The Colorado preschool learning foundations for math have been divided into four broad areas or strands.
Number Concepts and Quantities
Operations and Algebraic Thinking (Classification and Patterning)
Measurement & Data
Geometry & Spatial Sense
Mathematics Knowledge & Skills
The Mathematics Knowledge & Skills domain describes children’s abilities to understand numbers, quantity, and the relationships between them. Also important to this domain is a basic understanding of shapes, the position of shapes in space, patterns, and measurement. Many indicators described in this domain require children to make generalizations and think abstractly, which build cognitive skills that support early learning and are associated with positive outcomes. Some of the indicators and examples may not be appropriate until late in the 3–5 year age range. (Colorado Department of Human Services. (2021, October 26). Off to a great start. Colorado Early Learning & Development Guidelines. Retrieved August 19, 2022, from https://earlylearningco.org/)
Supporting Number Sense
The Number Sense strand refers to concepts of numbers and their relationships. It includes the development of counting skills, the understanding of quantities, recognizing ordering relations (which has more, fewer, or less), part-whole relationships, and a basic understanding of “adding to“ and “taking away” operations.
Figure 7.4: These children are exploring numbers and quantity as they play with dominos.[2]
1. Number Concepts and Quantities: The understanding that numbers represent quantities and have ordinal properties (number words represent a rank order, particular size, or position in a list).
Indicators
Children may.....
Examples
Children may....
1. Count verbally or sign to at least 20.
2. Instantly recognize, without counting, small quantities of up to five objects and say or sign the number.
3. Say or sign the number names in order when counting, pairing one number word that corresponds with one object, up to at least 10.
4. Use the number name of the last object counted to answer “How many?” questions for up to approximately 10 objects.
5. Accurately count as many as five objects in a scattered configuration or out of a collection of more than five objects.
6. Understand that each successive number name refers to a quantity that is one larger.
7. Identify whether the number of objects in one group is more than, less than, or the same as objects in another group for up to at least five objects.
8. Identify and use numbers related to order or position from first to fifth.
9. Associate a number of objects with a written numeral 0–5.
10. Recognize and, with support, write some numerals up to 10.
• Match a group of 1 to 10 objects with written and spoken numbers.
• Count, group, and sort objects and materials.
• Read stories, sing songs, and act out poems and finger plays that involve counting, numerals, and shapes.
• Match a group of one to five objects with written and spoken numbers.
• Copy a printed numeral using their own handwriting.
• Play games that involve matching numerals to numbers of objects, such as dots on cards
(Colorado Department of Human Services. (2021, October 26). Off to a great start. Colorado Early Learning & Development Guidelines. Retrieved August 19, 2022, from https://earlylearningco.org/)
Figure 7.5: 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 7.6: 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 Algebra and Functions (Classification and Patterning)
Obviously, preschool is not the time to teach traditional algebra, but this is the period when foundational algebraic concepts 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 7.7: As she built this tower, this young girl made a pattern with the colors red and blue.[6]
2. Operations and Algebraic Thinking: The use of numbers to describe relationships and solve problems.
Indicators
Children may...
Examples
Children may.....
1. Represent addition and subtraction in different ways, such as with fingers, objects, and drawings.
2. Solve addition and subtraction problems set in simple contexts. Add and subtract up to at least five to or from a given number to find a sum or difference up to 1.
3. With adult assistance, begin to use counting on (adding 1 or 2, for example) from the larger number for addition.
4. Fill in missing elements of simple patterns.
5. Duplicate simple patterns in a different location than demonstrated, such as making the same alternating color pattern with blocks at a table that was demonstrated on the rug. Extend patterns, such as making an eight-block tower of the same pattern that was demonstrated with four blocks.
6. Identify the core unit of sequentially repeating patterns, such as color in a sequence of alternating red and blue blocks.
• Use fingers on both hands to represent addition.
• Add a group of three and a group of two, counting “One, two three …” and then counting on “Four, five!” while keeping track using their fingers.
• Take three away from five, counting “Five, four, three … two!” while keeping track using their fingers.
• Say after receiving more crackers at snack time, “I had two and now I have four.”
• Predict what will happen when one more object is taken away from a group of five or fewer objects, and then verify their prediction by taking the object away and counting the remaining objects.
• Use art materials and other objects to create or replicate patterns (e.g., weaving, stringing beads, stacking blocks, or drawing repeating pictures).
• Recognize patterns in a story or song.
• Identify two blocks, one red and one blue, as the core unit of a longer pattern using alternating red and blue blocks.
• Sequence story cards to show beginning, middle, and end.
(Colorado Department of Human Services. (2021, October 26). Off to a great start. Colorado Early Learning & Development Guidelines. Retrieved August 19, 2022, from https://earlylearningco.org/)
Teachers can support children’s development of the 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 7.8: 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 for the child, sorting leaves by size. “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. Included in this strand is the child’s ability to compare and order objects by length, height, weight, or capacity; to use comparison vocabulary; and to 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 7.9: Tools such as this balance can help children compare weight.[9]
3. Measurement and Data: The understanding of attributes and relative properties of objects as related to size, capacity, and area
Indicators
Children may....
