purposefulness. According to the goal to be achieved there are: 'tasks to find', 'tasks to prove' and 'tasks to construct' (Polya, 1966). According to the level of thinking required for solving there are: tasks with lower cognitive demands that serve to develop procedural knowledge and skills and tasks with higher cognitive demands that serve to develop conceptual knowledge (Smit & Stein, 1998). According to purpose there are tasks that facilitate the transition from school to university level mathematics education, also known as "unusual tasks" (Breen et al., 2013), which are a type of tasks with higher cognitive demands. For learning and teaching the process of solving tasks, as well as for the process of proving Polya (1966) proposed four phases: (1) understanding the task, (2) devising a plan, (3) carrying out of the plan and (4) looking back.

Numerous studies in education confirm the multiple benefits of solving tasks with higher cognitive demands. For example, solving unusual tasks requires relying on previous experience, higher mental work, more patience and perseverance, searching for and establishing relationships between different concepts, etc., which results in flexibility in the thinking process and the development of mathematical thinking and reasoning. At the same time, they can serve as a real detector of difficulties and misunderstandings because when solving them, typical mistakes of students coming to the fore as well as their creativity (Leikin & Lev, 2007; Breen et al., 2013).

## Development of geometric thinking

Van Hiele's theoretical framework is widely accepted to monitor and ensure the development of geometric thinking, through three aspects. The first aspect refers to the process of developing abstract thinking, hierarchically through five levels: recognition, analysis, informal deduction, formal deduction and level of rigor. The second aspect refers to considering the characteristics of the five-part model: structure, principle of object duality and methods of thinking, communication, mutual (mis)understanding, age. The third aspect refers to the learning process in five stages that contribute to progress from one level of thinking to another: inquiry/information, directed orientation, explication, free orientation and integration (Van Hiele, 1986; Crowley, 1987). The results of education research confirm the effectiveness of the teaching strategy according to van Hiele's stages of learning: the stages provide an effective framework for clearly structuring teaching units and guiding students from one level to another (e.g. Dongawi, 2014), and their application improves students' geometric thinking, understanding geometric concepts, and consequently learning outcomes (e.g., Crowley, 1987; Teppo, 1991).

## Development of visual-spatial abilities

In recent years, the Cattell-Horn-Carroll (CHC) theory of cognitive abilities, which represents the most comprehensive structural model of intelligence, has been increasingly used. According to the CHC model, spatial ability is referred to visual processing (labeled Gv), and implies "the ability to generate, perceive, analyze, synthesize, store, retrieve, manipulate, transform, and think with visual patterns and stimuli" (Flanagan & Shauna, 2014, p. 7). According to Lohman (1993), individual differences in spatial abilities are realized in the speed of performing transformations, especially rotation, then in the skill of creating and retaining mental images, the amount of visual-spatial information that a person can maintain in an active state, and finally in the sophistication and flexibility of available strategies to solve certain tasks.

Through various types of research, a strong mutual connection between visual-spatial abilities and
purposefulness. According to the goal to be achieved there are: 'tasks to find', 'tasks to prove' and
'tasks to construct' (Polya, 1966). According to the level of thinking required for solving there are:
tasks with lower cognitive demands that serve to develop procedural knowledge and skills and tasks
with higher cognitive demands that serve to develop conceptual knowledge (Smit & Stein, 1998).
According to purpose there are tasks that facilitate the transition from school to university level
mathematics education, also known as "unusual tasks" (Breen et al., 2013), which are a type of tasks
with higher cognitive demands. For learning and teaching the process of solving tasks, as well as for
the process of proving Polya (1966) proposed four phases: (1) understanding the task, (2) devising a
plan, (3) carrying out of the plan and (4) looking back.

Numerous studies in education confirm the multiple benefits of solving tasks with higher cognitive demands. For example, solving unusual tasks requires relying on previous experience, higher mental work, more patience and perseverance, searching for and establishing relationships between different concepts, etc., which results in flexibility in the thinking process and the development of mathematical thinking and reasoning. At the same time, they can serve as a real detector of difficulties and misunderstandings because when solving them, typical mistakes of students coming to the fore as well as their creativity (Leikin & Lev, 2007; Breen et al., 2013).

Development of geometric thinking

Van Hiele's theoretical framework is widely accepted to monitor and ensure the development of geometric thinking, through three aspects. The first aspect refers to the process of developing abstract thinking, hierarchically through five levels: recognition, analysis, informal deduction, formal deduction and level of rigor. The second aspect refers to considering the characteristics of the five-part model: structure, principle of object duality and methods of thinking, communication, mutual (mis)understanding, age. The third aspect refers to the learning process in five stages that contribute to progress from one level of thinking to another: inquiry/information, directed orientation, explication, free orientation and integration (Van Hiele, 1986; Crowley, 1987). The results of education research confirm the effectiveness of the teaching strategy according to van Hiele's stages of learning: the stages provide an effective framework for clearly structuring teaching units and guiding students from one level to another (e.g. Dongawi, 2014), and their application improves students' geometric thinking, understanding geometric concepts, and consequently learning outcomes (e.g., Crowley, 1987; Teppo, 1991).

Development of visual-spatial abilities

In recent years, the Cattell-Horn-Carroll (CHC) theory of cognitive abilities, which represents the most comprehensive structural model of intelligence, has been increasingly used. According to the CHC model, spatial ability is referred to visual processing (labeled Gv), and implies "the ability to generate, perceive, analyze, synthesize, store, retrieve, manipulate, transform, and think with visual patterns and stimuli" (Flanagan & Shauna, 2014, p. 7). According to Lohman (1993), individual differences in spatial abilities are realized in the speed of performing transformations, especially rotation, then in the skill of creating and retaining mental images, the amount of visual-spatial information that a person can maintain in an active state, and finally in the sophistication and flexibility of available strategies to solve certain tasks.

Through various types of research, a strong mutual connection between visual-spatial abilities and
mathematical (especially geometric) achievements has been confirmed (e.g., Clements et al., 1997; Bruce & Hawes, 2015). In mathematics education research, the definition of visualization proposed by Arcavi (2003), which includes products and processes, is widely accepted. Visualization products include various types of mental images (Presmeg 1986), various types of external representations (Nakahara, 2007) as well as their mutuality. Visualization processes include the processes of creation and use of products, and differ with respect to the place, manner or purpose of creation/use. For successful work in geometric figures, Duval (1998) proposes a theoretical framework that considers four types of figure apprehension: perceptual, sequential, discursive and operational. The processing of the figure begins in a pure perceptual recognition of what is shown, and then, through mutual use of sequential, discursive and operational processing of the figure, the mathematical message that the figure represents in the appropriate context is revealed. To master the transition from pure perceptual apprehension to recognizing what is mathematically important, students need to know and develop different visualization processes, first separately and then in mutual coordination (Duval, 1995). In order to successfully learn geometry, it is necessary to develop the "geometric eye" through various types of tasks in which one should first gain certain practical experiences: by experimenting, measuring, drawing, etc., then make statements based on what has been observed and finally prove these statements by connecting them with definitions, axioms and theorems which are already known (Godfey 1903 according to Fujita & Jones, 2002).

## RESEARCH DESIGN AND QUESTIONS

The empirical research is a form of quasi-experimental research with two non-equivalent groups (Cohen et al., 2007) and was carried out in real-time during one semester during regular classes of Euclidean geometry with all enrolled students of the respective university. Both groups were tested at the beginning and at the end of the semester with three identical written tests. The experimental group participants were taught using an alternative approach, while the control group participants were taught in a traditional way.

The purpose of this research was to evaluate the chosen alternative approach in learning and teaching geometry at the tertiary level. The main goal was to examine whether an alternative approach leads to improvement of visual-spatial abilities and progress in geometric thinking, a greater tendency to visualize and consequently to better outcomes of learning geometry (or not). In accordance with the main goal, five groups of research tasks were set, and in this paper two groups are distinguished: (1) immediately before learning geometry, determine the relationship between the visual-spatial abilities, levels of geometric thinking and geometric prior knowledge of the participants, especially in the sense of a possible predictor progress, (2) after the teaching of geometry, determine whether the participants, who were taught with an alternative approach, made significant progress in the accumulated geometric knowledge and visualization skills and visual-spatial abilities compared to the participants who were taught in the traditional way (or not).

## METHODOLOGY

To obtain answers to the questions raised, the collected data were processed by integrating quantitative and qualitative methods because complex research questions can be more fully answered using a mixed methodology than individual analyzes separately.
mathematical (especially geometric) achievements has been confirmed (e.g., Clements et al., 1997; Bruce & Hawes, 2015). In mathematics education research, the definition of visualization proposed by Arcavi (2003), which includes products and processes, is widely accepted. Visualization products include various types of mental images (Presmeg 1986), various types of external representations (Nakahara, 2007) as well as their mutuality. Visualization processes include the processes of creation and use of products, and differ with respect to the place, manner or purpose of creation/use. For successful work in geometric figures, Duval (1998) proposes a theoretical framework that considers four types of figure apprehension: perceptual, sequential, discursive and operational. The processing of the figure begins in a pure perceptual recognition of what is shown, and then, through mutual use of sequential, discursive and operational processing of the figure, the mathematical message that the figure represents in the appropriate context is revealed. To master the transition from pure perceptual apprehension to recognizing what is mathematically important, students need to know and develop different visualization processes, first separately and then in mutual coordination (Duval, 1995). In order to successfully learn geometry, it is necessary to develop the "geometric eye" through various types of tasks in which one should first gain certain practical experiences: by experimenting, measuring, drawing, etc., then make statements based on what has been observed and finally prove these statements by connecting them with definitions, axioms and theorems which are already known (Godfey 1903 according to Fujita & Jones, 2002).

## RESEARCH DESIGN AND QUESTIONS

The empirical research is a form of quasi-experimental research with two non-equivalent groups (Cohen et al., 2007) and was carried out in real-time during one semester during regular classes of Euclidean geometry with all enrolled students of the respective university. Both groups were tested at the beginning and at the end of the semester with three identical written tests. The experimental group participants were taught using an alternative approach, while the control group participants were taught in a traditional way.

The purpose of this research was to evaluate the chosen alternative approach in learning and teaching geometry at the tertiary level. The main goal was to examine whether an alternative approach leads to improvement of visual-spatial abilities and progress in geometric thinking, a greater tendency to visualize and consequently to better outcomes of learning geometry (or not). In accordance with the main goal, five groups of research tasks were set, and in this paper two groups are distinguished: (1) immediately before learning geometry, determine the relationship between the visual-spatial abilities, levels of geometric thinking and geometric prior knowledge of the participants, especially in the sense of a possible predictor progress, (2) after the teaching of geometry, determine whether the participants, who were taught with an alternative approach, made significant progress in the accumulated geometric knowledge and visualization skills and visual-spatial abilities compared to the participants who were taught in the traditional way (or not).

## METHODOLOGY

To obtain answers to the questions raised, the collected data were processed by integrating quantitative and qualitative methods because complex research questions can be more fully answered using a mixed methodology than individual analyzes separately.
## Sample

The research includes all students of teacher studies at two universities in Croatia, who attended the geometry classes in the summer semester of 2015/2016. A total of 90 students participated: 52 students (average age 21.8) who attended geometry classes at one University made up the experimental group, and 38 students (average age 19.4) who attended geometry classes at another University made up the control group. These two groups were chosen because of the greatest congruence of the curriculum for learning geometry.

## Instrument

To collect data on the knowledge and skills of the participants immediately before learning and after learning geometry, three tests were used. To measure levels of geometric thinking according to van Hiele's model (VH test), the test from the CDASSG project was used with permission (Usiskin, 1982). A test designed for this research (GEO test) was used to measure knowledge of geometry and visualization skills. The final design of the GEO test was achieved after the results of a pilot study and consultation with three experts. Elementary geometric concepts, their definitions, visual representations, and applications were tested through 28 tasks, grouped into four parts. The SPAC test (Smith & Whetton, 1998) was used to measure a special factor of spatial abilities.

## Data collection and analysis

All of the three tests were conducted at the beginning of the summer semester in the first week and at the end of the semester in the last week. The evaluation of the VH test was carried out according to the proposal of Usiskin (1982): first, each correct answer was evaluated by levels, and then the entire test using mild (3 out of 5) and more strict (4 out of 5) criteria within the classic (C) and modified scale (M). The evaluation of the GEO test was carried out first for each of the four parts of which is composed, and then collectively: each correct solution 2 points, incorrect or incomplete 1 point, no answer 0 points. Due to space limitations, it is not possible to provide a detailed description of the GEO test in this paper. The SPAC test was conducted and evaluated by a psychologist according to strictly defined rules of psychological testing.

## Intervention

During the intervention in experimental group, two main teaching strategies dominated: (1) structuring teaching topics according to van Hiele's stages of learning and (2) teaching based on the visual-analytical method of directed observation. Both strategies were permeated by three-layered teaching by balancing three ways of expression: visual, linguistic and symbolic (VLS system and teaching for short; Figure 1). The teaching activities included: tangram activities, drawing, constructing, defining, making statements, testing the truth with proving or disproving, and solving problems. The selected activities were intertwined and built on each other through different topics, and were chosen with the aim of developing visual-spatial abilities, thinking processes and learning geometric concepts with understanding.

Traditionally, teaching in a control group means: first, the teacher teaches frontally, emphasizing theoretical description and symbolic writing, and then the participants do the tasks individually, usually imitating their teacher. Visualization is used, but to a lesser extent and only as an intermediary.
## Sample

The research includes all students of teacher studies at two universities in Croatia, who attended the geometry classes in the summer semester of 2015/2016. A total of 90 students participated: 52 students (average age 21.8) who attended geometry classes at one University made up the experimental group, and 38 students (average age 19.4) who attended geometry classes at another University made up the control group. These two groups were chosen because of the greatest congruence of the curriculum for learning geometry.

## Instrument

To collect data on the knowledge and skills of the participants immediately before learning and after learning geometry, three tests were used. To measure levels of geometric thinking according to van Hiele's model (VH test), the test from the CDASSG project was used with permission (Usiskin, 1982). A test designed for this research (GEO test) was used to measure knowledge of geometry and visualization skills. The final design of the GEO test was achieved after the results of a pilot study and consultation with three experts. Elementary geometric concepts, their definitions, visual representations, and applications were tested through 28 tasks, grouped into four parts. The SPAC test (Smith & Whetton, 1998) was used to measure a special factor of spatial abilities.

## Data collection and analysis

All of the three tests were conducted at the beginning of the summer semester in the first week and at the end of the semester in the last week. The evaluation of the VH test was carried out according to the proposal of Usiskin (1982): first, each correct answer was evaluated by levels, and then the entire test using mild (3 out of 5) and more strict (4 out of 5) criteria within the classic (C) and modified scale (M). The evaluation of the GEO test was carried out first for each of the four parts of which is composed, and then collectively: each correct solution 2 points, incorrect or incomplete 1 point, no answer 0 points. Due to space limitations, it is not possible to provide a detailed description of the GEO test in this paper. The SPAC test was conducted and evaluated by a psychologist according to strictly defined rules of psychological testing.

## Intervention

During the intervention in experimental group, two main teaching strategies dominated: (1) structuring teaching topics according to van Hiele's stages of learning and (2) teaching based on the visual-analytical method of directed observation. Both strategies were permeated by three-layered teaching by balancing three ways of expression: visual, linguistic and symbolic (VLS system and teaching for short; Figure 1). The teaching activities included: tangram activities, drawing, constructing, defining, making statements, testing the truth with proving or disproving, and solving problems. The selected activities were intertwined and built on each other through different topics, and were chosen with the aim of developing visual-spatial abilities, thinking processes and learning geometric concepts with understanding.

Traditionally, teaching in a control group means: first, the teacher teaches frontally, emphasizing theoretical description and symbolic writing, and then the participants do the tasks individually, usually imitating their teacher. Visualization is used, but to a lesser extent and only as an intermediary.
<figure><img src="image_55.png" /><figcaption>Figure 1. VLS system and VLS teaching</figcaption></figure>

## RESULTS AND DISCUSSION

Given that the data on the research participants was collected through three tests of different forms and purposes, mixed data processing was used and the analysis of the results and discussion were done through different aspects. Due to space limitations, this paper presents only some summative results that indicate the outcomes of learning geometry, achieved within the experimental and control groups and between them.

After the characteristics of the participants of the experimental and control groups were determined by analyzing the results of the three tests, a t-test was performed with the aim of determining the statistical significance of the difference between these two groups before the intervention (table 1).

<table><caption>Table 1. Results of the t-test for the experimental and control groups before the intervention</caption><thead><tr><th rowspan="2">Measuring</th><th colspan="3">Experimental group</th><th colspan="3">Control group</th><th colspan="3">t-test</th></tr><tr><th>N</th><th>M</th><th>SD</th><th>N</th><th>M</th><th>SD</th><th>t</th><th>df</th><th>p</th></tr></thead><tbody><tr><td>1CVH4</td><td>37</td><td>1.486</td><td>.768</td><td>30</td><td>1.567</td><td>.679</td><td>-.447</td><td>65</td><td>.656</td></tr><tr><td>1MVH4</td><td>38</td><td>1.500</td><td>.762</td><td>31</td><td>1.548</td><td>.675</td><td>-.276</td><td>67</td><td>.783</td></tr><tr><td>1GEO</td><td>52</td><td>5.962</td><td>2.368</td><td>34</td><td>5.676</td><td>2.889</td><td>.500</td><td>84</td><td>.618</td></tr><tr><td>1SPAC</td><td>46</td><td>55.609</td><td>10.697</td><td>31</td><td>58.419</td><td>9.113</td><td>-1.198</td><td>75</td><td>.235</td></tr></tbody></table>

Based on the results (Table 1) there is no statistically significant difference ($p > 0.05$) between the groups in any of the considered categories. Considering some parameters, the mean values of the control group are slightly better compared to the mean values of the experimental group (1CVH4, 1MVH4 and 1SPAC). Considering that there was no statistically significant difference between the research participants before the intervention, reliability was tested for each test on the whole data of both groups using Cronbach's alpha coefficient ($α$). According to the coefficient values the tests can be considered reliable measuring instruments: the VH test has satisfactory internal consistency ($α = 0.576$), the GEO test is really reliable ($α = 0.921$) and the SPAC test is a highly reliable instrument ($α = 0.754$) (Novak, 2020).

The statistical significance of the difference in achievements between the experimental and control groups after teaching, for all of the three tests, was tested in the t-test (Table 2). Based on the results the progress in the two tests was statistically significant in favor of the experimental group. The
<figure><img src="image_61.png" /><figcaption>Figure 1. VLS system and VLS teaching</figcaption></figure>

## RESULTS AND DISCUSSION

Given that the data on the research participants was collected through three tests of different forms and purposes, mixed data processing was used and the analysis of the results and discussion were done through different aspects. Due to space limitations, this paper presents only some summative results that indicate the outcomes of learning geometry, achieved within the experimental and control groups and between them.

After the characteristics of the participants of the experimental and control groups were determined by analyzing the results of the three tests, a t-test was performed with the aim of determining the statistical significance of the difference between these two groups before the intervention (table 1).

<table><caption>Table 1. Results of the t-test for the experimental and control groups before the intervention</caption><thead><tr><th rowspan="2">Measuring</th><th colspan="3">Experimental group</th><th colspan="3">Control group</th><th colspan="3">t-test</th></tr><tr><th>N</th><th>M</th><th>SD</th><th>N</th><th>M</th><th>SD</th><th>t</th><th>df</th><th>p</th></tr></thead><tbody><tr><td>1CVH4</td><td>37</td><td>1.486</td><td>.768</td><td>30</td><td>1.567</td><td>.679</td><td>-.447</td><td>65</td><td>.656</td></tr><tr><td>1MVH4</td><td>38</td><td>1.500</td><td>.762</td><td>31</td><td>1.548</td><td>.675</td><td>-.276</td><td>67</td><td>.783</td></tr><tr><td>1GEO</td><td>52</td><td>5.962</td><td>2.368</td><td>34</td><td>5.676</td><td>2.889</td><td>.500</td><td>84</td><td>.618</td></tr><tr><td>1SPAC</td><td>46</td><td>55.609</td><td>10.697</td><td>31</td><td>58.419</td><td>9.113</td><td>-1.198</td><td>75</td><td>.235</td></tr></tbody></table>

Based on the results (Table 1) there is no statistically significant difference (p > 0.05) between the groups in any of the considered categories. Considering some parameters, the mean values of the control group are slightly better compared to the mean values of the experimental group (1CVH4, 1MVH4 and 1SPAC). Considering that there was no statistically significant difference between the research participants before the intervention, reliability was tested for each test on the whole data of both groups using Cronbach's alpha coefficient (α). According to the coefficient values the tests can be considered reliable measuring instruments: the VH test has satisfactory internal consistency (α = 0.576), the GEO test is really reliable (α = 0.921) and the SPAC test is a highly reliable instrument (α = 0.754) (Novak, 2020).

