In Portugal, the curriculum reference document “Essential Mathematics Learning for the 1st Cycle of Basic Education” regarding Geometry and Measurement, refers to “the importance of the students’ contact with a wide range of shapes, related to figures in space and in the plane, with which they produce several operations, composing and decomposing, establishing spatial relations” [Ministério da Educação e Ciência (MEC), 2021, p. 11]. Geometry involves ways of thinking and representing systems that are used to analyze spatial, physical, and imaginary environments that support the conceptualization of those systems (Battista, 2007). Indeed, Geometry provides a favorable context for the development of intuition and spatial visualization (e.g., Abrantes et al., 1999; Presmeg, 2006). For this reason, its study should begin with concrete experiences and gradually expand to more formalized processes in order to develop the capacity for logical organization of thought. Several researchers argue that to promote geometric thinking geometric activities carried out with students should involve the composition and decomposition of objects, rotating objects in the plane and in space, and exploring models and their spatial relationships (e.g., Gonzato et al., 2011; Veloso, 2012). This idea is very present in the type of concrete geometry addressed in the early years of schooling where objects are physical and the argumentation presented by students is of a perceptual nature, considering the first experiences on geometric topics are crucial in the way students sustain the development of their geometric skill (Clements and Battista, 1992; Clements and Sarama, 2000; Parzysz and Jore, 2002).

For Duval (1998, 2012), the study of Geometry involves three kinds of cognitive processes that accomplish specific epistemological functions: (i) Visualization processes, relating to the visual representation of a geometric statement, the exploration or heuristic of a complex geometric situation; (ii) Construction processes using tools, where construction can function as a model in which representative actions and observed results are related to the mathematical objects being represented; (iii) Reasoning processes in relation to discursive processes to extend knowledge, support argumentation and interpretation. Although these processes can be carried out separately, they are closely linked and their synergy is cognitively necessary for the promotion of geometric thinking. Kuzniak et al. (2018) reinforce this idea by mentioning four skills used to describe geometric thinking: reasoning; representation; visualization; and operationalization. Those authors also mention the role of activity with materials, including their manipulation, in the construction of mathematical concepts, visualization and spatial skills, language, and reasoning. Thus, in learning Geometry and the consequent development of geometric thinking, especially in the early years of schooling, the spatial ability (or spatial sense) is understood as a set of skills related to the way students perceive the World around them and their ability to interpret, modify, and anticipate transformations in objects (Levine et al., 2012).

The present study is based on the work of Frostig et al. (1989). Those researchers define visual perception as the ability to recognize and discriminate visual stimuli, and to interpret them by associating them with previous experiences. In addition to seeing a figure or object, an analysis and interpretation of what is seen is performed, establishing relationships with the observer and other objects. These authors highlight the role of language in visual perception skills and also consider the development of visual perception associated to an appropriate sensory-motor development. They identify five capacities of visual perception: visual motor coordination; figure-background perception; perceptual constancy; perception of position in space; and perception of spatial relations. Later, these capacities were expanded on by the work of Hoffer (cited in Del Grande, 1990), integrating visual discrimination and visual memory. The following table shows the seven visual perception capacities and the respective descriptors of the child’s activity (Table 1).

In order to analyze the development of geometric thinking, several theories have contributed to an understanding of the teaching and learning processes. van Hiele (1986, 1999) stands out for defining a precise description of the development of this thinking, structured by five descriptive and qualitative levels of geometric, sequential and hierarchical reasoning, as follows:

Visualization or recognition, the student understands geometric figures in a global way by their appearance.

It should also be noted that the van Hiele model allows aspects, such as recursion, sequentiality, language, continuity, and location to be used. These aspects will allow the development of geometric thinking skills.

Geometry is an area favorable to the development of mathematical thinking since it is a privileged means to represent and give meaning to the world around us. In this sense, teaching Geometry promotes discovery and experimentation, and the development of students’ geometric thinking. One way to do this is through the use of manipulative materials that are often associated with the playful dimension of learning. These materials are part of a larger set called didactic materials, which Graells (2000) defines as all material that is created to facilitate teaching and student learning, classified into three categories: conventional materials, audio-visual materials, and new technologies. Manipulatives are considered by Graells (2000) to be conventional didactic materials, referring to them as being something that the student can manipulate in a physical way. A manipulative material is thus any concrete object that incorporates mathematical concepts, appeals to different senses, and can be touched, moved, rearranged, and manipulated by students (Graells, 2000). With similar reasoning, Swan and Marshall (2010) define a manipulative as “an object that can be handled by an individual in a sensory manner during which conscious and unconscious mathematical thinking will be fostered” (p. 14). More recently, Bartolini and Martignone (2020) consider that mathematics manipulatives are artifacts used in mathematics education that are manipulated by students to explore, learn, or investigate mathematical concepts or processes and to solve problems based on perceptual sensory evidence (e.g., visual, tactile). All of these definitions share the idea of mathematical experiences that allow the development of sensory skills, which are considered particularly relevant when learning Geometry.

