Refer to the Mathematics content area in the Kansas Early Learning Standards for specific indicators of how children of various ages meet the benchmarks and standards. The structure of the KELS for the area of mathematics is as follows:
Counting and Cardinality: know number names and counting sequence, count to identify the number of objects, compare numbers
Operations and Algebraic Thinking: understand addition as putting together and subtraction as taking from
Measurement and Data: describe and compare measurable attributes, classify objects and count the number of objects in each category
Geometry: identify and describe shapes, analyze, compare, create, and compose shapes (Kansas State Department of Education, 2024).
Supporting Number Concepts
The number concepts strand encompasses concepts related to numbers and their relationships. It includes the development of counting skills, understanding quantities, recognizing ordering relations (such as “more,” “fewer,” or “less”), part-whole relationships, and a basic understanding of “adding to“ and “taking away” operations.
Figure 9.5: These children are exploring numbers and quantity as they play with dominoes.[2]Figure 9.5: Image by Ian Joslin is licensed by CC-BY-4.0
Teachers can support children’s development of the number sense foundations with the following:
Observe children’s spontaneous counting and foster growth through scaffolding or modeling
Encourage counting during everyday interactions and routines
Include preschool children’s home language in counting activities, whenever possible
Ask questions that encourage purposeful counting
Foster one-to-one correspondence within the context of daily routines (such as setting the table)
Support preschool children’s ability to apply the counting procedure by
Providing a lot of objects to count
Starting with small sets
Modeling counting
Encouraging children to self-correct their counts
Consider adaptations for children with special needs
Use games, books, and other materials accessible to preschool children
Plan group activities focused on counting
Promote the use of comparison terms (more, same as, fewer, or less) through everyday interactions
Use everyday interactions and routines to illustrate and discuss addition and subtraction transformations (“adding to” results in more and “taking away” results in less)
Figure 9.6: Here is a material that supports children’s understanding of Arabic numerals and counting.[4]
Vignettes
Playing with cars on the rug, a child argued, “I have more: one, two, three, seven, nine, ten.” His friend replied, “No, I have more: one, two, three, four, five, six, seven.” The teacher intervened and asked, “How do you think we can find out who has more cars?” “I count,” said one of the children. The teacher suggested, “Let’s count together,” and she modeled counting together with the children. She put the cars in each set, in a row, and lined up the two sets against each other. The teacher pointed to each car while counting.
During snack time, Veronica asked: “Can I have two more crackers?” The teacher replied, “Yes, and I see you already have two crackers. When I give you two more, how many crackers will you have altogether?”[5]
Supporting Patterns and Relationships
Obviously, preschool is not the time to teach traditional algebra, but it is when foundational algebraic concepts begin to evolve and gradually develop. Children observe the environment and learn to recognize similarities and differences. They learn to sort, group, and classify objects. They learn to recognize ordering relations, such as large-to-small, and to identify patterns. They develop the ability to make predictions, form generalizations, and derive rules.
Figure 9.7: As she built this tower, this young girl made a pattern with red and blue colors.[6]
Teachers can support children’s development of algebra and functions foundations with the following:
Organize the classroom into different categorized storage areas to facilitate classification
Include materials and objects for sorting in the environment
Identify opportunities for sorting and classifying in everyday routines
Recognize and extend sorting in play
Plan for children at different levels
Integrate sorting into children’s current topic of interest and study
Point out patterns in the environment
Engage preschool children in conversations about patterns
Figure 9.8: Providing sets of materials in different colors sets up natural opportunities for classification and grouping.[8]
Vignettes
As part of a curriculum unit on the seasons, the children went for a nature walk and collected various types of leaves. During the walk and later in the classroom, the children explored the leaves and were encouraged to describe different attributes of the leaves such as shape (pointy, round, long, needle), size (small, tiny, wide, big), color (red, green, yellow, orange, brown) and texture (smooth, soft, hard, wet, dry, rough). Children were then asked by the teacher to sort the leaves: “Put leaves that belong together in groups.”