The statistical significance of the difference in achievements between the experimental and control groups after teaching, for all of the three tests, was tested in the t-test (Table 2). Based on the results the progress in the two tests was statistically significant in favor of the experimental group. The
statistical significance of 1% was on the GEO test (p < 0.001) and on all parameters of the VH test, except for the VH3 criterion where the statistical significance is 5%. The difference in achievements on the SPAC test is not statistically significant between the groups (p = 0.287). From the analysis by groups it is evident that progress was achieved within each group: in the experimental group it was statistically significant (p < 0.01), while in the control group it was not (p = 0.465).

<table><caption>Table 2. Comparison of achievement in all three tests between groups, after the intervention</caption><thead><tr><th rowspan="2">Test</th><th colspan="3">Experimental group</th><th colspan="3">Control group</th><th colspan="3">t-test</th></tr><tr><th>N</th><th>M</th><th>SD</th><th>N</th><th>M</th><th>SD</th><th>t</th><th>df</th><th>p</th></tr></thead><tbody><tr><td>2CVH4</td><td>38</td><td>2.026</td><td>1.078</td><td>30</td><td>1.267</td><td>.640</td><td>3.412</td><td>66</td><td>.001</td></tr><tr><td>2MVH4</td><td>38</td><td>2.026</td><td>1.078</td><td>30</td><td>1.267</td><td>.640</td><td>3.412</td><td>66</td><td>.001</td></tr><tr><td>2GEO</td><td>45</td><td>29.378</td><td>6.994</td><td>33</td><td>16.485</td><td>6.690</td><td>8.191</td><td>76</td><td>.000</td></tr><tr><td>2SPAC</td><td>44</td><td>59.659</td><td>9.906</td><td>26</td><td>57.077</td><td>9.398</td><td>1.074</td><td>68</td><td>.287</td></tr></tbody></table>

The presented results are in line with the results of other research in education: by mutual use of visual and analytical methods; by adapting teaching strategies to the participants' prior knowledge; by respecting the process of axiomatic construction of the deductive system; and by using different representations and their interconnections geometric achievements can be improved (e.g., Presmeg 1986; van Hiele, 1986; Bruce & Hawes, 2015).

## CONCLUSION

These results indicate that the geometry learning outcomes of pre-service primary education teachers and, thus, their preparation for work can be better when the teaching is adapted to their prior knowledge, and geometry is learned and taught through the systematic and mutual development of geometric knowledge, thinking and visual-spatial skills. In addition, these results can be an incentive for teachers at all levels of education to pay more attention to the three-layer teaching method, mutually and in harmony. It would be important to investigate what teaching outcomes would be achieved by applying this teaching strategy to groups with other characteristics.

## References

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. *Educational Studies in Mathematics*, 52(3), 215–241. https://doi.org/10.1023/A:1024312321077

Baranović, N. (2019). Pre-service primary education teachers’ knowledge of relationships among quadrilaterals, In J. Milinković & Z. Kadelburg (Eds.), *Research in mathematics education* (pp. 112–127). Mathematical Society of Serbia.

Breen, S., O’Shea, A., & Pfeiffer, K. (2013). The use of unfamiliar tasks in first year calculus courses to aid the transition from school to university mathematics. In *Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education* (pp. 2316–2325). ERME.

Bruce, C., Hawes. Z. (2014). The role of 2D and 3D mental rotation in mathematics for young children? What is it? Why is it important? And what can we do about it? *ZDM: The international journal on mathematics education*. https://doi.org/10.1007/s11858-014-0637-4

Clements, D.H., Battista, M.T., Sarama, J., Swaminathan, S. (1997). Development students' spatial thinking in a unit on geometric motions and area. *The elementary school journal*, 98(2).
statistical significance of 1% was on the GEO test (p < 0.001) and on all parameters of the VH test, except for the VH3 criterion where the statistical significance is 5%. The difference in achievements on the SPAC test is not statistically significant between the groups (p = 0.287). From the analysis by groups it is evident that progress was achieved within each group: in the experimental group it was statistically significant (p < 0.01), while in the control group it was not (p = 0.465).

<table><caption>Table 2. Comparison of achievement in all three tests between groups, after the intervention</caption><thead><tr><th rowspan="2">Test</th><th colspan="3">Experimental group</th><th colspan="3">Control group</th><th colspan="3">t-test</th></tr><tr><th>N</th><th>M</th><th>SD</th><th>N</th><th>M</th><th>SD</th><th>t</th><th>df</th><th>p</th></tr></thead><tbody><tr><td>2CVH4</td><td>38</td><td>2.026</td><td>1.078</td><td>30</td><td>1.267</td><td>.640</td><td>3.412</td><td>66</td><td>.001</td></tr><tr><td>2MVH4</td><td>38</td><td>2.026</td><td>1.078</td><td>30</td><td>1.267</td><td>.640</td><td>3.412</td><td>66</td><td>.001</td></tr><tr><td>2GEO</td><td>45</td><td>29.378</td><td>6.994</td><td>33</td><td>16.485</td><td>6.690</td><td>8.191</td><td>76</td><td>.000</td></tr><tr><td>2SPAC</td><td>44</td><td>59.659</td><td>9.906</td><td>26</td><td>57.077</td><td>9.398</td><td>1.074</td><td>68</td><td>.287</td></tr></tbody></table>

The presented results are in line with the results of other research in education: by mutual use of visual and analytical methods; by adapting teaching strategies to the participants' prior knowledge; by respecting the process of axiomatic construction of the deductive system; and by using different representations and their interconnections geometric achievements can be improved (e.g., Presmeg 1986; van Hiele, 1986; Bruce & Hawes, 2015).

## CONCLUSION

These results indicate that the geometry learning outcomes of pre-service primary education teachers and, thus, their preparation for work can be better when the teaching is adapted to their prior knowledge, and geometry is learned and taught through the systematic and mutual development of geometric knowledge, thinking and visual-spatial skills. In addition, these results can be an incentive for teachers at all levels of education to pay more attention to the three-layer teaching method, mutually and in harmony. It would be important to investigate what teaching outcomes would be achieved by applying this teaching strategy to groups with other characteristics.

## References

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. *Educational Studies in Mathematics*, 52(3), 215–241. https://doi.org/10.1023/A:1024312321077

Baranović, N. (2019). Pre-service primary education teachers' knowledge of relationships among quadrilaterals, In J. Milinković & Z. Kadelburg (Eds.), *Research in mathematics education* (pp. 112–127). Mathematical Society of Serbia.

Breen, S., O'Shea, A., & Pfeiffer, K. (2013). The use of unfamiliar tasks in first year calculus courses to aid the transition from school to university mathematics. In *Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education* (pp. 2316–2325). ERME.

Bruce, C., Hawes. Z. (2014). The role of 2D and 3D mental rotation in mathematics for young children? What is it? Why is it important? And what can we do about it? *ZDM: The international journal on mathematics education*. https://doi.org/10.1007/s11858-014-0637-4

Clements, D.H., Battista, M.T., Sarama, J., Swaminathan, S. (1997). Development students' spatial thinking in a unit on geometric motions and area. *The elementary school journal*, 98(2).
Cohen, L., Manion, L., Morrison, K. (2007). *Metode istraživanja u obrazovanju*. NAKLADA SLAP.

Crowley, M. L. (1987). The van Hiele model of development of geometric thought. In M. M. Lindquist (Ed.), *Learning and teaching geometry, K-12, 1-16, (Yearbook of the National Council of Teachers of Mathematics)* (pp. 1-16). National Council of Teachers of Mathematics.

De Villers, M. (2009). *To teach definitions in geometry or teach to define*. Available at: https://www.researchgate.net/publication/255605686

De Villiers, M. (2003). *Rethinking proof with The Geometer's Sketchpad*. Key Curriculum Press: Emeryville

Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processes. In R. Sutherland & J. Mason (Eds.), *Exploiting mental imagery with computers in mathematical education* (pp. 142-157). Springer

Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), *Perspectives on the Teaching of Geometry for the 21st century* (pp. 37-52). Kluwer.

Fischbein, E. & Nachlieli, T. (1998). Concepts and figures in geometrical reasoning. *International Journal of Science Education*, 20(10), 1193-1211. http://dx.doi.org/10.1080/0950069980201003

Flanagan, D. P., Shauna, G. D. (2014). The Cattell-Horn-Carroll theory of cognitive abilities. *Wiley Online Library*. Available at: https://doi.org/10.1002/9781118660584.ese0431

Fujita, T. and Jones, K. (2002). The bridge between practical and deductive geometry: Developing the "geometrical eye". In: A. D. Cockburn and E. Nardi (Eds), *Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education* (Vol 2, pp. 384-391). UEA.

Hammack, R. (2013). *Book of proof*. Virginia Commonwealth University.

Krutetskii, V. A. (1976). *The psychology of mathematical abilities in schoolchildren*. University of Chicago.

Leikin, R., & Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity [Paper presentation]. In *Proceedings of the 3rd Conference of the International Group for the Psychology of Mathematics Education* (Vol. 3, pp. 161-168). PME.

Lohman, D.F. (1993). Spatial ability and G. In I. Dennis, P. Tapsfield (Eds.), *Human abilities: Their nature and measurement*, (pp. 97-116).

Nakahara, T. (2007). Cultivating mathematical thinking through representation-Utilizing the representational. *Plenary paper*.

Novak, J. (2020). Pouzdanost mjerenja u psihologiji: Razvoj metode, zaluđenost Cronbachovim alfa koeficijentom i preporuke za ispravnu procjenu pouzdanosti. *Psihologijske Teme*, 29(2), 427-457. https://doi.org/10.31820/pt.29.2.11

PISA 2012 *Matematičke kompetencije za život* (2013). Nacionalni centar za vanjsko vrednovanje.

Polya, G. (1966). *Kako ću riješiti matematički zadatak*. Školska knjiga.

Presmeg, N.C.(1986). Visualization in high school mathematics. *For the Learning of Mathematics*, 6, 42-46.

Presmeg, N. (2014). Contemplating visualization as an epistemological learning tool in mathematics. ZDM *The international Journal on Mathematics Education*, 46, 151–157.

Smith, M. S., & Stein M. K. (1998). Selecting and creating mathematical tasks: From research to practice. *Mathematics Teaching in the Middle School*, 3, 344-350.

Thoma, A. & Nardi, E. (2018). Transition from School to University Mathematics: Manifestations of Unresolved Commognitive Conflict in First Year Students’ Examination Scripts. *International Journal of Research in Undergraduate Mathematics Education*, 4(1), 161–280.

TIMMS 2017 *Priručnik za unapređivanje nastave matematike s primjerima zadataka* TIMSS 2015.

Van Hiele, P. M. (1986). *Structure and Insight: A Theory of Mathematics Education*. Academic Press, Inc.
Cohen, L., Manion, L., Morrison, K. (2007). *Metode istraživanja u obrazovanju*. NAKLADA SLAP.

Crowley, M. L. (1987). The van Hiele model of development of geometric thought. In M. M. Lindquist (Ed.), *Learning and teaching geometry, K-12, 1-16, (Yearbook of the National Council of Teachers of Mathematics)* (pp. 1-16). National Council of Teachers of Mathematics.

De Villers, M. (2009). *To teach definitions in geometry or teach to define*. Available at: https://www.researchgate.net/publication/255605686

De Villiers, M. (2003). *Rethinking proof with The Geometer's Sketchpad*. Key Curriculum Press: Emeryville

Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processes. In R. Sutherland & J. Mason (Eds.), *Exploiting mental imagery with computers in mathematical education* (pp. 142-157). Springer

Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), *Perspectives on the Teaching of Geometry for the 21st century* (pp. 37-52). Kluwer.

Fischbein, E. & Nachlieli, T. (1998). Concepts and figures in geometrical reasoning. *International Journal of Science Education*, 20(10), 1193-1211. http://dx.doi.org/10.1080/0950069980201003

Flanagan, D. P., Shauna, G. D. (2014). The Cattell-Horn-Carroll theory of cognitive abilities. *Wiley Online Library*. Available at: https://doi.org/10.1002/9781118660584.ese0431

Fujita, T. and Jones, K. (2002). The bridge between practical and deductive geometry: Developing the "geometrical eye". In: A. D. Cockburn and E. Nardi (Eds), *Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education* (Vol 2, pp. 384-391). UEA.

Hammack, R. (2013). *Book of proof*. Virginia Commonwealth University.

Krutetskii, V. A. (1976). *The psychology of mathematical abilities in schoolchildren*. University of Chicago.

Leikin, R., & Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity [Paper presentation]. In *Proceedings of the 3rd Conference of the International Group for the Psychology of Mathematics Education* (Vol. 3, pp. 161-168). PME.

Lohman, D.F. (1993). Spatial ability and G. In I. Dennis, P. Tapsfield (Eds.), *Human abilities: Their nature and measurement*, (pp. 97-116).

Nakahara, T. (2007). Cultivating mathematical thinking through representation-Utilizing the representational. *Plenary paper*.

Novak, J. (2020). Pouzdanost mjerenja u psihologiji: Razvoj metode, zaluđenost Cronbachovim alfa koeficijentom i preporuke za ispravnu procjenu pouzdanosti. *Psihologijske Teme*, 29(2), 427-457. https://doi.org/10.31820/pt.29.2.11

PISA 2012 *Matematičke kompetencije za život* (2013). Nacionalni centar za vanjsko vrednovanje.

Polya, G. (1966). *Kako ću riješiti matematički zadatak*. Školska knjiga.

Presmeg, N.C.(1986). Visualization in high school mathematics. *For the Learning of Mathematics*, 6, 42-46.

Presmeg, N. (2014). Contemplating visualization as an epistemological learning tool in mathematics. ZDM *The international Journal on Mathematics Education*, 46, 151–157.

Smith, M. S., & Stein M. K. (1998). Selecting and creating mathematical tasks: From research to practice. *Mathematics Teaching in the Middle School*, 3, 344-350.

Thoma, A. & Nardi, E. (2018). Transition from School to University Mathematics: Manifestations of Unresolved Commognitive Conflict in First Year Students’ Examination Scripts. *International Journal of Research in Undergraduate Mathematics Education*, 4(1), 161–280.

TIMMS 2017 *Priručnik za unapređivanje nastave matematike s primjerima zadataka* TIMSS 2015.

Van Hiele, P. M. (1986). *Structure and Insight: A Theory of Mathematics Education*. Academic Press, Inc.
# GEOMETRY TEACHING FROM BABYLON TO THE COMPUTER ERA

Jaime Carvalho e Silva

University of Coimbra, Department of Mathematics, CMUC, 3000-143 Coimbra, Portugal

Geometry has been taught for millennia and we have rather detailed references of the geometry taught in Scribal Schools in Babylon, 4000 years ago. Today, with the computer revolution and Artificial Intelligence available, nobody seems to be proposing we stop teaching Geometry. But which Geometry should we teach, why and how? Which are the driving forces that influence each curriculum? How the new computing tools open new pathways for Geometry teaching? We will mention in some detail the controversies around the official Syllabus in Portugal in the last decades to try to identify which are the prevalent ideas in the curriculum and which are the different cultures involved in the decision process, from a purely abstract approach to a more applied approach, that answer to different societal and cultural views and mix the proposals of mathematicians, mathematics educators and teachers in the classroom. We observe that amid societal changes and advances in knowledge, Geometry is always focused on Problem Solving, use and exploration of diagrams to conjecture and achieve some kind of proof. We need to take intelligent paths to make the most of technology and not be a slave to it.

In order to mention the main ideas in some detail we have included diagrams, tables and the longer quotations in a web page available here: https://www.geogebra.org/m/cfnj6cb2

## THE SYLLABUS IN PORTUGAL

In the years 2021-2023 a new syllabus has been approved by the Portuguese Ministry of Education for the Mathematics courses of Basic Education (years 1 to 9) and Secondary Education (years 10 to 12), and this was done only for Mathematics. In Portugal the compulsory education includes all 12 years of school but there were no changes in other areas, as the Ministry of Education recognized there were unacceptable failure rates in Mathematics in Portugal. The Ministry began commissioning a special report to receive proposals to change the situation. The report *Recommendations for improving student learning in Mathematics* was published in 2020 (Carvalho e Silva, 2020) and was based on all the reports produced in Mathematics Education in Portugal in the last 30 years as well as on most international reports produced in this period, including international reports from OECD and UNESCO, as well as international assessments. It recommended a global revision of the mathematics curriculum in Portugal, that should be based on a certain number of principles, namely:

i) Universality, internal coherence, relevance, focus and higher cognitive level.

ii) Fulfill all the goals that justify the universality of the access to this course, taking into account the cultural, social and political dimensions of mathematics learning;

iii) Value understanding and look for an equilibrium of problem solving, mathematical reasoning, communication, connections, multiple representations, procedural fluency, creativity, digital literacy, reflection, resilience and individual and work group.
# GEOMETRY TEACHING FROM BABYLON TO THE COMPUTER ERA

Jaime Carvalho e Silva

University of Coimbra, Department of Mathematics, CMUC, 3000-143 Coimbra, Portugal

Geometry has been taught for millennia and we have rather detailed references of the geometry taught in Scribal Schools in Babylon, 4000 years ago. Today, with the computer revolution and Artificial Intelligence available, nobody seems to be proposing we stop teaching Geometry. But which Geometry should we teach, why and how? Which are the driving forces that influence each curriculum? How the new computing tools open new pathways for Geometry teaching? We will mention in some detail the controversies around the official Syllabus in Portugal in the last decades to try to identify which are the prevalent ideas in the curriculum and which are the different cultures involved in the decision process, from a purely abstract approach to a more applied approach, that answer to different societal and cultural views and mix the proposals of mathematicians, mathematics educators and teachers in the classroom. We observe that amid societal changes and advances in knowledge, Geometry is always focused on Problem Solving, use and exploration of diagrams to conjecture and achieve some kind of proof. We need to take intelligent paths to make the most of technology and not be a slave to it.

In order to mention the main ideas in some detail we have included diagrams, tables and the longer quotations in a web page available here: https://www.geogebra.org/m/cfnj6cb2

## THE SYLLABUS IN PORTUGAL

In the years 2021-2023 a new syllabus has been approved by the Portuguese Ministry of Education for the Mathematics courses of Basic Education (years 1 to 9) and Secondary Education (years 10 to 12), and this was done only for Mathematics. In Portugal the compulsory education includes all 12 years of school but there were no changes in other areas, as the Ministry of Education recognized there were unacceptable failure rates in Mathematics in Portugal. The Ministry began commissioning a special report to receive proposals to change the situation. The report *Recommendations for improving student learning in Mathematics* was published in 2020 (Carvalho e Silva, 2020) and was based on all the reports produced in Mathematics Education in Portugal in the last 30 years as well as on most international reports produced in this period, including international reports from OECD and UNESCO, as well as international assessments. It recommended a global revision of the mathematics curriculum in Portugal, that should be based on a certain number of principles, namely:

i) Universality, internal coherence, relevance, focus and higher cognitive level.

ii) Fulfill all the goals that justify the universality of the access to this course, taking into account the cultural, social and political dimensions of mathematics learning;

iii) Value understanding and look for an equilibrium of problem solving, mathematical reasoning, communication, connections, multiple representations, procedural fluency, creativity, digital literacy, reflection, resilience and individual and work group.
These are general principles but particular topics like Geometry are not discussed. New syllabuses
were produced in 2021 for Basic Education (Canavarro, 2021) and in 2023 for Secondary Education
(Carvalho e Silva, 2023). Other details that do not relate to Geometry will not be discussed here.

