When preschool children build in the block area, they typically build with a purpose. They may set out to build something specifically. This may change in the process but there is typically a clear process that occurs. Often, these constructions are representational. They are building something that they have experienced. An educator that is knowledgeable in mathematical concepts can anticipate what a preschooler may be exploring when they set out to build something specific. For example, if a child wishes to build their house with blocks, we know that they will construct walls, add some dimension of height, width, and/or depth, and there may be a roof enclosing. The child may focus on their perspective looking at the exterior of their house or they may focus on their perspective inside their house. With these options in mind, a teacher in this preschool classroom can anticipate math concepts of spatial awareness and dimensions of height, width, depth, as well as specific numbers of blocks that may be needed to reach a specific height/width/depth, as well as what may be built on the exterior or interior of the house that may be math-related.

In the following photos, we will see preschoolers building in the block area based on plans that they created beforehand by sketching on paper with markers.

This child has drawn the exterior of their house. What math concepts do you think the child is exploring in this drawing? How will this child use their mathematical understanding when building their house with blocks based on this plan?

This child then built her house with blocks based on what she had drawn.

While appearing like a simple block construction, this child has demonstrated cognitive abilities to recreate her house using a different medium, blocks, then her original drawing. Using only 7 blocks, she has accurately built a house that retains the original qualities of her drawing, particularly the arched doorway. This type of building is different from open-ended building. The child has an image already of what the completed building should look like, or specific qualities that it has (the arch in this case), and has this knowledge prior to building. In this way, the child knew that the walls would need to be higher than the door, and chose to use 2 blocks stacked on each side to create the walls, with a triangle on top. In this process, we can see the child choosing specific shapes to match her original drawn plan.

Similarly, as part of a small group activity, preschoolers drew pictures with markers documenting how they get to school typically. Jorge drew the following picture showing the school on the right, a car, and his house.

While all group members (Jorge, Maria, and Anna) drew a picture, this small group decided to use Jorge’s picture of the school as their plan. They used this plan to build the school using unit blocks. They began building, with discussion facilitated by a teacher who was focusing on mathematical concepts and using language reflecting these concepts.

They began building, creating walls and placing two square blocks representing windows on the bottom of the structure. However, with teacher guidance, they reasoned that there was not enough room to fit 2 windows and a door on the first level. They decided that the walls were too high as well and that only one block should be used for the height of the first level.

The new design above represented the first two floors. They decided that their drawing may not be accurate and that windows should start on the second level. We can notice a repeating AB pattern starting to take shape with the overall building design. There is a horizontal block followed by two vertical blocks. The group decided to put three square blocks representing windows on the second level. The teacher asked them why three and they explained that only three would fit. At this point, the teacher gave some direction by asking, “How many windows do there need to be?” This closed question gave the group some focus on how they should proceed. Maria told the group there were 16 windows. The teacher asked how she knew, and she replied that she had guessed. The group then counted the windows and found out that despite guessing, Maria was correct! There were 16 windows in the original plan. Based on this, they counting building focused on creating a building with 16 windows.

They continued building, following the pattern of 1 horizontal block, 2 vertical blocks, but they also continued a pattern of placing 3 square blocks as windows on each level. The decision that 3 windows was somewhat arbitrary, but they continued building based on this pattern.

At this point, they had a problem to solve. They could no longer build higher as they could not reach any higher, yet following their rule of 3 windows per level, they still needed to add windows to get to 16 total windows. With teacher guidance, they re-evaluated how many windows could fit on each level and discovered that they could safely place 6 windows on a level.

This is their finished school. They followed a pattern of six blocks per level and found out that they only needed 4 windows on the top level. In this activity, they explored many different mathematical concepts – number sense, spatial awareness, patterns, comparison & measurement, as well as cognitive reasoning and problem-solving. By planning this as a small group activity, their preschoolers were able to discuss issues amongst themselves and explain their ideas as they went along. While the teacher facilitated the overall activity, these children were able to think and explain themselves using mathematical language, demonstrating their understanding of math concepts.