The teacher asks Enrique, “Why did you put these leaves together and those leaves together?” Enrique responds, “They are same.” The teacher asks, “How are these the same?” Enrique points and says in Spanish, “Café aquí, amarillo aquí, y hojas rojas.” (“Brown here, yellow, here, and red leaves here.”). The teacher points to each group of leaves and says in English, “Great! Brown, yellow, and red leaves. What other ways can we sort the leaves? How about putting all the big leaves here and all the small ones there?” The teacher models sorting leaves by size for the child. “Where do you think this leaf would go?”
Pause to Reflect
Before reading this section, did you think that algebra was something that should be considered in the preschool classroom? Why or why not?
Supporting Measurement
The measurement strand involves comparing, ordering, and measuring things. This strand encompasses the child’s ability to compare and order objects by length, height, weight, or capacity, use comparison vocabulary, and begin to measure. Young children develop an intuitive notion of measurement through natural, everyday experiences. They explore and discover properties such as length, height, volume, and weight as they look for a longer block, measure who is taller, pour sand from a small bucket to a larger one, or try to pick up a heavy box and ask for help. They make comparisons to see which is longer, taller, heavier, larger, or smaller.
Figure 9.9: Tools such as this balance can help children compare weight.[9]
Teachers can support children’s development of measurement foundations with the following:
Provide opportunities to promote measurement concepts in the environment (things to measure and measurement tools)
Observe preschool children’s measurement concepts in everyday play and routines
Facilitate and reinforce measurement concepts in everyday play and routines by
Building the descriptive and comparative vocabulary
Asking questions to bring their attention to the measurable properties of objects
Challenging them to use measurement to solve problems
Provide opportunities to compare and order objects
Use literature to illustrate measurement concepts
Provide small-group activities using standard and nonstandard measurement
Encourage estimations of measurement
Encourage recording and documentation of measurements[10]
Figure 9.10: This boy is measuring the boulder with a yardstick[11]
Vignettes
As part of exploring and learning about growth, the children have planted sunflower seeds in the garden. A long stick was attached to each plant, and the teacher asked the children to mark the sunflower's height on it every week. Tracking sunflower growth has provided opportunities for comparison and measurement. For example, one week the teacher pointed to one of the sunflowers and explained to the children, “Last week when we measured this sunflower, it was up to here. It was seven inches long. This week it is up to here. How many more inches do you think it grew in the past week? What is your estimate?”
Children were encouraged to make estimates and then were invited to measure the growth of this sunflower. “How can we measure how much it has grown since last time?” Children had different ideas. Some children said, “You need a ruler.” Others said, “With this,” and pointed to a measuring tape. Over time, children also compared the sunflowers among themselves. On one occasion, the teacher helped a small group of children compare the heights of two flowers by using a string to represent the height of one flower, then laying it against the second flower.
Children enjoyed tracking the sunflowers’ growth and finding out, “Which sunflower is taller?” and “Which is taller?”—the child or the sunflower.[12]
Pause to Reflect
Involving children in measuring things that are meaningful to them is a great way to engage children in mathematics. Do you remember any measurement experiences from your childhood? This could be formal, like keeping track of your height on a wall or growth chart, or measuring the weight and/or length of a fish caught, or informal, such as recognizing that a new storage container holds more (volume) or noticing the length difference in a new pair of shoes.
Supporting Geometry and Spatial Sense
Geometry is the study of shapes and spatial relationships. Children enter preschool with a strong intuitive knowledge about shapes, spatial location, and transformations. They learn about geometry as they move in space and interact with objects in their environment. From infancy, they begin to form shape concepts as they explore their environment, observe shapes, and play with different objects. Before they can name and define shapes, very young children can match and classify objects by shape. During the preschool years, children develop a growing understanding of shape and spatial relationships. They learn the names of shapes and start to recognize the attributes of two- and three-dimensional shapes. They also develop an understanding of objects in relation to space, learning to describe an object’s location (e.g., on top, under), direction (e.g., from, up, down), and distance (e.g., near, far).