Geometry in Basic Education

Geometry occupies a substantial space in the new curriculum for Basic Schools. In the first four years the students begin their “development of spatial reasoning, with an emphasis on visualization and spatial orientation” (Canavarro, 2021); they use various types of materials and also technology (namely visual programming with Scratch and robots). In the second cycle of studies (years 5 and 6) the measure of angles and the study of triangles is introduced and dynamic geometry environments, such as GeoGebra are recommended. In the last three years of Basic Education (3rd cycle, years 7 to 9) the goal for Geometry is to “continue developing students' spatial reasoning, expanding their understanding of space”, as well as “the establishment of algebraic relationships from the study of geometric objects (...) accompanied by experience (where technology plays a fundamental role)” (Canavarro, 2021). Also geometric transformations are studied in a progressively more abstract and formal way.

Geometry in Secondary Education

Secondary Education in Portugal offers several paths of study that lead to further studies in Higher Education or offer a possibility of looking for a job immediately after school with a more profession oriented study. Students encounter six possible paths and in all of them we find some Geometry. The path *Mathematics Applied to Social Sciences* includes a chapter on Graph Theory and we will not discuss the details here (Carvalho e Silva, 2018). The Science and Technology path and the Economics path share the same Mathematics course spanning 3 years and 4.5 hours of class a week (Carvalho e Silva, 2023). This Mathematics course is called Mathematics A. The geometry topics studied in Mathematics A go from the notable points on the triangle with a synthetic perspective to matrices and geometric transformations, and include analytical geometry and trigonometry.

The chapter “Synthetic geometry in the plane” is a classical one but is new as such in the Portuguese curriculum. It is intended as a crossroad between classical geometry and experimentation with dynamical geometry systems, in order to “develop in students a taste for argumentation in general and proof as a central element of mathematics, for example with regard to the inscribed circle and the circumscribed circle” (Carvalho e Silva, 2023).

Analytic geometry is classic also but has a more algebraic flavor; in the curriculum students are encouraged to make software explorations (namely with *Geogebra 3D*) and work with mathematical modeling problems.

Implementation of the curricular change

We all know that an official document (intended curriculum) does not guarantee its implementation (enacted curriculum) even with good orientations (planned curriculum). Lots of countries have devised ways of connecting the intended curriculum to the enacted curriculum. We can quote the example of Costa Rica whose Reform has been very well documented by its mentor Angel Ruiz; this was a curricular change in mathematics only, that began in 2012. It developed original and innovative ideas and instruments, including blended courses for teachers, fully virtual courses with MOOC
These are general principles but particular topics like Geometry are not discussed. New syllabuses
were produced in 2021 for Basic Education (Canavarro, 2021) and in 2023 for Secondary Education
(Carvalho e Silva, 2023). Other details that do not relate to Geometry will not be discussed here.

Geometry in Basic Education

Geometry occupies a substantial space in the new curriculum for Basic Schools. In the first four years the students begin their “development of spatial reasoning, with an emphasis on visualization and spatial orientation” (Canavarro, 2021); they use various types of materials and also technology (namely visual programming with Scratch and robots). In the second cycle of studies (years 5 and 6) the measure of angles and the study of triangles is introduced and dynamic geometry environments, such as GeoGebra are recommended. In the last three years of Basic Education (3rd cycle, years 7 to 9) the goal for Geometry is to “continue developing students' spatial reasoning, expanding their understanding of space”, as well as “the establishment of algebraic relationships from the study of geometric objects (...) accompanied by experience (where technology plays a fundamental role)” (Canavarro, 2021). Also geometric transformations are studied in a progressively more abstract and formal way.

Geometry in Secondary Education

Secondary Education in Portugal offers several paths of study that lead to further studies in Higher Education or offer a possibility of looking for a job immediately after school with a more profession oriented study. Students encounter six possible paths and in all of them we find some Geometry. The path *Mathematics Applied to Social Sciences* includes a chapter on Graph Theory and we will not discuss the details here (Carvalho e Silva, 2018). The Science and Technology path and the Economics path share the same Mathematics course spanning 3 years and 4.5 hours of class a week (Carvalho e Silva, 2023). This Mathematics course is called Mathematics A. The geometry topics studied in Mathematics A go from the notable points on the triangle with a synthetic perspective to matrices and geometric transformations, and include analytical geometry and trigonometry.

The chapter “Synthetic geometry in the plane” is a classical one but is new as such in the Portuguese curriculum. It is intended as a crossroad between classical geometry and experimentation with dynamical geometry systems, in order to “develop in students a taste for argumentation in general and proof as a central element of mathematics, for example with regard to the inscribed circle and the circumscribed circle” (Carvalho e Silva, 2023).

Analytic geometric is classic also but has a more algebraic flavor; in the curriculum students are encouraged to make software explorations (namely with *Geogebra 3D*) and work with mathematical modeling problems.

Implementation of the curricular change

We all know that an official document (intended curriculum) does not guarantee its implementation (enacted curriculum) even with good orientations (planned curriculum). Lots of countries have devised ways of connecting the intended curriculum to the enacted curriculum. We can quote the example of Costa Rica whose Reform has been very well documented by its mentor Angel Ruiz; this was a curricular change in mathematics only, that began in 2012. It developed original and innovative ideas and instruments, including blended courses for teachers, fully virtual courses with MOOC
modality for high school teachers and students, and shorter virtual courses called Mini-MOOCs (Ruiz, 2015).

In Portugal several curricular changes were accompanied by a number of measures to work towards the coherence of the curriculum. Since 2021 several actions were taken: some classes (pilots) to anticipate the implementation of the new syllabus were run, tested materials were afterwards made available to the rest of the schools including comments from pilot classes, tasks were published online, continuous professional development courses were made in two phases: the first one to prepare schools leaders that in the sequence propose themselves continuous professional development courses to other teachers in all schools.

## Using the history of education tool

The options taken in the new syllabus were publicly discussed, there was a preliminary version for an open discussion and public sessions were held to discuss with teachers, associations and other partners. There was no consensus on a number of issues so the team writing the curriculum (including mathematicians, mathematics educators and experienced teachers) had to choose paths in a number of issues. Why these options were ultimately held? Are they promising? Will the global situation in mathematics education improve? The tools used to make choices in the curriculum were the experience of different countries, the most successful ones including international schools with its own curriculum, and past history of mathematics education, namely in Portugal. The educational system in Portugal was structured in 1772 and greatly revamped in 1836, so there is a long history of mathematics education in Portugal. Other countries and regions have an even longer history. What can we learn from our collective past experience teaching Geometry?

## HISTORY FLASHBACK ON GEOMETRY TEACHING

It is always useful to study which were the guiding principles and the school practice in past times. The Portuguese document *Recommendations for improving student learning in Mathematics* (Carvalho e Silva, 2020), produced before the last rewriting of the curriculum, made a summary of the main documents produced in Portugal in the last 30 years and discussed what happened in France, Finland, Estonia and Singapore. In the end a lot of experience was gained from the emphasis on problem solving, applications and mathematical modeling these countries have, and also on the use of technology namely with computational thinking with a large practical work in schools particularly in France (numerous classroom materials are available on the web also through the work of the well-known IREM).

Let's go back here some millennia.

## Teaching in Babylon

Scribal Schools from the Old Babylonian period are fascinating. Some 4000 years ago, we can find that “education went hand in hand with creative activity, supported by a very active milieu. (...) The most active centers were able to influence other, less creative ones, where the educational framework was less institutional. (...) These scribes conferred together and traveled.” (Proust, 2014). A lot of researchers have studied scribal schools and it is clear that Geometry was present from the beginning. As Friberg puts it: “to divide given parcels of land into shares according to some intricate set of rules, or dividing given amounts of food stuff into rations of various sizes according to some other intricate
modality for high school teachers and students, and shorter virtual courses called Mini-MOOCs (Ruiz, 2015).

In Portugal several curricular changes were accompanied by a number of measures to work towards the coherence of the curriculum. Since 2021 several actions were taken: some classes (pilots) to anticipate the implementation of the new syllabus were run, tested materials were afterwards made available to the rest of the schools including comments from pilot classes, tasks were published online, continuous professional development courses were made in two phases: the first one to prepare schools leaders that in the sequence propose themselves continuous professional development courses to other teachers in all schools.

## Using the history of education tool

The options taken in the new syllabus were publicly discussed, there was a preliminary version for an open discussion and public sessions were held to discuss with teachers, associations and other partners. There was no consensus on a number of issues so the team writing the curriculum (including mathematicians, mathematics educators and experienced teachers) had to choose paths in a number of issues. Why these options were ultimately held? Are they promising? Will the global situation in mathematics education improve? The tools used to make choices in the curriculum were the experience of different countries, the most successful ones including international schools with its own curriculum, and past history of mathematics education, namely in Portugal. The educational system in Portugal was structured in 1772 and greatly revamped in 1836, so there is a long history of mathematics education in Portugal. Other countries and regions have an even longer history. What can we learn from our collective past experience teaching Geometry?

## HISTORY FLASHBACK ON GEOMETRY TEACHING

It is always useful to study which were the guiding principles and the school practice in past times. The Portuguese document *Recommendations for improving student learning in Mathematics* (Carvalho e Silva, 2020), produced before the last rewriting of the curriculum, made a summary of the main documents produced in Portugal in the last 30 years and discussed what happened in France, Finland, Estonia and Singapore. In the end a lot of experience was gained from the emphasis on problem solving, applications and mathematical modeling these countries have, and also on the use of technology namely with computational thinking with a large practical work in schools particularly in France (numerous classroom materials are available on the web also through the work of the well-known IREM).

Let's go back here some millennia.

## Teaching in Babylon

Scribal Schools from the Old Babylonian period are fascinating. Some 4000 years ago, we can find that “education went hand in hand with creative activity, supported by a very active milieu. (...) The most active centers were able to influence other, less creative ones, where the educational framework was less institutional. (...) These scribes conferred together and traveled.” (Proust, 2014). A lot of researchers have studied scribal schools and it is clear that Geometry was present from the beginning. As Friberg puts it: “to divide given parcels of land into shares according to some intricate set of rules, or dividing given amounts of food stuff into rations of various sizes according to some other intricate
set of rules, must have been an important part of the metro-mathematical education given to the young scribes in the scribe schools” (Friberg, 2014). It is clear that Geometry, Arithmetic and Algebra were studied together. Geometrical diagrams were frequent in clay tablets.

## Euclid, the *Elements* and *Pseudaria*

As Robin Hartshorne points out “Throughout most of its history, Euclid's *Elements* has been the principal manual of geometry and indeed the required introduction to any of the sciences.” (Hartshorne, 2000). More than one thousand editions and translations are known. In the great reformation of the studies in Portugal, around 1772, Mathematics took center place at the University and so all students at the University had to study Geometry, Euclidian style, and the textbook used was a translation of the *Elements* in Portuguese.

A lost work attributed to Euclid, with a more marked pedagogical character, *Pseudaria*, never made into mainstream teaching and the knowledge of its content is almost completely lost because, possibly, as points out Fabio Acerbi, “the tradition of its pedagogical use was soon broken (...) as a consequence of a likely increase of the dogmatic character of mathematical teaching in post-Hellenistic times” (Acerbi, 2007).

## Multiple translations and uses of the *Elements*

As was already mentioned, more than one thousand editions and translations of Euclid's *Elements* are known. It was used at almost all schools in the world at some point. The first Chinese translation was made by the Jesuit Matteo Ricci (1552–1610), in 1607, when he was assigned to work in China (Joseph, 2011).

The contents and style of Euclid's *Elements* were imitated by numerous school textbooks for Basic and Secondary schools. The most used textbook in Geometry in Portugal for some 30 years, from 1944 to 1974, was the first in Portugal to use "the treatment of demonstrations in justified steps, as Americans and English use it, [and] it seems to us very advisable" (Paulo, 1944). The price to pay with this more formal approach, as the reviewer quoted wrote, is "that despite everything, sometimes the author makes an appeal to students' intuition, it is a shame that it is no longer done often".

## Modern Mathematics

The so called *Modern Mathematics Movement* changed completely the teaching of Mathematics in numerous countries including Portugal. The Portuguese mathematics curriculum for Basic and Secondary School saw a considerable reduction of the study of classical Geometry and some emphasis on the study of geometric transformations. Euclid's approach completely disappeared.

The coordinator of the modernization of Mathematics teaching in Portugal, mathematician José Sebastião e Silva (1914-1972), wrote that the teaching of Geometry should find an equilibrium between abstractions and the real world. He was very much influenced by Emma Castelnuovo (1913-2014).

As the 24th ICMI Study volume puts it:

*Classical synthetic geometry was completely eliminated and the main aim was not to study geometrical figures but to construct an algebraic tool to describe first the affine, then the Euclidian plane and space. Principal notions were projections, vectors, frames, transformations, etc.* (Gosztonyi et al., 2023, p. 53)
set of rules, must have been an important part of the metro-mathematical education given to the young scribes in the scribe schools” (Friberg, 2014). It is clear that Geometry, Arithmetic and Algebra were studied together. Geometrical diagrams were frequent in clay tablets.

## Euclid, the Elements and Pseudaria

As Robin Hartshorne points out “Throughout most of its history, Euclid's *Elements* has been the principal manual of geometry and indeed the required introduction to any of the sciences.” (Hartshorne, 2000). More than one thousand editions and translations are known. In the great reformation of the studies in Portugal, around 1772, Mathematics took center place at the University and so all students at the University had to study Geometry, Euclidian style, and the textbook used was a translation of the *Elements* in Portuguese.

A lost work attributed to Euclid, with a more marked pedagogical character, *Pseudaria*, never made into mainstream teaching and the knowledge of its content is almost completely lost because, possibly, as points out Fabio Acerbi, “the tradition of its pedagogical use was soon broken (...) as a consequence of a likely increase of the dogmatic character of mathematical teaching in post-Hellenistic times” (Acerbi, 2007).

## Multiple translations and uses of the Elements

As was already mentioned, more than one thousand editions and translations of Euclid's *Elements* are known. It was used at almost all schools in the world at some point. The first Chinese translation was made by the Jesuit Matteo Ricci (1552–1610), in 1607, when he was assigned to work in China (Joseph, 2011).

The contents and style of Euclid's *Elements* were imitated by numerous school textbooks for Basic and Secondary schools. The most used textbook in Geometry in Portugal for some 30 years, from 1944 to 1974, was the first in Portugal to use “the treatment of demonstrations in justified steps, as Americans and English use it, [and] it seems to us very advisable” (Paulo, 1944). The price to pay with this more formal approach, as the reviewer quoted wrote, is “that despite everything, sometimes the author makes an appeal to students' intuition, it is a shame that it is no longer done often”.

## Modern Mathematics

The so called *Modern Mathematics Movement* changed completely the teaching of Mathematics in numerous countries including Portugal. The Portuguese mathematics curriculum for Basic and Secondary School saw a considerable reduction of the study of classical Geometry and some emphasis on the study of geometric transformations. Euclid's approach completely disappeared.

The coordinator of the modernization of Mathematics teaching in Portugal, mathematician José Sebastião e Silva (1914-1972), wrote that the teaching of Geometry should find an equilibrium between abstractions and the real world. He was very much influenced by Emma Castelnuovo (1913-2014).

As the 24th ICMI Study volume puts it:

*Classical synthetic geometry was completely eliminated and the main aim was not to study geometrical figures but to construct an algebraic tool to describe first the affine, then the Euclidian plane and space. Principal notions were projections, vectors, frames, transformations, etc. (Gosztonyi et al., 2023, p. 53)*
## Tensions in the XXIst Century – Basic School

From 1990 to 2023, there were several changes in the mathematics curriculum for Basic School and Secondary School in Portugal. The reform of the educational system of 1990 introduced new syllabuses at all levels for all courses, but other changes were mainly concentrated on changes in Mathematics, in part because in the first TIMMS and PISA Portugal was in a very modest position.

The 1990 syllabus for mathematics in Basic School initiated an extensive study of geometric transformations in a rather intuitive and balanced way; but, with an overcrowded curriculum (reformers were very ambitious) geometric transformations tended to be omitted. The next change of the curriculum in basic school in 2007 took a different approach, more visual and connected to the real world, insisting in the development of the spatial sense of students, based on exploration, manipulation and experimentation with concrete materials. Small chains of deductions should be introduced in years 7 to 9 in topics like Parallel Lines, Similar Triangles, Pythagoras Theorem and Geometric Transformations (Breda et al, 2011).

But there were radical changes again in 2012-2013, and a highly structured approach, based on logic and set theory was introduced in Basic School with an extensive study of axiomatics in the 9th grade; numerous proofs became compulsory along the official "correct" use of the vocabulary of the axiomatic method and a rather detailed knowledge of the history of the axiomatization of Geometry. The axiomatization of Geometry with points, relation "a point is between two other points" and "pairs of points are equidistant" are mentioned as objects of study, as well as Hyperbolic Geometry. This syllabus was a huge failure, some minor modifications were introduced and in 2021 a new syllabus was approved by the Ministry of Education, somehow recovering the syllabus from 2007, that had been very well received by mathematics teachers at the time.

## Tensions in the XXIst Century – Secondary School

We will consider here the official documents for mathematics teaching at secondary school in Portugal on the course called "Mathematics A". There are big contrasts between the documents of 2013 and 2023. In 2013 the main goal was "the comprehension and hierarchization of mathematical concepts, the systematic study of their properties and the clear and precise argumentation, typical of this discipline" (Bivar et al. 2013). Mathematical modeling could "transmit to students a distorted vision of how one can, in fact, correctly apply Mathematics to the real world" (Bivar et al., 2013) and the use of intuition was discouraged as "conjectures formulated but not demonstrated are of limited interest" (Bivar et al., 2013). In contrast to this vision, the new documents of 2023 present the work in mathematics at secondary school as following:

(...) develop in students the ability to identify relevant mathematical concepts to solve real problems, apply appropriate mathematical procedures and interpret results in different contexts. Mathematical reasoning is the basis of the processes of understanding mathematical concepts and objects, which can and should be analyzed, represented and related in different ways. The formulation of hypotheses, testing of conjectures, deduction, generalization and abstraction are equally important, in the construction of logical arguments and conclusions, whose communication in an appropriate way is increasingly important in today's world. (Carvalho e Silva, 2023).

Which contents and methods are the most suitable for our century?
## Tensions in the XXIst Century – Basic School

From 1990 to 2023, there were several changes in the mathematics curriculum for Basic School and Secondary School in Portugal. The reform of the educational system of 1990 introduced new syllabuses at all levels for all courses, but other changes were mainly concentrated on changes in Mathematics, in part because in the first TIMMS and PISA Portugal was in a very modest position.

The 1990 syllabus for mathematics in Basic School initiated an extensive study of geometric transformations in a rather intuitive and balanced way; but, with an overcrowded curriculum (reformers were very ambitious) geometric transformations tended to be omitted. The next change of the curriculum in basic school in 2007 took a different approach, more visual and connected to the real world, insisting in the development of the spatial sense of students, based on exploration, manipulation and experimentation with concrete materials. Small chains of deductions should be introduced in years 7 to 9 in topics like Parallel Lines, Similar Triangles, Pythagoras Theorem and Geometric Transformations (Breda et al, 2011).

But there were radical changes again in 2012-2013, and a highly structured approach, based on logic and set theory was introduced in Basic School with an extensive study of axiomatics in the 9th grade; numerous proofs became compulsory along the official “correct” use of the vocabulary of the axiomatic method and a rather detailed knowledge of the history of the axiomatization of Geometry. The axiomatization of Geometry with points, relation “a point is between two other points” and “pairs of points are equidistant” are mentioned as objects of study, as well as Hyperbolic Geometry. This syllabus was a huge failure, some minor modifications were introduced and in 2021 a new syllabus was approved by the Ministry of Education, somehow recovering the syllabus from 2007, that had been very well received by mathematics teachers at the time.

## Tensions in the XXIst Century – Secondary School

We will consider here the official documents for mathematics teaching at secondary school in Portugal on the course called “Mathematics A”. There are big contrasts between the documents of 2013 and 2023. In 2013 the main goal was “the comprehension and hierarchization of mathematical concepts, the systematic study of their properties and the clear and precise argumentation, typical of this discipline” (Bivar et al. 2013). Mathematical modeling could “transmit to students a distorted vision of how one can, in fact, correctly apply Mathematics to the real world” (Bivar et al., 2013) and the use of intuition was discouraged as “conjectures formulated but not demonstrated are of limited interest” (Bivar et al., 2013). In contrast to this vision, the new documents of 2023 present the work in mathematics at secondary school as following:

(...) develop in students the ability to identify relevant mathematical concepts to solve real problems, apply appropriate mathematical procedures and interpret results in different contexts. Mathematical reasoning is the basis of the processes of understanding mathematical concepts and objects, which can and should be analyzed, represented and related in different ways. The formulation of hypotheses, testing of conjectures, deduction, generalization and abstraction are equally important, in the construction of logical arguments and conclusions, whose communication in an appropriate way is increasingly important in today's world. (Carvalho e Silva, 2023).