Blocks are a marvelous tool to support mathematical thinking and a well-stocked unit block area should be a staple in every preschool classroom. Young children should also have access to blocks throughout their day and I would encourage block play to occur at home with young children as well.

In this past 2 posts, I’ve discussed math in block play and we’ve explored the building process of a younger child. In this post, we’ll examine spatial awareness and patterns in block play and examine more block photos of preschool children’s constructions.

What do we see in the following photo that has evidence of mathematical concepts and thinking?

From this perspective, we see blocks arranged in horizontal and vertical lines, with a centered block on the top of this structure. We might not see significant evidence of mathematical thinking, however, when we change perspectives, what evidence do we see in the following photo?

From this angle, we see that this building has been built using a repeating pattern. Four separate towers are built in the corners to create the larger structure, with a roof built on top. In each tower, there is a repeating pattern of a horizontal block followed by two vertical blocks on each end. This repeating pattern is an intentional building choice, even if the child is unaware of what patterns actually are. A teacher in this situation would use language that focuses on repeating patterns, specifically the repeating AB pattern that this child has created, such as, “You made a pattern! It goes horizontal block, two vertical blocks, horizontal block, two vertical blocks.” This could lead to a discussion of horizontal and vertical as well.

We also see opportunities to discuss and use spatial awareness language. Spatial awareness is mathematical as it provides language that compares two or more objects. The child mentally constructs a set of blocks in relation to each and then uses comparative language to describe the set of blocks. Patterns additionally require children to mentally put objects together in sets to describe how the objects exist in relation to each other. For example, in the repeating pattern construction, for the child to describe the vertical blocks as “on top of” the horizontal block, he would need to mentally put the horizontal and vertical blocks into a set, and then would describe how they are different from each other, and where they physically are in relation to each other. Spatial awareness language includes phrases like on top of, next to, beneath, underneath, above, below, to the right/left of, inside, outside, through, &c. These words and phrases can serve the purpose of linking two or more objects and making a comparison between these objects.

What evidence of spatial awareness is there in the following photo?

We can guess that this child has created a building, likely an apartment building, where people live. She has created this building using an AB repeating pattern – two cylinder blocks, long rectangular block on top, &c. This pattern repeats for four cycles. In this situation, a teacher would be able to discuss the repeating pattern, using spatial awareness language, such as, “I see that the rectangular block is on top of two cylinders.” She could also ask the child to describe the building using spatial awareness language, such as “How did you create this building?” or “What is below/above the rectangular blocks or cylinders?”

These types of questions serve to focus the child on the mathematical aspects of her building. You may notice that the examples of questions given were closed questions, as opposed to open-ended questions. In this situation, knowing the child, a specific, well-crafted closed question will allow the child to engage cognitively and mathematically in reflecting upon what she has built. Open-ended questions certainly are important in early childhood years, but specific, closed questions used properly can engage the child in deeper mathematical thought at times. When teachers are attempting to engage children on a specific mathematical concept, a closed question may be more effective than an open-ended question in drawing attention to a math concept. Open-ended questions could then be used to elaborate the concept to allow the child to demonstrate her understanding.

Let’s examine one more photo. What types of mathematical concepts exist in this building? Is there evidence of patterns? Is there evidence of spatial awareness? What types of questions could you ask of the child in order to focus on specific math concepts related to spatial awareness? What other math concepts can be seen and what questions related to these concepts could be asked?

In this photo, we do see evidence of a part of a repeating pattern. On the second level of this building, we see long rectangular blocks, two square blocks, repeating for 2-½ cycles. This may be an unintentional aspect of the building process, but it can still be noted when discussing the building with the child. What is most striking to me is the symmetry in this building. The child has created a near mirror image structure, with two cylinders being the outliers in this symmetry. A teacher could discuss and ask questions based around this symmetry focusing on spatial awareness, the physical relationship of blocks in space to each other. In this case, a teacher may ask specific closed questions about where objects are, such as “Where are the two squares?” but could also use open-ended questions to gain explanations for why this child built this way, such as “Why did you create the building this way?”, hopefully leading to a discussion of symmetry.

