# Self-Learning Geometric Transformations: A Framework for the “Before and After” Style of Exercises

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometric Transformations

#### 2.1. Matrix Representation and Single Transformation

_{1}, V

_{2}, V

_{3}, obtaining the transformed vertices ${V}_{1}^{\prime}$, ${V}_{2}^{\prime}$, ${V}_{3}^{\prime}$.

#### 2.2. Composite Sequences

#### 2.3. Local and Global Coordinate Systems

#### 2.4. The “Before and After” Style of Learning Exercise

## 3. Method

#### 3.1. Base Conditions for the Definition of Learning Exercises

#### 3.1.1. A Framework for Characterizing Learning Exercises and Their Solutions

- T(2, 18, 14) ∙ Ry(−90°) ∙ Rz(30°) ∙ M(−1, −1, 1) ∙ S(1.5, 0.35, 1) ∙ T(−2, −4, −10)
- T(2, 18, 14) ∙ Rx(−30°) ∙ S(1, 0.35, 1.5) ∙ Ry(−90°) ∙ Rz(180°) ∙ T(−2, −4, −10)
- T(2, 18, 14) ∙ Ry(−90°) ∙ Rz(210°) ∙ S(1.5, 0.35, 1) ∙ T(−2, −4, −10)
- T(2, 18, 14) ∙ Rx(60°) ∙ S(1, 1.5, 0.35) ∙ Ry(−90°) ∙ Rz(−90°) ∙ T(−2, −4, −10)
- T(2, 18, 14) ∙ Ry(−90°) ∙ T(1.898, 2.712, −10) ∙ Rz(210°) ∙ S(1.5, 0.35, 1)
- T(12, 20.712, 15.898) ∙ Ry(−90°) ∙ Rz(210°) ∙ S(1.5, 0.35, 1)
- Rx(60°) ∙ T(12, 24.124, −9.988) ∙ S(1, 0.75, 0.0875) ∙ Ry(−90°) ∙ Rz(−90°) ∙ S(2, 4, 1)

- Each step in the composite sequence should be a “basic” one.
- The “basic” steps are translations, scales, reflections (mirror), rotation X, rotation Y, and rotation Z. Students understand better the scales and reflections as separate transformations, according to the visual effect on the object; despite the fact that they could be written as just one matrix if they are contiguous in a sequence (from a mathematical point of view). Additionally, rotations should be separated by axis because they become meaningless for students if they appear combined in a single rotation matrix. On the other hand, translations, scales, and reflections are well understood as basic steps, no matter if they involve one or more axes at once; for instance, T(4, 6, 2), S(2, 0.5, 1), or M(−1,−1, 1), are generally perceived well as a “single” step.
- Transformations in the sequence should appear in the TRS order, but they should be expressed as “basic” steps. It then becomes a T∙R
_{3}∙R_{2}∙R_{1}∙M∙S order, where R_{1}, R_{2}, and R_{3}are a permutation of R_{x}, R_{y}, and R_{z}(although not every exercise requires the three rotations). We call this sequence the “extended TRS” order. Additionally, if the object is out of the origin (in its initial conditions), the translation necessary to place it at the origin should appear as an additional step in the sequence. The extended TRS order becomes T_{1}∙R_{3}∙R_{2}∙R_{1}∙M∙S∙T_{0}, where T_{0}translates the object to the origin, and T_{1}translates it to its final location. - Numerical information provided in the exercise should be explicitly identifiable in the matrices’ content as much as possible. Students easily identify these numbers with the exercise’s description and requirements.
- If two or more equivalent sequences fit the “basic” steps principle and the extended TRS pattern, the sequence with fewer steps is the best candidate for the “preferred” solution.

