish between the mirror image of a complex object and its rotated image can be a challenging task also for a trained brain. In activities aimed to train spatial reasoning are often involved symmetry games as mirroring, identifying symmetries and asymmetries, or architectural examples [9].

1.3. Placing Our Research in the Broader Context of STEAM Education Research

A wider groundwork for our research was created by a long-term research project in STEAM education (STEAM stands for Science, Technology, Education, Art, and Mathematics) conducted at three primary schools in Slovakia. The aim of the project was to develop an interdisciplinary study program for the fifth and sixth grades of primary schools in the framework of the STEAM educational model. There were 20 activities (innovative task series) developed in mathematics (especially in geometry), 13 micro-projects in informatics (using computer visualisation), and 17 activities in art, which were connected in some way with visual perception and creativity. The partial results were published [10,11] or will be presented in relevant journals. The global evaluation of the main results reached in the development of spatial abilities in participating pupils during the research project realisation was published in Szabó et al. [12]. In Kmeťová and Nagyová Lehocká [13], a special transition between 2D and 3D perception was shown in geometry learning. In the current article, we focus on three geometric activities related to symmetry and spatial thinking. As part of the overall study, we try to explore whether there is a closer relationship between symmetry perception and spatial ability. We also compare the results of related tests by hierarchy trees using the statistical program CHIC [14].

From the above mentioned 20 task series in geometry, we chose 3 for our purpose. The first test is about planar and spatial symmetry connected with real life. The second one focuses on spatial orientation using 3D labyrinths. The third test uses Ubongo 3D geometric puzzle game parts for testing spatial imagining combined with views of structured objects constructed. The detailed characteristics of the three tests are described in the next chapter. In addition to the overall evaluation and comparison of results, we also present two case studies [15] about the thinking of selected individuals by comparing their works on symmetry and spatial orientation. During a detailed study of the achievements of some students, we concluded that it would be worthwhile to examine in depth the relationship between students’ spatial vision and symmetry perception.

2. Materials and Methods

2.1. Research Questions

Research questions are formulated according to the information above, as follows.

• Is there a strong relationship between spatial and planar symmetry perception in students?
• Is there any connection between symmetry perception and spatial ability globally?
• Is there a connection between mental rotation (and perhaps mental mirroring) and spatial orientation in the thinking of individual students?

2.2. Research Tools

In our research study, we have used three task series. In the following part, we describe their purpose.

2.2.1. Axial Symmetry Test

The first of the task series is about symmetries and symmetry-related geometric relationships where the axial symmetry has a fundamental role; we named it Axial Symmetry test (Appendix A/Supplementary Material). The tasks have teaching character; they show the way step by step for investigation in the field of geometric transformations. The first two tasks require drawing axes into the given picture followed by a task with applications of axial symmetry for different shapes. Before the fourth task, the orientation-changing property of axial symmetry is described and illustrated. Using this property in the fourth task, the children have to connect two axial symmetries and notice that the result has the original orientation. The next five tasks (5–9) are devoted to investigation where the final result is intended to be the knowledge that the changing orientation of the axially symmetric shapes during the transformation does not change the result shape. The difference is illustrated for children in Figure 1a,b, where the shape in (a) is axially symmetric, so its image is identical with the original, but the image of a left foot is a right foot—see (b).

Once the children have learned the properties of axial symmetry, they are encouraged to think out inscriptions that are not changed after mirroring. Tasks 10–14 contain mirror image exercises from everyday life, such as mirroring on the water surface or window glass. The demandingness of these exercises follows from easily exchangeable images of digital numbers 2 and 5 as it is illustrated in Figure 2.

The last part of the axial symmetry test was devoted to spatial imagination on a cube including symmetry and spatial motion, so tasks 15–17 were the most demanding and the least prosperous. In task 15, it was necessary to consider not only the numbers on the net of the cube but also their position after rotation (the correct answer is c). In task 16, children had to recognise that the imprint of letter B was a mirror picture of B, so the correct answer is d). Task 17 included both rotation in two directions and mirroring, the correct answer c) is shown in Figure 3.

2.2.2. Spatial Labyrinths

This test (Appendix B: Labyrinth/Supplementary Material) focuses on spatial imagination through motion in 3D space using layers of labyrinths. The introduction part of the test shows how to describe and depict the motion step-by-step (also up and down between the layers, i.e., floors of the given building). The idea of creating 3D labyrinths and the method of depicting the steps of solutions is based on the book [16].

