España 
Italia 
United States 
Brasil 
Chile
Ecuador
Mexico
Colombia
Perú
Summary and key conclusions:
- Learning geometry: should not be limited to memorizing stereotypical models (like an equilateral triangle resting on its base), but instead focus on exploring and analyzing geometric properties (sides, vertices).
- Space and shape in Early Childhood Education: these are the two big areas of this learning. “Shape” helps children learn about 2D shapes and 3D shapes, while “space” helps children locate themselves, orient themselves, and understand routes and maps.
- Geometric shapes and 3D shapes for children: the best way to bring geometry into an early childhood classroom and help students build these concepts is through using manipulatives. Using physical materials and everyday objects helps them understand the characteristics of shapes and 3D shapes.
- The importance of vocabulary: the teacher should act as a “math translator,” naturally introducing correct geometry vocabulary while children explore.
In early childhood education, mathematical knowledge of space and shape allows children to build their own mental model of the world they live in. It allows them to locate themselves in space and also recognize the geometric properties of the objects around them.
It is essential, therefore, that we give it significant emphasis in preschool and kindergarten classrooms. In fact, if we look at the content distribution proposed by the NCTM (2000), we see that for early childhood education (and through second grade), there is as much emphasis on geometry as there is on number sense.
Contents
How do children learn geometric shapes in early childhood education? Development and challenges
For many years geometry in early childhood education focused only on perceiving and recognizing a few 2D shapes, usually starting with triangles, squares, and circles. Also, these shapes were presented using stereotypical models, the clearest being the triangle, showing an equilateral triangle resting on one of its sides.
This led children to not identify other triangles when, for example, the position or side lengths changed, and the same happened with other geometric shapes (Dickson et al., 1991). Think, for example, about the confusion many adults have when we show them an image like the one below; they may not identify the shape on the right as a square, while there is no doubt about the one on the left.
The importance of exploring geometric properties
So, how does a child learn what a square is? Or, better yet, how do we know a square is a square? The best way to know is by analyzing its properties in depth: “How many sides and vertices does it have?”; “What are its angles like? What about its sides?”
This way of building knowledge through exploring, identifying, and analyzing the characteristics of shapes will be the central focus for organizing this content in early childhood education.
As the name suggests, the first lessons in this content domain are organized around two main topics: shape and space.
With regards to shape, we refer to recognizing geometric shapes and 3D shapes (two and three dimensions), as well as the relationships among them and geometric transformations. For space, we focus on recognition and spatial location, encouraging children to locate themselves and other objects in their environment.
Basic concepts of shape in early childhood education
In school, learning about “shape” means guiding children to observe, compare, and classify the physical properties of the objects they see in everyday life, in both two and three dimensions.
In most Early Childhood Education curricula, we can find that shape content is organized around general topics that could be summarized as follows (Edo, 2018, p. 266):
- Observing and recognizing similarities and differences among objects and materials: color, size, shape, and other properties.
- Identifying 3-dimensional shapes (sphere, cylinder, and prism) and 2D shapes (triangles, quadrilaterals, and circles) found in elements of their surroundings.
- Systematic exploration of some geometric shapes and 3D shapes to discover their properties and establish relationships among them.
Four areas for organizing content on geometric shapes
So, how do we organize this content in Innovamat? We organize it around 4 main areas: 3-dimensional shapes, 2-dimensional shapes, basic geometric elements, and isometric transformations and visualizations.
3-dimensional shapes (3D)
For 3-dimensional shapes we must keep in mind two learning trajectories. One that helps us recognize the elements of 3D shapes (faces, edges, vertices…), and another that helps us organize these 3D shapes into different groups based on their characteristics (polyhedra—prisms, pyramids,…—and solids of revolution—sphere, cylinder,…)
2-dimensional shapes (2D)
For 2-dimensional shapes, as in the previous case, we need to consider, on one hand, content related to exploration, elements, and characteristics of 2D shapes (identifying sides, vertices, angles…) and, on the other hand, content that helps us classify shapes by relating these attributes (recognizing triangles, quadrilaterals and their subgroups, hexagons,…)
Basic elements
As for basic elements, we will focus, on one hand, on identifying points and, on the other, on recognizing straight and curved lines, open and closed, as well as lines that cross, are parallel, or are perpendicular.
Transformations and visualizations
For transformations and visualizations, this content is, in fact, the intersection of Shape and Space.
For transformations in the plane, we will focus on isometric transformations, that is, those that involve a change in the position of a given shape but do not change its shape or size. In early childhood education, specifically, we talk about turns, translations, and symmetries.
For visualizations, we refer to the relationship between 2D and 3D, such as relating the shapes of flat faces on 3D shapes or representing different views of a 3D object in a plane.
Different views of the same object
Hands-on activities to work on shape in the early childhood classroom
It is important to keep in mind that to build this knowledge, we start by exploring and discovering the characteristics of 3D shapes and, from these, we discover 2D shapes.