Which contents and methods are the most suitable for our century?
## XXIst Century: a world full of smart technology

As some people said (no quote necessary) in a world of technology that draws nice graphs and images, why study so much Mathematics (Functions or Geometry)? In fact we are in a world that already produces algorithms that can make automatic deductions in Geometry and this has the potential of changing completely the way Geometry is taught. We face three types of essentially new challenges:

a) Dynamic Geometry Software (DGS);

b) Automated proof assistants, that can be Interactive Theorem Provers (ITP) or Automated Theorem provers (ATP);

c) Large language models (LLM) like ChatGPT.

DGS has been around for some time and has always been considered “rather efficient in the learning of geometry” (Balacheff, de La Tour, 2019); it is being continuously upgraded, and a surprising challenge is *Geogebra* that can run on smartphones with Augmented Reality (AR) tools; we are in a world where all students (at least from 12 years old) have a smartphone at a rate that surpasses 90%; which are the implications?

Beyond mechanizing computation, these technologies are now mechanizing mathematical reasoning and proofs with unprecedented consequences. DGS and ATP have been combined in a software with AR and used in an educational setting outdoors (Botana, Kovács, Martínez-Sevilla & Recio, 2019), (Botana, Kovács & Recio, 2020). In a recent paper Nuno Baeta and Pedro Quaresma describe an algorithm (a “Geometry Automated-Theorem-Prover”) that will “be able to efficiently prove a large set of geometric conjectures, producing readable proofs” and “will open its use by third-party programs, e.g. the dynamic geometry systems” (Baeta and Quaresma, 2023). These methods began being tested in secondary schools (Teles, Baeta and Quaresma, 2022) and, even if practical difficulties are enormous, one day they will surely be used regularly in classrooms. How?

Large language models are a recent acquisition to the world of pedagogical tools and have become very controversial because they produce mathematical errors frequently: the outputs a LLM gives are just words that have a high probability of showing up together in a sentence that was produced millions of times in the texts the LLM “read”. But LLM are already being used in school environments, associated to sophisticated mathematical software like Mathematica or as just a tool to give better feedback like in *Geogebra* as an *AI Math Assistant* (Hohenwarter, 2023) and *Khan Academy* as a text builder assistant. Will they help us teach Geometry better?

Po-Shen Loh, the coach of the American Math Olympiad team, calls the use of ChatGPT the “invasion” and thinks “the key to survival is knowing how to solve problems - and knowing which problems to solve” (Loh, 2023). He urges school to focus on “creativity, emotion and the stuff that distinguishes man from machine” (Loh, 2023).

## LESSONS FOR TODAY

In this extremely brief historical excursion, we see that in each period of time, Geometry had a different character but it always had some kind of relevance related to the reasoning that it conveyed and to the real life situations it was connected to; diagrams were always used from Babylon to Euclid and the Computer Era. Somehow, connecting diagrams to its abstract representation can lead to errors (like in *Pseudaria*) but this also stimulates a healthy or corrected reasoning, enabling students to be
## XXIst Century: a world full of smart technology

As some people said (no quote necessary) in a world of technology that draws nice graphs and images, why study so much Mathematics (Functions or Geometry)? In fact we are in a world that already produces algorithms that can make automatic deductions in Geometry and this has the potential of changing completely the way Geometry is taught. We face three types of essentially new challenges:

a) Dynamic Geometry Software (DGS);

b) Automated proof assistants, that can be Interactive Theorem Provers (ITP) or Automated Theorem provers (ATP);

c) Large language models (LLM) like ChatGPT.

DGS has been around for some time and has always been considered "rather efficient in the learning of geometry" (Balacheff, de La Tour, 2019); it is being continuously upgraded, and a surprising challenge is *Geogebra* that can run on smartphones with Augmented Reality (AR) tools; we are in a world where all students (at least from 12 years old) have a smartphone at a rate that surpasses 90%; which are the implications?

Beyond mechanizing computation, these technologies are now mechanizing mathematical reasoning and proofs with unprecedented consequences. DGS and ATP have been combined in a software with AR and used in an educational setting outdoors (Botana, Kovács, Martínez-Sevilla & Recio, 2019), (Botana, Kovács & Recio, 2020). In a recent paper Nuno Baeta and Pedro Quaresma describe an algorithm (a "Geometry Automated-Theorem-Prover") that will "be able to efficiently prove a large set of geometric conjectures, producing readable proofs" and "will open its use by third-party programs, e.g. the dynamic geometry systems" (Baeta and Quaresma, 2023). These methods began being tested in secondary schools (Teles, Baeta and Quaresma, 2022) and, even if practical difficulties are enormous, one day they will surely be used regularly in classrooms. How?

Large language models are a recent acquisition to the world of pedagogical tools and have become very controversial because they produce mathematical errors frequently: the outputs a LLM gives are just words that have a high probability of showing up together in a sentence that was produced millions of times in the texts the LLM "read". But LLM are already being used in school environments, associated to sophisticated mathematical software like Mathematica or as just a tool to give better feedback like in *Geogebra* as an *AI Math Assistant* (Hohenwarter, 2023) and *Khan Academy* as a text builder assistant. Will they help us teach Geometry better?

Po-Shen Loh, the coach of the American Math Olympiad team, calls the use of ChatGPT the "invasion" and thinks "the key to survival is knowing how to solve problems - and knowing which problems to solve" (Loh, 2023). He urges school to focus on "creativity, emotion and the stuff that distinguishes man from machine" (Loh, 2023).

## LESSONS FOR TODAY

In this extremely brief historical excursion, we see that in each period of time, Geometry had a different character but it always had some kind of relevance related to the reasoning that it conveyed and to the real life situations it was connected to; diagrams were always used from Babylon to Euclid and the Computer Era. Somehow, connecting diagrams to its abstract representation can lead to errors (like in *Pseudaria*) but this also stimulates a healthy or corrected reasoning, enabling students to be
able to deal with new problems, understanding them, looking for strategies, producing and criticizing conjectures and trying to prove them, learning how to deal with similar problems or problems that might in part benefit from the new strategies found.

The necessity to produce diagrams is a constant and our computer era is the most fertile in the availability of tools that enable us to produce a wide range of diagrams, static or dynamic. Pedagogical tools like *Geogebra 3D*, *GeoGebra AR* or *GeoGebra Discovery* (ATP) (Kovács et al., 2022) are powerful and simple to use and very promising for the classroom.

Teaching and learning Geometry did not change much for millennia. Archimedes, in his *Method*, tells Eratosthenes how to investigate Geometry problems with Mechanics and then prove the results obtained using Geometry and he said the investigative part is also useful for the proof of these results (Heath, 1912). Tools and scope and interplay with other areas have changed a lot in the present century but we must see what is now important in Geometry and find new ideas to continue teaching and learning Geometry in a meaningful way. As Gila Hanna and Xiaoheng Yan (2021) put it “there is a need for new approaches to teaching proof, ones that capitalize not only on newly-available technology but also on modern theories of teaching and learning”.

In the words of late President of ICMI, Miguel de Guzmán, when speaking about the impacts of new technology on mathematics teaching: “What is truly important will be the preparation [of students] for an **intelligent dialogue with the tools that already exist**, which some already have and others will have in a future that is almost present” (De Guzmán, 1992) (my emphasis).

We certainly need to reflect intensely on new and significant ways to teach Geometry in Basic and Secondary School, that use technology in an intelligent and meaningful way.

## Acknowledgements

This work was partially supported by the Centre for Mathematics of the University of Coimbra (funded by the Portuguese Government through FCT/MCTES, DOI 10.54499/UIDB/00324/2020).

## References

* Acerbi, F. (2008). Euclid's Pseudaria, *Arch. Hist. Exact Sci.* 62, 511–551.

* Baeta, N., & Quaresma, P. (2023). Towards a geometry deductive database prover. *Annals of Mathematics and Artificial Intelligence*, https://doi.org/10.1007/s10472-023-09839-0

* Balacheff, N., & Boy de La Tour, T. (2021). Proof technology and learning in mathematics: common issues and perspectives. In G. Hanna, D. Reid, & M. de Villiers. *Proof Technology in Mathematics Research and Teaching* (pp.349-365), Springer.

* Bivar, A., Grosso, C., Oliveira, F., & Timóteo, C. (2013). Programa de Matemática para o Ensino Básico. Lisboa: Ministério da Educação e Ciência.

* Bivar, A., Grosso, C., Oliveira, F., Timóteo, C., & Loura, L. (2014). Programa e Metas curriculares de Matemática A, Ensino Secundário. Lisboa: Ministério da Educação e Ciência.

* Botana, F., Kovács, Z., Martínez-Sevilla, A. & Recio, T. (2019). Automatically augmented reality with GeoGebra. In: T. Prodromou (Ed.), *Augmented reality in educational settings*, Brill-Sense. https://doi.org/10.1163/9789004408845

* Botana F., Kovács Z., & Recio T. (2020). Automatically augmented reality for outdoor mathematics. In: *Research on Outdoor STEM Education in the digiTal Age*. M. Ludwig, S. Jablonski, A. Caldeira & A.
able to deal with new problems, understanding them, looking for strategies, producing and criticizing conjectures and trying to prove them, learning how to deal with similar problems or problems that might in part benefit from the new strategies found.

The necessity to produce diagrams is a constant and our computer era is the most fertile in the availability of tools that enable us to produce a wide range of diagrams, static or dynamic. Pedagogical tools like *Geogebra 3D*, *GeoGebra AR* or *GeoGebra Discovery* (ATP) (Kovács et al., 2022) are powerful and simple to use and very promising for the classroom.

Teaching and learning Geometry did not change much for millennia. Archimedes, in his *Method*, tells Eratosthenes how to investigate Geometry problems with Mechanics and then prove the results obtained using Geometry and he said the investigative part is also useful for the proof of these results (Heath, 1912). Tools and scope and interplay with other areas have changed a lot in the present century but we must see what is now important in Geometry and find new ideas to continue teaching and learning Geometry in a meaningful way. As Gila Hanna and Xiaoheng Yan (2021) put it “there is a need for new approaches to teaching proof, ones that capitalize not only on newly-available technology but also on modern theories of teaching and learning”.

In the words of late President of ICMI, Miguel de Guzmán, when speaking about the impacts of new technology on mathematics teaching: “What is truly important will be the preparation [of students] for an **intelligent dialogue with the tools that already exist**, which some already have and others will have in a future that is almost present” (De Guzmán, 1992) (my emphasis).

We certainly need to reflect intensely on new and significant ways to teach Geometry in Basic and Secondary School, that use technology in an intelligent and meaningful way.

## Acknowledgements

This work was partially supported by the Centre for Mathematics of the University of Coimbra (funded by the Portuguese Government through FCT/MCTES, DOI 10.54499/UIDB/00324/2020).

## References

* Acerbi, F. (2008). Euclid's Pseudaria, *Arch. Hist. Exact Sci.* 62, 511–551.

* Baeta, N., & Quaresma, P. (2023). Towards a geometry deductive database prover. *Annals of Mathematics and Artificial Intelligence*, https://doi.org/10.1007/s10472-023-09839-0

* Balacheff, N., & Boy de La Tour, T. (2021). Proof technology and learning in mathematics: common issues and perspectives. In G. Hanna, D. Reid, & M. de Villiers. *Proof Technology in Mathematics Research and Teaching* (pp.349-365), Springer.

* Bivar, A., Grosso, C., Oliveira, F., & Timóteo, C. (2013). Programa de Matemática para o Ensino Básico. Lisboa: Ministério da Educação e Ciência.

* Bivar, A., Grosso, C., Oliveira, F., Timóteo, C., & Loura, L. (2014). Programa e Metas curriculares de Matemática A, Ensino Secundário. Lisboa: Ministério da Educação e Ciência.

* Botana, F., Kovács, Z., Martínez-Sevilla, A. & Recio, T. (2019). Automatically augmented reality with GeoGebra. In: T. Prodromou (Ed.), *Augmented reality in educational settings*, Brill-Sense. https://doi.org/10.1163/9789004408845

* Botana F., Kovács Z., & Recio T. (2020). Automatically augmented reality for outdoor mathematics. In: *Research on Outdoor STEM Education in the digiTal Age*. M. Ludwig, S. Jablonski, A. Caldeira & A.
Moura (Ed.). Conference Proceedings in Mathematics Education (6), pages 71-78.
https://doi.org/10.37626/GA9783959871440.0

Breda, A., Serrazina, L., Menezes, L., Sousa, H., & Oliveira, P. (2011). *Geometria e Medida no Ensino Básico*. DGIDC, ME, Lisboa.

Canavarro, A. P. (coord.), (2021). *Essential Learning of Mathematics in Basic Education*. DGE, 2021 [In Portuguese]

Carvalho e Silva, J. (2018). Secondary Mathematics for the Social Sciences. *Pre-Proceedings ICMI Study 24, School Mathematics Curriculum Reforms: Challenges, Changes and Opportunities* (pp. 309-316).

Carvalho e Silva, J. (coord.), (2020). *Recommendations for improving student learning in Mathematics*. DGE.
[In Portuguese]

Carvalho e Silva, J. (coord.), (2023). *Essential Learning. Coordination with the students' profile. 10th, 11th, 12th year. High school. Mathematics A*. December 2022. Portuguese Republic: Education. [In Portuguese]

De Guzmán, M. (1992). *Tendencias innovadoras en educación matemática*. Buenos Aires: Olimpiada Matemática Argentina. Available online on the *Catedra Miguel de Guzmán*: https://blogs.mat.ucm.es/catedramdeguzman/tendencias-innovadoras-en-educacion-matematica/

Friberg, J. (2014). Geometric Division Problems, Quadratic Equations, and Recursive Geometric Algorithms in Mesopotamian Mathematics. *Arch. Hist. Exact Sci.* 68:1–34.

Hanna, Gila & Yan, Xiaoheng (2021). Opening a discussion on teaching proof with automated theorem provers, *For the Learning of Mathematics* 41(3), 42-46 (November, 2021) FLM Publishing Association, New Westminster, BC, Canada (Preprint).

Hohenwarter, M. (2023). *Geogebra + AI. Presentation at the Congreso Internacional de GeoGebra, Córdoba, 9-12/11/2023*. https://congresogeogebra.org/

Gosztonyi, K., van den Heuvel-Panhuizen, M., Makinae, N., Shimizu, S., & van Zanten, M. (2023). International Co-operation and Influential Reforms. In: Shimizu, Y., Vithal, R. (eds) *Mathematics Curriculum Reforms Around the World*. New ICMI Study Series. Springer, Cham.

Hartshorne, R. (2000). Teaching Geometry According to Euclid. *Notices of the AMS*, vol. 47, 4, pp. 460-465.

Heath, T. L. (1912). *The Method of Archimedes*, Cambridge University Press.

Joseph, G. G. (2011). *The crest of the peacock: non-European roots of Mathematics*. Princeton University Press, 3rd ed.

Kovács, Z., Recio, T., & Vélez, M.P. (2022). Automated reasoning tools with GeoGebra: What are they? What are they good for? In P. R. Richard, M. P. Vélez, & S. van Vaerenbergh (Eds.), *Mathematics education in the age of artificial intelligence* (pp. 23–44), Springer. https://doi.org/10.1007/978-3-030-86909-0_2

Loh, Po-Shen (2023). The brilliant math coach teaching America's kids to outsmart AI. Interview to the *Wall Street Journal*. May 25, 2023.

Paulo, J. S. (1944). Review of Palma Fernandes - Elementos de Geometria, para o 1º, 2º e 3º anos dos Liceus. 1943, Livraria Cruz, Braga. *Gazeta de Matemática*, 20, 30-31. [In Portuguese]

Proust, C. (2014). Does a master always write for his students? Some evidence from old Babylonian scribal schools. *Scientific Sources and Teaching Contexts throughout History: Problems and Perspectives* (pp. 69-94). Springer.

Ruiz, A. (2015). Balance y perspectivas de la Reforma de la Educación Matemática en Costa Rica. *Cuadernos de Investigación y Formación en Educación Matemática*, 10(13) 15-33.

Teles, J., Santos, V., and Quaresma, P. (2022). A rule-based theorem prover: An introduction to proofs in secondary schools. In J. Marcos, W. Neuper and P. Quaresma (Eds.): *Theorem Proving Components for Educational Software* 2022 (ThEdu’22) EPTCS 375, 2023, pp. 24–37.
Moura (Ed.). Conference Proceedings in Mathematics Education (6), pages 71-78.
https://doi.org/10.37626/GA9783959871440.0

Breda, A., Serrazina, L., Menezes, L., Sousa, H., & Oliveira, P. (2011). *Geometria e Medida no Ensino Básico*. DGIDC, ME, Lisboa.

Canavarro, A. P. (coord.), (2021). *Essential Learning of Mathematics in Basic Education*. DGE, 2021 [In Portuguese]

Carvalho e Silva, J. (2018). Secondary Mathematics for the Social Sciences. *Pre-Proceedings ICMI Study 24, School Mathematics Curriculum Reforms: Challenges, Changes and Opportunities* (pp. 309-316).

Carvalho e Silva, J. (coord.), (2020). *Recommendations for improving student learning in Mathematics*. DGE.
[In Portuguese]

Carvalho e Silva, J. (coord.), (2023). *Essential Learning. Coordination with the students' profile. 10th, 11th, 12th year. High school. Mathematics A*. December 2022. Portuguese Republic: Education. [In Portuguese]

De Guzmán, M. (1992). *Tendencias innovadoras en educación matemática*. Buenos Aires: Olimpiada Matemática Argentina. Available online on the *Catedra Miguel de Guzmán*: https://blogs.mat.ucm.es/catedramdeguzman/tendencias-innovadoras-en-educacion-matematica/

Friberg, J. (2014). Geometric Division Problems, Quadratic Equations, and Recursive Geometric Algorithms in Mesopotamian Mathematics. *Arch. Hist. Exact Sci.* 68:1–34.

Hanna, Gila & Yan, Xiaoheng (2021). Opening a discussion on teaching proof with automated theorem provers, *For the Learning of Mathematics* 41(3), 42-46 (November, 2021) FLM Publishing Association, New Westminster, BC, Canada (Preprint).

Hohenwarter, M. (2023). *Geogebra + AI. Presentation at the Congreso Internacional de GeoGebra, Córdoba, 9-12/11/2023*. https://congresogeogebra.org/

Gosztonyi, K., van den Heuvel-Panhuizen, M., Makinae, N., Shimizu, S., & van Zanten, M. (2023). International Co-operation and Influential Reforms. In: Shimizu, Y., Vithal, R. (eds) *Mathematics Curriculum Reforms Around the World*. New ICMI Study Series. Springer, Cham.

Hartshorne, R. (2000). Teaching Geometry According to Euclid. *Notices of the AMS*, vol. 47, 4, pp. 460-465.

Heath, T. L. (1912). *The Method of Archimedes*, Cambridge University Press.

Joseph, G. G. (2011). *The crest of the peacock: non-European roots of Mathematics*. Princeton University Press, 3rd ed.

Kovács, Z., Recio, T., & Vélez, M.P. (2022). Automated reasoning tools with GeoGebra: What are they? What are they good for? In P. R. Richard, M. P. Vélez, & S. van Vaerenbergh (Eds.), *Mathematics education in the age of artificial intelligence* (pp. 23–44), Springer. https://doi.org/10.1007/978-3-030-86909-0_2

Loh, Po-Shen (2023). The brilliant math coach teaching America's kids to outsmart AI. Interview to the *Wall Street Journal*. May 25, 2023.

Paulo, J. S. (1944). Review of Palma Fernandes - Elementos de Geometria, para o 1º, 2º e 3º anos dos Liceus. 1943, Livraria Cruz, Braga. *Gazeta de Matemática*, 20, 30-31. [In Portuguese]

Proust, C. (2014). Does a master always write for his students? Some evidence from old Babylonian scribal schools. *Scientific Sources and Teaching Contexts throughout History: Problems and Perspectives* (pp. 69-94). Springer.