The Spatial Labyrinth test consists of 6 tasks of increasing difficulty. The tasks include finding the path from the entrance to the exit, finding at least two paths, finding the number of solutions of a given labyrinth, finding the shortest path, as well as designing a two-layer labyrinth.

2.2.3. Ubongo Test

The name of this test is derived from the 3D geometric puzzle game Ubongo. The Ubongo test (Appendix C: Ubongo/Supplementary Material) focuses on the part of spatial imagination that we need in order to construct prescribed shapes from given solid parts. We used four construction elements of Ubongo in the test tasks (Figure 4).

The test begins with the introductory part, where the used parts of the puzzle game are demonstrated on an example with a photo and exact description of the floor plan, top view, front view, and side view of the construction (Figure 5). We always used the left side view as a side view throughout the test.

In addition to the elements consisting of 2 red, 3 green, and 4 blue unit cubes mentioned in the demonstration, we also used a yellow element consisting of 5 yellow unit cubes in the tasks. Tasks 1 and 2 require laying out the structure given by the contours of its floor plan, as well as top, front, and side views. To make the task easier, the complete coloured front view is given. Task 3 contains two help questions: “How many unit cubes does the shape consist of? What colour elements should be used?” According to the floor plan, one can see that the first floor contains 6 unit cubes; according to the front and side views, the structure is a two-storey building without missing part on the second floor. It means that it consists of 12 unit cubes. To put together 12 unit cubes from the given elements, the only possibility is to use all the elements except the red one. The same approach is valid for the 4th and 5th tasks. The 6th task requires building a construction consisting of 14 unit cubes, which means using all the given elements. To make it easier, a clue for the green part is given in the floor plan.

2.3. Participants

The research was conducted on 71 students in the 6th grade, age 11–12 years (labelled as S01, S02, … S71), from 3 primary schools in Slovakia. The mathematics teaching implemented in the participating 3 classes was the usual education based on the National Program of Education (ISCED1) [17] used in Slovakia generally, with no extra teaching hours in mathematics. “As far as spatial geometry skills are concerned, the geometry part of the educational program contains constructions of basic shapes and exercises on constructing cube buildings according to a plan or picture and creating a plan of a given cube building. Students have to display the front view, right view, left view, and top view of the object” [13] (p. 5). As a pre-test, to establish the level of spatial skills of the participating students, a reliably evaluated spatial ability test was used [13,18,19]. The broader characteristics of the participants and the results of this pre-test are published in Kmeťová and Nagyová Lehocká [13].

2.4. Case Studies

In education, we often appreciate deep understanding of individual thinking more than some generalised knowledge about mean result of students’ thinking collected or acquired from large quantitative research studies because they convey different aspects. According to Bassey [15], quoting Cohen and Manion, they state that “Unlike the experimenter who manipulates variables to determine their causal significance or the surveyor who asks standardised questions of large, representative samples of individuals, the case study researcher typically observes the characteristics of an individual unit—a child, a class or a school” [15] (p. 24).

Case study is a universal term for an individual investigation. The case study is therefore a form of qualitative descriptive research. It only draws conclusions about a small group or an individual. In this case, we do not expect a universal, generalisable discovery, but the emphasis is placed on the deep exploration and description of the individual’s thinking [20]. The case study researcher may answer the question of why things occur as they do in the education process and investigates the emerging patterns [21].

2.4.1. Exploratory Case Study

Yin [22] distinguished three forms of case studies, which he named exploratory, explanatory, and descriptive case studies. The purpose of the exploratory case study is to determine the question and hypothesis of a subsequent investigation [22]. The exploratory case study can also discover a relationship between phenomena by direct observation, which satisfies the definition of the grounded theory approach according to Glaser [23], and refers to the emerging patterns [15,21,24].

2.4.2. Atomic Analysis

In our case studies, we followed the atomic analysis protocol [25]. The atomic analysis of written problem-solving steps aims to uncover and fully understand the solver´s thinking during the problem-solving process. According to Stehlíková [26], atomic analysis is based on two ideas, atomisation of the solving process and comparative analysis. Atomisation means that everything that the student wrote during the solution is broken down into the smallest components, the so-called atoms, i.e., small meaningful parts of the solution. The researcher tries to understand what thoughts guided the solver when he or she crossed something out, corrected it, or used different signs.