We’ve seen how block building can elicit mathematical language around patterns and spatial awareness and how a teacher’s questions can be specific to these math concepts as well as deepen a child’s thinking on these concepts. In the next post, we will see how intentional building, with child-created building plans, can showcase mathematical concepts.

I struggled as a teacher to use three-dimensional shape names. I still struggle with this and may need to look up three-dimensional names. What’s a cuboid and a rectangular prism? Most adults would know cube, sphere, and cylinder, but maybe not much more than that. In block play, some teachers, including myself, will refer to blocks by a two-dimensional name. This isn’t to say that we should be using specific names for blocks at all times. We should expose young children to these words when appropriate, but it shouldn’t inhibit a child’s active building with blocks. This is more to say that specific math-related vocabulary should be used in block play with young children because they can understand this vocabulary. This can include three-dimensional shape language, but it can also include descriptive and comparative language, such as tall-taller-tallest. This type of language should be used with young children of all ages.

This language describes concepts that children are actively exploring. A young child may not be able to explain specifically concepts of height and number sense, but is exploring these concepts when creating a tower. Young children need words that describe their actions. It is, therefore, the role of the educator to provide this vocabulary. For younger children, it likely is through parallel talk, and for older preschoolers, this may be a conversation with the adult posing mathematically-relevant questions as the child is building.

Let’s examine a younger child’s interaction and explorations in block play. In the following photos, a 22-month old child is exploring blocks as is using cylinders exclusively to build.

Before looking at a series of photos, let’s consider the following:

• To what extent does a 22-month old have mathematical knowledge?
• How can they demonstrate their understanding of this mathematical knowledge?
• How can an educator support math learning?
• How well can a 22-month old build with unit blocks?

As a quick side note, use unit blocks with young children as soon as they can manipulate them! Young children need to learn the physical qualities of blocks and how to use them. Some teachers of toddlers may be hesitant to use wooden blocks as they believe that children can hurt themselves with wooden unit blocks. They instead may use foam or cardboard blocks – these blocks certainly can be used but they are more difficult to build with as they are not sturdy like wooden blocks. Toddlers will be more successful at building with wooden blocks as they are less likely to get knocked over easily.

Now, let’s get back to our 22-month old building with cylinder blocks.

He is creating a row of blocks, standing them upright. In standing them upright, he is exploring qualities of a cylinder. While he cannot explain to an adult what these specific qualities are, he has discovered that cylinders need to be placed with the flat circle face on the floor, otherwise they may roll away. A teacher, in this case, would use parallel talk, using this mathematical language, to describe what he is doing, giving words to describe his actions. From this photo, he is also lining them up, so there is some order to the way he is building.

In this photo, he has continued building and has continued adding blocks to his row of blocks, using both larger and smaller cylinders. He has also added smaller cylinders on top of larger cylinders. Prior to this photo he attempted adding smaller cylinders on top of the same smaller cylinders and was unsuccessful. He then placed them on top of the larger cylinders and was successful. He explored size in relation to stability for stacking. Parallel talk would focus on these math concepts.

We now see that he has begun stacking larger cylinder blocks. His initial exploration of creating a row of cylinders has shifted and now he is focused on stacking cylinders. He has added cylinders on top of each other and has created a taller stack with each added block. A teacher’s parallel talk would now focus on adding more blocks and height as the stack grows taller. He has also stopped to look at what he has created, providing an opportunity for a teacher’s parallel talk to act as a reflection on the building process. A teacher could say, “You built a tower with cylinders! You used large cylinders and stacked them on top of each. You started on the ground and added cylinders on top and now it kept getting taller.”