Ruiz, A. (2015). Balance y perspectivas de la Reforma de la Educación Matemática en Costa Rica. *Cuadernos de Investigación y Formación en Educación Matemática*, 10(13) 15-33.

Teles, J., Santos, V., and Quaresma, P. (2022). A rule-based theorem prover: An introduction to proofs in secondary schools. In J. Marcos, W. Neuper and P. Quaresma (Eds.): *Theorem Proving Components for Educational Software* 2022 (ThEdu’22) EPTCS 375, 2023, pp. 24–37.
# DEFINITIONS OF PROSPECTIVE PRIMARY TEACHERS CONCERNING THE AREA OF 2D FIGURES

Sofia Caviedes¹, Genaro de Gamboa², Edelmira Badillo², Luis Pino Fan¹

¹ Universidad de Los Lagos, ² Universitat Autònoma de Barcelona

The aim of the present study is to characterize the personal definitions about the area of 2D figures of a group of prospective primary teachers. For this purpose, tools of the Ontosemiotic Approach to Mathematical Knowledge and Instruction are used. A content analysis is carried out to identify the emergence of different objects and processes in the definitions of prospective primary teachers. The results show that, for the most part, personal definitions of prospective primary teachers match institutionalized definitions of area as an object of reference. Likewise, the results show that the structure of the definitions provided is given by different objects and processes that allow describing the way in which prospective primary teachers use the different area partial meanings in their definitions. Finally, the results show tendencies of prospective primary teachers in the use of geometric, numerical, and algebraic representations to define area.

## INTRODUCTION

Definitions of geometric concepts are a fundamental component of teachers who teach mathematics (Ball et al., 2008; Zazkis & Leikin, 2008), as they reveal a series of logical relationships, for example, between propositions, didactic learning sequences, mathematical connections, or mathematical communication (Leikin & Winicki-Landman, 2001). The variety of ideas about the structure of a mathematical definition and the associated challenges, has led researchers to suggest that definitions should be explicitly addressed as part of initial and continuing teacher education (Leikin & Winicki-Landman, 2001; Zazkis & Leikin, 2008). This is because teachers play a determining role in the definitions used in the classroom and in the role given to these definitions in mathematical activity. However, research that addresses the knowledge that teachers have about mathematical definitions is scarce, pointing mostly to the difficulties associated with this process (Leikin & Winicki-Landman, 2001). Particularly, some studies show the difficulties of prospective and in-service teachers in defining geometric concepts, as quadrilaterals or polygons (e.g., Avcu, 2022; Fujita & Jones, 2007; Leikin & Zazkis, 2010; Miller, 2018). In explicit relation to the area of 2D figures, studies are more scarce and suggest that prospective teachers tend to define area as "length x width" (Livy et al., 2012), coinciding with students' definitions (e.g., Zacharos, 2006). Understanding the personal definitions of prospective primary teachers is a crucial step before designing targeted interventions, as it provides insights into the specific areas of difficulty and misconceptions, and therefore, allows to address the root causes of the difficulties outlined above. In this context, the aim of this paper is to characterize the personal definitions about the area of 2D figures of a group of prospective primary teachers (PPT). Personal definitions are understood as those interpretations made by PPTs of a formal definition of the area in the sense of Tall and Viner (1981). For this purpose, we take into consideration tools from the Ontosemiotic Approach to Mathematical Knowledge and Instruction, which we describe in the following section.
# DEFINITIONS OF PROSPECTIVE PRIMARY TEACHERS CONCERNING THE AREA OF 2D FIGURES

Sofia Caviedes¹, Genaro de Gamboa², Edelmira Badillo², Luis Pino Fan¹

¹ Universidad de Los Lagos, ² Universitat Autònoma de Barcelona

The aim of the present study is to characterize the personal definitions about the area of 2D figures of a group of prospective primary teachers. For this purpose, tools of the Ontosemiotic Approach to Mathematical Knowledge and Instruction are used. A content analysis is carried out to identify the emergence of different objects and processes in the definitions of prospective primary teachers. The results show that, for the most part, personal definitions of prospective primary teachers match institutionalized definitions of area as an object of reference. Likewise, the results show that the structure of the definitions provided is given by different objects and processes that allow describing the way in which prospective primary teachers use the different area partial meanings in their definitions. Finally, the results show tendencies of prospective primary teachers in the use of geometric, numerical, and algebraic representations to define area.

## INTRODUCTION

Definitions of geometric concepts are a fundamental component of teachers who teach mathematics (Ball et al., 2008; Zazkis & Leikin, 2008), as they reveal a series of logical relationships, for example, between propositions, didactic learning sequences, mathematical connections, or mathematical communication (Leikin & Winicki-Landman, 2001). The variety of ideas about the structure of a mathematical definition and the associated challenges, has led researchers to suggest that definitions should be explicitly addressed as part of initial and continuing teacher education (Leikin & Winicki-Landman, 2001; Zazkis & Leikin, 2008). This is because teachers play a determining role in the definitions used in the classroom and in the role given to these definitions in mathematical activity. However, research that addresses the knowledge that teachers have about mathematical definitions is scarce, pointing mostly to the difficulties associated with this process (Leikin & Winicki-Landman, 2001). Particularly, some studies show the difficulties of prospective and in-service teachers in defining geometric concepts, as quadrilaterals or polygons (e.g., Avcu, 2022; Fujita & Jones, 2007; Leikin & Zazkis, 2010; Miller, 2018). In explicit relation to the area of 2D figures, studies are more scarce and suggest that prospective teachers tend to define area as "length x width" (Livy et al., 2012), coinciding with students' definitions (e.g., Zacharos, 2006). Understanding the personal definitions of prospective primary teachers is a crucial step before designing targeted interventions, as it provides insights into the specific areas of difficulty and misconceptions, and therefore, allows to address the root causes of the difficulties outlined above. In this context, the aim of this paper is to characterize the personal definitions about the area of 2D figures of a group of prospective primary teachers (PPT). Personal definitions are understood as those interpretations made by PPTs of a formal definition of the area in the sense of Tall and Viner (1981). For this purpose, we take into consideration tools from the Ontosemiotic Approach to Mathematical Knowledge and Instruction, which we describe in the following section.
**Definitions in the teaching of geometry**

According to Tall and Vinner (1981) and Vinner (1983) the definition of a concept corresponds to a
verbal statement used to specify such concept, explaining it precisely and in a non-circular way. The
latter implies that the term to be defined should not be used in the definition (Zaslavsky & Shir, 2005).
Tall and Vinner (1981) make a classification of concept definitions, identifying a formal definition
of the concept and a personal definition of the concept. The formal definition of the concept
corresponds to that which is accepted by the mathematical community in general (Bingölbali &
Monaghan, 2008; Moore, 1994). To understand this type of definition, PPTs must generate their own
interpretations of the definition (Viholainen, 2008), which may change from one person to another
and from one context to another (Pinto & Tall, 2002). In this context of change, Tall and Vinner
(1981) point out that personal definitions correspond to a "discursive description that the learner uses
for his own explanation of his evoked concept image" (p. 152), this evoked image being the one that
is activated at a given moment. The authors point out that the personal definition of a concept may
be equivalent to the formal definition, or it may disagree and, consequently, be incomplete or
erroneous (Tall & Vinner, 1981). Thus, both definition itself and concept image come into play in the
process of defining, and the understanding of a geometric concept is determined by the link between
the concept image and the concept definition (Gutiérrez & Jaime, 2012). The concept image is often
in conflict with the formal definition of a concept (Tall & Vinner, 1981). For example, even if a
student has been taught, and is able to give a formal definition of a parallelogram as a quadrilateral
with two pairs of parallel opposite sides, he or she may not consider rectangles, squares, and
rhombuses as parallelograms, because the concept image they have of these is that not all angles
("amplitude") or sides ("length") can be equal. In the negotiation of this type of conflicts, the role of
definitions can be decisive (Vinner, 1991), since they allow establishing relationships between the
conditions (e.g., parallelism) that geometric figures must meet to be classified as parallelograms.

In line with the above, this paper delves into personal definitions that PPTs have about the area of 2D
figures. To do so, we use some of the tools of the Ontosemiotic Approach which attributes an essential
role to language and categorizes types of objects that emerge in mathematical activity.

**Ontosemiotic Approach (OSA)**

Godino et al. (2007) define mathematical practice as any action or manifestation performed by
someone to solve mathematical problems and communicate the solution to others. The set of such
practices is referred to as a system of practices, composed by operative facet (e.g., problems) and
discursive facet (e.g., arguments), which are mutually interrelated (Wilhemi et al., 2007). Since
defining mathematical objects involves “more than anything else the conflict between the structure
of mathematics, as conceived by professional mathematicians, and the cognitive processes of concept
acquisition” (Vinner, 1991, p.65), mathematical definitions represent a discursive component of
mathematical system of practices (e.g., the formalized language used in defining mathematical
concepts), which comes after the initial practical engagement or operative practices (Wilhemi et al.,
2007). In this sense, language is the articulating axis of the operative-discursive facets, allowing the
communication of procedures and meanings involved in mathematical activity (Wilhemi et al., 2007).
Operational and discursive practices can be attributed to an individual or shared within an institution
(Godino et al., 2007). In the first case, reference is made to the personal meaning of the object. In the
second case, reference is made to the corresponding institutional meaning. The personal meaning of
## Definitions in the teaching of geometry

According to Tall and Vinner (1981) and Vinner (1983) the definition of a concept corresponds to a verbal statement used to specify such concept, explaining it precisely and in a non-circular way. The latter implies that the term to be defined should not be used in the definition (Zaslavsky & Shir, 2005). Tall and Vinner (1981) make a classification of concept definitions, identifying a formal definition of the concept and a personal definition of the concept. The formal definition of the concept corresponds to that which is accepted by the mathematical community in general (Bingölbali & Monaghan, 2008; Moore, 1994). To understand this type of definition, PPTs must generate their own interpretations of the definition (Viholainen, 2008), which may change from one person to another and from one context to another (Pinto & Tall, 2002). In this context of change, Tall and Vinner (1981) point out that personal definitions correspond to a "discursive description that the learner uses for his own explanation of his evoked concept image" (p. 152), this evoked image being the one that is activated at a given moment. The authors point out that the personal definition of a concept may be equivalent to the formal definition, or it may disagree and, consequently, be incomplete or erroneous (Tall & Vinner, 1981). Thus, both definition itself and concept image come into play in the process of defining, and the understanding of a geometric concept is determined by the link between the concept image and the concept definition (Gutiérrez & Jaime, 2012). The concept image is often in conflict with the formal definition of a concept (Tall & Vinner, 1981). For example, even if a student has been taught, and is able to give a formal definition of a parallelogram as a quadrilateral with two pairs of parallel opposite sides, he or she may not consider rectangles, squares, and rhombuses as parallelograms, because the concept image they have of these is that not all angles ("amplitude") or sides ("length") can be equal. In the negotiation of this type of conflicts, the role of definitions can be decisive (Vinner, 1991), since they allow establishing relationships between the conditions (e.g., parallelism) that geometric figures must meet to be classified as parallelograms.

In line with the above, this paper delves into personal definitions that PPTs have about the area of 2D figures. To do so, we use some of the tools of the Ontosemiotic Approach which attributes an essential role to language and categorizes types of objects that emerge in mathematical activity.

## Ontosemiotic Approach (OSA)

Godino et al. (2007) define mathematical practice as any action or manifestation performed by someone to solve mathematical problems and communicate the solution to others. The set of such practices is referred to as a system of practices, composed by operative facet (e.g., problems) and discursive facet (e.g., arguments), which are mutually interrelated (Wilhemi et al., 2007). Since defining mathematical objects involves “more than anything else the conflict between the structure of mathematics, as conceived by professional mathematicians, and the cognitive processes of concept acquisition” (Vinner, 1991, p.65), mathematical definitions represent a discursive component of mathematical system of practices (e.g., the formalized language used in defining mathematical concepts), which comes after the initial practical engagement or operative practices (Wilhemi et al., 2007). In this sense, language is the articulating axis of the operative-discursive facets, allowing the communication of procedures and meanings involved in mathematical activity (Wilhemi et al., 2007). Operational and discursive practices can be attributed to an individual or shared within an institution (Godino et al., 2007). In the first case, reference is made to the personal meaning of the object. In the second case, reference is made to the corresponding institutional meaning. The personal meaning of
the object is conditioned by its relation/interpretation of the institutional or formal meaning of such
object (Tall & Vinner, 1981). A personal object is then an emergent of the system of personal
practices linked to a given problem situation. The description and analysis of mathematical activity
is conditioned by six primary objects (Godino et al., 2007): linguistic elements; problem-situations;
concepts; properties/propositions; procedures; arguments. The primary objects detailed in the OSA
emerge from systems of practices which underlines their complexity and the need to articulate them.
The later can be done through epistemic configurations (networks of institutional objects) or cognitive
configurations (networks of personal objects including cognitive constructs such as conceptual
images). Moreover, the relationships that can be established between primary objects take place
through different processes, such as definition, which can be considered from different dual facets
giving rise to other fundamental epistemic/cognitive processes in mathematical activity (Godino et
al., 2007): materialization-idealization (ostensive-non-ostensive), particularization-generalization
(intensive-extensive), decomposition-reification (unitary-systemic) and representation-meaning
(expression-content).

From the above-mentioned background, in this paper definition is understood as a process that
involves the cognitive configuration of PPTs and, thus, their personal meanings. Hence, it is accepted
that the personal definition of area constitutes the personal meaning of such an object from the
institutionalization-personalization duality (Godino et al., 2007).

METHOD

The study is situated in an interpretive paradigm and follows a qualitative approach (Cohen et al., 2007). A content analysis involving deductive coding is performed, being the unit of analysis the written protocols of PPTs. An epistemic configuration of area (Caviedes et al., 2021) is used to identify how the cognitive configuration that is activated in the definitions of PPTs matches up the institutional definition of area (in the sense of Tall and Vinner, 1981). The institutionalized definition of area is based on three partial meanings presented in the epistemic configuration mentioned above, which can be summarized as follows: *two-dimensional surface that can be covered with units of measurement and/or calculated by means of formulas*. Deductive coding makes it possible to identify the primary objects that account for the cognitive configuration of PPTs, and thus the personal meanings associated with area and the processes in their dual facets. The process of defining is then characterized by the gradual, systematic, and progressive emergence of different objects, processes, and meanings (Godino et al., 2007).

A semi-structured open-ended questionnaire was designed to be solved individually and in writing
by a group of 70 PPT in the third year of the Primary Education degree, in the 2020-21 academic year
at a public University in Spain. The questionnaire was part of an evaluated activity, had a total of 8
tasks and sought to problematize mathematical and didactic knowledge about area of PPTs (see
Caviedes et al., 2023a, 2023b). Although PPTs had worked on activities related to area (e.g., use of
different contexts and procedures to solve tasks, analysis of students' responses), they had not worked
on its definition. The questionnaire was applied by the professor in charge of the subject, in online
format due to the COVID-19 health contingency, and the PPTs had one week to answer it. Due to the
objective of this paper, evidence of the written protocols corresponding to Task 7 is presented, posing
the following problem situation: *If you had to introduce the geometric content of area in 5th grade,
how would you define area?* It's worth mentioning that PPTs were told to focus on *how* to define.
the object is conditioned by its relation/interpretation of the institutional or formal meaning of such
object (Tall & Vinner, 1981). A personal object is then an emergent of the system of personal
practices linked to a given problem situation. The description and analysis of mathematical activity
is conditioned by six primary objects (Godino et al., 2007): linguistic elements; problem-situations;
concepts; properties/propositions; procedures; arguments. The primary objects detailed in the OSA
emerge from systems of practices which underlines their complexity and the need to articulate them.
The later can be done through epistemic configurations (networks of institutional objects) or cognitive
configurations (networks of personal objects including cognitive constructs such as conceptual
images). Moreover, the relationships that can be established between primary objects take place
through different processes, such as definition, which can be considered from different dual facets
giving rise to other fundamental epistemic/cognitive processes in mathematical activity (Godino et
al., 2007): materialization-idealization (ostensive-non-ostensive), particularization-generalization
(intensive-extensive), decomposition-reification (unitary-systemic) and representation-meaning
(expression-content).

From the above-mentioned background, in this paper definition is understood as a process that
involves the cognitive configuration of PPTs and, thus, their personal meanings. Hence, it is accepted
that the personal definition of area constitutes the personal meaning of such an object from the
institutionalization-personalization duality (Godino et al., 2007).

METHOD

The study is situated in an interpretive paradigm and follows a qualitative approach (Cohen et al., 2007). A content analysis involving deductive coding is performed, being the unit of analysis the written protocols of PPTs. An epistemic configuration of area (Caviedes et al., 2021) is used to identify how the cognitive configuration that is activated in the definitions of PPTs matches up the institutional definition of area (in the sense of Tall and Vinner, 1981). The institutionalized definition of area is based on three partial meanings presented in the epistemic configuration mentioned above, which can be summarized as follows: *two-dimensional surface that can be covered with units of measurement and/or calculated by means of formulas*. Deductive coding makes it possible to identify the primary objects that account for the cognitive configuration of PPTs, and thus the personal meanings associated with area and the processes in their dual facets. The process of defining is then characterized by the gradual, systematic, and progressive emergence of different objects, processes, and meanings (Godino et al., 2007).

A semi-structured open-ended questionnaire was designed to be solved individually and in writing
by a group of 70 PPT in the third year of the Primary Education degree, in the 2020-21 academic year
at a public University in Spain. The questionnaire was part of an evaluated activity, had a total of 8
tasks and sought to problematize mathematical and didactic knowledge about area of PPTs (see
Caviedes et al., 2023a, 2023b). Although PPTs had worked on activities related to area (e.g., use of
different contexts and procedures to solve tasks, analysis of students' responses), they had not worked
on its definition. The questionnaire was applied by the professor in charge of the subject, in online
format due to the COVID-19 health contingency, and the PPTs had one week to answer it. Due to the
objective of this paper, evidence of the written protocols corresponding to Task 7 is presented, posing
the following problem situation: *If you had to introduce the geometric content of area in 5th grade,
how would you define area?* It's worth mentioning that PPTs were told to focus on *how* to define.
## Analysis

The analysis is carried out with the support of MAXQDAplus software. A system of a priori categories is established, given by the epistemic configuration presented in Caviedes et al. (2021) and by the processes of the OSA. The definitions provided by the PPTs allow inferring the emergence of primary objects associated with three partial meanings of area (Caviedes et al., 2021): (1) Sp1-area as space delimited by a closed line; (2) Sp2-area as two-dimensional units covering a surface; (3) Sp3-area as a product of two linear dimensions. The emergence of partial meanings allows us to identify the way in which personal definitions of PPTs enable the emergence of processes of representation/meaning and idealization/materialization. The first is associated with the expression-content duality, where a given representation informs about the structure of that object (Font & Rubio, 2017). The second process is associated with the ostensive-non-ostensive duality, since mathematical objects are in general not perceptible, but are used in mathematical practices through their associated ostensive (e.g., notations, graphs, etc.) (Font & Rubio, 2017). We identify 3 types of definitions: (1) definitions that involve two dual processes, (2) definitions that involve one dual process; (3) definitions that do not involve dual processes.

The definition of PPT 55 (Figure 1) allows inferring the emergence of a representation/meaning process, since PPT 55 attributes a content (area) to a rectangular surface, granting multiplicative meaning to the structure of rows and columns of the area model.

PPT 55: First, I would define it as the amount of space or surface area covered by a flat (two-dimensional) figure. The area is measured in square units of a fixed size, such as square centimeters, square inches, square miles, etc. Thus, if we delimit on the plan any figure, to find the area we count how many squares of a certain size will cover the region inside the polygon. So, an example would be the following, where we find a square formed by the following units (5x4):

<figure><img src="image_56.png" /></figure>

In this case, you can count the squares and get 20, so the area is 20 square units. However, this method can be ineffective if the rectangle is of larger dimensions or the units are smaller... in this case you can use multiplication, 5 x 4, since there are 4 rows of 5 squares.