The authors state the following practical advantages of the method [25]:

• The revealed way of thinking will make it possible to advise the student how to overcome obstacles in learning.
• The researcher, who repeatedly tries to penetrate the pupil’s thinking, increases the sensitivity to understanding the pupil’s actions.

We used this method to reveal the individual students thinking during geometry transformation tasks as well as spatial construction and spatial motion visualisation.

2.5. CHIC Analysis

The acronym CHIC analysis stands for Correspondence and Hierarchical Cluster analysis [14]. The relevant software package contains a package for Correspondence Analysis (CA) and Hierarchical Cluster Analysis (HCA). According to Marcos et al. [14], “CA is a multidimensional data analytic method, suitable for graphically exploring the association between two or more, non-metric variables without a priori hypotheses or assumptions” (p. 1). The additional use of HCA procedure offers the possibility of creation of a dendrogram for easy visual identification of higher- or lower-level connections between examined elements (in our case they are elementary steps of task-solving). By introducing implicative analysis into the package, now the name CHIC stands for Cohesive Hierarchical Implicative Classification (Classification Hiérarchique Implicative et Cohesitive in French original [27]), and the CHIC allows for the creation of similarity trees, hierarchy trees, and implication graphs [27,28].

Similarly, as other researchers have done [28,29], we used the CHIC statistical package to unravel hierarchical connections between parts of our research study data.

3. Data Analysis and Results

3.1. Quantitative Analysis of the Tests

The maximum number of points that could be obtained from the three tests mentioned above was 54, comprised of 23 from the Axial Symmetry test, 10 from the Labyrinth test, and 21 from the Ubongo test. None of the students reached the maximum number of points; the best result was 45 points. Figure 6 shows the distribution of the number of students in 10-point interval groups. The graph shows that some tasks were demanding or challenging for students, i.e., there are more students in the 0–10 points group than in the 40–50 points group.

The points achieved from the three tests separately are shown in Figure 7, Figure 8 and Figure 9.

The best score was achieved from the 3rd task of the Axial Symmetry test (70% successfulness, probably because similar exercises are parts of school lessons; see Appendix A/Supplementary Material) and the 2nd task of the Ubongo test. The 5th task of the Ubongo test proved to be the most challenging (see Appendix C/Supplementary Material); only 20% of the possible points were achieved.

3.2. The Case Studies

In this analysis, we are examining full solutions of two students, one of the best solutions and one mediocre solution, because we think those can also characterise the thinking of other students with similar results. No student gave fully correct solutions for all tasks. All students made some mistakes.

3.2.1. Atomic Analysis of a Good Solution

We divided all solutions of test tasks into elementary steps, each of which was evaluated with one point. The student labelled as S18 reached 45 points from the maximum possible 54 points, which was one of the best solutions.

The Axial Symmetry test evaluation (Appendix A)

Task 1. Drew the axes correctly (1 point).

Task 2. Missed axes in the black part of the picture (0 points).

Task 3. Constructed correctly (4 points).

The text after task 3 was an introduction to the next task; it showed the attribute of the mirroring image presented in the next task. Student S18 (as well as many others) ignored this text and only commented on the following two illustrative pictures with the words good, good. Missing the written information, he did not solve task 4 correctly.

Task 4: Constructed the mirror images of the given shapes but did not answer the questions concerned with mirroring and the order of axes in sequences of reflexions (1 point out of 2).

Task 5. Precisely constructed picture (1 point).

Task 6. Named the letters with required attributes but did not distinguish the horizontal and vertical axes (1 point out of 2).

Task 7. Precisely constructed reflexions (2 points).

Task 8. Correct answer (1 point).

Task 9. Understood the difference between the reflection of symmetric and non-symmetric shapes with a mistake (1 point). Student S18 was careless when he mirrored the letter K which was correctly reflected in solutions 7a) and 7b) (in Figure 10, where he gives an example of an axially symmetric word according to a previously provided pattern).

Task 10. The reflected numbers constructed on the blue part (according to the horizontal axis, see Figure 11) are incorrect. Student S18 made it automatically, without thinking about the details. He was able to apply axial symmetry in previous tasks, so we can see that this mistake arises from neglecting the known rules of symmetry in practical circumstances (0 points).