We develop a spiral curriculum that helps us connect this content and work on it with increasing depth. Therefore, we should alternate activities that help students explore objects (activities more focused on “visual recognition” of shapes) with others that help them analyze their properties.
So, little by little, we compare and connect these characteristics in order to, finally, organize and classify geometric shapes.
For example, we can propose the game shown in Image 7 (relating 2D shapes to the faces of 3D shapes) to identify which faces of different 3D shapes match specific 2D shapes.
And from here, ask questions: Which object has a face shaped like a circle? Is there more than one? Or, for example, ask students to build shapes using four different segments and then ask what they have in common and how they are different.
To do this in the classroom, we can begin with various situations:
- Consider everyday moments that help us analyze our environment and mathematize everyday situations from a geometric perspective.
- Provide manipulative materials and appropriate games that support exploration and visualization of all these mathematical ideas in a concrete way.
- Use other resources such as literature, visual arts, music, or technology resources.
The importance of using correct geometry vocabulary
As a final note, we do not want to forget the importance of using correct geometry vocabulary. It is natural for children to use their own words to describe geometric shapes and their characteristics, such as using the word “point” to refer to a “vertex.”
Although mastery of vocabulary should not be the main goal in early childhood education (Edo, 2018, p. 272), our role as teachers is to use correct geometry terms in any situation, even acting as “math translators” during conversations. This way, students will have their first contact with this vocabulary and will begin using it in meaningful contexts.
At the beginning of this text, we explained that, years ago, learning geometry focused on some 2D shapes, not only overlooking other shape content, but also setting aside focused practice for the second important topic we presented: space.
But how can we set aside this knowledge that we use in everyday life? How many of us have difficulty correctly interpreting a map? Do we struggle to give directions using right and left? Or even how hard it can be sometimes to make ourselves understood when we try to tell someone exactly where we left something on a shelf.
Developing spatial awareness in early childhood education
Working on space in early childhood is vital for children to learn to interpret their environment. This includes everything from understanding the limits of their own body to orienting themselves using external reference points.
One goal of this content domain is for children to learn to locate themselves and other objects in space, as well as to orient themselves and move through it. Specifically, in early childhood education, we talk about: relative positions and coordinates, and routes.
- Relative positions and coordinates. Younger children will begin by locating themselves in space, for example, inside or outside a box. Little by little, they will become able to locate objects and describe their positions in relation to their own body and even between objects, describing, for example, whether an object is in front of, behind, on top of, or under another object.
In this section, we also include using a system of rows and columns (coordinates) to locate items on a plane. One Innovamat activity example is the Relative Position in Space game presented in one of the math materials tabletop space proposals, in which children play at finding the Cricks in different scenes and complete a two-way table indicating their position.
- Routes. In this case, we refer to content focused on following and creating routes, identifying the start and the end. To do this, we need to orient ourselves and consider direction. We start with activities like gross motor circuits and, little by little, we can increase the challenge by including maps and floor plans (both representing and interpreting them).
Activities to support learning about space in early childhood education
To bring learning about space into the classroom, we start by offering experiences that allow children to locate themselves in their environment and build awareness of spatial relationships.
We begin with activities that support direct exploration, such as orientation games in which children must place objects in relation to their own body, like an obstacle course, or “in and out” activities, where children identify their position in space in relation to other objects.
Then, we can introduce the idea of relative position between two objects outside our own body. For example, in the following game children must identify the location of different characters (the Cricks) in a scene, using terms such as “in front of,” “behind,” “to the left,” or “to the right” and represent it in a coordinate table.
These types of activities allow children not only to learn about the position of objects, but also to develop skills for describing and communicating those positions, using appropriate mathematical language.
As children gain confidence in their ability to orient themselves, we can introduce more complex challenges, such as creating routes or interpreting and creating maps, which will allow them to connect physical experience with visual representations of space.
This gradual approach, moving from physical experience to abstract representation of space, gives children a deeper and more flexible understanding of how objects in their environment are organized and oriented.
References
Alsina, À. (2006). How to develop mathematical thinking from ages 0 to 6: instructional proposals.
Alsina, À. (2022). Learning trajectories for teaching mathematics ages 3 to 6.
Clements, D. H., & Sarama, J. (2015). Learning and teaching mathematics at an early age: The learning trajectories approach.
Dickson, L. Brown, M. Gibson, O. (1991) Learning mathematics. Ministry of Education and Science-Labor.
Edo, M. (2018) From identifying to analyzing geometric shapes. In M.C. Muñoz-Catalán and J. Carrillo (Eds.), Math methods for early childhood teachers (243-285). Editorial Paraninfo.
Van De Walle, J. A. & Lovin, L. H. (2005). Teaching student-centered mathematics. https://ci.nii.ac.jp/ncid/BB1085202X