<caption>Figure 1: Definition of PPT 55</caption>

The emergence of the concept of the square unit in the two-dimensional treatment of area (which can be expressed as the product of two lengths) is inferred, as PPT 55 mentions, superficially, that any figure (polygonal) can be measured using squares. Similarly, a process of idealization/materialization is observed, since PPT 55 evokes the properties of the units of measurement, which materializes the idea of area as a space that can be measured in square units and, subsequently, the algorithm that represents the area formula. In this sense, the idealization/materialization process allows giving a material sense to the area formula (the squares that cover the figure and the structure in rows and columns). From this process we infer the emergence of the concept of spatial structuring and the procedure of obtaining the area in an additive way, by means of graphic and symbolic representations.
## Analysis

The analysis is carried out with the support of MAXQDAplus software. A system of a priori categories is established, given by the epistemic configuration presented in Caviedes et al. (2021) and by the processes of the OSA. The definitions provided by the PPTs allow inferring the emergence of primary objects associated with three partial meanings of area (Caviedes et al., 2021): (1) Sp1-area as space delimited by a closed line; (2) Sp2-area as two-dimensional units covering a surface; (3) Sp3-area as a product of two linear dimensions. The emergence of partial meanings allows us to identify the way in which personal definitions of PPTs enable the emergence of processes of representation/meaning and idealization/materialization. The first is associated with the expression-content duality, where a given representation informs about the structure of that object (Font & Rubio, 2017). The second process is associated with the ostensive-non-ostensive duality, since mathematical objects are in general not perceptible, but are used in mathematical practices through their associated ostensive (e.g., notations, graphs, etc.) (Font & Rubio, 2017). We identify 3 types of definitions: (1) definitions that involve two dual processes, (2) definitions that involve one dual process; (3) definitions that do not involve dual processes.

The definition of PPT 55 (Figure 1) allows inferring the emergence of a representation/meaning process, since PPT 55 attributes a content (area) to a rectangular surface, granting multiplicative meaning to the structure of rows and columns of the area model.

**PPT 55:** First, I would define it as the amount of space or surface area covered by a flat (two-dimensional) figure. The area is measured in square units of a fixed size, such as square centimeters, square inches, square miles, etc. Thus, if we delimit on the plan any figure, to find the area we count how many squares of a certain size will cover the region inside the polygon. So, an example would be the following, where we find a square formed by the following units (5x4):

<figure><img src="image_62.png" /></figure>

In this case, you can count the squares and get 20, so the area is 20 square units. However, this method can be ineffective if the rectangle is of larger dimensions or the units are smaller... in this case you can use multiplication, 5 x 4, since there are 4 rows of 5 squares.

<caption>Figure 1: Definition of PPT 55</caption>

The emergence of the concept of the square unit in the two-dimensional treatment of area (which can be expressed as the product of two lengths) is inferred, as PPT 55 mentions, superficially, that any figure (polygonal) can be measured using squares. Similarly, a process of idealization/materialization is observed, since PPT 55 evokes the properties of the units of measurement, which materializes the idea of area as a space that can be measured in square units and, subsequently, the algorithm that represents the area formula. In this sense, the idealization/materialization process allows giving a material sense to the area formula (the squares that cover the figure and the structure in rows and columns). From this process we infer the emergence of the concept of spatial structuring and the procedure of obtaining the area in an additive way, by means of graphic and symbolic representations.
Thus, a personal meaning associated with Sp1, area as space delimited by a closed line; Sp2, area as the number of two-dimensional units that cover a surface; and Sp3, area as the product of two linear dimensions, is inferred.

Moreover, we infer the emergence of two concepts, the surface as an attribute that allows objects to be measured, which has length and width (two-dimensionality), and the square units of measurement, as a region of the plane delimited by a square and as a fixed quantity of a physical magnitude.

The definition of PPT 65 (Figure 2) allows inferring the emergence of a representation/meaning process. PPT 65 attributes a content (area) to a polygonal figure, giving meaning to the property of being a bounded/enclosed space with a surface extension that can be calculated. To exemplify this, PPT 65 makes use of a counterexample (non-delimited/enclosed surface). It is possible to infer that the personal definition of PPT 65 reduces the area to the surface of polygons, without contemplating, for example, concave or irregular surfaces. In this way, a bias that relates area to conventional geometric representations is evident. In addition, it is possible to infer a personal meaning associated with Sp1, area as space delimited by a closed line, through the emergence of the concept of surface as a quality that allows objects to be measured, which has length and width (two-dimensionality).

<figure><img src="image_57.png" /><figcaption>Figure 2: Definition of PPT 65</figcaption></figure>

The definition of PPT 56 (Figure 3) allows us to exemplify those definitions that do not show evidence of dual processes. It is possible to infer a personal meaning associated with Sp1, area as space delimited by a closed line and, likewise, the emergence of the concept of surface as a quality of two-dimensional objects. However, it is not possible to infer the emergence of dual processes, since the definition of PPT 56 does not make explicit the notion of closed or internal space of a two-dimensional figure or object, that is, there is no reference to the two-dimensionality of the area (beyond the use of the concept of surface), since a "thing" can have one, two or three dimensions.

**PPT 56:** To introduce the concept of area in the fifth-grade classroom, I would define it as follows: area is the surface occupied by something.

<caption>Figure 3: Definition of PPT 56</caption>

## RESULTS

Table 1 presents the two dual processes derived from the personal definitions of PPTs.
Thus, a personal meaning associated with Sp1, area as space delimited by a closed line; Sp2, area as the number of two-dimensional units that cover a surface; and Sp3, area as the product of two linear dimensions, is inferred.

Moreover, we infer the emergence of two concepts, the surface as an attribute that allows objects to be measured, which has length and width (two-dimensionality), and the square units of measurement, as a region of the plane delimited by a square and as a fixed quantity of a physical magnitude.

The definition of PPT 65 (Figure 2) allows inferring the emergence of a representation/meaning process. PPT 65 attributes a content (area) to a polygonal figure, giving meaning to the property of being a bounded/enclosed space with a surface extension that can be calculated. To exemplify this, PPT 65 makes use of a counterexample (non-delimited/enclosed surface). It is possible to infer that the personal definition of PPT 65 reduces the area to the surface of polygons, without contemplating, for example, concave or irregular surfaces. In this way, a bias that relates area to conventional geometric representations is evident. In addition, it is possible to infer a personal meaning associated with Sp1, area as space delimited by a closed line, through the emergence of the concept of surface as a quality that allows objects to be measured, which has length and width (two-dimensionality).

<figure><img src="image_63.png" /><figcaption>Figure 2: Definition of PPT 65</figcaption></figure>

The definition of PPT 56 (Figure 3) allows us to exemplify those definitions that do not show evidence of dual processes. It is possible to infer a personal meaning associated with Sp1, area as space delimited by a closed line and, likewise, the emergence of the concept of surface as a quality of two-dimensional objects. However, it is not possible to infer the emergence of dual processes, since the definition of PPT 56 does not make explicit the notion of closed or internal space of a two-dimensional figure or object, that is, there is no reference to the two-dimensionality of the area (beyond the use of the concept of surface), since a "thing" can have one, two or three dimensions.

**PPT 56:** To introduce the concept of area in the fifth-grade classroom, I would define it as follows: area is the surface occupied by something.

<caption>Figure 3: Definition of PPT 56</caption>

## RESULTS

Table 1 presents the two dual processes derived from the personal definitions of PPTs.
<table><caption>Table 1: Meanings and processes emerging from the definitions of PPTs (N=60). [**indicates the number of PPTs mobilizing two dual processes in their personal definitions; *indicates the number of PPTs mobilizing one dual process in their personal definitions]</caption><thead><tr><th>Emerging processes</th><th>Descriptor (Personal meaning)</th><th>Partial meanings</th><th>Frequency</th></tr></thead><tbody><tr><td rowspan="3">Idealization-materialization</td><td>(1) The emergence of the iteration procedure of units of measurement (standard and non-standard), and their respective properties, materializes the idea of area as a space that can be measured in square units.</td><td>Sp1, Sp2</td><td>1**</td></tr><tr><td>(2) The emergence of the product formula procedure, through the rectangular model of multiplication, materializes the rectangle formula by structuring rows and columns.</td><td>Sp1, Sp2, Sp3</td><td>2**</td></tr><tr><td></td><td>Sp2, Sp3</td><td>1**</td></tr><tr><td rowspan="4">Representation-meaning</td><td>(1) A content is attributed to the square polygonal surface, giving meaning to the property of being a delimited/closed space that can be decomposed into other figures.</td><td>Sp1</td><td>1*</td></tr><tr><td>(2) A content is attributed to a polygonal figure, giving meaning to the property of being a delimited/enclosed space with a surface area that can be measured.</td><td>Sp1</td><td>26*</td></tr><tr><td></td><td>Sp3</td><td>2*</td></tr><tr><td></td><td>Sp1, Sp2</td><td>6*</td></tr><tr><td></td><td>Sp1, Sp3</td><td>17*</td></tr><tr><td></td><td>Sp1, Sp2, Sp3</td><td>1**</td></tr><tr><td></td><td>(3) A content is attributed to the rectangular surface, giving multiplicative meaning to the row and column structure of the area model.</td><td>Sp1, Sp2, Sp3</td><td>3**</td></tr></tbody></table>

Each process is linked to one or more partial meanings of area (which constitute the institutional definition), and thus to specific primary objects which can be seen in detail in Caviedes et al. (2021). Table 1 shows that a majority of PPTs use the second representation/meaning process (attributing a content to a polygonal figure, giving meaning to the property of being a delimited/enclosed space with a measurable surface extension) based on Sp1. On the contrary, a minority of PPTs use the other processes of representation/ meaning, or the processes of idealization/materialization, which would allow them to make more complete personal definitions through the joint mobilization of the three partial meanings of the area. In addition, Table 1 also shows the relationship between meanings and processes. It is observed that the idealization/materialization processes occur only when two or three partial meanings of the area are connected. In the case of the representation/ meaning processes, it is observed that the first one (attributing a content to a square polygonal surface by decomposing it into other figures) occurs when mobilizing only Sp1. The second representation/ meaning process (attributing a content to a polygonal surface by measuring) occurs both by mobilizing the different partial meanings and by mobilizing only Sp1 and Sp3. The third representation/ meaning process (attributing a content to a rectangular polygonal surface by structuring rows and columns) occurs when Sp1, Sp2 and Sp3 are mobilized together.
<table><caption>Table 1: Meanings and processes emerging from the definitions of PPTs (N=60). [**indicates the number of PPTs mobilizing two dual processes in their personal definitions; *indicates the number of PPTs mobilizing one dual process in their personal definitions]</caption><thead><tr><th>Emerging processes</th><th>Descriptor (Personal meaning)</th><th>Partial meanings</th><th>Frequency</th></tr></thead><tbody><tr><td rowspan="3">Idealization-materialization</td><td>(1) The emergence of the iteration procedure of units of measurement (standard and non-standard), and their respective properties, materializes the idea of area as a space that can be measured in square units.</td><td>Sp1, Sp2</td><td>1**</td></tr><tr><td>(2) The emergence of the product formula procedure, through the rectangular model of multiplication, materializes the rectangle formula by structuring rows and columns.</td><td>Sp1, Sp2, Sp3</td><td>2**</td></tr><tr><td></td><td>Sp2, Sp3</td><td>1**</td></tr><tr><td rowspan="4">Representation-meaning</td><td>(1) A content is attributed to the square polygonal surface, giving meaning to the property of being a delimited/closed space that can be decomposed into other figures.</td><td>Sp1</td><td>1*</td></tr><tr><td>(2) A content is attributed to a polygonal figure, giving meaning to the property of being a delimited/enclosed space with a surface area that can be measured.</td><td>Sp1</td><td>26*</td></tr><tr><td></td><td>Sp3</td><td>2*</td></tr><tr><td></td><td>Sp1, Sp2</td><td>6*</td></tr><tr><td></td><td>Sp1, Sp3</td><td>17*</td></tr><tr><td></td><td>Sp1, Sp2, Sp3</td><td>1**</td></tr><tr><td></td><td>(3) A content is attributed to the rectangular surface, giving multiplicative meaning to the row and column structure of the area model.</td><td>Sp1, Sp2, Sp3</td><td>3**</td></tr></tbody></table>

Each process is linked to one or more partial meanings of area (which constitute the institutional definition), and thus to specific primary objects which can be seen in detail in Caviedes et al. (2021). Table 1 shows that a majority of PPTs use the second representation/meaning process (attributing a content to a polygonal figure, giving meaning to the property of being a delimited/enclosed space with a measurable surface extension) based on Sp1. On the contrary, a minority of PPTs use the other processes of representation/ meaning, or the processes of idealization/materialization, which would allow them to make more complete personal definitions through the joint mobilization of the three partial meanings of the area. In addition, Table 1 also shows the relationship between meanings and processes. It is observed that the idealization/materialization processes occur only when two or three partial meanings of the area are connected. In the case of the representation/ meaning processes, it is observed that the first one (attributing a content to a square polygonal surface by decomposing it into other figures) occurs when mobilizing only Sp1. The second representation/ meaning process (attributing a content to a polygonal surface by measuring) occurs both by mobilizing the different partial meanings and by mobilizing only Sp1 and Sp3. The third representation/ meaning process (attributing a content to a rectangular polygonal surface by structuring rows and columns) occurs when Sp1, Sp2 and Sp3 are mobilized together.
The definitions of PPTs present a structure that differs in the emergence of the mathematical primary objects corresponding to the three partial meanings of the area, and in the processes that these meanings make possible. This way, the personal meaning evidenced, and therefore the concept image, can be equivalent to the institutionalized meaning if Sp1, Sp2 and Sp3 are mobilized in the process of defining. The joint mobilization of these meanings would account for robust personal definitions. Isolated mobilization of partial meanings would account for incomplete personal definitions.

## CONCLUSIONS

The aim of this study was to characterize the personal definitions of the area of 2D figures of a group of PPT. The use of the configuration of objects and processes as a theoretical/analytical tool makes explicit the complexity underlying the process of defining (De Villiers et al., 2009), showing that this process involves describing objects and concepts, identifying properties of mathematical objects (Zaslavsky & Shir, 2005), communicating mathematical ideas, and coordinating different representations (Sinclair et al., 2012). The mobilization of different partial meanings makes it possible the emergence of certain processes that strengthen and complexify the definitions of PPTs. Hence, primary objects and processes account for the concept image that underlies personal definitions of PPTs (Tall & Vinner, 1981). Moreover, it is evidenced that PPTs concept image about the area of 2D figures can be put in correspondence with three partial meanings of the area (Caviedes et al., 2021), whose primary objects vary according to the intra-mathematical context in which they are framed. Therefore, this study considers the complexity of the process of defining (in terms of objects and processes) as a differentiating aspect of it and, at the same time, unifying, in the sense that it allows characterizing robust definitions through partial meanings of reference. In this sense, we agree that learning the process of defining should be an essential part of the learning process of mathematics in teacher training (Zaslavsky et al., 2003), since personal definition can have an impact on how they interpret the curriculum, what activities to propose and how to interpret students' mathematical practices. However, studies involving the design of teaching experiments based on these results are needed to strengthen the definitions and address the respective difficulties of PPTs.

## Acknowledgements

Study funded by ANID-Chile Strengthening of doctoral programmes n° 2022-86220016. PID2019-104964GB-I00 and GIPEAM, 2021 SGR 00159-AGAUR.

## References

- Avcu, R. (2022). Pre-service middle school mathematics teachers’ personal concept definitions of special quadrilaterals. *Mathematics Education Research Journal*, 1-46.

- Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? *Journal of Teacher Education*, 59(5), 389–407. https://doi.org/10.1177/0022487108324554

- Bingölbali, E., & Monaghan, J. (2008). Concept image revisited. *Educational Studies in Mathematics*, 68(1), 19–35. https://doi.org/10.1007/s10649-007-9112-2

- Caviedes, S., de Gamboa, G., & Badillo, E. (2021). Mathematical objects that configure the partial area meanings mobilized in task-solving. *International Journal of Mathematical Education in Science and Technology*, 54(6), 1092-1111. https://doi.org/10.1080/0020739X.2021.1991019

- Caviedes, S., de Gamboa, G., & Badillo, E. (2023a). Preservice teachers’ knowledge mobilized in solving area tasks. *Journal on Mathematics Education*, 14(1), 35–54. https://doi.org/10.22342/jme.v14i1.pp35-54
The definitions of PPTs present a structure that differs in the emergence of the mathematical primary objects corresponding to the three partial meanings of the area, and in the processes that these meanings make possible. This way, the personal meaning evidenced, and therefore the concept image, can be equivalent to the institutionalized meaning if Sp1, Sp2 and Sp3 are mobilized in the process of defining. The joint mobilization of these meanings would account for robust personal definitions. Isolated mobilization of partial meanings would account for incomplete personal definitions.

## CONCLUSIONS

The aim of this study was to characterize the personal definitions of the area of 2D figures of a group of PPT. The use of the configuration of objects and processes as a theoretical/analytical tool makes explicit the complexity underlying the process of defining (De Villiers et al., 2009), showing that this process involves describing objects and concepts, identifying properties of mathematical objects (Zaslavsky & Shir, 2005), communicating mathematical ideas, and coordinating different representations (Sinclair et al., 2012). The mobilization of different partial meanings makes it possible the emergence of certain processes that strengthen and complexify the definitions of PPTs. Hence, primary objects and processes account for the concept image that underlies personal definitions of PPTs (Tall & Vinner, 1981). Moreover, it is evidenced that PPTs concept image about the area of 2D figures can be put in correspondence with three partial meanings of the area (Caviedes et al., 2021), whose primary objects vary according to the intra-mathematical context in which they are framed. Therefore, this study considers the complexity of the process of defining (in terms of objects and processes) as a differentiating aspect of it and, at the same time, unifying, in the sense that it allows characterizing robust definitions through partial meanings of reference. In this sense, we agree that learning the process of defining should be an essential part of the learning process of mathematics in teacher training (Zaslavsky et al., 2003), since personal definition can have an impact on how they interpret the curriculum, what activities to propose and how to interpret students' mathematical practices. However, studies involving the design of teaching experiments based on these results are needed to strengthen the definitions and address the respective difficulties of PPTs.

## Acknowledgements

Study funded by ANID-Chile Strengthening of doctoral programmes n° 2022-86220016. PID2019-104964GB-I00 and GIPEAM, 2021 SGR 00159-AGAUR.

## References

Avcu, R. (2022). Pre-service middle school mathematics teachers’ personal concept definitions of special quadrilaterals. *Mathematics Education Research Journal*, 1-46.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? *Journal of Teacher Education*, 59(5), 389–407. https://doi.org/10.1177/0022487108324554

Bingölbali, E., & Monaghan, J. (2008). Concept image revisited. *Educational Studies in Mathematics*, 68(1), 19–35. https://doi.org/10.1007/s10649-007-9112-2

Caviedes, S., de Gamboa, G., & Badillo, E. (2021). Mathematical objects that configure the partial area meanings mobilized in task-solving. *International Journal of Mathematical Education in Science and Technology*, 54(6), 1092-1111. https://doi.org/10.1080/0020739X.2021.1991019

Caviedes, S., de Gamboa, G., & Badillo, E. (2023a). Preservice teachers’ knowledge mobilized in solving area tasks. *Journal on Mathematics Education*, 14(1), 35–54. https://doi.org/10.22342/jme.v14i1.pp35-54
Caviedes Barrera, S., de Gamboa Rojas, G., & Badillo Jiménez, E. (2023b). Mathematical and didactic knowledge of preservice primary teachers about the area of 2d figures. *Avances de Investigación en Educación Matemática*, (24), 1-20. https://doi.org/10.35763/aiem24.4076

De Villiers, M., Govender, R., & Patterson, N. (2009). Defining in geometry. En T. V. Craine, y R. Rubenstein (Eds.). *Understanding geometry for a changing world* (pp.189–203). National Council of Teachers of Mathematics.

Font, V., & Rubio, N. V. (2017). Procesos matemáticos en el enfoque ontosemiótico. En J. M. Contreras, P. Arteaga, G. R. Cañadas, M. M. Gea, B. Giacomone y M. M. López-Martín (Eds.), *Actas del Segundo Congreso International Virtual sobre el Enfoque Ontosemiótico del Conocimiento y la Instrucción Matemáticos*, (pp.1–27).

Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. *Zdm, 39*, 127-135. https://doi.org/10.1007/s11858-006-0004-1

Gutiérrez, Á., & Jaime, A. (2012). Reflections on the Teaching of Geometry in Primary and Secondary Schools. *Tecné, Episteme y Didaxis: TED, 32*, 55-70.