Manipulatives are not exclusively concrete (physical). With advances in technology it is possible to visualize and manipulate virtually, leading to the emergence of the concept of virtual manipulatives, introduced by Moyer-Packenham et al. (2002). As a result of the technological evolution of this type of material and research associated with its use, in 2016, the same authors further developed the concept and presented an updated definition of virtual manipulatives: “an interactive, technology-enabled visual representation of a dynamic mathematical object, including all of the programmable features that allow it to be manipulated, that presents opportunities for constructing mathematical knowledge” (Moyer-Packenham and Bolyard, 2016, p. 13). In an effort to distinguish concrete manipulatives from virtual ones, Bartolini and Martignone (2020) define the former as “physical artifacts that can be concretely handled by students and offer a large and deep set of sensory experiences” (p. 486), and the latter as “digital artifacts that resemble physical objects and can be manipulated, usually with a mouse, in a similar way as their authentic, concrete counterparts” (ibid.). It is important to note that virtual manipulatives are technologies and like any technology, by themselves, do not provide learning. It is the quality of engagement that the student has with technology that presents opportunities for effective math learning (Moyer-Packenham and Bolyard, 2016).

One manipulative, whether concrete or virtual, cannot be considered better than another in educational terms, since its choice by the teacher depends on several factors, such as what is more available, what suits the students’ interests, but above all the teacher themself and the vision they have about mathematics and how students learn (Kamii et al., 2001; Bartolini and Martignone, 2020). Furthermore, one of the fundamental ideas about using manipulatives, both concrete and virtual, is that they play a key role in the construction of mathematical knowledge, but they should be considered as a means and never as an end (Batista and Clements, 1994; Clements, 1999; Simon, 2022). According to these authors, the nature of a manipulative, by itself, does not constitute a mathematical idea, and it is necessary to design rich and powerful mathematical tasks where the use of these materials can be integrated.

In this study, one of the authors conducted a teaching experiment in the first grade class of a Portuguese primary school located in a predominantly urban area, focusing on the exploration of manipulatives, and with a duration of two lessons. The class consisted of 20 students, aged 6, of which 10 students were female and 10 were male. The class was heterogeneous since it included students with different learning abilities and paces, and no students with special educational needs. Of the 20 students in the class, 15 stated that they like the subject of mathematics and five that they did not like it, mainly because they considered that “the exercises are difficult,” “they do not like to do math,” or because “we only learn small numbers.” Regarding the students’ performance in their favorite subjects (Mathematics, Portuguese, or Environmental Studies) throughout the three terms of the school year, only one student received “Satisfactory” and “Insufficient” marks in Mathematics and Portuguese with the other students classified as “Good” and “Very Good.”

The teaching experiment was guided by indicators concerning the role of the student, the role of the teacher, and the use of manipulatives (Figure 1).

Based on these principles, the learning activities on geometric figures involved the exploration of manipulatives in solving the proposed tasks.

Data were collected from the audio and video recording of the lessons taught and the written records that the students produced.

The analysis of the data was based on the content analysis of the activities carried out by the students, some of them recorded in the form of dialogs resulting from their interaction with the teacher, following the works of Del Grande (1990) and Duval (1998, 2012). The information was analyzed using the following dimensions of analysis: (i) Figure background perception/perceptual constancy; (ii) Visual discrimination/visual memory; and (iii) Construction/composition and decomposition.

After the resolution of Task 1 and the class discussion on the “The forest of flat figures” story, the experiment focused on a free exploration of the logic blocks. This allowed students to become familiar with the material, interact and share ideas (Rêgo and Rêgo, 2006; Figure 3).

Once the free exploration of the material logic blocks was over, students were asked to solve Task 2 described below.

In solving the first question of this task, all students correctly placed the logic block pieces in their corresponding groups, as illustrated by the following dialog:

Initially, the students were confused by the fact that a square is also a rectangle, however when asked about the difference between them they stated that the sides (“these parts”) are “all the same” in the square. At an early stage, children should be encouraged to express their geometric understandings with their own vocabulary (Crowley, 1987).

In response to the question 2 of Task 2 (Build figures with the logic blocks and count geometric shapes), students built representations of a train and Pinocchio (inspired from Task 1), and counted the geometric figures in the constructions (Figure 4).

The manipulation and observation sparked the students’ interest and curiosity about the properties of geometric figures, as the following dialog shows:

Students showed they have acquired the capacity of visual discrimination (Del Grande, 1990), which can be seen in the moments when they are able to classify a set of objects according to a certain attribute, in this case the shape, according to the number of sides and vertices.