#### 3.1.2. Reducing Cases in Transformations’ Parameters

#### 3.1.3. Reducing the Number of Object Views to Specific Symmetries

#### 3.2. States Available for the Generation of Learning Exercises

#### 3.3. Automatic Exercises Generation and Solutions Feedback

#### 3.3.1. Automated Generation of Exercises with a Controlled Degree of Complexity

#### 3.3.2. Automatic Feedback on Solutions in Real-Time

#### 3.4. Software Tool 1: Geometric Transformations Visualizer

#### 3.4.1. System Architecture

#### 3.4.2. GTV Interface

#### 3.5. Software Tool 2 (Video Game): Geometric Transformations Cards

#### 3.5.1. Main Interface

- The goal zone: In this area, a cube is visualized after some random transformations are applied; this is the “goal”. The challenge is to figure out the correct ordering of the cards to build a sequence of steps to transform the object from its initial state to the “goal state”.
- The player zone: In this zone, when a new level starts, the cube appears in its initial state (without transformations). When the player drags cards from the deck into the sequence zone, the cube changes its state according to the transformations involved and the ordering of the cards.
- The deck: This zone is automatically populated with cards at the beginning of a round (an exercise). The game’s engine automatically chooses the type of cards and their number according to the complexity level at the time. Once the cards appear in the deck, the player can pick them up from this area and drop them into the sequence zone.
- The player dynamically decides the cards and their ordering during the round. The sequence is built step by step by moving cards from the deck into the sequence zone. The player can always change a partially built sequence by removing cards from the sequence zone and returning them to the deck.

#### 3.5.2. Visual Clues

#### 3.5.3. Game Mechanics

#### 3.5.4. Difficulty Levels

- The solution has a predefined number of slots, and the deck zone has a predefined number of cards. The arithmetic difference of these numbers is associated with the degree of difficulty. For example, there are only 6 permutations if the solution zone has three slots and the deck has three cards. Instead, there are 720 permutations if the sequence has three slots and the deck has ten cards. Each permutation represents a candidate solution (either correct or incorrect); that is, having a bigger number of options is associated with a greater overall exercise complexity.
- The complexity of a solution is directly related to its involved transformations. Lower complexities, for example, result after applying few (maybe only one) and easily perceived transformations, such as translations or very simple rotations. Instead, higher complexities arise by mixing scales, reflections, and rotations, in a specific order. Even with the same type of transformation, different degrees of difficulty can be defined. For example, rotations of 90 degrees are more easily perceived than rotations of 45 degrees.

#### 3.5.5. Cards Design

#### 3.6. The Empirical Test

#### 3.6.1. The Participants

#### 3.6.2. Research Design

- Case 1: Measure whether the non-CG and CG groups had a similar performance in a test evaluating GTs knowledge (the GTs Test in this study). The test was the same for both groups. We selected an independent t-test to compare whether there was statistical evidence that the mean score obtained in the GTs Test by both groups was significantly different. Regarding the estimation of the effect size, we calculated Cohen’s d for this case.
- Case 2: Measure whether the performance solving the exercises generated by the software tools and the performance on the GTs Test were correlated. We selected a bivariate analysis to test the strength of association between the scores obtained by each student in these tasks. This analysis was only performed on the non-CG group (experimental group).
- Case 3: Measure whether the study of GTs had an impact on the visual-spatial abilities of students. We selected a paired t-test (pre-test/post-test) to compare the performance of students in the Purdue Spatial Visualization Test (PSVT). The same analysis was independently performed on both the CG and the non-CG groups. The effect size, in this case, was estimated through Cohen’s d for each group.

#### 3.6.3. Recollection of Data

- Scores obtained in exercises generated by the software tool: We held four sessions of two hours each in the Computer Lab for the non-CG group to use the software tools. In the first session, they received twenty minutes of a general introduction to the subject of GTs, just to provide background but without any explanation of the methodology and formal treatment of transformations composition. After the introduction, they spent another ten minutes reviewing and understanding the GTVisualizer interface with the help of the instructor. Students were aware that during the rest of the session, they were not allowed to ask questions or clarify doubts about GTs, neither with fellow students nor with the instructor; their only “teacher” would be the software. For the rest of the session, they explored by themselves the software features, understood the matrices format, created their own sequences compositions, and solved exercises generated by the software. In the other three sessions, students used the second tool, the GTCards game, to try to solve the auto-generated challenges. The conditions of these last sessions were similar to those in the first one, with no communication with other students or the instructor to simulate a real scenario where they were alone at home using the software and learning GTs. At the end of each session, students were provided with a series of exercises that were previously generated by the software tools. All students solved the same exercises in a synchronous manner. Students were provided with answer sheets to record their responses.
- Scores obtained in the GTs Test: We wanted to compare the performance of both groups on the subject of GTs by answering a written test. We prepared 20 problems, including eight on general GTs knowledge and 12 about solving a series of “before and after” exercises, similar in structure and complexity to the exercises typically solved in the classroom in the CG course. Due to time constraints, the answers to the “before and after” exercises were of simple selection (given five possible answers) instead of performing calculations to get the transformed vertices, as usually is required in the regular course. We wanted to compare the “normal” learning strategies held in the classroom versus the proposed framework (materialized in the two software tools). Similar performance of students in both groups would suggest that tools are suitable to support the self-study of GTs and would be appropriate to be incorporated as supporting tools in a distance-learning syllabus of CG.
- Scores obtained in the PSVT: We applied the PSVT, developed by Guay [34], as a complementary test. PSTV is a multiple-choice test suitable for individuals 13 or older and has been used in engineering and sciences disciplines for more than 40 years to empirically measure the cognitive ability related to visual-spatial intelligence. The PSVT has been recognized [35] as a test with empirical reliability and validity evidence. We wanted to learn whether a similar impact on these abilities would be triggered by the “normal” coursework and study materials available for the CG students compared with the new software tools used by the non-CG students. We prepared a multimedia presentation with sixty problems; thirty problems on “rotations” and thirty problems on “views”, as supplied in the PSVT. Each slide in the self-paced animation corresponded to a single problem, and students had a fixed time (40 s) to synchronously solve each problem. The assigned time per problem was set according to the PSVT recommendation of twenty minutes per section.