Leikin, R., & Winicki-Landman, G. (2001). Defining as a vehicle for professional development of secondary school mathematics teachers. *Mathematics Teacher Education and Development, 3*, 62–73.

Livy, S., Muir, T., & Maher, N. (2012). How do they measure up? Primary pre-service teachers' mathematical knowledge of area and perimeter. *Mathematics Teacher Education and Development, 14*(2), 91-112.

Miller, S. M. (2018). An analysis of the form and content of quadrilateral definitions composed by novice pre-service teachers. *The Journal of Mathematical Behavior, 50*, 142–154.

Moore, R. C. (1994). Making the transition to formal proof. *Educational Studies in Mathematics, 27*(3), 249–266. https://doi.org/10.1007/BF01273731

Pinto, M., & Tall, D. (2002). Building formal mathematics on visual imagery: A case study and a theory. For the Learning of Mathematics, 22(1), 2–10.

Sinclair, N., Pimm, D., Skelin, M., & Zbiek, R. M. (2012). *Developing essential understanding of geometry for teaching mathematics in grades 6–8*. National Council of Teachers of Mathematics.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational Studies in Mathematics, 12*(2), 151–169.

Viholainen, A. (2008). Incoherence of a concept image and erroneous conclusions in the case of differentiability. *The Mathematics Enthusiast*, 5(2), 231–248. https://doi.org/10.54870/1551-3440.1104

Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293–305.

Wilhelmi, M. R., Godino, J. D., & Lacasta, E. (2007). Didactic effectiveness of mathematical definitions: The case of the absolute value. *International Electronic Journal of Mathematics Education, 2*(2), 72–90.

Zacharos, K. (2006). Prevailing educational practices for area measurement and students’ failure in measuring areas. *The Journal of Mathematical Behavior, 25*(3), 224-239.

Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. *Journal for Research in Mathematics Education, 36*(4), 317–346. https://doi.org/10.2307/30035043

Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. *Educational Studies in Mathematics, 69*, 131–148. https://doi.org/10.1007/s10649-008-9131-7
Caviedes Barrera, S., de Gamboa Rojas, G., & Badillo Jiménez, E. (2023b). Mathematical and didactic knowledge of preservice primary teachers about the area of 2d figures. *Avances de Investigación en Educación Matemática*, (24), 1-20. https://doi.org/10.35763/aiem24.4076

De Villiers, M., Govender, R., & Patterson, N. (2009). Defining in geometry. En T. V. Craine, y R. Rubenstein (Eds.). *Understanding geometry for a changing world* (pp.189–203). National Council of Teachers of Mathematics.

Font, V., & Rubio, N. V. (2017). Procesos matemáticos en el enfoque ontosemiótico. En J. M. Contreras, P. Arteaga, G. R. Cañadas, M. M. Gea, B. Giacomone y M. M. López-Martín (Eds.), *Actas del Segundo Congreso International Virtual sobre el Enfoque Ontosemiótico del Conocimiento y la Instrucción Matemáticos*, (pp.1–27).

Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. *Zdm*, **39**, 127-135. https://doi.org/10.1007/s11858-006-0004-1

Gutiérrez, Á., & Jaime, A. (2012). Reflections on the Teaching of Geometry in Primary and Secondary Schools. *Tecné, Episteme y Didaxis: TED*, **32**, 55-70.

Leikin, R., & Winicki-Landman, G. (2001). Defining as a vehicle for professional development of secondary school mathematics teachers. *Mathematics Teacher Education and Development*, **3**, 62–73.

Livy, S., Muir, T., & Maher, N. (2012). How do they measure up? Primary pre-service teachers' mathematical knowledge of area and perimeter. *Mathematics Teacher Education and Development, 14*(2), 91-112.

Miller, S. M. (2018). An analysis of the form and content of quadrilateral definitions composed by novice pre-service teachers. *The Journal of Mathematical Behavior, 50*, 142–154.

Moore, R. C. (1994). Making the transition to formal proof. *Educational Studies in Mathematics*, **27**(3), 249–266. https://doi.org/10.1007/BF01273731

Pinto, M., & Tall, D. (2002). Building formal mathematics on visual imagery: A case study and a theory. For the Learning of Mathematics, **22**(1), 2–10.

Sinclair, N., Pimm, D., Skelin, M., & Zbiek, R. M. (2012). *Developing essential understanding of geometry for teaching mathematics in grades 6–8*. National Council of Teachers of Mathematics.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational Studies in Mathematics*, **12**(2), 151–169.

Viholainen, A. (2008). Incoherence of a concept image and erroneous conclusions in the case of differentiability. *The Mathematics Enthusiast*, **5**(2), 231–248. https://doi.org/10.54870/1551-3440.1104

Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293–305.

Wilhelmi, M. R., Godino, J. D., & Lacasta, E. (2007). Didactic effectiveness of mathematical definitions: The case of the absolute value. *International Electronic Journal of Mathematics Education*, **2**(2), 72–90.

Zacharos, K. (2006). Prevailing educational practices for area measurement and students’ failure in measuring areas. *The Journal of Mathematical Behavior, 25*(3), 224-239.

Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. *Journal for Research in Mathematics Education, 36*(4), 317–346. https://doi.org/10.2307/30035043

Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. *Educational Studies in Mathematics, 69*, 131–148. https://doi.org/10.1007/s10649-008-9131-7
# PROBLEM TYPES IN GEOMETRY TEXTBOOKS: RUSSIA'S EXPERIENCE

Alexander Karp

Teachers College, Columbia University

This short paper is devoted to the analysis of problems offered in Russian geometry textbooks. These problems determine the types of activity prevalent in school work, which constitutes one of its important characteristics. We demonstrate that in Russian teaching practice, problems involving proofs and calculations continue to occupy an absolutely dominant position, and moreover, that students who are especially interested in mathematics are offered more problems involving proofs than “ordinary” students. We argue that the number of problem types (and consequently the types of activity in geometry classes) should be substantially increased in Russia, which requires specific work both on the part of textbook authors and on the part of teacher educators.

## INTRODUCTION: FORMULATING THE QUESTION

Much has been written in recent years about textbooks and about geometry textbooks in particular. We might mention, for example, the historical overview by Schubring (2023) or the proceedings of special conferences devoted to textbooks, such as Rezat et. al. (2019). At the same time, studies devoted to problems in textbooks are rare. As an example of what might be characterized as a typical study, we should mention the paper by Usiskin (2014), which addresses the investigation of some topic and mainly discusses why this topic is useful, to what extent it is present in textbooks, and so on. Of course, there exists a vast literature about problems (for example, the recent book edited by Leikin, 2023, to name just one study), but in this case the focus is usually specifically on the process of problem solving or problem posing, and furthermore, problems are often examined without any connection to the specific textbook. Studies about problems in textbooks also exist, of course: for example (to offer a list that in no way aspires to completeness), there are studies that to one or another degree address proofs in problems from school textbooks (Stylianides, 2009), or the organization of sets of problems and their interconnections (Karp, 2023), or the differential cognitive loads associated with problem solving, for example, in connection with the number of problems for which the textbook provides no direct recommendations (Jäder et al., 2020), and so on. In other words, what such studies usually focus on are characteristics that are important specifically for solving problems, not characteristics that are specific to their subject (in particular, geometry). Meanwhile, it is interesting to understand what it is that the hypothetical student must actually do in class, and what it is that this student must learn to do by studying the subject – and this is something that finds expression first and foremost in problems.

The classification of problems is quite a complicated matter, not least because the same problem can be solved in different ways. Below, in discussing problem types, we will take our cue from the assignment: from what exactly the student is asked to do, for example, to prove, to calculate, or to construct using compass and straightedge. It is clear that by limiting ourselves to such an examination,
# PROBLEM TYPES IN GEOMETRY TEXTBOOKS: RUSSIA'S EXPERIENCE

Alexander Karp

Teachers College, Columbia University

This short paper is devoted to the analysis of problems offered in Russian geometry textbooks. These problems determine the types of activity prevalent in school work, which constitutes one of its important characteristics. We demonstrate that in Russian teaching practice, problems involving proofs and calculations continue to occupy an absolutely dominant position, and moreover, that students who are especially interested in mathematics are offered more problems involving proofs than “ordinary” students. We argue that the number of problem types (and consequently the types of activity in geometry classes) should be substantially increased in Russia, which requires specific work both on the part of textbook authors and on the part of teacher educators.

## INTRODUCTION: FORMULATING THE QUESTION

Much has been written in recent years about textbooks and about geometry textbooks in particular. We might mention, for example, the historical overview by Schubring (2023) or the proceedings of special conferences devoted to textbooks, such as Rezat et. al. (2019). At the same time, studies devoted to problems in textbooks are rare. As an example of what might be characterized as a typical study, we should mention the paper by Usiskin (2014), which addresses the investigation of some topic and mainly discusses why this topic is useful, to what extent it is present in textbooks, and so on. Of course, there exists a vast literature about problems (for example, the recent book edited by Leikin, 2023, to name just one study), but in this case the focus is usually specifically on the process of problem solving or problem posing, and furthermore, problems are often examined without any connection to the specific textbook. Studies about problems in textbooks also exist, of course: for example (to offer a list that in no way aspires to completeness), there are studies that to one or another degree address proofs in problems from school textbooks (Stylianides, 2009), or the organization of sets of problems and their interconnections (Karp, 2023), or the differential cognitive loads associated with problem solving, for example, in connection with the number of problems for which the textbook provides no direct recommendations (Jäder et al., 2020), and so on. In other words, what such studies usually focus on are characteristics that are important specifically for solving problems, not characteristics that are specific to their subject (in particular, geometry). Meanwhile, it is interesting to understand what it is that the hypothetical student must actually do in class, and what it is that this student must learn to do by studying the subject – and this is something that finds expression first and foremost in problems.

The classification of problems is quite a complicated matter, not least because the same problem can be solved in different ways. Below, in discussing problem types, we will take our cue from the assignment: from what exactly the student is asked to do, for example, to prove, to calculate, or to construct using compass and straightedge. It is clear that by limiting ourselves to such an examination,
we significantly simplify the picture – a problem that asks a student to prove something can be very
simple or very difficult. Conversely, problems that ask students to construct something have usually
included proofs as well. In general, we must acknowledge at once that we make no claim to address
the full scope of the topic, if only because we will not analyze all textbooks or even all topics in
selected textbooks. The present study is a preliminary one, and we hope to continue it in several
directions, the most important of which is an international one: in the present study, we will confine
ourselves to textbooks from one country, but it would be interesting to analyze and compare different
countries from this point of view, as we believe the results might turn out to be very different.

Let us note that for quite a long time the study of geometry, as well as other mathematical disciplines,
practically did not involve independent problem solving at all. Høyrup (2014), a historian of medieval
mathematics, asks why medieval universities did not prepare major mathematicians, and sees the
explanation for this in their form of instruction, which was aimed at attaining wisdom, not making it.
They prepared not mathematicians, but religious thinkers. This tradition endured for a long time
beyond the Middle Ages, even if not in such an explicit form – problems began to be solved and
presented in textbooks, but they were few in number, and their role was minor – and in addition, it
was different than what it is today, aimed not at development, better assimilation, training, etc., but
once again at acquiring wisdom, except now it was somewhat differently structured (Karp, 2020).

In general, the answer to the question, "what must a student take away from a course in geometry?" has varied in different eras and for different segments of the population. I will permit myself to quote from a review, from 1849, of Geometric Notes for Young Women by V. Mikhelson: "V. Mikhelson had made of geometry everything that could possibly be made of it by the most tender feeling for young women – he has turned it into a light art, similar to embroidery or other female pursuits, of using geometric terms in appropriate situations" (Anon, 1849). This was achieved, of course, by means of a reduction of the level of difficulty, but also by the author's understanding of what exactly young female students were supposed to do.

It is natural to pose the question formulated above not only for different periods and different
segments of the population, but also in relation to presumed levels of mathematical ability.
Accordingly, the present study attempt to address the following research question: What types of
problems (in the sense established above) are represented in "ordinary" and in more "advanced"
Russian geometry textbooks, and what is the relation between problems of different types in such
textbooks.

Let us repeat that the methodological problem of determining the type to which one or another
problem belongs will be solved based on the formulation given in the problem itself ("prove," "find,"
"draw," etc.). The identification of new types, that is, instances in which several problems can be
united in one group, will be discussed separately below.

ON RUSSIAN (SOVIET) GEOMETRY TEXTBOOKS

We will be analyzing Russian geometry textbooks, and therefore we need to say a few words about
their specific characteristics. In what follows we rely on the study by Karp and Werner (2011). The
Russian course in geometry took shape in the nineteenth century and has not undergone too many
changes: it has remained traditional in the sense that, just as it did back then, it still represents a
systematic and methodical exposition of plane geometry in grades 7-9 and of solid geometry in grades
we significantly simplify the picture – a problem that asks a student to prove something can be very
simple or very difficult. Conversely, problems that ask students to construct something have usually
included proofs as well. In general, we must acknowledge at once that we make no claim to address
the full scope of the topic, if only because we will not analyze all textbooks or even all topics in
selected textbooks. The present study is a preliminary one, and we hope to continue it in several
directions, the most important of which is an international one: in the present study, we will confine
ourselves to textbooks from one country, but it would be interesting to analyze and compare different
countries from this point of view, as we believe the results might turn out to be very different.

Let us note that for quite a long time the study of geometry, as well as other mathematical disciplines,
practically did not involve independent problem solving at all. Høyrup (2014), a historian of medieval
mathematics, asks why medieval universities did not prepare major mathematicians, and sees the
explanation for this in their form of instruction, which was aimed at attaining wisdom, not making it.
They prepared not mathematicians, but religious thinkers. This tradition endured for a long time
beyond the Middle Ages, even if not in such an explicit form – problems began to be solved and
presented in textbooks, but they were few in number, and their role was minor – and in addition, it
was different than what it is today, aimed not at development, better assimilation, training, etc., but
once again at acquiring wisdom, except now it was somewhat differently structured (Karp, 2020).

In general, the answer to the question, "what must a student take away from a course in geometry?" has varied in different eras and for different segments of the population. I will permit myself to quote from a review, from 1849, of Geometric Notes for Young Women by V. Mikhelson: "V. Mikhelson had made of geometry everything that could possibly be made of it by the most tender feeling for young women – he has turned it into a light art, similar to embroidery or other female pursuits, of using geometric terms in appropriate situations" (Anon, 1849). This was achieved, of course, by means of a reduction of the level of difficulty, but also by the author's understanding of what exactly young female students were supposed to do.

It is natural to pose the question formulated above not only for different periods and different
segments of the population, but also in relation to presumed levels of mathematical ability.
Accordingly, the present study attempt to address the following research question: What types of
problems (in the sense established above) are represented in "ordinary" and in more "advanced"
Russian geometry textbooks, and what is the relation between problems of different types in such
textbooks.

Let us repeat that the methodological problem of determining the type to which one or another
problem belongs will be solved based on the formulation given in the problem itself ("prove," "find,"
"draw," etc.). The identification of new types, that is, instances in which several problems can be
united in one group, will be discussed separately below.

ON RUSSIAN (SOVIET) GEOMETRY TEXTBOOKS

We will be analyzing Russian geometry textbooks, and therefore we need to say a few words about
their specific characteristics. In what follows we rely on the study by Karp and Werner (2011). The
Russian course in geometry took shape in the nineteenth century and has not undergone too many
changes: it has remained traditional in the sense that, just as it did back then, it still represents a
systematic and methodical exposition of plane geometry in grades 7-9 and of solid geometry in grades
10-11. We should emphasize at once, therefore, that the Russian course is one that is taught over five years for 2-3 hours per week, rather than a course taught for only one year for five days a week, as it is in the USA, for example. The textbooks contain many proofs and it is expected that practically every assertion will be proven (in reality, of course, this is not quite the case – a number of fundamental assertions are accepted without proofs, and the list of axioms, which are referred to by precisely this word and are simply stated or “swallowed,” is more extensive than it needs to be).

Our investigation is made easier by the fact that the USSR followed a single textbook policy, in other words, the same textbook was used across the country. Subsequently, even already a couple of years before the collapse of the USSR, different textbooks began to be permitted, but in any case their number was and remains small (textbooks go through state expert review and must be approved in order to be admitted to the schools; the teachers are expected to teach using the textbook, although the extent to which what happens in class corresponds to the textbook has hardly been monitored in any way in recent decades). Consequently, in analyzing, say, the textbooks by Atanasyan, et al. (2004), we can be confident that they provide a realistic understanding about what actually goes on in Russian schools – it is a certainty that they are used by substantially over one half of the students in the country.

We will focus on textbooks that are used today or that were used very recently. Even so, to convey a sense of the tradition, we should say that from the 1930s until the mid-1970s, schools used Andrey Kiselev's textbooks, which were first published in 1892, and Nikolai Rybkin's problem book. In his introduction to his textbook, Kiselev himself explained that: "The book is supplied with a considerable number of exercises, comprised in part of certain theorems that did not make it into the text but are of interest, but mainly of compass and straightedge problems and calculation problems." This had been even more the case with the textbook by Avgust Davidov, which dominated the market during the pre-Kiselev period and remained in use in schools for over a half-century. It is noteworthy that initially, this textbook contained only problems involving proofs and construction. Only later were problems involving calculations added. And both the former and the latter remained relatively few in number: in the entire course on plane and solid geometry, there were 256 problems involving proofs and construction, and 337 problems involving calculations (Karp, 2015). Note that later books, written under the influence of the international reform movement, were usually characterized by far greater variety. For example, the geometry textbook by Alexander Kulisher from 1922 also contains problems that require students to draw something, or to cut something out, or even to describe what they feel with their hands when touching one or another geometric object – in this case, the influence of the idea of the informal study of geometry is evident. However, it is clear that the success of such innovative approaches at that time was very limited (Karp, 2012). But let us turn to the present.

## TWO CONTEMPORARY TEXTBOOKS

We will first discuss two widely used geometry textbooks – the already mentioned textbook by Atanasyan, et al. and the textbook by Pogorelov (for example, Pogorelov, 1985). Both textbooks appeared in the late 1970s and during a certain period practically divided the whole Russian market between them. Subsequently, the textbooks of A.V. Pogorelov, who died in 2002, became less popular, but continued to be actively used later as well. Over the years, certain changes appeared in these textbooks (for example, the numeration of grade levels changed, and grades 6-10 became grades 7-11), but in terms of their approaches to problems, there were, in our view, no noticeable changes.
10-11. We should emphasize at once, therefore, that the Russian course is one that is taught over five years for 2-3 hours per week, rather than a course taught for only one year for five days a week, as it is in the USA, for example. The textbooks contain many proofs and it is expected that practically every assertion will be proven (in reality, of course, this is not quite the case – a number of fundamental assertions are accepted without proofs, and the list of axioms, which are referred to by precisely this word and are simply stated or “swallowed,” is more extensive than it needs to be).

Our investigation is made easier by the fact that the USSR followed a single textbook policy, in other words, the same textbook was used across the country. Subsequently, even already a couple of years before the collapse of the USSR, different textbooks began to be permitted, but in any case their number was and remains small (textbooks go through state expert review and must be approved in order to be admitted to the schools; the teachers are expected to teach using the textbook, although the extent to which what happens in class corresponds to the textbook has hardly been monitored in any way in recent decades). Consequently, in analyzing, say, the textbooks by Atanasyan, et al. (2004), we can be confident that they provide a realistic understanding about what actually goes on in Russian schools – it is a certainty that they are used by substantially over one half of the students in the country.

We will focus on textbooks that are used today or that were used very recently. Even so, to convey a sense of the tradition, we should say that from the 1930s until the mid-1970s, schools used Andrey Kiselev's textbooks, which were first published in 1892, and Nikolai Rybkin's problem book. In his introduction to his textbook, Kiselev himself explained that: "The book is supplied with a considerable number of exercises, comprised in part of certain theorems that did not make it into the text but are of interest, but mainly of compass and straightedge problems and calculation problems." This had been even more the case with the textbook by Avgust Davidov, which dominated the market during the pre-Kiselev period and remained in use in schools for over a half-century. It is noteworthy that initially, this textbook contained only problems involving proofs and construction. Only later were problems involving calculations added. And both the former and the latter remained relatively few in number: in the entire course on plane and solid geometry, there were 256 problems involving proofs and construction, and 337 problems involving calculations (Karp, 2015). Note that later books, written under the influence of the international reform movement, were usually characterized by far greater variety. For example, the geometry textbook by Alexander Kulisher from 1922 also contains problems that require students to draw something, or to cut something out, or even to describe what they feel with their hands when touching one or another geometric object – in this case, the influence of the idea of the informal study of geometry is evident. However, it is clear that the success of such innovative approaches at that time was very limited (Karp, 2012). But let us turn to the present.