Task 11. Once student S18 thought that the reflection worked as simply as in task 10, he applied the same simplified but incorrect method in the next tasks. The correct solution of task 11 (Appendix A/Supplementary Material) would be 22:51, but he wrote 12:55, as he supposed according to the previous task, ignoring that the axis of the required reflection is horizontal (0 points).

Task 12. Here the axis of reflection was vertical, but the student again neglected the details of reflecting objects and did not recognise that the digit 2 mirrors into digit 5 and vice versa. He thought that this was such a simple exercise that it was enough to rewrite the numbers in the opposite order. Seemingly, he even forgot to switch the order of digits in the place of seconds—for the first attempt he wrote 2 instead of 4 (see Figure 12, 0 points).

Task 13. This task was easier because it did not contain the confusing digits 2 and 5. To make the difference between digits 6 and 9 in the mirror was not so challenging (1 point). Almost all the students answered it correctly.

Task 14. This task offers an opportunity to return to the previous answers and to correct them if the respondent realises the rules he or she is required to apply. Student S18 did not grasp this opportunity (0 points).

Task 15. This task required the student to visualise spatial motion, to imagine the position of numbers on the net during the motion in three directions (Appendix A). The correct answer is c); student S18 chose net b) where the numbers were in a different order (0 points).

Task 16. Student S18 chose the wrong answer a), probably influenced by the previous question, where the cube was wrapped into the net (0 points). In this task, the correct answer was d) since the question referred to the imprint left by the paint, which means mirrored letters on the white table.

Task 17. Student S18 chose the incorrect answer d), where letter Z is under the letter R, contrary to the given picture of the cube (0 points). The correct answer c) is illustrated in Figure 3.

Student S18 scored 13 points out of 23 in the Axial Symmetry test. His mistakes arose from carelessness in reading the text and quick decision-making without considering details when applying theoretical rules to practical problems.

The Labyrinth test evaluation (Appendix B/Supplementary Material).

Tasks 1–6. Student S18 understood the task questions and oriented in the given 3D labyrinths properly. He was able to imagine the spatial motion and visualise his solution by drawing the routes into the given plans of the buildings. Figure 13 shows his solution of task 5. He also designed a correct two-storey labyrinth with exactly one solution (Figure 14), so he achieved 10 points for Test L, which was the maximum.

• The Ubongo test evaluation (Appendix C)

Tasks 1–6. All the tasks were answered correctly, which means the correct colouring of the given blank views. Student S18 had first built the required constructions, then accordingly coloured the floor plan, top view, front view, and side view. Figure 15 shows his solution of task 6, the floor plan, top view, and front view of the construction from the given Ubongo elements (Figure 4).

Student S18 collected all the possible 21 points for the Ubongo test.

3.2.2. Atomic Analysis of a Mediocre Solution

Student S37 scored 30 points from the possible 54, so we consider his solution to be average or mediocre.

The Axial Symmetry test evaluation

Task 1. The axes are correctly drawn (1 point).

Task 2. Student S37 found only 2 axes out of 6 (0 points).

Task 3. Three of four reflections were constructed correctly (3 points). Figure 16 shows the wrong construction 3b).

Task 4. The mirror images are constructed, but the questions concerned with mirroring and the order of axes in sequences of reflexions are not answered (1 point out of 2).

Task 5. Correct answer (1 point).

Task 6. Named letters with required attributes, correctly distinguished them according the horizontal and vertical axes (2 points).

Task 7. Precisely constructed reflexions (2 points).

Task 8. Correct answer (1 point).

Task 9. No answer (0 points).

Task 10. The student wrote down the answer not recognising that the mirror image of 2 is 5 and the mirror image of 5 is 2 in this digital representation (Figure 17a, 0 points).

Task 11. Without considering axial symmetry properties, the student wrote the previous time, 12:55, into the picture (see Figure 17b, 0 points).

Task 12. The student repeatedly neglected the details of the mirroring digits. He answered 14:25:42 instead of 14:52:45 (0 points).

Task 13. Student S37 realised that the order of digits is changing in mirroring, so he tried to change the order of digits in the mirror image according to the horizontal axis (Figure 18). He began writing 64 to the place of minutes, which he corrected to 46 but he left changed the order of the last two digits. The solution for the mirror image according to the vertical axis was correct (1 point out of 2).

Task 14–17. No answers (0 points).

Student S37 scored 11 points out of a total of 23 points for the Axial Symmetry test. He was confused by the changing orientation of shapes and could not clearly overview which properties are invariant in axial symmetry.