Examples
Children may....
1. Use comparative language, such as shortest, heavier, biggest, or later.
2. Compare or order up to five objects based on their measurable attributes, such as height or weight.
3. Measure using the same unit, such as putting together snap cubes to see how tall a book is.
• Sort objects by physical characteristics such as a color or size.
• Group objects according to their size, using standard and nonstandard forms of measurement (e.g., height, weight, length, color, or brightness).
• Explore various processes and units for measurement and begin to notice different results of one method or another.
(Colorado Department of Human Services. (2021, October 26). Off to a great start. Colorado Early Learning & Development Guidelines. Retrieved August 19, 2022, from https://earlylearningco.org/)
Teachers can support children’s development of the measurement foundations with the following:
Provide opportunities to promote measurement concepts in the environment (things to measure and measure 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 7.10: This boy is measuring the boulder with a yardstick[11]
Vignettes
As part of exploring and learning the concept of growth, the children have planted sunflower seeds in the garden. A long stick was attached to each plant, and the teacher asked that every week the children mark on the stick the height of the sunflower. Tracking the growth of sunflowers has generated comparison and measurement experiences. 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 were also comparing the sunflowers one to another. On one occasion, the teacher helped a small group of children compare the height of two flowers by using a string to represent the height of one flower and then laying the string 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 measurements 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
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 are able to match and classify objects based on 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 7.11: By using their bodies to make a triangle these children are working with shapes and spatial understanding.[13]
4. Geometry and Spatial Sense: The understanding of shapes, their properties, and how objects are related to one another.
Indicators
Children may....
Examples
Children may....
1. Name and describe shapes in terms of length of sides, number of sides, and number of angles/ corners.
2. Correctly name basic shapes (circle, square, rectangle, triangle) regardless of size and orientation.
3. Analyze, compare, and sort two-and three dimensional shapes and objects in different sizes. Describe their similarities, differences, and other attributes, such as size and shape.
4. Compose simple shapes to form larger shapes.
5. Understand and use language related to directionality, order, and the position of objects, including up/down and in front/behind.
6. Correctly follow directions involving their own position in space, such as “Stand up” and “Move forward.”*.1 Identify simple two-dimensional shapes, such as a circle and square.
• Match, sort, group, and name basic shapes found outside or in the classroom.
• Use pattern tiles to make shapes out of other shapes, such as putting two squares side-by-side to make a non-square rectangle.
• Put away blocks and/or tiles into different containers based on the number or length of sides.
• Use the vocabulary of geometry and position to describe shapes within the room and surrounding environment.
• Understand relational directions, such as “Please put a mat under each plate.”
(Colorado Department of Human Services. (2021, October 26). Off to a great start. Colorado Early Learning & Development Guidelines. Retrieved August 19, 2022, from https://earlylearningco.org/)
Teachers can support children’s development of the 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 7.12: 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 from home different sized boxes and figures or characters 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 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 different shapes and materials, and were proud of it.
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 kept adding more pieces to the structure and added different elements to decorate the castle.[16]
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, to apply mathematical knowledge in different problem-solving situations, and to come up with different solutions. Mathematical reasoning is natural to most young children as they explore the environment and make sense of the world around them.
Logic & Reasoning
The Logic & Reasoning domain describes children’s ability to think through problems and apply strategies for solving them. Such strategies require the ability to make connections among events or ideas, such as cause and effect relationships and comparisons. Likewise, the ability to think abstractly, or symbolically, about their world allows children to better understand the world around them. Such critical thinking skills are essential to children’s early learning and also to their ability to understand and adapt to a wide range of situations at home and in the community. (Colorado Department of Human Services. (2021, October 26). Off to a great start. Colorado Early Learning & Development Guidelines. Retrieved August 19, 2022, from https://earlylearningco.org/)
1. Reasoning and Problem-Solving: The ability to recognize, understand, and analyze a problem and draw on knowledge or experience to seek solutions to a problem
Indicators
Children may....
Examples
Children may....
1. Seek multiple solutions to a question, task, or problem.
2. Recognize cause and effect relationships.
3. Classify, compare, and contrast objects, events, and experiences.
4. Use past knowledge to build new knowledge.
5. Identify problems and search for solutions by asking questions during collaborative explorations of the topic; begin to state facts about the topic.
• Make suggestions to generate ideas.
• Make predictions, including hypotheses about cause or effect.
• Act out and talk about experiences.
• Talk about activities of yesterday, today, and tomorrow.
• Talk about what they are learning.
• Try different approaches to solve a problem.
• Differentiate between questions and statements.
(Colorado Department of Human Services. (2021, October 26). Off to a great start. Colorado Early Learning & Development Guidelines. Retrieved August 19, 2022, from https://earlylearningco.org/)
Figure 7.13: 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 here only three. 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?”