## TWO CONTEMPORARY TEXTBOOKS

We will first discuss two widely used geometry textbooks – the already mentioned textbook by Atanasyan, et al. and the textbook by Pogorelov (for example, Pogorelov, 1985). Both textbooks appeared in the late 1970s and during a certain period practically divided the whole Russian market between them. Subsequently, the textbooks of A.V. Pogorelov, who died in 2002, became less popular, but continued to be actively used later as well. Over the years, certain changes appeared in these textbooks (for example, the numeration of grade levels changed, and grades 6-10 became grades 7-11), but in terms of their approaches to problems, there were, in our view, no noticeable changes.
Below, we will limit ourselves to analyzing one section, Quadrilaterals. The topics covered in both textbooks in this case are practically identical, apart from the fact that this section in Pogorelov's textbook includes a paragraph devoted to the so-called Thales's theorem about equal line segments intercepted by parallel lines on the rays of an angle. In Atanasyan's textbook, this theorem is given as a problem (although its solution is immediately given as well). The results of the analysis are shown in Table 1.

<table><caption>Table 1: Problem types in textbooks by Atanasyan, et al. and Pogorelov</caption><thead><tr><th>Assignment</th><th>Atanasyan</th><th>Pogorelov</th></tr></thead><tbody><tr><td>Prove</td><td>22</td><td>17</td></tr><tr><td>Calculate</td><td>22</td><td>40</td></tr><tr><td>Construct</td><td>10</td><td>8</td></tr><tr><td>Draw</td><td>0</td><td>1</td></tr><tr><td>Analyze a drawing</td><td>3</td><td>1</td></tr><tr><td>Investigate</td><td>1</td><td>1</td></tr><tr><td>Count the variants</td><td>3</td><td>1</td></tr><tr><td>Total</td><td>61</td><td>69</td></tr></tbody></table>

Let us explain the categories introduced above. While the category “Construct” includes problems that require students to carry out a rigorous straightedge and compass construction which is usually unique, under the category “Draw” we include assignments that require students, for example, to construct any parallelogram. Under the category “Count the variants” we include problems that require students, for example, to determine how many parallelograms can be constructed with vertices in three given points that are not on the same straight line. “Analyze a drawing” includes, for example, a problem that requires students to select that letter in a picture which has an axis of symmetry. Lastly, “Investigate” may include quite different assignments, which require students to verify (sometimes by proving, sometimes by providing an example) one or another assertion, for example, that any quadrilateral whose diagonals are equal and perpendicular is a square.

## TWO OTHER, LESS POPULAR TEXTBOOKS

We will now analyze two textbooks that were published in much smaller editions – one of them, Atanasyan, et al. (1996), consisted of supplementary chapters to the textbook by the same authors cited above, intended for schools and classes with an advanced course of study in mathematics (sometimes loosely called schools for the mathematically gifted). This textbook included certain additional theoretical topics and many additional problems. Another textbook, Smirnova and Smirnov (2001), was intended for use in “ordinary” schools, but represents an attempt to make a more “humanities-friendly” textbook, that is, to take into account the varied interests and inclinations of the students. This textbook, too, was published in relatively small editions. In both cases, as above, we will analyze the section devoted to quadrilaterals – in the textbook by Atanasyan, et al. (1996), this section constitutes a separate chapter; in the textbook by Smirnova and Smirnov (2001), it consists of several paragraphs.

Note that the problem we included in the category “The World around us” requires students to explain situations from the surrounding world – specifically, why a certain axis occupies a vertical position.
Below, we will limit ourselves to analyzing one section, Quadrilaterals. The topics covered in both textbooks in this case are practically identical, apart from the fact that this section in Pogorelov's textbook includes a paragraph devoted to the so-called Thales's theorem about equal line segments intercepted by parallel lines on the rays of an angle. In Atanasyan's textbook, this theorem is given as a problem (although its solution is immediately given as well). The results of the analysis are shown in Table 1.

<table><caption>Table 1: Problem types in textbooks by Atanasyan, et al. and Pogorelov</caption><thead><tr><th>Assignment</th><th>Atanasyan</th><th>Pogorelov</th></tr></thead><tbody><tr><td>Prove</td><td>22</td><td>17</td></tr><tr><td>Calculate</td><td>22</td><td>40</td></tr><tr><td>Construct</td><td>10</td><td>8</td></tr><tr><td>Draw</td><td>0</td><td>1</td></tr><tr><td>Analyze a drawing</td><td>3</td><td>1</td></tr><tr><td>Investigate</td><td>1</td><td>1</td></tr><tr><td>Count the variants</td><td>3</td><td>1</td></tr><tr><td>Total</td><td>61</td><td>69</td></tr></tbody></table>

Let us explain the categories introduced above. While the category “Construct” includes problems that require students to carry out a rigorous straightedge and compass construction which is usually unique, under the category “Draw” we include assignments that require students, for example, to construct any parallelogram. Under the category “Count the variants” we include problems that require students, for example, to determine how many parallelograms can be constructed with vertices in three given points that are not on the same straight line. “Analyze a drawing” includes, for example, a problem that requires students to select that letter in a picture which has an axis of symmetry. Lastly, “Investigate” may include quite different assignments, which require students to verify (sometimes by proving, sometimes by providing an example) one or another assertion, for example, that any quadrilateral whose diagonals are equal and perpendicular is a square.

## TWO OTHER, LESS POPULAR TEXTBOOKS

We will now analyze two textbooks that were published in much smaller editions – one of them, Atanasyan, et al. (1996), consisted of supplementary chapters to the textbook by the same authors cited above, intended for schools and classes with an advanced course of study in mathematics (sometimes loosely called schools for the mathematically gifted). This textbook included certain additional theoretical topics and many additional problems. Another textbook, Smirnova and Smirnov (2001), was intended for use in “ordinary” schools, but represents an attempt to make a more “humanities-friendly” textbook, that is, to take into account the varied interests and inclinations of the students. This textbook, too, was published in relatively small editions. In both cases, as above, we will analyze the section devoted to quadrilaterals – in the textbook by Atanasyan, et al. (1996), this section constitutes a separate chapter; in the textbook by Smirnova and Smirnov (2001), it consists of several paragraphs.

Note that the problem we included in the category “The World around us” requires students to explain situations from the surrounding world – specifically, why a certain axis occupies a vertical position.
<table><caption>Table 2: Problem types in textbooks by Atanasyan, et al. and Smirnova and Smirnov.</caption><thead><tr><td>Assignment</td><td>Atanasyan, et al</td><td>Smirnovs</td></tr></thead><tbody><tr><td>Prove</td><td>66</td><td>24</td></tr><tr><td>Calculate</td><td>6</td><td>37</td></tr><tr><td>Construct</td><td>6</td><td>9</td></tr><tr><td>The World around us</td><td>0</td><td>1</td></tr><tr><td>Investigate</td><td>20</td><td>6</td></tr><tr><td>Count the variants</td><td>2</td><td>1</td></tr><tr><td>Total</td><td>100</td><td>78</td></tr></tbody></table>

## TWO SOLID GEOMETRY TEXTBOOKS

Let us analyze two more textbooks, in solid geometry, by Atanasyan et al. (2000) and Sharygin (1999). The former remains the most widely used solid geometry textbook in the country, while the latter is distinguished by the author's methodological approach. We will analyze the chapters devoted to straight lines and planes in space, which are similar in content (without taking into account additional problems given separately in the textbooks).

<table><caption>Table 3: Problem types in textbooks by Atanasyan et al. and Sharygin.</caption><thead><tr><td>Assignment</td><td>Atanasyan et al.</td><td>Sharygin</td></tr></thead><tbody><tr><td>Prove</td><td>83</td><td>37</td></tr><tr><td>Calculate</td><td>44</td><td>40</td></tr><tr><td>Construct</td><td>13</td><td>5</td></tr><tr><td>The World around us</td><td>2</td><td>0</td></tr><tr><td>Investigate</td><td>10</td><td>6</td></tr><tr><td>Analyze the Drawing</td><td>1</td><td>0</td></tr><tr><td>Count the variants</td><td>0</td><td>2</td></tr><tr><td>Total</td><td>153</td><td>90</td></tr></tbody></table>

## DISCUSSION

Naturally, each textbook possesses individual traits that are characteristic specifically of its authors, and there is no reason to attribute them to all textbooks that are addressed to the corresponding group of students. Textbooks in Russia go through a many-stage review process, and if an author today decides to stop at something easy and pleasant, like embroidery, their book will simply not be allowed in the schools; but within the bounds of the permissible, sufficiently many possibilities remain. Nonetheless, an analysis of textbooks used by many millions of students makes it possible to draw certain conclusions about the existing reality – about what exactly educators want to teach students in the course in geometry.

Evidently, two things are demanded: the ability to perform geometric calculations and the ability to carry out proofs; everything else is less important. Moreover, the great majority of the supplementary problems for students interested in mathematics are devoted specifically to proofs; but “ordinary” students are very often given such problems as well. In this study, it is not our aim to address comparisons with other countries, but clearly the frequently encountered pessimism with regard to the “ordinary” student’s ability to carry out proofs is unjustified. (We might recall a similar discussion when Krutetskii refuted Thorndike, who had claimed that the standard course in algebra was not
<table><caption>Table 2: Problem types in textbooks by Atanasyan, et al. and Smirnova and Smirnov.</caption><thead><tr><td>Assignment</td><td>Atanasyan, et al</td><td>Smirnovs</td></tr></thead><tbody><tr><td>Prove</td><td>66</td><td>24</td></tr><tr><td>Calculate</td><td>6</td><td>37</td></tr><tr><td>Construct</td><td>6</td><td>9</td></tr><tr><td>The World around us</td><td>0</td><td>1</td></tr><tr><td>Investigate</td><td>20</td><td>6</td></tr><tr><td>Count the variants</td><td>2</td><td>1</td></tr><tr><td>Total</td><td>100</td><td>78</td></tr></tbody></table>

## TWO SOLID GEOMETRY TEXTBOOKS

Let us analyze two more textbooks, in solid geometry, by Atanasyan et al. (2000) and Sharygin (1999). The former remains the most widely used solid geometry textbook in the country, while the latter is distinguished by the author's methodological approach. We will analyze the chapters devoted to straight lines and planes in space, which are similar in content (without taking into account additional problems given separately in the textbooks).

<table><caption>Table 3: Problem types in textbooks by Atanasyan et al. and Sharygin.</caption><thead><tr><td>Assignment</td><td>Atanasyan et al.</td><td>Sharygin</td></tr></thead><tbody><tr><td>Prove</td><td>83</td><td>37</td></tr><tr><td>Calculate</td><td>44</td><td>40</td></tr><tr><td>Construct</td><td>13</td><td>5</td></tr><tr><td>The World around us</td><td>2</td><td>0</td></tr><tr><td>Investigate</td><td>10</td><td>6</td></tr><tr><td>Analyze the Drawing</td><td>1</td><td>0</td></tr><tr><td>Count the variants</td><td>0</td><td>2</td></tr><tr><td>Total</td><td>153</td><td>90</td></tr></tbody></table>

## DISCUSSION

Naturally, each textbook possesses individual traits that are characteristic specifically of its authors, and there is no reason to attribute them to all textbooks that are addressed to the corresponding group of students. Textbooks in Russia go through a many-stage review process, and if an author today decides to stop at something easy and pleasant, like embroidery, their book will simply not be allowed in the schools; but within the bounds of the permissible, sufficiently many possibilities remain. Nonetheless, an analysis of textbooks used by many millions of students makes it possible to draw certain conclusions about the existing reality – about what exactly educators want to teach students in the course in geometry.

Evidently, two things are demanded: the ability to perform geometric calculations and the ability to carry out proofs; everything else is less important. Moreover, the great majority of the supplementary problems for students interested in mathematics are devoted specifically to proofs; but “ordinary” students are very often given such problems as well. In this study, it is not our aim to address comparisons with other countries, but clearly the frequently encountered pessimism with regard to the “ordinary” student’s ability to carry out proofs is unjustified. (We might recall a similar discussion when Krutetskii refuted Thorndike, who had claimed that the standard course in algebra was not
accessible to a great proportion of students: Krutetskii (1976) showed that in the USSR it was successfully assimilated, if not by everyone, then by a far greater percentage of students than was to be expected according to Thorndike.)

Straightedge and compass construction problems have clearly receded into the background – they are noticeably fewer in number. Moreover, going into a deeper analysis of the problems, we should point out that an assignment to construct a parallelogram based on its sides and angle is by no means the same thing as the classic straightedge and compass construction problems with their many steps, analysis, and unobvious proof. This has disappeared from the course in geometry.

On the other hand, what has clearly appeared and is more noticeably represented in textbooks for schools with an advanced course of study in mathematics are problems that we characterized as investigative, which require students to determine whether something can be the case, or to verify the correctness of various assertions. In “ordinary” textbooks, such “open” problems exist, but in very small numbers (which in our view is explained by a certain conservatism on the part of the authors and by the fact that such problems remain unfamiliar to teachers). At the same time, in the textbook by Smirnova and Smirnov (2001), which is aimed at a more “humanities-oriented” student, the number of such problems is somewhat higher – informal geometry and the ability to imagine one or another construction (or to understand that it might not exist) are necessary and accessible to everyone.

We should mention one other characteristic of the materials we have analyzed. They contain almost no problems involving the application of geometry or real-world problems. It may be objected, of course, that this is not the case in all sections – to repeat, we did not analyze all sections, or all textbooks – but on the whole, their absence is telling. In general, we believe that the analysis we have carried out makes it possible to discern important characteristics of the teaching of geometry in Russia with its positive and negative aspects – an unquestionable conservatism, but at the same time the preservation of a high level of substantiation-by-proof and reasoning. To repeat, it would be interesting to investigate the geometry textbooks of other countries as well, and to compare the obtained results.

## SOME PRACTICAL CONCLUSIONS

Textbook authors are usually aware of what specific types of problems they offer in their textbooks. Below are several types of problems identified by Valery Ryzhik, one of the authors of the textbooks in geometry written under the supervision of the well-known geometer and academician Alexander Alexandrov (citation based on Karp, & Werner, 2011):

* *Looking.* Students are taught to interpret information presented in visual form, and students’ spatial (2-dimensional and 3-dimensional) imaginations are developed.

* *Drawing.* Students develop their spatial thinking skills.

* *Investigating.* Problems whose conditions or possible results may contain some uncertainty, incompleteness, and ambiguity.

* *Applying geometry.* Problems from outside mathematics that must be translated into mathematical language.

As we can see, this list includes problem types that we have already encountered as well as new ones. For example, the type that is labeled “Investigating” differs somewhat from the type that was called
accessible to a great proportion of students: Krutetskii (1976) showed that in the USSR it was successfully assimilated, if not by everyone, then by a far greater percentage of students than was to be expected according to Thorndike.)

Straightedge and compass construction problems have clearly receded into the background – they are noticeably fewer in number. Moreover, going into a deeper analysis of the problems, we should point out that an assignment to construct a parallelogram based on its sides and angle is by no means the same thing as the classic straightedge and compass construction problems with their many steps, analysis, and unobvious proof. This has disappeared from the course in geometry.

On the other hand, what has clearly appeared and is more noticeably represented in textbooks for schools with an advanced course of study in mathematics are problems that we characterized as investigative, which require students to determine whether something can be the case, or to verify the correctness of various assertions. In “ordinary” textbooks, such “open” problems exist, but in very small numbers (which in our view is explained by a certain conservatism on the part of the authors and by the fact that such problems remain unfamiliar to teachers). At the same time, in the textbook by Smirnova and Smirnov (2001), which is aimed at a more “humanities-oriented” student, the number of such problems is somewhat higher – informal geometry and the ability to imagine one or another construction (or to understand that it might not exist) are necessary and accessible to everyone.

We should mention one other characteristic of the materials we have analyzed. They contain almost no problems involving the application of geometry or real-world problems. It may be objected, of course, that this is not the case in all sections – to repeat, we did not analyze all sections, or all textbooks – but on the whole, their absence is telling. In general, we believe that the analysis we have carried out makes it possible to discern important characteristics of the teaching of geometry in Russia with its positive and negative aspects – an unquestionable conservatism, but at the same time the preservation of a high level of substantiation-by-proof and reasoning. To repeat, it would be interesting to investigate the geometry textbooks of other countries as well, and to compare the obtained results.

## SOME PRACTICAL CONCLUSIONS

Textbook authors are usually aware of what specific types of problems they offer in their textbooks. Below are several types of problems identified by Valery Ryzhik, one of the authors of the textbooks in geometry written under the supervision of the well-known geometer and academician Alexander Alexandrov (citation based on Karp, & Werner, 2011):

* *Looking.* Students are taught to interpret information presented in visual form, and students’ spatial (2-dimensional and 3-dimensional) imaginations are developed.

* *Drawing.* Students develop their spatial thinking skills.

* *Investigating.* Problems whose conditions or possible results may contain some uncertainty, incompleteness, and ambiguity.

* *Applying geometry.* Problems from outside mathematics that must be translated into mathematical language.

As we can see, this list includes problem types that we have already encountered as well as new ones. For example, the type that is labeled “Investigating” differs somewhat from the type that was called
so by us. The author quoted expressed the view that, for example, it is useful to give students "impossible" problems, such as computing the area of a right triangle, given one of its legs and the projections of both of its legs onto the hypotenuse (it is evident that there are too many givens here). This, in his view, develops their capacity for observation and for working in the real world, where problems are not necessarily correctly posed. The type "Looking" lends itself especially to the study of three-dimensional geometry, in which it is important to be able to reach a conclusion by looking at a drawing.

Note that in addition to the types listed above, there are also others, and combinations of the types represented above are possible as well. As an example, consider the following problem (Werner & Karp, 2019, p. 336):

Consider four points in space, A, B, C, D. We would like to construct all planes that contain at least three of these points. It is not true that these four points could be chosen such that it would be possible to construct:

a) only one plane;

b) only two planes;

c) only four planes;

d) infinitely many planes.

This is a kind of multiple-choice problem, but what it requires of the student is not only the answer, but also the reasoning behind it. In other words, the student must imagine (draw!) such positions for the four points for which it will be possible to construct one, four, or infinitely many planes, and then to prove that it will be impossible to construct only two planes. This is a kind of investigation problem, involving various kinds of activity.

Working teachers, in our view, must be familiar with different types of activities in which students are involved in the process of studying geometry, without limiting themselves to the classical types. Our conception of geometry today is broader than it was two thousand years ago. Without giving up classical problems involving proofs and calculations, we should broaden the palette of possibilities – which poses new problems both for textbook authors and for teacher educators.

## References

* Anon. (1849). [Review]. *Otechestvennye zapiski, LXIV*, Part VI, p. 45.

* Atanasyan, L.S., Butuzov, V.F., Kadomtsev, S.B., Shestakov, S.A., & Yudina, I.I. (1996). *Geometriya 8. Dopolnitelnye glavy k shkol'nomu uchebniku* [Geometry 8. Additional Chapters for the school textbook]. Prosveschenie.

* Atanasyan, L.S., Butuzov, V.F., Kadomtsev, S.B., Kiseleva, L.S., & Poznyak, E. G. (2000). *Geometriya 10-11*. [Geometry 10-11]. Prosveschenie.

* Atanasyan, L.S., Butuzov, V.F., Kadomtsev, S.B., Poznyak, E. G., & Yudina, I.I. (2004). *Geometriya 7-9* [Geometry 7-9]. Prosveschenie.

* Høyrup, J. (2014). *Mathematics Education in the European Middle Ages*. In A. Karp & G. Schubring (Eds.), *Handbook on the history of mathematics education* (pp. 109-127). Springer.

* Jäder, J., Lithner, J., & Sidenvall, J. (2020). Mathematical problem solving in textbooks from twelve countries. *International Journal of Mathematical Education in Science & Technology, 51*(7), 1120-1136.