• The Labyrinth test evaluation

Tasks 1–2. Correct solutions (2 points).

Task 3. Only one correct solution out of two (1 point).

Task 4. Both solutions were founded (2 points).

Task 5. The given solution is not feasible (see Figure 19, 0 points).

Task 6. No answer (0 points).

Student S37 scored 5 points out of 10 in the Labyrinth test. He was able to find his way around a two-storey building, but the three-storey house in task 5 was a big challenge for him.

• The Ubongo test evaluation.

Task 1. Student S37 found the right structure and coloured the floor plan and top view correctly, but he changed the left-side view to the right-side view (Figure 20, 2 points out of 3).

Task 2. Correct solution; in this case he has given the left side view correctly (3 points).

Task 3. Correct solution up to the changing of the left-side view to the right-side view again (3 points out of 4).

Task 4. According to the given blank views, the construction consists of 12 unit cubes; it means the usage of elements from 3 green, 4 blue, and 5 yellow unit cubes. The student tried to use the red, blue, and yellow parts, so his solution given by four coloured views was not feasible (0 points).

Task 5. Correct solution (4 points).

Task 6. According to the floor plan and top view given by the student, we can see that he constructed the required building properly (as it is in Figure 15); however, the given front view makes no sense (Figure 21, 2 points out of 3).

Student S37 scored 14 points out of 21 from the Ubongo test.

During the atomic analysis of students´ written work, we came to the realisation that, like in the case of student S37 above, some mistakes occurred simultaneously in symmetry problems (Test A) and in construction viewing problems (Test U). Specifically, we observed a connection between errors in axial symmetry applications for spatial problems and types of mistakes such as changing views upside down or left-side view instead of the right-side view. In some cases, we also noticed that only one part of the view was wrongly changed to a mirror image, while the other part was correct. For example, students S36 and S44 incorrectly solved task 11 in Test A and also made an error in solving task 6 in test U when they interchanged the left and right sides of the front view. (Figure 22 and Figure 23).

The repeated occurrence of similar simultaneous mistakes led us to a grounded theory hypothesis that there was a connection between perception of symmetry and spatial reasoning in 11–12-year-old students. The same idea was formulated in The Robertson Program [9], based on individual observations. Consequently, we decided to test our hypothesis with CHIC statistical analysis [14].

3.3. The CHIC Statistical Analysis

Input data for the CHIC statistical software are the so-called didactic variables consisting of points 0 or 1 for each elementary step of the solution for each task. The elementary steps are denoted as Ai.j for Axial Symmetry test, Li.j for Labyrinth test, and Ui.j for Ubongo test, where i is the number identifying the task and j is the number identifying the step in it. The one-step tasks are labelled with Ai, Li, or Ui, where i is the task number in the respective test. Results of students (labelled as S01, S02, … S71) are organised into the variables table, the so-called indicator matrix [14]. Aware of the sensibility of CHIC statistics to input data [27], we excluded all the students´ results with missing tests A, L, or U (for students who were not present at school during some of the testing). We got an indicator matrix consisting of 47 rows (for 47 students) and 54 columns (for 54 test points).

3.3.1. CHIC Results for Tests A, L, and U

CHIC analysis has not revealed any connections between the results of tests Axial Symmetry (test A) and Labyrinth (test L), nor between the Axial Symmetry and Ubongo (test U). The most interesting results are presented in graphs describing the results obtained (implicative tree, implicative graph).

The implicative tree for the Axial Symmetry test, Labyrinth test and Ubongo test contains no connection between the test results of the others. Relationships were shown only within the individual tests A, L, or U. Since the tasks in the different tests are not related to each other, we displayed only the relevant part of the implication tree and the relationship in the form of implication and equivalence in Figure 24. Implications highlighted in red are more significant levels between variables than the following or previous levels. There is an implication L3.1. and L1 (cohesion 1), which means that if a student gave one of two solutions for task 3 in the Labyrinth test, then he had solved task 1 also. Another more significant relationship is between the variables U4.1 and group of variables $\left(U4.4 \Rightarrow \left(U4.2\iff U4.3\right)\right)$ also with cohesion 1 at each level. This shows that if a student coloured the floor plan in the task 4 of the Ubongo test correctly, then he or she gave the full solution of the task correctly.