Figure 9.11: By using their bodies to make a triangle, these children are working with shapes and spatial understanding.[13]
Teachers can support children’s development of geometry foundations with the following:
Refer to shapes and encourage the use of shape names in everyday interactions
Engage preschool children in conversations about shapes, including both
Two-dimensional shapes (such as circles, squares, and triangles)
Three-dimensional shapes (such as spheres, cubes, and cones)
Provide materials that encourage preschool children to explore and manipulate shapes in space
Include books, games, and other learning materials with shape-related themes in the preschool environment
Provide preschool children with playful opportunities to explore and represent shapes in a variety of ways
Present preschool children with many different examples of a type of shape
Provide materials and equipment to promote spatial sense
Support preschool children’s spatial sense in everyday interactions
Provide preschool children with planned experiences to promote the understanding of spatial sense, including
Figure 9.12: Building with pattern blocks such as these promotes geometry.[15]
Vignettes
The teacher had noticed that several children in her group had shown a strong interest in castles. They built castles in the block area, in the sandbox, and even looked for castles in fairy tale books when visiting the library. The teacher suggested that the group build a big castle outside. They started by gathering the materials. The children brought different-sized boxes and figures or characters from home to be included in the castle. The teacher also offered big cylinders, cones, building blocks, construction boards, and other materials. The children made different suggestions: “Put all the big boxes here and the small ones on top of them.” ”I put it above this for the roof.” “We can use these for the tower.”
The teacher described their ideas using the names of shapes and spatial terms: “So you want to put the small square blocks on top of the big rectangle blocks?” “Are you suggesting using the cylinders to build the tower?” The children enjoyed building the structure using various shapes and materials, and they were proud of their work.
During circle time, the teacher invited children to describe the castle and how it was built. “Look at the castle you built. Can you tell me what it looks like?” Children were encouraged to use spatial words and the names of shapes in their talk. The activity evolved into a long-term project. The children continued adding pieces to the structure and incorporated various elements to decorate the castle.[16]
Supporting Data Collection and Organization
In preschool mathematics, the area of data collection and organization introduces young children to foundational concepts of gathering, sorting, and interpreting information. At this stage, activities are typically hands-on and exploratory, allowing children to collect data through simple observations, surveys, or classification tasks (e.g., counting how many children prefer apples vs. bananas). They then organize this data using concrete materials like charts, pictographs, or manipulatives. These experiences help children begin to recognize patterns, make comparisons, and draw basic conclusions, all of which are essential skills for more advanced mathematical thinking. For early childhood educators, fostering these skills supports the development of logical reasoning, problem-solving, and early analytical abilities.
Supporting Mathematical Reasoning
Mathematical reasoning is a key process in learning and developing mathematical knowledge in all areas of mathematics, including number and operations, classification, patterning, measurement, and geometry. It involves the ability to think and reason logically, apply mathematical knowledge in various problem-solving situations, and devise multiple solutions. Mathematical reasoning is natural to most young children as they explore the environment and make sense of the world around them.
Figure 9.13: This boy uses mathematical reasoning when he constructs his train tracks.[17]
Teachers can support children’s development of the mathematical reasoning foundations with the following:
Identify and create opportunities for mathematical reasoning through both spontaneous interactions and planned experiences
Pose meaningful questions that promote investigation and inquiry, and challenge children to think through a problem and come up with a solution
Support preschool children in reasoning mathematically by providing clues, encouragement, and modeling, as needed[18]
Vignettes
The children cleaned up the play yard before going back inside. The teacher, Ms. Denise, had noticed that not all the shovels were picked up from the sandbox. Ms. Denise asked for help, saying, “We need all five shovels back in the box so our toys aren’t lost. I see only three here. We need more shovels in the box. How many more shovels do we need?” The teacher had noticed that Ling Wa, one of the older preschool children in the group, was counting her fingers, trying to find out how many shovels were missing.
Ling Wa suddenly said, “Ms. Denise, we need two more.”
Ms. Denise went further, asking, “Do you think we need two more shovels?” How did you figure that out?”