## 4. Results

_{113}= 8.01 and p < 0.001, so we rejected the null hypothesis. Cohen’s d (effect size) was calculated as 0.491 from the means and standard deviations in Figure 15.

_{19}= 3.73 and p < 0.003, so we also rejected the null hypothesis. Cohen’s d (effect size) was calculated as 0.502 from the means and standard deviations in Figure 16.

## 5. Discussion

## 6. Conclusions and Future Work

- As mentioned in the results section, a long-term test with larger sample sizes and encompassing more academic terms (semesters) should be performed to validate the current results.
- Diverse computational techniques can be experimented with to “tune” the software tools. For example: (1) Adjust the difficulty degree associated with exercises to increase the “naturality” of how students perceive the increasing complexity as they advance their learning. (2) The real-time feedback when student builds composite sequences can be improved. (3) The video game can be enhanced with strategies to increase engagement and fun without sacrificing its main “serious” objective.
- Convert current tools from prototypes to more “consumable” learning tools with embedded analytics and higher availability. This change would provide helpful feedback on the learning outcomes in a broader spectrum with students with diverse needs and backgrounds.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Transformations matrices: (

**a**) Translation, (

**b**) Scale, (

**c**) Mirror or reflection, (

**d**) Rotation on X axis, (

**e**) Rotation on Y axis, (

**f**) Rotation on Z axis.

**Figure 7.**Restrictions placed on symmetric views: (

**a**) Chosen views with rotations only in multiples of 45 degrees; (

**b**) Comparison of loss of symmetry in four views (7, 8, 9, and 10).

**Figure 14.**Examples of card design. (

**a**) Rotation of minus 90 degrees on Y-axis; (

**b**) Reflection on X-axis; (

**c**) Scale factor of 0.5 on Z-axis; (

**d**) Translation of −1 on X-axis; (

**e**) Translation of one unit on X-axis and one unit on Y-axis.

Transformations | Axes | Options | Total Choices |
---|---|---|---|

Rotations | X, Y, Z | Multiples of 45 degrees (positive/negative) | Rx(45°), Rx(90°), Rx(135°), Rx(180°) Ry(45°), Ry(90°), Ry(135°), Ry(180°) Rz(45°), Rz(90°), Rz(135°), Rz(180°) |

Reflections (Mirrors) | X, Y, Z | All | Mx My Mz |

Scales | X, Y, Z | Double, Half | Sx(0.5), Sx(2.0) Sy(0.5), Sy(2.0) Sz(0.5), Sz(2.0) |

Translations | X, Y, Z | Fixed distance (positive/negative) | Tx(1.0) Ty(1.0) Tz(1.0) |

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**MDPI and ACS Style**

González-Campos, J.S.; Arnedo-Moreno, J.; Sánchez-Navarro, J.
Self-Learning Geometric Transformations: A Framework for the “Before and After” Style of Exercises. *Mathematics* **2022**, *10*, 1859.
https://doi.org/10.3390/math10111859

**AMA Style**

González-Campos JS, Arnedo-Moreno J, Sánchez-Navarro J.
Self-Learning Geometric Transformations: A Framework for the “Before and After” Style of Exercises. *Mathematics*. 2022; 10(11):1859.
https://doi.org/10.3390/math10111859

**Chicago/Turabian Style**

González-Campos, José Saúl, Joan Arnedo-Moreno, and Jordi Sánchez-Navarro.
2022. "Self-Learning Geometric Transformations: A Framework for the “Before and After” Style of Exercises" *Mathematics* 10, no. 11: 1859.
https://doi.org/10.3390/math10111859