He has now added a fifth cylinder as is attempting to add a sixth. At first, he chooses a smaller cylinder then changes his mind and adds another larger cylinder. In doing this, he has shown that he knows the difference between the two sizes of cylinders in addition to continuing to explore height. At this point, he stopped constructing as he could no longer add blocks on top of his stack.

In this series of photos we see the building process of a 22-month old and how a teacher can support mathematical knowledge through their own engagement in the child’s building process. A teacher in this interaction needs to be attentive to what the child is interested in building and what math concepts match different types of building, such as the math concepts that may be different between creating a row of blocks versus creating a tower by stacking blocks (length v. height).

We see in this interaction that a 22-month old can definitely explore math concepts, and can successfully build with wooden unit blocks. Mathematical can and should be used when working with younger children as it gives them specific vocabulary to describe their actions. We will explore in a later post the building process of an older preschooler.

As a preschool teacher, the block area was my favorite area in my classroom and I spent a good portion of my time working with preschoolers as they built with blocks. Unit blocks specifically have enormous value in their use in the preschool classroom. Once children are familiar with the qualities of these blocks, they can create elaborate constructions, typically based in daily experiences, and used for representational purposes. These constructions and representations, both final products and the process of building, have huge potential for teaching and learning early mathematics.

Blocks are a very constructive material. Their dimensions are set and cannot be changed, unlike creating with clay or play dough. Typical wooden blocks also allow children to build with creativity as they are not bound to blocks needing to fit together a specific way, such as with Legos which interlock and must be in a specific position. This allows children to build with an infinite number of possible constructions.

Unit blocks are inherently mathematical. The standard unit block, a rectangle block or rectangular prism, can be seen as the base of the unit block system. Two half-unit blocks, square face blocks, laid lengthwise make up the same dimensions of a unit block. Triangle-faced blocks likewise can be put together to equal a standard unit block. These equalities are the basis of algebraic thinking, where equal parts are compared, as well as an understanding of geometry and number operations.

We can see these visually in the photos above. While young children may happen upon these relationships, it is our role as adults and educators of children to understand the mathematical implications of blocks and know how to support children’s mathematical learning through block play. Let’s examine the photo below.

A group of children created this during free choice time in a preschool classroom. What do you see in this photo? What math concepts do you think these children may have been explicitly or implicitly exploring? How do you think they constructed this? Why do you think they constructed this?

At first glance, it may appear that these children were randomly building, putting blocks together without a plan. However, upon a closer examination, we begin to see patterns of construction. These children were building their neighborhood in Chicago. We can see a roadway at the bottom, with triangle blocks representing cars on a bridge. Above the roadway, we see rows of buildings. There are shorter buildings and taller buildings, buildings with short triangular roofs and some with steep, tall sloping roofs. We can imagine the mathematical language that the children were using with each other, and the mathematical language that the preschool teacher was using to support mathematical learning. How many buildings do we need in the neighborhood? Let’s add a short house next to the tall building! There’s not enough cars on the road, we need more! How many cars should there be? How many buildings fit on this street? How many more triangle blocks do we need? We have a long rectangle block on the side of the road, do we need more blocks for the other side of the street? How many short rectangle blocks do we need to have the same length as the longest rectangle block?

All of these questions, responses, and statements focus on mathematical knowledge. Children can explore number sense and operations by adding and removing blocks, and by counting and quantifying the blocks in portions of their constructions. They can explore geometry by putting shapes together to create other shapes and by understanding what three-dimensional shapes are called. Algebraic thinking can be explored as children see equalities and inequalities in their constructions. They can represent objects in the real world and can plan and recreate their experiences. This re-creation of their experience can utilize mathematical thinking by educators prompting questions about these experiences, such as, “How tall was the building?”

In the next few blog posts, I will be further exploring math in block play. In the meantime, get out some blocks and play! Blocks are fun for all ages. Test out your own understanding of math concepts in your own block play.