From the next implicative graph as well, the connections between the three tests were not obvious at all. Figure 25 shows the relationships between variables that had an intensity of 100%. These are the solutions to the problems marked U4 and U6, and between the different steps the students took to solve them. This intensity is between the variables U6.2, U6.1, U6.3 and U4.3, U4.2, U4.1, U4.4 (Figure 25). It means that the CHIC analysis revealed strong implications only within individual tasks, namely the strongest one in task 4 and 6 of the Ubongo test.

3.3.2. CHIC Results for Tests A and U

In the next analysis, we compared the results only from the Axial Symmetry and Ubongo tests (tests A and U, respectively). The exclusion of the Labyrinth test did not create any new, interesting relationships between the steps that students took to solve the tasks. A selection of interesting relationships is presented in the following implicative graph and implicative tree.

All the relationships between the variables shown in Figure 26 have cohesion 1. The strongest implication is the one between the variable U4.1 and the group of variables U4.4, U4.2, U4.3. Another strong equivalence is between the variables U6.1 and U6.2. This shows that there is an equivalence between having a correct floor plan and top view in the solution of task 6 of the Ubongo test and that if the front view is correct, then probably the floor plan and the top view are also correct in the solution of the given task.

The only connection between the two tests is the one shown in Figure 27. Relationships shown in red have an intensity of 100%, and those shown in blue have an intensity of 99%. The relationship between variables U5.4 and U5.1 indicates that if a student constructed the building in task 5 in the Ubongo test, then mostly he also drew the views correctly.

Only variables A7.1 and U5.4 are linked in the Axial Symmetry and Ubongo tests. This means that the only strong connection that CHIC analysis could find between the Axial Symmetry and Ubongo tests is the connection between task 7 in test A (which is about the construction of a mirror image) and task 5 in test U, where some students mistakenly changed the front view by mirroring or rotating it (e.g., Figure 28).

Overall, the results of CHIC analysis did not fully confirm our grounded theory hypothesis; they show statistically only one connection (mentioned above) between the perception of symmetry and spatial reasoning in problem solving.

4. Conclusions

Our research investigated the interconnection between geometric transformations, mental rotation, spatial ability, and visualisation in the geometric thinking of 11–12-year-old students. Through a series of three targeted tests—axial symmetry, spatial labyrinths, and spatial constructions—we sought to understand how these competencies are linked. By analysing the test results, a grounded theory hypothesis was created, which assumes a relationship between the perception of symmetry and spatial thinking. This hypothesis was tested by CHIC statistical analysis. Interestingly, the Axial Symmetry test, the Labyrinths test, and the Ubongo test were not associated with each other in the CHIC analysis. It turned out that, contrary to our previous assumption based on individual observations, there were no close relationships between symmetry perception and spatial reasoning. These results were based on individual observations also observed by The Robertson Program [9].

As for research questions, first, regarding the relationship between spatial and planar symmetry perception, our findings suggest that while individual cases demonstrate a strong connection, this is not uniformly evident across the entire group of students in the study. Detailed atomic analysis revealed cases where symmetry perception and spatial reasoning intersect, but these occurrences were not widespread.

Second, regarding the global connection between symmetry perception and spatial ability, our findings from the CHIC statistical analysis suggest no significant relationship. The results did not confirm a broad linkage between symmetry tasks and spatial reasoning capabilities. This emphasizes that while students may understand plane symmetry relations, it doesn’t universally translate into enhanced spatial abilities.

Third, evaluating the connection between mental rotation (and mental mirroring) and spatial orientation in students’ thinking revealed more nuanced results. Although our initial hypothesis posited a significant connection, the statistical analysis reflected that only specific tasks showed this linkage. In particular, the correlation between errors in symmetry tasks and spatial orientation tasks suggests that accurate mental rotation and mirroring are based on well-developed plane geometric transformation skills.

In conclusion, while there are observable cases of a relationship between symmetry perception and spatial reasoning, these connections are not pervasive among all students participating in the study. Our results highlight the complexity of spatial ability development and suggest that targeted education on plane geometric transformations could enhance overall spatial reasoning skills. The CHIC analysis did not fully confirm our hypothesis, showing only a limited connection between symmetry perception and spatial reasoning in problem solving. This underlines the need for continued research and refined educational strategies to support diverse cognitive development in geometry learning. Future studies should aim to further clarify these complex relationships and improve educational interventions to better equip students with critical thinking and problem-solving skills necessary for success in STEAM disciplines.