In question 3 of Task 2, students were asked about everyday objects that they associate with geometric shapes.

In the examples presented by the students, given their young age, the researcher felt that it was not appropriate to distinguish two-dimensional figures from three-dimensional figures. For learning to be meaningful, students must be able to relate to it.

To make the poster in question 4 of Task 2 students were distributed into four groups and each group was assigned a geometric figure and various representations of everyday objects. The correspondence between the geometric shape and the objects was evidenced in a poster (Figure 5).

To synthesize what they had learned about the shape of the geometric figures under study students were asked to solve Task 3, which consisted of verbally describing shapes and properties using appropriate language to develop their visual memory capacity.

Students were able to correctly draw the intended figure, as well as correctly describe it by stating its characteristics:

The tactile perception of some characteristics of geometric figures was complemented by their visual translation through an activity called “Geometric Twister” performed in a large group outside of the classroom. In this activity, students had to place their hands or feet on a mat that contained the geometric figures already discussed (Figure 6), also working on memory about laterality concepts (left and right).

This activity showed that the students recognized the geometric shapes, independently of size and position, which allowed them to verify their capacity of visual discrimination and visual memory of the figures, supported by empirical indicators for these capacity:

Compares the various shapes of the logic block pieces, finding similarities, and differences.

For the construction of geometric figures, students began by exploring the Geoplane in constructing a triangle (Figure 7).

In order to understand if students identified triangles from their characteristics and not by memorization, the teacher engaged the students in dialogs such as the following:

Following Duval (1998), we can see in the previous dialog that the representation of triangles using the Geoplane supports the process of visualization of their characteristics, as seen in the students’ arguments, allowing the expansion of geometric knowledge about the triangular figure. This was followed by the construction of triangles with the student’s choice of Tangram pieces (Figure 8). The challenge consisted of experimenting if students could get a new triangle from either two triangle pieces, or one square piece and one triangle piece.

According to the National Council of Teachers of Mathematics (NCTM) (2014), effective mathematics teaching should involve using relevant questions to evaluate and enhance students’ sense-making about mathematical ideas and relationships.

We proceeded in the same way for the construction of the rectangle and the square (Figure 9).

When constructing rectangles on the Geoplane, some students were asked what characteristics they identified in this figure.

In order to translate these characteristics, the students were asked to construct rectangles by composition with Tangram pieces (Figure 10).

The aim of constructing squares on the Geoplane (Figure 9) was for students to discover that all sides are equal because they have the same length. Simply saying that they were all the same could be due to memorization, as the below excerpt from a dialog with a student about their constructions illustrates.

As the pins served as guides for the student’s construction of the squares on the Geoplane, they were then asked to construct squares using the Tangram pieces instead (Figure 11).

During this construction, students were asked whether the characteristics they had identified with the constructions on the Geoplane still held:

These activities allowed students to explore various situations, taking advantage of the dynamic nature of the Geoplane and Tangram. These manipulatives enhance exploration and discussion of the characteristics of geometric shapes rather than memorization of their names.

After the constructions, students were given moments to reflect on what they had accomplished by moving from the Geoplane construction to dotted paper (Figure 12).

Overall, the students showed no difficulty in moving from constructions of rectangles and squares to dotted paper. Although the line was not very firm, their reproduction on the dotted paper was faithful to the construction on the Geoplane. For triangles, the transition to dotted paper from the construction on the Geoplane was more difficult and their reproduction on the dotted paper was not faithful to their construction on the Geoplane, although the students displayed concern in joining ‘pin’ to ‘pin’ when drawing the proposed figures. Regarding Tangram, Figure 13 illustrates some of the students’ reproductions on paper of their Tangram constructions.

In the representation of triangles, squares and rectangles on paper, in general students do not present difficulties despite an often imprecise line, and their reproduction is faithful to the constructions with Tangram. Moving to paper is a way for students to reflect on what they have learned and go beyond simple manipulation. In the activities performed, students distinguish geometric shapes in any position and of any size by enumerating their properties, developing their capacity of construction and composition and decomposition of geometric figures, through the following indicators:

Builds triangles, squares and rectangles using the Geoplane.

The above activities, involving the construction and composition/decomposition of figures, supported geometric thinking about various plane figures (Gonzato et al., 2011; Veloso, 2012). For example, the construction of a square using two triangles involved a certain composition from the student: placing the two triangles so that the new figure had all equal sides and right angles (Figures 13C,D). For constructing a triangle, on the other hand, those same pieces needed a different arrangement (Figures 13A,B). In the construction of the rectangle (in the example shown in the same image) the student used three Tangram pieces, one square and two triangles, with a specific composition of these pieces (Figures 13E,F). The dialog presented by the students was supported by the drawing of the parts used.